JP4664994B2 - Calculation base axis setting device, calculation base axis setting method, program, and recording medium - Google Patents

Description

  The present invention relates to a quantum error correction coding technique, and more particularly, to a technique for performing error correction on noise composed of a classic mixture of symmetrical noise and random noise.

<Quantum error correction code>
The quantum computer handles a continuum. Therefore, it is very difficult to completely remove the error. On the other hand, in a meaningful quantum calculation, a very high calculation accuracy is often required for the input length. Therefore, in the research on the realization of quantum computers, it is one of very important problems to make a device that is less susceptible to noise and has less error.

  As a method for reducing the error, in addition to the research on the physical device itself, there has been research on effectively reducing the error by handling the generated error while correcting it. A typical example is a so-called stabilizer code (fixed subgroup code) (see Non-Patent Document 1, for example). Stabilizer code refers to the whole quantum code that can be written in a certain format, and the quantum error correction method by this is a general theory of error correction method when there is no information about noise except that the noise is small to some extent. . On the other hand, when information about noise is known to some extent, error correction is performed on the decoherence-free subspaces, and the known symmetry of noise is used to correct the noise. There is also a method of coding a quantum state in an unaffected region (for example, see Non-Patent Documents 2 and 3).

  Both theories completely correct errors in a given system. For example, in the case of a stabilizer code, when noise occurs only in a predetermined number or less of qubits in designing the code, this can be completely corrected. In addition, the decoherence subspace has a structure in which the influence of noise does not occur in the logical qubit in the first place when only the noise assumed in advance occurs.

<System noise and quantum error correction code>
Many experimental systems assume that the system noise is a classical mixture of some kind of symmetric noise that is expected and random noise that is totally unpredictable. For this reason, the quantum error correction code needs to be resistant to random noise and robust against symmetric noise. Such noise can be completely eliminated with respect to symmetrical noise by using the logical qubits configured in the decoherence subspace as physical qubits, and symmetric noise can be completely eliminated, and error correction can be performed to some extent as a whole. be able to.
D. Gottesman, “Theory of fault-tolerant quantum computation”, Phys. Rev. A57 (1988) 127. DA Lidar, IL Chuang and KB Whaley, “Decoherence-Free Subspaces for Quantum Computation”, Phys. Rev. Lett. 81 (1998) 2594. D. Bacon, J. Kempe, DA Lidar, and KB Whaley, “Universal Fault-Tolerant Quantum Computation on Decoherence-Free Subspaces”, Phys. Rev. Lett. 85, (2000) 1758.

  However, when a stabilizer code is assembled using the logical qubits configured in the decoherence subspace as physical qubits, the number of physical qubits constituting one logical qubit inevitably increases. Furthermore, one operation on a logical qubit becomes an operation on a large number of physical qubits. Therefore, if the amount of symmetric noise is not so large, these drawbacks may outweigh the advantage that symmetric noise can be completely eliminated.

  The present invention has been made in view of such a point, and performs error correction for noise consisting of a classical mixture of symmetrical noise and random noise using fewer physical qubits than in the past. It aims at providing the technology that can be.

に基づき、設定されたユニタリー行列V_jを採用するか否かを判定し、その判定結果を出力する。なお、テンソル積演算子を(×)とし、ユニタリー行列V_j(j∈{1,...,n})のテンソル積からなる行列をV=V₁(×) …(×)V_nとし、スタビライザー符号によってｋ(n＞k≧1)個の論理量子ビットに符号化されたn個の物理量子ビットの量子状態を示す列ベクトルである状態ベクトルを|Φ_L ^(k)>とし、それと複素共役な行ベクトルを<Φ_L ^(k)|とし、n個の物理量子ビットの密度行列を行列ρとし、当該スタビライザー符号の誤り訂正に対応するCPTP写像をCとし、 (h_x, h_y, h_z)を要素とする３次元ベクトルをHとし、３次元ベクトルHの大きさをh=|H|とし、p(h)を To solve the problem in the present invention, with respect to a common reference calculation basis axis for each physical qubit, j [j∈ {1, ..., n} , n ≧ 2 ] Calculation of th physical qubit Q _j Set the 2 × 2 unitary matrix V _j to indicate the rotation of the base axis and set fidelity

Based on the above, it is determined whether or not to use the set unitary matrix V _j , and the determination result is output. Note that the tensor product operator is (×), and the matrix composed of tensor products of the unitary matrix V _j (j∈ {1, ..., n}) is V = V ₁ (×)… (×) V _n Let | Φ _L ^(k) > be a state vector that is a column vector indicating the quantum state of n physical qubits encoded into k (n> k ≧ 1) logical qubits by a stabilizer code, and The complex conjugate row vector is <Φ _L ^(k) |, the density matrix of n physical qubits is the matrix ρ, and the CPTP map corresponding to the error correction of the stabilizer code is C, and (h _x , h _y , h _z ) as an element, a three-dimensional vector is H, the size of the three-dimensional vector H is h = | H |, and p (h) is

を満たすhの関数とし、2×2の単位行列をIとし、虚数単位をiとし、ＸをパウリＸ行列とし、ＹをパウリＹ行列とし、ＺをパウリＺ行列とし、S=(X, Y, Z)とし、h_x・X+h_y・Y+h_z・ZをH・Sとし、Hに対するユニタリー行列I・cos h+i(H・S /h)sin hをU(H)とし、n個のユニタリー行列U(H)のテンソル積からなる行列をU(H)^(×)nとし、行列βの共役転置行列をβ⁺とし、qを正の実数とし、対称性のあるノイズによる物理量子ビット毎に無相関で一様な作用素をE_q(・)とし、

とし、行列Vに従って計算基底軸を回転させた場合における行列V⁺ E_q(VρV⁺)Vを行列RE_q(ρ)とし、

And the 2 × 2 unit matrix is I, the imaginary unit is i, X is a Pauli X matrix, Y is a Pauli Y matrix, Z is a Pauli Z matrix, and S = (X, Y , Z), h _x・ X + h _y・ Y + h _z・ Z is H ・ S, unitary matrix I ・ cos h + i (H ・ S / h) sin h for H is U (H) , A matrix consisting of tensor products of n unitary matrices U (H) is U (H) ^{(×) n} , the conjugate transpose matrix of matrix β is β ⁺ , q is a positive real number, and symmetric noise E _q (・) is an uncorrelated and uniform operator for each physical qubit by

And the matrix V ⁺ E _q (VρV ⁺ ) V when the calculation base axis is rotated according to the matrix V is the matrix RE _q (ρ),

を満たし、任意の関数fと、上記スタビライザー符号によってｋ個の論理量子ビットに符号化されたn個の物理量子ビットの量子状態を示す列ベクトルが張るベクトル空間VSの元である要素数2ⁿのベクトルを当該ベクトル空間VSに写像する2ⁿ×2ⁿの任意のユニタリー行列AUと、に対して

を満たす、上記ベクトル空間VS上での積分とする。

The number of elements 2 ⁿ that is an element of a vector space VS spanned by an arbitrary function f and a column vector indicating quantum states of n physical qubits encoded into k logical qubits by the stabilizer code For any 2 ⁿ × 2 ⁿ unitary matrix AU that maps the vector of

Which satisfies the above-mentioned integration in the vector space VS.

  The operation of the present invention will be described below. By changing the calculation base axis of each physical qubit, the symmetry of noise may be effectively broken. In this case, it is possible to effectively prevent the symmetrical noise from strengthening by synchronizing with the physical qubit. As a result, symmetric noise can be appropriately error-corrected only by the stabilizer code in the same way as random noise.

  However, if the calculation base axis of each physical qubit is arbitrarily changed, the effective noise symmetry is not lost for any stabilizer code, the calculation base axis of each changed physical qubit and Depending on the specifically adopted stabilizer code and the nature of the symmetric noise, the influence of the symmetric noise may not be effectively reduced.

  Therefore, in the present invention, the direction of the calculation base axis of each physical qubit is changed for each physical qubit, and the fidelity in the case of encoding by applying a specific stabilizer code is analytically evaluated to be symmetrical. Only when the reduction of the influence of noise is confirmed, the direction of the calculation base axis of each changed physical qubit is adopted.

  In the present invention, it is possible to perform error correction for noise consisting of a classic mixture of symmetrical noise and random noise using fewer physical qubits than in the past.

The best mode for carrying out the present invention will be described below with reference to the drawings.
[Definition of symbols and terms]
First, symbols and terms are defined.

であることを意味する。 [General symbol]
First, general symbols used below are defined.
i: Imaginary unit.
I: 2 × 2 identity matrix.
α ^* : Value that is complex conjugate with the value α.
β ⁺ : conjugate transpose matrix of matrix β.
(×): Tensor product operator.
β ^{(×) α} : tensor product of α matrices β.
| η |: The magnitude of the vector η.
O (α): Big-o notation (order). F (x) = O (g (x)) for a single variable real function f (x), g (x)

It means that.

であることを意味する。

o (α): Small-o notation. F (x) = o (g (x)) for a single variable real function f (x), g (x)

It means that.

[Quantum bit]
Next, quantum bits are defined. A physical qubit means a quantum two-level system (a system that takes two quantum mechanical states). For example, atomic nuclear spins, electron spins in quantum dots, photon polarization, and the like correspond to physical qubits. In the following, the number of physical qubits used in quantum computation is n (n ≧ 2), and the j [j∈ {1, ..., n}, n ≧ 2] -th physical qubit is denoted as Q _j .
The calculation base axis of the physical qubit means a three-dimensional coordinate axis for defining two quantum mechanical states of each physical qubit. For example, the general measurement axis of spin is the Z axis of the calculation base axis.

  The logical qubit means a basic unit in quantum computation composed of a plurality of physical qubits. In the following, it is assumed that k (k ≧ 1) logical qubits are configured using n physical qubits by a stabilizer code.

[Quantum state]
Next, the quantum state notation will be defined.
Each physical qubit can assume any superposition state corresponding to the binary value of the classical bit. The superposition state means the property that each physical qubit can take a state corresponding to the binary value of the classical bit at the same time. This state is a basis vector indicating each quantum state corresponding to the classical bit value. It is expressed by a linear combination of For example, n physical qubits can take t = 2 ⁿ superposed states of φ ₁ , φ ₂ ,..., Φ _t , and their quantum state φ (state vector) is quantum state φ _1, φ _2, as ..., phi basis vectors indicating _t respectively _{| φ 1>, | φ 2} >, ..., | is expressed by a linear combination of phi _t>.

と表現される。 φ = c ₁ | φ ₁ > + c ₂ | φ ₂ > +… + c _t | φ _t > (c ₁ ² + c ₂ ² +… + c _t ² = 1)… (1)
Here, c ₁ , c ₂ ,..., C _t are coefficients of each quantum state φ ₁ , φ ₂ ,..., Φ _t represented by complex numbers, and the probability that their squares take each quantum state Indicates. Also, | δ> means a column vector (vertical vector) indicating a quantum state, and the basis vectors | φ ₁ >, | φ ₂ >, ..., | φ _t > corresponding to n physical qubits are respectively This is a t-dimensional vector. Furthermore, equation (1) omits the basis vectors.

It is expressed.

また、|φ><φ’|は、以下の行列を意味する。

<Δ | means a row vector (horizontal vector) complex with | δ>, and <φ | is expressed as follows.
<φ | = (c ₁ ^* , c ₂ ^* , ..., c _t ^* )… (3)
If <φ ′ | = (b ₁ ^* , b ₂ ^* ,..., B _t ^* ), <φ | φ ′> means the following inner product.

Moreover, | φ><φ ′ | means the following matrix.

In this embodiment, a general quantum state of n physical qubits is expressed as Φ ^(n), and its state vector is expressed as | Φ ⁽ⁿ⁾ >. In particular, a state vector which is a column vector indicating the quantum state Φ _L ^(k) of n physical qubits encoded into k logical qubits by a stabilizer code is denoted as | Φ _L ^(k) > The complex conjugate row vector is expressed as <Φ _L ^(k) |. A general density matrix of n physical qubits is denoted as a matrix ρ.

[Operation gate]
Next, an operation gate for the physical qubit is defined.
X: Pauli that transforms the quantum state of _one qubit whose quantum state is h ₁ | 0> + h ₂ | 1> (h ₁ and h ₂ are complex numbers) into h ₂ | 0> + h ₁ | 1> X gate. When the Pauli X gate is expressed as a matrix (Pauli matrix), it is as follows (the matrix expression of the Pauli X gate is referred to as a Pauli X matrix).

Ｚ：量子状態がh₁|0>+h₂|1>である１つの量子ビットの量子状態をh₁|0>-h₂|1>に変換するパウリＺゲート。パウリＺゲートを行列表現すると以下のようになる（パウリＺゲートの行列表現をパウリＺ行列と呼ぶことにする）。 Y: Pauli Y gate that converts the quantum state of _one qubit whose quantum state is h ₁ | 0> + h ₂ | 1> into -i · h ₂ | 0> + i · h ₁ | 1>. The matrix representation of the Pauli Y gate is as follows (the matrix representation of the Pauli Y gate is referred to as the Pauli Y matrix).

Z: A Pauli Z gate that converts the quantum state of _one qubit whose quantum state is h ₁ | 0> + h ₂ | 1> into h ₁ | 0> −h ₂ | 1>. The matrix representation of the Pauli Z gate is as follows (the matrix representation of the Pauli Z gate is referred to as the Pauli Z matrix).

I：無変換を示す。これを行列表現すると以下の2×2の単位行列になる。

I: Indicates no conversion. When this is expressed in a matrix, the following 2 × 2 unit matrix is obtained.

[Stabilizer code]
A stabilizer code is a state vector indicating the quantum state of a physical qubit encoded into a logical qubit, which is stabilized by each element of a stabilizer (stabilizer: fixed subgroup) that is a set of operators shown below. This means a technology that enables logical error correction to random noise by configuring logical qubits to extend vector space (for example, Non-Patent Document 1, Nielsen and Chuang, "Quantum Computation and Quantum Information", Cambridge University) See Press etc.). This stabilizer code enables quantum error correction for random noise. The outline of the stabilizer code is shown below.

First, define all n-fold tensor products consisting of combinations of Pauli matrices X, Y, Z with coefficient factors ± 1, ± i and unit matrix I as Pauli groups G _n that work on n physical qubits To do. This shows the total noise generated in n physical qubits.
G _n ≡ {σ ₁ (×) σ ₂ (×)… (×) σ _n | σ _j ∈ {± I, ± iI, ± X, ± iX, ± Y, ± iY, ± Z, ± iZ}, j∈ {1, ..., n}}… (6)

A commutative group formed by the entire noise that is effectively non-converted in a given code space CS is called a stabilizer. That is, for all | φ> ∈CS, the eigenvalue is 1 (g _r | φ> = | φ>), and nk generators g _r (r∈r) of the mutually independent and commutative Pauli group G _n A subgroup S∈G _n consisting of {1, ..., nk}) is called a stabilizer. As described above, n represents the number of physical qubits, and k represents the number of logical qubits. Furthermore, the generators g _r ∈ S are independent from each other. A vector that is an eigen vector for all S generators other than the generator g _r ∈ S is an eigen vector for the generator g _r ∈ S. The relationship that does not hold holds for all r∈ {1, ..., nk}.

Further, since the generators of the stabilizer S are independent from each other, generators other than the generator (other generators of the stabilizer S) cannot be expressed only by the product of the generators of the stabilizer S. However, the other of the stabilizer S by the product of one or more generators of the stabilizer S and any one of the complements of the stabilizer S in the Pauli group G _n (except for the element consisting of the n-fold tensor product of I) It is possible to represent the origin of That is, for the generators g _r and g _μ ∈ S of the stabilizer S

を満たす、パウリ群G_nにおけるスタビライザーSの補集合の元g(r)∈G_n-Sが存在する。そして、すべてのr∈{1,...,n-k}についてそれぞれ式(7)を満たすg(r)を求めた場合に、単位行列I以外の演算子（X,Y,Zの何れか）の数が最も少ないg(r)の当該単位行列I以外の演算子の数をdとする。例えば、この単位行列I以外の演算子の数が最も少ないg(r)が、X(×)Z(×)I(×)…(×)Iであったとするとd=2となる。

There exists an element g (r) ∈G _n -S of the complement set of the stabilizer S in the Pauli group G _n that satisfies When g (r) satisfying equation (7) is found for all r∈ {1, ..., nk}, an operator other than unit matrix I (any of X, Y, Z) Let d be the number of operators other than the unit matrix I of g (r) with the smallest number of. For example, if g (r) having the smallest number of operators other than the unit matrix I is X (×) Z (×) I (×)... (×) I, d = 2.

In addition, as described above, encoding by configuring logical qubits so as to configure a code space CS that defines a stabilizer is called a stabilizer code. A stabilizer code corresponding to n, k, d as described above is referred to as an [n, k, d] stabilizer code. That is, the state vector | Φ _L ^(k) > indicating the quantum state of n physical qubits encoded into k logical qubits by the stabilizer code becomes | Φ _L ^(k) > ∈CS. .

In the case where the logical qubit is configured in this way, if no error has occurred in the physical qubit, the eigenvalue for the state vector | Φ _L ^(k) > is 1 for all generators g _r ∈S ( g _r | Φ _L ^(k) > = | Φ _L ^(k) >). On the other hand, if an error has occurred in any physical qubit, the eigenvalue for the state vector | Φ _L ^(k) > for any generator g _r ∈S is −1 (g _r | Φ _L ^{( k)} > =-| Φ _L ^(k) >). That is, when the state vector | Φ _L ^(k) > is observed at each generator g _r ∈ S of the stabilizer S and all of the observed values are 1, it can be seen that no error has occurred in the physical qubit. If the observed value of -1 is -1, it can be seen that an error has occurred in one of the physical qubits.

If the number of physical qubits in which an error has occurred is equal to or less than d / 2, the physical qubit in which an error has occurred and the error can be identified, and error correction can be performed. This error correction is performed by applying an operation to correct the specified error (Pauli matrix X, Y, Z operation with coefficient factors ± 1, ± i) to the physical qubit where the error occurs. Done. The mapping corresponding to this operation is called a CPTP (completely positive trace preserving) mapping and is expressed as C. C (ρ) means the following.

が成り立つことが言える。なお、このような行列E_q(ρ)の一般表現は以下のようになる。 In addition, the property that a stabilizer code can completely correct an error that has occurred in a finite number of physical qubits due to noise is the result that `` if any noise acts on the physical qubits if the noise is smaller than a certain amount, '' Even so, a decrease in the fidelity of the logical bit is necessarily smaller than a decrease in the fidelity of the physical qubit. That is, E _q (·) is an uncorrelated and uniform operator for each physical qubit due to noise acting on the quantum state Φ ⁽ⁿ⁾ , q is a positive real parameter,

Can be said to hold. The general expression of such a matrix E _q (ρ) is as follows.

を満たし、任意の関数fと2ⁿ×2ⁿの任意のユニタリー行列AUとに対して

For any function f and any 2 ⁿ × 2 ⁿ unitary matrix AU

を満たし、任意の関数fと、特定のスタビライザー符号によってｋ個の論理量子ビットに符号化されたn個の物理量子ビットの量子状態を示す列ベクトルが張るベクトル空間VSの元である要素数2ⁿのベクトルを当該ベクトル空間VSに写像する2ⁿ×2ⁿの任意のユニタリー行列AUと、に対して

を満たす、上記ベクトル空間VS上での積分とする。

The number of elements that is an element of the vector space VS spanned by an arbitrary function f and a column vector indicating quantum states of n physical qubits encoded into k logical qubits by a specific stabilizer code 2 ^For any 2 ⁿ × 2 ⁿ unitary matrix AU that maps ⁿ vectors to the vector space VS,

Which satisfies the above-mentioned integration in the vector space VS.

とする。なお、Hは(h_x, h_y, h_z)を要素とする３次元ベクトルであり、hは３次元ベクトルHの大きさであり、p(h)は

を満たすhの関数である。また、U(H)^(×)nは、n個のユニタリー行列U(H)のテンソル積からなる行列であり、U(H)は、Hに対するユニタリー行列
U(H)=I・cos h+i(H・S /h)sin h …(16)
であり、H・Sは、h_x・X+h_y・Y+h_z・Zを意味する。なお、S=(X, Y, Z)であり、前述のように、ＸはパウリＸ行列であり、ＹはパウリＹ行列であり、ＺはパウリＺ行列である。 [Symmetric noise]
Also, ^let E _q (•) be an uncorrelated and uniform operator for each physical qubit due to symmetric noise acting on the quantum state Φ ⁽ⁿ⁾ , and let the operator E _q (•) act on the density matrix ρ. Result

And H is a three-dimensional vector whose elements are (h _x , h _y , h _z ), h is the size of the three-dimensional vector H, and p (h) is

This is a function of h that satisfies U (H) ^{(×) n} is a matrix composed of tensor products of n unitary matrices U (H), and U (H) is a unitary matrix for H.
U (H) = I ・ cos h + i (H ・ S / h) sin h… (16)
And H · S means h _x · X + h _y · Y + h _z · Z. Note that S = (X, Y, Z), and as described above, X is a Pauli X matrix, Y is a Pauli Y matrix, and Z is a Pauli Z matrix.

  Such a noise is expected to be very likely to occur in a scalable quantum computer when viewed on a time scale sufficiently smaller than the magnitude of the energy of interaction with the heat bath.

〔principle〕
Next, the principle of this embodiment will be described.

  In general, the direction of the calculation base axis of each physical qubit (for example, the Z axis for defining the direction of spin) is set to be the same for all physical qubits. However, when symmetrical noise acts on each physical qubit whose calculation base axis is set in the same direction, the noise acting on each physical qubit is strengthened by synchronizing with each other, and the influence of the noise increases. End up. For this reason, sufficient quantum error correction may not be possible with only the stabilizer code. In this embodiment, the direction of the calculation base axis of each physical qubit that is normally set in the same direction is independently changed for each physical qubit, and the stabilizer code is configured using those physical qubits. To solve.

Changing the calculation base axis of the physical qubit is to change the method of implementation on the physical qubit of the observation, the operation on the logical qubit, and the implementation method of the logical qubit on the physical qubit. Equivalent to. Specifically, first, the same reference calculation base axis is set for each physical qubit, and the j [j∈ {1, ..., n}, n ≧ 2] -th physical qubit Q _j A matrix consisting of tensor products of unitary matrix V _j (j∈ {1, ..., n}), where V _j is a 2 × 2 unitary matrix that indicates the rotation of the calculation base axis (rotation relative to the reference calculation base axis) Is V = V ₁ (×)... (×) V _n . In this case, the observation of the j th physical qubit by the operator indicated by the matrix A is changed to the observation of the j th physical qubit by the operator indicated by the matrix V _j AV _j ⁺ , and is indicated by the matrix A The operation on the jth physical qubit by the operator is changed to the operation on the jth physical qubit by the operator indicated by the matrix V _j AV _j ⁺ . In addition, an operation (for example, C-NOT operation) on two physical qubits Q _j , Q _m [m∈ {1, ..., n}, m ≠ j] indicated by the matrix B is {V _j ( ×) I} {I (× ) V m} B {I (×) V m} + {V j (×) I} + two physical qubit Q _j indicated, Q _m [m ∈ {1, ..., n}, m ≠ j]. Further, the state vector | Φ _L ^(k) > of n physical qubits encoded in the logical qubit is changed to V | Φ _L ^(k) >.

Effectively, such a change does not change the quantum code, and the matrix expressed by Equation (14)
RE _q (ρ) = V ⁺ E _q (VρV ⁺ ) V… (17)
It can be said that it is equivalent to changing to a matrix expressed by. If the noise symmetry is effectively lost by this change, the noise acting on each physical qubit will not be intensified by synchronization, and the influence of the symmetric noise can be kept small.

However, when the unitary matrices V _j and V are arbitrarily set, the effective noise symmetry is not lost for any stabilizer code, but the unitary matrices V _j and V and the stabilizers specifically adopted Depending on the sign and the nature of the symmetric noise, the influence of the symmetric noise may not be reduced.

Therefore, in this embodiment, fidelity (symmetric with respect to the state without quantum error) when the direction of the calculation base axis of each physical qubit is independently changed for each physical qubit and coding is performed by applying a specific stabilizer code. The computational basis of each modified physical qubit is only evaluated analytically by evaluating the fidelity of the state after correcting the quantum error due to characteristic noise and reducing the effects of symmetric noise. Adopt the direction of the axis. Specifically, the order terms in equation (9) are evaluated specifically as a function of V = V ₁ (×)… (×) V _n , and all j∈ {1, ..., n} The unitary matrix V _j and V = V ₁ (×)… (×) V _n are used only when the fidelity is larger than when V _j = I (when V is a unit matrix of 2 ⁿ × 2 ⁿ ). adopt. More specifically,
RE _q (| Φ _L ^(k) ><Φ _L ^(k) |) = V ⁺ E _q (V | Φ _L ^(k) ><Φ _L ^(k) | V ⁺ ) V… (18)
Fidelity

に基づき、ユニタリー行列V_j及びV=V₁(×) …(×)V_nを採用するか否かを判定する。さらに具体的には、スタビライザー符号を[n,k,d] スタビライザー符号とし、h_x=h・sinθ・cosν，h_y=h・sinθ・sinν，h_z=h・cosθとし、

とし、

Based on the above, it is determined whether or not the unitary matrix V _j and V = V ₁ (×)... (×) V _n are adopted. More specifically, the stabilizer code is an [n, k, d] stabilizer code, h _x = h · sinθ · cosν, h _y = h · sinθ · sinν, h _z = h · cosθ,

age,

とし、式（１６）のユニタリー行列U(H)とH=h・H_jとした当該ユニタリー行列U(h・H_j)とユニタリー行列V_jに対してU(h・H_j)= V_j U(H) V_j ⁺を満たす３次元ベクトルをH_jとした場合における、フィデリティ

の値を算出し、当該算出値に基づき、ユニタリー行列V_j及びV=V₁(×) …(×)V_nを採用するか否かを判定する。

And U (h · H _j ) = V _{j for the} unitary matrix U (h · H _j ) and the unitary matrix V _j with unitary matrix U (H) and H = h · H _j in equation (16) Fidelity when a three-dimensional vector satisfying U (H) V _j ⁺ is H _j

Then, based on the calculated value, it is determined whether or not to adopt the unitary matrix V _j and V = V ₁ (×) (×) V _n .

[First Embodiment]
Next, a first embodiment of the present invention will be described.
<Configuration>
FIG. 1 is a block diagram illustrating a functional configuration of a calculation base axis setting device 10 according to the first embodiment.
As shown in FIG. 1, the calculation base axis setting device 10 of this embodiment includes a storage unit 11, an input unit 12, a calculation base axis setting unit 13, a calculation base axis evaluation unit 14, an output unit 16, and a control. Part 17. The calculation base axis setting device 10 executes each process under the control of the control unit 17.
FIG. 2 is a block diagram illustrating a hardware configuration of the calculation base axis setting device 10 of the first embodiment.

  As illustrated in FIG. 2, the calculation base axis setting device 10 of the present embodiment includes a CPU (Central Processing Unit) 10a, an output unit 10b, an output unit 10c, a RAM (Random Access Memory) 10d, and a ROM (Read Only Memory) 10e. And an auxiliary storage device 10f and a bus 10g. The CPU 10a includes a control unit 10aa, a calculation unit 10ab, and a register 10ac, and executes various calculation processes according to various programs read into the register 10ac. The RAM 10d has a program area 10da for storing a predetermined program and a data area 10db for storing various data. The auxiliary storage device 10f includes, for example, a program area 10fa that stores a predetermined program and a data area 10fb that stores various data. The bus 10g connects the CPU 10a, the output unit 10b, the communication unit 10c, the RAM 10d, the ROM 10e, and the auxiliary storage device 10f so that information can be exchanged.

  The CPU 10a writes the program stored in the program area 10fa of the auxiliary storage device 10f in the program area 10da of the RAM 10d in accordance with the read OS (Operating System) program. Similarly, the CPU 10a writes various data stored in the data area 10fb of the auxiliary storage device 10f to the data area 10db of the RAM 10d. The address on the RAM 10d where the program and data are written is stored in the register 10ac of the CPU 10a. The control unit 10ab of the CPU 10a sequentially reads these addresses stored in the register 10ac, reads a program and data from the area on the RAM 10d indicated by the read address, and causes the calculation unit 10ab to sequentially execute the operation indicated by the program. The calculation result is stored in the register 10ac.

  With this configuration, the functional configuration of the calculation base axis setting device 10 illustrated in FIG. 1 is realized. That is, the storage unit 11 in FIG. 1 is one of the data area 10fb in the auxiliary storage device 10f in FIG. 2, the data area 10db in the RAM 10d, the register 10ac in the CPU 10a, other buffer memory or cache memory, or the like. It corresponds to a storage area. Further, the input unit 12 and the output unit 16 in FIG. 1 correspond to the input unit 10b and the output unit 10c in FIG. 2 that are driven under the control of the CPU 10a in which the above-described program is read. Further, the calculation base setting unit 13, the calculation base axis evaluation unit 14, and the control unit 17 in FIG. 1 are each a CPU 10a in which the above-described program is read.

<Processing>
Next, the processing of this embodiment will be described.
In the present embodiment, a process applied when encoding n physical qubits into k (n> k ≧ 1) logical qubits using a specific [n, k, d] stabilizer code will be described.
FIG. 3 is a flowchart for explaining the processing of the first embodiment.

  First, n pieces encoded into k (n> k ≧ 1) logical qubits by a specific [n, k, d] stabilizer code are input to the input unit 12 of the calculation base axis setting device 10 (FIG. 1). Code space information CS for specifying a vector space spanned by a column vector indicating the quantum state of the physical qubits of this and a CPTP map C corresponding to error correction of the [n, k, d] stabilizer code are input, and these Data is stored in the storage unit 11 (step S1). As the code space information CS, for example, a state vector of a physical qubit indicating a logical base vector of a logical qubit encoded by the specific [n, k, d] stabilizer code can be exemplified.

Next, the calculation base axis setting unit 13 sets the j [j∈ {1,..., N}, n ≧ 2] -th physical qubit Q _{j to} the reference calculation base axis common to each physical qubit. A 2 × 2 unitary matrix V _j indicating the rotation of the calculation base axis is set. Note that the calculation base axes of this embodiment are the X axis, the Y axis, and the Z axis in a three-dimensional space.

Each unitary matrix V _j is desirably set independently for each physical quantum bit Q _j . This is because the possibility of reducing the influence of symmetrical noise is generally increased. The setting of the unitary matrix V _j, for example, a random 2 × 2 unitary matrix by n number generation, to them may be performed by each unitary matrix V _j, preset 2 It is also possible to select n unitary matrices from the × 2 unitary matrix and use them as each unitary matrix V _j . Also, for example, the fastest descent method is applied to the loop of steps S2 to S4, the initial matrix of each unitary matrix V _j is arbitrarily set, and each unitary matrix is set according to the update method defined by the fastest descent method. V _j may be updated.

Next, the calculation base axis evaluation unit 14 reads the code space information CS, the CPTP map C, and each unitary matrix V _j set in step S2 from the storage unit 11. Then, the calculation base axis evaluation unit 14 uses these to calculate based on the fidelity shown in Equation (19) (the fidelity of the state after correcting the quantum error due to symmetric noise with respect to the state without the quantum error). It is determined whether or not the unitary matrix V _j set by the base axis setting unit 13 is adopted, and the determination result is output to the control unit 17 (step S3). Hereinafter, the determination method of step S3 is illustrated.

[Determination Method 1 in Step S3]
The calculation base axis evaluation unit 14 in this example employs a unitary matrix V _j having a greater fidelity than when V _j = I for all j∈ {1,..., N}. That is, the matrix V that can reduce the influence of symmetric noise is selected by rotating the calculation base axis according to the matrix V.

Specifically, first, (A) the calculation base axis evaluation unit 14 receives the code space information CS, the CPTP mapping C, and each unitary matrix V _j set in step S2 read from the storage unit 11 as inputs. The value EV (V) of (21) is calculated. (B) Next, the calculation base axis evaluation unit 14 receives the code space information CS, the CPTP map C, and V _j = I (all j∈ {1,..., N}) as inputs, ) Value RV is calculated. (C) Then, the calculation base axis evaluation unit 14 determines to adopt each unitary matrix V _j set in step S2 when the value EV is smaller than the value RV, and performs step when the value EV is equal to or larger than the value RV. It is determined that each unitary matrix V _j set in S _< b> 2 is not adopted, and the determination result is output to the control unit 17.

[Determination method 2 in step S3]
The calculation base axis evaluation unit 14 in this example evaluates each unitary matrix V _j using the fastest descent method. In this case, for example, the code space information CS and the CPTP map C input in step S1 are substituted, and 3n pieces (3n is the degree of freedom of n unitary matrices V _j ) for specifying each unitary matrix V _j. EV (vx ^{(V) is} a function of equation (21) with a vector vx ^(V) = (vx ₁ ^(V) , vx ₂ ^(V) ,..., ^Vx _3n ^(V) ) consisting of elements of ). Then, the first-order partial differential of EV (vx ^(V) ) for each vx ₁ ^(V) , vx ₂ ^(V) , ..., vx _3n ^(V) is expressed as ∂EV (vx) / ∂vx ₁ ^{(V )} , EV (vx) / ∂vx ₂ ^(V) , ..., ∂EV (vx) / ∂vx _3n ^(V) . In this example, the calculation base axis evaluation unit 14 obtains each first-order partial differential value ∂EV (vx) / ∂vx ₁ ^(V) , EV (vx) / at the time of each unitary matrix V _j set in step S2. ∂vx ₂ ^(V) , ..., ∂EV (vx) / ∂vx _3n ^(V) is calculated, and the absolute values of all these first-order partial differential values are within the specified range ( Or, when all the first-order partial differential values are 0), it is determined that each unitary matrix V _j set in step S2 is adopted. Otherwise, each unitary matrix V _j set in step S2 is determined. It determines with not employ | adopting and outputs a determination result to the control part 17. FIG.

If it is determined that each unitary matrix V _j set in step S2 is not adopted, the vector vx ^(V) = (vx ₁ ^(V) , corresponding to each unitary matrix V _j set in step S2 vx ₂ ^(V) , ..., vx _3n ^(V) ) and the corresponding SP ・ (∂EV (vx) / ∂vx ₁ ^(V) , EV (vx) / ∂vx ₂ ^{(V )} , ..., ∂EV (vx) / ∂vx _3n ^(V) ) Each unitary matrix set by subtracting values (SP is a positive step parameter) in step S2 of the next loop It becomes a vector corresponding to V _j (end of the example of the determination method in step S3).

Next, when the determination result output from the calculation base axis evaluation unit 14 is that “the set unitary matrix V _j is not adopted”, the control unit 17 returns the process to step S2. The loop of steps S2 to S4 is executed again. On the other hand, when the determination result output from the calculation base axis evaluation unit 14 indicates that “the set unitary matrix V _j is adopted”, the control unit 17 advances the processing to step S5.
In step S5, each unitary matrix V _j (jε {1,..., N}) finally calculated in step S2 is read from the storage unit 11 and sent to the output unit 16. The output unit 16 These are output (step S5).

<Example of Stein code>
Next, a case where a Steane code (for example, see AM Steane, “Active Stabilization, Quantum Computation, and Quantum State Synthesis”, Phys. Rev. Lett. 78 (1997) 2252, etc.) is used as a stabilizer code is illustrated. . The Stein code is classified as a [7,1,3] stabilizer code. A logical qubit basis vector (| 0 _L >, | 1 _L >) constituted by a Stein code can be expressed as follows using a physical qubit basis vector. In the following description, normalization of the coefficient of the basis vector of each physical qubit is omitted. Also, description of the term of the basis vector of the physical qubit whose coefficient is 0 is omitted.
| 0 _L > = | 0000000> + | 0001111> + | 0110011> + | 0111100> + | 1010101> + | 1011010>
+ | 1100110> + | 1101001>
| 1 _L > = | 1111111> + | 1110000> + | 1001100> + | 1000011> + | 0101010> + | 0100101>
+ | 0011001> + | 0010110>

The quantum state in this case is expressed as follows using a physical qubit basis vector.
Φ _L ^(k) = h ₁ | 0 _L > + h ₂ | 1 _L >
= h ₁ (| 0000000> + | 0001111> + | 0110011> + | 0111100> + | 1010101> + | 1011010>
+ | 1100110> + | 1101001>) + h ₂ (| 1111111> + | 1110000> + | 1001100> + | 1000011> + | 0101010>
+ | 0100101> + | 0011001> + | 0010110>)… (22)
Then, the state vector in this example | Φ _L ^(k)> is a 2 ^{7-dimensional} column vector the coefficients of the basis vectors of the formula (22) (coefficient including 0) and element. Also, code space information CS of this embodiment, for example, logical basis vectors of a logical qubit above | 0 _L>, | showing 1 _L>, respectively, is 2 ^{7-dimensional} state vector of physical qubits.

The generator of the Stein code stabilizer is as follows.
g ₁ = I (×) I (×) I (×) X (×) X (×) X (×) X
g ₂ = I (×) X (×) X (×) I (×) I (×) X (×) X
g ₃ = X (×) I (×) X (×) I (×) X (×) I (×) X… (23)
g ₄ = I (×) I (×) I (×) Z (×) Z (×) Z (×) Z
g ₅ = I (×) Z (×) Z (×) I (×) I (×) Z (×) Z
g ₆ = Z (×) I (×) Z (×) I (×) Z (×) I (×) Z

In the error correction of the Stein code, the quantum state Φ is observed by the six generators of the equation (23), and the observation result (g ₁ | Φ> = δ ₁ | Φ>, g ₂ | Φ> = δ _2). | Φ>, ..., g ₆ | Φ> = δ ₆ | Φ> where δ ₁ , δ ₂ ,…, δ ₆ ) are all 1, it is determined that there is no error and no error correction is performed. . On the other hand, if any of the observation results (δ ₁ , δ ₂ ,..., Δ ₆ ) is −1, the X and Z operators are appropriately actuated accordingly. Specifically, the (1-δ ₃ ) / 2 + (1-δ ₂ ) +2 (1-δ ₂ ) th physical qubit (j = (1-δ ₃ ) / 2 + (1- The Z operator is applied to the physical qubit Q _j ) in the case of δ ₂ ) +2 (1-δ ₂ ), and (1-δ ₆ ) / 2 + (1-δ ₅ ) +2 ( 1−δ ₄ ) th physical qubit (physical qubit Q _{j in} the case of j = (1-δ ₆ ) / 2 + (1-δ ₅ ) +2 (1-δ ₄ )) Make it work. Note that the CPTP map C (ρ) in the case of a Stein code is a function in which R _τ in equation (8) is set as follows.

R _τ = rx _τ1・ rz _τ2 (24)
rx ₀ = 2 ^-3・ (g ₁ + I ^{(×) 7} ) (g ₂ + I ^{(×) 7} ) (g ₃ + I ^{(×) 7} )
rx ₁ = 2 ^-3・ (g ₁ + I ^{(×) 7} ) (g ₂ + I ^{(×) 7} ) (g ₃ -I ^{(×) 7} ) ・ Z (×) I (×) I (×) I (×) I (×) I (×) I
rx ₂ = 2 ^-3・ (g ₁ + I ^{(×) 7} ) (g ₂ -I ^{(×) 7} ) (g ₃ + I ^{(×) 7} ) ・ I (×) Z (×) I (×) I (×) I (×) I (×) I
rx ₃ = 2 ^-3・ (g ₁ + I ^{(×) 7} ) (g ₂ -I ^{(×) 7} ) (g ₃ -I ^{(×) 7} ) ・ I (×) I (×) Z (×) I (×) I (×) I (×) I
rx ₄ = 2 ^-3・ (g ₁ -I ^{(×) 7} ) (g ₂ + I ^{(×) 7} ) (g ₃ + I ^{(×) 7} ) ・ I (×) I (×) I (×) Z (×) I (×) I (×) I
rx ₅ = 2 ^-3・ (g ₁ -I ^{(×) 7} ) (g ₂ + I ^{(×) 7} ) (g ₃ -I ^{(×) 7} ) ・ I (×) I (×) I (×) I (×) Z (×) I (×) I
rx ₆ = 2 ^-3・ (g ₁ -I ^{(×) 7} ) (g ₂ -I ^{(×) 7} ) (g ₃ + I ^{(×) 7} ) ・ I (×) I (×) I (×) I (×) I (×) Z (×) I
rx ₇ = 2 ^-3・ (g ₁ -I ^{(×) 7} ) (g ₂ -I ^{(×) 7} ) (g ₃ -I ^{(×) 7} ) ・ I (×) I (×) I (×) I (×) I (×) I (×) Z
rz ₀ = 2 ^-3・ (g ₄ + I ^{(×) 7} ) (g ₅ + I ^{(×) 7} ) (g ₆ + I ^{(×) 7} )
rz ₁ = 2 ^-3・ (g ₄ + I ^{(×) 7} ) (g ₅ + I ^{(×) 7} ) (g ₆ -I ^{(×) 7} ) ・ Z (×) I (×) I (×) I (×) I (×) I (×) I
rz ₂ = 2 ^-3・ (g ₄ + I ^{(×) 7} ) (g ₅ -I ^{(×) 7} ) (g ₆ + I ^{(×) 7} ) ・ I (×) Z (×) I (×) I (×) I (×) I (×) I
rz ₃ = 2 ^-3・ (g ₄ + I ^{(×) 7} ) (g ₅ -I ^{(×) 7} ) (g ₆ -I ^{(×) 7} ) ・ I (×) I (×) Z (×) I (×) I (×) I (×) I
rz ₄ = 2 ^-3・ (g ₄ -I ^{(×) 7} ) (g ₅ + I ^{(×) 7} ) (g ₆ + I ^{(×) 7} ) ・ I (×) I (×) I (×) Z (×) I (×) I (×) I
rz ₅ = 2 ^-3・ (g ₁ -I ^{(×) 7} ) (g ₅ + I ^{(×) 7} ) (g ₆ -I ^{(×) 7} ) ・ I (×) I (×) I (×) I (×) Z (×) I (×) I
rz ₆ = 2 ^-3・ (g ₄ -I ^{(×) 7} ) (g ₅ -I ^{(×) 7} ) (g ₆ + I ^{(×) 7} ) ・ I (×) I (×) I (×) I (×) I (×) Z (×) I
rz ₇ = 2 ^-3・ (g ₄ -I ^{(×) 7} ) (g ₅ -I ^{(×) 7} ) (g ₆ -I ^{(×) 7} ) ・ I (×) I (×) I (×) I (×) I (×) I (×) Z
However, τ = 1,..., 64, and each R _τ is each matrix product represented by the equation (24) for all combinations of the matrix rx _τ1 and the matrix rz _τ2 described above.

In this case, the function f (H _j ) included in the equation (21) has a maximum value of 28.9333 when H _j = H (corresponding to unitary matrix V _j = I for all j). Therefore, if a unitary matrix V _j other than I is selected for at least any j in step S2, the noise resistance against the quantum code is improved, and a determination is made in step S4 that the matrix V is adopted.

とすることで、上述の式（２１）の値は7.472となる。よって、これらのユニタリー行列V_jを用いてスティーン符号を変更することにより、すべてのjについてV_j=Iである場合に比べ、４倍ほどノイズに対する耐性が向上する。 In step S2,

By doing so, the value of the above equation (21) is 7.472. Therefore, by changing the Stein code using these unitary matrices V _j , the resistance to noise is improved by about four times compared to the case where V _j = I for all j.

[Second Embodiment]
The present embodiment is an application example of the first embodiment, and uses a unitary matrix V _j determined to be adopted by the same processing as that of the first embodiment and adds a process of changing the stabilizer code. Below, it demonstrates centering around difference with 1st Embodiment, and abbreviate | omits description about the matter which is common in 1st Embodiment.

<Configuration>
FIG. 4 is a block diagram illustrating a functional configuration of the calculation base axis setting device 20 of the second embodiment. In FIG. 4, portions common to the calculation base axis setting device 10 of the first embodiment are denoted by the same reference numerals as in FIG. 1.

  As shown in FIG. 4, the calculation base axis setting device 20 of this embodiment includes a storage unit 11, an input unit 12, a calculation base axis setting unit 13, a calculation base axis evaluation unit 14, and a quantum circuit information correction unit 25. And an output unit 16 and a control unit 17. The calculation base axis setting device 20 executes each process under the control of the control unit 17. As in the first embodiment, the calculation base axis setting device 20 of the present embodiment is also a device configured by reading a predetermined program into a known computer, and the quantum circuit changing unit 25 reads the predetermined program. This corresponds to the CPU.

<Processing>
Next, the processing of this embodiment will be described.
Also in this embodiment, a process applied when encoding n physical qubits into k (n> k ≧ 1) logical qubits using a specific [n, k, d] stabilizer code will be described.
FIG. 5 is a flowchart for explaining the processing of the second embodiment.
First, n pieces encoded into k (n> k ≧ 1) logical qubits by a specific [n, k, d] stabilizer code are input to the input unit 12 of the calculation base axis setting device 20 (FIG. 4). Code space information CS for specifying a vector space spanned by a column vector indicating the quantum state of the physical qubits of this and a CPTP map C corresponding to error correction of the [n, k, d] stabilizer code are input, and these Data is stored in the storage unit 11 (step S11).

Further, in this embodiment, the input unit 12 of the calculation base axis setting device 20 is configured by the above-described stabilizer code when V _j = I for all j∈ {1,..., N}. A state vector | Φ> indicating an initial quantum state set in the quantum circuit, and a matrix A and / or two physical qubits indicating an operation (including observation) on one physical qubit Q _j included in the quantum circuit Reference quantum circuit information QC including a matrix B indicating operations for Q _j , Q _m [m∈ {1,..., N}, m ≠ j] is input, and the reference quantum circuit information QC is stored in the storage unit 11 (step S12). Note that the operation on one physical qubit Q _j, 2 a physical qubit Q _j, by the C-NOT operation on Q _m, it is possible to constitute all of the quantum computation gate. In addition, the quantum circuit means a quantum calculation mechanism that applies a plurality of operations in a predetermined order to a plurality of physical qubits set to an initial quantum state.

Next, the process of step S2, S3 of 1st Embodiment is performed (step S13, S14).
Next, when the determination result output from the calculation base axis evaluation unit 14 in step S14 indicates that “the set unitary matrix V _j is not adopted”, the control unit 17 performs the process. Returning to S13, the loop of steps S13 to S15 is executed again. On the other hand, when the determination result output from the calculation base axis evaluation unit 14 indicates that “the set unitary matrix V _j is adopted”, the control unit 17 advances the processing to step S16.

In step S16, the quantum circuit information correction unit 25 uses the unitary matrix V _j set by the calculation base axis setting unit 14 to change the reference quantum circuit information QC to the corrected quantum circuit information RQC. That is, the quantum circuit information correcting unit 25, from the storage unit 11, and the unitary matrix V _j which is set in the calculation base axis setting unit 14 reads a reference quantization circuit information QC, using the unitary matrix V _j, reference quantization The state vector | Φ> indicating the initial quantum state included in the circuit information QC is changed to V | Φ>, the matrix A indicating the operation for one physical quantum bit Q _j is changed to the matrix V _j AV _j ⁺ , and 2 A matrix B indicating an operation on two physical qubits Q _j , Q _m [m∈ {1, ..., n}, m ≠ j],
{V _j (×) I} {I (×) V _m } B {I (×) V _m } ⁺ {V _j (×) I} ⁺
The modified quantum circuit information RQC is generated, and the modified quantum circuit information RQC is stored in the storage unit 11 (step S16).

  Thereafter, the modified quantum circuit information RQC read from the storage unit 11 is sent to the output unit 16, and the output unit 16 outputs the modified quantum circuit information RQC. By setting the initial quantum state and operation gate of the quantum computer based on the modified quantum circuit information RQC output in this way, a quantum computer with high noise resistance can be configured. In addition, as a physical system of a quantum computer that realizes physical qubit, quantum state initialization processing, operation gate, observation, etc., for example, a method using each spin in a liquid (Gershenfield, Chuang, Bulk spin resonance quantum computation, Science, 275; 350, 1997), method using nuclear spin in silicon crystal (BE Kane, A silicon-based nuclear spin quantum computer, Nature 393, 133, 1998), method using electron spin in quantum dot (D Loss and DP DiVincenzo, Quantum computation with quantum dots, Physical Review A 57, 120-126, 1998), method using photons (Y. Nakamura, M. Kitagawa, K. Igeta, In 3-rd Proc. Asia-Pacific Phys. Comf., World Scientific, Singapore, 1988). In addition, the realization method of quantum computers for each physical system is detailed in “M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Chapter 7 Physical Realization”.

Needless to say, the present invention is not limited to the above-described embodiment, and can be appropriately changed without departing from the spirit of the present invention.
Further, when the above-described configuration is realized by a computer, processing contents of functions that each device should have are described by a program. And a processing function is implement | achieved on a computer by running this program with a 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, an error correction function of a quantum computer can be exemplified.

FIG. 1 is a block diagram showing a functional configuration of the calculation base axis setting device of the first embodiment. FIG. 2 is a block diagram illustrating a hardware configuration of the calculation base axis setting device according to the first embodiment. FIG. 3 is a flowchart for explaining the processing of the first embodiment. FIG. 4 is a block diagram illustrating a functional configuration of the calculation base axis setting device of the second embodiment. FIG. 5 is a flowchart for explaining the processing of the second embodiment.

Explanation of symbols

10,20 Calculation base setting device

Claims (8)

A calculation base axis setting device for setting a calculation base axis of physical qubits constituting a quantum error correction code,
A storage unit;
2 × 2 indicating the rotation of the calculation base axis of the j [j∈ {1, ..., n}, n ≧ 2] th physical qubit Q _j with respect to the reference calculation base axis common to each physical qubit set the unitary matrix V _j, and calculates the base axis setting unit for storing the unitary matrix V _j in the storage unit,
Read the unitary matrix V _j set by the calculation base axis setting unit from the storage unit, set the tensor product operator to (×), and use the tensor of the unitary matrix V _j (j∈ {1, ..., n}) The matrix consisting of products is V = V ₁ (×)... (×) V _n, and quantum states of n physical qubits encoded into k (n> k ≧ 1) logical qubits by a stabilizer code. The state vector, which is a column vector indicating, is | Φ _L ^(k) >, the complex conjugate row vector is <Φ _L ^(k) |, the density matrix of n physical qubits is the matrix ρ, and the stabilizer The CPTP map corresponding to the error correction of the code is C, the three-dimensional vector having (h _x , h _y , h _z ) as the element is H, the size of the three-dimensional vector H is h = | H |, and p (h)

Where 2 × 2 unit matrix is I, imaginary unit is i, X is Pauli X matrix, Y is Pauli Y matrix, Z is Pauli Z matrix, and S = (X, Y , Z), h _x・ X + h _y・ Y + h _z・ Z is H ・ S, unitary matrix I ・ cos h + i (H ・ S / h) sin h for H is U (H) , A matrix consisting of tensor products of n unitary matrices U (H) is U (H) ^{(×) n} , the conjugate transpose matrix of matrix β is β ⁺ , q is a positive real number, and symmetric noise E _q (・) is an uncorrelated and uniform operator for each physical qubit by

And the matrix V ⁺ E _q (VρV ⁺ ) V when the calculation base axis is rotated according to the matrix V is the matrix RE _q (ρ),

The number of elements 2 ⁿ that is an element of a vector space VS spanned by an arbitrary function f and a column vector indicating quantum states of n physical qubits encoded into k logical qubits by the stabilizer code For any 2 ⁿ × 2 ⁿ unitary matrix AU that maps the vector of

Fidelity in the case of integration on the vector space VS that satisfies

Based on the calculation base axis evaluation unit that determines whether to adopt the unitary matrix V _j set in the calculation base axis setting unit, and outputs the determination result;
A calculation base axis setting device characterized by comprising: The calculation base axis setting device according to claim 1,
The calculation base axis evaluation unit is
Adopt a unitary matrix V _j where the fidelity is larger than when V _j = I for all j∈ {1, ..., n}.
A calculation base axis setting device characterized by that. The calculation base axis setting device according to claim 1 or 2,
The stabilizer code is
[n, k, d] is a stabilizer code,
The calculation base axis evaluation unit is
h _x = h ・ sinθ ・ cosν, h _y = h ・ sinθ ・ sinν, h _z = h ・ cosθ,

age,

And, a small O-notation and o, the unitary matrix U (H) and H = h · H _j and the corresponding unitary matrix U (h · H _j) and U with respect to unitary matrix V _j (h · H _j) = V _j U (H) Let H _{j be a} three-dimensional vector that satisfies V _j ⁺ , and set the smallest integer greater than or equal to δ

Fidelity

Determine whether to adopt the unitary matrix V _j set in the calculation base axis setting unit based on the calculated value,
A calculation base axis setting device characterized by that. The calculation base axis setting device according to any one of claims 1 to 3,
The calculation base axis setting unit is
Set unitary matrix V _j independently for each physical qubit Q _j ,
A calculation base axis setting device characterized by that. The calculation base axis setting device according to any one of claims 1 to 4,
A state vector | Φ> indicating an initial quantum state set in the quantum circuit configured by the stabilizer code when V _j = I for all j∈ {1, ..., n}, and Matrix A indicating operation on one physical qubit Q _{j included in the} quantum circuit and / or operation on two physical qubits Q _j , Q _m [m∈ {1, ..., n}, m ≠ j] An input unit for receiving input of reference quantum circuit information QC including a matrix B indicating
The unitary matrix V _j set by the calculation base axis setting unit is read from the storage unit, and using the unitary matrix V _j , a state vector | Φ> indicating the initial quantum state included in the reference quantum circuit information QC is obtained. V | Φ>, the matrix A is changed to the matrix V _j AV _j ⁺ , and the matrix B is changed to
{V _j (×) I} {I (×) V _m } B {I (×) V _m } ⁺ {V _j (×) I} ⁺
To generate modified quantum circuit information RQC, and further includes a quantum circuit information modification unit that stores the modified quantum circuit information RQC in the storage unit,
A calculation base axis setting device characterized by that. A calculation base axis setting method for setting a calculation base axis of physical qubits constituting a quantum error correction code,
The calculation base axis setting unit of the quantum error correction coding device is configured to generate a j [j∈ {1, ..., n}, n ≧ 2] -th physical qubit with respect to a reference calculation base axis common to the physical qubits. A calculation base axis setting process of setting a 2 × 2 unitary matrix V _j indicating rotation of the calculation base axis of Q _j and storing the unitary matrix V _j in a storage unit;
The calculation base axis evaluation unit of the quantum error correction encoder reads the unitary matrix V _j set by the calculation base axis setting unit from the storage unit, sets the tensor product operator to (×), and sets the unitary matrix V _j ( The matrix consisting of tensor products of j∈ {1, ..., n}) is V = V ₁ (×)… (×) V _n, and k (n> k ≧ 1) logical qubits by the stabilizer code The state vector, which is a column vector indicating the quantum state of n physical qubits encoded in, is defined as | Φ _L ^(k) > and the complex conjugate row vector as <Φ _L ^(k) |, where n The density matrix of the physical qubits of is the matrix ρ, the CPTP map corresponding to the error correction of the stabilizer code is C, the three-dimensional vector whose elements are (h _x , h _y , h _z ) is H, Let the magnitude of the vector H be h = | H | and p (h)

And the 2 × 2 unit matrix is I, the imaginary unit is i, X is a Pauli X matrix, Y is a Pauli Y matrix, Z is a Pauli Z matrix, and S = (X, Y , Z), h _x・ X + h _y・ Y + h _z・ Z is H ・ S, unitary matrix I ・ cos h + i (H ・ S / h) sin h for H is U (H) , A matrix consisting of tensor products of n unitary matrices U (H) is U (H) ^{(×) n} , the conjugate transpose matrix of matrix β is β ⁺ , q is a positive real number, and symmetric noise E _q (・) is an uncorrelated and uniform operator for each physical qubit by

And the matrix V ⁺ E _q (VρV ⁺ ) V when the calculation base axis is rotated according to the matrix V is the matrix RE _q (ρ),

The number of elements 2 ⁿ that is an element of a vector space VS spanned by an arbitrary function f and a column vector indicating quantum states of n physical qubits encoded into k logical qubits by the stabilizer code For any 2 ⁿ × 2 ⁿ unitary matrix AU that maps the vector of

Fidelity in the case of integration on the vector space VS that satisfies

Based on the calculation base axis evaluation process of determining whether to adopt the unitary matrix V _j set in the calculation base axis setting process, and outputting the determination result,
A calculation base axis setting method characterized by comprising:   A program for causing a computer to function as the calculation base axis setting device according to any one of claims 1 to 5.   A computer-readable recording medium storing the program according to claim 7.

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