JP2018022446A - Quantum information processor - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a technology allowing a trial wave function formed of a single arrangement state function to be efficiently generated.SOLUTION: A quantum information processor of an aspect of the present invention generates an N-electron-system spin eigenfunction in which the number of β spin electrons is N, of N unpaired electrons, as a trial wave function formed of a single arrangement state function. The quantum information processor includes N quantum bits, and a quantum arithmetic unit. The quantum arithmetic unit executes processing of (a) rotating the 2i-th quantum bit by a rotation angle θ=-π/4 while increasing a variable i from 1 to Nin increments of one, and (b) applying a control NOT calculation to the first quantum bit to the 2i-1-th quantum bit using the 2i-th quantum bit as the control reference.SELECTED DRAWING: Figure 8

Description

  The present invention relates to a technology of a quantum information processing apparatus.

  Exactly solving the Schroedinger equation governing the physical and chemical properties of atoms and molecules is the ultimate goal in the chemical and physical fields. If the Schrodinger equation can be solved exactly and the electronic wave function, which is the information function of microscopic particles, can be obtained, the molecular structure, chemical reaction path, spectroscopic properties, molecular properties, etc. can be accurately predicted. . However, except for special cases such as one-electron systems, an exact analytical solution of the Schroedinger equation cannot be obtained.

The FCI (Full Configuration Interaction) method is known as a technique for obtaining an optimal solution of the Schroedinger equation in a space formed by a given basis function based on a variational method. The calculation amount of the FCI method depends exponentially on the number of molecular electrons and the number of basis functions used for wave function expansion. For this reason, the calculation of the FCI method is still reported only for small molecules such as nitrogen molecules (N ₂ ), CN molecules, and the like.

  In 1982, Feynman was able to simulate physical and matter systems governed by the laws of quantum mechanics on a classical computer (a modern computer operating in classical mechanics / electromagnetism). Although time is required, he pointed out that the simulation can be done efficiently by using a quantum computer (a computer whose principle is the principle of quantum mechanics). In 2005, Aspuru-Guzik et al. Proposed an algorithm that can perform FCI calculation efficiently by using a quantum computer (Non-patent Document 1). The speedup of this algorithm originates in the quantum phase estimation algorithm, and it has been shown that calculations that require exponential time can be performed in polynomial time in a classical computer.

  The success probability of the quantum phase estimation algorithm is proportional to the square of the overlap of the trial wave function prepared for calculation and the FCI wave function that is the answer to be obtained. Therefore, it is important to develop a method for efficiently generating a trial wave function having a large overlap with the FCI wave function. In a general closed-shell singlet molecule, it is known that an HF (Hartley-Fock) wave function having a single slater determinant has a large overlap with the FCI wave function and is a good trial wave function. In 2010, the HF wave function was used as a trial wave function, and a demonstration experiment of FCI / STO-3G calculation of a hydrogen molecule based on a quantum phase estimation algorithm was carried out with a quantum optical system (Non-patent Document 2) and a nuclear magnetic resonance method ( Non-patent document 3).

  Up to now, an adiabatic state preparation method (ASP method) has been proposed as a method for improving the trial wave function starting from the HF wave function. However, the ASP method has a high calculation cost, and development of a method for generating a trial wave function having a good open-shell molecule more easily is desired.

Aspuru-Guzik, A., Dutoi, A. D., Love, P. & Head-Gordon, M. Simulated quantum computation of molecular energies. Science 309, 1704-1707 (2005). Lanyon, BP; Whitfield, JD; Gillett, GG; Goggin, ME; Almeida, MP; Kassal, I .; Biamonte, JD; Mohseni, M .; Powell, BJ; Barbieri, M .; Aspuru-Guzik, A .; White, AG Towards quantum chemistry on a quantum computer.Nature Chemistry 2, 106−111 (2010). Du, J .; Xu, N .; Peng, X .; Wang, P .; Wu, S .; Lu, D. NMR implementation of a molecular hydrogen quantum simulation with adiabatic state preparation.Physical Review Letters 104, 030502 (2010 ).

  In FCI calculations based on quantum computers published in 2005, wave function information is stored in qubits using a direct mapping method. In the direct mapping method, the qubit is | 1> if electrons are occupied in the spin orbit, and the qubit is | 0> if the electrons are not occupied. It is. The electronic Hamiltonian of a molecule is described by the following equation 1 using the generation and annihilation operators based on the second quantization.

In the direct mapping method, the number of electrons occupied by spin orbits is stored in qubits. However, in order to satisfy the anti-symmetry of the wave function derived from the fact that the electrons are fermions, the Jordan-Wigner transformation is applied, and the generation / annihilation operators are expressed by the following equations 2 and 3. It is expressed as follows.

Since the electronic Hamiltonian represented by the above equation 1 includes many non-commutative operators, the Trotter-Suzuki decomposition is generally applied to simulate the time evolution of the wave function.

Here, a quantum circuit that performs FCI calculation based on the direct mapping method, the second quantization, and the quantum phase estimation algorithm takes a maximum O (N _orb ¹¹ ) operation time when the number of spin orbits is N _orb . (Experience is O (N _orb ⁹ ) calculation time).

In addition, in order to apply FCI calculation based on quantum phase estimation to physically and chemically important systems, it is important to develop a method that efficiently generates a trial wave function Ψ ₀ that has a large overlap with the true wave function. It is. This problem is particularly important for open-shell molecules that contain β-spin as an unpaired electron. This is because the FCI wave function of open-shell molecules has a strong multi-configuration, and the overlap with the HF wave function becomes exponentially smaller with respect to the number of β spins (N _β ). In such a case, if the HF wave function is used as the trial wave function and the FCI calculation based on the quantum phase estimation is performed, the exponential number of iterations are repeated. End up.

As a trial wave function that can have a large overlap with the FCI wave function in an open-shell molecule, there is an arrangement state function that is a simultaneous eigenfunction of electron spin operators S _z and S ² (hereinafter referred to as a spin eigenfunction). The configuration state function is represented by a linear combination of an exponential number of Slater determinants with respect to the number of unpaired electrons. For this reason, in processing by a general classical computer, it takes time to calculate an exponential function to prepare the arrangement state function.

  In one aspect, the present invention has been made in view of such a situation, and an object of the present invention is to provide a technique capable of efficiently generating a trial wave function including a single arrangement state function. It is to be.

  The present invention employs the following configuration in order to solve the above-described problems.

That is, the quantum information processing device according to one aspect of the present invention provides an N-electron spin in which N _β electrons are N _β of N unpaired electrons as a trial wave function including a single arrangement state function. a quantum information processing apparatus for generating an eigenfunction, and N qubits, while incrementing by one the variable i from 1 to N _beta, the rotation angle with respect to (a) 2i th qubit theta = - a quantum operation unit that performs π / 4 rotation, (a) performs a control NOT operation on the 1st to 2i−1 qubits using the 2i-th qubit as a control bit, and executes these processes And comprising. According to this configuration, a spin eigenfunction of an N electron system as a trial wave function composed of a single arrangement state function can be generated within a polynomial operation time of O (N _β ² ).

  In addition, as another form of the quantum information processing apparatus according to each of the above forms, an information processing method for realizing each of the above configurations, a program, or a computer recording such a program Further, it may be a storage medium that can be read by other devices, machines and the like. Here, the computer-readable recording medium is a medium that stores information such as programs by electrical, magnetic, optical, mechanical, or chemical action.

For example, in the quantum information processing method according to one aspect of the present invention, a quantum computer or a classical computer that enables simulation of a quantum computer can perform (a) a rotation angle θ = A process of performing a rotation of −π / 4, and (a) a process of performing a CNOT operation on the first to 2i−1 qubits using the 2i-th qubit as a control bit, and setting the variable i to 1 To N _β by incrementing by 1 as a trial wave function consisting of a single configuration state function, an N-electron spin having N _β electrons out of N unpaired electrons. This is an information processing method for generating an eigenfunction.

In addition, for example, a quantum information processing program according to one aspect of the present invention provides a quantum computer or a classical computer that can simulate processing of the quantum computer by (a) a rotation angle with respect to the 2i-th qubit. a process of performing rotation of θ = −π / 4, (a) a process of performing a CNOT operation on the first to 2i−1 qubits using the 2i-th qubit as a control bit, and a variable i By repeatedly executing it while incrementing from 1 to N _β by one, as a trial wave function consisting of a single configuration state function, an N-electron system in which N _β electrons are N _β electrons out of N unpaired electrons A program for generating a spin eigenfunction.

  According to the present invention, a trial wave function including a single arrangement state function can be efficiently generated.

FIG. 1 shows an example of an arrangement state function. FIG. 2 is a branch diagram illustrating the relationship between the spin quantum number and the number of electrons. FIG. 3 shows nine paths in which the number of electrons is 6 and the spin quantum number S is 1 in FIG. FIG. 4 is a schematic diagram showing a six-electron spin model. FIG. 5 shows an example in which the different paths in FIG. 2 represent the same electronic structure. FIG. 6 shows an example of a quantum computer (quantum information processing apparatus) according to the present embodiment. FIG. 7 shows an example of a processing procedure for generating a trial wave function including a single arrangement state function according to the present embodiment. FIG. 8 shows an example of a quantum circuit that can execute the processing shown in FIG. FIG. 9 shows a cluster of hydrogen atoms used in the numerical calculation example.

  Hereinafter, an embodiment according to an aspect of the present invention (hereinafter, also referred to as “this embodiment”) will be described with reference to the drawings. However, the present embodiment described below is merely an example of the present invention in all points, and is not intended to limit the scope thereof. It goes without saying that various improvements and modifications can be made without departing from the scope of the present invention. That is, in implementing the present invention, a specific configuration according to the embodiment may be adopted as appropriate. Although data appearing in this embodiment is described in a natural language, more specifically, it is specified by a pseudo language, a command, a parameter, a machine language, or the like that can be recognized by a computer.

[Method of expressing arrangement state function]
First, a method for expressing an arrangement state function will be described with reference to FIG. Ψ _CSF shown in FIG. 1 represents a wave function composed of one arrangement state function (number of unpaired electrons N = 4, spin quantum number S = 0, magnetic quantum number M _s = 0), and is shown in parentheses. The arrangement corresponds to one slater determinant. The horizontal lines in parentheses indicate molecular orbitals, and the upward and downward arrows indicate electrons whose spin coordinates are α and β, respectively.

  In quantum chemistry calculations, the wave function of a molecule is represented by a linear combination of Slater determinants developed in molecular orbitals. In quantum chemistry calculations by a quantum computer, a direct mapping method (DM) in which a spin orbit corresponds to a qubit, and if the spin orbit is occupied by an electron, the quantum bit is | 1> and if not occupied, | 0> is assumed. Method) is generally used.

  However, as illustrated in FIG. 1, the configuration state function is a Slater determinant having a different electron arrangement only in a molecular orbital (one-electron occupied orbital) in which one electron occupies only one of the α orbital or β orbital. It is described by a linear combination of That is, the electron arrangement of the two-electron occupied orbitals and the unoccupied orbitals is the same in all Slater determinants. Therefore, in order to construct a spin eigenfunction as an arrangement state function, a slater determinant that differs only in the spin coordinates of electrons that occupy one electron occupation orbit may be appropriately linearly combined.

  That is, in order to generate the arrangement state function, it is only necessary to focus on one electron occupation orbit. Therefore, in this embodiment, as a method of embedding wave function information (spin eigenfunction) in a qubit (logical qubit 10 described later), when the spin coordinate of an electron occupying one electron occupation orbit is α. When | 0> and β, a spin coordinate mapping method (SCM method) of | 1> is used.

  In order to perform quantum phase estimation, it is necessary to re-express the wave function by the direct mapping method. Note that the one-electron occupied orbital representation in the spin coordinate mapping method is consistent with the β-spin orbital representation in the one-electron occupied orbital in the direct mapping method. On the other hand, the expression of the α spin orbit in one electron occupation orbit in the direct mapping method is in agreement with the expression in which | 0> and | 1> in the spin coordinate mapping method are interchanged. Therefore, a qubit corresponding to the β spin orbit of one electron occupation orbit in the direct mapping method can be used for the spin orbit mapping method. Further, the α spin orbit of one electron occupation orbit in the direct mapping method is generated by a control NOT (CNOT) operation using a qubit representing the corresponding β spin orbit as a control bit for the qubit initialized to | 1>. be able to.

  The quantum phase estimation is a method of reading the eigenvalue of the unitary operator U as the phase φ (0 ≦ φ <1) using a quantum computer. In order to perform FCI calculation by quantum phase estimation, the calculation shown in the following Equation 4 is performed with U = −iHt / h.

Here, H is a Hamiltonian of the system, Ψ is a wave function, and E is an energy eigenvalue. φ is a phase determined by the quantum computer, uses j qubits, and is 0. φ ₁ φ ₂ ... φ _j If Ψ is a true wave function that is an eigenfunction of Hamiltonian, quantum phase estimation can be performed deterministically. However, since it is generally difficult to prepare a true wave function, quantum phase estimation is performed using a trial wave function. I do. In this case, the quantum phase estimation is stochastic, and the success probability is proportional to the square of the overlap of the trial wave function and the true wave function.

  The CNOT operation is one of basic two qubit operations in a quantum computer. Specifically, in the CNOT operation, one qubit is a control bit, the other qubit is a target bit, and when the control qubit is | 0>, no operation is performed on the target qubit, When the control qubit is | 1>, a NOT operation is performed on the target qubit. When a CNOT operation is performed on a general state composed of two qubits with the first qubit as the control bit and the subsequent qubit as the target bit, the quantum state changes as shown in Equation 5 below. Equation 5 is a commonly known CNOT operation. When the control qubit is | 1>, no operation is performed on the target qubit, and when the control qubit is | 0>, the target quantum A two-qubit operation that performs a NOT operation on a bit is also called a CNOT operation.

[Method of generating spin eigenfunctions based on angular momentum synthesis]
Next, a method for generating a spin eigenfunction based on a synthesis rule of angular momentum will be described. The spin eigenfunction of the (k + 1) electron system is to synthesize one electron spin angular momentum into the spin eigenfunction of the k electron system using the following equations 6 and 7 derived from the angular momentum synthesis rule. Can be generated. Here, attention is paid to a spin eigenfunction where the magnetic quantum number M _s = spin quantum number S. For nonrelativistic Schlesinger over all energies in the Staudinger equation does not depend on the magnetic quantum number M _s, generality is not lost even if only the magnetic quantum number M _s = spin quantum number S.

Here, | Ψ (k, S, S)> is a spin eigenfunction of a k-electron system (k unpaired electrons) having a spin quantum number S and a magnetic quantum number M _s = S. Further, | α> and | β> are spin eigenfunctions of a one-electron system defined by the following equations 8 and 9, respectively.

If | ψ (k, S, S)> and | ψ (k, S, S−1)> can be obtained from these equations, a spin eigenfunction of the (k + 1) electron system can be generated. . Although details will be described later, when the spin quantum number S is ½, using the spin coordinate mapping method, the spin eigenfunction | Ψ (k, S = 1/2, M _s = −1 / 2). )> Can be generated from | Ψ (k, S = 1/2, M _s = + 1/2)> by a simple quantum operation. Therefore, the spin eigenfunction having an arbitrary number of electrons can be sequentially generated by repeatedly using the above equations 6 and 7.

  Next, the relationship between the order of synthesizing angular momentum and the spin eigenfunction will be described with reference to FIGS. FIG. 2 is a branch diagram illustrating the relationship between the spin quantum number and the number of electrons. FIG. 3 shows nine paths in which the number of electrons is 6 and the spin quantum number S is 1 in FIG. The spin eigenfunction is characterized by the order in which the angular momentum is synthesized. One path in the branch diagram shown in FIG. 2 corresponds to one spin eigenfunction.

  The numbers described at the vertices (circles) represent the number of spin eigenfunctions having the corresponding number of electrons N and the spin quantum number S, and the number of paths that can reach the vertex starting from the origin. Match. The line extending from the apex to the upper right corresponds to the case where the spin quantum number increases by ½ when one electron spin angular momentum is synthesized into a spin eigenfunction. On the other hand, the line extending from the apex to the lower right corresponds to a case where the spin quantum number decreases by ½ when one electron spin angular momentum is synthesized into a spin eigenfunction. For example, as illustrated in FIG. 3, there are nine independent paths that reach the apex where the number of electrons N is 6 and the spin quantum number S is 1. That is, there are nine spin eigenfunctions where the number of electrons N is 6 and the spin quantum number S is 1.

  Next, the relationship between the order of synthesizing the angular momentum and the spin site numbering will be described with reference to FIGS. FIG. 4 is a schematic view illustrating a six-electron spin model. FIG. 5 shows an example in which different paths on the branch diagram of FIG. 2 represent the same electronic structure. Here, as a specific example of the process of generating the spin eigenfunction using the angular momentum synthesis rule, the triplet state (S = 1) composed of 6 electron spins shown in FIG. 3 is taken up. In FIG. 3, an arrow represents an electron spin, and its direction schematically shows the spin arrangement. If the two arrows are pointing in the same direction, a ferromagnetic interaction is acting between the two electrons, and if the two arrows are pointing in the opposite direction, an antiferromagnetic interaction is acting between the two electrons. Show.

  When the 6-electron spin eigenfunction illustrated in FIG. 4 is generated, spin site numbering can be arbitrarily performed. Therefore, by appropriately numbering the spin sites, different paths on the branch diagram can represent the same electronic structure as illustrated in FIG. FIG. 5 shows an example of the numbering of spin sites corresponding to three different paths that reach the vertex illustrated in FIG. From this, if the spin eigenfunction can be generated for any one of the possible paths on the bifurcation diagram, the spin sites are numbered so as to correspond to the spin eigenfunction generation order in that path. Thus, a spin eigenfunction having a large overlap with a true wave function can be generated in an open shell molecule whose approximate electron spin structure is known.

[M _s sign inversion operator]
Next, the M _s sign inversion operator used in this embodiment will be described. The key step in sequentially generating spin eigenfunctions using Equation 6 and Equation 7 is to generate | ψ (k, S, S-1)> from | ψ (k, S, S)>. It is a step. By utilizing the above-described arbitrary spin site numbering, in the bifurcation diagram illustrated in FIG. 2, in the first 2N _β step, between the spin quantum numbers S = 1/2 and S = 0. In the remaining N-2N _β steps, attention is paid only to the path shown in FIG. 3 and FIG. 5 (ix) where the spin quantum number S is increased. N _β represents the number of electrons occupying the β spin orbit among N unpaired electrons. The number N _β of electrons in this β spin is equal to or less than the number N _{α of} electrons occupying the α spin orbit (N = N _α + N _β , N _α ≧ N _β ).

In this focused path, the synthesis of the electron spin angular momentum that reduces the spin quantum number S (a line extending in the lower right in the figure) appears only between the spin quantum numbers S = 1/2 and S = 0. It has the characteristics. That is, the above formula 7 is applied only to the spin eigenfunction having the spin quantum number S = 1/2, and | ψ (k, S, S)> to | ψ (k, S, S−1). > Is generated from | Ψ (k, S = 1/2, M _s = 1/2)> to | Ψ (k, S = 1/2, M _s = −1 / 2)>. Replaced with an operation. This operation can be achieved by using the M _s sign inversion operator (M _s SIGN_FLIP (i)) defined by Equation 10 below.

This operator M _s _SIGN_FLIP (i) reverses the coordinates of all electron spins of the spin eigenfunction of the i-electron system (| α> ← → | β>), thereby obtaining the absolute sign of the magnetic quantum number M _s. Has the effect of reversing. When the spin coordinate mapping method is used, the M _s sign inversion operator can be described using the σ _x operator as shown in the following Expression 11.

Here, σ _x (p) is a Pauli operator acting on the p-th qubit, and corresponds to a NOT operation. In terms of the spin coordinate mapping method, the M _s sign inversion operator acts as an operator that performs a NOT operation on the 1st to i-th qubits. Therefore, the M _s sign inversion operator is a unitary operator and can be executed on the quantum computer. Further, when the above formula 7 is applied to the spin eigenfunction of the spin quantum number S = 1/2, the above formula 7 is expressed as the following formula 12.

This operation is performed only when the (k + 1) th qubit is a control qubit and the (k + 1) th qubit is | 0>, and the M _s sign inversion operation is performed on each of the first to kth qubits. This can be achieved by operating a child (hereinafter, this operation is referred to as a “Control_M _s SIGN_FLIP operation”). That, Control_M _s _SIGN_FLIP operation is accomplished by performing a CNOT operation on the (k + 1) -th control bits qubits, each qubit from the first to k-th. Using this Control_M _s _SIGN_FLIP operation, among the process of sequentially generating a spin eigenfunctions with the number 6 and number 7, a spin eigenfunctions with number 7 | Ψ (k, S = 1/2, M S = 1/2)> to generate a spin eigenfunction | Ψ (k + 1, S = 0, M _S = 0)>. Note that the spin eigenfunctions with number 6 | Ψ (k, S, M S)> spin eigenfunctions from | Ψ (k + 1, S + 1/2, M S +1/2) operation that generates a> is the spin coordinates mapping When the method is used, this is achieved by initializing the (k + 1) th qubit to | 0>.

[Example of hardware configuration of quantum computer]
Next, a quantum computer that performs the quantum operation will be described with reference to FIG. FIG. 6 shows an example of the quantum computer 1 according to this embodiment. As illustrated in FIG. 6, the quantum computer 1 according to the present embodiment includes a plurality of logical qubits 10 each including a plurality of physical qubits 11 and a quantum error correction circuit 12, and each logical qubit 10. And a quantum operation unit 13 that performs a quantum operation on each physical qubit 11. This quantum computer 1 corresponds to a “quantum information processing apparatus” of the present invention.

  In the quantum computer 1, each operation is executed as a quantum operation governed by the laws of quantum mechanics. Each physical qubit 11 constituting the logical qubit 10 is a basic element that holds information (per qubit) and can take an arbitrary linear combination of | 0> and | 1>. Each physical qubit 11 is realized by, for example, a semiconductor quantum dot, a nuclear spin or an electron spin in a molecule, a nitrogen vacancy (lattice defect) in a diamond, an optical quantum circuit, a superconducting quantum circuit, an ion trap, or the like. The quantum operation unit 13 is appropriately configured according to a method for realizing the physical qubit 11 so that a quantum operation process can be performed on the physical qubit 11.

  Each physical qubit 11 that is a quantum entity has an inherently constant quantum error (attenuation of quantum properties), and therefore, when used directly for practical information transmission and complex large-scale quantum computation, a defect occurs. There is. In order to cope with this, the logical qubit 10 includes a quantum error correction circuit 12. The quantum error correction circuit 12 includes a plurality of physical qubits, and is appropriately configured so that the quantum error of each physical qubit 11 can be corrected. Thereby, the logical qubit 10 can hold quantum information corresponding to the number of physical qubits 11 while extending the coherence time.

  In addition, regarding the specific hardware configuration of the quantum computer 1, components can be omitted, replaced, and added as appropriate according to the embodiment. The hardware system of the quantum computer 1 may be, for example, a quantum annealing system or a quantum gate system. Examples of quantum annealing quantum computers include D-Wave machines sold by D-Wave.

  In addition, the quantum information processing apparatus may be configured by a classical computer as long as the above quantum operation can be simulated by a simulated annealing method or the like. In this case, the quantum information processing apparatus may be a general-purpose information processing apparatus such as a personal computer in addition to the information processing apparatus designed for exclusive use.

[Operation example]
Next, an operation example of the quantum computer 1 will be described with reference to FIG. FIG. 7 shows that the quantum computer 1 generates a spin eigenfunction of an N-electron system in which N _β electrons out of N unpaired electrons are N _β as a trial wave function composed of a single configuration state function. A processing procedure is schematically illustrated. Note that the processing procedure described below is merely an example, and the processing of each step may be changed as much as possible. Further, in the processing procedure described below, steps can be omitted, replaced, and added as appropriate according to the embodiment.

  First, as an initial operation, the quantum computer 1 starts up in a state where no information is stored in each logical qubit 10. Subsequently, the quantum computer 1 executes reading of the program, etc., sets 1 to the variable i, and starts processing related to the quantum operation. In the following quantum operation, the quantum computer 1 uses N logical qubits 10. In the present embodiment, the logical qubit 10 corresponds to the “qubit” of the present invention.

  In step S101, the quantum computer 1 initializes the (2i-1) th and 2ith logical qubits 10 and adds them to the quantum computation region, for example, as shown in the following Equation 13. At this time, the first logical qubit 10 represents | ψ (1, 1/2, 1/2)>. The initialization of the (2i-1) -th logical qubit uses the equation (6) above to (2i-2) synthesize one electron spin angular momentum to the spin eigenfunction of the electron system, and the spin quantum number This corresponds to the process of increasing S by 1/2.

  In next step S <b> 102, the quantum computer 1 applies a rotation operation with a rotation angle θ = −π / 4 to the 2i-th logical qubit 10 by the quantum operation unit 13. For example, the quantum operation unit 13 performs a rotation process with a rotation angle θ = −π / 4 by the operation shown in the following Expression 14.

In the next steps S103 and S104, the quantum computer 1 sets the variable j to 1 by the quantum operation unit 13, and performs the control _Ms sign inversion operation. That is, the quantum computer 1 performs a CNOT operation on the j-th logical qubit 10 using the 2i-th logical qubit 10 as a control qubit. The quantum computer 1 repeatedly executes this CNOT operation while incrementing the variable j by 1 from 1 to 2i-1. In other words, the quantum computer 1 repeatedly performs steps S103 and S104 to perform a CNOT operation on the first to 2i−1 logical qubits 10 using the 2i-th logical qubit 10 as a control bit. . This process uses the above formula 7 to synthesize one electron spin angular momentum with the spin eigenfunction of (2i-1) electron system and reduce the spin quantum number S by 1/2. Equivalent to.

In the next step S105, the quantum computer 1 determines whether or not the variable i matches the number N _β of electrons of β spin. If the variable i does not match _Nβ , the quantum computer 1 adds 1 to the variable i and repeats the processing from step S101. Thus, a quantum computer 1, the quantum computation unit 13, while incrementing by one the variable i from 1 to N _beta, repeats the process from step S101 to step S104. On the other hand, when the variable i matches _Nβ , the quantum computer 1 stops the repetition process and proceeds to the next step S106.

In the next step S106, the quantum computer 1 initializes the remaining (N−N _β ) logical qubits 10 and adds them to the quantum operation region, for example, as shown in the following formula 15. As a result, N logical qubits 10 added to the quantum operation region have N _β spin electrons out of N unpaired electrons as a trial wave function composed of a single arrangement state function. The spin eigenfunction of the N electron system is expressed and stored by the spin coordinate mapping method. The quantum computer 1 appropriately reads out and uses the trial wave function from the N logical qubits 10 included in the quantum operation region.

  In the quantum operation related to the above processing procedure, N logical qubits 10 are used. Therefore, in each logical qubit 10, coherency can be maintained for a relatively long time by the action of the quantum error correction circuit 12. In the above processing procedure, initialization is performed immediately before each logical qubit 10 is used. Therefore, the time for holding information in each logical qubit 10 can be shortened, and the influence of quantum errors can be reduced. However, it is not necessary to be limited to such an example, and in the above embodiment, the quantum operation may be performed using N physical qubits 11. In this case, each physical qubit 11 corresponds to a “qubit” of the present invention. The quantum computer 1 may initialize all N logical qubits 10 when starting the processing.

<Features>
As described above, according to the present embodiment, a trial wave function including a single arrangement state function is generated based on a synthesis rule of angular momentum, which is a basic theorem in quantum mechanics. One feature is that the arrangement state function can be generated on the logical qubit 10 within the polynomial time by the above processing procedure. Further, as another feature, the processing procedure can be executed by a very simple quantum operation such as steps S101 to S105 by appropriately selecting a procedure for synthesizing the angular momentum of unpaired electrons.

This will be described in detail with reference to FIG. FIG. 8 shows an example of a quantum operation circuit capable of executing the processing procedure of FIG. As apparent from the above procedure and 8, a quantum computer 1, before generating the arrangement functions, and - [pi] / 4 rotation processing for one logical qubit 10 N _beta times, N _beta ² times CNOT operation Process. That is, a quantum computer 1, for N electron system having an electron N _beta number of beta spin, the quantum computation may be performed within a polynomial time O (N _β ^2).

  The Slater determinant constituting the configuration state function differs in the electron configuration only in the molecular orbitals occupied by unpaired electrons, which are called one-electron occupation orbitals. Other molecular orbitals are occupied or unoccupied by two electrons and are not important for the generation of configuration state functions. In the above embodiment, in order to generate an arrangement state function on the quantum computer 1 using this property, a spin coordinate mapping method focusing on the spin coordinates of unpaired electrons occupying one electron occupied orbital is used. A spin eigenfunction is generated in a subspace consisting only of occupied orbitals. As a result, it is possible to generate the arrangement state function more easily than the case of using the direct mapping method often used for FCI calculation using the quantum phase estimation algorithm.

[Numeric calculation example]
FIG. 6 illustrates cluster models (hereinafter referred to as 5H cluster, 9H cluster, and 13H cluster, respectively) each consisting of 5, 9, and 13 hydrogen atoms. A numerical simulation was performed on the cluster model illustrated in FIG. 9 with respect to the overlap between the trial wave function consisting of a single arrangement state function and the FCI wave function by simulation using a classical computer.

  When FCI / STO-3G calculation was performed for each cluster, four slater determinants of 4 in the 5H cluster, 16 in the 9H cluster, and 64 in the 13H cluster were approximately 0.469, 0.220, and 0.103, respectively. It was found to have the same expansion coefficient. It was also found that the overlap between the FCI wave function and the HF wave function decreases rapidly as the system size increases. On the other hand, when the overlap between the FCI / STO-3G wave function and the trial wave function generated by the above embodiment is calculated, the expansion coefficients of 0.939, 0.881, and 0.828 are obtained in the 5H cluster, 9H cluster, and 13H cluster, respectively. The configuration state function was calculated as the main configuration, and it was shown that the 13-electron system also has a quantum phase estimation success probability of about 70%. Thereby, according to this embodiment, it was shown that the trial wave function which consists of a single arrangement | positioning state function can be produced | generated efficiently.

1 ... Quantum computer,
10: Logic qubit,
DESCRIPTION OF SYMBOLS 11 ... Physical quantum bit, 12 ... Quantum error correction circuit, 13 ... Quantum arithmetic unit

Claims (1)

A quantum information processing device that generates a spin eigenfunction of an N-electron system having N _β electrons out of N unpaired electrons as a trial wave function composed of a single configuration state function,
N qubits,
While incrementing by one the variable i from 1 to N _beta, the rotation of the rotation angle θ = -π / 4 is performed, as control (A) 2i th qubit with respect to (A) 2i th qubit, A quantum operation unit that performs a control NOT operation on the first to 2i−1 qubits, and that performs processing;
Comprising
Quantum information processing device.