JP5700827B2 - Quantum circuit generation apparatus, method, program, and recording medium - Google Patents

Description

  The present invention relates to a technique for generating a quantum circuit capable of executing a quantum Fourier transform on a symmetry group in polynomial time.

Since the discovery of a quantum algorithm that performs factoring at high speed by Shower in 1994, research on quantum algorithms has been actively conducted all over the world. In particular, unresolved problems proposed at the end of Shore's papers have attracted attention, and many researchers have competed to find fast algorithms for those problems. Among them, the most famous problem is the graph isomorphism determination problem. The graph isomorphism determination problem is a problem in which, when two graphs are given, it is determined whether there exists a mapping between vertices of the graphs that preserves edges. For example, in the example of FIG. 1, G ₁ and G ₂ are the same type, but they and G ₃ are not the same type. The graph isomorphism determination problem is a kind of pattern matching. If a fast algorithm is found, its application range is very wide.

  The graph isomorphism determination problem is more difficult than factorization, and no fast algorithm has been found on current computers. On the other hand, the graph isomorphism determination problem is easier than the NP complete problem, which is difficult to solve even with a quantum computer, and a fast quantum algorithm may be found.

The most important quantum circuits in algorithms Shore is, {0,1, ..., n- 1} is a quantum Fourier transform on Z _n is a cyclic group having the based. The quantum Fourier transform has an efficient quantum circuit, which plays an important role in the efficient execution of factorization. This is because the quantum Fourier transform is from serve to retrieve cycle information subgroup of Z _n. The most straightforward way to extend the algorithm Shore the graph isomorphism problem is to replace the quantum Fourier transform on Z _n in the algorithm of Shore quantum Fourier transform on the symmetric group S _n. Here, the symmetric group S _n represents the whole one-to-one correspondence from a set of n different characters to a given natural number n. If the numbers 1,2, ..., n are assigned to n characters, it is a group created by the substitution of {1,2, ..., n-1, n}. S _n has n! Elements, and considers the product of the permutation as a group operation.

The quantum Fourier transform on S _n is shown for the first time can be performed in polynomial time is Beals (e.g., see Non-Patent Document 1.).

  Later, Moore et al. Proved the existence of efficient quantum circuits for many transformations including quantum Fourier transformations on non-commutative groups (see, for example, Non-Patent Document 2). An efficient quantum circuit here means that there exists a quantum circuit that can be configured with a polynomial size O (poly (n)) for the number of input qubits n.

Robert Beals, “Quantum computation of Fourier transforms over symmetric groups”, In ACM, editor, Proceedings of the twenty-ninth annual ACM Symposium on the Theory of Computing: El Paso, Texas, May 4--6, 1997, pp. 48 --53, New York, NY, USA, 1997. ACM Press. C. Moore and D. Rockmore, “Generic quantum fourier transforms”, Technical Report quant-ph / 0304064, Quantum Physics e-Print Archive, 2003.

However, Non-Patent Documents 1 and 2, efficient against quantum Fourier transform on the symmetric group S _n, i.e. although it has been shown theoretically that there is a quantum circuit polynomial size, specific quantum The circuit configuration method is not shown. In order to execute operations on the quantum computer, it is necessary to configure a specific quantum circuit.

The purpose of this invention, a quantum circuit generation method for generating a quantum circuit polynomial size for performing quantum Fourier transform on the symmetric group S _n, is to provide a program and a recording medium.

In the quantum circuit generation device according to an aspect of the present invention, n is a predetermined integer equal to or greater than 3, and an _n-1 pieces corresponding to an _n-1 nodes of the ( _n-1 ) th hierarchy in the blater diagram the irreducible representation _{ρ i (i = 1,2, ...} , a n-1) and then, the t _i number of irreducible representations corresponding respectively to t _i-number of child nodes of the irreducible representations ρ _i ρ _ij ( j = 1,2, ..., a t _i), irreducible representation [rho _i, d the number of dimensions of the representation space of the [rho _ij respectively _.rho.i, as _d ρij, number of boxes correspond to the respective standard Young Release of n An orthonormal basis calculation unit for calculating an orthonormal basis which is an adapted gel fund Zetterin base, and b = 0,1,2, ..., n-1 and g _b = (1,2, ..., n) ^b ∈ A matrix representation ρ _ij representing g _b using an orthonormal basis corresponding to the standard Young's board of n irreducible representations ρ _ij each corresponding to n nodes of the nth layer in the literary diagram as S _n in direct sum of (g _b) That performs a process of generating a matrix representation ρ (g _b) for each b, the generated matrix representation _{ρ (g 0), ρ (} g 1), ..., the column matrix constituting the ρ (g _n-1) The process of constructing a matrix M _ρi of d _ρi・ n × d _ρi・ n by multiplying by a predetermined value sequentially from the column matrix located at the upper left is performed for each i, and d _ρ1 M _ρ1 , d _ρ2 of _M ρ2, ..., d the direct sum of _ρan-1 pieces of M _ρan-1 and matrix G _n, a n-order square matrix and I _n, the quantum Fourier transform on the symmetric group S _n-1 F _Sn-1 and then, the (×) and the tensor product, a replacement for converting the P _n-1 and _{(I n (×) F Sn} -1) to _{(F Sn-1 (×)} I n), of ^a-† matrix, and conjugate transposition, including a quantum circuit generator to _{G n P n-1 (I} n (×) F Sn-1) P n-1 † the on symmetric group S _n quantum Fourier transform F _Sn.

It is possible to generate a quantum circuit polynomial size for performing quantum Fourier transform on the symmetric group S _n.

The figure for demonstrating the graph isomorphism determination problem. The figure for demonstrating a young figure. The figure for demonstrating a Young board. The figure for demonstrating a standard young figure. Diagram illustrating a quantum circuit quantum Fourier transform F _s3 on symmetric group S _3. The figure for demonstrating a blater diagram. The figure for demonstrating the property of a blateri diagram. The figure for _{demonstrating} the structure of M ( _{rho) i} . The block diagram for demonstrating the structure of the quantum circuit generation apparatus of embodiment. The figure for demonstrating Vandermonde determinant. The figure for demonstrating the method of calculation of an orthonormal basis. The figure for demonstrating the method of calculation of an orthonormal basis. The figure for demonstrating the method of calculation of an orthonormal basis. The figure for demonstrating the method of calculation of an orthonormal basis. The figure for demonstrating the method of calculation of an orthonormal basis. The figure for demonstrating the method of calculation of an orthonormal basis. The flowchart of the algorithm for calculating _Mρi . diagram for explaining the structure of ρ (g _i). Diagram for explaining the structure of G _4. Diagram illustrating a quantum circuit quantum Fourier transform F _s4 on symmetric group S _4. The flowchart for demonstrating the quantum circuit production | generation apparatus of embodiment.

  First, background knowledge for understanding the present invention will be described.

Before defining the quantum Fourier transform on S _n (where n is a natural number), the basics of the representation theory of symmetric groups will be explained using examples of S ₂ and S ₃ . G is a group. For an appropriate vector space V, the mapping ρ: G → U (V)
Is commutative with group operations (ie when the mapping is homomorphic), ρ is called an expression and V is called an expression space. When W ⊆ V becomes invariant with ρ (G), W is called an invariant subspace. V (whole space) and {0} are obviously universally invariant, so they are called trivial invariant subspaces. When there is only an obvious invariant subspace, ρ is called an irreducible expression.

For example, let S _{3 be} g ₀ = (1), g ₁ = (1,2,3), g ₂ = (1,3,2), g ₃ = (1,2), g ₄ = (2 , 3), g ₅ = (1,3). A symmetric group is a set based on all sort operations. g ₀ = (1) represents a unit element, that is, an operation that does not change the order. (a, b, c) means an operation of shifting the order back one by one from a to b, b to c, c to a. (a, b) is an operation for exchanging a and b.

  At this time,

Ρ defined by is an expression because it is a homomorphic map, but is not irreducible. Because

Because is a non-trivial invariant subspace. After described as irreducible representations of S ₃ is only three, the above expression does not enter therein.

In general, irreducible representation of S _n may be enumerated by Young tableau (Young diagram). The details of _Sn 's representation theory using Young's figures are handed over to technical books, and here, only the minimum explanation necessary for understanding the invention is given. A young figure is a figure in which square boxes are arranged at the top left. FIG. 2 lists all the Young figures when the number of boxes is 2,3,4. As represented by the equation of λ in FIG. 2, each Young figure is described in a form in which the number of boxes for each column is arranged in a list from the top. Looking at the example of n = 3, λ = (3) corresponds to a Young figure with three boxes in a row, λ = (2,1) has two in the first row, and one in the second row Λ = (1,1,1) corresponds to a Young figure with one box in the first row, one in the second row, and one box in the third row .

It is known that the irreducible representation of S _n corresponds one-to-one to a Young figure with n boxes. The reason why there are three irreducible representations of S ₃ is that there are three types of Young figures when n = 3.

  A figure in which numbers are written in each box of a young figure is called a Young tableau. A Young board is illustrated in FIG. Among all the Young's discs, the numbers that are written are larger as they go down and larger toward the right for all rows and columns. A standard Young disc is illustrated in FIG.

It is known that the number of standard Young discs for each Young figure matches the number of dimensions of the expression space of the expression corresponding to the Young figure. The number of dimensions of the expression space for the expression _ρ is represented by d _ρ . Then, the following formula is established.

(+) When the [rho a direct sum of the overall representation of S _n, (+) [rho can be expressed by the unitary matrix in the space of sigma _[rho d _[rho dimension. For example, when n = 2, the number of Young figures is 2 and the number of standard Young discs is 1 + 1 = 2, so each element of S ₂ can be represented by a 2 × 2 matrix.

Further, in the case of n = 3, the number of young figure 3, the number of standard Young board is because 1 + 2 + 1 = 4, the original S _n can be expressed by 4 × 4 matrix. Each matrix is a direct sum of 1 × 1, 2 × 2, and 1 × 1 block matrices.

Next, the quantum Fourier transform on the symmetry group will be described. When the elements of the matrix of ρ (g ₁ ), ρ (g ₂ ),..., Ρ (g ₅ ) expressed by the above 4 × 4 are multiplied by a constant, a quantum Fourier transform is obtained. The quantum Fourier transform F _S2 on the symmetric group S ₂ and the quantum Fourier transform F _S3 on the symmetric group S ₃ are specifically written as follows.

F _S2 is equal to the Hadamard transform. F _S3 can be rewritten as follows.

The quantum circuit of F _S3 is shown in FIG. Here, the first line represents qubit and the second line represents qutrit. Qubit is an element that expresses the superposition state of two states of | 0> and | 1> with one element, and qutrit has three elements of | 0>, | 1>, and | 2> with one element. It is an element that expresses the overlapping state. Further, the second gate indicated by a white circle in FIG. 5 is 3 × 3 placed on the second line on the second bit which is the target bit when the first bit which is the control bit is | 0>. This means that an operation represented by a matrix is performed. Further, the third gate indicated by a black circle in FIG. 5 is a so-called Controlled rotation gate, and the second bit that is the target bit is the second bit when the first bit that is the control bit is | 1>. This means that an operation represented by a 3 × 3 matrix placed on the line is performed.

The quantum Fourier transform F _sn for n of 4 or more can be recursively defined by a blatter diagram, which is a diagram illustrating a system of Young figures.

  Each Young figure has a Young figure as a child with a larger number of one box as a child, so that the Young figure with one box is the apex and the nth node has n boxes. It can be expressed as a figure composed of figures. This is called a blater diagram. A bratter diagram is illustrated in FIG.

As already described, irreducible representation of S _n is a one-to-one correspondence to the Young tableau with n box. So, if you look at the n-th stage of the bra Teri diagram, irreducible representation of S _n are listed. The numbers given to each node in FIG. 6 are the total number of paths from the vertex to each node, with the vertex being 1, and it is known that the number matches the number of dimensions in the expression space of each Young figure. . Therefore, the number written at the node of the Young figure corresponding to the expression ρ _i is represented by d _ρi which is the same symbol as the number of dimensions in the expression space.

  The following holds for the number of each node in the blater diagram:

Consider the case of n = 4. FIG. 7 shows the irreducible representations ρ ₁ , ρ ₂ , ρ _{3 of} S ₃ and their child nodes ρ ₁₁ , ρ ₁₂ , ρ ₂₁ , ρ ₂₂ , ρ ₂₃ , ρ ₃₁ , ρ ₃₂ . The child notes of ρ ₁ are ρ ₁₁ and ρ ₁₂ , the child nodes of ρ ₂ are ρ ₂₁ , ρ ₂₂ and ρ ₂₃ , and the child nodes of ρ ₃ are ρ ₃₁ and ρ ₃₂ . Looking at ρ ₂ , the left side of equation (1) is d _ρ2 · n = 2 · 4 = 8, and the right side of equation (1) is Σ _{j = 1} ³ d _ρ2j = 3 + 2 + 3 = 8. Can be confirmed.

The size information of the matrix necessary for executing the quantum Fourier transform F _Sn can be obtained from the number of each node of the blater diagram. Quantum Fourier transform F _Sn on symmetric group S _n is known to be decomposed in the following manner. In the following matrix, the component of the blank part is 0. The same applies to other matrices.

Here, G _n is a block diagonal matrix as shown in FIG. P _n-1 is a substitution that converts I _n (×) F _Sn-1 to F _Sn-1 (×) I _n . From this equation, the quantum Fourier transform F _Sn can be expressed by the product of G _n ,..., G ₃ . I _n is an n-dimensional unit matrix, (×) is a tensor product, and ^† is a conjugate transpose of the matrix.

Since F _S2 is a Hadamard transform, the quantum Fourier transform F _Sn can be performed if G _n , G _n−1 ,..., G ₃ can be performed. Therefore, in the present invention, a specific matrix of G _n , G _n−1 ,..., G ₃ is constructed.

Matrix representation of G _n is, S _n of the original ^{(1,2, ..., n) i} (i = 0,1, ..., n-1) determined by the matrix representation of. (1,2, ..., n) ⁱ is a cyclic permutation that shifts to the right by i. For example, (1,2, ..., n) ¹ = (1,2,3, ..., n), and if n is an odd number, (1,2, ..., n) ² = (1,3, 5, ..., 2,4, ...) and n is an even number, (1,2, ..., n) ² = (1,3,5, ..., n-1) (2,4,6 ..., n). In order to construct a quantum circuit of _Gn, a matrix representation must first be determined. Then, in order to represent a matrix of any original S _n, it is first necessary to determine the orthonormal basis vectors of representation space.

  As shown in FIG. 9, the quantum circuit generation device includes an input unit 1, a battery diagram acquisition unit 2, an orthonormal basis calculation unit 3, a quantum circuit generation unit 4, a basic gate decomposition unit 5, and an output unit 6. The quantum circuit generation device performs processing of each step of the quantum circuit generation method illustrated in FIG.

The input unit 1 acquires the order of the symmetric group. The input unit 1 is an input device such as a keyboard and a mouse, for example, and the order of the symmetric group is input by the user using these input devices. When generating a quantum circuit of the quantum Fourier transform F _Sn for S _n , the order n is input. Information about the order n is transmitted to each unit of the quantum circuit generation device.

Bra Teri diagram acquiring unit 2 acquires information about the bra Terri diagram for symmetric group S _n, as shown illustrated in FIG. The battery diagram acquisition unit 2 may read information about the battery diagram from the battery diagram 21 in which information about the battery diagram is stored in advance, and appropriately calculates information about the battery diagram as needed. May be. Information about the battery diagram is transmitted to the orthonormal basis calculation unit 3 and the quantum circuit generation unit 4.

  The orthonormal basis calculation unit 3 calculates an orthonormal basis which is an adapted gel'fand-Tsetlin Basis (step S1). The calculated orthonormal basis constitutes an orthonormal basis vector of the expression space. Information about the calculated orthonormal basis is transmitted to the quantum circuit generation unit 4.

The quantum circuit generation unit 4 obtains a matrix representation from the orthonormal basis calculated by the orthonormal basis calculation unit 3, generates a quantum circuit for G _m (m = 3,4,..., N), and generates a symmetric group. generating a quantum circuit quantum Fourier transform F _Sn on S _n (step S2). Information about the generated quantum circuit is transmitted to the basic gate decomposition unit 5.

  The basic gate decomposition unit 5 converts the quantum circuit obtained by the quantum circuit generation unit 4 into a series of basic operations on the quantum computer (step S3). A well-known technique can be used for this part. Information about the sequence of basic operations on the converted quantum computer is transmitted to the output unit 6.

The quantum circuit output from the quantum circuit generation unit 4 is an arithmetic sequence realized by combining a plurality of types of elements such as qubit, qutrit, and quatrit. When executed on an actual quantum computer, it is realistic to unify all of them into the same element, for example, a qubit, and convert them into a series of basic gates that perform operations between 1 qubit or 2 qubits on the qubit. D _ρi · n × d _ρi · n matrix (i = 1,2, ..., a) which is the target bit operation in the control unitary operation corresponding to G _m (m = 3,4, ..., n) _n-1 ) and the above-mentioned unusual exchange operations are embedded in a larger 2 ^m × 2 ^m matrix, for example, using a decomposition method such as that described in Ref. It may be converted.

[Reference 1] Y. Nakajima, Y. Kawano, and H. Sekigawa, A New Algorithm for Producing Quantum Circuits using KAK Decompositions, Quantum Information and Computation, Vol. 6, No. 1, pp. 67-80, 2006.
The output unit 6 is an output device such as a display and a printer, and outputs information about a series of basic operations on the quantum computer to make the user perceive.

  Hereinafter, the orthonormal basis calculation unit 3 and the quantum circuit generation unit 4 which are main parts of the present invention will be described in detail.

First, the adapted Gel'fand-Tsetlin Basis generated by the orthonormal basis calculation unit 3 will be described. To determine any original matrix representation of S _n must first define the orthonormal basis vectors of representation space. Since the method of taking orthonormal basis vectors in the expression space is not unique, the matrix representation differs depending on the way of taking the basis vectors. However, in order to make an efficient quantum circuit later when the quantum circuit is constructed, it is necessary to select an adapted gel fund zetterin base. Here, “adapted Gel'fand-Tsetlin Basis” of ρ is orthonormal basis v ₁ ,..., V _dρ that satisfies the following conditions.

  ρ has a one-to-one correspondence with the Young figure. The expression corresponding to the parent node of ρ in the blateri diagram is

And From the nature of the blater diagram,

Is established. If ρ is an expression of S _n , an arbitrary g∈S _n is expressed by a d _ρ · d _ρ matrix when an appropriate orthonormal basis is determined. On the other hand, since ρ _i (i = 1, 2,..., A _n-1 ) is an expression of S _n-1 , an arbitrary g∈S _n-1 determines the d _ρi · d _ρi matrix when an appropriate orthonormal basis is determined. It is expressed by At this time, for any g∈S _n

An orthonormal basis for which is established is called an adapted gel fund Zetterin basis. The existence of an adapted gel fund zetterin basis is guaranteed but not unique. Using the adapted gel fund Zetterin basis, the element of Sn _-1 is represented by a block diagonal matrix.

By choosing Adapted Gelfand Tsue' endothelin basal appropriate, any original S _n is represented by _d ρ · d ρ matrix. Thus, if any of the original matrix representation of S _n is determined, the quantum Fourier transform on S _n can be defined as follows accordingly. [ρ (g)] _{α, β} is the [α, β] component of ρ (g).

Next, the Spechet polynomial will be described. Vandermonde determinant Δ (x ₁ ,…, x _n )

It is defined as FIG. 10 shows a Vandermonde determinant Δ (x ₁ ) of a Young figure composed of 1 × 1 boxes, a Vandermonde determinant Δ (x ₁ , x ₂ ) of a Young figure composed of 2 × 1 boxes, Young figure Vandermonde determinant Δ (x ₁ , x ₂ , x ₃ ) composed of × 1 boxes and Young figure Vandermonde determinant Δ (x ₁ , x ₂ , x composed of 4 × 1 boxes ₃ , x ₄ ).

  For a Young disc T with m rows and 1 columns, the Specht polynomial Δ (T) is

For a general Young board T with m rows and i columns, the Specht polynomial Δ (T) is

It is defined as

Where T _j is a Young's board from which the j-th column of T is taken out, and Δ (T _j ) is a Vandermonde determinant with a number written on the Young board as a subscript. For example, if T consists of 3 columns and the numbers written on T are 1, 4, 5 from the top of the first column, 2, 3 from the top of the second column, 6, 7 from the top of the third column,

It becomes.

the g to the original S _n. For a Specht polynomial Δ (T) corresponding to a Young's board T of size n, g (Δ (T)) is rewritten as subscript _i of x i to g (i) in Δ (T) . An entire polynomial that can be written by a linear combination with a coefficient as a whole of the entire Specht polynomial corresponding to a Young board with a Young figure of size n fixed and a number from 1 to n written on it is a finite-dimensional vector over complex numbers It becomes space V.

It is known that the whole of the Specht polynomial corresponding to the standard Young's board can be taken as the basis of V. Further, since the operation of the G∈S _n is naturally extended to linear transformation and V, the mapping from S _n to U (V) is obtained. This mapping is the homomorphic, a representation of S _n to the representation space of the V. That this representation is irreducible, irreducible representation of S _n are known to be exhausted by the representation size is made to this for all Young tableau of n. However, the basis consisting of the whole Specht polynomial corresponding to the standard Young's disc is not an adapted gel fund zetterin basis.

  In this embodiment, the Gram-Schmidt orthonormalization method is applied to the Specht polynomial by the following procedure in order to obtain the adapted gel fund Zetterin basis. The basis consisting of the polynomial thus obtained is the adapted gel fund Zetterin basis.

  1. The orthonormal basis calculation unit 3 reads a Young figure having n boxes. It is assumed that the young figures are arranged in a line in the following order.

2. The orthonormal basis calculation unit 3 sequentially extracts the read young figures one by one from the top and performs the following operations.
(1) Expand all standard Young's discs for the extracted Young figure and arrange them in a line according to the whole order defined above. That is, the one with the largest order is placed behind.
(2) Take the standard Young's board in order from the back and apply the Gram-Schmidt orthonormalization method to the corresponding Specht polynomial.
(3) The polynomial obtained by the Gram-Schmidt orthonormalization method is stored in the storage unit. These polynomials constitute orthonormal basis vectors for generating matrix expressions ρ (g ₁ ), ρ (g ₂ ),..., Ρ (g _n ).

  In this way, the orthonormal basis calculation unit 3 calculates an orthonormal basis that is an adapted gel fund Zetterin basis corresponding to each of the n standard Young's discs.

The entire order of young figures will be described. For two different Young discs T ₁ and T ₂ with numbers in the same Young figure with n boxes, T ₁ <T ₂ means that “a certain i (1 ≦ i ≦ n) exists. For all j (i <j ≦ n), the column to which j belongs is the same for T ₁ and T ₂ and the column to which i belongs at T ₁ is to the left of the column to which i belongs at T ₂ Is defined. In this way, “<” is the total order of the set of Young discs in which numbers are entered in Young figures with n boxes.

Additional orthonormal basis the calculation method will be described an example of applying the S _4. Assume that x _i and x _j (j ≠ i) are independent of each other.

  1. As shown in FIG. 11, there are five Young figures having four boxes. They are arranged in the order shown in FIG.

2. First, the following operation is performed on λ ₁ .
(1) There is only one standard Young disk for λ ₁ as shown in FIG.
(2) The corresponding Specht polynomial is 1. If Gram-Schmidt's orthonormalization method is applied to this, v ₁ = 1.

3. Next, the following operation is performed on λ ₂ .
(1) There are three standard Young's discs for λ ₂ as shown in FIG. The order is smaller toward the left.
(2) The Specht polynomials corresponding to these are as follows.

  The Gram-Schmidt orthonormalization method applied to these Specht polynomials in order from the right is as follows.

4). The operation for λ ₃ is as follows.
(1) There are two standard Young's discs for λ ₃ as shown in FIG. The order is smaller toward the left.
(2) Specht polynomials corresponding to these are as follows.

  The Gram-Schmidt orthonormalization method applied to these Specht polynomials in order from the right is as follows.

5. It includes the following operation for the λ _4.
(1) There are three standard Young's discs for λ ₄ as shown in FIG. The order is smaller toward the left.

  The Gram-Schmidt orthonormalization method applied to these Specht polynomials in order from the back is as follows.

6). Finally, the operation for λ ₅ is as follows.
(1) There is only one standard Young disk for λ ₅ as shown in FIG.
(2) The corresponding Specht polynomial is:

  When the Gram-Schmidt orthogonalization method is applied to this, the result is as follows.

Next, the quantum circuit generation unit 4 will be described. Quantum circuits generating unit 4, as follows, constitute a quantum circuit of the matrix G _n of the Fourier transform F _Sn on S _n by using the orthonormal basis of S _n calculated in orthonormal basis computation unit 3.

Each representation of S _n-1

And then, the child nodes of (a _n-1 is the number of representations of S _n-1), bra Terri diagram on at _{ρ i (i = 1,2, ...} , a n-1)

And As shown in FIG. 7, there is an overlap in ρ _ij (j = 1, 2,..., T _i ). For example, in the example of FIG. 7, two notations ρ ₁₂ and ρ ₂₁ are associated with the second node from the left of the fourth hierarchy from the top of the battery diagram. The nodes indicated by ρ ₁₂ and ρ ₂₁ are the same.

Let S _{n be} g ₀ = (1,2,3, ..., n) ⁰ , g ₁ = (1,2,3, ..., n) ¹ , g ₂ = (1,2,3, ..., n) Let ² , ..., g _n-1 = (1,2,3, ..., n) ^n-1 .

At this time, G _n, as illustrated in FIG. 8,

It is expressed as a direct sum of For each i,

Are all the same. This matrix is written as M _ρi . If M _ρi for each i can be determined, G _n is obtained.

M _ρi is _composed of ρ (g _b ) (b = 0, 1,..., N−1). ρ (g _b ) represents g _b using the orthonormal basis corresponding to the standard Young's board of n irreducible representations ρ _ij respectively corresponding to n nodes in the nth layer in the blateri diagram This is a matrix representation that is the direct sum of the matrix representation ρ _ij (g _b ). FIG. 18 schematically shows ρ (g _i ) when n = 4. In FIG. 18, ρ (g _b ) is expressed as ρ (g _i ).

The matrix representation ρ _ij (g _b ) is constructed using orthonormal basis corresponding to the standard Young's board for the irreducible representation ρ _ij . Specifically, the matrix representation ρ _ij (g _b), the original g _b of S _n, irreducible representation _{ρ ij (j = 1,2, ...} , t i) orthonormal corresponding to the standard Young Release of It is constructed by acting on the vector that the base constitutes. Applying the element g _{b of} S _n to the vector means that the permutation defined by g _n is applied to the vector. For example, when g _n = (1,2) and 1 and 2 are compatible, it means that x ₁ and x ₂ in the vector formed by the orthonormal basis are interchanged. The orthonormal basis corresponding to the standard Young's board for the irreducible expression ρ _{ij is} expressed as v _{i, k} (k = 1,..., K). At this time, the vector v ^→ formed by the orthonormal basis v _{i, k} (k = 1, ..., K) corresponding to the standard Young's board for the irreducible expression ρ _ij is v ^→ = (v _{i, 1} , v _{i , 2} ,…, v _{i, K} ) ^T. v ^→ to the vector by the action of g _b and v ^{→ *.} At this time, the matrix expression ρ _ij (g _b ) is a matrix that satisfies the relationship of v ^{→ *} = ρ _ij (g _b ) · v ^→ .・^T is a transposed matrix.

First, the quantum circuit generation unit 4 _{obtains the} matrix representation ρ _ij (g _b ) for each (i, j) pair, and takes the direct sum of the obtained ρ _ij (g _b ) as ρ (g _b ). . The direct sum means adding a matrix from the upper left to the lower right as exemplified in FIG.

In this way, the quantum circuit generation unit 4 generates a matrix representation ρ (g _b ) for each b (b = 0, 1,..., N−1). The upper part of FIG. 19 schematically represents ρ (g _b ) (b = 0, 1,..., 3) when n = 4.

The quantum circuit generation unit 4 sequentially selects the column matrix constituting the generated matrix representation ρ (g ₀ ), ρ (g ₁ ),..., Ρ (g _n−1 ) from the column matrix positioned at the upper left. _Are combined for each i to form a matrix M _ρi of d _ρi · n × d _ρi · n.

Then, the quantum circuit generation unit 4 sets the direct sum of d _ρ1 M _ρ1 , d _ρ2 M _ρ2 ,..., D _ρan-1 M _ρan-1 as a matrix G _n .

In FIG. 19, when n = 4, ρ (g ₀ ), ρ (g ₁ ), ρ (g ₂ ), ρ (g ₃ ) constitute M _ρ1 , M _ρ2 , M _ρ3 , and these M It is the figure which showed typically a mode that _G4 was comprised from _{( rho ) 1} , M _{( rho ) 2} , M ( _{rho ) 3} . In the case of n = 4, as shown in FIG. 19, G ₄ includes 4 × 4, 8 × 8, 8 × 8, and 4 × 4 matrices M _ρ1 , M _ρ2 , M _ρ2 , and M _{ρ3 in} order from the top. It will be in a diagonal line.

As shown in FIG. 19, each M _ρi is composed of a combination of four matrixes with the same hatching (actually, the matrixes are combined after multiplication by a predetermined value). Leftmost column matrix of the hatching of _M ρi is the same column matrix is derived from the column matrix of the same hatching of ρ (g _0), 2 th column matrix from the inner left of the hatching of _M ρi is the same column matrix ρ (g ₁₎ derived from the column matrix of the same hatching, the second column matrix from the inner right hatching M _.rho.i same column matrix is derived from the column matrix of the same hatching ρ (g _{2), M} ρi The rightmost column matrix among the column matrices with the same hatching is derived from the column matrix with the same hatching of ρ (g ₃ ).

For example, as shown in FIG. 19, the leftmost column matrixes constituting the first row of M _.rho.1 derived from ρ (g ₀₎ of ρ ₁₁ (g _0), the first row of M _.rho.1 The second column matrix from the left is derived from ρ ₁₁ (g ₁ ) of ρ (g ₁ ), and the second column matrix from the right constituting the first row of M _ρ1 is ρ (g ₂ ) derived from ρ ₁₁ (g _2), the last column matrixes constituting the first row of M _.rho.1 from ρ (g ₃₎ of ρ ₁₁ (g _3).

When n is larger than 4, G _n can be calculated in the same manner as in FIG.

Specifically, the quantum circuit generation unit 4 calculates the matrix G _n based on the following algorithm.

The x-th column matrix of ρ _ij (g _b ) (b = 0, 1,..., n) is written as ρ _ij (g _b ) [x]. ρ _ij (g _b ) [x] is a d _ρij × 1 matrix. In addition, M ρi [k, _l], and of _M ρi

D _ρij × 1 submatrix with elements at the beginning

. M _ρi [k, l] (i = 1,2,…, a _n−1 , k = 1,2,…, t _i , l = 1,2,… d _ρi · n) is expressed as ρ _ij (g _b ) [x] (i = 1,2,…, a _n-1 , j = 1,2,…, t _i , b = 0,1,…, n, x = 1,2,…, d _ρij ) Can be obtained as follows. FIG. 17 is a flowchart corresponding to the following procedure.

_Let x _ρij be an integer variable for the expression ρ _{ij of} S _n .
[Step 1] _Set x _ρij = 1 for all ρ _ij . (i = 1,2, ..., a _n-1 , j = 1,2, ..., t _i )
[Step 2] Set i = 1.
[Step 3] Set k = 1.
[Step 4] While increasing the value of l by 1 from l = 1 to d _ρi · n,

And if l is divisible by n, x _ρik = x _ρik +1.
[Step 5] If k <moving as t _i If k = k + 1 in step 4. If you go to the k = t _i if step 6.
[Step 6] If i <a _n-1 , move to Step 3 with i = i + 1. If i = a _n-1 , end.

When n = 3, g _i is represented by a 4 × 4 block diagonal matrix.

From here, the above algorithm is used to define a 6 × 6 matrix as follows.

Then, if F _S3 = G ₃ (F _S2 (×) I ₃ ) is calculated, the quantum Fourier transform F _s3 on the symmetric group S ₃ is obtained.

Once the matrix decomposition of Equation (2) is obtained, a quantum circuit can be created therefrom. G _n is

This is a series of control unitary gates. Here, d _ρi is the i-th number from the left of the (n-1) th stage of the _blateri diagram, and increases exponentially as i increases.

For example, the circuit shown in FIG. 20 is an example of an F _S4 quantum circuit. The first, second, and third lines represent qubit, qutrit, and quatrit, respectively. A quatrit is an element that expresses a superposed state of four states of | 0>, | 1>, | 2>, and | 3> with one element.

The last ♦ in the second line indicates the control operation when | i ₂ > = | 2>. The symbol XX on the left side is not a normal exchange operation.

The symbol “x-x” on the right side means the operation represented by the transpose matrix.

[Modifications, etc.]
Data exchange between the units of the quantum circuit generation device may be performed directly or via a storage unit (not shown).

  In addition, the present invention is not limited to the above-described embodiment. For example, the various processes described above are not only executed in time series according to the description, but may also be executed in parallel or individually as required by the processing capability of the apparatus that executes the processes.

  Further, when the above-described configuration is realized by a computer, the processing content of each unit that each device should have is described by a program. Each part is realized on the computer by executing this program on the computer.

  The program describing the processing contents can be recorded on a computer-readable recording medium. As the computer-readable recording medium, for example, any recording medium such as a magnetic recording device, an optical disk, a magneto-optical recording medium, and a semiconductor memory may be used.

  Needless to say, other modifications are possible without departing from the spirit of the present invention.

Claims (5)

Let n be a predetermined integer of 3 or more, and let a _n-1 irreducible representations corresponding to a _n-1 nodes in the _n-1st hierarchy in the blater diagram be ρ _i (i = 1, 2, ..., and a _n-1), t _i-number of the irreducible representation ρ _ij (j = 1,2 respectively corresponding to t _i-number of child nodes of the irreducible representation [rho _i, ..., t _i) and then , The dimensionality of the representation space of the irreducible representations ρ _i and ρ _ij are d _ρi and d _ρij , respectively.
An orthonormal basis calculation unit for calculating an orthonormal basis that is an adapted gel fund Zetterin basis corresponding to each of the standard Young's discs with n boxes;
b = 0,1,2, ..., and _{n-1, g b = (} 1,2, ..., n) as ^b ∈S _n, respectively corresponding to the n nodes of the n-th hierarchy in bra Teri diagram n A process of generating a matrix representation ρ (g _b ) that is a direct sum of the matrix representation ρ _ij (g _b ) representing g _b using the orthonormal basis corresponding to the standard Young's board of the irreducible representations ρ _ij For each b, the column matrix constituting the generated matrix representation ρ (g ₀ ), ρ (g ₁ ),..., Ρ (g _n−1 ) Multiply and combine to form a matrix M _ρi of d _ρi · n × d _ρi · n for each i, d _ρ1 M _ρ1 , d _ρ2 M _ρ2 , ..., d _ρan-1 the direct sum of _{M ρan-1} and matrix G _n, a n-order square matrix and I _n, the quantum Fourier transform on the symmetric group S _n-1 and F _Sn-1, and tensor product of (×), P _{n -1} is a permutation that transforms (I _n (×) F _Sn-1 ) into (F _Sn-1 (×) I _n ), ^† is a conjugate transpose of a matrix, and G _n P _n-1 (I _n (×) F _Sn-1 ) And a quantum circuit generator for the P _n-1 ^† Quantum Fourier transform F _Sn on symmetric group S _n,
A quantum circuit generation device including: The quantum circuit generation device according to claim 1,
The orthonormal basis calculation unit generates a Specht polynomial corresponding to each of the standard Young's boards, and makes the generated Specht polynomial orthogonal to each other by Gram Schmidt's orthonormalization method to form the orthonormal basis ,
Quantum circuit generator. Let n be a predetermined integer of 3 or more, and let a _n-1 irreducible representations corresponding to a _n-1 nodes in the _n-1st hierarchy in the blater diagram be ρ _i (i = 1, 2, ..., and a _n-1), t _i-number of the irreducible representation ρ _ij (j = 1,2 respectively corresponding to t _i-number of child nodes of the irreducible representation [rho _i, ..., t _i) and then , The dimensionality of the representation space of the irreducible representations ρ _i and ρ _ij are d _ρi and d _ρij , respectively.
An orthonormal basis calculation step, wherein the orthonormal basis calculation unit calculates an orthonormal basis that is an adapted gel fund Zetterin basis corresponding to each of the standard Young's discs with n boxes;
The quantum circuit generation unit sets b = 0,1,2, ..., n-1 and g _b = (1,2, ..., n) ^b ∈ S _n. A matrix representation ρ (g that is the direct sum of the matrix representation ρ _ij (g _b ) representing g _b using the above orthonormal basis corresponding to the standard Young's board of n irreducible representations ρ _ij corresponding to each node the process of generating the _b) is performed for each b, the generated matrix representation _{ρ (g 0), ρ (} g 1), ..., the column matrix which is located the column matrix in the upper left constituting the ρ (g _{n-1) Are} sequentially multiplied by predetermined values and combined to form a matrix M _ρi of d _ρi · n × d _ρi · n for each i, d _ρ1 M _ρ1 , d _ρ2 M _ρ2,. a straight sum of _{d ρan-1} pieces of _{M ρan-1} and matrix G _n, a n-order square matrix and I _n, the quantum Fourier transform on the symmetric group S _n-1 and F _Sn-1, the (×) A tensor product, P _n-1 is a permutation that transforms (I _n (×) F _Sn-1 ) to (F _Sn-1 (×) I _n ), ^† is a conjugate transpose of a matrix, And quantum circuits generating step to _{G n P n-1 (I} n (×) F Sn-1) P n-1 † the on symmetric group S _n quantum Fourier transform F _Sn,
A quantum circuit generating method including:   The program for functioning a computer as each part of the quantum circuit generation apparatus of Claim 1 or 2.   A computer-readable recording medium on which the program of claim 4 is recorded.

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