CN108025911B

CN108025911B - System and method for solving 3SAT using quantum computer

Info

Publication number: CN108025911B
Application number: CN201680052327.0A
Authority: CN
Inventors: 扎卡里·B·沃尔特斯
Original assignee: Zha KaliBWoertesi
Current assignee: Zha KaliBWoertesi
Priority date: 2015-09-11
Filing date: 2016-09-12
Publication date: 2022-04-19
Anticipated expiration: 2036-09-12
Also published as: EP3341325A1; CN108025911A; EP3341325A4

Abstract

A method for solving NP-complete problem 3SAT and other computational problems that can be reduced to 3 SAT. Whenever they exist, quantum mechanical operations are performed on a finite number of quantum mechanical bits, or "qubits," in such a way as to concentrate the probabilities in the state in which a given 3SAT problem is solved. The concentration of probabilities is achieved by generalizing the traditional reversible model of quantum computation to include irreversible operations that map one density matrix to another.

Description

System and method for solving 3SAT using quantum computer

Cross Reference to Related Applications

This patent application claims the benefit of united states patent application 15/263,240 filed on 12/9/2016, united states patent application 15/263,240 claims the benefit of united states provisional patent application 62/249,438 filed on 2/11/2015, and united states provisional patent application 62/217,432 filed on 11/2015, both of which are incorporated herein by reference.

Background

1. Field of the invention

The present invention relates generally to computing, and in particular to quantum computing.

2. Description of the related Art

Quantum computers solve computational problems by performing physical transformations on finite quantum mechanical systems to cause the systems to occupy states with desirable properties. Quantum computers do not operate on classical bits that must deterministically assume a value of 1 or 0, but on quantum mechanical "qubits" that can exist in two states arbitrarily superimposed. It is known that quantum computers can solve some computational problems-especially simulations of other quantum systems-far faster than their classical counterparts. However, it is not known that there may be a class of problems that address significant acceleration.

By propositional logical formulation in conjunctive normal form or "CNF

A set of Boolean variables x₁，x₂，x₃.., assigning true or false constitutes 3 variables boolean per clause that satisfy the problem or 3 SAT. Here, the variables (x) within the same clause_n) Or negated form thereof

Linked by the OR (V-shaped) operator, while different clauses are linked by the AND (A) operator. The assignment must satisfy all clauses to solve the problem.

3SAT is known in computer science as NP-complete [8, 4 ]. This means that it has the following properties: any problem whose solution can be examined in polynomial time (class "NP") can be reduced to a 3SAT problem in polynomial time. Thus, the polynomial time solution for 3SAT can also be used to solve any problem in NP in polynomial time. It is believed, but not proven, that algorithms on classical computers cannot solve NP-complete problems in polynomial time; the question of whether this is true is one of the most well-known problems in mathematics [3 ].

The specification here consists of a mapping of an instance of one problem to an instance of another problem, in such a way that the solution of the second instance can be mapped to the solution of the first instance. Computational problems consist of a series of problem parameters, plus a statement of the nature that must be satisfied to solve the problem [4 ]. An instance of a problem is obtained by specifying specific values for all problem parameters.

The solving algorithm of the computational problem determines whether a particular instance has a solution or is proven to have no solution, but does not necessarily produce a solution. As long as some solutions exist, a constructive solution algorithm will produce at least one valid solution. The probabilistic solving algorithm determines whether a solution exists or does not exist and the probability of error is not greater than some value e between 0 and 1/2, which is conventionally chosen to be 1/3. The probabilistic algorithm may be run multiple times to increase the certainty that a given instance has or does not have a solution.

Summary of The Invention

The present invention comprises a novel quantum computer that allows irreversible physical changes to occur in a quantum mechanical system, a set of physical transformations, or "gates" that change the state of the system in a logically meaningful way, and an algorithm that transforms quantum probabilities from physical states that fail to solve some 3SAT problem of interest to a state or set of states that solve the 3SAT problem of interest. If there is at least one solution state, the algorithm exponentially decays the total no solution probability such that the probability of occupying the solution state becomes arbitrarily close to 1.

The new quantum computer, the quantum gates used by the new quantum computer, and the algorithms for concentrating the probabilities in the solution state all depart from the traditional principle of quantum computing, which considers that any operation performed by a computer must be reversible in the sense of having a well-defined inverse operation that restores the original wave function. For the sake of clarity, the present novel computer will be referred to as an "irreversible" quantum computer. An irreversible quantum computer does not map one wave function to another wave function as in a conventional quantum computer, but rather maps one density matrix to another density matrix. As indicated by name, this mapping may have an inverse that is not well defined.

The general approach for achieving this goal is to non-coherently transition populations (population) from any initial state to a state or set of states that satisfy some 3SAT problem of interest. This is achieved by constructing a decimation gate-essentially an error correction loop-that switches the population from a state that invalidates individual logical clauses to a state that satisfies those clauses. In pseudo code, the algorithm can be described as:

each clause do in the For problem

Encoding the satisfiability of clauses in temporary grab bits

If clause does not satisfy then

Using grab bits to perform controlled rotation of variables in clauses

end if

The temporary grab bit is cleared by the measurement.

end for

Here, the controlled rotation causes population loss for a state that invalidates a particular clause. A state that invalidates many clauses loses the population faster than a state that invalidates only some, and the solution state that satisfies all clauses acts as a population container.

Because quantum computers operate by physically transforming finite quantum mechanical systems, all of the methods described herein have a second use as a means of quantum mechanical state preparation. This has value in the physical sciences where it is often necessary to prepare a wave function at a particular initial state to observe a desired physical effect.

The present application also relates to the following aspects:

1) a method for performing irreversible quantum computation, comprising one or more of the following steps:

performing a unitary operation; or

Performing an insert operation; or

A delete operation is performed.

2) The method according to 1), comprising the additional step of:

encoding the output of the logical operation or any property representable as a logical operation in the state of the first bit;

controlling a state of one or more other bits using a state of the first bit; and is

Storing, deleting or further modifying the state of the first bit.

3) A quantum computer, comprising:

an algorithm installed on the computer, the algorithm configured to focus probabilities in a solution state;

a quantum computer function that maps the first density matrix to the second density matrix; and

the mapping function does not have a well-defined inverse.

4) A method for changing the probability distribution of states in a quantum system concentrates probabilities in a state or set of states that satisfy a logical formula.

5) The method of 4), wherein a parameter such as an identification of the extraction gate, an extraction angle corresponding to a particular extraction gate, an application sequence of different gates, or a number of times a particular extraction gate is applied is varied between iterations.

Brief Description of Drawings

For a more complete understanding of the present invention, reference is made to the following description and accompanying drawings, in which:

FIG. 1-insert operation couples a system of interest 1 to a new bit 2 prepared in the desired state and not entangled with existing bits in the system;

FIG. 2-the delete operator measures the state of the selected bit, projecting the NxN density matrix 3 onto the (N-1) × (N-1) sub-matrices. Where the dashed lines indicate measurements along a certain axis;

FIG. 3-Single bit rotation operator wrap around Single bit 4

The shaft rotates. Here, the straight arrow indicates the state of the bit in the xz plane of the Bloch sphere in question, and the curved arrow indicates the direction of rotation. True is selected to be equal to state | + z>Agree and false is chosen as | -z>. The directions of the z-axis and x-axis in this figure are also used in the other figures;

figure 4-the two bit rotation operator controls the rotation of bit 6 around the y-axis using the projection of bit 5 on the desired axis. Here, state 5 is

Orthographic projection on the axis surrounds bit 6

The shaft rotates counterclockwise;

figure 5-the two bit rotation operator controls the rotation of bit 6 around the y-axis using the projection of bit 5 on the desired axis. Here, state 5 is

The negative projection on the axis surrounds the bit 6

The shaft rotates clockwise;

figure 6-measurement operator measures the projection of state 3 on the desired axis and projects it onto the eigenstates corresponding to the measured values. The dashed line here indicates the measuring axis;

FIG. 7-Quantum circuit diagram for the self-controlled revolving gate SCROT;

FIG. 8-autonomous revolving Gate SCROT uses the projection of a bit onto a desired axis to control itself around

The shaft rotates. In I, control bits to

The orthographic projection on the axis causes the capture bit 8 to rotate counterclockwise by pi/2. In II, grab bit to

The orthographic projection on the axis rotates the control bits counterclockwise. In III, the following is

The orthogonal axes measure the state of the capture bits, resulting in irreversible operations on the control bits. Because along the and "true axis (axis of truth)"

The measurement is made on an axis orthogonal to the axis,this information does not yield information about the state of the control bits;

FIG. 9-self-controlled revolving Gate SCROT uses the projection of bits on the desired axis to control itself around

The shaft rotates. In I, control bits to

The negative projection on the axis rotates the capture bit 8 clockwise by pi/2. In II, grab bit to

The negative projection on the axis rotates the control bit clockwise. In III, the following is

The orthogonal axes measure the state of the capture bits, resulting in irreversible operations on the control bits. Because along the and "true axis"

The axes orthogonal to the axis are measured, so this information does not yield information about the state of the control bits;

fig. 10-a quantum circuit diagram for a QOR gate;

the fig. 11-QOR gate uses the states of

control bits

9 and 10 to control the state of capture bit 11 such that the final state of bit 11 is true if 9 or 10 is true, and false otherwise. The grab bit 12 is a temporary grab bit that is used to determine whether an even or odd number of control bits is true. In I, the states of 9 and 10 control the rotation of

bits

11 and 12. In II, 12 is in

The projection on the axis controls the rotation of the bit 11. In III, the state of bit 11 is rotated by π. In IV, along

The axis measures the state of bit 12, so that no information is produced about whether the odd or even number of control bits is true;

the fig. 12-QOR gate uses the states of

control bits

bits

11 and 12. In II, 12 is in

the fig. 13-QOR gate uses the states of

control bits

bits

11 and 12. In II, 12 is in

FIG. 14-Quantum circuit diagram for QAND gate;

FIG. 15-QAND gate uses the states of

control bits

13 and 14 to control the state of capture bit 15 such that if 13 and 14 are true, the final state of bit 15 is true, and notIt is false. The grab bit 16 is a temporary grab bit that is used to determine whether the even or odd number of control bits is true. In I, the states of 13 and 14 control the rotation of

bits

15 and 16. In II, 16 is in

The projection on the axis controls the rotation of the bit 15. In III, the state of bit 15 is rotated by π. In IV, along

The state of the axis measurement bit 16, so as not to produce information as to whether the odd or even number of control bits is true;

the fig. 16-QAND gate uses the states of

control bits

13 and 14 to control the state of capture bit 15 such that the final state of bit 15 is true if 13 and 14 are true and false otherwise. The grab bit 16 is a temporary grab bit that is used to determine whether the even or odd number of control bits is true. In I, the states of 13 and 14 control the rotation of

bits

15 and 16. In II, 16 is in

the FIG. 17-QAND gate uses the states of

control bits

bits

15 and 16. In II, 16 to

The projection on the axis controls the rotation of the bit 15. In III, bit 15The state is rotated by pi. In IV, along

FIG. 18-Quantum circuit diagram for QANDOR gate;

the fig. 19-QANDOR gate uses the states of

control bits

17 and 18 to control the state of capture bit 19 such that the final state of 19 is true if 17 and 18 are true, false if 17 and 18 are false, and the final state of 19 is false, otherwise an overlap of true and false. The grab bit 20 is a temporary grab bit that is used to determine whether the even or odd number of control bits is true. In I, the states of 17 and 18 control the rotation of

bits

19 and 20. In II, 20 to

The projection on the axis controls the rotation of the bits 19. In III, the state of bit 19 is rotated by π. In IV, along

The axis measures the state of the bits 20, so that no information is produced about whether the odd or even number of control bits is true;

the figure 20-QANDOR gate uses the states of the

control bits

17 and 18 to control the state of the capture bit 19 such that the final state of 19 is true if 17 and 18 are true, the final state of 19 is false if 17 and 18 are false, and there is an overlap of true and false otherwise. The grab bit 20 is a temporary grab bit that is used to determine whether the even or odd number of control bits is true. In I, the states of 17 and 18 control the rotation of

bits

19 and 20. In II, 20 to

Axial measurement bit 2A state of 0, thereby generating no information as to whether the odd or even number of control bits is true;

the FIG. 21-QANDOR gate uses the states of

control bits

17 and 18 to control the state of capture bit 19 such that if 17 and 18 are true, the final state of 19 is true, if 17 and 18 are false, the final state of 19 is false, otherwise there is an overlap of true and false. The grab bit 20 is a temporary grab bit that is used to determine whether the even or odd number of control bits is true. In I, the states of 17 and 18 control the rotation of

bits

19 and 20. In II, 20 is in

FIG. 22-Quantum circuit diagram for a QXOR gate;

the figure 23-QXOR gate uses the states of

control bits

21 and 22 to control the state of the grab bit 23 such that if 21XOR 22 is true, grab bit 23 is true, otherwise it is false. In I, 21 to

The orthographic projection on the axis results in a counterclockwise rotation of pi. In II, 22 to

The orthographic projection on the axis results in a counterclockwise rotation of pi. In III, a counter-clockwise pi is generated to

A negative projection on the axis;

24-QXOR gate uses the states of

control bits

21 and 22 to control the state of capture bit 23, capture bit 23 being true if 21XOR 22 is true, and false otherwise. In I, 21 to

The orthographic projection on the axis results in a counterclockwise rotation of 23 by pi. In II, 22 to

The negative projection on the axis does not cause a net rotation of 23. In III, 23 is rotated counter-clockwise by π to yield

An on-axis orthographic projection;

the figure 25-QXOR gate uses the states of

control bits

21 and 22 to control the state of capture bit 23, which is true if 21XOR 22 is true and false otherwise. In I, 21 to

The negative projection on the axis does not cause a net rotation of 23. In II, 22 to

A negative projection on the axis;

FIG. 26-Quantum circuit diagram for 3OR gate;

the fig. 27-3 OR gate combines two QOR gates to produce a bit that is true when either one of the

input bits

24, 25 OR 25 is true, and false otherwise. Here, the QOR gates using 24 and 25 as inputs produce output bits 27. The second QOR gate uses 27 and 26 to generate output bits 28. If no information about the states of 24 and 25 is desired, the temporary grab bit 27 may be deleted by a measurement orthogonal to the true axis, or the temporary grab bit 27 may be deleted by a measurement orthogonal to another axis to perform a partial measurement on the states of 24 and 25;

figure 28-the complex gate takes the state of bits 29 and 30 as input and produces bit 31 as output. The input bit may be a variable bit or the output of another gate. After the use of the composite gate, the state of the input bits may be deleted or retained for further use;

FIG. 29-Quantum circuit diagram for DEC gates; and

FIG. 30-DEC gate extracts populations that do not satisfy the state of certain clauses. The variable bits 32, 33, and 34 control the state of the capture bits 35 and 36 through a 3OR gate shown in fig. 27. The state of the variable bit is then rotated by a controlled rotation (dashed line) using the state of bit 36. If the clause is satisfied, the variable bit is not rotated. If the clause is not satisfied, the variable bit will be rotated by the decimation angle θ. The grab bits 35 and 36 may be deleted or reserved for further use.

Description of The Preferred Embodiment

I. Introduction and Environment

As required, detailed aspects of the present invention are disclosed herein; however, it is to be understood that the disclosed aspects are merely exemplary of the invention, which can be embodied in various forms. Therefore, specific structural and functional details disclosed herein are not to be interpreted as limiting, but merely as a basis for teaching one skilled in the art how to variously employ the present invention in virtually any appropriately detailed structure and as a representative basis.

For convenience, certain terminology will be used in the following description for reference only and not for limitation. The terminology will include the words specifically mentioned, derivatives thereof, and words of similar import.

Irreversible quantum computation

In 1985, Deutsch [1, 2] demonstrated that any unitary operation of a small set of one-bit and two-bit operators, or "gates", on N qubits, can be generalized in the sense that any precision can be approximated by a quantum circuit consisting of only these gates.

The unitary operator U has the property that its inverse is equal to its hermitian conjugate

Wherein

Is hermitian conjugate and I is identity. A unitary may act on a wave function to produce another wave function

ψ′＝Uψ (3)

Or act on a density matrix to produce another density matrix

As indicated by the name "universal", the core principle of conventional quantum computation is that any computation consists of a unitary operation applied to a wave function, which can be prepared in a desired initial state. For example, see [11]p171 (". there is a small set of general purpose gates, that is, any quantum computation can be represented by these gates anyway), or [ 9]]p36 ("since quantum computers must operate reversibly to develop their magic power (except for the measurement gates), they are usually designed to operate with both input and output registers.. except for n + m in the input and output registers, the computation process will usually require many qubits, but we should now ignore these extra qubits, looking at whether the computation of f will transform U unimportant anyway_fN + m qubits applied to the input and output registers. ").

The algorithm described below operates on a novel quantum computer, an "irreversible" quantum computer. As indicated by its name, an irreversible computer performs an operation that maps an initial density matrix to a final density matrix, which in principle cannot be inverted to restore the initial state of the system. This is in sharp contrast to the unitary operation U performed by a conventional "reversible" quantum computer, which has a well-defined inverse

The distinction between reversibility and irreversibility is a fundamental physical distinction,it has been approved for more than a century in the context of thermodynamics [12]。

While the fundamental equations of physics are reversible, irreversible operations can be performed on a system of interest by allowing the system of interest to interact with another system that functions as an environment [13 ]. When two systems interact, the state of the system is entangled with the state of the environment, such that the states of the two systems cannot be described independently. A reduced density matrix describing only the system can be obtained from a density matrix describing both the system and the environment together by tracking all degrees of freedom of the environment.

Where the subscript "e" represents the state of the environment, "s" is the state of the system and "se" is the state of the system coupled to the environment. Generally, tracking environmental degrees of freedom in this way will produce a reduced density matrix that is correlated with the system's density matrix before interacting with the environment through irreversible transformation — it is not possible to recover the initial density matrix given uniquely the final reduced density matrix.

In contrast, conventional quantum computers perform a unitary operation U on the system of interest in such a way that there is a reversible transformation that restores the initial state. The state of the system after any unitary operation is given by:

if the system of interest interacts with another system, such as the surrounding environment, the calculations are performed in such a way that the state of the additional system remains entangled with the state of the system of interest [11, 9] so that the density matrix can be made at

Before, and during, the calculation

And then split into the product of the system density matrix and the ambient density matrix.

Because the density matrix for the system and environment can be written in such a way that tracking the state of the environment results

Here, the final reduced density matrix is related to the initial density matrix by a unitary transformation, and there is a well-defined inverse operation to restore the initial system

It can be seen that reversible quantum operations are a special case of irreversible quantum operations, in which the final state of the system plus the environment can be divided into a system state and an environment state.

To achieve the irreversibility, two new operations are required. The insert operation I shown in fig. 1 introduces new qubits in an initial state selected by the user that were not previously intertwined with the bits in the system of interest. This projects the N × N density matrix onto the (N +1) × (N +1) density matrix. The delete operation D shown in fig. 2 measures the state of the qubit in the system. This projects the measured qubits onto the eigenstates of the measurement operators and the NxN density matrix onto the (N-1) × (N-1) sub-matrices. The deleted bits can be immediately reinserted into the problem if needed, keeping the size of the density matrix constant.

The second deletion method consists of dividing the entire system into deleted bits d and remaining systems s, and leaving the deleted bits unaffected after deletion without measuring their state. Then, performing any measurements on the system s will track the state of the deleted bits. This deletion method does not yield information about the state of the system, but can entangle the deleted bits with the bits of the system, making them unsuitable for reinsertion at a later stage of the algorithm.

Both insert operators and delete operators formally extend the capabilities of quantum computers by allowing operations that change the number of bits in the system of interest. However, reversible quantum computing differs from irreversible quantum computing not formally, but physically. As indicated by

equations

9 and 5, the irreversible computation requires information not contained in the system of interest to recover the initial density matrix, while the reversible computation is inverted using only the information contained in the system.

Thus, although insert and delete operations are used, quantum error correction, which introduces redundant bits beyond those required to describe the system and perform intermediate measurements, is an example of reversible computation, with the aim of keeping the state of the system from becoming entangled with the state of the surrounding environment. In contrast, if the density matrix for the system plus environment cannot be decomposed into the product of the density matrix describing the system and the density matrix describing the environment, performing a unitary operation that wraps n system and m environment bits represents an irreversible calculation. The use of natural decoherence processing to effect irreversible changes to the system density matrix is a special case of the second case, where a unitary operation entangling the system and environment is generated due to natural processing rather than user-induced operations.

Rotating and measuring door

A quantum gate is a series of quantum operations applied to a subset of some bits of the system of interest to achieve some desired physical change. Irreversible gates have the additional property that it is not possible to restore the original state of the system; for example, multiple initial states may map to the same final state. This is a particularly desirable property in the context of 3SAT, where the number of potential states may be exponentially larger than a subset of states having a desired property such as solving a problem of interest.

A common method of representing individual qubits is to map two state systems on the surface of a Bloch sphere. The qubits can be projected onto a certain selected axis using Pauli spin operators, or rotated around this axis

And

although in general qubits can be mapped to any vector on a Bloch sphere, a preferred embodiment of this algorithm utilizes bits that are always confined to the xz plane, such that a single bit is rotated only around the y-axis. Since the algorithm does not require information encoded by another polar angle, it can be freely used for other purposes, such as quantum error correction. If the qubit is present in

A positive projection on the axis is selected as true and if there is a negative projection, it is selected as false.

a. One and two bit revolving door

Bit n is rotated in the xz plane by a single bit operator

Which is illustrated in fig. 3.

The projection of bit n on axis i can be used to rotate bit m by two bit operators

Wherein the selection of the symbol depends on X_nOr

Whether the value of (d) is used to control rotation. A positive projection on axis i causes a counterclockwise rotation, which is shown in fig. 4, and a negative projection causes a clockwise rotation, which is shown in fig. 5.

Unless otherwise stated, operators and gates defined in this document will be defined such that the state of a variable, rather than its negatives, serves as a control parameter. If it is more convenient to use the negation of the variable as a control parameter, this can be achieved by changing the sign within the exponent.

To simplify symbols, it is useful to define controlled rotation operators

Wherein bit m is rotated by angle θ if bit n has a positive projection on axis i, and bit m remains unchanged if bit n has a negative projection on this axis.

Because of equation 16

Operators may be difficult to physically implement, so controlled rotation may be more directly implemented using out-of-plane rotation. Rotation about the x-axis is defined as

Equation 16 can also be written as:

a second variant of conditional rotation is to perform the rotation when the control bit is false and to keep the target bit from rotating when the control bit is true. Defining:

it can be seen that

b. Measuring door

By using projection operators

Projection onto eigenstate | γ > of the spin matrix to measure the projection of bit n onto axis i

σ_i(θ，φ)＝cos(θ)σ_z+sin(θ)cos(φ)σ_x+sin(θ)sin(φ)φ_y (21)，

Wherein gamma is_iIs associated with the eigen state | gamma_i>The associated eigenvalues. This operator is illustrated in fig. 6.

Following the rules of quantum mechanics, the projection of a measurement bit n onto axis i causes the bit to occupy an eigenstate of the measurement operator-where the eigenstate has a positive (| γ) along the axis being measured_i>＝|1/2>) Or negative (| γ)_i>＝|-1/2>) And (5) projecting. If the state of the bit being measured is entangled with the states of other bits, this measurement may give partial information about the state of the bit in question. Conversely, in some cases, the measurement axis may be selected so as to not provide information about the state of other bits. Here, "partial measurement" means that some information about the state of one or more bits is collected, but not enough information to uniquely determine their quantum state.

IV. irreversible door

The general design of an irreversible gate defined below is to introduce a temporary grab bit initialized to true (| + z >), and then perform a quantum operation such that the bit has a positive projection on a certain axis if the logical expression evaluates to true and a negative projection on the same axis if the expression evaluates to false. This axis is named "true axis". The state of the grab bit can then be used as an input to a new gate, or to control the rotation of other variables in the problem. When no longer needed, the information contained in the grab bit is corrupted by measurements orthogonal to the true axis.

The selection of which axis is used as the "true axis" is entirely arbitrary. However, the choice of measuring the grab bit along an axis orthogonal to this is physically meaningful. The projection of the measurement bits onto an axis oriented at an angle θ relative to the true axis yields partial information as to whether the logical expression is satisfied, with no information change at all when θ ═ π/2 to complete the information when θ ═ 0. Thus, all irreversible gates defined in this section have the general nature in which the state of the grab bit is measured along an axis oriented at an angle θ ≠ π/2 relative to the true axis.

Another physically meaningful departure from this model is to project the grab bit onto state | α > if the expression evaluates to true, and project the grab bit onto state | β > if the expression evaluates to false, where | α > and | β > are not orthogonal. In this case, it is impossible to define the true axis. However, the state of the grab bit can still be used as an input to a new gate or act on other variables in the problem in such a way as to produce an irreversible operation on the density matrix.

QCOPY door

The quantum copy gate introduces a new bit s in the initial state | + z > and rotates it so that it copies the state of the existing bit c

QNOT gate

The quantum NOT gate introduces a new bit s in the initial state | + z > and rotates it to be directly opposite the existing bit c

c. Self-control rotation

The use of temporary grab bits allows the state control for the bit to rotate itself or for conditional rotation. Rotation of a bit n according to its own projection onto an axis i can be achieved by introducing a grab bit s, initializing to true, and applying two-bit rotations and measurements to produce a self-controlled rotation

Here, the state of bit n is used to rotate the grab bit, which is then used to rotate the bit of state n. If bit n is originally true, then after the first step, the state of the grab bit is | + x >, so that the second step rotates bit n by angle θ. If bit n is originally false, the first step rotates the grab bit to | -x >, and the second step rotates bit n by- θ. Since the state of bit n is now encoded in the projection of the grabbed bit onto the x-axis- "true axis" -measured along the axis orthogonal thereto, e.g. the z-axis, the rotation is made irreversible without measuring any information about the state of bit n. This gate is graphically illustrated in fig. 7, 8 and 9, showing the evolution of the individual bits in the case of a quantum circuit diagram plus a self-controlled bit with a positive projection and a negative projection, respectively, on the control axis.

Writing density matrix operation, and mapping the first two steps as follows:

and

wherein the order of the states is | s, n>＝|+z,+z>，|+z,-z>，|-z,+z>，|-z,-z>. Edge of

The axis measurement grab bit selects the top left or bottom right 2 x 2 sub-matrix. Regardless of the measurement result, if it is in

With a positive projection on the axis, bit n is rotated by angle θ, and if it has a negative projection, bit n is rotated by angle- θ. By the following formula except that

Outside shaft

To control the rotation of bits

d. Irreversible quantum OR gate

More complex state operations may be made possible by encoding the output of the logical operation into a state that captures bits. Definition of

Here, the second grab bit is initialized to true, and remains true if either or both of the control bits are true, otherwise it is rotated to true. As previously described, the negation of the variable may control the state of the grab bit by an appropriate set of sign changes. At its sideBefore measurement, capture bit s₁In that

On-axis projection encoding an odd or even number of input variables c₁And c₂Whether it is true. If only one is true, s₂Is rotated by an additional pi/2 so that TT, TF and FT produce s₂To

The same projection on the axis. As previously described, a negation of the variable may be used to control the state of the grab bit by an appropriate set of sign changes. This gate is graphically illustrated in fig. 10, 11, 12, 13, showing the quantum circuit diagram and the evolution of the individual bits in the case when the two input bits are true, one is true and the other is false and both are false, respectively. Table 0.1 gives the state of the variable and grab bit before the QOR gate and after the application gate but before any grab bit is measured.

Construction of quantum NOR gates by omitting the final rotation of the capture bit

TABLE 0.1QOR gate at s₂Is coded in the state of (c)₁∨c₂) To output of (c). Passing QOR operator and c₁And c₂Before the state of (2) is entangled, a bit s is grabbed₁And s₂Is initialized to true (| + z)>). Here | T>＝|+z>And | F>＝|-z>。

e. Irreversible quantum AND gate

Using a method similar to QOR gates, irreversible AND gates can be obtained by changing the angle of the SCR gate from π/2 to- π/2.

This gate is graphically illustrated in fig. 14, 15, 16 and 17, showing the quantum circuit diagram and the evolution of the individual bits in the case of two input bits being true, one being true and one being false and both being false, respectively.

Construction of quantum NAND gates by omitting the final rotation of the capture bits

f. Irreversible gate ANDOR gate combining AND AND OR characters

A gate combining characters of AND OR can be obtained by continuously varying the angle of the SCR gate between-pi/2 AND pi/2.

This gate is graphically illustrated in fig. 18, 19, 20 and 21, showing the quantum circuit diagram and the evolution of the individual bits in the case when two input bits are true, one is true and one is false and both are false, respectively. Here, the grab bit is initialized to true, remains true if both inputs are true, and toggles to false if both inputs are false. If one input is true and the other is false, the grab bit will occupy the superposition of true and false

|s>＝cos(π/4-θ/2)|+z>+sin(π/4-θ/2|-z> (34)。

Gates that combine NAND and NOR characters into QANDOR in the same way can be constructed by omitting the final rotation of the capture bit

g. Irreversible quantum XOR gate

Using a method similar to QOR and QAND gates, irreversible XOR gates can be obtained by

This gate is graphically illustrated in fig. 22, 23, 24, 25, showing the quantum circuit diagram and the evolution of the individual bits when two input bits are true, one is true and one is false and both are false, respectively.

Quantum XNOR gates obtained by omitting the final rotation of the capture bit

TABLE 0.23 OR Gate at s₄Pair under the state (c)₁∨c₂∨c₃) Is encoded, wherein s₁And s₂Is the grab bit corresponding to the first QOR gate in equation 38, and s₃And s₄Is a grab bit corresponding to the second QOR gate. Before entanglement with the state of the input bits, each grab bit is initialized to true (| + z)>). Here the state of all four grab bits is given before the measurement, where | T>＝|+z>And | F>＝|-z>。

h. Irreversible quantum 3OR gate

Using two QOR gates and an intermediate grab bit allows the second grab bit to encode whether a particular quantum state satisfies three variable logic clauses

Initializing two grab bits as true, s if the clause is satisfied₂Hold true, otherwise rotate it false. The door is shown graphically in fig. 26 and 27. More generally, the outputs of the link QOR, QAND and QANDOR gates produce a composite gate, which is shown graphically in FIG. 28. Coding variable m₁Or m₂Bit s of whether true or not₁By using the equation as in equation 38

Is projected to

The axes are deleted or reserved for use in subsequent logical operations. Table 0.2 gives the state of both the variable and the grab bit before applying any rotation and after applying a rotation but before measuring any grab bit.

i. Irreversible quantum extraction gate

The 3SAT problem can only be solved by satisfying the state of each clause in the problem. Since the 3OR gate encodes the capture bit s₂Is satisfied, it can be used to achieve a step-by-step extraction of the population by conditionally rotating gates on all variables contained in clause M without satisfying any state of the clause.

If the clause is satisfied, the variable X_m1、X_m2And X_m3The state of (c) is unaffected. If not, each variable is rotated about the y-axis by an angle θ. This decimation operator reduces the population of original states by rotating the state of each variable to its opposite value, and increases the number of variables other than the variables in clause MThe same number of states as the original state. Measure along

The grab bit of the axis makes this cluster transfer irreversible without giving any information about the satisfiability of the clause. This door is shown in graphical form in fig. 29 and 30.

When tracking the state of the grab bit after measurement, all entries where the state of a particular grab bit is not the same on the left-and right-vector side will sum to zero. Writing

It can be seen that

And is

In particular, the measurement grab bit s₄Eliminate cross terms

Etc., wherein the right vector side appears due to satisfying the initial state of the clause and is represented by the initial state | FFF due to the DEC operator>Rotate to make leftThe sagittal side appears or vice versa. Because of state | FFF>Is to generate s₄All cross terms that occur due to conditional rotation of the decimation gate contribute zero to the reduced density matrix.

A simple example of an extraction process is illustrated by using the extraction process to flip the state of a single variable. Here state | α > evaluates to true for a particular clause, while state | α' > evaluates to false. Except for the state of variable m, | α > and | α' > are the same. Because the satisfiability of the clause is encoded in the state of the grab variable, the density matrix after the 3OR operator but before the conditional rotation can be written as:

wherein the state ordering is | T, α >, | T, α '>, | F, α >, | F, α' >.

When according to CR^s→mUsing the state of the grab bit to rotate the variable bit x_mIn the state of (1), the density matrix becomes

Transfer operator of population from alpha' to alpha

Is completed, measured along with

Orthogonal axes ("true axes") to produce states of grab bits

Wherein the transfer of the population depends on the initial population of alpha' and the rotation angle, but not on the off-diagonal coherence term ρ_αα′Or rho_α′α. This rotation angle may also be referred to as the "extraction angle" because it controls the degree to which the initial state population is extracted.

Finally, tracking the state of the capture bits produces a new reduced density matrix

Wherein the transfer of the population depends on the rotation angle and the initial population of α', but not on the off-diagonal coherence term ρ_αα′Or rho_α′α。

j. Irreversible global extraction gate

This extraction process can now be applied to each clause in the question to create a global extraction gate

Wherein

Giving the order in which clauses were extracted and θ_MThe decimation angle for clause M is given. By construction, the state satisfying each clause is not changed by SATDEC, and each non-solution state is extracted by at least one clause.

V. algorithm for solving 3SAT

As described above, the application of the SATDEC gate corresponds to the application of a decimation DEC gate corresponding to each logical clause in the 3SAT problem of interest.

The probability change of state s due to a single application of SATDEC is given by

Wherein the transition matrix T is parametrically dependent on the order of the clauses and the associated rotation angle:

sin associated with transfer of population²(θ) proceeds to a leading step (which is quadratic in the decimation angle. The rate of convergence to equilibrium is given by the slowest eigenvector attenuation. Please note that because of T_s'→sIs asymmetric, so its eigenvectors are not orthogonal.

Conservation of the total probability means that any eigenstates with non-zero eigenvalues must have traces of zero, while for all s, P_sThe limit of ≦ 1 means that all non-zero eigenvalues must be negative. Since the solution state s is never decimated, T is for all s_s*→s＝0。

If there is at least one solution state s, T_s→s'Must contain at least one solution state. Is included in eigenvectors that cannot solve the problem

Must invalidate at least one clause and be projected onto the state s' by the decimation algorithm, defined as equal to s for variables outside the clause and s for variables inside the clause. If s 'is not a solution, it must in turn invalidate a clause and project it onto s ", s" equals s' for variables outside the new clause and s "equals s for variables inside it. For a limited number of clauses, the process must end with s or other solution states. In particular, there are no eigenvectors with zero eigenvalues consisting of only solution-free states.

a. Converge on equilibrium

The convergence speed for the decimation algorithm is limited by the presence of eigenvectors with small eigenvalues. Let theta₀Is the minimum extraction angle corresponding to a clause in the question of interest. The eigenvectors of fast decay are defined as having a characteristic that may be equal to or greater than

And a slowly decaying eigenvector is defined as

Although there is no minimum magnitude for such eigenvalues, the requirement that λ be small uniquely determines slowly decaying eigenvectors. Writing equation 51 as an eigenvalue equation

Relative to T_s→s'Is negligible unless T is present for all s_s→s'0-that is, unless s is the solution state.

A population of solution-free states for slowly decaying eigenvectors can be constructed by setting the following equation for all solution-free states s and s

And requires ∑_s

P

_s1. The population of solution states for this eigenvector may be constructed by setting the following for all solution states s

Requires Δ P for all s_s*＝-λP_s*And sigma_s*P_s*Is-1. It should be noted that if no solution exists, equation 53 defines an equilibrium probability distribution for the no solution state.

Since the probability distribution for slowly decaying eigenstates is by choosing the extraction angle

The bottleneck determined, and therefore the probability of slow decay, may be changed at each iteration of the SATDEC

But is interrupted. Since such a probability distribution may decay very slowly, it will be referred to as a "quasi-equilibrium distribution".

The quasi-balanced eigenvectors can be constructed in two steps. First, equation 53 is solved for the solution-free population and normalized so that ∑ is_sP_sFor all solution states s, a population for solution states is constructed by setting the following formula

And requires Δ P for all s_s*＝-νP_s*And sigma_s*P_s*-1, so that the complete intrinsic distribution is traceless.

Because of the fact that

The condition (c) uniquely determines the probability distribution for the slowly decaying eigendistribution, so the reverse holds: any unsolved eigendistributions are not determined by equations 53 and 55, but must have a distribution that may be equivalent to or greater than

The decay rate of (c).

b. Attenuation of solution-free probability

For a given extraction angle

Repeated application of SATDEC results in the probability distribution evolving in two time dimensions. The probability distribution at a given time can be written as a sum over the eigendistributions

Wherein

Is a quasi-equilibrium eigen-distribution defined by equations 53 and 55 and

is the remaining intrinsic profile, wherein the decay rate may be equal to or greater than

Repeated application of the same transition matrix results in a fast "blow" in which the non-solution probabilities contained in the fast decaying intrinsic distribution are quickly transitioned to a set of solution states, followed by long pauses in which the non-solution probabilities approach a quasi-equilibrium distribution, resulting in slow transitions to solution states.

The slow decay of the quasi-equilibrium distribution can be accelerated by changing the decimation angle corresponding to different clauses. Since the slowly decaying eigen-distribution is given by equation 53

Is determined, and is thus varied

The original quasi-equilibrium distribution is projected non-adiabatically into the new intrinsic base, wherein the quasi-equilibrium distributions may be substantially different. If it is not

Is for the transfer matrix T⁽¹⁾Slowly decaying intrinsic distribution of, and

is for the transfer matrix T⁽²⁾Slowly decaying intrinsic distribution of

Is for the transfer matrix T⁽²⁾Of fast decay, wherein

From the transfer matrix T⁽¹⁾To T⁽²⁾Is varied with the coefficient A_nDecays to zero resulting in a second blow.

The coefficient A may be constructed by approximating the decay rate of a slowly decaying eigen-distribution to zero_VSo that

T⁽¹⁾V⁽¹⁾＝V⁽¹⁾ (58)

T⁽²⁾V⁽²⁾＝V⁽²⁾ (59)。

Let Δ T be T⁽²⁾-T⁽¹⁾And Δ V ═ V⁽²⁾-V⁽¹⁾So that equation 57 can be written as:

at T⁽²⁾By expanding population differences in the intrinsic base of

After that

So that is at

And

the ejection is largest when the least similar.

c. Iterative of iterations

A simple process of transferring the population out of the slowly decaying eigen-distribution is to randomly change the extraction angle corresponding to a particular clause for one iteration of SATDEC, and then revert to the original extraction angle to recover the original transfer matrix for the next iteration. Order to

For a transition matrix corresponding to the following settings

For each clause C, and

wherein

And σ_cAre chosen to be 1 or-1 and have equal probability.

The transfer matrix T ═ pi due to the individual clauses C_CT^(C)Write the product of the transition matrix, notice T^(C)Is theta_CThe square of (a) is,

the change of state s of the slowly decaying intrinsic distribution due to this change in the transfer matrix can be written as

Wherein, is Δ V^(C)Is T^(C)The first order correction of offset change in (1), and Δ V^(R)A new quasi-equilibrium distribution is implemented that obeys the residual correction of equation 53.

For the leading step, the population change of state s due to a small change in the extraction angle is constructed by setting the following equation

So that small terms of multiplication are ignored

After that time, the user can use the device,

subtracting these terms from equation 66 yields a linear system for other corrections

With the right hand side having the desired value 0.

The second step of the reciprocal iteration changes all the decimation angles back to their original values, so that

ΔT^(C)→-ΔT^(C) (70)

And is

And will decay slowly intrinsic distributions

Projection to T⁽¹⁾In the intrinsic group (c).

Both changes in the transition matrix result in a loss of total solution-free probability as the population transitions from slowly decaying to rapidly decaying eigendistributions. The coefficient of the slowly decaying intrinsic distribution after the second transfer is given by

Order to

And is

So that

Then, the user can use the device to perform the operation,

because of σ_CAnd σ_C'Is an independent random variable with an average value of 0, so

And is

Wherein

Is the number of clauses invalidated by state s, so that for each non-solution state

Whereby it follows the expected value

So that the loss of population from slowly decaying eigendistributions due to the repeated iteration cycles is controlled by a user-controlled parameter η. Is selected by

Ensure that all non-solution probabilities are in relation to

A larger attenuation in the time dimension.

Working example of reciprocal iteration

By solving for where | TTT>The 3SAT problem, which is three variables of the unique solution, obtains a simple example of a reciprocal iteration. This corresponds to performing the DEC operation for clauses 0-6 of Table 0.3. Different from each | TTT>Each state of (a) invalidates a single clause and for theta_CThe leading step in (c), which is extracted by moving the population to three states where a single variable has been flipped. Select θ for all clauses_C＝θ₀0.1 and includes

Table 0.3 each of the eight three variable states invalidates a single clause. For the leading step, the DEC operator for this clause reduces the population of the state that invalidates the clause and fills in three states other than this state by a single variable.

Entry of the leader order in θ yields the transition matrix

Where the state ordering is taken from table 0.3. The eigenvalues and the eigendistributions of this transfer matrix are given in table 0.4.

TABLE 0.4T⁽¹⁾Eigenvalues and eigendistributions of (c).

Setting eta equal to 0.2 and

changing the transition matrix into

And the eigenvalues and the eigendistributions are given by table 0.5.

TABLE 0.5T⁽²⁾Eigenvalues and eigendistributions of (c).

Note that in both Table 0.4 and Table 0.5, the same applies to

And V⁽²⁾Relative to the decay rate

Smaller, simultaneous intrinsic distribution

May be equivalent to this value or greater than this value.

Transfer matrix from T⁽¹⁾Change to T⁽²⁾Will V⁽¹⁾Projected into the new intrinsic base(s),

once rapidly decayedIntrinsic distribution of

Has decayed to zero, returns to the original transition matrix and has V⁽²⁾Projected onto the original intrinsic base

So that

Is 1.005 x 0.956 ≈ 0.96 ≈ 1- η²。

a. Algorithm runtime

Restoring fast fading eigenvectors having

Or greater magnitude and selecting

The number of SATDEC iterations requires the implementation of an overall solution probability alpha approximation

Because each iteration requires a constant number of operations per clause, the runtime for the algorithm as a whole scales linearly with the number of clauses in the problem.

Measuring the corresponding edge of each bit

Projection of the axis's variables and restoration of the | + z values corresponding to true | + z>And corresponding to false | -z>Potential solutions to the 3SAT problem of interest are generated, which can be checked for correctness using classical computers simply and unambiguously. If the problem is not solvable, the potential solution will always be invalid. If the problem is solvable, then this isThe process will generate a valid solution with a probability of α and an invalid solution with a probability of 1- α. Following conventional probabilistic algorithms, N therefore needs to be selected_IterationSo that 1-alpha is not more than 1/3.

Comparison with existing algorithms

a. And

algorithmic comparison

The algorithm described hereinabove is similar in spirit to

Probabilistic classical algorithm [14]. In that

In the algorithm, the value of each variable is initially chosen randomly. Each clause is then tested for satisfaction and any clause failure is resolved by randomly flipping the state of one of the variables in the clause. If it is at test 3N_VThe specification is found to be unsatisfied after the clause, and the process is restarted using the new random initial state.

The difference between the two algorithms comes from the quantum nature of the system being run. In current algorithms, the system consists of an incoherent superposition of many different states, while classical algorithms are constrained to occupy a single state at a time.

The algorithm of (a) may be simulated after each decimation operation by measuring the state of each variable bit along the true axis, thereby causing the system to occupy a single state. Because current algorithms do not transfer all probabilities out of the initial state, the step of decimation immediately following the measurement can be repeated until one of the variable bits changes.

Here, the measurement behavior fundamentally changes the behavior of the algorithm. With each stale clause treated as a branch point, current algorithms transfer a small fraction of the probability to occupy the initial state along each branch, and

the algorithm will shift the entire probability along a single branch chosen at random. In the language of computer science, the current algorithm performs a first search for the parallel width of the solution state, and

the algorithm performs a random depth first search-random walk (random walk).

The flow of the population due to a single DEC gate in the current algorithm can be classically simulated by summing all random walks and all possible initial states, weighted by the initial probability of occupying a state and the probability of adopting a particular walk. However, this classical simulation algorithm is very inefficient. Although quantum algorithms require a constant number of operations to implement the DEC gates, the classical algorithm requires a constant number of operations to all

The initial states are summed. Thus, the existence of a polynomial time quantum algorithm does not imply the existence of a polynomial time classical simulation algorithm.

b. Compared to fixed point quantum search

The algorithm described above has the property that the population can be transferred to the solution state, but never leaves. Because the solution state satisfies all clauses, it is a fixed point of iteration. Conceptually, this is in comparison with Tulsi et al's fixed-point quantum search algorithm [5,15,10,7,6]Similarly, where a single oracle bit is used to rotate the bits by | s that satisfy the problem>And solving state | s>Source state | s in a composed 2d subspace>. Such as [15 ]]The fixed point search described in (1) has the desired property of ensuring that the initial state evolves towards the target state even if the algorithm is not run to completion, and any errors due to incomplete transformations in the previous iteration will be cleared by the subsequent iteration as long as the state is still present in the problem of the defined space. However, existing fixed point searches have heretofore not provided speed relative to non-fixed point algorithmsAnd (4) degree advantage. [15]Generate a run time comparable to a classical search for [10 ]]Run-time from Grover search algorithm

With continuous change of run-time characteristics (where n is the total number of states) to classical search with variation of damping parameters

The present algorithm differs from the existing fixed-point algorithm by using information specific to 3SAT for general cases that are not suitable for chaotic database searches-an efficient solution must satisfy the conditions for each clause in the problem. It does not use a single oracle bit to encode the satisfaction of the entire problem, but rather uses multiple bits that encode the satisfaction of individual clauses. Rather than directly transitioning the population from the source state to the target state, the state population that the extractor uses to invalidate a particular clause is reduced, thereby transitioning the population to other states that differ only by the variables contained in the clause. In this way, the flow of populations mimics a breadth-first search for states that satisfy the problem of interest. The final transition of the population to this solution state occurs only indirectly and may include transitions to many intermediate states through many different paths. The rate of population reduction due to the invalidation of individual clauses is a user-controlled parameter with no a priori maxima.

Each of these differences arises because of the additional problem-specific information provided by pure search 3SAT against an unordered database, and they collectively account for differences in convergence time. As shown in equation 52 and discussed below, the population loss of the smallest part due to the failure of an individual clause defines an important time measure of the decay of the unsolved population distribution. The condition that the eigen-profile decays slower than this uniquely defines a slowly decaying quasi-equilibrium eigen-profile. In "back and forth iterations," changes that alter a fraction of population losses due to individual clause failures project a slowly decaying quasi-equilibrium distribution to a rapidly decaying intrinsic distribution, increasing the speed at which the population transitions to a solution state set.

It will be understood that while certain embodiments and/or aspects of the invention have been illustrated and described, the invention is not so limited and encompasses various other embodiments and aspects.

Claims

1. A computer-implemented method for performing irreversible quantum computation on an irreversible quantum computer, the irreversible quantum computer comprising a gate having a temporary grab bit that is initialized to true and has a positive projection on a true axis if a logical expression evaluates to true and a negative projection on the true axis if the logical expression evaluates to false, the method comprising:

using the state of the grab bit as an input to a new gate, or to control the rotation of other variables;

corrupting the information contained in the grab bit by a measurement orthogonal to the true axis when information is no longer needed;

performing one or more operations selected from the group consisting of unitary, insert, and delete operations;

encoding a property of an output capable of being represented as a logical operation or other property capable of being represented as a logical operation in a state of a first bit;

storing, deleting or modifying the state of the first bit; and is

A method of changing the probability distribution of states in a quantity subsystem that concentrates probabilities in a state or set of states that satisfy a logical formula, and wherein a parameter selected from the group consisting of an identification of a decimation gate, a decimation angle corresponding to a particular decimation gate, an order of application of different gates, or a number of times a particular decimation gate is applied is varied between iterations.

2. An irreversible quantum computer, comprising:

a gate having a temporary grab bit initialized to true and having a positive projection on a true axis if a logical expression evaluates to true and a negative projection on the true axis if the logical expression evaluates to false;

an algorithm installed on the irreversible quantum computer, the algorithm configured for concentrating probabilities in solution states and varying a method of probability distribution of states in a quantity subsystem that concentrates probabilities in a state or set of states that satisfy a logical formula, and wherein a parameter selected from the group consisting of an identification of a draw gate, a draw angle corresponding to a particular draw gate, an order of application of different gates, or a number of times a particular draw gate is applied is varied between iterations;

an irreversible quantum computer function that maps the first density matrix to the second density matrix;

the mapping function does not have a well-defined inverse;

the second density matrix describes the system obtained from the first density matrix describing system and environment together, only by tracking all environmental degrees of freedom using the following algorithm:

where subscript "e" represents the state of the environment, subscript "s" represents the state of the system and subscript "se" represents the state of the system coupled to the environment.

3. A computer-readable medium, in which a computer program is stored which is operative to perform all the steps of claim 1 when the computer program is executed on a non-reciprocal quantum computer.