JP2005269105A - Method and device for correcting quantum error, and method and device for sharing quantum information - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To share secret quantum information and to detect/correct an error occurring on a quantum line during downloading of the quantum information. <P>SOLUTION: Arbitrary quantum information using qubit of quantum two-state system is block encoded with quantum error correction codes at an information source. A plurality of block qubits constituting one block are distributed to a plurality of persons through a quantum communication line. At the information source, a plurality of entangled auxiliary qubit systems to be used at the time of error correction are constituted and distributed to respective persons through the quantum communication line. Using the tensor product of 2×2 Pauli matrix acting on each qubit as an error detection operator, error of the block qubit and auxiliary qubit due to decoherence on the quantum communication line is detected during downloading of the qubit from the information source to each person and then that error is corrected. <P>COPYRIGHT: (C)2005,JPO&NCIPI

Description

  The present invention relates to a quantum error correction method and correction apparatus, and an information processing method and apparatus for secretly sharing quantum information among a plurality of persons.

  Research on techniques related to quantum error correction and techniques for secretly sharing quantum information is underway. Therefore, first, conventional techniques in these fields will be described. A model in which a quantum two-state system (qubit) undergoes phase relaxation will be described.

[Quantum error correction method]
One of the problems to be overcome for realizing a quantum computer is how to handle errors that occur in quantum data. Quantum mechanical information (state) is usually considered much more fragile than classical information. In addition, it has been shown that it is impossible in principle to make a perfect replica of an arbitrary quantum state, and it is possible to prepare for multiple damages of certain quantum information to prepare for damage due to errors. No (non-patent document 1).

  Therefore, two methods, a quantum error correction code (Non-Patent Document 2) and a fault-tolerant calculation method (Non-Patent Document 3), have been proposed as methods for removing errors in the quantum computer. Here, we first describe the features of quantum error and the above two error elimination methods, and then, as specific examples of quantum error correction codes, Shor's 9-qubit code, DiVincenzo, Shor's 5-qubit code (non-patent Reference 4) is introduced. In addition, fault-tolerant error correction operations are explained using 5-qubit code.

[Error in quantum data]
The following two are considered as causes of data errors in the quantum computer (Non-patent Document 3).
・ Decoherence Quantum computation is easily affected by the interaction with the external environment, and even a slight disturbance in the computer's unexpected state may greatly affect the computation results. Such a phenomenon is called decoherence. The most effective way to prevent decoherence is to isolate the computer's internal system from the external environment, but this has technical limitations.
-Quantum gate execution accuracy Since quantum gates when performing quantum computation are physical elements, there is a limit to execution accuracy. This error accumulates and greatly affects the calculation result.

このような変換は、一般性を失うことなく次のように書き直せる(非特許文献２)。

ただし、環境系Ｅの状態を表す|ａ₀〉_E、|ａ₁〉_E、|ａ₂〉_E、|ａ₃〉_Eは、一般に規格化されておらず、互いに直交し合うとも限らない。|0〉_Q、|1〉_Qはqubit系Ｑの正規直交基底で、qubitの任意の状態は|ψ〉_Q＝ｃ₀|0〉_Q＋ｃ₁|1〉_Qで表される。今後、混乱が生じない場合、添え字Ｅ，Ｑを省略することがある。結合系ＱＥの相互作用は、初期状態が系Ｑと系Ｅの状態の直積(productstate)であっても、ユニタリー発展後は一般に、系Ｑ，Ｅの直積では書き表せない、もつれ合った状態(entangled state)にしてしまう。 Here, we will specifically see how the error acts on the quantum state. Suppose that an external environment system | e> _E interacts with a single qubit | ψ> _Q , resulting in an arbitrary unitary evolution U _QE .

Such conversion can be rewritten as follows without losing generality (Non-Patent Document 2).

However, | a ₀ > _E , | a ₁ > _E , | a ₂ > _E , and | a ₃ > _E representing the state of the environmental system E are generally not standardized and may not be orthogonal to each other. _{| 0> Q, | 1>} Q is orthonormal basis of qubit system Q, any state of the qubit is | represented by _{1> Q | ψ> Q =} c 0 | 0> Q + c 1. In the future, the subscripts E and Q may be omitted if there is no confusion. The interaction of coupled system QE is an entangled state that cannot be expressed by the direct product of systems Q and E after unitary development, even if the initial state is the product state of system Q and system E. entangled state).

ただし、｛σ_x, σ_y,σ_z｝は、qubit系Ｑに作用するPauli行列で、

という基底に対して、

と表現されるとする。これを見ると、1-qubitの任意のデコヒーレンスは、｛Ｉ,σ_x, σ_y, σ_z｝の項に分解できることが分かる。三つのPauli行列は、それぞれ、σ_xがビット反転エラー(|0〉←→|1〉)、σ_zが位相反転エラー(|0〉→−|0〉，|1〉→−|1〉)、σ_y＝iσ_xσ_zがビット、位相エラーの混合を表している。実はこのように任意のデコヒーレンスがPauli行列に分解できる性質は、1-qubitだけでなく複数のqubitについても成立する(非特許文献５)。このことを以下で見てみる。 The unitary development of qubit and environment E can be written as follows.

Where {σ _x , σ _y , σ _z } is a Pauli matrix acting on the qubit system Q,

For the basis

It is assumed that From this, it can be seen that any 1-qubit decoherence can be decomposed into {I, σ _x , σ _y , σ _z } terms. In the three Pauli matrices, σ _x is a bit inversion error (| 0> ← → | 1>) and σ _z is a phase inversion error (| 0> → − | 0>, | 1> → − | 1>) , Σ _y = iσ _x σ _z represents a mixture of bits and phase errors. In fact, the property that an arbitrary decoherence can be decomposed into a Pauli matrix as described above is valid not only for 1-qubits but also for a plurality of qubits (Non-Patent Document 5). Let's look at this below.

ただし、

と表される。｛|μ〉_E｝は系Ｅの正規完全直交基底で、Σ|μ〉_EE〈μ|＝Ｉ_Eとする。このとき、

が成立する。例えば、(2)の場合、

となる。(2)，(3)は、しばしばKraus表現と呼ばれる(非特許文献６)。 Consider a situation where multiple qubits interact with environmental systems and receive correlated errors between qubits. | ψ> _Q is an arbitrary state of the n-qubit system Q, | e> _E is an initial state of the environment system E, and U _QE is an arbitrary unitary transformation that acts on the coupled system QE. _How U _QE works is

However,

It is expressed. {| Μ> _E } is a normal perfect orthogonal basis of the system E, and Σ | μ> _EE <μ | = _IE . At this time,

Is established. For example, in the case of (2)

It becomes. (2) and (3) are often called Kraus expressions (Non-Patent Document 6).

の複素数を係数とする和で書き表される。よって、

として、

ただし、

と表される。従って、n-qubitが環境系から任意の相互作用を受ける場合、エラーは各qubitにσ_x(ビット反転エラー)、σ_z(位相反転エラー)、σ_y＝iσ_xσ_z(ビット、位相エラーの混合)の三種類のどれかの演算子が作用されたものと考えてよい。 A _μ ^Q is an arbitrary 2 ⁿ × 2 ⁿ complex matrix that satisfies (3) and acts on the system Q. In general, any 2 ⁿ × 2 ⁿ complex matrix is a tensor product of Pauli matrices acting on n qubits,

Is expressed as a sum with the complex number of. Therefore,

As

However,

It is expressed. Therefore, when n-qubit receives any interaction from the environment, the error is σ _x (bit inversion error), σ _z (phase inversion error), σ _y = iσ _x σ _z (bit, phase error) It can be considered that one of the three types of operators) is acted upon.

[Removal of quantum error]
Many quantum error correction codes for correcting quantum errors by appropriate encoding have been proposed so far (Non-Patent Document 2). In addition, a fault-tolerant calculation method in which further improvements are added to the quantum error correction code has been proposed (Non-Patent Document 3). Therefore, the characteristics of these two methods will be described.

  The purpose of the quantum error correction code is to reduce or eliminate errors in static scenes such as storage and transmission of quantum data. Block information of k pieces of qubits is encoded with n (> k) pieces of qubits so that errors caused by up to t pieces of qubits in one block can be corrected. Since Shor first announced the 9-qubit code (n = 9, k = 1, t = 1) and proposed the principle of quantum error correction, various codes have been announced.

Until now, many quantum error correction codes that have been proposed basically depend on the following method. The encoded n qubits are combined with the environment system and subjected to arbitrary unitary transformation, and become entangled. Therefore, appropriate partial observations are performed on n qubits to cause state contraction. This partial observation corresponds to, for example, writing and observing n-qubit parity information in an auxiliary qubit prepared separately from the 9-qubit of the block. The observation result represents the error received in the state after contraction, and according to this, unitary transformation equivalent to error correction is applied to n qubits. In this way, in the quantum error correction code proposed by Shor, for the state inside the quantum computer,
Some qubit observations → It is assumed that the process of strong observation of the remaining qubit state contraction can be performed any number of times within the decoherence time.

  As described above, when a plurality of qubits are unitary developed by combining with the environment, each qubit receives an error represented by a Pauli matrix. Quantum error correction code is formulated as a problem that divides the n-qubit vector space into subspaces so that the codeword state and the error codeword state generated by the Pauli matrix are orthogonal to each other. Is done.

  Since being formulated as such a mathematical problem, many quantum error correction codes have been devised. In particular, Calderbank, Shor, and Steane 7-qubit codes (n = 7, k = 1, t = 1) are extended to more general methods of constructing quantum error correction codes from classical error correction codes. It is possible and various quantum codes can be designed as required (Non-patent Document 7). The DiVincenzo, Shor 5-qubit code (n = 5, k = 1, t = 1) constitutes one block as a code that can encode 1-qubit information and remove 1-qubit errors. The number of qubits is the smallest code (Non-Patent Document 4).

The fault-tolerant calculation method is intended to remove and reduce errors in a dynamic scene where quantum data is operated by a gate. More precisely, regarding quantum error correction codes,
1. How to handle errors that occur during error correction operations,
2. How to gate encoded n-qubit blocks without decoding each one,
The purpose is to solve these two problems.

  First, the first problem is explained. In the quantum error correction code, normally, parity information is written in an auxiliary qubit for error removal, and this is observed. If an error occurs in the auxiliary qubit, not only will the parity detection result be incorrect, but the error will also be transmitted to the n-qubit codeword block body as a reaction. In fault-tolerant calculation, the expansion of the reaction caused by this error is suppressed as much as possible, and the error in the block body is designed not to exceed the number of errors that can be removed.

  The second problem has the following meaning. Originally, the quantum computation algorithm is expressed by a combination of gate operation operations on qubits before being encoded into an error correction code. Therefore, when a qubit is encoded into a block, a gate operation must be performed on the block. At this time, if a gate operation is performed across a plurality of qubits in a block, there is a possibility that the error is enlarged. The gate operation must be designed so that errors in the block are kept within a removable range.

The first problem will be explained later using the 5-qubit code as an example. The method for solving the second problem will be omitted because it is very complicated. If Shor is a fault-tolerant calculation using the 7-qubit code of Calderbank, Shor, and Steane, the error rate per quantum gate can be reduced to Ο (1 / log ⁴ s). It was shown that the quantum gate operation can be performed correctly.

[Shor 9-qubit error correction code]
It is possible to correct a 1-qubit error (decoherence) by encoding an arbitrary 1-qubit state into a 9-qubit block with the quantum error correction code that was first announced (Non-Patent Document 2). ).

  First, make the following assumptions: Each of the nine qubits interacts with different environmental systems independently. This is an independent particle approximation often done in physics. In addition, as explained before, the strong observation process of writing 9 qubit information into the auxiliary qubit and observing the auxiliary qubit to contract the 9-qubit state is what happens within the decoherence time. It can be repeated at any time.

とブロック化する。 Encoding is performed as follows. An arbitrary state of 1-qubit | ψ> = c ₀ | 0> + c ₁ | 1>,

And block.

Here, as shown in FIG. 10, it is assumed that an error has occurred in the first qubit | ψ> _Q. If an error occurs in another qubit, the same argument can be made as follows. FIG. 10 shows a situation where, for example, an arbitrary unitary development occurs due to the interaction of the photon | e> _E from the outside with the first | ψ> _Q of nine ions trapped linearly. The state of 1-qubit decoherence at this time is represented by (1).

ただし、次の関係が成立することに注意する。

そこで、9-qubit全体でのデコヒーレンスは、

のように与えられる。 Let's look at the state of decoherence in the entire 9-qubit. In order to improve the prospect of formula deformation, the following eight states are defined.

Note that the following relationship holds.

Therefore, decoherence in the entire 9-qubit is

Is given as follows.

From this, if the environmental system E is successfully contracted to either (| a ₀ > ± | a ₃ >) or (| a ₁ > ± | a ₂ >), σ _x , σ _z , σ _z σ It can be seen that the state can be restored to {| φ ₀ >, | φ ₁ >} by applying _x to the first qubit. However, if the 9-qubits that make up the block are observed directly, the state of the block will be broken. Therefore, in addition to the nine qubits used for encoding, an auxiliary qubit is prepared, and information about errors is written and observed in this auxiliary qubit, thereby reducing the state. The information written in the auxiliary qubit relates to how the error occurs and must be completely independent of the encoded information. As a result, the information encoded by the observation of the auxiliary qubit is not destroyed.

で与えられる。オペレーターの働きを見るために、3-qubitを一組として次の計算をしてみる。

これは、

が|Φ^±〉に対して、固有値が１であることを示している。図１２に、これらの結果をまとめておく。これより、Ｍ₁，…，Ｍ₆が、各3-qubitの組が|Φ^±〉、|Ψ^(1,±)〉、|Ψ^(2,±)〉、|Ψ^(3,±)〉のどれであるかを識別し、Ｍ₇，Ｍ₈が、±の逆転している3-qubitの組を識別することが分かる。 In order to detect an error, eight operators shown in FIG. 11 are prepared (Non-Patent Document 8). The white space represents the identity operator, for example

Given in. To see the operator's work, try the following calculation using 3-qubit as a set.

this is,

Indicates that the eigenvalue is 1 for | Φ ^± >. FIG. 12 summarizes these results. From this, M ₁ ,..., M ₆ are each a set of 3-qubits | Φ ^± >, | Ψ ^{(1, ±)} >, | Ψ ^{(2, ±)} >, | Ψ ^{(3, ±)} >. It can be seen that M ₇ and M ₈ identify a set of 3-qubits that are reversed by ±.

で、固有値Ｍ₇＝−１が飛び出す。9-qubitの状態にＭ₁，…，Ｍ₈を作用させたときの固有値を、次の様に補助qubitに書き込む。

ただし、ここでは、演算子Ｍ_iの固有値もＭ_iと書いている。次に、(Ｍ₁，…，Ｍ₈)の固有値を観測する。もし(Ｍ₁，…，Ｍ₈)＝(0，0，0，0，0，0，1，1)が観測されたなら、(6)の状態は第二項に収縮する。観測結果により、ブロック中の１番目のqubitにσ_zエラーが作用していることが分かる。そこで、１番目のqubitにσ_zを作用させてエラー訂正する。 Consider that M ₁ ,..., M ₈ act on the decohered 9-qubit in (6). For example, if M ₇ is applied to the second term,

Thus, the eigenvalue M ₇ = −1 pops out. The eigenvalues when M ₁ ,..., M ₈ are applied to the 9-qubit state are written in the auxiliary qubit as follows.

However, here, the eigenvalue of the operator M _i is also written as M _i . Next, the eigenvalues of (M ₁ ,..., M ₈ ) are observed. If (M ₁ ,..., M ₈ ) = (0, ₀ , 0, 0, ₀ , 0, ₁ , ₁ ) is observed, the state of (6) contracts to the second term. From the observation results, it can be seen that the σ _z error is acting on the first qubit in the block. Therefore, error correction is performed by applying σ _z to the first qubit.

より、ブロック中の１番目のqubitにσ_zエラーが生じた状態と、２番目のqubitにσ_zエラーが生じた状態は、同じで見分けが付かないことに気づく。しかし、エラー訂正は可能である。つまり、このコードはσ_zのエラー状態が縮退しているのである。このようなコードを特に、縮退コードと呼ぶことがある。 M ₁ ,..., M ₈ are Hermitian operators, and eigenvectors having different eigenvalue pairs are orthogonal to each other. Therefore, it is possible to completely identify an encoding rule in which an error has occurred by a set of eigenvalues. By the way, if you look closely at (5),

Thus, it is noticed that the state in which the σ _z error occurs in the first qubit in the block and the state in which the σ _z error occurs in the second qubit are the same and cannot be distinguished. However, error correction is possible. In other words, this code is a degenerate error state of σ _z . Such a code is sometimes called a degenerate code.

より、１番目、２番目のqubitのビット値をmod 2で足し合わせたもの(パリティ)と考えて良い。そこで、図１４のようにゲート・ネットワークを組めば、補助qubitにＭ₁の固有値を書き込むことが出来る。ただし、図１４で使われているゲートは、controlled-NOT(C-NOT)と呼ばれるもので(図１３)、制御qubitが|1〉のときに限り標的qubitを|0〉←→|1〉反転させる。C-NOTゲートは2-qubit間にもつれ合いを生じさせる基本的なゲートの一つである。 Consider a specific gate network for error correction. For example, when M _{1 is applied} to a 9-qubit block, the eigenvalue is

From this, it can be considered that the bit value of the first and second qubits is added by mod 2 (parity). Therefore, if a gate network is formed as shown in FIG. 14, the eigenvalue of M ₁ can be written in the auxiliary qubit. However, the gate used in FIG. 14 is called controlled-NOT (C-NOT) (FIG. 13), and the target qubit is set to | 0> ← → | 1> only when the control qubit is | 1>. Invert. The C-NOT gate is one of the basic gates that cause entanglement between 2-qubits.

を定義すると、

という関係が得られる。9-qubitの任意の状態を|ψ〉として、

となるので、ブロック中の先頭の６個のqubitについて、それぞれHadamard変換してからパリティ検査すると良い。ゲート・ネットワークは図１５で与えられる。 The eigenvalue of M ₇ should be as follows: Hadamard transform for 1-qubit,

Define

The relationship is obtained. Any state of 9-qubit is defined as | ψ>.

Therefore, it is preferable to perform parity check after Hadamard conversion for the first six qubits in the block. The gate network is given in FIG.

と書き下す。|Ψ_good〉はエラーの生じていない項、|Ψ_bad〉はエラーの生じた項を表す。ここで、物理的な根拠はないが、|Ｒ₀〉_Eが、|Ｒ₁〉_E，|Ｒ₂〉_E，|Ｒ₃〉_Eと直交すると仮定する。これにより、1-qubitのデコヒーレンスの確率が評価しやすくなる。まず、〈Ψ_good|Ψ_bad〉＝０が成立する。さらに、〈Ｒ₀|Ｒ₀〉＝1-pとして、〈Ψ_good|Ψ_good〉＝1-p、〈Ψ_bad|Ψ_bad〉＝pが得られる。これは、各qubitのデコヒーレンスする確率を、古典的な意味で0≦p≦1と考えて良いということである。 Consider again the physical meaning of 9-qubit coding. As mentioned at the beginning, for the sake of simplicity, we think that each qubit interacts independently with a different environmental system. The unitary development of one qubit and the corresponding environment, referring to (1),

Write down. | Ψ _good > represents a term in which no error has occurred, and | Ψ _bad > represents a term in which an error has occurred. Here, although there is no physical basis, it is assumed that | R ₀ > _E is orthogonal to | R ₁ > _E , | R ₂ > _E , and | R ₃ > _E. This makes it easier to evaluate the probability of 1-qubit decoherence. First, <Ψ _good | Ψ _bad > = 0 holds. Furthermore, assuming that <R ₀ | R ₀ > = 1-p, <Ψ _good | Ψ _good > = 1-p and <Ψ _bad | Ψ _bad > = p are obtained. This means that the decoherence probability of each qubit can be considered as 0 ≦ p ≦ 1 in the classical sense.

と書ける。ただし、wt(i)は、i=(i₁,…,i₉)で“0”でない成分の個数(x，y，zのどれかを値として持つ成分の数)としている。また、(7)の三行目の最初の和を取っている部分を|correctable〉、二つ目の和を取っている部分を|uncorrectable〉としている。|uncorrectable〉がエラー訂正不可能な部分であり、この確率振幅が十分に小さいとき、エラー訂正は有効に働くと言える。前の1-qubitあたりのデコヒーレンス確率pを使うと、エラー訂正不可能な確率は、36p²＋Ο(p³)となる。符号化したほうが、しない場合よりも有利になるためには、36p²＜pが成立しなくてはならない。従って、p＜(1/36)のとき、この9-qubit符号化は意味を持つ。 Next, consider the state of decoherence in the entire block (FIG. 16). If the state of the α-th qubit is | ψ _α > and the initial state of the corresponding environmental system is | e _α >,

Can be written. However, wt (i) is the number of components that are not “0” with i = (i ₁ ,..., I ₉ ) (the number of components having any of x, y, and z as values). In addition, the first summing part of the third row in (7) is | correctable>, and the second summing part is | uncorrectable>. | uncorrectable> is the part that cannot be corrected, and when this probability amplitude is sufficiently small, it can be said that error correction works effectively. Using the previous decoherence probability p per 1-qubit, the probability that the error cannot be corrected is 36p ² + Ο (p ³ ). In order for encoding to be more advantageous than not, 36p ² <p must hold. Therefore, this 9-qubit encoding is meaningful when p <(1/36).

[DiVincenzo-Shor 5-qubit code]
This code encodes 1-qubit information and removes any 1-qubit error. It is considered that the number of qubits constituting a block is the smallest (5-qubit). (Non-Patent Document 4).

ただし、

このような符号化をＳ表現と呼ぶことにする。上の式では規格化定数が省かれており、〈ｃ_i|ｃ_j〉＝δ_ijが成立するとする。|ｃ₁〉は、|ｃ₀〉の各qubitにσ_xを作用させて、bit値を反転させたものであることに注意する。 Any 1-qubit state | ξ> = α | 0> + β | 1> is encoded with 5-qubit as follows.

However,

Such encoding will be referred to as S representation. In the above equation, the normalization constant is omitted, and it is assumed that <c _i | c _j > = δ _ij holds. Note that | c ₁ > is the bit value inverted by applying σ _x to each qubit of | c ₀ >.

このコードでは、符号語|ｃ₀〉±|ｃ₁〉と、符号語に上のエラーが作用した状態の合わせて３２個の状態が全て互いに直交し合っており、４個のパリティ検査演算子でエラーが完全に識別できる。このことを以下に順を追って説明する。 I want to make sure that this encoding removes any 1-qubit errors. The error for the i-th qubit (i = 0,1,2,3,4) is written as X _i = σ _x , Y _i = σ _y , Z _i = σ _z . The following 16 errors are considered.

In this code, the code word | c ₀ > ± | c ₁ > and the state in which the above error is applied to the code word, all 32 states are orthogonal to each other, and four parity check operators Can fully identify the error. This will be explained step by step below.

上の、|ｃ'₀〉、|ｃ'₁〉を見ると、どちらも０，１，２，４番目のqubitのパリティが偶であることに気づく。そこで、

という符号化を、Ｌ₃表現と呼ぶことにする。０，１，２，４番目のqubitのパリティを検出する演算子として、

を考えると、|ｃ'₀〉、|ｃ'₁〉はＭ₃を作用させても不変である。 First, the code word of the S expression (| c ₀ > ± | c ₁ >) is invariant to permutation of a 5-qubit circular cycle. Next, Hadamard transformation H is applied to the 0th and 1st qubits of the S expression.

Looking at | c ′ ₀ > and | c ′ ₁ > above, you can see that the parity of the ₀ , ₁ , 2, and 4th qubits is even. there,

The coding that will be referred to as L ₃ expression. As an operator to detect the parity of the 0, 1, 2, 4th qubit,

Therefore, | c ′ ₀ > and | c ′ ₁ > are unchanged even when M ₃ is applied.

Ｌ₀表現を

Ｌ₁表現を

と定める。 In the same manner, the L ₄ expression is

L ₀ expression

L ₁ expression

It is determined.

となる。｛Ｋ₃，Ｋ₄，Ｋ₀，Ｋ₁｝は円循環の関係になっており、Ｓ表現の基底の線形結合|ψ〉＝α(|ｃ₀〉＋|ｃ₁〉)＋β(|ｃ₀〉−|ｃ₁〉)を不変に保つ(固有値が１という意味である)。実は、これらの四つの演算子で１６種類のエラーが識別可能である。 M ₃ , M ₄ , M ₀ and M ₁ are considered based on the basis of the S expression.

It becomes. {K ₃ , K ₄ , K ₀ , K ₁ } have a circular relationship, and the linear combination of the basis of the S expression | ψ> = α (| c ₀ > + | c ₁ >) + β (| c ₀ > − | c ₁ >) is kept unchanged (meaning that the eigenvalue is 1). In fact, these four operators can identify 16 types of errors.

より、

で、Ｋ₁，Ｋ₃の固有値は−１，１である。１６種類のエラーに対する固有値の組は図１７のとおりで、全て異なっているのが分かる。 This can be seen in the following example. Assume that a σ _z ⁽⁴⁾ error occurs in the encoded information | ψ> of the S expression.

Than,

The eigenvalues of K ₁ and K ₃ are −1 and 1. The sets of eigenvalues for the 16 types of errors are as shown in FIG. 17 and are all different.

The parity check operators K ₃ , K ₄ , K ₀ , and K ₁ are all Hermite, and the states of different eigenvalue pairs are orthogonal to each other. Accordingly, the 2 ⁵ = 32 dimensional Hilbert space spanned by 5-qubit is divided into a total of 32 mutually orthogonal one-dimensional subspaces of two codewords and 16 kinds of 1-qubit errors that occur in them. . This code does not degenerate between states with different errors in the codeword. Such a code may be referred to as a non-degenerate code.

[Fault tolerant error detection]
Here, of the two purposes of the fault-tolerant calculation method described above, what to do when an error occurs during an error correction operation is discussed (Non-Patent Document 3). Prior to this, a 5-qubit code will be described as an example, but the present invention can also be applied to other error correction codes.

The errors handled here are due to both decoherence due to the interaction between the qubit and the environment, and quantum gate operation errors. All of these errors were caused by errors of σ _x , σ _y , and σ _z when the quantum gate was executed. Generality is not lost even if it thinks. These errors ignite to other qubits by gate operations that cause entanglement. To see this, consider the case where a σ _z error acts on the target qubit immediately before C-NOT is applied (FIG. 18). In this case, it is equivalent to the σ _z error acting on both the control qubit and the target qubit after C-NOT is applied. This can be interpreted as an error that occurred in 1-qubit propagated to two qubits by a C-NOT gate and expanded.

First, let's consider what kind of inconvenience occurs when parity detection is performed on 5-qubit without any ingenuity. As shown in FIG. 19, it is assumed that the eigenvalue of M ₃ is obtained using only one auxiliary qubit. If an error occurs in the second C-NOT gate, the error is transmitted not only to the auxiliary qubit but also to the three qubits in the block. Since this code can only correct 1-qubit errors, it is beyond the range of errors. Therefore, such a method is not good.

を用意する。ただし、wt(b)は二進数列ｂの全bit値の和で、これをｂのweightと呼ぶ。|Φ〉はweightが偶の基底の重ね合わせである。図２０のように、ブロック|ψ〉と補助qubit|Φ〉の各qubitに対して、C-NOTを作用させる。|Φ〉の中の1-qubitでエラーが発生しても、ブロックに伝わるエラーの数は１個で抑えられる。これは除去可能な範囲である。 Instead of FIG. 19, consider the following method. 4-qubit status, as auxiliary qubit

Prepare. However, wt (b) is the sum of all bit values of the binary sequence b, and is called the weight of b. | Φ> is a superposition of bases whose weight is even. As shown in FIG. 20, C-NOT is applied to each qubit of block | ψ> and auxiliary qubit | Φ>. Even if an error occurs in 1-qubit in | Φ>, the number of errors transmitted to the block can be reduced to one. This is a removable range.

  A mechanism capable of parity detection will be described with reference to FIG. If no error occurs during the parity detection operation, the codeword | ψ> has only two kinds of parity, even or odd, and the auxiliary qubit | Φ> always has an even parity. Even if the parity detection operation is performed with C-NOT as shown in FIG. 20, no entanglement occurs between | ψ> and | Φ>. Therefore, even when | Φ> is observed, the state does not shrink at | ψ>. This can be easily understood by thinking as follows.

でもつれ合いは生じない。|Φ〉の各qubitを｛|0〉,|1〉｝の基底で観測して、|ｂ₀〉(wt(b₀)＝even)を得たとする。これは、|Φ〉が|ｂ₀〉に収縮したことを意味するが、このとき|ψ〉に状態の収縮は起こらない。 First, if both | ψ> and | Φ> have an even parity, the even number of qubits in | Φ> are inverted by | 0> ← → | 1> in four C-NOT operations. Since all terms with an even weight are overlapped with equal weight, the state is unchanged as a whole. Therefore,

No entanglement occurs. Assume that | b ₀ > (wt (b ₀ ) = even) is obtained by observing each qubit of | Φ> on the basis of {| 0>, | 1>}. This means that | Φ> contracted to | b ₀ >, but no state contraction occurs at | ψ>.

となる。補助qubitの観測で|Φ'〉はweightが奇な基底の一つに収縮するが、|ψ〉に収縮は起こらず、何の影響も与えない。 On the other hand, if the parity of | ψ> is odd, the auxiliary qubit | Φ> is a C-NOT operation and the odd number of qubits is inverted by | 0> ← → | 1>'>, But no entanglement,

It becomes. In the observation of the auxiliary qubit, | Φ '> shrinks to one of the odd weight bases, but | ψ> does not shrink and has no effect.

程度である。パリティ検出を誤る確率を０＜t＜１で抑えたいとすると、

より、繰り返し回数は、

と評価できる。logｔというオーダーで考えると非常に効率的といえる。 Even with the method using | Φ> for the auxiliary qubit, there is a case where the parity detection is erroneous, so the majority must be made by repeating the measurement several times. Suppose that a classical probability 0 <ε <1 is given that an error occurs in the auxiliary qubit system during one parity check. The probability of making a majority decision with M detections and getting the wrong parity is

Degree. If we want to suppress the probability of erroneous parity detection with 0 <t <1,

Therefore, the number of repetitions is

Can be evaluated. Considering the order of logt, it can be said that it is very efficient.

  If an error occurs in any one qubit in the block during the parity detection process, there is a possibility that the parity is erroneously detected, but at least the number of errors in the block is suppressed to one, and it is still Error removal is possible. There is no problem if the parity detection is repeated several times and the detection result is determined by majority vote.

  Here, it is assumed that only one error occurs in the entire 5-qubit block and auxiliary qubit. This means that the time at which one error occurs in all qubits is τ, and storage and error correction by a code word must be performed within the time τ. If the quantum information is to be stored for a long time, these operations are repeated with a period τ.

(9)の一行目から二行目への変換で、状態にもつれ合いが生じたことが見て取れる。この際、図２１からも明らかなように、一番目のqubitにエラーが発生すると、残りのqubitにもエラーが伝わってしまう。このように、直積状態(product state)から、もつれ合った状態(entangled state)への変換をフォールト・トレラント化するのは難しい。 Let's consider how to prepare | Φ>. If | Φ> is simply created from the initial state | 0000> by the basic gate operation, the procedure is as follows (FIG. 21).

(9) It can be seen that the transformation from the first line to the second line caused entanglement in the state. At this time, as is apparent from FIG. 21, if an error occurs in the first qubit, the error is also transmitted to the remaining qubits. Thus, it is difficult to make the conversion from the product state to the entangled state fault tolerant.

と思って良い。上の式には任意の相対位相エラーｅ^iδが残っており、各qubitにHadamard変換を施すと、

となるが、再度、ランダムなパリティ検査を行えば、高い確率で|Φ〉を得ることが可能となる。 However, this problem can be avoided by the following method. First, at the stage where the cat state in the second row in (9) is completed, any two qubits are selected and a parity check is performed on the bit value. If the parity is odd, the 4-qubit is discarded and recreated. If you repeat the parity check several times,

You can think. In the above equation, there is an arbitrary relative phase error e ^iδ , and when Hadamard transform is applied to each qubit,

However, if random parity check is performed again, | Φ> can be obtained with high probability.

Finally, FIG. 22 shows a 5-qubit code fault tolerant error correction gate network. The auxiliary qubit for measuring the eigenvalues of M ₃ , M ₄ , M ₀ , and M ₁ is represented by a single line in the figure, but it has been explained so far using 4-qubit | Φ>. Suppose that you observe with the method that you have done. Note also that during operation, the post-sign representation changes to L ₃ , L ₄ , L ₀ , L ₁ . When the four eigenvalues are obtained, the position and type of error are determined by a classical computer or the like, and error correction is performed.

[How to share quantum information in secret]
A method has been proposed in which quantum information is secretly shared among multiple people using a quantum error correction code (Non-patent Document 9). Here, this method is explained concretely by using the 5-qubit code of Shor-DiVincenzo.

で、α，βに関する情報は何も得られない。ただし、Ｉ₄は４×４単位行列とする。 Coding using any 1-qubit state α | 0> + β | 1> (| α | ² + | β | ² = 1) to obtain | ψ> = α | c ₀ > + β | c ₁ > Suppose. Next, it is assumed that five qubits constituting one block are given to five people. In this case, even if two people with the 0th and 1st qubits jointly observe the qubit at their hands,

No information about α and β can be obtained. However, I ₄ is a 4 × 4 unit matrix.

が成立し、五人の内の任意の二人が共同で観測しても、|ψ〉に関する情報は一切得られない。また、上の結果より、明らかにｉ番目のqubitの密度演算子は、

ただしＩ₂は２×２単位行列となり、一人が手元のqubitを観測しても何も情報は得られない。 In fact, in this case

Even if any two of the five members observe together, no information about | ψ> is obtained. From the above result, the density operator of the i-th qubit is clearly

However, I ₂ is a 2 × 2 unit matrix, and no information can be obtained even if one person observes the qubit at hand.

  However, conversely, if any three or more of the five perform joint operations, the original quantum information can be completely restored. The feature of this method is that the original information cannot be obtained at all or can be completely obtained depending on the number of people who cooperate to extract the information. Such secret sharing of quantum information by the 5-qubit code is called ((3, 5)) threshold scheme.

  The ((3, 5)) threshold scheme means that even if any 1-qubit is discarded, the original encoded information can be restored with the remaining 4-qubit. Such a state gives a ((3, 4)) threshold scheme. The previous ((3, 5)) threshold scheme encoded quantum information given in the pure state of 1-qubit into the pure state of 5-qubit, whereas in the ((3, 4)) threshold scheme, 4- It will be encoded into a mixed state of qubit.

に収縮する。確率1/2で|0〉が得られた場合、

に収縮する。従って、1-qubitの量子情報α|0〉＋β|1〉は、

に符号化される。 ((3,4)) Let's take a closer look at the threshold scheme. When the 0th qubit in (8) is observed and | 0> is obtained with probability 1/2, the codeword is

Shrink to. If | 0> is obtained with probability 1/2,

Shrink to. Therefore, 1-qubit quantum information α | 0> + β | 1>

Is encoded.

で与えられる。しかし、｛|φ₀〉，|φ₁〉｝を使った符号、|ψ'〉＝α|φ₀〉＋β|φ₁〉では、

となり、((3，4))threshold schemeを実現しない。

をエラー訂正コードとして見ると、次のようになる。符号語の部分空間を不変にする演算子は、

の三つで、エラーの作用した符合語に対する固有値は図２３で与えられる。これにより、((3，4))threshold schemeは1-qubitのエラー検出が可能と分かる。 For example, for the above ρ, the leading 2-qubit density operator is

Given in. However, in the code using {| φ ₀ >, | φ ₁ >}, | ψ ′> = α | φ ₀ > + β | φ ₁ >

And ((3,4)) threshold scheme is not realized.

As an error correction code, it is as follows. The operator that makes the subspace of the codeword invariant is

In FIG. 23, the eigenvalues for the code word affected by the error are given in FIG. This shows that the ((3, 4)) threshold scheme can detect 1-qubit errors.

[Phase relaxation model]
In this specification, a case is considered in which phase relaxation (phase error) occurs in the qubit due to the interaction between the qubit system and the environment system in a quantum communication line or the like. Therefore, a model in which such an error occurs is shown.

ただし、

は輻射場の生成、消滅演算子とする。上の式より、［σ_z，Ｈ］＝０は明らかで、qubit系において位相のデコヒーレンスのみ起こると分かる。これは次のように考えれば良い。1-qubitの密度行列は、一般に、

ただしａ_i(i=x,y,z)は実数、と書ける。Ｉ，σ_zの項はＨで引き起こされるユニタリー変換、

で不変であり、このモデルではσ_x，σ_yの項のみ変化が引き起こされる。これにより、位相緩和(デコヒーレンス)のみ生じ、密度行列の対角成分の変化である散逸は起こらない。時刻t=0でのqubit系Ｑと環境系Ｒの密度行列を、

ただし、

とする。これは、環境系Ｒとして、熱平衡状態にある輻射場を考えていることに相当する。 As the Hamiltonian of the qubit system Q and the environment system (heat bath) R, the following is considered (Non-patent Document 10).

However,

Is an operator for generating and annihilating the radiation field. From the above equation, [σ _z , H] = 0 is clear, and it can be seen that only phase decoherence occurs in the qubit system. This can be considered as follows. The 1-qubit density matrix is generally

However, a _i (i = x, y, z) can be written as a real number. I, σ _z terms are unitary transformations caused by H,

In this model, only the terms σ _x and σ _y are changed. As a result, only phase relaxation (decoherence) occurs, and dissipation that is a change in the diagonal component of the density matrix does not occur. The density matrix of qubit system Q and environment system R at time t = 0

However,

And This is equivalent to considering a radiation field in a thermal equilibrium state as the environmental system R.

ただし、Ｐは時間順序演算子で、

とする。計算により、

ただし、

が得られる。 Such a system is easy to handle if an interaction display is taken.

Where P is a time order operator and

And By calculation

However,

Is obtained.

で与えられる。(10)の結果を代入すると、

ただし、

となる。これは、次のように書き換え可能である、

ただし、

としている。これより、qubit系がσ_zエラーのみ受けていることが分かる。
W.K.Wootters, W.H.Zurek, 'A single quantum cannot be cloned', Nature, 299, p.802-803(1982) P.W.Shor, 'Scheme for reducing decoherence in quantum computer memory', Phys. Rev. A 52, 2493(1995) P.W.Shor,'Fault-tolerant quantum computation',Proc.37th Annual Symposium on Foundations of Computer Science,IEEE Computer Soc.Press,1996,pp.56-65 D.P.DiVincenzo and P.W.Shor, 'Fault-Tolerant Error Correction with Efficient Quantum Codes', Phys. Rev.Lett. 77, 3260(1996), C.H.Bennett, D.P.DiVincenzo, J.A.Smolin and W.K.Wootters, 'Mixed-state entanglement and quantum error correction', Phys. Rev.A 54,3824(1996) A.Ekertand C.Macchiavello, 'Error Correction for Quantum Communication', Phys.Rev.Lett.77,2585(1996),J.Preskill and A.Kitaev, lecture notes §7, CaliforniaInstitute ofTechnology,(1997-1999),http://www.theory.caltech.edu/people/preskill/ph229/ B.Schumacher, 'Sebding entanglement through noisyquantum channel', Phys. Rev.A 54, 2614(1996) A.R.Calderbank and P.W.Shor, 'Good quantum error-correcting codes exist', Phys. Rev.A 54,1098(1996), A.Steane,'Multiple particle interference and quantum error correction', Proc.R.Soc.Lond.A 452,2551(1996), A.Steane, 'Simple quantum error-correcting codes', Phys. Rev.A 54,4741(1996) D.Gottesman，Ph.D.thesis，California Institute of Technology，LANL quantum Physics archive quant-ph/9705052 R.Cleve, D.Gottesman and H-K.Lo, 'How to Share a Quantum Secret', Phys. Rev. Lett. 83,648(1999), J.Preskill and A.Kitaev,lecture notes §7, California Institute of Technology,(1997-1999), http://www.theory.caltech.edu/people/preskill/ph229/ G.M.Palma, K-A.Suominen and A.K.Ekert, Quantum Computers and Dissipation', LANL quantum physics archive quant-ph/9702001 The density matrix of qubit at time t is

Given in. Substituting the result of (10)

However,

It becomes. This can be rewritten as

However,

It is said. From this, it can be seen that the qubit system receives only σ _z error.
WKWootters, WHZurek, 'A single quantum cannot be cloned', Nature, 299, p.802-803 (1982) PWShor, 'Scheme for reducing decoherence in quantum computer memory', Phys. Rev. A 52, 2493 (1995) PWShor, 'Fault-tolerant quantum computation', Proc. 37th Annual Symposium on Foundations of Computer Science, IEEE Computer Soc. Press, 1996, pp. 56-65 DPDiVincenzo and PWShor, 'Fault-Tolerant Error Correction with Efficient Quantum Codes', Phys. Rev. Lett. 77, 3260 (1996), CHBennett, DPDiVincenzo, JASmolin and WKWootters, 'Mixed-state entanglement and quantum error correction', Phys. Rev. A 54,3824 (1996) A. Ekertand C. Macchiavello, 'Error Correction for Quantum Communication', Phys. Rev. Lett. 77, 2585 (1996), J. Preskill and A. Kitaev, lecture notes §7, California Institute of Technology, (1997-1999), http://www.theory.caltech.edu/people/preskill/ph229/ B. Schumacher, 'Sebding entanglement through noisyquantum channel', Phys. Rev. A 54, 2614 (1996) ARCalderbank and PWShor, 'Good quantum error-correcting codes exist', Phys. Rev. A 54, 1098 (1996), A. Steane, 'Multiple particle interference and quantum error correction', Proc. R. Soc. Lond. A 452 , 2551 (1996), A. Steane, 'Simple quantum error-correcting codes', Phys. Rev. A 54,4741 (1996) D. Gottesman, Ph.D. thesis, California Institute of Technology, LANL quantum Physics archive quant-ph / 9705052 R. Cleve, D. Gottesman and HK.Lo, 'How to Share a Quantum Secret', Phys. Rev. Lett. 83,648 (1999), J. Preskill and A. Kitaev, lecture notes §7, California Institute of Technology, (1997-1999), http://www.theory.caltech.edu/people/preskill/ph229/ GMPalma, KA.Suominen and AKEkert, Quantum Computers and Dissipation ', LANL quantum physics archive quant-ph / 9702001

  When using a quantum error correction code to share secret information among multiple people, select how many people will share the secret and how many people will need to cooperate to restore the information completely. The encoding method to be different is different.

Also, when considering the procedure until information is shared, it is assumed that block coding is performed by an information source that emits secret information, and each qubit constituting the block is distributed to each person through a quantum communication line. In an actual quantum line, decoherence occurs due to interaction with the environment, and the delivered qubit receives an error. Therefore, it is desirable to perform error detection and correction when the qubit is delivered.
An object of the present invention is to provide an encoding method for sharing secret quantum information and a method for detecting and correcting an error received in a quantum line for some special cases.

In order to achieve the above object, a quantum error correction method according to the present invention comprises:
Arbitrary quantum information using qubit that is a quantum two-state system is block-coded with a quantum error correction code at an information source, and a plurality of blocks qubit constituting one block are distributed to a plurality of people through a quantum communication line.
In the information source, a plurality of tangled auxiliary qubit systems used for error correction are configured, and these are distributed to each person through the quantum communication line,
The block due to decoherence in a quantum communication line during delivery of a qubit from the information source to each person using a tensor product of 2 × 2 Pauli matrix acting on each qubit as an error detection operator of the quantum error correction code Detect and correct qubit and auxiliary qubit errors.

In addition, the quantum error correction device of the present invention that achieves the above-described object,
The first distribution in which arbitrary quantum information using qubit, which is a quantum two-state system, is block-coded with a quantum error correction code at an information source, and a plurality of blocks qubit constituting one block are distributed to a plurality of people through a quantum communication line Means,
In the information source, a plurality of entangled auxiliary qubit systems used in error correction are configured, and the second distribution means distributes these to each of the plurality of persons through the quantum communication line;
The block due to decoherence in a quantum communication line during delivery of a qubit from the information source to each person using a tensor product of 2 × 2 Pauli matrix acting on each qubit as an error detection operator of the quantum error correction code Detecting / correcting means for detecting and correcting qubit and auxiliary qubit errors.

In addition, the quantum information sharing method according to the present invention for achieving the above object is as follows:
Arbitrary quantum information using qubit that is a quantum two-state system is block-coded with a quantum error correction code at an information source, and a plurality of blocks qubit constituting one block are distributed to a plurality of people through a quantum communication line.
In the information source, a plurality of tangled auxiliary qubit systems used for error correction are configured, and by distributing these to each person through the quantum communication line, arbitrary quantum information is shared by a plurality of persons,
The block qubit and the auxiliary qubit use a tensor product of 2 × 2 Pauli matrix acting on each qubit as an error detection operator of the quantum error correction code, so that a quantum between delivering the qubit from the information source to each person is used. It is possible to detect and correct errors due to decoherence in a communication line.

Furthermore, a quantum information sharing apparatus according to the present invention for achieving the above object is as follows.
The first distribution in which arbitrary quantum information using qubit, which is a quantum two-state system, is block-coded with a quantum error correction code at an information source, and a plurality of blocks qubit constituting one block are distributed to a plurality of people through a quantum communication line Means,
The information source comprises a plurality of tangled auxiliary qubit systems used in error correction, and a second distribution means for distributing them to each of the plurality of persons through the quantum communication line,
The block qubit and the auxiliary qubit use a tensor product of 2 × 2 Pauli matrix acting on each qubit as an error detection operator of the quantum error correction code, so that a quantum between delivering the qubit from the information source to each person is used. It is possible to detect and correct errors due to decoherence in the communication line.

  According to the present invention, it is possible to share secret quantum information and to detect and correct an error received on the quantum line during the distribution of the quantum information.

  Hereinafter, preferred embodiments of the present invention will be described with reference to the accompanying drawings.

  In the following first embodiment, four people share secrets, one person can obtain nothing about quantum information, two people can obtain only partial information, and three or more can share information. An encoding method that can be completely restored will be described. Also, in the second embodiment, 6 people share a secret, and 2 or less people get nothing about quantum information, and 3 people can get the original information completely by the combination of cooperating people The coding method can be divided into either one of them, which cannot be obtained at all, and the information can be completely restored when there are four or more people. Further, in each embodiment, an operator for detecting an error in these codes and an eigenvalue when these operators are applied to a code word that has received an error are checked, and error detection and correction are performed. Configure a gate network to be fault tolerant. The entangled state of a plurality of qubits used as auxiliary qubits in the gate network is composed of information sources, and each qubit is delivered to each person through a quantum communication line. For this reason, both the block qubit and the auxiliary qubit receive an error in the middle of the line, but errors are detected and corrected as a whole.

  As described above, when error detection and correction are performed when qubits are delivered to each person through the quantum communication line, for example, two people bring each other's qubits and perform unitary transformation across 2-qubits. It is not possible to cause entanglement between 2-qubits. Multiple people who have qubits are allowed only two types of operations: communicating through classical communication lines and performing arbitrary unitary transformations on each person's qubit. If the fault-tolerant error correction described above is performed, the error can be detected and corrected in such a situation.

(First embodiment)
The first embodiment describes a quantum error correction method and correction apparatus for sharing quantum information among four persons, and a method and apparatus for secretly sharing quantum information among a plurality of persons.

  That is, an arbitrary quantum information input means, storage means, calculation means, and output means using a quantum two-state system (qubit) are prepared, and arbitrary quantum information is block-coded with a quantum error correction code, and one block By distributing multiple quantum two-state systems (qubits) that make up qubits to multiple people, when the original quantum information is shared and stored secretly, the quantum communication line between delivering the qubits from the information source to each person When detecting and correcting errors due to decoherence in the network, and removing errors due to decoherence in the quantum communication line during the delivery of qubits from the information source to each person, multiple people are contacted using the classical communication line However, each unit has a fault-tolerant property that can be detected and corrected by applying arbitrary unitary transformations to the qubits held by each person. A quantum error correction method and a correction apparatus are disclosed, which corrects this when receiving a signal, and detects a phase error when one qubit in an auxiliary qubit receives a phase error. .

  Further, in the above configuration, as a quantum error correction code for block encoding arbitrary quantum information, 1-qubit is encoded with 4-qubit, and each qubit is distributed to four people to share secrets, One person can obtain nothing about quantum information, two can obtain only partial information, and three or more can fully restore information. A secret sharing method and apparatus will be described.

ただし、規格化定数は省略しており、〈ｃ_i|ｃ_j〉＝δ_ijとする。この4-qubitコードは、エラーがσ_zタイプに限られるなら、1-qubitエラーが訂正可能である。しかし、任意のエラーが発生する状況では、1-qubitのσ_xエラーが検出可能であるのみで、1-qubitのσ_yエラーとσ_zエラーを区別できない。従って、このコードは位相エラーのみ発生する系では有効と考えられる。さらに、このコードには、以下で議論するように、いくつかの特徴が見られる。 Consider the following 4-qubit code:

However, the normalization constant is omitted, and <c _i | c _j > = δ _{ij is} assumed. This 4-qubit code can correct a 1-qubit error if the error is limited to the σ _z type. However, in a situation where an arbitrary error occurs, only the 1-qubit σ _x error can be detected, and the 1-qubit σ _y error cannot be distinguished from the σ _z error. Therefore, this code is considered effective in a system in which only a phase error occurs. In addition, the code has several features, as discussed below.

ただし、

これより、｛|ｃ₀〉，|ｃ₁〉｝を不変にする演算子が三つ見付かったことになる。Ｋ_iとエラーを表すPauli行列との交換、反交換関係を調べることにする。

これより、例えば、符号化された任意の状態|ψ〉＝α|ｃ₀〉＋β|ｃ₁〉の０番目のqubitに位相エラーσ_z ⁽⁰⁾が作用した場合、Ｋ₀|ψ〉＝|ψ〉に注意すると、

で、固有値−１が得られる。図２にエラーとＫ₀，Ｋ₁，Ｋ₂の固有値の関係をまとめておく。これを見ると、1-qubitのσ_xタイプのエラーが検出可能なこと、また、σ_zとσ_yのエラーの区別が付かないことが分かる。これは、

という事情による。 First, we will investigate in detail what errors can be corrected. Focus on the following relationship:

However,

From this, three operators that make {| c ₀ >, | c ₁ >} invariant have been found. Let us examine the exchange and anti-exchange relations between K _i and Pauli matrices representing errors.

Thus, for example, when the phase error σ _z ⁽⁰⁾ acts on the ^0th qubit of an encoded arbitrary state | ψ> = α | c ₀ > + β | c ₁ >, K ₀ | ψ> = Note that | ψ>

Thus, the eigenvalue −1 is obtained. FIG. 2 summarizes the relationship between the error and the eigenvalues of K ₀ , K ₁ , and K ₂ . From this, it can be seen that 1-qubit σ _x type errors can be detected, and that σ _z and σ _y errors cannot be distinguished. this is,

It depends on the circumstances.

で与えられる。これは、4-qubit中の１個のqubitだけを観測しても、量子情報(α，βについての情報)を全く得ることが出来ないことを意味する。 This 4-qubit code is incomplete in terms of error correction, but has good properties in terms of secret sharing. Consider observing any 1-qubit in a 4-qubit code. If the 4-qubit density operator is ρ = | ψ><ψ |, where | ψ> = α | c ₀ > + β | c ₁ >, | α | ² + | β | ² = 1, The 1-qubit density operator is

Given in. This means that even if only one qubit in 4-qubit is observed, quantum information (information about α and β) cannot be obtained at all.

また、

で、二人の観測者は|α|²，|β|²を統計的な情報として得ることが出来る。０番目と２番目のqubitを観測する場合、

また、

で、二人の観測者はIm(αβ^*)を統計的な情報として得ることが出来る。特に、もともとの情報がα＝０またはβ＝０の古典情報だった場合、ρ^(0,1)，ρ^(0,3)，ρ^(2,3)，ρ^(1,2)は観測によって完全に知ることが出来る。 However, when two qubits are observed, the situation is different. When observing the 0th and 1st qubit,

Also,

Thus, the two observers can obtain | α | ² and | β | ² as statistical information. When observing the 0th and 2nd qubits,

Also,

Thus, the two observers can obtain Im (αβ ^* ) as statistical information. In particular, if the original information was classical information with α = 0 or β = 0, ρ ^(0,1) , ρ ^(0,3) , ρ ^(2,3) , ρ ^(1,2) You can know completely.

と変換され、０番目のqubitに元々の量子情報が復元される。 Therefore, if arbitrary quantum information is encoded with this 4-qubit code and each of the four people has 1-qubit, secrets can be shared. Even if one person observes the quantum state without the permission of the other three, no secret information can be obtained. This also means that information can be retrieved with the consent of the three people. Moreover, if two people agree, information can also be partially destroyed. FIG. 3 shows a decoding procedure by three persons having the 0th, 1st and 2nd qubits. In the middle of the network, a gate called a controlled-controlled σ _z (CC-σ _z ) gate that applies σ _z to the target qubit is used only when the two controlled qubits are both | 1>. This

And the original quantum information is restored to the 0th qubit.

では、|ψ〉＝α|φ₀〉＋β|φ₁〉、|α|²＋|β|²＝１、ρ＝|ψ〉〈ψ|とすると、1-qubitの密度演算子は、

で与えられる。これは、一人が手元のqubitを観測しても、何らかの情報が得られることを示している。ただし、元の情報が古典的であるならば、秘密は守られる。 This kind of property that can be used for secret sharing is unique to this code in 4-qubit. Even with the same 4-qubit code

Then, if | ψ> = α | φ ₀ > + β | φ ₁ >, | α | ² + | β | ² = 1, ρ = | ψ><ψ |, the density operator of 1-qubit is

Given in. This shows that even if one person observes the qubit at hand, some information can be obtained. However, if the original information is classic, the secret is preserved.

と符号化される。 Here, the secret sharing procedure will be considered more specifically (FIG. 5). Suppose that a certain information source provides expensive information that you want to keep secret. If the information is quantum information, it is impossible to make a complete copy, and it is immediately encoded into 4-qubit as it is. For example, the gate network shown in FIG. As a result, any state α | 0> + β | 1> of 1-qubit is

Is encoded.

Next, four qubits are distributed to four people through quantum communication lines. Here, a phase error (σ _z error) occurs in the line. However, in the information source, the encoding device, and the device that operates four qubits, no error occurs (or the error occurrence probability is negligible). (Small) Think about the situation.

  Immediately after the delivery of qubit, the four must work together to correct the error. At this time, the four people can unitary transform their qubit and send classical information to each other, but operations that cause entanglement between two qubits of different people, for example, the qubit of two people Suppose you can't do C-NOT. This is the same as fault-tolerant error correction.

とする。ここで、

という関係を利用している。 FIG. 1 shows a network for error correction of this code in a fault tolerant manner. However,

And here,

Is used.

It is considered that the four people A, B, C, and D have the 0, 1, 2, and 3rd qubits. The auxiliary qubit for writing the eigenvalue of M _i also uses a superposition | Φ> of weights consisting of a plurality of qubits, and each qubit is shared by A, B, C, and D. If | Φ> is also created from information sources and distributed to each person using a line as shown in FIG. 5, the error that occurs in the auxiliary qubit can be considered as the σ _z type.

Here, it is assumed that the σ _z error occurs only in 1-qubit of the total of 4-qubit and auxiliary qubit. Consider the error correction procedure. If a σ _z error occurs in the auxiliary qubit during the parity check of M ₀ and M ₁ , one σ _x error is transmitted to the 4-qubit code. This can be detected by repeating the parity check once again.

でσ_yエラーとなる。ブロック内にσ_yエラーが生じると、σ_zエラーとの区別が出来ず取り扱いが難しそうに思えるが、パリティ検査を二回繰り返し、一回目と二回目の検査結果を比較すると問題は回避できる。 The case where the σ _z error of the auxiliary qubit is transmitted to the 4-qubit in the block by the parity check of M ₂ , can be considered as follows. When an error is transmitted to the first and second qubits, a σ _z error always occurs. On the other hand, if an error is transmitted to the 0th and 3rd qubits,

Causes a σ _y error. When a σ _y error occurs in a block, it seems difficult to handle it because it cannot be distinguished from a σ _z error, but the problem can be avoided by repeating the parity check twice and comparing the first and second check results.

For example, assume that (M ₀ , M ₁ , M ₂ = ( ₁ , ₁ , 1) is obtained in the _first inspection, and σ _y ⁽³⁾ is generated as an error in the M ₂ measurement at this time. (M ₀ , M ₁ , M ₂ = (1, -1, 1) is obtained. Therefore, in the 4-qubit block, either σ _z ⁽¹⁾ or σ _y ⁽³⁾ is obtained. Although it is known that an error has occurred, it cannot be specified as either, but by comparing the results of the first and second parity checks, the error is transmitted from the auxiliary qubit at least in the first M ₂ measurement. I understand that.

  From the above considerations, errors transmitted from the auxiliary qubit to the 4-qubit block cannot be corrected, but can always be detected. Such parity check is repeated, and the result is communicated between the four parties through the classical communication line, and the error is identified. If an error is detected but cannot be corrected, each qubit is discarded and the encoded qubit is received from the information source again.

(Second Embodiment)
In the present embodiment, a method and apparatus that enable six people to share quantum information in a secret manner will be described.

  That is, an arbitrary quantum information input means, storage means, calculation means, and output means using a quantum two-state system (qubit) are prepared, and arbitrary quantum information is block-coded with a quantum error correction code, and one block By distributing multiple quantum two-state systems (qubits) that make up the qubit to multiple people, when the original quantum information is shared and stored secretly, the quantum communication line between delivering the qubit from the information source to each person When detecting and correcting errors due to decoherence in the network, and removing errors due to decoherence in the quantum communication line during the delivery of qubits from the information source to each person, multiple people are contacted using the classical communication line However, each unit has a fault-tolerant property that can be detected and corrected by applying arbitrary unitary transformations to the qubits at hand. A quantum error correction method and a correction apparatus characterized by detecting an error when each qubit receives an error will be described.

  Furthermore, as a quantum error correction code for block coding of arbitrary quantum information, 1-qubit is encoded with 6-qubit, each qubit is distributed to 6 people, and secret is shared. There is nothing about quantum information, and the three people can either get the original information completely or not at all depending on the combination of cooperating people, and if it is four or more, the information is completely restored A method and apparatus for secretly sharing quantum information among a plurality of persons will be described.

ただし、規格化定数は省略している。これら四つの状態は互いに直交し合う。これらの6-qubitコードは1-qubitのエラーが検出可能である。さらに、これらのコードを上手く利用すれば、６人が秘密の共有を行い、任意の二人が共同で秘密情報を取り出すのを無効にすることができる。 Consider the following two 6-qubit codes:

However, normalization constants are omitted. These four states are orthogonal to each other. These 6-qubit codes can detect 1-qubit errors. Furthermore, if these codes are used successfully, 6 people can share secrets, and any two people can disable the extraction of secret information jointly.

次の関係が成立する。

これより、

を不変にする演算子が５個見付かったことになる。図６にエラーとＫ_i(i=0,…,4)の固有値の関係をまとめておく。 First, we will investigate in detail the nature of error correction. Prepare the following five operators.

The following relationship holds.

Than this,

5 operators have been found to make the invariant. FIG. 6 summarizes the relationship between errors and eigenvalues of K _i (i = 0,..., 4).

で符号化する。6-qubit中の任意の2-qubitの密度演算子は、

となる。これは、6-qubit中の２個のqubitだけを観測しても、量子情報(α，βについての情報)を全く得ることが出来ないことを意味する。また、上の結果より、6-qubit中の任意の1-qubitの密度演算子は、

となる。6-qubit中の3-qubitの密度演算子については、

および

が成立する。(15)の場合、α，βの情報を完全に取り出すことが出来る。 Consider sharing secrets with the following encoding: 1-qubit quantum information | ψ> = α | 0> + β | 1>

Encode with Any 2-qubit density operator in 6-qubit is

It becomes. This means that quantum information (information about α and β) cannot be obtained at all even if only two qubits in 6-qubit are observed. From the above result, the density operator of any 1-qubit in 6-qubit is

It becomes. For the 3-qubit density operator in 6-qubit,

and

Is established. In the case of (15), the information of α and β can be completely extracted.

  Therefore, if arbitrary quantum information is encoded with this 6-qubit mixed state code and each of the six people has 1-qubit, secrets can be shared. Even if up to two people observe the quantum state without the permission of the rest, no secret information can be obtained. This also means that information can be retrieved with the consent of at least four people. For example, in order for four people having 0th, 1st, 2nd and 3rd qubits to jointly extract confidential information, the operation shown in the gate network of FIG. 7 may be performed. When three people observe jointly, there are combinations that can completely extract confidential information and combinations that cannot be extracted at all.

の二種類のコードが1/2ずつの確率で混じっている。一つ目の符号は例えば図８(i)のゲート・ネットワークを組めば良い。これにより、1-qubitの任意の状態は、

と符号化される。二つ目の符号は図８(ii)のゲート・ネットワークを組めば良い。図８(i)，図８(ii)の二つのネットワークのどちらかで確率1/2で符号化すれば(13)が得られる。 Finally, an encoding method using the 6-qubit code will be described. This code

The two types of codes are mixed with a probability of 1/2. For example, the first code may be a gate network shown in FIG. With this, any state of 1-qubit is

Is encoded. The second code may be formed by the gate network shown in FIG. 8 (ii). If encoding is performed with probability 1/2 in one of the two networks of FIG. 8 (i) and FIG. 8 (ii), (13) is obtained.

(Third embodiment)
In this embodiment, arbitrary quantum information input means, storage means, calculation means, and output means using the quantum two-state system (qubit) of the first embodiment are described. In the information processing process using the quantum two-state system (qubit) in the second embodiment, the information input means, storage means, calculation means, and output means are commonly used.

  In addition, as the quantum two-state system (qubit) used, spins of electrons, nucleons, etc., especially electrons in solids, spins of nucleons, spins of nucleons in polymer compounds, and ground and excited states of ions A quantum error correction method and correction apparatus, and a method and apparatus for secretly sharing quantum information among a plurality of persons will be described.

  The qubit used in the previous first and second embodiments has no essential difference in any physical system as long as it is a quantum system that realizes a Hilbert space based on two states that are orthogonal to each other. . Examples of qubit candidates include spins of electrons and nucleons, particularly electrons in solids, spins of nucleons, spins of nucleons in polymer compounds, and ground and excited states of ions.

(16)の第二項はZeeman項、第三項はスピン−スピン相互作用項を表している。ここで、磁場がｚ軸を向いていて(Ｂ＝(0,0,B₀))、また、スピン−スピン相互作用がZeeman効果によるエネルギー準位のずれより十分に小さいとする。すなわち、

とする。ｚ方向の強い磁場のために、二つのスピンはｚ方向にそろった方がエネルギーが低く抑えられ、第三項はｚ成分のスカラー相互作用に近似できる。ハミルトニアンは、

ただし、

となる。ここで、ω_A−ω_Bはわずかに異なるとし、そのずれを、Δε＝|ω_A−ω_B|とする。エネルギー準位のずれをまとめると、図９のようになる。 For example, consider the nucleon spin. The Hamiltonian of two nuclear spins in a uniform static magnetic field B (vector) is given by

The second term of (16) represents the Zeeman term, and the third term represents the spin-spin interaction term. Here, it is assumed that the magnetic field is directed to the z-axis (B = (0,0, B ₀ )) and that the spin-spin interaction is sufficiently smaller than the energy level shift due to the Zeeman effect. That is,

And Because of the strong magnetic field in the z direction, the energy of the two spins aligned in the z direction is kept low, and the third term can be approximated by a z-component scalar interaction. Hamiltonian

However,

It becomes. Here, it is assumed that ω _A −ω _B is slightly different, and the deviation is Δε = | ω _A −ω _B |. The energy level shift is summarized as shown in FIG.

  When the spin is diagonalized in the z direction and the downward direction is | 1> and the upward direction is | 0>, the nuclear spin system can be regarded as a qubit system. If multiple nuclear spins are prepared, multiple qubit systems are prepared. If it is desired to input | 0 ... 0> as an initial state to this qubit system, a sufficiently strong uniform static magnetic field B = (0, 0, B) in the z-axis direction may be applied to the entire nuclear spin system. This aligns all nuclear spins in the direction of the magnetic field. Therefore, control by the interaction between the external magnetic field and the nuclear spin (Zeeman effect) becomes the information input means in this case.

  Once the quantum state of the nuclear spin is determined, the state is retained at least within the decoherence time. Therefore, this may be considered as information storage means.

  The calculation means is as follows. All operations performed on the qubit system are unitary transformations on the Hilbert space that the qubit system stretches. Arbitrary unitary transformation for multiple qubits can be realized by combining arbitrary unitary transformation of 1-qubit and C-NOT gate between 2-qubits. These unitary transformations can be executed by irradiating nuclear spins with electromagnetic waves (laser pulses) having a frequency corresponding to energy slightly shifted from the energy level difference between the states shown in FIG. In nuclear spins irradiated with electromagnetic waves, Bloch oscillations occur between levels with energy differences corresponding to frequencies. If the intensity of the laser pulse and the irradiation time are adjusted, the target state can be freely changed.

の周波数の電磁波を照射すれば、|↓↓〉，|↓↑〉との間でユニタリー回転が実行できる。共鳴パルスを上手く組み合わせれば、C-NOTも実行できる。従って、外部磁場と核スピンの相互作用(Zeeman効果)、および、スピン−スピン相互作用によるエネルギー準位のずれを利用したBloch振動が、この場合の情報の演算手段となる。 For example,

Unitary rotation can be executed between | ↓↓> and | ↓ ↑>. If the resonance pulses are combined well, C-NOT can also be executed. Therefore, Bloch oscillation using the interaction between the external magnetic field and the nuclear spin (Zeeman effect) and the energy level shift due to the spin-spin interaction is the information calculation means in this case.

  Obtaining output from qubit means observing the z-axis direction of nuclear spins. Various methods for doing this have also been devised. The most easily understood method is called the Stern-Gerlach device, which is often given as an example in the introduction of quantum mechanics. The nuclear spin to be measured is passed through a non-uniform magnetic field as a beam. ”Translated by Shigenobu Sunagawa, Iwanami Shoten (1979)). Assume that the magnetic field and its gradient are oriented in the z direction, and the nucleon beam is incident perpendicular to the magnetic field. The nucleon beam is split by the z-direction component of the spin, and if this is detected, the z-axis direction of the nuclear spin is observed. Also, a method of observing nuclear spin information by replacing it with electron spin was proposed in the literature (BEKane, 'A silicon-based nuclear spin quantum computer', Nature, 393, p.133-137 (1998)). Has been. Therefore, these are regarded as information output means.

  The above has been described by taking nuclear spins as an example, but even when other physical systems are used as qubits, it is possible to prepare information input means, storage means, calculation means, and output means, and the physical used there There are many similarities to this explanation.

  In the literature (JICirac and P. Zoller, 'Quantum Computations with Cold Trapped Ions', Phys. Rev. Lett. 74, 4091 (1995)) Has been discussed. In this case, n ions are captured linearly, and the ground state and the first excited state of each ion are regarded as qubit {| 0>, | 1>}. The operation of the quantum gate is realized by irradiating each ion with a laser from the outside.

  The ions trapped in a straight line interact with each other, and each ion oscillates around the equilibrium point. When this vibration mode is quantized, it becomes a phonon and can be used as an auxiliary qubit. In the previous document, a method for executing the C-NOT gate by skillfully using this phonon mode has been proposed.

  As described above, according to the first embodiment (and the third embodiment), arbitrary quantum information input means, storage means, calculation means, and output means using a quantum two-state system (qubit). When the arbitrary quantum information is block-encoded with the quantum error correction code, and a plurality of qubits constituting one block are distributed to a plurality of people, the original quantum information is secretly shared and stored. Acts on each qubit as an error detection operator for a block-coded quantum error correction code in order to detect and correct errors due to decoherence in the quantum communication line during the delivery of the qubit from the information source to each person 2 X2Pauli matrix tensor product is prepared. In addition, in the information source, multiple entangled auxiliary qubit systems used for error correction are configured, distributed to each person via quantum communication lines, block qubit, auxiliary qubit Detecting both errors, characterized by correcting, quantum error correction method, apparatus and method and apparatus for shared secret among the plurality of persons quantum information can be realized.

  In addition, according to the first embodiment, when removing an error due to decoherence in the quantum communication line during the delivery of a qubit from the information source to each person, the Hamming weight of the bit string in the information source is used as an auxiliary qubit. Are constructed by superimposing all the bases that are even with the same coefficient, delivering these qubits to each person via quantum communication lines, and communicating with each other using classical communication lines among each person, Quantum error correction method, device, and quantum information are secretly shared among multiple people by performing fault detection and correction error-tolerant by applying arbitrary unitary transformation to qubit held by a person A sharing method and apparatus can be realized.

で符号化し、各qubitを４人に配布することで秘密の共有を行い、１人では量子情報について何も得られず、２人では部分的な情報しか得られず、３人以上だと情報を完全に復元することが出来ることを特徴とする、量子エラー訂正方法、装置並びに量子情報を複数人の間で秘密に共有する方法及び装置が実現できる。 Further, according to the first embodiment, as a quantum error correction code for block-coding arbitrary quantum information, 1-qubit is changed to 4-qubit,

It is encoded with, and each qubit is distributed to 4 people to share secrets, 1 person can get nothing about quantum information, 2 people can get only partial information, 3 people or more information Can be completely restored, and a quantum error correction method and apparatus, and a method and apparatus for secretly sharing quantum information among a plurality of persons can be realized.

  In addition, according to the first embodiment, as a quantum error correction code for block encoding arbitrary quantum information, 1-qubit is encoded with 4-qubit, and one qubit in the block is a phase error in the quantum communication line. 10. The quantum error correction method and device according to claim 9, wherein when a qubit in the auxiliary qubit receives a phase error, the phase error is detected when the qubit receives a phase error. In addition, a method and apparatus for secretly sharing quantum information among a plurality of people can be realized.

の二種類のコードを等しい古典統計的な確率で発生させた混合状態で符号化し、各qubitを６人に配布することで秘密の共有を行い、２人以下では量子情報について何も得られず、３人では協力する人の組み合わせによって元々の情報が完全に得られるか、全く得られないかのどちらかに分かれ、４人以上だと情報を完全に復元することが出来ることを特徴とする、量子エラー訂正方法、装置並びに量子情報を複数人の間で秘密に共有する方法及び装置が実現できる。 Further, according to the second embodiment, as a quantum error correction code for block-coding arbitrary quantum information, 1-qubit is changed to 6-qubit,

The two types of codes are encoded in a mixed state generated with equal classical statistical probabilities, and each qubit is distributed to 6 people to share secrets. With 2 or less people, nothing can be obtained about quantum information It is characterized in that the original information can be obtained completely or not at all by the combination of three people who cooperate, and the information can be completely restored when there are four or more people In addition, a quantum error correction method and apparatus, and a method and apparatus for secretly sharing quantum information among a plurality of persons can be realized.

  Further, according to the second embodiment, as a quantum error correction code for block encoding arbitrary quantum information, 1-qubit is a mixture of two types of 6-qubit codes generated with equal classical statistical probability. Quantum error correction method, apparatus, and quantum information of a plurality of persons, characterized by detecting when one of the qubits in the block and the auxiliary qubit has received an error. A method and apparatus for secretly sharing between files can be realized.

  Further, according to the third embodiment, as the quantum two-state system (qubit) to be used, spins of electrons, nucleons, etc., in particular, electrons in solids, spins of nucleons, spins of nucleons in polymer compounds, In addition, a quantum error correction method and apparatus, and a method and apparatus for secretly sharing quantum information among a plurality of persons, characterized by utilizing the ground state and excited state of ions, can be realized.

It is a figure showing the fault-tolerant error correction network of 4-qubit code used in the 1st Embodiment of this invention. It is a figure showing the error in 4 qubit code and the eigenvalues of K ₀ , K ₁ , K ₂ used in the first embodiment of the present invention. It is a figure showing the decoding network by the 0th, 1st, 2nd qubit in the 4-qubit code used in the first embodiment of the present invention. It is a figure showing the encoding network in 4-qubit code | cord | chord used in the 1st Embodiment of this invention. It is a figure showing the procedure of the secret sharing in 4-qubit code | cord | chord used in the 1st Embodiment of this invention. Used in the second embodiment of the present invention, errors and K _i in 6-qubit code (i = 0, ..., 4 ) is a view showing eigenvalues. It is a figure showing the decoding network by the 0, 1, 2, 3rd qubit in 6-qubit code | cord | chord used by the 2nd Embodiment of this invention. It is a figure showing the encoding network in 6-qubit code used in the 2nd Embodiment of this invention. It is a figure showing the energy shift by a Zeeman term and a scalar coupling term in two nuclear spin systems used in the embodiment of the present invention. It is a diagram showing the interaction between photons and qubits in a general 9-qubit code. It is a figure showing the operator who detects the error in a general 9-qubit code. It is a figure showing the operator's eigenket and eigenvalue in a general 9-qubit code. It is a figure showing the network showing a general controlled-NOT gate. General, is a diagram showing the network that detects the eigenvalues of M ₁ in 9-qubit code. General, is a diagram showing the network that detects the eigenvalues of M ₇ with 9-qubit code. It is a figure showing the interaction between each qubit and the environment in the general independent particle approximation of 9-qubit code. It is a figure showing 16 kinds of errors in a general 5-qubit code and eigenvalues of {K ₃ , K ₄ , K ₀ , K ₁ }. It is a figure showing the reaction of the error from a general auxiliary qubit. It is a figure showing the parity detection which used only one auxiliary qubit with a general 5-qubit code. It is a figure showing the fault-tolerant parity detection in a general 5-qubit code. It is a figure showing the network which makes a general auxiliary qubit | Φ>. FIG. 2 is a diagram illustrating a general network of fault correction in a general 5-qubit code with fault tolerance. It is a figure showing the error specific value of a general ((3,4)) threshold scheme.

Claims (16)

Arbitrary quantum information using qubit that is a quantum two-state system is block-coded with a quantum error correction code at an information source, and a plurality of blocks qubit constituting one block are distributed to a plurality of people through a quantum communication line.
In the information source, a plurality of tangled auxiliary qubit systems used for error correction are configured, and these are distributed to each person through the quantum communication line,
The block due to decoherence in a quantum communication line during delivery of a qubit from the information source to each person using a tensor product of 2 × 2 Pauli matrix acting on each qubit as an error detection operator of the quantum error correction code A quantum error correction method characterized by detecting and correcting qubit and auxiliary qubit errors. The plurality of auxiliary qubits are qubits obtained by configuring a state in which all bases whose Hamming weights of the bit string are even in the information source are overlapped with an equal coefficient,
It is characterized by performing fault detection and correction in a fault-tolerant manner by allowing an arbitrary unitary transformation to act on the block qubit held by each person while communicating with each other using a classical communication line. The quantum error correction method according to claim 1. Block encoding using the quantum error correction code is:

Block coding from 1-qubit to 4-qubit represented by
3. The quantum error correction method according to claim 2, wherein arbitrary quantum information is block-encoded using the quantum error correction code, and each qubit is distributed to four persons. The above detection and correction of block qubit and auxiliary qubit errors are corrected when one qubit in the block receives a phase error in the quantum communication line, and one qubit in the auxiliary qubit also corrects the phase error. The quantum error correction method according to claim 3, wherein the quantum error correction method detects when the error is received. The block encoding using the quantum error correction code is a block encoding from 1-qubit to 6-qubit.

Is a block coding using two types of codes in a mixed state generated with equal classical statistical probability,
3. The quantum error correction method according to claim 2, wherein arbitrary quantum information is encoded using the quantum error correction code, and each qubit is distributed to six persons. 6. Arbitrary quantum information is block-encoded with the quantum error correction code, and when a block qubit and one qubit in an auxiliary qubit receive an error in a quantum communication line, this is detected. The quantum error correction method described in 1. Quantum two-state systems (qubits) to be used are spins of electrons, nucleons, etc., especially electrons in solids, spins of nucleons, spins of nucleons in polymer compounds, and ground and excited states of ions The quantum error correction method according to any one of claims 2 to 6. The first distribution in which arbitrary quantum information using qubit, which is a quantum two-state system, is block-coded with a quantum error correction code at an information source, and a plurality of blocks qubit constituting one block are distributed to a plurality of people through a quantum communication line Means,
In the information source, a plurality of entangled auxiliary qubit systems used in error correction are configured, and the second distribution means distributes these to each of the plurality of persons through the quantum communication line;
The block due to decoherence in a quantum communication line during delivery of a qubit from the information source to each person using a tensor product of 2 × 2 Pauli matrix acting on each qubit as an error detection operator of the quantum error correction code A quantum error correction apparatus comprising: detection / correction means for detecting and correcting errors of qubits and auxiliary qubits. The plurality of auxiliary qubits are qubits obtained by configuring a state in which all bases whose Hamming weights of the bit string are even in the information source are overlapped with an equal coefficient,
The detection / correction means performs fault detection / correction by applying an arbitrary unitary transformation to the block qubit held by each person while communicating with each other using a classical communication line. 9. The quantum error correction apparatus according to claim 8, wherein the quantum error correction apparatus is tolerant. Block encoding using the quantum error correction code in the first distribution means is:

Block coding from 1-qubit to 4-qubit represented by
The quantum error correction apparatus according to claim 9, wherein arbitrary quantum information is block-encoded using the quantum error correction code, and each qubit is distributed to four persons. The detection / correction means corrects when one qubit in the block qubit receives a phase error in the quantum communication line, and corrects this when one qubit in the auxiliary qubit receives a phase error. The quantum error correction apparatus according to claim 10, wherein the quantum error correction apparatus detects the quantum error correction apparatus. Block encoding using the quantum error correction code in the first cup new means is block encoding from 1-qubit to 6-qubit.

Is a block coding using two types of codes in a mixed state generated with equal classical statistical probability,
The quantum error correction apparatus according to claim 9, wherein arbitrary quantum information is encoded using the quantum error correction code, and each qubit is distributed to six persons. 13. The quantum error correction apparatus according to claim 12, wherein the detection / correction means detects a block qubit and one qubit in the auxiliary qubit received an error in the quantum communication line. Quantum two-state systems (qubits) to be used are spins of electrons and nucleons, especially electrons in solids, spins of nucleons, spins of nucleons in polymer compounds, and ground and excited states of ions The quantum error correction apparatus according to claim 9, wherein the quantum error correction apparatus is provided. Arbitrary quantum information using qubit that is a quantum two-state system is block-coded with a quantum error correction code at an information source, and a plurality of blocks qubit constituting one block are distributed to a plurality of people through a quantum communication line.
In the information source, a plurality of tangled auxiliary qubit systems used for error correction are configured, and by distributing these to each person through the quantum communication line, arbitrary quantum information is shared by a plurality of persons,
The block qubit and the auxiliary qubit use a tensor product of 2 × 2 Pauli matrix acting on each qubit as an error detection operator of the quantum error correction code, so that a quantum between delivering the qubit from the information source to each person is used. A quantum information sharing method characterized in that an error due to decoherence in a communication line can be detected and corrected. The first distribution in which arbitrary quantum information using qubit, which is a quantum two-state system, is block-coded with a quantum error correction code at an information source, and a plurality of blocks qubit constituting one block are distributed to a plurality of people through a quantum communication line Means,
The information source comprises a plurality of tangled auxiliary qubit systems used in error correction, and a second distribution means for distributing them to each of the plurality of persons through the quantum communication line,
The block qubit and the auxiliary qubit use a tensor product of 2 × 2 Pauli matrix acting on each qubit as an error detection operator of the quantum error correction code, so that a quantum between delivering the qubit from the information source to each person is used. A quantum information sharing apparatus capable of detecting and correcting errors due to decoherence in a communication line.

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