JP2002042104A - Control system and control method using quantum soft computing - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a methodology programming the algorithm for solving a problem by using a quantum logic and to provide the algorithm. SOLUTION: A quantum logic program can be executed on a quantum computer. This algorithm can be executed on a non-quantum computer by using the non-quantum computer simulating the quantum computer. The principle of the quantum computation of concept, function, superposition, entanglement, quantum interference and others (massive parallelism feasible by these principles) can be utilized at its maximum by the non-quantum computer by using this algorithm without developing the hardware for the quantum computer.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a control system based on quantum soft computing, and to programming of a quantum computation algorithm and use of the quantum computation algorithm.

[0002]

2. Description of the Related Art Mathematics and physics have always strived and developed in their respective fields, with good influence on each other. The calculation was made by Newton and Leibniz to understand and explain the laws of kinetics of body motion. In general, geometry and physics have been successfully co-existing for a long time. That is, while classical mechanics and Newton's universal gravitational force are based on Euclidean geometry, Einstein's theory of general relativity is a non-Euclidean geometry, Riemannian geometry (an important insight taken from mathematics into physics). Is the basis. Physics and geometry are still very closely linked, but today information theory and quantum physics are among the most strikingly related. It seems that mathematics is going to be more "physical".

Considering the origin of computation from the viewpoint of information processing, computation based on the laws of classical physics and computation based on quantum mechanics (first performed by Feymann and Deutsch) , It is completely different.
Computing seems like the best item (eg, speed) that will get forever, but suddenly amazing, as it gets cheaper. In the last few decades, information processing capacity has grown 10 million times faster than the information processing capacity of the human nervous system, which was achieved four billion years after life was born on Earth. . However, computational theory and techniques have remained stagnant for over 50 years on Turing machine models of computation. And Turing machine models have a number of difficult and indeterminate problems that cannot be solved even today.

[0004] The use of a quantum computer can solve such a difficult problem. Unfortunately, there is currently no algorithm to "program" a quantum computer. Arithmetic calculation of a quantum computer is similar to that of a conventional computer, and includes quantum hardware (physical realization of the computer itself, for example, a quantum gate and the like) and quantum software (hardware (The calculation algorithm realized by). Until now, quantum software algorithms used to solve problems on quantum computers have been developed in an ad-hoc manner without using actual structures or programming methodologies. For example, Shore's algorithm is a typical example.

This method is somewhat similar to designing a conventional logic circuit without using a Carnot diagram. When a logic circuit designer sets an input and an output corresponding to the input, the logic circuit designer may be able to design a complex logic circuit using a NAND gate without using a Karnaugh diagram. Unfortunately, designers will have to design logic circuits with more or less intuition and trial and error. The Carnot diagram makes it possible to quickly design a logic circuit that performs the desired logical calculation because the structure and algorithm for performing logical operations (AND, OR, etc.) are clear.

[0006] The lack of a methodology for programming and designing a quantum computer has severely impaired its usefulness. It also limits the usefulness of the quantum principles underlying quantum logic used for quantum computing, such as superposition, entanglement and interference. These quantum principles are suitable for problem solving not used in conventional computers.

[0007] These quantum principles can be used on conventional computers in much the same way as the evolutionary genetic principles currently used in gene optimizers. Nature has devised an effective way to optimize large-scale linear systems during the course of evolution. Gene-optimizers running on computers simulate natural evolutionary processes to efficiently solve many previously difficult optimization problems.
Nature uses quantum mechanics to solve problems such as optimization problems, search problems, and reference selection problems, using quantum logic. However, quantum principles and quantum logic could not be used in conventional computers because there was no way to program algorithms using quantum logic.

SUMMARY OF THE INVENTION The present invention solves these and other problems by providing a methodology and algorithm for programming algorithms to solve problems using quantum logic. Quantum logic programs can be "executed" on a quantum computer. Also, the algorithm can be "run" on a non-quantum computer using a non-quantum computer that simulates the quantum computer. With this algorithm, the concepts, functions and superposition, entanglement, quantum interference, etc. of quantum computation (and the massive quantum parallelism enabled by these principles) do not develop hardware for quantum computers. But you can take advantage of the advantages of non-quantum computers.

In one embodiment, the quantum programming method is used with a control-based gene search algorithm. Conventional gene search algorithms search for an optimal solution in a single space. Quantum search algorithms provide a global search for optimal solutions in multiple spaces. In one embodiment, the algorithm design for quantum soft computing is performed by encoding the input function f into a unitary matrix operator U _F. Next, the operator U _F is incorporated into the quantum gate G. Where G
Is a unitary matrix. Apply gate G to the first canonical basis vector to generate a basis vector. After that, the basis vectors are measured. These steps are repeated as many times as necessary to generate a set of measured basis vectors. Decoding the measured basis vectors,
Convert to output vector.

In one embodiment, encoding to U _F includes:
A mapping table of f is a single mapping function F, a mapping table of _{F is} a mapping table of UF,
And it is included to convert the mapping table of U _F to U _F. 1
In an embodiment, the basis vector Shannon
Is minimized. In one embodiment, an intelligent control system having a quantum search algorithm that reduces Shannon's entropy uses a fitness function that is configured to minimize the rate of entropy generation of the controlled machine. And a local optimizer to find the local solution. Quantum search algorithms are used to search for local solutions with the goal of finding a global solution using a fitness function that is configured to minimize Shannon's entropy.

In one embodiment, the global solution includes fuzzy neural network weights. In one embodiment, the fuzzy neural network is configured to handle a fuzzy controller, and the fuzzy controller includes a PI controlling a mechanical device.
The control weight is provided to the D controller. In one embodiment, the quantum search algorithm is developed according to a fitness function selected to minimize Shannon's entropy.

In one embodiment, the quantum search algorithm is developed by minimizing Heisenberg's uncertainty and minimizing Shannon's entropy. In one embodiment, the quantum search algorithm applies an entanglement operator to generate a plurality of correlation state vectors from a plurality of input state vectors, and applies an interference operator to the correlation state vectors to generate an intelligent state vector. It is expanded by generating. Here, the intelligent state vector has a smaller classical entropy than the correlation state vector.

In one embodiment, a first transformation is applied to the initial state to improve the properties of the partial optimal solution,
Perform global optimization by creating a coherent superposition of ground states. A second transform using a reversible transform is applied to the coherent superposition to create a coherent output state. A third transform is applied to the coherent output states, creating output state interference.
The global solution is then selected from the output state interferences. In one embodiment, the first transformation is a Hadamard rotation. 1
In the embodiment, each ground state can be represented by a qubit. 1
In an embodiment, the second transformation is a solution of the Schrodinger equation. In one embodiment, the third transform is a quantum fast Fourier transform. In one embodiment,
The selection is made to find the maximum probability. In one embodiment, the superposition of input states includes a collection of local solutions of a global fitness function.

DETAILED DESCRIPTION INTRODUCTION In classical control theory, it is assumed that all machinery is controlled by a linear system. Unfortunately, in the real world, this assumption is rare. Most machinery is highly non-linear and in most cases cannot be controlled by simple algorithms. Currently developed control systems perform non-linear control using the concept of soft computing, for example, a gene analyzer, a fuzzy neural network, or the like. Control systems have evolved over time and adapted to changes in machinery and / or operating environment.

FIG. 1A shows a control system 100 for controlling a mechanical device 104 by soft computing. The entropy generation calculator 106 controls the mechanical device 104
Calculate the entropy production rate. The output of the entropy generation calculator 106 is a gene analyzer (GA) 1
07 (GA is also called gene optimizer). The GA 107 searches for a set of control weights that generates the minimum entropy. The weights are passed to a fuzzy neural network (FNN) 108. The output of the FNN is passed to the fuzzy controller 109. The output of fuzzy controller 109 is a set of gain schedules for conventional PID controller 103 that controls mechanical device 104.

[0016] In soft computing systems based on gene analyzers (GAs), in most cases, control laws that have the same meaning as in classical control do not actually exist. The control is performed based on physical control rules such as generation of minimum entropy. Since the stability of the GA 107 is guaranteed when used in combination with the feedback, robust control can be performed using this physical control law. However, robust control is not always optimal control.

The GA 107 seeks to find a global optimum solution of the target solution space. Stochastic disturbances in the machinery can "jump" the GA 107 into another solution space. For example, if you want to control the vehicle suspension,
Changing the road condition forcibly changes the GA 107 to another solution space. Genetic algorithms are global search algorithms based on natural genes and natural selection mechanisms. In the gene search, each design variable is represented by a bit string having a finite length, and these finite bit strings are concatenated into one bit string. Possible solutions are coded into groups of bits. Subsequently, genetic transformations similar to biological replication and evolution are used to improve and modify the encoded solution. Usually three main operators are used in gene search: replication, crossover and mutation.

[0018] Duplication produces better matching members,
This is a process in which the search is biased in a direction to eliminate incompatible members. First, the fitness of the individual is assigned to each character string of the group. It is a simple method to assign a selection probability to each member based on the degree of fitness and select a member of the initial group to be duplicated. Next, a new population pool of the same size as the original size is generated using members having a high average goodness of fit.

In the duplication process, a plurality of dominantly designed copies are simply included in the population. In the crossover process, the design characteristics are exchanged between members of the collective pool to improve the fitness of the next generation. Crossover is performed by selecting two parental character strings to be crossed, randomly selecting two positions on the character string, and exchanging the character strings of 0 and 1 at these selected positions.

Mutations protect the genetic search process by losing valuable genetic material during replication and crossover. In the process of mutation, simply select several members from the population pool according to the probability of mutation, and at the position of the randomly selected mutation in the selected string,
Change 0 to 1 or vice versa. GA10
FIG. 1B shows an outline of the process of searching for a gene used by No. 7. First, the group of strings is converted to decimal code and then sent to a fitness step 160 to calculate the fitness of all strings. A biased roulette 106 (where each character string is a roulette slot of a size proportional to its fitness) is created. The rotation of the weighted roulette makes duplicate candidates. By this method, a character string having a high degree of matching has many descendants in the succeeding generation. After the character string to be duplicated is selected, a duplicate of the character string is created based on the fitness. Then, it is input to the hybridization pool 162 for the next genetic operation. After replication, a new population of strings is generated using the evolution process of crossover 163 and mutation 164. Finally, as explained above, all genetic processes are performed iteratively until an optimal solution is found.

As described above, although the mechanism of gene search is simple, there are some important differences from the conventional method, and this difference is an advantage of the method of the present invention. Genetic algorithms only evaluate functions and do not require derivatives. While the derivative contributes to a fast convergence to the optimal value, the function evaluation also leads to a search for a local optimal point. Furthermore, unlike the conventional scheme, where the search is continued from one point to another, the search is continued from a plurality of points in the design space to another set of design points. Is a good probability of indicating a global minimum. Also, the genetic algorithm works on the design variables to be encoded, not on the variables themselves. This allows for the extension of genetic algorithms to a design space consisting of a mixture of continuous, discrete, and integer variables.

As explained above, GA 107 searches for a global optimum in a single solution space. However, it is desirable to search for a global optimum in a multiple solution space to find a "universal" optimal solution. A quantum search algorithm (as described below) can simultaneously search multiple spaces. FIG. 2 shows the quantum search algorithm (QSA) 202 as GA 107 and F
The modified control system is arranged between the NNs 108. The QSA 108 searches a plurality of solution spaces at the same time and finds a universal optimal solution, that is, one optimal solution considering all the solution spaces.

The quantum algorithm is a global random search algorithm based on the principles, rules and quantum effects of quantum mechanics. In the quantum search, each design variable is represented by a linear and finite superposition of classical initial states. This initial state is a series of basic unitary steps that manipulate the initial quantum state | i> (input) so that the measurement of the final state of the system is correctly output. Beginning with basic classical preprocessing, we apply the following quantum experiments. That is, first superimpose all possible states, calculate the classical function, apply the quantum Fourier transform, and finally perform the measurement. Depending on the results obtained, a choice is made between repeatedly performing similar quantum experiments or completing the process with classical post-processing calculations. Usually, three important operators are used in the quantum search algorithm. These operators are linear superposition (coherent states), entanglement and interference.

Linear superposition is closely related to the well-known mathematical principle of linear combination of vectors.
A quantum system can be represented by a wave function ψ in Hilbert space. The Hilbert space is represented by the ground state | φ _i >, and the system is a quantum state

[0025]

(Equation 1)

## EQU2 ## | Ψ> is referred to as a linear superposition of the ground states | φ _i >, and the coefficient c _i is generally a complex number. Here, the bracket symbol of Dirac is used. Ket |> is analogous to a column vector, and bra <| is analogous to the complex conjugate transpose of ket. In quantum mechanics, the Hilbert space and its basis are physically explained, but this explanation may clearly be the most counterintuitive theory. Contrary to intuition, the state of the system is represented at the microscopic level as a wave function, a linear superposition of all ground states (ie, in a sense, the system is simultaneously Has become). However, at a macroscopic or classical level, the system consists of only a single ground state. For example, at the quantum level, one electron is a superposition of many different energies. But this cannot be a classic realm. This is analogous to a coin being flipped in both its ground state (front and back) while the coin is in the air during a flip. When the coin lands, the wave function "settles" into one of the classic states, either front or back. As long as the coin is in the air, no one can tell whether it is upside down or upside down, only the probability can be calculated. Wave function of quantum mechanicsψ
The same is true for the probability that the system "settles" in each of its ground states.

[0027] Coffeence and decoherence are closely related to the concept of linear superposition. A quantum system is said to be coherent if it is a linear superposition of its ground states. If the system is in a linear superposition of states interacting with its environment, quantum mechanical conclusions will destroy the superposition. The decrease in coherence is called decoherence and is determined by the wave function. The coefficient c _i is called the probability amplitude. | c _i | ² is the probability of | ψ> contracting to the state | φ _i >, which is the probability of interaction with the environment (ie, measurement) causing decoherence.
Wave functions represent real physical systems that must contract to exactly one ground state. Therefore, the sum of the probabilities determined by the amplitude c _i must be 1. The essential constraint is the unitary condition

[0028]

(Equation 2)

It is said that The probability that a quantum state degenerates to an eigenstate (ground state) is calculated using Dirac's notation
| <φ _i | ψ> | ² and is similar to the dot product (projection) of two vectors. For example, consider a discrete physical variable called spin. The simplest spin system is
This is a two-state system called a spin 1/2 system. The ground state is represented by spin-up and spin-down. In this simplest system, the wave function has a distribution of two values (up and down) and the coherent state is a linear superposition of spin-up and spin-down. While the system retains its quantum coherence, the system is neither spin-up nor spin-down. In a sense, they are both spin-up and spin-down at the same time. Of course,
Classically, it must be in either state, and if the system becomes decoherent, it will either spin up or spin down.

The Hilbert space operator represents a method of changing a wave function into another wave function. The operator is
Usually represented by a matrix acting on a vector. Using the operator, the eigenvalue equation is

[0031]

(Equation 3)

## EQU1 ## In this equation, a _i represents the eigenvalue of the operator A. The solution of such an equation is called an eigenstate and is used to construct the basis of the Hilbert space. In quantum formalism, all properties can be expressed by operators. The eigenstate of an operator indicates the basis of the Hilbert space related to its property. In addition, the eigenvalue indicates a quantum having a value of the property. It is important that the operator of quantum mechanics is a linear operator and that it is unitary which satisfies A * A = AA * = I. Where I is the identity operator and A * is the complex conjugate transpose of A (or called the conjugate operator).

Interference is a well-known wave phenomenon.
The peaks of the phase waves reinforce each other, while if the phases do not match, they decompose. This is a phenomenon common to all wave dynamics from water to light. Also, the well-known two-slit experiment shows empirically that interference hits a quantum mechanical stochastic wave at the quantum level. Entanglement is the potential of a quantum state that represents a classically unexplained correlation. From a computational point of view, entanglement looks sufficiently intuitive. Quantum systems with different correlations (for example,
It can be easily explained that they can be present in different particles).
For example, if one particle is in a spin-up state, another particle is in a spin-down state. One particle is in both states because the quantum states exist as superpositions and their correlations are also superposed. If the superposition is broken, an appropriate correlation is made instantaneously between the particles. This communication is important in entanglement. Communication occurs instantaneously, even if the particles are far away. Einstein
Said it was "a ghostly remote action."

From a computational point of view, Hamming
The superposed quantum states of only the ground states that are as far apart as possible with respect to the distance have a huge entanglement. Furthermore, as long as the interference has certain quantum properties, the entanglement is a complete quantum phenomenon with no classically similar.

A quantum network is one of a plurality of models of quantum computation. Others include quantum Turing machines and quantum cellular automata. In the quantum network model, each unitary operator is modeled as a quantum logic gate affecting a plurality of qubits (qubits). A qubit exists in a superposition of states. Thus, the quantum logic gate acts on all superposition states simultaneously, acting on the qubit. This is the quantum parallelism. (The term quantum logic gate is used in the sense of a completely schematic method of describing the evolution of a quantum system over time. The term "gate" refers to a method by which quantum computation resembles a classical logic network. It does not mean that it can be physically realized by the following.) Due to the entanglement (quantum correlation) between particles, the state of each particle cannot express the state of the system at all. Instead, the state of the particle is a complex superposition of all 2 ⁿ ground states. Therefore, the description requires 2 ⁿ complex coefficients. The exponential size of the Hilbert space is a factor in quantum computation. It is attractive that exponential parallelism means exponential computational power. However, it will not be described here. In other words, exponential parallelism can be seen in classical calculations as well. The problem is how to extract exponential information out of the system. In quantum computation, a system must be observed in order to extract quantum information.
The measurement process causes the breakdown of the famous wave function. Quite simply, it means that an exponential number of possible states is mapped into one state after the measurement. Thus, an exponential amount of computational information is lost.

To take advantage of exponential parallelism, it is necessary to combine it with interference. Using interference, exponential number calculations in parallel cancel each other, just like destruction by wave and light interference. I want to erase only the relevant calculations and cancel out the remaining calculations. The combination of exponential parallelism and interference powers quantum computing and plays an important role in quantum algorithms. In fact, the Fourier transform describes interference and exponentiality.

In classical computing and digital electronics, a series of basic operations (eg, ANDS, OR
And operations such as NOT). These operations are used for classic bit array operations. Operations are fundamental in the sense that they only operate on a few bits (one or two bits) simultaneously. Operations are a set of results and operators,
It may be convenient to refer to a matrix, instruction, step or gate operation. In quantum computation, a series of basic operations (for example, control NOT and qubit rotation)
Which operates on qubits instead of classical bits. The quantum series of basic operations can be largely illustrated by qubit circuits. In quantum computation, in most cases, a unitary operator U representing expansion of a qubit array is known.

When a quantum search algorithm utilizing exponential parallelism inherent to a quantum system is used, the decision making process of the control system can be performed. FIG. 3 compares the structures of the GA and the QSA algorithm. As shown in FIG. 3, in the GA search, the solution space 301 is led to an initial position (input) 302. The first position 302 is encoded into a bit string using a binary encoding scheme 310. GA operators, such as selection 303, crossover 304, and mutation 305, are applied to the encoded string to generate a population. A fitness function 306 (eg, a fitness function based on minimal entropy generation or other desirable properties) is used to determine a global optimum for space 301.

In the QSA shown in contrast to FIG. 3, N grouped solution spaces 350 are used to create an initial position (input) 351. Quantum operators, such as superposition 352, entanglement 353, and interference 354, act on the first position and make a measurement. The superposition is generated using the Hadamard transform 361 (one bit operation). Using control NOT operation 362 (2-bit operation), an entanglement is created. Using the quantum Fourier transform (QFT) 363, interference is generated. Using the quantum operator, a universal optimum that satisfies the entire space of the group 350 is determined.

As described above, the classic selection process is almost
Similar to the generation of a superposition of quantum processing. Also, classic crossover processing is almost similar to the entanglement of quantum processing. Furthermore, classical mutation processing is almost similar to the interference of quantum processing. FIG. 4 shows the general structure of a QSA (eg, QSA 202). 400 conceptual layers, 401 structural layers, 40 hardware layers
2, and the software layer is denoted by 403. In the concept layer 400, the initial state 410 is passed to a processing unit 420 that generates a state superposition. The state superposition is passed to a processing unit 430 which provides a unitary operator U _f for the superposition. The output of the processing unit 430 is passed to a solution unit 440 that calculates the solution interference. The output of the solution unit 440 is passed to the observation / measurement unit 460.

In the structure layer, the input is encoded as a continuous quantum bit (qubit) in an initial state (eg, a logically zero state). The input is passed to a Hadamard transform matrix 421 to generate a superposition. Superposition of the matrix 421 is passed to the operator U _f. Here, U _f is a solution of the Schrodinger equation of the processing unit 431. The output of the processing unit, which is the solution of the Schrodinger equation, is the quantum fast Fourier transform (Q
FFT) 441. The output of QFFT 441 is
It is passed to the transformation matrix 451. The output of the transformation matrix 451 is
The solution has the maximum probability amplitude 461.

The overlay 420 in the hardware layer is created by the rotating gate 422. The operator U _f is implemented as a series of basic gate operations 432. QFF
T441 is implemented as a series of Hadamard and Replace (P) operator gates. Then, the transformation matrix 451
Are implemented using a rotating gate 452. FIG.
Shows an architecture of the QSA including a series of processes from the initial state to the generation of the superposition. Entanglement,
It is applied to superposition using the quantum parallelism inherent in entangled coherent quantum systems. Parallelism is lost when interference is introduced and solution superposition using QFFT is performed. FIG. 5 shows a comparison of these processes in three operations of a classical two-slit experiment, a logical quantum operation, and a quantum search operation.

In a classic two-slit experiment, the light source 501
First produces superimposed particles. This is similar to a quantum algorithm operation that applies a Hadamard (rotating gate) to a qubit initialized to an eigenstate. In the two-slit experiment, the entanglement was 50 slits.
Produced using particles passing through 2. This corresponds to the process of superposition operation using the unitary operator U _f.

In a two-slit experiment, interference is created when entangled particles reach a photographic film placed behind a slit that creates interference fringes (superposition of solutions). This corresponds to QFFT. Finally, the choice of the target solution corresponds to choosing the largest probability from the QFFT (ie the brightest line created on the film).

FIG. 6 illustrates the use of QSA in conjunction with GA 605. In FIG. 6, the generator in the initial state 604 works in association with the GA 605 to form a set of initial states. In some cases, it works in association with the fuzzy neural network 603 to create a set of initial states.
The initial state is passed to a Hadamard transform 602 to create a superposition of the classical state 601. The superposition of classical states is passed to a processing unit 606 that performs entanglement using an operator such as control NOT. The output of the processing unit 606 is Q
It is passed to an interference unit 607 that calculates the entangled interference using FFT. The output of the interference unit 607 is passed to the measurement / observation unit 608. The measurement / observation unit 608 selects a target solution from the superposition of the solutions calculated by the interference unit 607.

The output of the observation / measurement unit 608 is
9 passed. The determination unit 609 determines, for example, the initial state 60
4 determine the input of the generator. In some cases, a new fitness function of the GA 605 is determined. Also,
The determination unit 609 provides the data to the decoding unit 610 and receives the data from the decoding unit 610. The decoding unit 610 can communicate with sensors, other control systems, users, and the like.

The basis of quantum computation is obtained from quantum information theory. In quantum information theory, information is encoded into the state of a physical system, and calculations are actually performed on physically feasible devices. With quantum constraints,
Two examples showing the effect of the method of developing a validity model of a physical object are shown. In the first example, Shannon's classic data transmission channel C _c

[0048]

(Equation 4)

And the quantum data transmission channel C _q

[0050]

(Equation 5)

Is compared with Here, P is the strength of the input signal of the channel, and N is the strength of the noise. In the case of N → 0, limC _c → ∞, limC _q → lnP, and a validity model of a noisy information transmission channel is obtained. 2
In the second example, we consider the identification of a mathematical model of a controlled object with quantum constraints. In a classic controlled object,
[X, y] = 0 (when exchangeable), the following equation

[0052]

(Equation 6)

Is as follows. On the other hand, in the case of a control target based on the quantum correlation of model identification, [x, y] = ih (when exchange is not possible),

[0054]

(Equation 7)

Is as follows. In the case of quantum, the invalidity of the first-order Fredform integral equation maps to the correctness of the second-order Fredform integral equation. This is the input signal x
It means that a slight error occurs in the measurement of the output signal y (t) of the equation (2.2) for validating the structure of the dynamical system k (t) with (t) or a slight error. I have. In the case of the classical equation (2.1), a slight error in the incorrect model identification as well as in the measurement of the output signal results in a large error in the identification signal.

Quantum computation is based on the principles of quantum superposition, interference and entanglement. The quantum state is always
It has components that correspond to some or all of the classically possible states. This quantum effect is called a superposed state. Computers that obey quantum rules can process different inputs using massive parallelism and produce a superposition of the outputs. In this case, the quantum computer is a physical machine and can accept input states that represent a coherent superposition of many possible and different inputs. Then, after receipt, those input states can be developed into a superposition of the corresponding outputs. Quantum entanglement can be used to encode data into a superposition of a plurality of previously prepared non-trivial multi-particles of the ground state. Then, by using quantum interference (dynamic process), the superposition of intermediate particles can be changed in a predetermined manner, and the initial quantum state (input) can be expanded to the final state (output). Quantum computers use quantum interference in different computation paths to improve correct results and suppress erroneous computation results. When viewing quantum computation as multi-particle interference, a common tendency to demonstrate quantum algorithms can be identified. Multiparticle interference is essentially a quantum process, as there is no similarity (unlike single particle interference) in classical interference.

Classic computers are based on quantum physics, but are not completely quantum. Classical computers do not use the "quantity" of matter, which is actually a problem at the information theory level. That is, in a classical computer, information is macroscopically recorded in a two-level system. The wires that carry current in a computer are in two basic states. One is a state in which no current flows, that is, a state of logical "0",
The other is in a state where current is flowing, that is, a state of logical "1". These two states represent bit information. All calculations are based on the logical manipulation of bits using logic gates acting on the wires representing these bits. However, as described here, a quantum computer can be simulated on a classical computer.

Instead of wires and currents, quantum computers use, for example, the electronic state of atoms to record information. For example, two quantum states are ground state | 0
> And excited state | 1> (using Dirac notation). Since atoms obey the laws of quantum mechanics, the most common electronic state is the superposition of these two basic states and is expressed as | ψ ₁ > = c ₁ | 0> + c ₂ | 1>. This electronic state is, for short, a qubit,
It is called a qubit. In addition to the 0 and 1 states, the qubit is in all states "between".
For two classic bits, there can be four cases, 00,01,10,11. This is generally | ψ ₁ > = c ₁
| 00> + c ₂ | 01> + c ₃ | 10> + c ₄ | 11> in comparison with two qubits. For example,

[0059]

(Equation 8)

In the case of the famous Einstein-Podolski-Rosen (EP
R)) State

[0061]

(Equation 9)

Is obtained. The two qubits in this state represent a degree of correlation that is impossible in classical physics. Therefore, the Bell inequality that does not satisfy all local (ie, classical) states does not hold. This phenomenon is called entanglement and is the basis of accurate quantum computation. By using a large number of entangled qubits locally, the computational speed of a quantum computer can be significantly improved compared to the computational speed of a classical computer. Thus, classical and quantum computation differ in the way they encode and manipulate information. That is, whether the logic is based on classical logic (Boolean logic) or quantum logic plays an important role.

The quantum computer can efficiently solve problems such as factorization and database search. Quantum search algorithms (QSAs) can be used to solve problems that cannot be solved by classical algorithms. In one embodiment, quantum mechanical algorithms combine with efficient database algorithms that take advantage of the special properties of databases.

General Structure of Quantum Algorithm In one embodiment, a quantum algorithm represented by a quantum circuit is converted into a corresponding programmable quantum gate. This gate can be represented by a matrix operator, and when applied to the vector representation of the input state of the quantum register, the generated result is a vector representation of the output state of the quantum register as shown in FIG.

The quantum computation is based on three operators acting on the quantum coherent state. These operators are
Superposition, entanglement and interference. A coherent state represents an evolved state of minimal uncertainty (according to Heisenberg, a quantum state of "maximum classical properties"),
It can be expressed as a solution to the corresponding Schrodinger equation. The Hadamard transform produces a superposition from classical states. Quantum operators, such as CNOT, create strong superposition. The quantum fast Fourier transform provides interference. Many operations in quantum computation are efficiently realized by controlled phase adjustment, replacement, superposition of transforms, and generalization of phase adjustment to block matrix transformation. These operations generalize the operators used within quantum search algorithms implemented on classical computers. Next, Deutsch-Jozs
The use of this technique in the general case of simulating on a classical computer will be described according to an example based on the algorithm of a) and the Grover algorithm.

When the problem solved by the quantum algorithm is given by a function f: [0, 1] ⁿ → [0, 1] ^m , f
Find the nature of And FIG. 8 shows the structure of the quantum algorithm in a high-level representation. In FIG. 8, an input 801 represented by a function f is passed to an encoder 802. The output of encoder 802 is operator U _f 803. The operator 803 is passed to the quantum unit 804. The output 804 of the quantum part is a set of basis vectors 805. The basis vector 805 is passed to the decoder 806. Answer 807
Is the output of the decoder 806. Input 801 and output 8
07 creates a bit string layer. The encoder 802 and the decoder 806 create a mapping table and an interpretation space layer. Operator 80
3. The quantum part 804 and the basis vector 805 exist in the complex Hilbert space.

The input of the quantum algorithm is a function f that maps a bit sequence into a bit sequence. This function can be represented by a mapping table that defines the mapping of all strings. Function f
Is first encoded into a unitary matrix operator U _F depending on the nature of f. In a sense, the unitary operator U _F
Is to compute f when its input and output strings are encoded into canonical basis vectors in complex Hilbert space. U _F uses f to map the vector code of all strings to the vector code of its image. The complex square matrix U _F is unitary if and only if its inverse matches the conjugate transpose U _F ^-1 = U _F. Unitary matrices are always invertible and preserve the norm of the vector.

After the matrix operator U _F has been generated, U _F is
It is incorporated in a quantum gate G that is a unitary matrix. The structure of the unitary matrix depends on the form of the matrix U _F and the problem to be solved. Quantum gates are central to quantum algorithms. The quantum gate operates on the first canonical basis vector to form a complex linear combination of the basis vectors (superposition).
Produces the output of This overlay contains all the information needed to answer the first question.

After the superposition is generated, measurement is performed to extract information. Measurement in quantum mechanics is a non-deterministic operation that produces as output only one of the input superposition basis vectors. The probabilities of all basis vectors that are the output of the measurement depend on the complex coefficient (probability amplitude) of the complex linear combination that is input. The partial action of the quantum gate and the measurement is passed to the quantum unit 804. The quantum unit 804 is repeated k times to create a set of k base vectors. Since measurement is a non-deterministic operation,
These basis vectors need not be identical. Then, one piece of information necessary for answering the question is encoded.

At the end of the algorithm, an interpretation of the collected basis vectors is performed to obtain an answer to the first problem which indicates a particular probability. FIG. 9 shows the operation of the encoder unit 802. As shown in FIG. 9, the function f is encoded into a matrix U _F in three steps. First, the function f: [0, 1] ⁿ →
The mapping table of [0,1] ^m is a single mapping function F: [0,1] ^{n + m} → [0,
1] It is converted into a mapping table of ^{n + m} . Where the single-shot function F is

[0071]

(Equation 10)

Is satisfied. The single-shot function is derived from the requirement that U _F be unitary. Since unitary operators are reversible, two different inputs cannot be mapped to the same output. U _F is a representation of F in a matrix,
F is assumed to be single-shot. Since f can be non-single-shot, the direct use of the matrix representation of the function f results in a non-unitary matrix. Therefore, the single shooting property is satisfied by increasing the number of bits and considering the function F instead of the function f. The function f is (y0, ..., y
_m-1 ) = (0, ..., 0), and m of the last output character string
By reading these values, it is always calculated from F.

The mapping table of the function F in the second step of FIG.

[0074]

[Equation 11]

According [0075], it is converted to the mapping table of the U _F. Sign mapping

[0076]

(Equation 12)

Represents a target complex Hilbert space)
Is

[0078]

(Equation 13)

Is satisfied. The symbol τ maps a bit value to a two-dimensional complex vector belonging to the C ² standard. Using the tensor product, τ maps an ^n- dimensional bit sequence of a general state into a 2 ^n- dimensional vector, and turns this state into an n-bit combined state that makes up a register. All bit states are converted to corresponding two-dimensional basis vectors. Next, the state of the string is set to the corresponding 2 by assembling all the bit vectors using the tensor product.
Maps to an ^n- dimensional basis vector. Thus, the tensor product corresponds to the vector of the state coupling.

Finally, in the third step of FIG. 9, the mapping table of U _F

[0081]

[Equation 14]

Is converted to U _F using This rule can be understood by considering the vectors | i> and | j> as column vectors. When these vectors is associated with a standard group, U _F defines a mapping replacement row of the unit matrix. In general, row | j> is mapped to row | i>. FIG. 10 shows the operation of the quantum unit 804. The center of the quantum unit 804 is the quantum gate 1002. The quantum gate is a matrix U
Depends on the nature of _F.

The matrix operator U _F is an output of the encoder unit 802 and an input of the quantum unit 804. In the quantum unit 804, the matrix operator U _F is first incorporated into a quantum gate G, which is a more complicated gate. Unitary matrix G
Is applied k times to the first canonical basis vector | i> of 2 ^{n + m} dimensions. Each time, a complex superposition G | 0. . 01. . 1> is measured, so that
One basis vector | x _i > is created. The measured basis vectors {| x ₁ >, .., | x _k >} are collected together. This collection is the output of the quantum unit 804.
The "intelligence" of this algorithm lies in its ability to assemble a quantum gate that can extract the information needed to determine the properties required for f and store it in a collection of output vectors.

To represent a quantum gate, it is convenient to use a diagram called a quantum circuit. FIG. 11 shows an example of a quantum circuit.
Shown in Each rectangle is associated with an n × n matrix. Here, n is the number of lines entering and exiting the rectangle. For example, a rectangular U _F is related to the matrix U _F. Quantum circuits are high-level representations of gates. Quantum circuits can be organized into corresponding gate matrices using transformation rules. These conversion rules are shown in FIGS. 12A-12.
Shown in F.

The decoder section 806 interprets the basis vectors collected after repeating the execution of the quantum section 804. Decoding the basis vector means reconverting the basis vector into a bit string. Then, if the basis vector contains the answer to the problem already started, the bit string is interpreted directly. That is, a bit string is used as a coefficient vector of a certain equation system to obtain a solution to be searched.

As an example of the above algorithm, it is convenient to develop a quantum gate representation of the Deutsche-Yoser (DJ) algorithm. This gate uses the matrix operator U
_This is realized according to the method developed in connection with FIG. 8 representing the structure of _F. The DJ algorithm is described as follows. Fixed value or equilibrium function f: {0, 1} ⁿ → {0,
Given 1｝, determine whether f is fixed or balanced (this problem is very similar to Deutsche's problem, but Deutsche's problem is n> 1
Was generalized to the case).

A. First, consider the DJ algorithm encoder for the special case of n = 2. Therefore,

[0088]

(Equation 15)

Consider the case (1). In this case, the mapping table of F is

[0090]

[Table 1]

Is given by The mapping table of f is input to the encoder unit, and it is encoded into the matrix operator U _F. U _F operates in complex Hilbert space. The function f is expressed by the following equation

[0092]

(Equation 16)

Is encoded into a single-shot function F according to Therefore, the mapping table of F

[0094]

[Table 2]

Is obtained. The mapping table of F

[0096]

[Equation 17]

[0097] Use is encoded into the mapping table of the U _F.
Here, τ is the code mapping defined above. Therefore, the mapping table of U _F is

[0098]

[Table 3]

## EQU10 ## Based on a mapping table of U _F, compute the corresponding matrix operator. This matrix is a rule

[0100]

(Equation 18)

Is obtained. Therefore, U _F is

[0102]

[Table 4]

The matrix is as follows. Using the tensor product of the matrix, U _F is

[0104]

[Equation 19]

Can be expressed by here,

[0106]

(Equation 20)

Is a tensor product, I is a quadratic identity matrix, C
Is

[0108]

(Equation 21)

Represents a NOT matrix defined by The matrix C inverts the basis vector. That is, this matrix converts vector | 0> to | 1> and | 1> to | 0>. The vector obtained when applying the matrix U _F to the tensor product of three two-dimensional vectors is obtained by applying the matrix I to each of the first two vectors of the input vector, and obtaining the matrix C
Is applied to the third vector of the input vectors, resulting in a tensor product of the three vectors. That is, U
_F is the first two of the tensor products of the input vector
One vector is stored as it is, and the third vector is inverted (the effect of C). This effect corresponds to the constraints shown in mapping table of U _F.

B. next,

[0111]

(Equation 22)

Consider the case (1). In this case, the mapping table of f is

[0113]

[Table 5]

Is given by Mapping table of F

[0115]

[Table 6]

Is obtained. A mapping table of F and encoded map table U _F, mapping table of U _F

[0117]

[Table 7]

Is obtained. This mapping table can be easily converted into a matrix. All vectors are stored, corresponding matrix is two ^{3-dimensional} matrix. That is, the matrix corresponding to U _F

[0119]

[Table 8]

Is obtained. This matrix, using the matrix tensor product,

[0121]

(Equation 23)

It can be expressed by U _F independently expands all the two-dimensional basis vectors forming the tensor product of the input vectors. None of the vectors affect other vectors. For example, the equilibrium function

[0123]

(Equation 24)

Consider the following. In this case, the mapping table of f is

[0125]

[Table 9]

Is as follows. Performing the calculations described above yields the following mapping table:

[0127]

[Table 10]

Represents a single-shot function F (where f is encoded). Then encodes the mapping table of F to the mapping table of the U _F, mapping table of U _F

[0129]

[Table 11]

Is obtained. Therefore, the matrix corresponding to U _F

[0131]

[Table 12]

Is obtained. This matrix cannot be expressed as a tensor product of small matrices. That is, when represented by a block matrix,

[0133]

[Table 13]

Is obtained. It can be seen that the matrix operator acting on the third vector forming the tensor product of the input vectors depends on the values of the first two vectors. For example, if the first two vectors are | 0> and | 0>, the operator acting on the third vector is an identity matrix. If the first two vectors are | 0> and | 1>, the third expansion is determined by the matrix C. Thus, this operator is tangling, that is, correlating the vector of tensor products.

C. Consider a general function of n = 2. In this general case, the mapping table for f is

[0136]

[Table 14]

Is given by Where f _i ∈ ｛0,
1｝, i = 00, 01, 10, and 11. If f is a constant,

[0138]

(Equation 25)

The following holds. If f is balanced |
_{{F i: f i = 0} } | = | {f i: f i = 1} | is satisfied.
The single-map function F (where f is encoded) is the following mapping table

[0140]

[Table 15]

Can be represented by Then encodes the mapping table of F to the mapping table of the U _F, mapping table of U _F

[0142]

[Table 16]

Is obtained. The matrix corresponding to U _F is expressed in the following general form using a block matrix

[0144]

[Table 17]

Can be expressed by Here, if f _i = 0, M _i = I,
In the case of f _i = 1, M _i = C. Where i = 00,0
1, 10, and 11. This matrix is transformed by applying a zero operator to the third vector of the input vector to produce a zero probability amplitude when mapping the first two vectors of the input vector to another vector. Therefore, the first two vectors do not always change. On the other hand, when the first two vectors of the input vector are mapped to themselves, the operator belongs to the set where M _i {I, C} holds, and the operator is the third vector. Applied to If all M _i match, the operator U _F encodes the value function. Otherwise, the operator U _F encodes a non-definite function, and if | [M _i : M _i = I] | = | [M _i : M _i = C] | I have.

In the general case where D.n> 0, the mapping table of the input function f is

[0147]

[Table 18]

Is given by Where f _i ∈ ｛0,1｝,
i∈ (0,1) ⁿ holds. If f is a constant

[0149]

(Equation 26)

Is obtained. If f is balanced, | [f _i :
_{f i = 0] | = |} [f i: f i = 1] | become. The mapping table of the corresponding single-shot function F is

[0151]

[Table 19]

Is obtained. Then encodes the mapping table of F to the mapping table of the U _F, mapping table of U _F

[0153]

[Table 20]

Is obtained. The matrix corresponding to U _F is expressed in the following general form using a block matrix

[0155]

[Table 21]

Can be expressed by Here, when f _i = 0, M _i =
I, and if f _i = 1, then M _i = C. Where i∈
{0, 1} ⁿ . This matrix applies the operator M _i {I, C} to the last vector, leaving the first n vectors intact. If all M _i coincides with I or C, matrix encodes the value function

[0157]

[Equation 27]

Can be expressed by In this case, entanglement is created
Not. Otherwise | Condition | _i: M_i= I｝ | =
｜ ｛M_i: M_i= C｝ | is satisfied, f is balanced
And the operator creates a vector correlation. Next, Enko
Matrix U which is the output of_FIs the quantum algorithm of the DJ algorithm
Embedded in the chart. This gate is shown in FIG.
Is represented using a quantum circuit. 12C identity
Using rules, the circuit of FIG. 13 is shown in FIG.
Circuit.

If f is a constant and its value is 1, the matrix operator U _F is

[0160]

[Equation 28]

Is represented by Thus, as shown in FIG. 12A, U _F operates on n + 1 two-dimensional vectors simultaneously forming a tensor product of the input vectors, n
Decomposed into +1 small operators. The resulting circuit is shown in FIG.
Shown in Using FIG. 12B to determine partial gates acting on all two-dimensional vectors of the input, FIG. 16 is obtained. All input vectors expand independently. This is because the operator U _F does not make any correlation. Thus, all expansions of the input vector can be analyzed separately. Since this circuit has M · I = M, it can be simply represented as shown in FIG. Since H ² = I, the circuit is as shown in FIG.

Next, the effect of the operator acting on all of the vectors

[0163]

(Equation 29)

Consider the following. Using the results of these operators in conjunction with the operations shown in FIGS. 12D and 12C, a circuit representation as shown in FIG. 19 is obtained. Thus, if f is a constant with the value 1, the first n vectors are saved. A similar analysis is performed for the constant function of value 0. In this case, U _F

[0165]

[Equation 30]

Are represented by FIG. 20 shows the last circuit. Also in this case, the first n input vectors are retained. Therefore, the output values of the first n input vectors after the operation of the quantum gate are | 0> as they are. Operating the circuit of FIG. 14 results in a gate that generally implements the DJ algorithm shown in FIG. FIG.
Is expanded to the final circuit shown in FIG.

If n = 2, U _F is

[0168]

[Table 22]

The format is as follows. Here, M _i {I, C}. Here, i = 00, 01, 10, and 11. Quantum gate

[0170]

[Equation 31]

Then,

[0172]

[Table 23]

[0173]

[Table 24]

In the general case where n> 0, we have U _F
Is

[0175]

[Table 25]

The format is as follows. Here, M _i {I, C}. Here, i∈ ｛0,1｝ ⁿ . Quantum gate

[0177]

(Equation 32)

Then,

[0179]

[Table 26]

Is obtained. This table uses the bit string operator " ^. " This operator represents the parity of the bits obtained by ANDing the bits of two strings. This operator, given two bit strings x and y of length n, gives

[0181]

[Equation 33]

Is defined as Here, the symbol "." Used between the two bits indicates the logical operator AND. To show that the matrix ^{n + 1} H is actually obtained in the above table,

[0183]

[Equation 34]

It suffices to prove that the following holds. Proof is made by induction. When n = 1, the following equation

[0185]

(Equation 35)

The following holds. If n> 1, the following equation

[0187]

[Equation 36]

Is obtained. The matrix ^{n + 1} H is obtained from ⁿ H by the tensor product. Similarly, the matrix

[0189]

(37)

Is calculated,

[0191]

[Table 27]

[0192]

[Table 28]

Is obtained. This operator is exclusive with the input vector | 0. . 01>, only the first column of gate G is calculated. Therefore, only the first column is shown in the following table showing the calculation results.

[0194]

[Table 29]

[0195] If f is a constant, it can be seen that all matrices M _i are the same. This is

[0196]

(38)

The following is obtained. This means that the sum of +1 in this sum is -1
Because it is equal to the number of Therefore, the input vector |
0. . 01> is a vector | 0. . 00> and | 0. . 01
> Are superimposed. If f is balanced,
The number M _i = I is equal to the number M _i = C. this is,

[0198]

[Equation 39]

Is obtained. Therefore,

[0200]

(Equation 40)

The following is obtained. For the equilibrium function, the input vector |
0. . 01> is a vector | 0. .
00> or | 0. . It can be seen that the image cannot be mapped to the superposition including <01>. The quantum part ends with the measurement. The measurable outputs and their probabilities can be determined. The results are shown in the following table.

[0202]

[Table 30]

In the set of AB, all elements of A belong and elements of B do not belong. This set may be denoted as A / B. The quantum part is repeated only once in the Deutsche-Yosa algorithm. Thus, the last acquisition is performed on only one vector. In Deutsche's algorithm, when the last basis vector is measured,
It is necessary to interpret the basis vectors to determine whether f is constant or balanced. The obtained vector is | 0. . If 0>, the function is constant; otherwise, the function is balanced. That is, when f is a constant, the gate G outputs the measured basis vector | 0. . 00>
And | 0. . 01> only have a non-zero probability amplitude exclusively. Also, when f is balanced, gate G creates a vector such that these two vectors have a zero coefficient in a linear combination of the basis vectors. In this way, the resulting vector is decoded and the following solution to the Deutsche-Yosa problem is obtained.

[0204]

[Table 31]

The algorithm of Grover described next is a modification of the algorithm of Deutsche-Yosa.
Grover's algorithm is

[0206]

[Table 32]

Can be expressed by The Deutsche-Yosa algorithm classifies input functions into two types and determines to which question the input function belongs. Although Grover's algorithm handles 2 ⁿ classes of input functions, the shape of the problem is similar for these difficult problems (each function of the type described is considered a class). In the following, a special function of n = 2 is first considered for a clear description. Next, a general case where n = 2 will be described. Finally, a general case where n> 0 will be described.

First, consider the case where n = 2 f (01) = 1. In this case, the mapping table of f is

[0209]

[Table 33]

Is defined as The function f is encoded into a single-shot function F created as described in connection with FIG. The function f is expressed by the following equation

[0211]

[Equation 41]

Is encoded into a single-shot function F according to next,
The mapping table of F is

[0213]

[Table 34]

Is obtained. Next, the mapping table of F

[0215]

(Equation 42)

[0216] encoded into the mapping table of U _F using. Where τ is the code mapping described in connection with FIG. Therefore, the mapping table of U _F is

[0219]

[Table 35]

The following is obtained. From the mapping table of U _F, compute the corresponding matrix operator. This matrix is a rule

[0219]

[Equation 43]

Obtained by using As a result, the corresponding matrix operator

[0221]

[Table 36]

Is obtained. If the first vector is | 0> and the second vector is | 1>, the first and second input basis vectors forming the tensor product of the input vectors are left as they are, and the third input basis is Invert the vector. This is consistent with the constraints of the U _F described above.

Next, the following general formula, which is a more general case, is used.

[0224]

[Equation 44]

Consider the following. The corresponding matrix operator is

[0226]

[Table 37]

Is as follows. From the case of n = 2 of n> 1 is a shortcut that generalize the operator U _F in. The operator C on the main diagonal of the block matrix is based on the correspondence of the cell named by the vector | x>. Where x
Is a bit string obtained by mapping the vector | x> by f. Therefore,

[0228]

[Table 38]

Is obtained. The matrix U _F which is the output of the encoder
Is incorporated in the quantum gate. This gate is shown in the quantum circuit of FIG. The operator D _n is said to be an n-order diffusion matrix and plays a role of interference in this algorithm. QFT _{n of} Shore's algorithm and Deutsche-Yosa algorithm and Simon
It plays the same role as ⁿ H of the algorithm of n). This matrix is

[0230]

[Table 39]

Is defined as Using the transformation shown in FIG. 12C, the circuit of FIG. 23 is combined into the circuit of FIG. For example, U _F

[0232]

[Table 40]

Consider the case of Next, the quantum gate

[0234]

[Equation 45]

Then, the case is calculated. In this case, each term in the equation is

[0236]

[Table 41]

Is obtained. If h = 1, then

[0238]

[Table 42]

Is obtained. In one example, the operator ³ H is
Let the first canonical basis vector | 001> be the superposition of all basis vectors with the same (real) coefficient in absolute value. However, this coefficient has a positive sign if the last vector is | 0>, and a negative sign otherwise. Operator U
_F makes a correlation. That is, the operator U _F inverts the third vector when the first two vectors are | 0> and | 1>. Finally,

[0240]

[Equation 46]

Create interference. That is, all basis vectors

[0242]

[Equation 47]

On the other hand, the operator U _{F inverts} the sign of its initial probability amplitude α ′ _x0x1y0 ,

[0244]

[Equation 48]

Average of all probability amplitudes of a vector of the form

[0246]

[Equation 49]

The output probability amplitude α '
Calculate _x0x1y0 . In the example

[0248]

[Equation 50]

Is as follows. For example, if the basis vector is | 00
0>

[0250]

(Equation 51)

Is obtained. In general, for n = 2, U _F is

[0252]

[Table 43]

Is obtained. Quantum gate in this general case

[0254]

(Equation 52)

Is

[0256]

[Table 44]

The following is obtained. Next, when G is applied to the vector | 001>,

[0258]

(Equation 53)

Consider the following. When the output vectors are measured and the first two 2D basis vectors of the resulting tensor product are decoded,

[0260]

[Table 45]

The result of the result probability is obtained. In the general case where n> 0, U _F is

[0262]

[Table 46]

The format is as follows. Quantum gate

[0264]

(Equation 54)

Is

[0266]

[Table 47]

Is as follows. For example, if h = 1,

[0268]

[Table 48]

Is obtained.

[0270]

[Equation 55]

Thus, this column is

[0272]

[Table 49]

The following is obtained. Therefore,

[0274]

[Table 50]

Is obtained. Next, the vector | 1> is added to the matrix operator {[−1+ (2 ⁿ −1) / 2 ⁿ⁻¹ ] I + C /
2 ^n-1 ｝ H / 2 ^{n / 2} and matrix operator ｛[(2 ⁿ -1) / 2 ^n-1
When I + [-1 + 1 / ^2n-1 ] C] H / ^{2n / 2} is applied,
Respectively,

[0276]

[Equation 56]

Is obtained. Therefore,

[0278]

[Equation 57]

Is obtained. Expressing this equation as a block vector,

[0280]

[Table 51]

Is obtained. Operator on vectors of this form

[0282]

[Equation 58]

Is applied. Vectors of this form

[0284]

[Table 52]

It is assumed that Here, α and β are [2 ⁿ -1]
It is a real number that satisfies α ² + β ² = 1. The result of applying the operator is

[0286]

[Table 53]

Is obtained. Vector of this form G _{h = 1} |
0. . 01>, h times operator

[0288]

[Equation 59]

By applying, the coefficient at the t time becomes

[0290]

[Equation 60]

[0291] Thus, as β increases, α decreases. When the output vector from the Glover quantum gate is measured as in the Deutsche-Yosa algorithm, the output is interpreted.

[0292]

[Equation 61]

Is obtained. This step is relatively simple. That is, by selecting a large h, the searched vector with a probability close to 1

[0294]

(Equation 62)

Is sufficient. After finding the vectors, the first n basis vectors of the obtained tensor product are decoded into binary values, and the final solution is a string

[0296]

[Equation 63]

Is obtained. Information theory Next, the development of the quantum algorithm from the viewpoint of information theory will be described. Here, a complex vector input to the quantum gate is considered as an information source at both the classical level and the quantum level. Shannon's entropy _Hsh is a classical information unit. Hilbert space

[0298]

[Equation 64]

Consider a complex vector having an absolute value of 1. Where Hil _Qk is two-dimensional for all k and is a complex linear combination of basis vectors.

[0300]

[Equation 65]

[0301] Next, the base

[0302]

[Equation 66]

The state | ψ> of Shannon's entropy

[0304]

[Equation 67]

[0305] here,

[0306]

[Equation 68]

[0307] is a vector | think the probability of measuring the _{i 1, i 2 ... i n} >. Von Neumann's entropy is used to measure information stored in quantum correlation. Let ρ = | ψ><ψ | be the density matrix associated with the state | ψ>,

[0308]

[Equation 69]

[0309] next,

[0310]

[Equation 70]

The following is defined. Here, Tr [ _1, ... _{, n} ]
_-T (…) is a partial trace operator. The entropy of von Neumann of qubit j of | ψ> is

[0312]

[Equation 71]

Is defined. Also,

[0314]

[Equation 72]

The following definition is also used. These quantities are shown in FIG. 25 using the Wenn diagram. The measure of entropy is different from the measure of most physical quantities.
In quantum mechanics, we must distinguish between observables and states. Observed quantities (position, momentum, etc.) are mathematically described by a self-conjugate operator in Hilbert space. A state (generally mixed) is represented by a density matrix ρ (0, that is, a Hermitian operator of a trace Tr (ρ) = 1. The expected value of the observed amount A of the state ρ is <A> = Tr (ρ
A). Entropy is not an observable. Thus, no operator has the property that its expected value in a state is its entropy. To be precise, operators are functions of state. The physical entropy is given by the Janis relation between the information-theoretic entropy and the physical entropy using Boltzmann's constant k _B.

[0316]

[Equation 73]

A particular value of can be associated with a quantum object. The classical limit of entropy

[0318]

[Equation 74]

Can mathematically represent a coherent state.
Here, it is possible to measure only the probability of finding a particle in a minimal uncertain state centered on a classical value, that is, a coherent state.

[0320]

[Equation 75]

In the general case of

[0322]

[Equation 76]

In this case, | z> = W (z) | 0> is set to the coherent state, and q and p are the expected values of the position or the momentum, respectively. | 0> is clearly a wave function in the configuration space

[0324]

[Equation 77]

Is given by W (z) is a unitary operator

[0326]

[Equation 78]

Where Q and P represent position or momentum operators, respectively. Next, a classical density distribution corresponding to the density matrix ρ is defined by ρ (Z): = <z | ρ | z>. For all functions f (z),

[0328]

[Expression 79]

At most one density matrix ρ exists. S (x) due to concaveness: = ｛− xlnx (x>
0); 0 (x = 0)}, S (<z | ρ | z>)
(<Z | ρ | z>, so that

[0330]

[Equation 80]

Since the following holds,

[0332]

[Equation 81]

The relationship becomes true. More generally, for any convex (concave) function f,

[0334]

(Equation 82)

The following holds. by the continuity of ρ (z)

[0336]

[Equation 83]

Then, for all z, S (<z |
ρ | z>) = (<z | S (ρ) | z>). Then S
All | z> must be eigenvectors of ρ for the narrow sense concaveity of (•). But it is impossible, so

[0338]

[Equation 84]

The following holds. Classic entropy is not invariant in all unitary transformations. That is, for all U

[0340]

[Equation 85]

Cannot be said. However, this relationship only applies to restricted classes. For example, U = W (z
₀ ), the following equation

[0342]

[Equation 86]

The following holds. Also, this argument is dz = d
In the case of z '(canonical transformation), UW that determines the time of the phase factor
It operates on all the unitary U for which (z) = W (z ′) holds. | ψ> is a pure state (unit vector) and ρ =
If | ψ><ψ |, then ρ (z) = | <ψ | z> | ² and

[0344]

[Equation 87]

The following holds. By substituting | ψ> = | z ₀ >,

[0346]

[Equation 88]

Is obtained. On the other hand, there is a pure state of any high classical entropy. This only needs to show that when ε> 0, a unit vector | ψ> that satisfies <ψ | z><ε can be obtained for all z. In pure state, the well-known inequality

[0348]

[Equation 89]

The following holds. The minimum entropy state of a classical system can be represented exactly by the density matrix | z >><z |, and the result is

[0350]

[Equation 90]

It is considered that

[0352]

[Equation 91]

In order to reduce Sup | ρ (z) |
Must approach one. Otherwise, the above inequality will be very large for the entropy of classical systems. Then, if Sup | ρ (z) | is exactly equal to 1, continuity will result in ρ (z ₀ ) = 1, ie, <z ₀
| Ρ | z _0> = 1 is z ₀ to satisfy will be present to.

[0354]

(Equation 92)

Thus, this is

[0356]

[Equation 93]

[0357] It is shown that On the other hand, Tr
Since (ρ) = 1 and the other eigenvalues of ρ must be 0, ρ = | z ₀ >><z ₀ | holds. The coherent state in which the potential of the harmonic oscillator is the least uncertain is minimized by adding the constraint that the ground state is a member of the set. ) Can be defined as a coherent state. The minimally uncertain coherent state is considered to be as close as possible to the classical state. Beyond the limits of the harmonic oscillator system, coherent states can be expanded for quantum (Schrodinger) systems of general potential and general Lie symmetry. These coherent states are referred to as (general) minimally indeterminate coherent states and (general) distance operator coherent states. A different generalization exists in the coherent state of the harmonic oscillator system. This is the concept of "squeezed state". (Squeezing is the contraction of orthogonal fluctuations below the level associated with vacuum.) One-mode harmonic oscillator (Schrodinger's cat state)
The even and odd coherent states of are representative of non-classical states. Schrodinger's cat state has properties similar to the squeezed state. That is, the squeezed vacuum state and the even coherent state include the Fock state of even-numbered photons.

In quantum mechanics, two non-commutative observables cannot be measured simultaneously with any precision. This fact, often referred to as Heisenberg's uncertainty principle, is a fundamental limitation and is not related to either defects in existing real measuring equipment or experimental errors in observations. It is rather a property inherent to the quantum state itself. The uncertainty principle offers a (paradoxically sufficient) way to avoid multiple interpretation problems. The uncertainty principle specified for a given pair of observables expresses its mathematical manifestation as an uncertainty relationship. Basically non-commutative observables (ie, position and momentum

[0359]

[Equation 94]

The strict first derivative of the uncertainty relation of

[0361]

[Equation 95]

The following holds. This is a consequence of the nature of the Fourier transform linking the wave functions of the system in the representation of position and momentum. Expressing the quantum uncertainty relationship (UR) in terms of entropy, or information ("entropy UR" -EUR), helps in this regard. The normal "standard UR" (of standard deviation) is

[0363]

[Equation 96]

Is as follows. (The second term of this inequality is the covariance of observables A and B,

[0365]

(97)

It should be noted that A and B are
Entropy inequality

[0367]

[Equation 98]

Is the observed quantity of the state | ψ>, which is an information format that more appropriately expresses the uncertainty principle

[0369]

[Equation 99]

[0370] Given two non-commutative observables, we can derive its uncertainty relationship. States that satisfy the inequality equality are said to be intelligent states. )
For example, an arbitrary continuous parameter λ, which is a generator of a parametric expansion, and an arbitrary Hermitian observable A
Consider (λ). In that case, the UR is

[0371]

[Equation 100]

The following is obtained. here,

[0373]

[Equation 101]

Is the average of the uncertainty parameters of the observables,

[0375]

[Equation 102]

Is the measured distance of A's conjugate variable space. This formula generalizing UR would hold for position-momentum, phase-number or any combination. If the initial state and the final state are orthogonal,

[0377]

[Equation 103]

All states that can be expressed by the equation

[0379]

[Equation 104]

It has been found that only an intelligent state that satisfies is satisfied. However, if the initial state and the final state are not orthogonal, the state does not satisfy this equation. In this case, the generator A of the parametric expansion
Is divided into two parts, A ₀ + A _l ,

[0381]

[Equation 105]

All the states that can be expressed by are intelligent states of non-orthogonal initial state and final state. Where A ₀ is a normalized eigenvector that degenerates the spectrum {a ₀ } of the set I of quantum numbers

[0383]

[Equation 106]

Where A ₁ is a matrix element (A ₁ ) _ii = 0 = 0 (A ₁ ) _jj and (A ₁ ) _ij = (A ₁ )
_ji = have a _1. It is useful to compare the various features of "maximum information" and show their relevance to "minimum uncertainty". The following discussion is mainly for the sake of simplicity, mainly in the "simple case" observable (the smallest non-trivial Boolean

[0385]

[Equation 107]

[0386] However, generality is not lost. The amount of the problem is one effect, E: Iψ
(E) = Eψln (Eψ) + E′ψln (E′ψ), E
'= Information on IE. The incompatibility or mismatch of the (unknown) properties E and F generally does not allow the probability of measuring or preparing them both at the same time. In particular, E
= E ^Q (X), F = F ^P (Y) is the spectral projection of the position and momentum associated with the bounded measurable sets X, Y,

[0387]

[Equation 108]

The following equation holds:

[0389]

(Equation 109)

Is equivalent to Therefore, the determination of the "specific" position and momentum are mutually exclusive, and the problem arises as to how much "uncertainty" the position and momentum can be known at the same time. Consider the maximum binding knowledge, that is, some reasonable property of the binding information. In this case, the above expression is equivalent to

[0391]

[Equation 110]

[0392] "State of maximum information" can be defined using three types of values. First, the maximum value of the expression E {+ F} is handled, and the corresponding "state of maximum information" will be systematically described below. E of any pair of effects E and F
Not only {+ F}, but E {F} and I {(E)
Similarly, the problem of the maximum value for + Iψ (F) will be described. In particular, we show that each quantity can be maximized only if there is a condition that results in a minimum uncertainty product of the UR. Further, the projection of the maximum of Iψ (E) + Iψ (F) is (if present)

[0393]

(Equation 111)

This corresponds to the projection of the maximum value of the quantity. Eψ + F
For the maximum value of ψ, <ψ | E | ψ> + <ψ | F | ψ>
Since the variation of −λ <ψ | ψ> must be zero, the following equation: (E + F) | ψ> = (Eψ + Fψ) |
ψ> holds. Taking the increment for E or F and calculating its expected value,

[0395]

[Equation 112]

[0396]

[0397]

[Equation 113]

Becomes the minimum UR. Similarly, the product Eψ
When the maximum value of Fψ is obtained, (FψE + EψF) | ψ> =
2Eψ ・ Fψ | ψ>,

[0399]

[Equation 114]

In the case of

[0401]

[Equation 115]

Becomes the minimum UR again. Finally, the maximum information sum Iψ (E) + Iψ (F) is

[0403]

[Equation 116]

[0404] This is realized in a state where the above is satisfied.
In general, this equation has all stationary points, for example,

[0405]

[Formula 117]

That is, it includes the coupling eigenstate. One of the objectives here is to find the state of the maximum information about the positive results of E and F,

[0407]

[Equation 118]

Assume that Next, this equation is

[0409]

[Equation 119]

[0410]

[0411]

[Equation 120]

However, it becomes the minimum uncertainty product of UR again. The three concepts of maximum information are invariant as long as they exhibit a minimum uncertainty product. For example, if E and F are the spectral projections of position and momentum, respectively, E = E ^Q (X), F = F ^P
(Y). The sum of probabilities, E {+ F}, is given by X and Y as bounded measurable sets:

[0413]

[Equation 121]

In the state of ψ = ψ _min that holds, the maximum is shown. here

[0415]

[Equation 122]

Is the largest eigenvalue of the compact operator (FEF), and g ₀ is the corresponding

[0417]

[Equation 123]

This is an eigenvector satisfying the following. From the above description, it is clear that ψ _min must be an (E + F) eigenstate. Also, this is directly known by the following method.

[0419]

[Equation 124]

[0420] Then,

[0421]

[Equation 125]

Is obtained. ψ _min is the symmetric form

[0423]

[Equation 126]

Can be represented by The value of ψ _min is divided into three quantities (Eψ · Fψ), (Eψ + Fψ) and (Iψ (E)
+ Iψ (F)) is maximized. And ψ
_min minimizes the uncertain product △ ψE · △ ψF. Therefore, the maximum information (minimum entropy) and the minimum uncertainty are
It is made up of intelligent coherence and will match again. The next new presentation describes the role of changing the entropy of quantum algorithms as an information data flow process and how the amount of information in classical and quantum systems changes the variability of quantum algorithms. Next, a qualitative axiom explanation of the evolution of the information flow fluctuations of the quantum algorithm is provided.

(1) While the quantum algorithm is being executed, the information amount (information content) of a successful result increases. (2) The amount of information is a fitness function for successful result recognition. It then derives a measure of the accuracy of successful results.
In this case, the minimal principle of classical / quantum system entropy corresponds to the successful recognition of intelligent output states of quantum algorithm computation.

(3) When the entropy of the classical system of the output vector is small, the order and order of this output information become relatively large, and the output of the intelligent state measurement process of the quantum algorithm gives necessary information. Well solve the first problem. These three information axioms mean that the algorithm can automatically guarantee the convergence of the amount of information for the required accuracy. This is used to provide reliable and stable results for fault tolerant calculations.

As an example of the use of entropy using the quantum algorithm, n = 3 Deutsche-Yosa (DJ)
Consider the following example of an algorithm. Figures 27, 28 and 29
Are input by the following operators, respectively. The operators used in FIG.

[0428]

[Equation 127]

Is as follows. The operators used in FIG.

[0430]

[Equation 128]

Is as follows. Operators used in FIG. 29 is a U _F = ⁴ I. 28 to 29 show the variation of the DJ algorithm. At each step, we observe the Shannon and von Neumann entropy values. All steps follow the general schematic of FIG. 26 and apply the corresponding quantum operators. Examining FIGS. 28 to 29, the entropy of the classical and quantum systems changes as shown below after superposition, entanglement and interference have occurred.

The input vector is a basis vector, and the classical information in this state is 0. The input vector is a product of n tensors of a two-dimensional basis vector. Therefore, the entropy of von Neumann of the qubits constituting the basis vector is also all zero. The superposition operator ⁴ H increases the classical Shannon entropy from a minimum of 0 to a maximum of 4. However, from the viewpoint of von Neumann's entropy of the quantum system, the entropy has not changed.

The entanglement operator is a classical unitary operator, and maps a basis vector to a different basis vector without changing the classical information of the system. But,
The entanglement operator correlates different binary vectors of the tensor product describing the state of the system. This correlation is then described by von Neumann's entropy in different parts of the system. Even when the entanglement operator makes a correlation, the quantum information of the entire system is always 0 because the vector representing the correlation is in a pure state. On the contrary, the internal values of mutual information and conditional entropy can be positive or negative. That is, the internal value encodes quantum information necessary for decoding the property required for the operator U _F. The state of the system before and after the operation of the entanglement operator cannot be distinguished from classical information because Shannon's entropy has not changed.
Only when a quantum information technique is used can the difference between these two states be revealed.

[0434] Interference operator so maintains the encoded information is used to identify the U _F as value or equilibrium operator does not change the status of the quantum information. Conversely, interference operators reduce the entropy of classical systems making quantum information available. And
By interference, the vector obtains a minimization of the entropy of the classical system. That is, by definition, such vectors are intelligent. This is because the vector represents the coherent output state of the QA calculation using the minimum entropy uncertainty relationship (EUR) as a successful result.

When FIG. 27 and FIG. 28 are compared, FIG.
It can be seen that the entanglement operator effectively makes the quantum correlation of the different parts of the system. On the contrary, FIG.
8, the general state is represented by a tensor product of the basis vectors. Therefore, no quantum correlation is included.
The interference operator of FIG. 27 reduces the entropy of a classical system by one bit. In contrast, the interference operator of FIG. 28 reduces the entropy of a classical system by three bits.

The existence of a quantum correlation is denoted as the resistance (immunity) of the system, altering the entropy of the classical system and defining the internal intelligent potential of the quantum algorithm. The result of FIG. 29 is similar to the result of FIG. The entanglement operator of FIG. 29 makes no correlation. This is, as shown in FIG. 30, an arbitrary binary constant k
F (x) = k · x or f (x) = ¬ (k ·
x) holds the function f: Common to all {0,1} ⁿ → {0,1} linear operator U _F to realize a ^m. These functions of the equilibrium and fixed-value input sets minimize the "difference" between the highest and lowest information values shown in the Wen diagrams of FIGS. 27-31 to zero.

[0437] Interference effect, the mapping of the properties of U _F, nature reveal intelligent state. In contrast, other equilibrium functions are mapped to lower intelligent states. This intelligent state has a high entropy vector of the classical system. This means that it is an unsuccessful result, as shown in FIG. From a quantum information theory perspective, the Deutsche-Yosa algorithm has a special structure of input sets of functions. This structure is shown in FIG.

As a next example, Shore's algorithm is handled from information theory. The operators that implement the two input functions with periods 2 and 4, respectively, are: FIG.
The operators used in are

[0439]

[Equation 129]

(N = 3). The operator used in FIG.

[0441]

[Equation 130]

(N = 2). 34 and 35 show the development of the quantum algorithm when these operators are applied. In FIG. 34, the entanglement operator makes a quantum correlation of the vectors 3, 4 and 5. That is, the correlation identifies the period of the input function. The interference operator preserves the quantum correlation, but moves the correlation from vector (3, 4, and 5) to vector (1, 4, and 5). This movement maintains the period of the encoded input function. However, movement reduces the entropy of classical systems from three bits to two bits, since it allows access to periodic information that produces intelligent states, ie, states that contain all the necessary quantum information. . However, as a qualitative measure of free energy, the minimum entropy of classical systems is maintained.

In FIG. 35, the entanglement operator produces a strong correlation between vectors 1, 2, 3, and 4. That is, the correlation identifies the input function with the largest period (and therefore the largest entanglement). The interference operator preserves the correlation, but does not reduce the entropy of the classical system because the entanglement is too large (very high resistance). As shown in FIG. 33, Shore's algorithm has a special structure of an input space, that is, a periodic function. All functions are represented by their capacities and create entanglements of quanta. The entanglement of the quantum depends on its period. This structure is shown in FIG.

FIGS. 36 to 38 show the information analysis of Grover's algorithm. The operator that encodes the input function is
Next formula

[0445]

[Equation 131]

[0446] FIG. 36 shows a general iterative algorithm for information analysis of Grover's algorithm. FIGS. 37 and 38 show two iterations of this algorithm. As shown in FIGS. 37 and 38, each entanglement operator increases the correlation of different qubits. Each iteration of the interference operator reduces the entropy of the classical system, but as a side effect, the interference operator destroys some of the quantum correlations measured by von Neumann entropy.

When the Grover algorithm is repeated a plurality of times, an intelligent state is created. In all iterations, the first searched function is encoded by entanglement, but the information encoded by the interference operator is somewhat destroyed, so the need to have and access the encoded information Multiple iterations are needed to hide the gender.

The Deutsche and Deutsche-Yosa algorithm is a decision algorithm. The Simon, Shore, and Grover algorithms are search algorithms. The principle of classical (quantum) entropy minimization of the output of a quantum algorithm means that it is provided by intelligent output states. The decision algorithm is
It provides intelligent output state recognition with classical system entropy smaller or larger than that corresponding to the state of the quantum search algorithm. The quantum search algorithm is based on a combination of the minimum entropy of a classical system and the minimum entropy of a quantum system. The ability to combine these values characterizes the intelligence of the quantum search algorithm

[Brief description of the drawings]

FIG. 1A is a block diagram showing a control system combining soft computing and a gene search algorithm.

FIG. 1B is a block diagram showing a gene search process.

FIG. 2 is a block diagram illustrating a control system that combines soft computing and a quantum search algorithm.

FIG. 3 is a block diagram showing the structure of a classical genetic algorithm and a quantum search algorithm for global optimization.

FIG. 4 is a block diagram illustrating a general structure of a quantum search algorithm.

FIG. 5 is a block diagram illustrating a quantum network of a quantum search algorithm.

FIG. 6 is a block diagram of a quantum search algorithm.

FIG. 7 illustrates a gating technique for simulating a quantum algorithm using a classical computer.

FIG. 8 shows a programming diagram of a quantum algorithm.

FIG. 9 illustrates a structure of the quantum encoder illustrated in FIG. 8;

FIG. 10 illustrates a structure of a quantum unit illustrated in FIG. 8;

FIG. 11 illustrates an example of a quantum circuit.

FIG. 12A shows a quantum circuit for tensor product conversion.

FIG. 12B shows a quantum circuit for dot product conversion.

FIG. 12C shows a quantum circuit of identity transformation.

FIG. 12D illustrates a quantum circuit of propagation.

FIG. 12E shows a repeated quantum circuit.

FIG. 12F shows an input / output quantum circuit.

FIG. 13 shows a first representation of a quantum circuit of a Deutsche-Yoser quantum gate.

FIG. 14 shows a second representation of a quantum circuit of a Deutsche-Yoser quantum gate.

FIG. 15 shows a constant value function circuit of value 1—first circuit.

FIG. 16 shows a circuit of a constant function of value 1—a second circuit.

FIG. 17 shows a circuit of a constant function of value 1—third circuit.

FIG. 18 shows a circuit of a constant function of value 1—fourth circuit.

FIG. 19 shows a circuit of a constant function of value 1—fifth circuit.

FIG. 20 shows a constant function of value 0.

FIG. 21 shows the development of a DJ quantum gate.

FIG. 22 shows a final DJ quantum gate.

FIG. 23 shows a quantum circuit of Grover's quantum gate.

FIG. 24 shows the final circuit of Grover's quantum gate.

FIG. 25 is a Wen diagram showing the entropy and mutual information of a quantum system.

FIG. 26 is a general circuit diagram of a quantum unit.

Represents information analysis algorithms Yoza - Figure 27] Deutscher first operator U _F.

Represents information analysis algorithms Yoza - Figure 28] Deutscher second operator U _F.

It represents information analysis Yoza algorithms - Figure 29] Deutscher third operator U _F.

FIG. 30 shows an information analysis of the Deutsche-Yosa algorithm of a linear function.

FIG. 31 shows an information analysis of the Deutsche-Yoser algorithm for the nonlinear equilibrium function.

FIG. 32 shows a quantum information structure of the input space of Deutsche-Yoser.

FIG. 33 illustrates a quantum information structure of the Shore input space.

[Figure 34] represents information analysis Shore algorithm first operator U _F.

[Figure 35] represents information analysis Shore algorithm second operator U _F.

FIG. 36 illustrates an information analysis of a general iterative Grover algorithm.

FIG. 37 shows the information analysis of the first iteration of Grover's algorithm.

FIG. 38 illustrates an information analysis of the second iteration of Grover's algorithm.

──────────────────────────────────────────────────続 き Continued on the front page (51) Int.Cl. ⁷ Identification symbol FI Theme coat ゛ (Reference) G06F 9/44 550 G06F 9/44 550C 17/10 17/10 Z

Claims (45)

[Claims] 1. Encoding an input function f into a unitary matrix operator U _F , incorporating the operator U _F into a quantum gate G, where G represents a unitary matrix.
Applying the quantum gate G to the initial vector to form a basis vector, and measuring the basis vector and repeating the above steps of applying and measuring k times (where k satisfies 0 <k) And decoding the basis vector, the decoding comprising transforming the basis vector into an output vector, the method of quantum algorithm design in quantum soft computing. Wherein said coding comprises the steps of converting a mapping table of f in single shooting function F, and converting the function F to the mapping table of the operator U _F, wherein the U _F the method of claim 1, the mapping table including the steps of converting the operator U _F described above. 3. The method of claim 1, further configured to minimize Shannon entropy of the basis vectors of claim 1. 4. A method for determining one or more local solutions using a fitness function that is configured to minimize the rate of entropy generation of the controlled machine. With the goal of finding a global solution using a gene optimizer and a fitness function that is configured to minimize Shannon's entropy,
An intelligent control system, comprising: a quantum search algorithm configured to search for the local solution; and a quantum search algorithm configured to minimize Shannon's entropy. 5. The intelligent control system according to claim 4, wherein said global solution includes fuzzy neural network weights. 6. The fuzzy neural network is configured to handle a fuzzy controller, the fuzzy controller is configured to provide weights for controlling a PID controller, and the PID controller is configured to control the fuzzy controller. 5. The intelligent control system according to claim 4, wherein the intelligent control system is configured to control a mechanical device. 7. A method for developing a quantum search algorithm that includes the step of selecting a fitness function that is configured to minimize Shannon's entropy. 8. A method for developing a quantum search algorithm that includes the step of minimizing Heisenberg's uncertainty and Shannon's entropy. 9. Applying an entanglement operator to generate a plurality of correlation state vectors from a plurality of input state vectors, and applying an interference operator to said correlation state vector to generate an intelligent state vector Generating an intelligent state vector having a classical system entropy that is small compared to said correlated state vector. 10. applying a first transform to an initial state to create a coherent superposition of ground states;
Applying a second transform using a reversible transform to the coherent superposition to create a coherent output state, and applying a third transform to the coherent output state to create output state interference And a step of selecting a global solution from the interference of the output state. A method of global optimization for improving characteristics of a partial optimal solution. 11. The method of claim 1, wherein the first transformation is Hadamard (Ha
11. The method of claim 10, wherein the method is a damard) rotation. 12. The method of claim 10, wherein each of said ground states is represented using a qubit. 13. The method of claim 10, wherein said second transform is a Schrodinger solution. 14. The method according to claim 10, wherein said third transform is a quantum fast Fourier transform. 15. The method of claim 10, wherein said selecting step includes determining a maximum probability. 16. The method of claim 10, wherein said superposition of input states comprises a collection of local solutions of a global fitness function. 17. An encoder module configured to encode an input function into a unitary matrix operator, and an embedded module configured to integrate said unitary matrix operator into a quantum gate. ,
A processing module configured to apply the quantum gate to an initial vector to produce a basis vector; a measurement module configured to measure the basis vector; and the basis vector. And a decoder configured to convert the basis vector into an output vector. 18. A first conversion module for converting the mapping table of the input function into a single mapping function, wherein the encoder converts the single mapping function into a mapping table of the unitary matrix operator. 18. The apparatus of claim 17, comprising: a third transformation module; and a third transformation module for transforming the mapping table of the unitary matrix operator into the unitary matrix operator. 19. The apparatus according to claim 1, further comprising a module for minimizing Shannon entropy of said basis vector.
An apparatus according to claim 7. 20. Optimizing a plurality of local solutions using a fitness function that is configured to minimize the rate of entropy generation of a controlled machine; and minimizing Shannon's entropy Searching using a quantum search algorithm to search for the local solution, with the aim of finding a global solution using a fitness function that:
A method of intelligent control including a quantum search algorithm that is configured to minimize Shannon's entropy. 21. The method of claim 20, wherein said global solution includes fuzzy neural network weights. 22. Handling the fuzzy controller, providing control weights to the PID controller from the fuzzy controller, and using the PID controller to control the controlled mechanical device. 22. The method of claim 21, comprising: 23. A module for calculating entropy of a quantum system, a module for calculating entropy of a classical system, and a module for searching a solution space of a solution that reduces both entropy of the quantum system and entropy of the classical system. An apparatus for developing a quantum search algorithm including: 24. A first module for applying an entanglement operator for generating a plurality of correlation state vectors from a plurality of input state vectors, and an intelligent state having a classical entropy smaller than the correlation state vector. A second module for applying an interference operator to said correlation state vector to generate a vector. 25. A global optimizer for improving properties of a suboptimal solution, said optimizer including computer software loaded into memory,
The software includes a first module that applies a first transform to an initial state to produce a coherent superposition of ground states, and a second module that uses a reversible transform to apply a second transform to the coherent superposition. A second module for generating a coherent output state of the first and second modules, a third module for applying a third transform to the plurality of coherent output states to generate an output state interference, and selecting a global solution from the output state interference. And a fourth module. 26. The optimizer according to claim 25, wherein said first transform is a Hadamard rotation. 27. The optimizer according to claim 25, wherein each of said ground states is represented using a qubit. 28. The optimizer according to claim 25, wherein said second transform is based on a solution of the Schrodinger equation. 29. The optimizer according to claim 25, wherein said third transform is a quantum fast Fourier transform. 30. The optimizer according to claim 25, wherein said fourth module is configured to determine a maximum probability. 31. The superposition of input states includes a collection of local solutions to a global fitness function.
Optimizer as described in. 32. An input function f is defined by a unitary matrix operator U _F
Means for integrating the operator U _F into the quantum gate G; applying the quantum gate G to a plurality of initial vectors to form a plurality of basis vectors; measuring and measuring the basis vectors An apparatus for quantum soft computing, comprising: means for creating a vector; and means for decoding said measurement vector into an output vector. 33. The means for encoding converts a mapping table of f into a single mapping function F and converts the function F to the operator U _F
33. The apparatus of claim 32, converting the mapping table of U _F to the mapping table of U _F and the operator U _F. 34. The apparatus according to claim 32, further comprising means for minimizing the entropy of said basis vectors. 35. A means for optimizing a plurality of local solutions using a fitness function configured to minimize the rate of entropy generation of a controlled machine, and minimizing Shannon's entropy. Quantum means for searching for the local solution for the purpose of finding a global solution using a fitness function that is configured to minimize Shannon's entropy. Intelligent control system including a quantum search algorithm. 36. The intelligent control system of claim 35, wherein said global solution includes fuzzy neural network weights. 37. Further, using the global solution, PI
36. The intelligent control system of claim 35, including means for handling a fuzzy controller providing D controller weights. 38. An apparatus for deriving a quantum search algorithm comprising: a gene optimizer based on a fitness function; and means for selecting said fitness function that minimizes the entropy of a classical system and the entropy of a quantum system. 39. An apparatus for developing a quantum search algorithm including means for minimizing entropy of a quantum system and means for minimizing entropy of a classical system. 40. A means for applying an entanglement operator to produce a plurality of correlation state vectors from a plurality of input state vectors, and applying an interference operator to said correlation state vector and making said correlation state vector smaller than said correlation state vector. Means for generating an intelligent state vector having entropy of classical system. 41. A means for applying a first transform to an initial state to produce a coherent superposition of ground states, and applying a second transform using a reversible transform to said coherent superposition to produce a coherent output. A suboptimal solution comprising: means for creating a state; means for applying a third transform to the coherent output state to create output state interference; and means for selecting a global solution from the output state interference. Global optimization device to improve characteristics. 42. The apparatus according to claim 41, wherein said second transform is a solution of the Schrodinger equation. 43. The apparatus according to claim 41, wherein said third transform is a quantum fast Fourier transform. 44. The apparatus according to claim 41, wherein said selecting means determines a maximum probability. 45. The apparatus of claim 41, wherein said superposition of input states comprises a collection of local solutions to a global fitness function.