JP2005332242A - Method, apparatus and program for decomposing unitary matrix, and recording medium - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To realize a process for automatically decomposing a unitary matrix into basic gates on a computer, by applying Cartan decomposition recursively to the unitary matrix. <P>SOLUTION: First, Lie algebra expression f [where f is unitary of a set su (2<SP>n</SP>)]of a special unitary matrix F for decomposition [F is an element of a set SU (2<SP>n</SP>)] is extracted. The above Lie algebra expression f is then decomposed into elements k<SB>1</SB>, m<SB>1</SB>[k<SB>1</SB>is an element of a set k, while m<SB>1</SB>is an element of a set m] included in two orthogonal vector spaces k, m, through Cartan decomposition. Next, a special unitary matrix U<SB>m</SB>for diagonalizing U<SB>h</SB>×exp (m<SB>1</SB>)×U<SB>h</SB>to a diagonal matrix D<SB>h</SB>is calculated. Here, U<SB>h</SB>indicates a matrix which simultaneously diagonalizes the entire bases of the Cartan subalgebra h (n), and the diagonal matrix D<SB>h</SB>satisfies [D<SB>h</SB>is an element of a set (U<SB>h</SB>×exp (h<SB>1</SB>)×U<SB>h</SB>)] (h<SB>1</SB>is unitary of a set h (n)). Then, the diagonal matrix D<SB>h</SB>is decomposed into a product of Toffoli gates TG<SB>h</SB>, and each Toffoli gate TG<SB>h</SB>is decomposed into a product of the basic gates BG<SB>h</SB>. Thereafter, using exp (k<SB>1</SB>) and U<SB>m</SB>as new input matrices, the similar process is repeated recursively. <P>COPYRIGHT: (C)2006,JPO&NCIPI

Description

  The present invention relates to a unitary matrix decomposition method for decomposing an arbitrary unitary matrix into basic gates, a unitary matrix decomposition apparatus, a unitary matrix decomposition program for realizing the function thereof by a computer, and a recording medium storing the unitary matrix decomposition program.

One of the mathematical methods for decomposing a matrix is Cartan decomposition (see, for example, Non-Patent Document 1). In Cartan decomposition, a special unitary matrix belonging to the Lie group (G = SU (2 ⁿ )) is targeted for decomposition, and matrix decomposition is performed using its Lie algebraic expression (g = su (2 ⁿ )). It is. Here, the Lie group (SU (2 ⁿ )) and the Lie algebra expression (su (2 ⁿ )) are a set of 2 ⁿ × 2 ⁿ matrices, and exp (su (2 ⁿ )) = SU (2 ⁿ ) Are in a relationship. Also, su (2 ⁿ ) is the entire strained Hermite matrix. When this special unitary matrix is realized by a quantum circuit, n corresponds to the number of input qubits to this quantum circuit. The special unitary matrix means a unitary matrix having a determinant value of 1.

で定義されるが、非特許文献１においてＮ.Ｋｈａｎｅｊａは、パウリ（Ｐａｕｌｉ）のスピン行列

を用い、リー群（ＳＵ（２^ｎ））に対応するリー代数表現（ｓｕ（２^ｎ））を定義している。以下ではまず、このＮ.Ｋｈａｎｅｊａによるｓｕ（２^ｎ）の定義について説明する。 Where exp (a) for matrix a is

In NPL 1, N. Khaneja is a Pauli spin matrix.

Is used to define the Lie algebra expression (su (2 ⁿ )) corresponding to the Lie group (SU (2 ⁿ )). In the following, the definition of su (2 ⁿ ) by N. Khaneja will be described first.

<Definition of su (2 ⁿ ) by N. Khaneja>
First, an orthogonal basis {iB _s } in the Lie algebra expression (su (2 ⁿ )) is taken in the following form (i is an imaginary unit).

で定義される。なお、このＩ_αは式(6)の右辺の先頭からｋ番目に位置する。また、αはｘ，ｙ，ｚの何れかであり、Ｉ_αは上述の式(2)〜(4)の何れかとなる。さらに、Ｅは２×２の単位行列であり、

はテンソル積を意味する。また、式(5)のｑはＢ_ｓの行列式の値が１となるように選択される値である。 Where I _kα is

Defined by Note that I _α is located at the k-th position from the beginning of the right side of Equation (6). Further, α is any one of x, y, and z, and I _α is any one of the above formulas (2) to (4). Furthermore, E is a 2 × 2 identity matrix,

Means tensor product. Further, q of formula (5) is the value that the value of the determinant of B _s is selected to be 1.

と書ける。なお、ｓｐａｎ{＊}は、{＊}で定義される要素を基底とするベクトル空間を意味する。 Then, the Lie algebra expression (su (2 ⁿ )) is

Can be written. Note that span {*} means a vector space based on the element defined by {*}.

に分解する。ここでは、式(5)で定義した直交基底を用い、ｋとｍを以下のように定義する。 <Cartan decomposition by N. Khaneja>
Next, cartan decomposition described in Non-Patent Document 1 will be described.
In Cartan decomposition, a given Lie algebraic expression g = su (2 ⁿ ) is a direct sum of two orthogonal vector spaces (k and m = k ^⊥ ).

Disassembled into Here, the orthogonal basis defined by the equation (5) is used, and k and m are defined as follows.

ｎ≧３の場合、このｋとｍがそれぞれ作る空間は、図９のような格子状の行列となる。なお、図９において色の付いていない部分には０が、色のついている部分には任意の値がそれぞれ入る。
そして、これらのリー代数表現ｇ、ｋ、ｍにそれぞれ対応するリー群Ｇ，Ｋ＝ｅｘｐ（ｋ）、Ｍ＝ｅｘｐ（ｍ）では、
Ｇ＝ＫＭ …(10)
が成り立つ。これはリー群Ｇ＝ＳＵ（２^ｎ）の任意の元が、Ｋの元とＭの元との積に分解できることを意味する。

In the case of n ≧ 3, the spaces created by k and m are lattice-like matrices as shown in FIG. In FIG. 9, 0 is entered in the uncolored portion, and an arbitrary value is entered in the colored portion.
And in Lie groups G, K = exp (k), M = exp (m) respectively corresponding to these Lie algebra expressions g, k, m,
G = KM (10)
Holds. This means that an arbitrary element of the Lie group G = SU (2 ⁿ ) can be decomposed into a product of an element of K and an element of M.

ここで、Ｈ＝ｅｘｐ（ｈ（ｎ））⊂Ｇとすると、式(10)のＭ＝ｅｘｐ（ｍ）は、さらに
Ｍ＝ＫＨＫ …(12)
と変換できる。なお、式(12)におけるＫは、式(10)のＫと同じである。
従って、この式(12)より、式(10)に示すリー群Ｇは、
G=KM=K(KHK)=KHK ,(K=SU(2^n-1),H=exp(h(n))⊂G) …(13)
と表すことができる。すなわち、リー群Ｇの任意の元は、それよりも次元の低い、Ｋの元と、（ｇ，ｋ）の組に対するカルタン部分代数ｈ（ｎ）に対応するリー群の元との積に分解できる。 Next, a cartan partial algebra h (n) for a set of (g, k) is defined. The Cartan partial algebra h (n) is a maximal commutative partial algebra including the vector space m. Here, the cartan partial algebra h (n) is defined as follows.

Here, when H = exp (h (n)) ⊂G, M = exp (m) in the equation (10) is further calculated as M = KHK (12)
And can be converted. Note that K in equation (12) is the same as K in equation (10).
Therefore, from this equation (12), the Lie group G shown in equation (10) is
G = KM = K (KHK) = KHK, (K = SU (2 ^n-1 ), H = exp (h (n)) ⊂G)… (13)
It can be expressed as. That is, an arbitrary element of the Lie group G is decomposed into a product of a lower dimension K element and a Lie group element corresponding to the Cartan partial algebra h (n) for the set of (g, k). it can.

As an industrial application field of the above-described Cartan decomposition, for example, decomposition of a unitary matrix such as a unitary transformation quantum circuit can be exemplified. That is, when an arbitrary unitary matrix is realized by an actual quantum circuit, the unitary matrix must be decomposed into basic gates. If each element decomposed by the Cartan decomposition is a new input source and this Cartan decomposition can be repeated recursively, any unitary matrix can be systematically decomposed into basic gates. Is also possible.
N. Khaneja, S. Glaster, “Cartan Decomposition of SU (2n), constructive controllability of spin systems and universal quantum computing,” quant-ph / 0010100,2000.

However, heretofore, a specific method for independently extracting the elements of these Lie groups without destroying the properties of K and H from the decomposition result by the cartan decomposition has not been known. Therefore, conventionally, it has not been possible to systematically realize a process of recursively applying Cartan decomposition to a unitary matrix corresponding to a quantum circuit or the like and decomposing the unitary matrix into basic gates.
The present invention has been made in view of such a point, and provides a technique for realizing, on a computer, processing for recursively applying Cartan decomposition to a unitary matrix and automatically decomposing the unitary matrix into basic gates. The purpose is to do.

<First invention>
In order to solve the above-described problem, in the first aspect of the present invention, first, a Lie algebraic expression fε which is a natural logarithm of a 2 ⁿ × 2 ⁿ special unitary matrix FεSU (2 ⁿ ) to be decomposed. The matrix representation conversion unit extracts su (2 ⁿ ), and the extracted Lie algebra representation f is converted into the element k included in the two orthogonal vector spaces k and m by Cartan decomposition. ₁ that is an element of the set K, decomposed into _{m 1 ∈m.}
Next, the diagonal matrix calculation means calculates a special unitary matrix U _m that diagonalizes U _h · exp (m ₁ ) · U _h into a diagonal matrix D _h . U _h means a matrix that simultaneously diagonalizes all the bases of the Cartan partial algebra h (n), which is a maximal commutative partial algebra included in the vector space m, and the diagonal matrix D _h is a Cartan part. This means a matrix obtained by diagonalizing the Lie group exp (h (n)) corresponding to the algebra h (n) by the matrix U _h (D _h εU _h · exp (h (n)) · U _h ).

が成り立つ。また、ｋ_１，ｍ_１に対応するリー群Ｋ_１＝ｅｘｐ（ｋ_１）及びＭ_１＝ｅｘｐ（ｍ_１）は、Ｆ＝Ｋ_１Ｍ_１の関係を満たす。 Then, the diagonal matrix decomposition means decomposes the diagonal matrix D _h into products of toffoli gates TG _h , and the basic gate decomposition means decomposes each tori gate TG _h into products of basic gates BG _h. , The decomposition element storage means converts exp (k ₁ ), U _h , U _m , U _m ^* and BG _h into F decomposition elements (F = exp (k ₁ ) · (U _h · U _m · BG _h · U _m ^* · U _h )). It should be noted that the “tolygate” of the present invention is a concept including a matrix that becomes a torigate by a simple operation such as replacement of rows.
Since the in Cartan decomposing means, the Lie algebra expressions f, orthogonal vector space k, the element k ₁ that is an element of the set K contained in m, is decomposed into m ₁ ∈m,

Holds. _Also, Lie groups corresponding to _{k 1, m 1 K 1 =} exp (k 1) and _{M 1} = exp _(m 1) satisfy the relationship of _{F =} K _{1 M} 1.

As shown in the above equation (12), this M ₁ is
M ₁ = K ₂ · exp (h ₁ ) · K ₃ (14)
(H ₁ εh (n), K ₂ , K ₃ εK = exp (k)).
Here, if M ₁ = K ₂ · exp (h ₁ ) · K ₃ is diagonalized by the matrix U _h ,
U _h · M ₁ · U _h = (U _h · K ₂ · U _h ) (U _h · exp (h ₁ ) · U _h ) (U _h · K ₃ · U _h ) (15)
It becomes. U _h means a matrix that diagonalizes all of the bases of h (n) at the same time. Therefore, U _h is also a matrix that diagonalizes h ₁ , and further, the definition of the above formula (1) Thus, the Lie group exp (h ₁ ) is also a matrix that is diagonalized. Since D _h = U _h · exp (h (n)) · U _h , the above equation (15) is
_{U h · M 1 · U h} = U h · K 2 · U h · D h · U h · K 3 · U h ... (16)
It becomes.

In the diagonal matrix calculation means, a special unitary matrix U _m that diagonalizes U _h · exp (m ₁ ) · U _h into a diagonal matrix D _h is calculated, but M ₁ = exp (m ₁ )
U _m ^* (U _h · M ₁ · U _h ) U _m = D _{h
U h · M 1 · U h} = U m · D h · U m * ... (17)
Holds. Here, in order for equations (16) and (17) to hold,
K ₂ = U _h · U _m · U _h (18)
K ₃ = U _h · U _m ^* · U _h (19)
If it is enough.

Then, substituting this equation (18) (19) into the above equation (14),
M ₁ = K ₂ · exp (h ₁ ) · K _{3
= (U h · U m ·} U h) · exp (h 1) · (U h · U m * · U h)
= U _h · U _m · (U _h · exp (h ₁ ) · U _h ) · U _m ^* · U _h
= U _h · U _m · D _h · U _m ^* · U _h
It becomes.
That is, from this equation, the special unitary matrix F to be decomposed is
F = K ₁ M ₁
= K ₁・ U _h・ U _m・ D _h・ U _m ^*・ U _h (20)
It can be seen that it can be decomposed.

In this equation (20), the quantum circuit of the matrix U _h is known (described later). The diagonal matrix D _h can be decomposed into a product of the basic gate BG _h by the diagonal matrix decomposition means and the basic gate decomposition means. Therefore, if the remaining K ₁ and U _m can be decomposed to the basic gate, F can be decomposed into the basic gate. Therefore, next, the process of the present invention is recursively repeated with these K ₁ and U _m as special unitary matrices to be decomposed. Incidentally, the U _m from the U _m are those diagonal matrix calculation means has calculated, it is easy to identify as a special unitary matrix is decomposed, also if U _m can be identified, the remaining K It is also easy to extract ₁ from equation (20). Thereby, F can be decomposed | disassembled to a basic gate level.

<Second invention>
In order to solve the above problem, in the second aspect of the present invention, first, a Lie algebraic expression fε which is a natural logarithm of a 2 ⁿ × 2 ⁿ special unitary matrix FεSU (2 ⁿ ) to be decomposed. The matrix representation conversion unit extracts su (2 ⁿ ), and the extracted Lie algebra representation f is converted into the element k included in the two orthogonal vector spaces k and m by Cartan decomposition. ₁ that is an element of the set K, decomposed into _{m 1 ∈m.}
Next, a special unitary matrix U ₁ that diagonalizes m ₁ into a diagonal matrix D ₁ is calculated by a diagonal matrix calculation means, and a diagonal matrix decomposition means converts exp (D ₁ ) to a toffoli gate. decomposing the product of TG _1, the basic gate decomposing means, the respective Toffoli gate TG ₁ is decomposed into a product of the basic gate BG _1, the resolution element storing means, said _{exp (k 1), U 1} , U 1 ^{* And} BG ₁ are stored in the storage means as F decomposition elements (F = exp (k ₁ ) · (U ₁ · BG ₁ · U ₁ ^* )).

が成り立つ。また、ｋ_１，ｍ_１に対応するリー群Ｋ_１＝ｅｘｐ（ｋ_１）及びＭ_１＝ｅｘｐ（ｍ_１）は、Ｆ＝Ｋ_１Ｍ_１の関係を満たす。 Here, in the Cartan decomposition unit, the Lie algebra expressions f, orthogonal vector space k, because it decomposed into m, k ₁ that is an element of the set K, When m ₁ ∈m,

Holds. _Also, Lie groups corresponding to _{k 1, m 1 K 1 =} exp (k 1) and _{M 1} = exp _(m 1) satisfy the relationship of _{F =} K _{1 M} 1.

Further, the diagonal matrix calculation means calculates a special unitary matrix U ₁ that diagonalizes m ₁ into a diagonal matrix D ₁ . Then, from the definition of the aforementioned formula (1), the special unitary matrix _{U 1} is a matrix that diagonalization to _{m 1} of Lie groups _M 1 = exp _{(m 1)} a diagonal matrix exp _{(D 1)} . Therefore, using this special unitary matrix U ₁ , M ₁ is
M ₁ = U ₁ · exp (D ₁ ) · U ₁ ^* (21)
Can be written.
Therefore, the special unitary matrix F to be decomposed is
F = K ₁ M ₁
= K ₁・ U ₁・ exp (D ₁ ) ・ U ₁ ^* (22)
It can be seen that it can be decomposed.

In this equation (22), the diagonal matrix D ₁ can be decomposed into the product of the basic gate BG ₁ by the diagonal matrix decomposition means and the basic gate decomposition means. Therefore, if the remaining K ₁ and U ₁ can be decomposed to the basic gate, F can be decomposed into the basic gate. Therefore, next, the process of the present invention is recursively repeated using these K ₁ and U ₁ as special unitary matrices to be decomposed. Incidentally, the U ₁ from U ₁ are those diagonal matrix calculation means has calculated, it is easy to identify as a special unitary matrix is decomposed, also if U ₁ can be identified, the remaining K It is also easy to extract ₁ from equation (22). Thereby, F can be decomposed | disassembled to a basic gate level.

  As described above, in the present invention, a part of the result of cartan decomposition is replaced with a diagonal matrix, so that it is possible to apply cartan decomposition recursively. As a result, it is possible to realize processing for automatically decomposing an arbitrary unitary matrix into basic gates on a computer.

Hereinafter, embodiments of the present invention will be described with reference to the drawings.
[First Embodiment]
First, a first embodiment of the present invention will be described.
<Configuration>
First, the configuration of the unitary matrix decomposition apparatus according to this embodiment will be described.
FIG. 1 is a block diagram for explaining the configuration of a unitary matrix decomposition apparatus 1 according to this embodiment.
The unitary matrix decomposition apparatus 1 is constructed by causing a known computer to execute a predetermined program, and includes an input unit 2, a main storage device 3 (corresponding to “storage means”), a structure determination unit 4, a matrix expression conversion unit 5, It has a Cartan decomposition unit 6, a diagonal matrix calculation unit 7, an M decomposition unit, a diagonal matrix decomposition unit 9, a basic gate decomposition unit 10, a decomposition element storage unit 11, and an output unit 12.

<Main memory>
The main storage device 3 corresponds to a 2 ⁿ × 2 ⁿ special unitary matrix (SU (2 ⁿ )) and a Lie algebraic expression (su (2 ⁿ )) that is a natural logarithm of the special unitary matrix. Store with attachments. Then, the matrix to be decomposed is replaced with the product form after the decomposition.

<Structure determination unit>
The structure determination unit 4 determines whether or not the special unitary matrix stored in the main storage device 3 is a basic gate, extracts the special unitary matrix determined not to be a basic gate from the main storage device 3, and extracts the extracted special unitary matrix. It is determined whether or not the unitary matrix is included in SU (4).
Here, when it is determined that the extracted special unitary matrix is included in the SU (4), the structure determination unit 4 passes the special unitary matrix to the basic gate decomposition unit 10. On the other hand, when it is determined that the extracted special unitary matrix is not included in SU (4), the structure determination unit 4 passes the special unitary matrix to the matrix representation conversion unit 5. The special unitary matrix passed to the matrix representation conversion unit 5 is a special unitary matrix of n ≧ 3 excluding the basic gate among the special unitary matrices stored in the main storage device 3 (“the special unit matrix to be decomposed” Equivalent to unitary matrix F). The special unitary matrix included in SU (4) (the special unitary matrix is n = 2) is not limited to the case where the special unitary matrix is a 4 × 4 matrix, and 0 in each row or column. The case where the maximum value of the number of elements other than is 4 is also included.

<Matrix representation conversion unit>
Whether the matrix representation conversion unit 5 stores the Lie algebraic representation fεsu (2 ⁿ ) corresponding to the special unitary matrix FεSU (2 ⁿ ) to be decomposed received from the structure determination unit 4 in the main storage device 3. Judge whether or not.
Here, when it is determined that the corresponding Lie algebra expression f is stored, the matrix expression conversion unit 5 uses the Lie algebra expression f∈su () corresponding to the special unitary matrix F∈SU (2 ⁿ ) to be decomposed. 2 ⁿ ) is extracted from the main memory 3. On the other hand, when it is determined that the corresponding Lie algebra representation f is not stored, the matrix representation conversion unit 5 calls the diagonal matrix calculation unit 7 and diagonalizes the F into the diagonal matrix D ₂ . ₂ is calculated, and the calculation result of U ₂ · log (D ₂ ) · U ₂ ^* is stored in the main storage device 3 as the Lie algebra expression f corresponding to F.

<Diagonal matrix calculator>
The diagonal matrix calculation unit 7 obtains a unitary matrix that diagonalizes the input matrix. Then, the unitary matrix U to be diagonalized and the diagonal matrix D diagonalized thereby are output. Specifically, for example, this calculation is performed by the following algorithm.
[Algorithm 1: When input matrix size is small]
1. Eigenvalue det (A−λI) = 0 is solved exactly to obtain eigenvalue λεC. A denotes an m (= 2 ⁿ ) -order input matrix, I denotes a unit matrix, and det (...) Denotes a determinant value.

2. Substituting each obtained eigenvalue λ into Aν = λν and solving this equation, eigenvectors ν∈C ^m and ν ≠ 0 corresponding to each eigenvalue λ are obtained.
3. The obtained eigenvalues _{λ 1, λ 2, ...,} λ m, and eigenvectors [nu ₁ corresponding to these eigenvalues, ν _2, ..., from [nu _m,
U = (ν ₁ , ν ₂ ,..., Ν _m )
D = diag (λ ₁ , λ ₂ ,..., Λ _m )
Is calculated. _{Incidentally, diag (λ 1, λ 2} , ..., λ m) _{are, λ 1, λ 2, ...} , means a diagonal matrix with the lambda _m diagonal components.

[Algorithm 2: When the input matrix size is large]
1. The input matrix A is converted into a Hermite matrix B (a matrix of B ^* = B).
2. Convert B to Hermitian tridiagonal matrix J (similar transformation).
3. Solve the eigen equation det (J-λI) for J. Here, since the eigenvalue of J is equal to B and J is a tridiagonal matrix, a solution can be obtained more easily than directly solving the eigenequation from B.
4. The eigenvalue λεC of A is calculated from the obtained eigenvalue of B.
5). The subsequent steps are the same as those of algorithm 1.

The following method can be used for conversion to the Hermitian matrix in “1.” of algorithm 2.
(a) When the input matrix A is a unitary matrix, if B = A + A ^* , B becomes a Hermitian matrix. In this case, if the eigenvalues of A alpha ₁ and the eigenvalues beta ₁ of B _{is, β 1 = α 1 + α} 1 * (α 1 * is, alpha ₁ complex conjugate) is (in the process of "4." Correspondence).
(b) When the input matrix A is a distorted Hermitian matrix, if B = iA, B becomes a Hermitian matrix. In this case, if the eigenvalues of A alpha ₁ and the eigenvalues beta ₁ of B is β ₁ = -iα ₁ (corresponding to the processing of "4").
In addition to the above method, an eigenvalue can be obtained by using an algorithm that solves a well-known eigenvalue problem such as QR method or power method by performing appropriate similarity transformation (preprocessing). it can.

<Basic gate disassembly unit>
The basic gate decomposition unit 10 converts the matrix (n = 2 special unitary matrix excluding the basic gate from the special unitary matrix stored in the main storage device 3) included in the torigate or SU (4) to SU. (2) Decomposes into CNOT product (basic gate product).
<Cartan decomposition part>
Cartan decomposition unit 6 decomposes the received Lie algebra expressions f, by Cartan (Cartan) decomposition, two orthogonal vector space k, the element _{k 1} that is an element of the set K contained in m, to the direct sum of _{m 1 ∈m.} Then, the decomposed m ₁ is passed to the M decomposition unit 8. Thereafter, the decomposition result returned from the M decomposition unit 8 is transferred to the decomposition element storage unit 11 together with k.

<M disassembly unit>
The M decomposition unit 8 passes the received m to the diagonal matrix calculation unit 7, and passes U _h · exp (m ₁ ) · U _h to the diagonal matrix D _h (D _h εU _h · exp (h (n)) -A special unitary matrix U _m to be diagonalized is calculated in U _h ). The M decomposition unit 8 receives the special unitary matrix U _m and the diagonal matrix D _h calculated by the diagonal matrix calculation unit 7, and expresses exp (m ₁ ) as U _h · U _m · D _h · U _m. ^* _Replace with Uh. Further, the M decomposition unit 8 passes the diagonal matrix D _h to the diagonal matrix decomposition unit 9, and replaces the diagonal matrix D _h with the decomposition result returned from the diagonal matrix decomposition unit 9, and replaces them with U _h , U _m and U _m ^* are returned to the cartan decomposition unit 6.

より、
h₁=i(α₁σ₁+α₂σ₂+α₃σ₃+α₄σ₄)∈h(3)
と書ける。この場合、ｈ（３）の基底σ_１，σ_２，σ_３，σ_４のすべてを同時対角化する行列、すなわち、任意のｈ_１∈ｈ（３）に対して、
U_h ^*・h(n)・U_h=α₁・U_h ^*・σ₁・U_h+α₂・U_h ^*・σ₂・U_h+α₃・U_h ^*・σ₃・U_h+α₄・U_h ^*・σ₄・U_h=D_h
が成り立つような行列Ｕ_ｈが存在し、その行列Ｕ_ｈは以下のようになる。 [Description of Matrix U _h ]
As described above, U _h is a matrix that simultaneously diagonalizes all the bases of the Cartan partial algebra h (n), which is a maximal commutative partial algebra included in the vector space m. That is, a matrix U _h that simultaneously diagonalizes all of them exists in the basis of the cartan partial algebra h (n) defined by the above-described equation (11). For example, considering the case of g∈su (2 ³ ), the Cartan partial algebra h (n) for the set of (g, k) is

Than,
h ₁ = i (α ₁ σ ₁ + α ₂ σ ₂ + α ₃ σ ₃ + α ₄ σ ₄ ) ∈h (3)
Can be written. In this case, for a matrix that simultaneously diagonalizes all the bases σ ₁ , σ ₂ , σ ₃ , and σ ₄ of h (3), that is, for any h ₁ ∈h (3),
U _h ^*・ h (n) ・ U _h = α ₁・ U _h ^*・ σ ₁・ U _h + α ₂・ U _h ^*・ σ ₂・ U _h + α ₃・ U _h ^*・ σ ₃・ U _h + α ₄・ U _h ^*・ σ ₄・ U _h = D _h
There exists a matrix U _h that holds, and the matrix U _h is as follows.

ここで、U_h ^*・σ_ｐ・U_h（ｐ＝１，２，３，４）を計算すると、それぞれ対角行列が求まることから、同時対角化できることがわかる。

Here, when U _h ^* · σ _p · U _h (p = 1, 2, 3, 4) is calculated, a diagonal matrix is obtained, and it can be seen that simultaneous diagonalization is possible.

Such a matrix U _h can be regularly configured as shown in FIG. 3 when n ≧ 3, and U _h ^* = U _h ⁻¹ = U _h holds. That is, when n = 3, as shown in FIG. 3A, two CNOT gates 21 and 22 and a Hadamard (Hadamard) arranged between the CNOT gates 21 and 22 on the target bit path. ) The gate 31 and the Hadamard gate 32-1 arranged on the third qubit path other than these target bits and control bits form a matrix U _{h (3)} (U _{h (β)} is meaning n = β matrix U _h ). Further, when n = 4, as shown in FIG. 3B, the matrix U _{h (4)} is further obtained by adding the Hadamard gate 32-2 arranged on the fourth qubit path. Composed. When this is generalized, as shown in FIG. 3C, two CNOT gates 21 and 22 and Hadamard gates 31 disposed between the CNOT gates 21 and 22 on the target bit path The Hadamard gates 32-1 to _n constitute a matrix U _{h (n)} (end of description of the matrix U _h ).

<Diagonal matrix decomposition unit>
The diagonal matrix decomposition unit 9 decomposes a 2 ⁿ × 2 ⁿ diagonal matrix into a product of 2 ⁿ⁻¹ n qubit toffoli gates. In addition, the “toligate” here includes a matrix that becomes a shape of the torigate by a simple operation.
For example, a diagonal matrix when n = 3

の場合、対角行列分解部９は、この対角行列Ｄを４個の３量子ビットのトフォリゲートの積に分解する。すなわち、Ｄ＝Ｆ_１Ｆ_２Ｆ_３Ｆ_４

と分解する。

In this case, the diagonal matrix decomposition unit 9 decomposes the diagonal matrix D into a product of four 3-qubit torigates. That is, D = F ₁ F ₂ F ₃ F ₄

And disassemble.

Here, F ₄ is a 3-qubit torigate itself. F ₁ , F ₂ , and F ₃ can be realized by a 3-qubit torigate if the rows are exchanged. For example, in the case of F ₁ , first, the upper half and the lower half are exchanged (0 and 1 of the first qubit are inverted), and then the upper half and the lower half of the lower half are further exchanged (second By reversing 0 and 1 of the qubit), it becomes a form of a 3-qubit torigate. In the present invention, such things as F ₁ , F ₂ , and F ₃ are also referred to as “toforigates”.
When the diagonal matrix decomposition unit 9 decomposes the diagonal matrix into torigates, it next calls the basic gate decomposition unit 10 to decompose this torigate into basic gates. Then, the decomposition result is returned to the M decomposition unit 8.

<Basic gate disassembly unit>
The basic gate decomposition unit 10 decomposes the special unitary matrix and the torigate included in SU (4) into products of basic gates using row / column replacement and diagonalization. Note that an n-qubit torigate can be realized by a basic gate of O (n ² ) (for example, b “A. Barenco et al., Elementary gates for quantum computation, Phys. Rev. A52, pp. 3457- 3467, 1995. ”and“ C. Moore and M. Nilsson, parallel quantum computation and quantum codes, SIAM J. Comp, Vol.31, No.3, pp.799-815, 2001 ”).

<Disassembly element storage unit>
The decomposition element storage unit 11 stores exp (k ₁ ), U _h , U _m , U _m ^*, and the basic gate BG _h into F decomposition elements (F = exp (k ₁ ) · (U _h · U _m · BG ^{_{h · U m * · U h}} = exp (m 1))) as stored in the main memory 3 _{(where, k 1 ∈k, m 1 ∈m} ).
<Processing>
Next, a unitary matrix decomposition method according to this embodiment will be described.
FIG. 2 is a flowchart for explaining a unitary matrix decomposition method according to this embodiment. Hereinafter, the unitary matrix decomposition processing according to the present embodiment will be described with reference to FIGS. 1 and 2.
First, an arbitrary unitary matrix is input from the input unit 2. The input unitary matrix is converted into a special unitary matrix FεSU (2 ⁿ ) in the input unit 2 and stored in the main storage device 3 as a matrix list F.

Next, the structure determination unit 4 determines whether all the special unitary matrices F (the matrix F included in the matrix list of the main storage device 3) stored in the main storage device 3 are basic gates (step). S1). Here, if it is determined that all the special unitary matrices F stored in the main storage device 3 are basic gates, the processing is terminated.
On the other hand, when it is determined that the special unitary matrix F that is not a basic gate exists in the matrix list, the structure determination unit 4 reads (extracts) the special unitary matrix F that is not the first basic gate in the matrix list from the main storage device 3. (Step S2). The structure determination unit 4 that extracted the special unitary matrix F determines whether or not the extracted special unitary matrix is included in SU (4) (step S3). Here, when it is determined that the extracted special unitary matrix is included in SU (4) (when it is determined that n <3), the structure determination unit 4 converts the special unitary matrix F into the basic gate decomposition unit. 10, the basic gate decomposition unit 10 replaces the special unitary matrix F with the product of the basic gates and stores it in the main storage device 3 and ends the process.

On the other hand, when it is determined that the extracted special unitary matrix is not included in SU (4) (when it is determined that n ≧ 3), the structure determination unit 4 determines that the special unitary matrix F (“with decomposition target”). (Corresponding to a certain special unitary matrix F)) is sent to the matrix representation conversion unit 5. Receiving this, the matrix representation conversion unit 5 determines whether or not the Lie algebra representation f corresponding to the special unitary matrix F that is the decomposition target is calculated (whether or not it is stored in the main storage device 3). (Step S5). If it is determined that the corresponding Lie algebra expression f has not been calculated, the matrix expression conversion unit 5 obtains this Lie algebra expression f and stores it in the main storage device 3 (step S6). Specifically, for example, first, the matrix representation conversion unit 5 calls the diagonal matrix calculation unit 7 to calculate the matrix U ₂ that diagonalizes the special unitary matrix F to be decomposed into the diagonal matrix D ₂ . A matrix U ₂ and a diagonal matrix D ₂ are received. Then, the matrix representation conversion unit 5
log (F) = log (U ₂ · D ₂ · U ₂ ^* )
= U ₂ · log (D ₂ ) · U ₂ ^*
Is stored in the main storage device 3 as the Lie algebra expression f corresponding to F. Thus, by performing by decomposing F diagonal matrices D ₂ log operation, it is possible to reduce the amount of calculation than calculating a direct log (F). Note that log is a natural logarithm corresponding to the exponential operation defined by equation (1).

On the other hand, if it is determined in step S5 that the corresponding Lie algebra expression f has been calculated, the matrix expression conversion unit 5 does not calculate the Lie algebra expression f.
After that, the matrix representation conversion unit 5 extracts the Lie algebra representation fεsu (2 ⁿ ) corresponding to the special unitary matrix FεSU (2 ⁿ ) to be decomposed from the main storage device 3 (step S7). This is sent to the disassembly unit 6. Cartan decomposition unit 6 decomposes the sent Lie algebra expressions f, by Cartan decomposition, two orthogonal vector space k, the element _{k 1} that is an element of the set K contained in m, to _{m 1 ∈m,} corresponding F = exp The relationship of replacing (f) with exp (k ₁ ) · exp (m ₁ ) is stored (step S8). In this cartane decomposition, as described above (FIG. 9), the elements of each decomposed matrix may be taken in a lattice shape.

The m ₁ decomposed in the cartan decomposition unit 6 is sent to the M decomposition unit 8, and the M decomposition unit 8 sends this to the diagonal matrix calculation unit 7. The diagonal matrix calculation unit _{7, U h · exp (m 1} ) · U h diagonalizing the diagonal matrix _{D h} was calculated special unitary matrix _{U m,} the diagonal matrix _{D h} and special unitary matrix U _m is returned to the M decomposition unit 8. The M decomposition unit 8 stores a relationship in which exp (m ₁ ) is replaced with U _h , U _m , D _h , U _m ^*, and U _h (step S9).
Next, the M decomposition unit 8 sends the diagonal matrix D _h to the diagonal matrix decomposition unit 9, and the diagonal matrix decomposition unit 9 decomposes the diagonal matrix D _h into a product of the torigate TG _h (step). S10). Diagonal matrix decomposition unit 9 stores the relationships to replace diagonal matrix D _h the product of Toffoli gates TG _h, sends these Toffoli gate TG _h to the basic gate decomposition unit 10. The basic gate decomposition unit 10 decomposes each sent torigate TG _h into a product of the basic gate BG _h (step S11), and replaces the relationship of replacing the torigate TG _h with the product of the basic gate BG _{h as} a diagonal. It returns to the matrix decomposition unit 9.

The diagonal matrix decomposition unit 9 uses the stored “relationship of replacing the diagonal matrix D _h with the product of the torigate TG _h ” and uses the product of the basic gate BG _h received from the basic gate decomposition unit 10 as a pair. replacing the diagonal matrix _{D h.} Then, the diagonal matrix D _h replaced by the product of the basic gate BG _h is returned to the M decomposition unit 8.
The M decomposition unit 8 is sent from the diagonal matrix decomposition unit 9 using the stored “relationship of exp (m ₁ ) with U _h , U _m , D _h , U _m ^* , U _h ”. U _h · U _m · D _h · U _m ^* · U _h is replaced with the product of the basic gate BG _h (= D _h ), and the replacement result exp (m ₁ ) = U _h · U _m · BG _h · U _m ^* Return U _h to the cartan decomposition unit 6.

Cartan decomposition unit 6, using the "relation replaced by F = exp a _{(f) exp (k 1)} · exp (m 1) " to save had been, exp sent from M decomposition section 8 _(m 1) = U _h · U _m · BG _h · U _m ^* · U _h replaces F = exp (f), and the replacement result F = exp (k ₁ ) · U _h · U _m · BG _h · U _m ^* · U _h is sent to the disassembly element storage unit 11.
The decomposition element storage unit 11 converts exp (k ₁ ), k ₁ , U _h , U _m , U _m ^*, and BG _h into F decomposition elements (F = exp (k ₁ ) · (U _h · U _m · BG _h · U _m ^* · U _h )) and stored in the main memory 3 (step S12).

Here, the quantum circuit configuration of U _h stored in the main memory 3 is known as described above (FIG. 3). In addition, of course, it is also known circuit configuration of the basic gate BG _h. Therefore, if exp (k ₁ ) and U _m can be decomposed into U _h or BG _h , F is decomposed into a realizable quantum circuit. Therefore, thereafter, the exp (k ₁ ) and U _m are set as a new special unitary matrix FεSU (2 ⁿ ), and the above-described steps S1 to S12 are recursively repeated. Thereby, all matrices are decomposed into realizable quantum circuits.
<Decomposition of Fourier transform matrix of 3 qubits>
Next, a specific example of this embodiment will be described. In this example, decomposition of a matrix of Fourier transform of 3 qubits (n = 3) is performed.

Ｅは２×２の単位行列で、ＮはＮＯＴゲート、Ｈはアダマールゲートである。また、Ｃ_１は第１量子ビットが１のときだけ、第２量子ビットの状態を反転させる制御ＮＯＴ（ＣＮＯＴ）ゲートで、Ｃ_２はＣＮＯＴの逆で、第１量子ビットが０ときだけ、第２量子ビットの状態を反転させるゲートである。また、Ｓは第１量子ビットと第２量子ビットの状態を入れ替えるＳＷＡＰゲートである。 <Basic matrix>
First, the basic matrix used for the decomposition of this example is shown.

E is a 2 × 2 unit matrix, N is a NOT gate, and H is a Hadamard gate. Further, only when _{C 1} first qubit is 1, the control NOT (CNOT) gate for inverting the state of the second qubit, _{C 2} is the inverse of CNOT, only when the first qubit is zero, the It is a gate that inverts the state of 2 qubits. S is a SWAP gate that switches the state of the first qubit and the second qubit.

<Input>
The matrix U _F ∈ SU (8) of the Fourier transform of 3 qubits is the following matrix.

である。 First, this matrix U _F (corresponding to “F”) is input to the input unit 2 (FIG. 1) and stored in the main memory 3. The matrix _{U F} is not basic gate (step S1), and not included in SU (4) (Step S3). Further, at this time, even not calculated Lie algebra expressions corresponding to the matrix U _F (step S4). Therefore, first, the matrix representation conversion unit 5 calls the diagonal matrix calculation unit 7 to obtain the Lie algebra representation f corresponding to the matrix U _F (steps S5 and S6). That is, since U _F = exp (f), the matrix representation conversion unit 5 uses a matrix U ₁ that diagonalizes U _F into a diagonal matrix D _f .
U _F = U ₁ · D _f · U ₁ ^* = exp (f)
f = log (U _F ) = log (U ₁ · D _f · U ₁ ^* ) = U ₁ · log (D _f ) · U ₁ ^*
Thus, the Lie algebra expression f is obtained. Here, the matrix U ₁ to realize the diagonalization is a matrix obtained by arranging eigenvectors of U _F vertically to meet the nature of the unitary matrix. That is,

It is.

である。このように算出されたＬｉｅ代数表現ｆは、行列Ｕ_Ｆに対応付けられて主記憶装置３に格納される。 And the Lie algebraic expression f corresponding to the matrix U _F is

It is. Thus calculated Lie algebra expressions f is stored in association with the matrix U _F in the main storage device 3.

という２つの行列ｋ_１，ｍ_１の直和に分解する（ステップＳ８）。この分解は、図９のように格子状に要素を抽出することによってできる。すなわち、

となる。そして、Ｋ₁＝ｅｘｐ（ｋ_１），Ｍ_１＝ｅｘｐ（ｍ_１）より、Ｕ_Ｆ＝Ｋ₁・Ｍ_１の分解が求まる。 Next, the matrix representation conversion unit 5 extracts a Lie algebra expressions f corresponding to the matrix U _F from the main memory 3 (step S7), and sends it to the Cartan decomposition unit 6. The cartan decomposition unit 6 uses the cartan decomposition to convert this f into

Are decomposed into a direct sum of the two matrices k ₁ and m ₁ (step S8). This decomposition can be performed by extracting elements in a grid pattern as shown in FIG. That is,

It becomes. Then, from K ₁ = exp (k ₁ ) and M ₁ = exp (m ₁ ), the decomposition of U _F = K ₁ · M ₁ is obtained.

M ₁ calculated by the Cartan decomposition unit 6 is sent to the M decomposition unit 8, and the M decomposition unit 8 calls the diagonal matrix calculation unit 7, and M ₁ = exp (m ₁ ) M ₁ = U _h. replaced by ^{_{U m · D h · U m}} * · U h ( step S9). Here, U _h is a matrix represented by the aforementioned equation (23), and can be realized by the quantum circuit of FIG. D _h is a diagonal matrix in which the eigenvalues of U _h , M _1, and U _h are diagonal components,
D _h = diag (i, -i, i, -i, 1, 1, 1, 1) (24)
It becomes.

である。 This diagonal matrix D _h is decomposed by the diagonal matrix decomposition unit 9 into a product of four 3-qubit torigates as described in the explanation of <diagonal matrix decomposition unit> _{Since 3} and F ₄ are unit matrices, they are decomposed into products of two 3-qubit torigates (step S10). The basic gate decomposition unit 10 further decomposes the product into the following basic gate products (step S11).

It is.

である。
そして、以上のような対角行列Ｄ_ｈの基本ゲートの積、行列Ｕ_ｍ、Ｕ_ｍ ^＊、Ｕ_ｈ、Ｋ₁＝ｅｘｐ（ｋ_１），ｋ_１は、分解要素格納部１１において主記憶装置３に格納される（ステップＳ１２）。 The matrix U _m for realizing such diagonalization is

It is.
The product of the basic gates of the diagonal matrix D _h as described above, the matrices U _m , U _m ^* , U _h , K ₁ = exp (k ₁ ), and k ₁ are stored in the main storage device in the decomposition element storage unit 11. 3 (step S12).

<Decomposition of U _m >
Next, since U _m and K ₁ are not basic gates (step S1), the structure determination unit 4 first reads the matrix U _m from the main memory 3 (step S2). Here, as shown in Expression (25), the maximum number of elements other than 0 included in each row or column is 4. Therefore, this matrix U _m is included in SU (4), and the structure determination unit 4 sends this matrix U _m to the basic gate decomposition unit 10. Then, the basic gate decomposition unit 10 replaces this matrix U _m with the product of the basic gates as follows (step S4).

ここで、Ｕ_４は行と列を入れ替える行列であり、基本ゲート分解部１０は、トフォリゲートＴ_８を用いて、さらにこのＵ_４を以下のように構成する。

Here, U ₄ is a matrix in which the rows and columns are interchanged, and the basic gate decomposition unit 10 further configures this U ₄ using the torigate T ₈ as follows.

このＴ₈は、第１量子ビットと第２量子ビットの状態が共に１の場合にのみ、第３量子ビットの状態を反転させる制御ＮＯＴゲートである。

This T ₈ is a control NOT gate that inverts the state of the third qubit only when the states of the first qubit and the second qubit are both 1.

The basic gate decomposition unit 10 decomposes the T ₅ to a tensor product of the following matrices.

以上のような分解結果は主記憶装置３に格納される。 <Disassembly of U ₆ >
Furthermore, the basic gate decomposition unit 10 decomposes the U ₆ in the following form.

The decomposition results as described above are stored in the main storage device 3.

<Decomposition into K ₁ = K2 · M2>
Then, because it is not yet K ₁ is basic gate (step S1), K ₁ is read from the main memory 3 in the configuration determining unit 4 (step S2). Since K _{1 is} also included in SU (4), the structure determination unit 4 sends it to the K ₁ basic gate decomposition unit 10. The basic gate decomposition unit 10, as follows, replacing the K ₁ by the product of the basic gate (step S4).

となる。従って、

First, the basic gate decomposition unit 10 extracts a Lie algebra expression k ₁ corresponding to K ₁ = exp (k ₁ ) from the main storage device 3. Then, the basic gate decomposition unit 10 decomposes this Lie algebra expression k ₁ into a direct sum of k ₂ and m ₂ by Cartan decomposition. Specifically, for example, k ₂ is obtained by extracting the element in FIG. 4A from k ₁ , and m ₂ is obtained by extracting the element in FIG. 4B. Then

It becomes. Therefore,

となる。

It becomes.

<Decomposition of _{K 2>}
Then the basic gate decomposition unit 10, a K ₂ represented by the above formula (26) is rewritten to the following form.

ここで、Ｕ_２は、Ｋ_２の行や列を入れ替えて作られた行列であり、Ｓ_１がその行と列の入れ替えを表している。具体的には、

であり、このＳ_１の操作は、
１.第２量子ビットと第３量子ビットを入れ替える。
２.第１量子ビットと第２量子ビットを入れ替える。
３.第１量子ビットの否定（ＮＯＴ）をとる。
という操作から構成されている。

Here, U ₂ is a matrix created by replacing the rows and columns of K ₂ , and S ₁ represents the replacement of the rows and columns. In particular,

, And the operation of the _{S 1} is,
1. Swap the second and third qubits.
2. Swap the first qubit and the second qubit.
3. Take the negation (NOT) of the first qubit.
It consists of the operation.

さらに、基本ゲート分解部１０は、式(29)のＵ_３を、以下の形に分解する。 <Decomposition of _{U 2>}
Next, the basic gate decomposition unit 10 calls the diagonal matrix calculation unit 7 to decompose U ₂ represented by the above equation (28) into a matrix product including the diagonal matrix D ₂ as follows.

Further, the basic gate decomposition unit 10 decomposes U ₃ in the equation (29) into the following form.

次に基本ゲート分解部１０は、式(30)のＤ_２を、以下の形に分解する。

Then basic gate decomposition unit 10 decomposes the D ₂ of formula (30), the following form.

ここで、Ｔ_６は２ビットのトフォリゲートであり、これをsingle-qubit rotationsとCNOTに分解する手法は既に知られている（例えば、ｂ「A.Barenco et al., Elementary gates for quantum computation, Phys. Rev. A52, pp.3457-3467,1995.」や「C. Moore and M. Nilsson, parallel quantum computation and quantum codes, SIAM J. Comp, Vol.31, No.3, pp.799-815, 2001.」を参照。）。
そして、以上のような分解結果は主記憶装置３に格納される。 <Decomposition of _{M 2>}
Further, the basic gate decomposition unit 10 decomposes M ₂ represented by the above-described equation (27) into the following form.

Here, T ₆ is a 2-bit torigate, and a technique for decomposing it into single-qubit rotations and CNOT is already known (for example, b “A. Barenco et al., Elementary gates for quantum computation”). , Phys. Rev. A52, pp.3457-3467, 1995. '' and `` C. Moore and M. Nilsson, parallel quantum computation and quantum codes, SIAM J. Comp, Vol.31, No.3, pp.799- 815, 2001.).
The decomposition results as described above are stored in the main storage device 3.

また、図５の（ａ）は、上記の式(31)に示すＵ_Ｆのカルタン分解により求まる量子回路を示している。この図に示すように、Ｕ_Ｆの量子回路は、Ｕ_ｈ回路４１、Ｕ_ｍ ^＊回路４２、Ｄ_ｈ回路４３、Ｕ_ｍ回路４４、Ｕ_ｈ回路４５、Ｍ_２回路４６、Ｋ_２回路４７を直列に配置して構成される。
図５の（ｂ）は、Ｄ_ｈ回路４３を基本ゲートに分解した構成（式(35)）を示している。この図に示すように、Ｄ_ｈ回路４３は、４つのＮＯＴゲート４３ａ〜４３ｄと２つの制御ユニタリ（Ｔ_９）ゲート４３ｅ，４３ｆによって構成される。 The decomposition of the found Fuorier transform by the above processing is as follows.

Further, (a) in FIG. 5 shows a quantum circuit obtained by Cartan decomposition of U _F shown in the above equation (31). As shown in this figure, the quantum circuit of U _F includes a U _h circuit 41, a U _m ^* circuit 42, a D _h circuit 43, a U _m circuit 44, a U _h circuit 45, an M ₂ circuit 46, and a K ₂ circuit 47. It is arranged in series.
FIG. 5B shows a configuration (formula (35)) in which the _Dh circuit 43 is disassembled into basic gates. As shown in FIG, _{D h} circuit 43, four NOT gates 43a~43d and two control unitary _{(T 9)} gate 43e, constituted by 43f.

(C) in FIG. 5 shows a configuration obtained by decomposing the M ₂ circuit 46 to the basic gate (formula (33)). As shown in FIG, _{M 2} circuit 46, two SWAP gates 46a, 46b and constituted by a single control unitary gate _(T 6) 46c.
FIG. 6A shows a configuration (formula (34)) in which the _Um circuit 44 is disassembled into basic gates. As shown in FIG., _{U m} circuit 44, eight NOT gates 44A to 44H, 4 one control unitary gate _(T 8) _44i~44p, 1 single control unitary gate _(T 1) 44k, 1 single control unitary gate (T ₇ ) 44m, two CNOT gates (C ₁ ) 44q, 44r, one NCNOT gate (C ₂ : a gate that inverts the state of the second qubit only when the first qubit is 0) 44s, The Hadamard gate (H) 44t, 44u is composed of one SWAP gate (S) 44v.

(B) of FIG. 6 shows a configuration in which decompose K ₂ circuit 47 to the basic gate (formula (32)). As shown in FIG, _{K 2} circuit 47, four SWAP gate (S) 47a-47d, four NOT gates 47E～47h, 2 two CNOT gate _{(C 1) 47i, 47j,} 2 two control unitary gate ( T ₁ ) 47m, 47s, three control unitary gates (T ₂ ) 47k, 47n, 47r, one control unitary gate (T ₃ ) 47q, and one control unitary gate (T ₄ ) 47p.

[Second Embodiment]
Next, a second embodiment of the present invention will be described.
This embodiment is a modification of the first embodiment. In this embodiment, the unitary matrix is decomposed by using a matrix that diagonalizes a Lie algebra-represented matrix without using the above-mentioned simultaneous diagonalization matrix. Below, it demonstrates centering around difference with 1st Embodiment, and abbreviate | omits description about the matter which is common in 1st Embodiment.

<Configuration>
First, the configuration of the unitary matrix decomposition apparatus according to this embodiment will be described.
FIG. 7 is a block diagram for explaining the configuration of the unitary matrix decomposition apparatus 100 according to this embodiment. In FIG. 7, the functional components common to the unitary matrix decomposition apparatus 1 (FIG. 1) of the first embodiment are denoted by the same reference numerals as in FIG.
The difference of the configuration of the unitary matrix decomposition apparatus 100 from the first embodiment is that an m decomposition unit 108 is provided instead of the M decomposition unit 8 of the unitary matrix decomposition apparatus 1 of the first embodiment. .

<M disassembly unit>
The m decomposition unit 108 passes the received m ₁ to the diagonal matrix calculation unit 7 and calculates a special unitary matrix U ₁ that diagonalizes m ₁ into the diagonal matrix D ₁ . Then, the m decomposition unit 108 receives the special unitary matrix U ₁ calculated by the diagonal matrix calculation unit 7 and the diagonal matrix D ₁ and converts exp (m ₁ ) to U ₁ · exp (D ₁ ) · U. ^Replace with ₁ ^* . In addition, the m decomposition unit 108 passes the diagonal matrix D ₁ to the diagonal matrix decomposition unit 9, replaces the diagonal matrix D ₁ with the decomposition result returned from the diagonal matrix decomposition unit 9, U ₁ , Return U ₁ ^* to the cartan decomposition unit 6.

<Processing>
Next, a unitary matrix decomposition method according to this embodiment will be described.
FIG. 8 is a flowchart for explaining the unitary matrix decomposition method in this embodiment. The unitary matrix decomposition / decomposition processing in this embodiment will be described below with reference to FIGS.
First, an arbitrary unitary matrix is input from the input unit 2. The input unitary matrix is converted into a special unitary matrix FεSU (2 ⁿ ) in the input unit 2 and stored in the main storage device 3 as a matrix list F.

Next, the structure determination unit 4 determines whether all the special unitary matrices F (the matrix F included in the matrix list of the main storage device 3) stored in the main storage device 3 are basic gates (step). S21). Here, if it is determined that all the special unitary matrices F stored in the main storage device 3 are basic gates, the processing is terminated.
On the other hand, when it is determined that the special unitary matrix F that is not a basic gate exists in the matrix list, the structure determination unit 4 reads (extracts) the special unitary matrix F that is not the first basic gate in the matrix list from the main storage device 3. (Step S22). The structure determination unit 4 that has extracted the special unitary matrix F determines whether or not the extracted special unitary matrix is included in SU (4) (n ≧ 3) (step S23). Here, when it is determined that the extracted special unitary matrix is included in SU (4) (when it is determined that n <3), the structure determination unit 4 converts the special unitary matrix F into the basic gate decomposition unit. 10, the basic gate decomposition unit 10 replaces the special unitary matrix F with the product of the basic gates and stores it in the main storage device 3 and ends the process.

  On the other hand, when it is determined that the extracted special unitary matrix is not included in SU (4) (when it is determined that n ≧ 3), the structure determination unit 4 determines that the special unitary matrix F (“with decomposition target”). (Corresponding to a certain special unitary matrix F)) is sent to the matrix expression conversion unit 5. Receiving this, the matrix expression conversion unit 5 determines whether or not the Lie algebra expression f corresponding to the special unitary matrix F that is the decomposition target is calculated (whether or not it is stored in the main storage device 3). (Step S25). Here, when it is determined that the corresponding Lie algebra expression f has not been calculated, the matrix expression conversion unit 5 obtains this Lie algebra expression f and stores it in the main storage device 3 as in the first embodiment. Store (step S26). On the other hand, if it is determined in step S25 that the corresponding Lie algebra expression f has been calculated, the matrix expression conversion unit 5 does not calculate the Lie algebra expression f.

Thereafter, the matrix representation conversion unit 5 extracts the Lie algebra representation fεsu (2 ⁿ ) corresponding to the special unitary matrix FεSU (2 ⁿ ) to be decomposed from the main storage device 3 (step S27), This is sent to the disassembly unit 6. Cartan decomposition unit 6 decomposes the sent Lie algebra expressions f, by Cartan decomposition, two orthogonal vector space k, the element _{k 1} that is an element of the set K contained in m, to _{m 1 ∈m,} corresponding F = exp The relationship of replacing (f) with exp (k ₁ ) · exp (m ₁ ) is stored (step S28).
The m ₁ decomposed in the cartan decomposition unit 6 is sent to the m decomposition unit 108, and the m decomposition unit 108 sends it to the diagonal matrix calculation unit 7. The diagonal matrix calculation unit 7 calculates the special unitary matrix U ₁ to diagonalized the m ₁ a diagonal matrix D _1, the diagonal matrix D ₁ and special unitary matrix U ₁ to m decomposition unit 108 return. The m decomposition unit 108 stores the relationship of replacing exp (m ₁ ) with U ₁ · exp (D ₁ ) · U ₁ ^* (step S29).

Next, the m decomposing unit 108 sends the diagonal matrix exp (D ₁ ) to the diagonal matrix decomposing unit 9, and the diagonal matrix decomposing unit 9 sends the diagonal matrix exp (D ₁ ) to the torigate TG _h . The product is decomposed (step S30). Diagonal matrix decomposition unit 9 stores the relationships to replace diagonal matrix exp a (D ₁₎ by the product of Toffoli gates TG _h, it sends these Toffoli gate TG _h to the basic gate decomposition unit 10. The basic gate decomposition unit 10 decomposes each sent torigate TG _h into a product of the basic gate BG _h (step S31), and replaces the relationship of replacing the torigate TG _h with the product of the basic gate BG _{h as} a diagonal. It returns to the matrix decomposition unit 9.

The diagonal matrix decomposition unit 9 uses the stored “relationship of replacing the diagonal matrix exp (D ₁ ) with the product of the torigate TG _h ” and stores the basic gate BG _h received from the basic gate decomposition unit 10. Replace the diagonal matrix exp (D ₁ ) with the product. Then, the diagonal matrix exp (D ₁ ) replaced with the product of the basic gate BG _h is returned to the m decomposition unit 108.
The m decomposition unit 18 uses the saved “relationship to replace exp (m ₁ ) with U ₁ · exp (D ₁ ) · U ₁ ^* ”, and the basic gate BG sent from the diagonal matrix decomposition unit 9. The product of _h (= exp (D ₁ )) replaces U ₁ · exp (D ₁ ) · U ₁ ^* , and the replacement result exp (m ₁ ) = U ₁ · BG _h · U ₁ ^* Return to.

Cartan decomposition unit 6, the saved "the _{F = exp (f) exp (} k 1) · replaced relationship exp _{(m 1)"} used, it exp sent from m decomposing section 108 _(m 1) F = exp (f) is replaced with = U ₁ · BG _h · U ₁ ^* , and the replacement result F = exp (k ₁ ) · U ₁ · BG _h · U ₁ ^* is sent to the disassembly element storage unit 11.
The decomposition element storage unit 11 converts exp (k ₁ ), k ₁ , U ₁ , U ₁ ^*, and BG ₁ into F decomposition elements (F = exp (k ₁ ) · (U ₁ · BG ₁ · U _1). ^* = Exp (m ₁ ))) and stored in the main memory 3 (step S32).

Here, the circuit configuration of the basic gate BG _h stored in the main memory 3 are also known. Therefore, if exp (k) and U ₁ can be decomposed into BG _h , F is decomposed into a realizable quantum circuit. Therefore, thereafter, the above-described steps S21 to S32 are recursively repeated with exp (k ₁ ) and U ₁ as a new special unitary matrix FεSU (2 ⁿ ). Thereby, all matrices are decomposed into realizable quantum circuits.
The present invention is not limited to the above-described embodiments, and it goes without saying that other modifications can be made as appropriate without departing from the spirit of the present invention.
Further, when the above configuration is realized by a computer, the processing contents of functions that each device should have are described by a program. The processing functions are realized on the computer by executing the program on the computer.

The program describing the processing contents can be recorded on a computer-readable recording medium. The computer-readable recording medium may be any medium such as a magnetic recording device, an optical disk, a magneto-optical recording medium, or a semiconductor memory. Specifically, for example, the magnetic recording device may be a hard disk device or a flexible Discs, magnetic tapes, etc. as optical disks, DVD (Digital Versatile Disc), DVD-RAM (Random Access Memory), CD-ROM (Compact Disc Read Only Memory), CD-R (Recordable) / RW (ReWritable), etc. As the magneto-optical recording medium, MO (Magneto-Optical disc) or the like can be used, and as the semiconductor memory, EEP-ROM (Electronically Erasable and Programmable-Read Only Memory) or the like can be used.
The program is distributed by selling, transferring, or lending a portable recording medium such as a DVD or CD-ROM in which the program is recorded. Furthermore, the program may be distributed by storing the program in a storage device of the server computer and transferring the program from the server computer to another computer via a network.

A computer that executes such a program first stores, for example, a program recorded on a portable recording medium or a program transferred from a server computer in its own storage device. When executing the process, the computer reads a program stored in its own recording medium and executes a process according to the read program. As another execution form of the program, the computer may directly read the program from a portable recording medium and execute processing according to the program, and the program is transferred from the server computer to the computer. Each time, the processing according to the received program may be executed sequentially. Also, the program is not transferred from the server computer to the computer, and the above-described processing is executed by a so-called ASP (Application Service Provider) type service that realizes the processing function only by the execution instruction and result acquisition. It is good. Note that the program in this embodiment includes information that is used for processing by an electronic computer and that conforms to the program (data that is not a direct command to the computer but has a property that defines the processing of the computer).
In this embodiment, the present apparatus is configured by executing a predetermined program on a computer. However, at least a part of these processing contents may be realized by hardware.

  As an industrial application field of the present invention, a design support field of an arbitrary unitary transformation quantum circuit constituting a quantum computer can be exemplified. By applying the present invention, an arbitrary unitary transformation quantum circuit can be easily decomposed into basic gates such as CNOT.

The block diagram for demonstrating the structure of the unitary matrix decomposition | disassembly apparatus in 1st Embodiment. The flowchart for demonstrating the unitary matrix decomposition | disassembly method in 1st Embodiment. Shows a quantum circuit matrix U _h that all joint diagonalization of underlying Cartan subalgebra h (n). The figure for demonstrating the method of cartan decomposition | disassembly. The figure which showed the quantum circuit corresponding to a decomposition | disassembly result. The figure which showed the quantum circuit corresponding to a decomposition | disassembly result. The block diagram for demonstrating the structure of the unitary matrix decomposition | disassembly apparatus in 2nd Embodiment. The flowchart for demonstrating the unitary matrix decomposition | disassembly method in 2nd Embodiment. The figure for demonstrating the method of cartan decomposition | disassembly.

Explanation of symbols

1,100 unitary matrix decomposition device 2 input unit 3 main storage device (storage means)
4 Structure determination unit (structure determination means)
5 Matrix representation conversion unit (Matrix representation conversion means)
6 Cartan decomposition part (Cartan decomposition means)
7 Diagonal matrix calculator (diagonal matrix calculation means)
8 M Decomposition Unit 9 Diagonal Matrix Decomposition Unit (Diagonal Matrix Decomposition Unit)
10 Basic gate disassembly section (Basic gate disassembly means)
11 Decomposition element storage (decomposition element storage means)
12 Output unit 108 m Disassembly unit

Claims (10)

A unitary matrix decomposition method for decomposing a unitary matrix,
A matrix representation conversion means for extracting a Lie algebraic representation fεsu (2 ⁿ ) that is a natural logarithm of a 2 ⁿ × 2 ⁿ special unitary matrix FεSU (2 ⁿ ) to be decomposed;
The extracted the Lie algebra expressions f, Cartan decomposition means, by Cartan (Cartan) decomposition, two orthogonal vector space k, the element _{k 1} that is an element of the set K contained in m, and decomposing the _{m 1 ∈m,}
U _h · exp (m ₁ ) · U _h (U _h is a matrix that simultaneously diagonalizes all the bases of the Cartan partial algebra h (n) that is a maximal commutative partial algebra included in the vector space m). The diagonal matrix calculating means calculates a special unitary matrix U _m that is diagonalized into a diagonal matrix D _h (D _h ∈ U _h · exp (h (n)) · U _h );
Diagonal matrix decomposition means for decomposing the diagonal matrix D _h into a product of toffoli gates TG _h ;
Basic gate decomposition means for decomposing each of the torigates TG _h into a product of the basic gates BG _h ;
The decomposition element storage means converts the exp (k ₁ ), U _h , U _m , U _m ^*, and BG _h into the F decomposition element (F = exp (k ₁ ) · (U _h · U _m · BG _h • U _m ^* · U _h )) in the storage means;
A unitary matrix decomposition method characterized by comprising: A unitary matrix decomposition method for decomposing a unitary matrix,
A matrix representation conversion means for extracting a Lie algebraic representation fεsu (2 ⁿ ) that is a natural logarithm of a 2 ⁿ × 2 ⁿ special unitary matrix FεSU (2 ⁿ ) to be decomposed;
The extracted the Lie algebra expressions f, Cartan decomposition means, by Cartan (Cartan) decomposition, two orthogonal vector space k, the element _{k 1} that is an element of the set K contained in m, and decomposing the _{m 1 ∈m,}
a diagonal matrix calculation means calculating a special unitary matrix U ₁ that diagonalizes m ₁ into a diagonal matrix D ₁ ;
Diagonal matrix decomposition means decomposes exp (D ₁ ) into a product of toffoli gates TG ₁ ;
A basic gate decomposition means for decomposing each of the torigate gates TG ₁ into a product of the basic gates BG ₁ ;
The decomposition element storage means converts exp (k ₁ ), U ₁ , U ₁ ^*, and BG ₁ into F decomposition elements (F = exp (k ₁ ) · (U ₁ · BG ₁ · U ₁ ^* )). And storing in the storage means as
A unitary matrix decomposition method characterized by comprising: The unitary matrix decomposition method according to claim 1 or 2,
The structure determining means determining whether all the special unitary matrices stored in the storage means are basic gates; and
A step of extracting a special unitary matrix determined not to be a basic gate from the storage means by the structure determination means;
A step in which the structure determining means determines whether or not the extracted special unitary matrix is n ≧ 3;
Further comprising
The special unitary matrix F to be decomposed is
In the structure determination means, a special unitary matrix determined as n ≧ 3.
A unitary matrix decomposition method. A unitary matrix decomposition apparatus that decomposes a unitary matrix,
Storage means for storing a 2 ⁿ × 2 ⁿ special unitary matrix (SU (2 ⁿ )) and a Lie algebraic representation (su (2 ⁿ )) that is a natural logarithm of the special unitary matrix in association with each other. When,
Matrix expression conversion means for extracting the Lie algebra expression f∈su (2 ⁿ ) corresponding to the special unitary matrix F∈SU (2 ⁿ ) to be decomposed from the storage means;
The extracted the Lie algebra expressions f, by Cartan (Cartan) decomposition, two orthogonal vector space k, the element _{k 1} that is an element of the set K contained in m, and Cartan decomposition means for decomposing the _{m 1 ∈m,}
U _h · exp (m ₁ ) · U _h (U _h is a matrix that simultaneously diagonalizes all the bases of the Cartan partial algebra h (n) that is a maximal commutative partial algebra included in the vector space m). A diagonal matrix calculation means for calculating a special unitary matrix U _m diagonalized to a diagonal matrix D _h (D _h ∈ U _h · exp (h (n)) · U _h );
Diagonal matrix decomposition means for decomposing the diagonal matrix D _h into products of toffoli gates TG _h ;
Basic gate disassembling means for resolving each torigate TG _h into a product of basic gates BG _h ;
The above exp (k ₁ ), U _h , U _m , U _m ^* and BG _h are divided into the decomposition elements of F (F = exp (k ₁ ) · (U _h · U _m · BG _h · U _m ^* · U _h = exp (m ₁ ))) in the storage means,
A unitary matrix decomposition apparatus characterized by A unitary matrix decomposition apparatus that decomposes a unitary matrix,
Storage means for storing a 2 ⁿ × 2 ⁿ special unitary matrix (SU (2 ⁿ )) and a Lie algebraic representation (su (2 ⁿ )) that is a natural logarithm of the special unitary matrix in association with each other. When,
Matrix expression conversion means for extracting the Lie algebra expression f∈su (2 ⁿ ) corresponding to the special unitary matrix F∈SU (2 ⁿ ) to be decomposed from the storage means;
The extracted the Lie algebra expressions f, by Cartan (Cartan) decomposition, two orthogonal vector space k, the element _{k 1} that is an element of the set K contained in m, and Cartan decomposition means for decomposing the _{m 1 ∈m,}
diagonal matrix calculation means for calculating a special unitary matrix U ₁ that diagonalizes m ₁ into a diagonal matrix D ₁ ;
diagonal matrix decomposition means for decomposing exp (D ₁ ) into products of toffoli gates TG ₁ ;
Basic gate disassembling means for disassembling each torigate TG ₁ into a product of the basic gate BG ₁ ;
The above exp (k ₁ ), U ₁ , U ₁ ^* and BG ₁ are converted into the decomposition elements of F (F = exp (k ₁ ) · (U ₁ · BG ₁ · U ₁ ^* = exp (m ₁ ))) Decomposition element storage means for storing in the storage means as
A unitary matrix decomposition apparatus characterized by The unitary matrix decomposition apparatus according to claim 4 or 5,
The special unitary matrix F to be decomposed is
Of the special unitary matrix stored in the storage means, a special unitary matrix of n ≧ 3 excluding the basic gate,
A unitary matrix decomposition apparatus. The unitary matrix decomposition apparatus according to claim 6,
The basic gate disassembly means is
Further, among the special unitary matrices stored in the storage means, the unit unit that decomposes the n = 2 special unitary matrix excluding the basic gate into the product of the basic gate BG ₂ .
A unitary matrix decomposition apparatus. The unitary matrix decomposition apparatus according to claim 4 or 5,
The diagonal matrix calculation means includes:
When the Lie algebraic expression fεsu (2 ⁿ ) corresponding to the special unitary matrix FεSU (2 ⁿ ) to be decomposed is not stored in the storage means, the F is diagonalized into the diagonal matrix D ₂ Means for further calculating the matrix U ₂ ;
The matrix representation conversion means is
Furthermore, it is means for storing the operation result of U ₂ · log (D ₂ ) · U ₂ ^{* in} the storage means as a Lie algebra expression f corresponding to F.
A unitary matrix decomposition apparatus.   A unitary matrix decomposition program for causing a computer to function as the unitary matrix decomposition apparatus according to claim 4.   A computer-readable recording medium storing the unitary matrix decomposition program according to claim 9.

Images