WO2022269712A1 - 複数量子ビットオブザーバブルのパーティショニング方法、複数量子ビットオブザーバブルのパーティショニングプログラム、および情報処理装置 - Google Patents
複数量子ビットオブザーバブルのパーティショニング方法、複数量子ビットオブザーバブルのパーティショニングプログラム、および情報処理装置 Download PDFInfo
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Definitions
- the present invention relates to a multi-qubit observable partitioning method, a multi-qubit observable partitioning program, and an information processing apparatus.
- Quantum computers use electronic information called quantum bits.
- a qubit can have a superposition of '0' and '1'. It is also possible to create entangled states with multiple qubits. Quantum computers can use these superposition and entanglement states to compute all possibilities in parallel, allowing them to find solutions to complex problems in a short amount of time.
- Quantum computers can solve problems much faster than classical computers.
- Algorithms for solving problems with quantum computers are called quantum algorithms, and algorithms for solving problems with classical computers are called classical algorithms.
- the classical algorithm requires about N N steps (exponential time) to find a solution
- the quantum algorithm requires about N 2 steps (polynomial time) to find a solution (N is a natural number indicating the size of the problem).
- Examples of problems that can be calculated in a short time by quantum computers include Deutsch-Jozsa's algorithm and Shor's algorithm.
- the quantum state it is necessary to read out not only the z-axis of the Bloch sphere, but also the x- and y-axis values.
- the x-axis or y-axis values can be measured by performing an operation to convert those values to the z-axis before measurement and reading the values after the conversion operation. Note that the superposition state of the qubits is destroyed when the qubits are measured. Therefore, in order to fully know the quantum state, it is necessary to repeatedly execute quantum calculations for each measurement of the x-axis, y-axis, and z-axis.
- the state of one or more qubits is represented using observables. To completely know the state of n qubits, it is sufficient to measure the expected values of 4 n ⁇ 1 observables (n is a natural number). By finding the expected value for each observable, we can fully know the state of the qubit. When obtaining the expected value for each observable, one quantum circuit is basically created for each observable. Quantum computers take 10 4 -10 5 measurements per observable to reduce statistical errors.
- Simultaneous measurement of the expected values of multiple observables generates a set (partition) of simultaneously measurable observables. Each observable is contained in some partition. Simultaneous measurement of expected values by a quantum computer is performed for each predetermined number of observables included in the same partition. In this case, the smaller the number of partitions, the more efficient the measurement of the qubit state. However, when the number of observables to be measured increases, the amount of calculation required to generate partitions containing more observables becomes enormous, making it difficult to reduce the number of partitions.
- this case aims to reduce the number of partitions that collect simultaneously measurable observables.
- a method for partitioning multiple qubit observables in which a computer performs the following processes.
- the computer determines whether simultaneous measurement of expected values is possible for each pair of two observables among the plurality of observables representing the states of the plurality of quantum bits.
- the computer then includes a variable indicating whether to include each of the plurality of observables in a first partition that is a subset of the plurality of observables, wherein pairs of observables that cannot be measured simultaneously are included in the first partition.
- the value of the Hamiltonian increases, and the value of the Hamiltonian decreases as the number of observables included in the first partition increases.
- the computer generates a first partition including one or more observables based on the values of variables corresponding to each of the plurality of observables obtained by searching the ground state of the Ising model.
- the number of partitions collecting concurrently measurable observables can be reduced.
- FIG. 4 is a diagram illustrating an example of a partitioning method for multiple-qubit observables according to the first embodiment; It is a figure which shows an example of a system configuration.
- 1 is a diagram showing an example of classical computer hardware;
- FIG. It is a figure explaining the simultaneous measurement method of an observable.
- It is a figure which shows the example of determination of exchange relationship.
- FIG. 4 is a diagram showing an example of partitioning; 1 is a block diagram showing an example of the functionality of a classical computer;
- FIG. It is a figure which shows an example of the process which an exchange relationship determination part performs.
- 10 is a diagram illustrating an example of repeated partition generation processing; It is a figure which shows an example of expected value measurement of an observable using the quantum circuit for simultaneous measurements.
- 4 is a flow chart showing an example of an observable measurement procedure; 10 is a flowchart (1/2) showing an example of a procedure of partitioning processing; 2 is a flowchart (2/2) showing an example of a procedure of partitioning processing;
- FIG. 10 is a diagram showing comparison results of the number of partitions for each partitioning method;
- FIG. 10 is a diagram showing a comparison result of calculation time for each partitioning method;
- the first embodiment is a multi-qubit observable partitioning method that uses the Ising model to reduce the number of partitions that collect simultaneously measurable observables.
- FIG. 1 is a diagram showing an example of a partitioning method for multiple-qubit observables according to the first embodiment.
- FIG. 1 shows an example in which the information processing apparatus 10 executes the partitioning method for the multiple-qubit observable.
- the information processing apparatus 10 can implement a multi-qubit observable partitioning method, for example, by executing a multi-qubit observable partitioning program.
- the information processing device 10 has a storage unit 11 and a processing unit 12 .
- the storage unit 11 is, for example, a memory or a storage device that the information processing device 10 has.
- the processing unit 12 is, for example, a processor or an arithmetic circuit included in the information processing device 10 .
- the information processing device 10 partitions a plurality of observables representing the states of quantum bits.
- the information processing device 10 is connected to the annealing type computer 1, and determines observables to be included in the partition using the annealing type computer 1 during partitioning.
- the storage unit 11 stores the observable group 2 .
- Observable group 2 includes a plurality of observables to be measured. In the example of FIG. 1, circles in observable group 2 represent observables. Observables to be measured are determined according to the problem to be solved using a quantum computer. For example, 4 n ⁇ 1 observables are included in the observable group 2 when the state of n qubits is desired to be known completely.
- the processing unit 12 performs partitioning to divide the observable group 2 into a plurality of partitions.
- the processing unit 12 generates partitions one by one. In the example of FIG. 1 , the processing unit 12 first creates the first partition 4 and creates the second partition 5 based on observables not included in the partition 4 . Specifically, the processing unit 12 performs the following processing.
- the processing unit 12 determines whether simultaneous measurement is possible for each pair of two observables among the plurality of observables included in the observable group 2 . For example, the processing unit 12 determines that simultaneous measurement is possible if the order of tensor products of observables can be exchanged. In the example of FIG. 1, lines connect pairs of simultaneously measurable observables.
- the processing unit 12 also generates an Ising model 3.
- the Ising model 3 includes variables indicating whether or not to include each of the observables in the first partition 4, which is a subset of the observables. Also, in the Ising model 3, the value of the Hamiltonian increases when the first partition 4 includes a pair of observables whose order of tensor products cannot be exchanged. Furthermore, in the Ising model 3, the value of the Hamiltonian decreases as the number of observables included in the first partition 4 increases.
- the processing unit 12 generates a first partition 4 containing one or more observables based on the values of the variables of each of the plurality of observables obtained by the Ising model 3 by searching for the ground state.
- the ground state of the Ising model 3 is the state in which the Hamiltonian is the minimum value.
- the processing unit 12 generates the first partition 4 including observables corresponding to the variables indicated to be included in the first partition 4 when the Ising model 3 is in the ground state.
- the Ising model 3 becomes the ground state when the pair of observables whose expected values cannot be measured simultaneously is not included in the first partition 4 . Therefore, any pair of observables contained in the generated first partition 4 is capable of simultaneous measurement of expected values. Further, the value of the Hamiltonian of the Ising model 3 decreases as the number of observables included in the first partition 4 increases. Therefore, when the Ising model 3 becomes the ground state, the number of variables included in the first partition 4 becomes maximum. That is, by generating the first partition 4 based on the Ising model 3 of the ground state, more observables can be included in the first partition 4 .
- the processing unit 12 repeats the partition generation process many times until all observables are included in one of the partitions. For example, after generating the first partition 4 , the processing unit 12 limits variables of the Ising model 3 to variables corresponding to remaining observables that are not included in the generated first partition 4 . Then, the processing unit 12 includes one or more remaining observables based on the values of the variables of the remaining observables obtained by searching the ground state of the Ising model 3 including the variables corresponding to the remaining observables. Create a second partition 5 . The processing unit 12 repeats such second partition generation processing until all observables are included in one of the partitions.
- the processing unit 12 may cause the annealing computer 1 to search for the ground state of the Ising model 3 .
- the processing unit 12 instructs the annealing computer 1 to search for the ground state of the Ising model 3, and obtains from the annealing computer 1 the values of the variables of each of the plurality of observables when the Ising model 3 is in the ground state. .
- the generated partitions 4 and 5 can be used to improve the efficiency of quantum bit state measurement by a gate-type quantum computer (not shown).
- the processing unit 12 selects a predetermined number of target observables from the same partition of the generated partitions 4 and 5 .
- the processing unit 12 generates a quantum circuit that simultaneously measures expected values of the selected target observables.
- the processing unit 12 instructs the gate-type quantum computer to simultaneously measure the expected value of the target observable based on the generated quantum circuit.
- Ising model 3 For partitioning, the number of partitions is kept small. Therefore, many observables can be measured simultaneously with other observables. As a result, it becomes possible to efficiently measure the expected value of the observable by the gate-type quantum computer.
- a second embodiment is a computer system for efficiently measuring the complete state of a qubit.
- FIG. 2 is a diagram showing an example of the system configuration.
- a classical computer 100 is connected to an annealing computer 200 and a gate quantum computer 300 .
- Classical computer 100 is, for example, a von Neumann computer.
- the annealing computer 200 is a non-Von Neumann computer for calculating the ground state of the Ising model.
- the annealing type computer 200 may be either one using a superconducting quantum circuit or one that reproduces a quantum phenomenon with a semiconductor circuit.
- the gate quantum computer 300 is a non-von Neumann computer that can solve general-purpose problems by manipulating quantum gates.
- the classical computer 100 controls the annealing computer 200 and the gate quantum computer 300 to perform quantum computation. At that time, the classical computer 100 classifies the observables of the problem to be computed into one of the partitions. The classification process into partitions is called partitioning. For example, the classical computer 100 generates an Ising model for generating partitions from the information of a set of simultaneously measurable observables, and causes the annealing computer 200 to calculate the ground state of the Ising model. The classical computer 100 then controls the gate quantum computer 300 to simultaneously measure the expected values of two or more observables included in the same partition.
- FIG. 3 is a diagram showing an example of the hardware of a classical computer.
- a classical computer 100 is entirely controlled by a processor 101 .
- a memory 102 and a plurality of peripheral devices are connected to the processor 101 via a bus 109 .
- Processor 101 may be a multiprocessor.
- the processor 101 is, for example, a CPU (Central Processing Unit), an MPU (Micro Processing Unit), or a DSP (Digital Signal Processor).
- processor 101 executing a program may be realized by an electronic circuit such as ASIC (Application Specific Integrated Circuit) or PLD (Programmable Logic Device).
- ASIC Application Specific Integrated Circuit
- PLD Programmable Logic Device
- the memory 102 is used as the main storage device of the classical computer 100.
- the memory 102 temporarily stores at least part of an OS (Operating System) program and application programs to be executed by the processor 101 .
- the memory 102 stores various data used for processing by the processor 101 .
- a volatile semiconductor memory device such as RAM (Random Access Memory) is used.
- Peripheral devices connected to the bus 109 include a storage device 103 , a GPU (Graphics Processing Unit) 104 , an input interface 105 , an optical drive device 106 , a device connection interface 107 and a network interface 108 .
- a storage device 103 a storage device 103 , a GPU (Graphics Processing Unit) 104 , an input interface 105 , an optical drive device 106 , a device connection interface 107 and a network interface 108 .
- GPU Graphics Processing Unit
- the storage device 103 electrically or magnetically writes data to and reads data from a built-in recording medium.
- the storage device 103 is used as an auxiliary storage device for the computer.
- the storage device 103 stores an OS program, application programs, and various data.
- an HDD Hard Disk Drive
- an SSD Solid State Drive
- the GPU 104 is an arithmetic unit that performs image processing, and is also called a graphics controller.
- a monitor 21 is connected to the GPU 104 .
- the GPU 104 displays an image on the screen of the monitor 21 according to instructions from the processor 101 .
- Examples of the monitor 21 include a display device using an organic EL (Electro Luminescence), a liquid crystal display device, and the like.
- a keyboard 22 and a mouse 23 are connected to the input interface 105 .
- the input interface 105 transmits signals sent from the keyboard 22 and mouse 23 to the processor 101 .
- the mouse 23 is an example of a pointing device, and other pointing devices can also be used.
- Other pointing devices include touch panels, tablets, touchpads, trackballs, and the like.
- the optical drive device 106 reads data recorded on the optical disc 24 or writes data to the optical disc 24 using laser light or the like.
- the optical disc 24 is a portable recording medium on which data is recorded so as to be readable by light reflection.
- the optical disc 24 includes DVD (Digital Versatile Disc), DVD-RAM, CD-ROM (Compact Disc Read Only Memory), CD-R (Recordable)/RW (ReWritable), and the like.
- the device connection interface 107 is a communication interface for connecting peripheral devices to the classical computer 100 .
- the device connection interface 107 can be connected to the memory device 25 and the memory reader/writer 26 .
- the memory device 25 is a recording medium equipped with a communication function with the device connection interface 107 .
- the memory reader/writer 26 is a device that writes data to the memory card 27 or reads data from the memory card 27 .
- the memory card 27 is a card-type recording medium.
- the network interface 108 is connected to the network 20.
- Network interface 108 transmits and receives data to and from annealing computer 200 or gate quantum computer 300 via network 20 .
- the network interface 108 is a wired communication interface that is connected by a cable to a wired communication device such as a switch or router.
- the network interface 108 may be a wireless communication interface that communicates with a wireless communication device such as a base station or an access point via radio waves.
- the classical computer 100 can implement the processing functions of the second embodiment with the above hardware.
- the information processing apparatus 10 shown in the first embodiment can also be realized by hardware similar to the classical computer 100 shown in FIG.
- the classical computer 100 implements the processing functions of the second embodiment, for example, by executing a program recorded on a computer-readable recording medium.
- a program describing the processing content to be executed by the classical computer 100 can be recorded in various recording media.
- a program to be executed by the classical computer 100 can be stored in the storage device 103 .
- the processor 101 loads at least part of the program in the storage device 103 into the memory 102 and executes the program.
- the program to be executed by the classical computer 100 can also be recorded in a portable recording medium such as the optical disk 24, memory device 25, memory card 27, or the like.
- a program stored in a portable recording medium can be executed after being installed in the storage device 103 under the control of the processor 101, for example.
- the processor 101 can read and execute the program directly from the portable recording medium.
- the quantum state can be expressed using the density operator ⁇ .
- ⁇ can be expressed as a 2 n ⁇ 2 n matrix (where n is the number of qubits).
- the density operator ⁇ can be represented using multiple observables (64 for 3 qubits). Each observable becomes a linearly independent basis that constitutes ⁇ , which represents the quantum state. That is, ⁇ in the case of 3 qubits is represented by the following equation.
- ⁇ ⁇ 0 I 0 I 1 I 2 + ⁇ 1 X 0 I 1 I 2 + ⁇ 2 Y 0 I 1 I 2 + ⁇ 3 Z 0 I 1 I 2 + ⁇ 4 I 0 X 1 I 2 + ⁇ 5 X 0 X 1 I 2 + ... + ⁇ 62 y 0 Z 1 Z 2 + ⁇ 63 Z 0 Z 1 Z 2 ⁇ 0 , . . . , ⁇ 63 are real coefficients.
- the subscripted X, Y, Z are Pauli operators and are 2 ⁇ 2 Pauli matrices ( ⁇ x , ⁇ y , ⁇ z ). The numerical value of the subscript is the number of the quantum bit to be measured.
- the subscripted I is the identity operator, a 2 ⁇ 2 identity matrix.
- the identity operator is also a kind of Pauli operator.
- An observable is represented by an array of Pauli operators. This array of Pauli operators is called a Pauli string.
- a Pauli string indicates the tensor product of the Pauli operators represented by the Pauli string.
- the expected value of the k (natural number)-th term observable is ⁇ k . For example, the expected value of X 0 X 1 I 2 is ⁇ 5 .
- the number of observables to be measured to fully know the state of n qubits is 4 n ⁇ 1.
- the state of the qubit can be fully known by measuring three observables: ⁇ X, Y, Z ⁇ .
- n 2 , there are 15 observables measuring the expected value : ⁇ I0X1 , I0Y1 , I0Z1 , X0I1 , X0X1 , X0Y1 , X0 Z 1 , Y 0 I 1 , Y 0 X 1 , Y 0 Y 1 , Y 0 Z 1 , .
- FIG. 4 is a diagram for explaining a method of simultaneously measuring observables.
- ⁇ > of 3 qubits is obtained.
- the three observables "I 0 Y 1 X 2 , Z 0 Z 1 Z 2 , X 0 I 1 X 2 " of the quantum circuit 30 corresponding to the problem to be solved.
- Horizontal lines in quantum circuit 30 correspond to qubits.
- quantum circuits 31 to 33 corresponding to each of the three observables are generated. Rectangular symbols shown on the horizontal lines of the quantum circuits 31 to 33 are quantum gates acting on respective quantum bits.
- the "S” rectangle indicates an S gate.
- the "X” rectangle indicates an X gate.
- the “H” rectangle indicates an H gate (Hadamard gate). The state of each qubit is measured at the position where the measurement symbol of each qubit is indicated.
- the quantum circuit 31 in order to calculate the expected value of the observable "I0Y1X2", three quantum bits are used to calculate the expected values of "I0 " , " Y1 " , and "X2". Measure. “I 0 ” corresponding to the 0th qubit is the identity operator, and no expected value measurement is required. For the first quantum bit, S gate, H gate, and X gate operations are performed. This allows us to measure the expected value of Y 1 from the first qubit. For the second qubit, an H-gate operation is performed. This allows us to measure the expected value of X 2 from the first qubit. Based on these measurements, the expected value of the observable "I 0 Y 1 X 2 " can be obtained.
- the quantum circuit 33 in order to calculate the expected value of the observable " Z0Z1Z2 ", three quantum bits are used to measure the expected values of " Z0 ", " Z1 " and “ Z2 ". do.
- the operation by the quantum gate on the output side of the original quantum circuit 30 is not performed, and "Z 0 ", “Z 1 ", “Z 2 " expectations can be measured. Based on these measurements, the expected value of the observable "Z 0 Z 1 Z 2 " can be obtained.
- the quantum circuits 31 to 33 obtain the expected value of one observable based on the measurement results of the three qubits. Therefore, in order to obtain expected values of three observables, three quantum circuits 31 to 33 measure the state of each quantum bit.
- the 0th quantum bit is first provided with an H gate, the first quantum bit is the control bit, and the second quantum bit is the target bit.
- a gate is provided.
- a SWAP gate is provided for the 0th quantum bit and the 1st quantum bit.
- an S gate is provided for the 0th qubit, and a CZ gate with the 1st qubit as the control bit and the 2nd qubit as the target bit.
- each qubit is provided with an H-gate.
- Measuring the 0th qubit of quantum circuit 34 yields the expected value of the observable "I 0 Y 1 X 2 ". Measuring the first qubit of quantum circuit 34 yields the expected value of the observable "Z 0 Z 1 Z 2 ". Measuring the second qubit of quantum circuit 34 yields the expected value of the observable "X 0 I 1 X 2 ".
- the expected value of one observable can be measured from one qubit, and the expected values of three observables can be measured simultaneously. In this way, the three observables individually measured using the quantum circuits 31 to 33 can be measured simultaneously by one quantum circuit 34.
- FIG. 1 the expected value of one observable can be measured from one qubit, and the expected values of three observables can be measured simultaneously.
- FIG. 5 is a diagram showing an example of exchange relationship determination.
- the commutative/anti-commutative correspondence table 35 indicates whether each combination of operators is commutative or anti-commutative.
- a symbol indicating whether the operator shown in the row and the operator shown in the column is commutative or anti-commutative is shown at the intersection of the row and the column.
- a "+" sign indicates commutativity and a "-" sign indicates anti-commutation.
- changing the order of the product does not change the value, but in the anti-commutative case, changing the order of the product reverses the sign.
- I 0 Y 1 X 2 and Z 0 Z 1 Z 2 it is determined whether each pair of I 0 and Z 0 , Y 1 and Z 1 , X 2 and Z 2 is commutative or anti-commutative. .
- I 0 Z 0 ” (commutative)
- FIG. 6 is a diagram showing an example of partitioning. For example, for 14 observables 41-54, exchange relationships with other observables are determined. In the example of FIG. 6, a set of observables having a commutative exchange relationship are connected by lines. Partitions 61 and 62 are generated based on the exchange relationship between the observables. In all combinations of observables belonging to the partition 61, the exchange relation is commutative. Similarly, all combinations of observables belonging to the partition 62 have commutative relations.
- FIG. 7 is a diagram showing an example of partitioning.
- partitioning When classifying each partition of the observable group 71 into one of a plurality of partitions, there are a plurality of ways of partitioning. For example, it can be divided into three partitions 72a-72c. It is also possible to divide into six partitions 73a-73f.
- a small number of partitions means a large number of observables belonging to one partition.
- the processing for measuring all observables becomes more efficient.
- the observable group is partitioned so that the number of partitions in the classical computer 100 is reduced.
- the classical computer 100 uses the annealing type computer 200 to realize partitioning in a short time.
- FIG. 8 is a block diagram showing an example of the functions of a classical computer.
- the classical computer 100 has a commutation relation determination unit 110 , a partitioning unit 120 , a quantum circuit generation unit 130 and an expected value acquisition unit 140 .
- the exchange relationship determination unit 110 determines exchange relationships for all combinations of observables to be measured.
- the partitioning unit 120 partitions the observable group based on the exchange relationship between the observables.
- the partitioning unit 120 performs partitioning using the annealing computer 200, for example.
- the partitioning unit 120 creates an Ising model such that the energy decreases as the number of observables included in the partition increases.
- the partitioning unit 120 then instructs the annealing computer 200 to search for the ground state of the created Ising model.
- the annealing-type computer 200 then responds with observable information belonging to the partition.
- the quantum circuit generation unit 130 generates a quantum circuit for measuring observables. For example, the quantum circuit generation unit 130 selects a predetermined number of observables for simultaneous measurement from the same partition, and generates a quantum circuit corresponding to the combination of the selected observables.
- the expected value acquisition unit 140 controls the gate quantum computer 300 to measure observables using the generated quantum circuit.
- the function of each element in the classical computer 100 shown in FIG. 8 can be realized, for example, by causing the computer to execute a program module corresponding to that element.
- FIG. 9 is a diagram depicting an example of processing performed by an exchange relationship determination unit;
- the exchange relationship determination unit 110 lists observables that are elements of the observable group 40 to be measured. Observables are represented by Pauli strings (tensor products of Pauli operators).
- the exchange relationship determination unit 110 determines the exchange relationship for all pairs that select two observables from the observable group 40 .
- the exchange relationship determination unit 110 stores information that associates pairs of observables determined to be commutative. In the example of FIG. 9, pairs of observables determined to be commutative are connected by lines. The commutation relation between observables that are not connected by wires is anticommutative.
- the partitioning unit 120 After the exchange relationship determination unit 110 finishes determining the exchange relationship, the partitioning unit 120 performs partitioning.
- the partitioning unit 120 repeats, for example, partition generation processing that includes as many observables as possible.
- FIG. 10 is a diagram illustrating an example of partition generation processing.
- the partitioning unit 120 generates an Ising model 81 using ⁇ i ⁇ .
- the Ising model 81 is represented by the following formula.
- f( ⁇ i ⁇ ) in equation (1) corresponds to the Hamiltonian.
- the lower bound of m depends on the number of observables of interest. For example, when the number of observables is about 10, the lower limit of m is about "0.25". If the number of observables is about "5000", m can be lowered to about "0.05". If the value of m is increased, the solution will converge more quickly in the solution search. Therefore, when priority is given to calculation accuracy, it is appropriate to set the value of m to a value close to the lower limit.
- the second term on the right side becomes a higher value as the number of pairs of anticommutative observables included in the partition elements increases.
- f( ⁇ i ⁇ ) takes the minimum value in equation (1) of the Ising model 81, the second term on the right side is required to be zero.
- the partitioning unit 120 causes the annealing computer 200 to calculate ⁇ i ⁇ that minimizes f( ⁇ i ⁇ ).
- Equation (1) is a combinatorial optimization problem for finding a combination of values of ⁇ i ⁇ that minimizes f( ⁇ i ⁇ ), and can be calculated at high speed by using the annealing computer 200 .
- the partitioning unit 120 After creating one partition from the observable group A 0 , the partitioning unit 120 assumes that the subgroup constituting the created partition is B 0 .
- the partitioning unit 120 repeats the partition creation process for the subgroup whose elements are observables that are not included in any partition.
- FIG. 11 is a diagram illustrating an example of repeated partition generation processing.
- the annealing computer 200 is caused to generate the partition 61 having the maximum number of elements based on the Ising model 81 using all the observables in the observable group 40 as elements.
- Annealing computer 200 responds with, for example, ⁇ ⁇ 1 , .
- the partitioning unit 120 instructs the annealing computer 200 to generate partitions based on subgroups whose elements are observables not included in the partition 61 .
- the annealing computer 200 generates the partition 62 with the maximum number of elements according to the instruction from the partitioning section 120 .
- the quantum circuit generation unit 130 After the partitioning is completed, the quantum circuit generation unit 130 generates a quantum circuit for measuring the complete state of the quantum bits.
- the expected value acquisition unit 140 then controls the gate quantum computer 300 to acquire the expected value of the observable based on the generated quantum circuit.
- FIG. 12 is a diagram showing an example of observable expected value measurement using a quantum circuit for simultaneous measurement.
- the quantum circuit generation unit 130 extracts elements corresponding to observables that are simultaneously measured from one of the plurality of partitions 82a, 82b, .
- the gate quantum computer 300 can perform simultaneous measurements with 4 qubits.
- the quantum circuit generator 130 extracts sets of four elements from each partition.
- the quantum circuit generator 130 then generates a quantum circuit for each combination of the extracted elements. For example, when "YYXX, YXXY, XYYX, XXYY" is extracted, the quantum circuit 83 is generated.
- the generated quantum circuit 83 has a configuration in which a measurement quantum circuit 83b that performs operations for simultaneous measurement of a plurality of observables is added to the output side of a unitary gate 83a that performs operations according to the problem to be solved. ing.
- the measurement quantum circuit 83b is registered in advance in the quantum circuit generation unit 130 in association with, for example, a combination of observables to be measured. That is, the quantum circuit generation unit 130 stores quantum circuits for simultaneously measuring the expected values of the observables shown in the combination pattern in association with the combination pattern of the observables. Then, the quantum circuit generation unit 130 extracts quantum circuits corresponding to the combination of observables indicated by the elements extracted from the partitions from the group of quantum circuits registered in advance.
- the quantum circuit generation unit 130 connects the measurement quantum circuit 83b corresponding to the observable combination to the output side of the unitary gate 83a corresponding to the problem to be solved, and inputs the quantum circuit 83 to the gate-type quantum computer
- the expected value acquisition unit 140 transmits the quantum circuit generated for each observable combination to the gate quantum computer 300 .
- Gate-type quantum computer 300 measures the expected value of the observable according to the received quantum circuit. For example, the gate quantum computer 300 repeats measurement of the state of each quantum bit using the received quantum circuit multiple times. Then, the gate quantum computer 300 calculates the expected value of the corresponding observable based on the results of multiple measurements of the state of each quantum bit.
- the gate-type quantum computer 300 transmits the input measurement results 84a, 84b, . . . to the classical computer 100 for each quantum circuit.
- the measurement results 84a, 84b, . . . indicate the expected value of the state of each quantum bit of the corresponding quantum circuit (the expected value of the observable corresponding to the corresponding quantum bit).
- the expected value acquiring unit 140 takes the expected values of the quantum bits indicated by the measurement results 84a, 84b, .
- Obtaining the expected values of all observables gives the density operator ⁇ , which indicates the complete state (states of the X-, Y-, and Z-axes) of the output of the unitary gate 83a corresponding to the problem to be solved. .
- the expected value acquisition unit 140 outputs the solution of the problem to be solved, which is obtained based on the density operator ⁇ , as a calculation result.
- FIG. 13 is a flow chart showing an example of an observable measurement procedure. The processing shown in FIG. 13 will be described below along with the step numbers.
- the exchange relationship determination unit 110 lists observables that are elements of an observable group according to the problem to be solved. [Step S102] The exchange relationship determination unit 110 determines exchange relationships between observables included as elements in an observable group. For example, the exchange relationship determination unit 110 generates all combinations (pairs of observables) that select two observables included as elements of the observable group. Then, for each observable pair, the commutative relationship determination unit 110 determines whether the observable pair is commutative or anticommutative based on the commutative/anticommutative relationship between the Pauli operators included in each observable. judge.
- the exchange relationship determination unit 110 creates an Ising model in which the value of the Hamiltonian decreases as the number of observables included in the partition increases.
- the partitioning unit 120 partitions the observable group. Partitioning produces one or more partitions. Details of the partitioning process will be described later (see FIG. 14).
- the quantum circuit generation unit 130 selects a predetermined number of observables whose expected values have not yet been calculated from the same partition. [Step S106] The quantum circuit generation unit 130 generates a quantum circuit for simultaneously measuring a plurality of selected observables.
- the expected value acquiring unit 140 transmits the quantum circuit to the gate-type quantum computer 300 and instructs calculation according to the quantum circuit.
- the gate-type quantum computer 300 repeats the calculation according to the quantum circuit a predetermined number of times, and calculates the expected value of the quantum bit corresponding to each selected observable.
- the gate quantum computer 300 transmits the obtained expected value to the expected value acquisition unit 140 .
- the expected value acquisition unit 140 acquires the expected value of the observable from the gate quantum computer 300 .
- the quantum circuit generator 130 determines whether there is an unselected observable. If there is an unselected observable, quantum circuit generation unit 130 advances the process to step S105. Further, when all observables have been selected, the quantum circuit generation unit 130 advances the process to step S109.
- the expected value obtaining unit 140 obtains the solution of the problem to be solved based on the expected value of the observable to be measured, and outputs the obtained solution. In this way, by performing partitioning, a partition is generated by collecting simultaneously measurable observables, and the expected values of the observables included in the same partition are simultaneously measured.
- FIG. 14 is a flowchart (1/2) showing an example of a partitioning process procedure. The processing shown in FIG. 14 will be described below according to the step numbers.
- Step S201 The partitioning unit 120 sets the number of observables to N (N is a natural number), and sets each observable to ⁇ P 1 , P 2 , . . . , P N ⁇ . Each observable is represented by a Pauli string.
- Step S202 The partitioning unit 120 generates an N ⁇ N square matrix C in which all element values are “0”.
- Step S203 The partitioning unit 120 counts up the loop variable n1 one by one from 1, and repeats the processing of steps S204 to S207 until it reaches N.
- Step S204 The partitioning unit 120 counts up the loop variable n2 from n1 by one, and repeats the processing of steps S205 and S206 until it reaches N.
- Step S207 When the processes of steps S205 and S206 are completed for all n2 from n1 to N, the partitioning unit 120 advances the process to step S208.
- Step S208 When the processes of steps S204 to S207 are completed for all n1 from 1 to N, the partitioning unit 120 advances the process to step S211 (see FIG. 15).
- FIG. 15 is a flowchart (2/2) showing an example of the partitioning process procedure. The processing shown in FIG. 15 will be described below according to the step numbers.
- the partitioning unit 120 generates an Ising model.
- the Ising model is represented by the following formula.
- Step S214 The partitioning unit 120 transmits the Ising model to the annealing computer 200 and instructs to search for the ground state of the Ising model.
- the annealing computer 200 generates a binary variable set ⁇ (n) that minimizes f( ⁇ (n) ) in the Ising model according to the instructions.
- the partitioning unit 120 acquires from the annealing computer 200 the binary variable set ⁇ (n) that minimizes f( ⁇ (n) ).
- the partitioning unit 120 defines a set P (n) of observables included in the partition as follows.
- the set P (n) includes observables corresponding to variables whose value is "1" in the binary variable set ⁇ ( n) that minimizes f( ⁇ (n) ).
- the partitioning unit 120 defines ⁇ (n+1) as shown in the following equation (4).
- ⁇ (n+1) includes variables whose value is “0” in the binary variable group ⁇ (n) that minimizes f( ⁇ (n) ).
- the partitioning unit 120 determines whether ⁇ (n+1) is an empty set. The partitioning unit 120 ends the partitioning process if ⁇ (n+1) is an empty set. If ⁇ (n+1) is not an empty set, the partitioning unit 120 advances the process to step S217.
- FIG. 16 is a diagram showing comparison results of the number of partitions for each partitioning method.
- a graph 91 in FIG. 16 represents the relationship between the number of observables and the number of partitions for each partitioning method.
- the graph 91 has the number of observables on the horizontal axis and the number of partitions on the vertical axis.
- BronK QWC “BronK QWC”, “BopanaH QWC”, “BopanaH GC”, and “OpenF QWC” are calculation methods that perform partitioning without using the Ising model.
- BronK and “BoppanaH” are abbreviations for the Bron-Kerbosch algorithm and the Boppana-Halldorsson algorithm, respectively.
- OpenF is an abbreviation for a method using Open Fermion, which is a PYTHON (registered trademark) package.
- Ising model use GC Full-tomography
- Ising model use GC VQE-observable
- partitioning unit 120 shown in the second embodiment.
- Full-tomography in “Using the Ising model GC” is an example of measuring all observables to fully understand the quantum state.
- VQE-observable is an example in which partitioning is performed on an observable used for calculation of a Variational Quantum Eigensolver (VQE).
- FIG. 17 is a diagram showing a comparison result of calculation times for each partitioning method.
- a graph 92 in FIG. 17 represents the relationship between the number of observables and computation time for each partitioning scheme.
- the horizontal axis is the number of observables
- the vertical axis is the calculation time required for partitioning.
- the annealing computer 200 is used to search for the ground state of the Ising model, but the classical computer 100 can also search for the ground state of the Ising model.
- the classical computer 100 can search for the ground state of the Ising model by performing software-based simulated annealing. Searching for the ground state of the Ising model by simulated annealing by software requires more processing time than when the annealing computer 200 is used, but it is possible to reduce the number of partitions.
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Abstract
Description
コンピュータは、複数の量子ビットの状態を表す複数のオブザーバブルのうちの2つのオブザーバブルの対ごとに、期待値の同時測定が可能か否かを判定する。次にコンピュータは、複数のオブザーバブルの部分集合である第1のパーティションに複数のオブザーバブルそれぞれを含めるか否かを示す変数を含み、同時測定できないオブザーバブルの対が第1のパーティションに含まれるとハミルトニアンの値が大きくなり、第1のパーティションに含めるオブザーバブル数が多いほどハミルトニアンの値が小さくなるイジングモデルを生成する。そしてコンピュータは、イジングモデルの基底状態の探索により得られた複数のオブザーバブルそれぞれに対応する変数の値に基づいて、1以上のオブザーバブルを含む第1のパーティションを生成する。
本発明の上記および他の目的、特徴および利点は本発明の例として好ましい実施の形態を表す添付の図面と関連した以下の説明により明らかになるであろう。
〔第1の実施の形態〕
まず第1の実施の形態について説明する。第1の実施の形態は、イジングモデルを利用して同時測定可能なオブザーバブルを集めたパーティションの数を低減する複数量子ビットオブザーバブルのパーティショニング方法である。
次に第2の実施の形態について説明する。第2の実施の形態は、量子ビットの完全な状態の測定を効率的に行うコンピュータシステムである。
ρ=λ0I0I1I2+λ1X0I1I2+λ2Y0I1I2+λ3Z0I1I2+λ4 I0X1I2+λ5X0X1I2+・・・+λ62y0Z1Z2+λ63Z0Z1Z2
λ0,・・・,λ63は実数の係数である。添字付きのX,Y,Zはパウリ演算子であり、2×2のパウリ行列(σx,σy,σz)である。添字の数値は、測定対象の量子ビットの番号である。添字付きのIは恒等演算子であり、2×2の単位行列である。恒等演算子もパウリ演算子の一種である。
図4は、オブザーバブルの同時測定方法を説明する図である。図4の例では、3量子ビットの量子状態|ψ>を求める場合を想定している。解くべき問題に対応する量子回路30の3つのオブザーバブル「I0Y1X2、Z0Z1Z2、X0I1X2」を測定するものとする。量子回路30の横線が、量子ビットに対応する。
図6は、パーティショニングの一例を示す図である。例えば14個のオブザーバブル41~54について、他のオブザーバブルとの間の交換関係が判断される。図6の例では、可換の交換関係を有するオブザーバブルの組が線で接続されている。そしてオブザーバブル間の交換関係に基づいてパーティション61,62が生成される。パーティション61に属するオブザーバブルのすべての組み合わせにおいて交換関係は可換となる。同様にパーティション62に属するオブザーバブルのすべての組み合わせにおいて交換関係は可換となる。
図7は、パーティショニングの例を示す図である。オブザーバブル群71の各パーティションを複数のパーティションのいずれかに分類するとき、パーティションの分け方は複数通り存在する。例えば3個のパーティション72a~72cに分けることができる。また6個のパーティション73a~73fに分けることも可能である。
パーティショニング部120は、オブザーバブル間の交換関係に基づいて、オブザーバブル群のパーティショニングを行う。パーティショニング部120は、例えばアニーリング型コンピュータ200を利用してパーティショニングを実施する。アニーリング型コンピュータ200を利用する場合、パーティショニング部120は、パーティションに含まれるオブザーバブル数が多いほどエネルギーが低くなるようなイジングモデルを作成する。そしてパーティショニング部120は、作成したイジングモデルの基底状態の探索をアニーリング型コンピュータ200に指示する。するとアニーリング型コンピュータ200からは、パーティションに属するオブザーバブルの情報が応答される。
なお、図8に示した古典コンピュータ100内の各要素の機能は、例えば、その要素に対応するプログラムモジュールをコンピュータに実行させることで実現することができる。
図9は、交換関係判定部が行う処理の一例を示す図である。交換関係判定部110は、測定するオブザーバブル群40の要素となるオブザーバブルを列挙する。オブザーバブルは、パウリ文字列(パウリ演算子のテンソル積)で表される。交換関係判定部110は、オブザーバブル群40から2つのオブザーバブルを選択するすべての対について、交換関係を判定する。交換関係判定部110は、可換と判定したオブザーバブルの対を関連付ける情報を記憶する。図9の例では、可換と判定されたオブザーバブルの対が線で接続されている。線で接続されていないオブザーバブル間の交換関係は反可換である。
図13は、オブザーバブル測定手順の一例を示すフローチャートである。以下、図13に示す処理をステップ番号に沿って説明する。
[ステップS102]交換関係判定部110は、オブザーバブル群に要素として含まれるオブザーバブル間の交換関係を判定する。例えば交換関係判定部110は、オブザーバブル群の要素として含まれるオブザーバブルを2つ選択するすべての組み合わせ(オブザーバブルの対)を生成する。そして交換関係判定部110は、オブザーバブルの対ごとに、各オブザーバブルに含まれるパウリ演算子間の可換・反可換の関係に基づいて、オブザーバブルの対が可換か反可換かを判定する。
[ステップS104]パーティショニング部120は、オブザーバブル群のパーティショニングを行う。パーティショニングにより、1以上のパーティションが生成される。パーティショニング処理の詳細は後述する(図14参照)。
[ステップS106]量子回路生成部130は、選択した複数のオブザーバブルを同時測定するための量子回路を生成する。
[ステップS109]量子回路生成部130は、未選択のオブザーバブルがあるか否かを判断する。量子回路生成部130は、未選択のオブザーバブルがあれば処理をステップS105に進める。また量子回路生成部130は、すべてのオブザーバブルが選択済みとなった場合、処理をステップS109に進める。
このようにして、パーティショニングを行うことで同時測定可能なオブザーバブルを集めたパーティションが生成され、同一のパーティションに含まれるオブザーバブルの期待値が同時測定される。
図14は、パーティショニング処理の手順の一例を示すフローチャート(1/2)である。以下、図14に示す処理をステップ番号に沿って説明する。
[ステップS203]パーティショニング部120は、ループ変数n1を1から1ずつカウントアップし、NになるまでステップS204~S207の処理を繰り返す。
[ステップS205]パーティショニング部120は、n1番目のオブザーバブルPn1とn2番目のオブザーバブルPn2との対が可換(Pn1Pn2=Pn2Pn1)か反可換(Pn1Pn2=-Pn2Pn1)かを判断する。パーティショニング部120は、可換であれば処理をステップS207に進める。またパーティショニング部120は、反可換であれば処理をステップS206に進める。
[ステップS211]パーティショニング部120は、バイナリ変数群χ(1)={χ1,χ2,・・・,χN}を定義する。
[ステップS213]パーティショニング部120は、イジングモデルを生成する。イジングモデルは以下の式で表される。
[ステップS214]パーティショニング部120は、アニーリング型コンピュータ200にイジングモデルを送信し、イジングモデルの基底状態の探索を指示する。アニーリング型コンピュータ200は、指示に従って、イジングモデルにおけるf(χ(n))が最小となるバイナリ変数群χ(n)を生成する。
[ステップS216]パーティショニング部120は、パーティションに含まれるオブザーバブルの集合P(n)を以下のように定義する。
[ステップS217]パーティショニング部120は、χ(n+1)を以下の式(4)に示すように定義する。
[ステップS218]パーティショニング部120は、χ(n+1)が空集合か否かを判断する。パーティショニング部120は、χ(n+1)が空集合であればパーティショニング処理を終了する。またパーティショニング部120は、χ(n+1)が空集合でなければ処理をステップS217に進める。
このようにして、いずれのパーティションにも含まれていないオブザーバブルに基づいて、可能な限り多くのオブザーバブルを含むパーティションが繰り返し生成される。その結果、生成されるパーティション数の削減が可能となる。しかもアニーリング型コンピュータ200を利用して高速にパーティションを作成することができる。
第2の実施の形態では、イジングモデルの基底状態の探索にアニーリング型コンピュータ200を利用しているが、古典コンピュータ100がイジングモデルの基底状態の探索を行うことも可能である。例えば古典コンピュータ100は、ソフトウェアによるシミュレーテッドアニーリングを実行することでイジングモデルの基底状態を探索可能である。ソフトウェアによるシミュレーテッドアニーリングでイジングモデルの基底状態を探索するとアニーリング型コンピュータ200を利用した場合よりも処理時間はかかるが、パーティション数を低減することは可能である。
2 オブザーバブル群
3 イジングモデル
4 第1のパーティション
5 第2のパーティション
10 情報処理装置
11 記憶部
12 処理部
Claims (7)
- コンピュータが、
複数の量子ビットの状態を表す複数のオブザーバブルのうちの2つのオブザーバブルの対ごとに、期待値の同時測定が可能か否かを判定し、
前記複数のオブザーバブルの部分集合である第1のパーティションに前記複数のオブザーバブルそれぞれを含めるか否かを示す変数を含み、同時測定できないオブザーバブルの対が前記第1のパーティションに含まれるとハミルトニアンの値が大きくなり、前記第1のパーティションに含めるオブザーバブル数が多いほど前記ハミルトニアンの値が小さくなるイジングモデルを生成し、
前記イジングモデルの基底状態の探索により得られた前記複数のオブザーバブルそれぞれに対応する変数の値に基づいて、1以上のオブザーバブルを含む前記第1のパーティションを生成する、
複数量子ビットオブザーバブルのパーティショニング方法。 - 前記コンピュータが、さらに、
前記イジングモデルの変数を、生成した前記第1のパーティションに含まれない残存のオブザーバブルに対応する変数に限定して、前記イジングモデルの基底状態を探索することで得られた前記残存のオブザーバブルそれぞれの変数の値に基づいて、1以上の前記残存のオブザーバブルを含む第2のパーティションを生成する、
請求項1記載の複数量子ビットオブザーバブルのパーティショニング方法。 - 前記第2のパーティションの生成を、前記第1のパーティションまたは生成済みの前記第2のパーティションのいずれにも含まれないオブザーバブルがなくなるまで繰り返す、
請求項2記載の複数量子ビットオブザーバブルのパーティショニング方法。 - 前記第1のパーティションの生成では、前記イジングモデルの基底状態の探索をアニーリング型コンピュータに指示し、前記アニーリング型コンピュータから前記イジングモデルの基底状態の探索結果として前記複数のオブザーバブルそれぞれの変数の値を取得する、
請求項1から3までのいずれかに記載の複数量子ビットオブザーバブルのパーティショニング方法。 - 前記コンピュータが、さらに、
前記第1のパーティションから所定数の対象オブザーバブルを選択し、
選択した前記対象オブザーバブルの期待値を同時測定する量子回路を生成し、
ゲート型量子コンピュータに、生成した前記量子回路に基づく前記対象オブザーバブルの期待値の同時測定を指示する、
請求項1から4までのいずれかに記載の複数量子ビットオブザーバブルのパーティショニング方法。 - コンピュータに、
複数の量子ビットの状態を表す複数のオブザーバブルのうちの2つのオブザーバブルの対ごとに、期待値の同時測定が可能か否かを判定し、
前記複数のオブザーバブルの部分集合である第1のパーティションに前記複数のオブザーバブルそれぞれを含めるか否かを示す変数を含み、同時測定できないオブザーバブルの対が前記第1のパーティションに含まれるとハミルトニアンの値が大きくなり、前記第1のパーティションに含めるオブザーバブル数が多いほど前記ハミルトニアンの値が小さくなるイジングモデルを生成し、
前記イジングモデルの基底状態の探索により得られた前記複数のオブザーバブルそれぞれに対応する変数の値に基づいて、1以上のオブザーバブルを含む前記第1のパーティションを生成する、
処理を実行させる複数量子ビットオブザーバブルのパーティショニングプログラム。 - 複数の量子ビットの状態を表す複数のオブザーバブルのうちの2つのオブザーバブルの対ごとに、期待値の同時測定が可能か否かを判定し、前記複数のオブザーバブルの部分集合である第1のパーティションに前記複数のオブザーバブルそれぞれを含めるか否かを示す変数を含み、同時測定できないオブザーバブルの対が前記第1のパーティションに含まれるとハミルトニアンの値が大きくなり、前記第1のパーティションに含めるオブザーバブル数が多いほど前記ハミルトニアンの値が小さくなるイジングモデルを生成し、前記イジングモデルの基底状態の探索により得られた前記複数のオブザーバブルそれぞれに対応する変数の値に基づいて、1以上のオブザーバブルを含む前記第1のパーティションを生成する処理部、
を有する情報処理装置。
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