Decoding device, decoding method, program, and quantum computer

Abstract

A decoding device 1 decodes measurement results of a concatenated quantum code in which a plurality of quantum codes are concatenated, the decoding device including: a level 1 minimum distance candidate calculation unit 11 that, when the number of concatenated quantum codes is L, obtains a minimum distance, which is the minimum value of the Hamming distance between a code word and a measurement result, for each level 1 code block, and selects an encoded bit string corresponding to the minimum distance of each level 1 code block as a minimum distance candidate for each level 1 code block; a level i minimum distance candidate calculation unit 14 that, for i = 2,..., L, obtains the minimum distance for each level i code block by using a level (i-1) minimum distance and minimum distance candidate, for each level i encoded block consisting of a plurality of level (i-1) encoded blocks, and selects an encoded bit string corresponding to the minimum distance of each level i code block as a minimum distance candidate for each level i code block; and an output unit 20 that selects and outputs one of the level L minimum distance candidates.

Description

Decoding device, decoding method, program, and quantum computer

[0001] The present invention relates to a decoding device, a decoding method, a program, and a quantum computer.

[0002] Quantum error-correcting codes are known as a technology for correcting errors in quantum computers. In conventional quantum error correction, one or a small number of logical qubits are often encoded into a large number of physical qubits. However, in this case, as the code size increases, the encoding rate (also simply referred to as "rate"), defined as the number of logical qubits / the number of physical qubits, becomes very small, making it very inefficient. In other words, in this case, there is a problem that many physical qubits are required. Therefore, in recent years, high-rate quantum codes, which encode a large number of logical qubits together, in other words, as a single code block, have been attracting attention. As an example, concatenated quantum Hamming codes, which are obtained by concatenating high-rate quantum Hamming codes, are known (see, for example, Non-Patent Document 1). However, a method for decoding such concatenated high-rate quantum codes efficiently with high performance has not yet been known.

[0003] H. Yamasaki and M. Koashi, arXiv:2207.08826 (2022)E. Knill, Nature 434, 39 (2005)H. Goto and H. Uchikawa, Sci. Rep. 3, 2044 (2013)R. Chao and BW Reichardt, Phys. Rev. Lett. 121, 050502 (2018)H.Goto,Sci.Rep.4, 7501 (2014)

[0004] Conventional decoding methods for concatenated quantum codes include hard-decision decoding (see, for example, Non-Patent Document 2) and soft-decision decoding (see, for example, Non-Patent Document 3).

[0005] In hard-decision decoding of concatenated quantum codes, if hard-decision decoding is possible for the quantum codes used at each concatenation level, hard-decision decoding is performed starting from the lowest level (level 0 corresponds to the physical qubits, and the encoded qubits at the highest level correspond to the logical qubits). This can be performed efficiently even for concatenated high-rate quantum codes, but hard-decision decoding generally has poor decoding performance (relatively high probability of decoding failure).

[0006] On the other hand, soft-decision decoding of concatenated quantum codes is known as a decoding method with higher performance than hard-decision decoding. However, conventional soft-decision decoding can only be efficiently performed when encoding one or a small number of logical qubits, and cannot be directly applied to concatenated high-rate quantum codes. In other words, soft-decision decoding calculates the probabilities corresponding to all combinations of bit values ​​of logical qubits, but in high-rate quantum codes, the number of combinations of bit values ​​of logical qubits increases exponentially with the number of logical qubits, so it is not possible to efficiently calculate all the corresponding probabilities.

[0007] The present invention has been made in view of the above circumstances, and its object is to provide a technique capable of decoding concatenated high-rate quantum codes efficiently with high performance.

[0008] In order to solve the above problem, a decoding device of one embodiment of the present invention is a decoding device that decodes the measurement result of a concatenated quantum code in which multiple quantum codes are concatenated, and is equipped with: a level 1 minimum distance candidate calculation unit that, when the number of concatenations of the concatenated quantum code is L, calculates a minimum distance, which is the smallest value of the Hamming distance between the code word and the measurement result, for each code block of level 1, and selects an encoded bit sequence corresponding to the minimum distance of each code block of level 1 as a minimum distance candidate for each code block of level 1; a level i minimum distance candidate calculation unit that, for i = 2, ..., L, for each code block of level i consisting of multiple code blocks of level (i-1), calculates the minimum distance of each code block of level i using the minimum distance and minimum distance candidates of level (i-1), and selects an encoded bit sequence corresponding to the minimum distance of each code block of level i as a minimum distance candidate for each code block of level i; and an output unit that selects and outputs one of the minimum distance candidates of level L.

[0009] In one embodiment, n ｉ Let s be the number of qubits in the quantum code at level i. ｉ When the number of parity checks corresponding to the Z stabilizer of the quantum code at level i is set as (n ｉ -s ｉ ) minimum distance candidates at level (i-1) and s ｉ Using the parity check conditions, the remaining s ｉ The coded bit sequences of the level (i-1) code blocks may be calculated.

[0010] In one embodiment, for i=3,...,L, k ｉ is the number of encoded quantum bits of the quantum code at level i, the level i minimum distance candidate calculation unit is ｉ In calculating the minimum distance of the coded bit strings of the code blocks of level (i-1), ｉ－１ From the minimum distance candidates of the code blocks of level (i-2), ｉ－１ -k ｉ－１ -s ｉ－１ ) minimum distance candidates are selected, and the remaining (k ｉ－１ +s ｉ－１) coded bit sequences of level (i-2) are coded bit sequences of level (i-1) code blocks and s ｉ－１ The minimum distance of the coded bit strings of each code block at level (i-1) may be calculated based on the parity check conditions.

[0011] In one embodiment, if the total number of minimum distance candidates at level (i-1) is greater than a predetermined threshold, the level i minimum distance candidate calculation unit may select candidates in order from the code block at level (i-1) with the largest number of minimum distance candidates so that the total number of candidates is less than or equal to the threshold.

[0012] In one embodiment, if the total number of minimum distance candidates at level (i-2) is greater than a predetermined threshold, the level i minimum distance candidate calculation unit may select candidates in order from the code block at level (i-2) with the largest number of minimum distance candidates so that the total number of candidates is less than or equal to the threshold.

[0013] In one embodiment, for a natural number L' between 1 and L, the level 1 minimum distance candidate calculation unit may output a failure if any one of the number of minimum distance candidates at the (L-L'+1) levels between L' and L is not 1.

[0014] In one embodiment, if the measurement result also includes an erasure error, the level 1 minimum distance candidate calculation unit may calculate the Hamming distance after removing the bits corresponding to the erasure error.

[0015] Another aspect of the present invention is a decoding method for decoding a measurement result of a concatenated quantum code obtained by concatenating a plurality of quantum codes. This method includes: a level 1 minimum distance candidate calculation step of calculating, where L is the number of concatenations of the concatenated quantum code, the minimum value of the Hamming distance between a codeword and a measurement result, i.e., the minimum distance, for each code block of level 1, and selecting an encoded bit sequence corresponding to the minimum distance for each code block of level 1 as a minimum distance candidate for each code block of level 1; a level i minimum distance candidate calculation step of calculating, for i=2, ..., L, the minimum distance for each code block of level i, which is composed of a plurality of code blocks of level (i-1), using the minimum distance and minimum distance candidates for level (i-1), and selecting an encoded bit sequence corresponding to the minimum distance for each code block of level i as a minimum distance candidate for level i; and a selection step of selecting and outputting one of the minimum distance candidates for level L.

[0016] Yet another aspect of the present invention is a program for decoding a measurement result of a concatenated quantum code obtained by concatenating a plurality of quantum codes. This program causes a computer to execute the following steps: a level-1 minimum distance candidate calculation step of calculating, where L is the number of concatenations of the concatenated quantum code, the minimum value of the Hamming distance between a codeword and a measurement result, i.e., the minimum distance, for each code block of level 1, and selecting an encoded bit sequence corresponding to the minimum distance for each code block of level 1 as a minimum distance candidate for each code block of level 1; a level-i minimum distance candidate calculation step of calculating, for i=2, ..., L, the minimum distance for each code block of level i, which is composed of a plurality of code blocks of level (i-1), using the minimum distance and minimum distance candidates for level (i-1), and selecting an encoded bit sequence corresponding to the minimum distance for each code block of level i as a minimum distance candidate for level i; and a selection step of selecting and outputting one of the minimum distance candidates for level L.

[0017] Yet another aspect of the present invention is a quantum computer, which includes the above-described decryption device.

[0018] Any combination of the above components, and conversion of the present disclosure into an apparatus, method, system, recording medium, computer program, etc., are also effective aspects of the present invention.

[0019] According to the present invention, concatenated high-rate quantum codes can be decoded efficiently with high performance.

[0020] 7 is a diagram showing the configuration of a concatenated quantum code when both level 1 and level 2 are [[6,4,2]]. It is a diagram showing the processing flow of error-correction teleportation. It is a flowchart showing the decoding method of the present invention for a level 3 concatenated quantum code. It is a flowchart showing the decoding method of the present invention for a general level L concatenated quantum code. It is a flowchart showing the processing procedure of subroutine S3 of FIG. 3 for a level 3 concatenated [[6,4,2]] quantum code. It is a flowchart showing the processing procedure of subroutine S312 of FIG. 6 for a level 3 concatenated [[6,4,2]] quantum code. It is a flowchart showing the processing procedure of subroutine S3121 of FIG. 7 for a level 3 concatenated [[6,4,2]] quantum code. It is a schematic diagram of a quantum circuit of an encoder that generates any coded state of [[6,4,2]]. It is a schematic diagram of a quantum circuit of an encoder that generates the zero state of [[6,4,2]]. It is a diagram showing the execution of a coding CNOT gate for [[6,4,2]]. 1 is a diagram illustrating the execution of a coded Hadamard gate for [[6,4,2]]. It is a diagram illustrating the results of a numerical simulation of the first embodiment. It is a flowchart illustrating a processing procedure for level 2 of the second embodiment. It is a diagram illustrating the results of a numerical simulation of the second embodiment. It is a flowchart illustrating a processing procedure for level 1 of the third embodiment. It is a diagram illustrating the results of a numerical simulation of the third embodiment. It is a diagram illustrating an example of an error-resilient level 1 zero-state encoder for a concatenated [[6,4,2]] code. It is a diagram illustrating an example of an error-resilient level 2 zero-state encoder for a concatenated [[6,4,2]] code. It is a diagram illustrating an example of X-error detection (EDX) and Z-error detection (EDZ) at level 1. It is a diagram illustrating an example of X-error detection (EDX) and Z-error detection (EDZ) at level 2. It is a diagram illustrating an example of an error-resilient level 4 coded zero-state encoder for a concatenated [[6,4,2]] code. It is a diagram illustrating the results of a numerical simulation evaluating the performance of error-correction teleportation in a circuit error model using the decoding method of the present invention. FIG. 2 is a schematic functional block diagram of a quantum computer 30 according to the present embodiment.1 illustrates a method for performing a logical Hadamard gate on some logical qubits in a code block using an encoding +0 state. 2 illustrates a method for generating one level 1 encoding +0 state of a concatenated [[6,4,2]] code in an error-resilient manner. 3 illustrates a method for generating one level 1 encoding +0 state of a concatenated [[6,4,2]] code in an error-resilient manner. 4 illustrates a method for generating one level 1 encoding +0 state of a concatenated [[6,4,2]] code in an error-resilient manner. 5 illustrates a swap gate for level 1 encoding qubits of a concatenated [[6,4,2]] code. 6 illustrates a state verification for generating an error-resilient level 2 encoding +0 state. 7 illustrates a method for performing a logical phase gate S. 8 illustrates an error-resilient |YYYY>. Ｌ 1 shows a diagram of state verification in preparation for . 2 shows a diagram of how to implement a logical CNOT gate between different code blocks in an error-resilient manner. 3 shows a diagram of how to implement a logical CNOT gate within the same code block in an error-resilient manner. 4 shows a diagram of how to implement a logical CNOT gate within the same code block in an error-resilient manner. 5 shows a diagram of how to implement a logical CNOT gate within the same code block in an error-resilient manner. １２ 00> Ｌ１ FIG. 13 is a diagram showing an error-resilient generation method of one level 1 coded BB state of a concatenated [[6,4,2]] code. FIG. 14 is a diagram showing an error-resilient generation method of a coded B0 state using a coded BB state. FIG. 15 is a diagram showing an error-resilient generation method of one level 2 coded BB state of a concatenated [[6,4,2]] code. FIG. 16 is a diagram showing an error-resilient generation method of one level 2 coded BB state of a concatenated [[6,4,2]] code. FIG. 17 is a functional block diagram of a decoding device according to a first embodiment. FIG. 18 is a flowchart showing the processing steps of a decoding method according to an eighth embodiment.

[0021] The present disclosure will be described below with reference to the drawings based on preferred embodiments. The embodiments are illustrative and do not limit the invention, and all features and combinations thereof described in the embodiments are not necessarily essential to the invention. The same or equivalent components, parts, and processes shown in each drawing are designated by the same reference numerals, and redundant description will be omitted where appropriate. The scale and shape of each part shown in each drawing are set for convenience to facilitate explanation and should not be interpreted as limiting unless otherwise specified. Furthermore, when terms such as "first" and "second" are used in this specification or claims, unless otherwise specified, these terms do not represent any order or importance, but are merely used to distinguish one configuration from another. Furthermore, some components that are not important for explaining the embodiments are omitted from each drawing.

[0022] Before describing specific embodiments, basic knowledge will be described. First, quantum codes and concatenated quantum codes will be described. A quantum code with a code distance of d that encodes k quantum bits into n quantum bits is represented as [[n,k,d]]. A concatenated quantum code (hereinafter referred to as a "level-2 concatenated quantum code") in the case where [[n1,k1,d1]] is used in level 1 and [[n2,k2,d2]] is used in level 2 is [[n2n1,k2k1,d2d1]]. This level-2 concatenated quantum code is constructed as follows. k2k1 level-2 encoded quantum bits are arranged in a k2×k1 matrix {Q (2) i,j |i=1,…,k2, j=1,…,k1}, and the k1 level-1 encoding qubits of each of the n2 level-1 code blocks that make up these are expressed as an n2 × k1 matrix {Q (1) i,j |i=1,…,n2, j=1,…,k1}, and the n2n1 physical qubits that make up these are represented as an n2×n1 matrix {Q (0) i,j |i=1,…,n2, j=1,…,n1}. Then, the j-th column vector V (2) j ={Q (2) i,j|i=1,…,k2} is the j-th column vector V of level 1 by [[n2,k2,d2]] (1) j ={Q (1) i,j |i=1,…,n2}. Similarly, the i-th row vector U (1) i ={Q (1) i,j |j=1,…,k1} is the i-th row vector U of level 0 (physical qubit) by [[n1,k1,d1]] (0) i ={Q (0) i,j |j=1,...,n1}. Figure 1 shows the structure when both level 1 and level 2 are [[6,4,2]].

[0023] Similarly, if [[n1,k1,d1]] is used for level 1, [[n2,k2,d2]] is used for level 2, and [[n3,k3,d3]] is used for level 3, the concatenated quantum code (hereinafter referred to as "level 3 concatenated quantum code") is [[n3n2n1,k3k2k1,d3d2d1]]. This level 3 concatenated quantum code is constructed as follows. Let us consider k3k2k1 level 3 encoding qubits as k3×k2×k1 tensor {Q (3) i,j,l |i=1,…,k3, j=1,…,k2, l=1,…k1}, and the k2k1 level-2 encoding qubits of the n3 level-2 code blocks that make up these are represented as n3×k2×k1 tensors {Q (2) i,j,l |i=1,…,n3, j=1,…,k2, l=1,…k1}, and the k1 level-1 encoding qubits of each of the n3n2 level-1 code blocks that make up these are represented as an n3×n2×k1 tensor {Q (1) i,j,l |i=1,…,n3, j=1,…,n2, l=1,…,k1}, and the n3n2n1 physical qubits that make up these are represented as n3×n2×n1 tensors {Q (0) i,j,l |i=1,…,n3, j=1,…,n2, l=1,…n1}. Then, the (j,l)-th vector W(3) j,l ={Q (3) i,j,l |i=1,…,k3} is the (j,l)-th vector W at level 2 by [[n3,k3,d3]] (2) j,l ={Q (2) i,j,l |i=1,…,n3}. Similarly, the (i,l)-th vector V (2) j ={Q (2) i,j,l |j=1,…,k2} is the (i,l)-th vector V at level 1 by [[n2,k2,d2]] (1) j ={Q (1) i,j,l |j=1,…,n2}. Furthermore, the (i,j)-th vector U (1) i,j ={Q (1) i,j,l |l=1,…,k1} is the (i,j)-th vector U (0) i,j ={Q (0) i,j,l |l=1,...,n1}. The structure of concatenated quantum codes at levels 4 and above is similar.

[0024] In this disclosure, we assume that the quantum code is a CSS code. In the CSS code, s Z Z stabilizer operators and s X Define the code space as the joint eigenspace of eigenvalue 1 with the X stabilizer operators. This allows us to define the code space as k=ns for [[n,k,d]] CSS codes. Z -s X holds. The computational basis states of the k encoding qubits |b1 (1) …b K (1) > L (b i (1)=0,1,i=1,…,k) is defined as the simultaneous eigenstates (in the code space) of k coded Z operators that commute with all X stabilizer operators and are independent of all Z stabilizer operators (eigenvalue +1 corresponds to bit value 0, eigenvalue -1 corresponds to bit value 1). Each |b1 (1) …b K (1) > L is 2 sX n-qubit computational basis states |b1 (0) …b n (0) >(b i (0) =0,1,i=1,…,n) and the Z-basis measurement (0 / 1 measurement M 0 / 1 ) and the above 2 sX bit string b1 (0) …b n (0) The computational basis state of the encoding qubit |b1 (1) …b K (1) > L The bit string b1 corresponding to (1) …b k (1) is called an "encoded bit string," and the two sX n-qubit computational basis states |b1 (0) …b n (0) >The bit string b1 (0) …b n (0) is encoded as bit string b1 (1) …b k (1) We call the corresponding bit string a "codeword." We also define the distance between the bit string resulting from the measurement of n qubits and the encoded bit string as the minimum Hamming distance (the number of bits that differ in value) between the codeword corresponding to the encoded bit string and the bit string resulting from the measurement.

[0025] Knill's error-correcting teleportation is used as a quantum error correction method (e.g., Non-Patent Documents 2 and 3). Figure 2 shows the processing flow of error-correcting teleportation. In error-correcting teleportation, each physical quantum bit constituting two code blocks is measured in the Z basis at the end, the bit strings of the measurement results are decoded to estimate the measurement values ​​of the encoded bit strings of each code block, and the results are fed forward. Therefore, decoding in this case means estimating the measurement values ​​of the encoded bit strings from the bit strings of the measurement results of the physical quantum bits.

[0026] Hard-decision decoding is a conventional decoding method for concatenated quantum codes (see, for example, Non-Patent Documents 2 and 3). Hereinafter, with regard to hard-decision decoding, a level-2 concatenated quantum code [[n2n1,k2k1,d2d1]] is described, which uses [[n1,k1,d1]] for level 1 and [[n2,k2,d2]] for level 2. It is assumed that hard-decision decoding of [[n1,k1,d1]] and [[n2,k2,d2]] can be easily performed. First, hard-decision decoding of [[n1,k1,d1]] is performed on the bit string resulting from the measurement of the physical quantum bits for each of n2 level-1 code blocks, thereby obtaining an encoded bit string for each level-1 code block. Next, hard-decision decoding of [[n2,k2,d2]] is performed on the obtained level-1 encoded bit string, thereby finally obtaining a level-2 encoded bit string. Since the coded bit sequence for each level is determined uniquely in the processing at each level, the computational complexity can be kept low and it can be applied to concatenated high-rate quantum codes. However, hard-decision decoding generally has poor decoding performance (the probability of decoding failure is relatively high).

[0027] Soft-decision coding is a conventional method that offers higher performance than hard-decision decoding (see, for example, Non-Patent Document 3). As with the above, regarding soft-decision coding, we will explain level-2 concatenated quantum code [[n2n1,k2k1,d2d1]], which uses [[n1,k1,d1]] for level 1 and [[n2,k2,d2]] for level 2. In soft-decision decoding, first, a posteriori probability of all coded bit sequences of [[n1,k1,d1]] is calculated for each of n2 level-1 code blocks using the bit sequence of the measurement result of the physical quantum bit and the error probability of the physical quantum bit. Here, the coded bit sequence of [[n1,k1,d1]] is 2 k1 There are 22 pieces in total. k1 Next, the posterior probability of the obtained level 1 coded bit sequence is used to calculate the posterior probability of all coded bit sequences of the level 2 concatenated quantum code [[n2n1,k2k1,d2d1]]. Then, the coded bit sequence with the maximum posterior probability is returned. This is called block MAP decoding, since it calculates the maximum a posteriori probability (MAP) of the code block. The coded bit sequence of [[n2n1,k2k1,d2d1]] has 2 k2×k1 There are 2 k2×k1 In conventional coding of one or a small number of quantum bits, k1, k2, and k1k2 are all small numbers, and the above probability calculations can be performed efficiently. However, in the case of concatenated high-rate quantum codes, the number of coded bit sequences increases exponentially with the number of coded quantum bits, making the above probability calculations inefficient. In other words, conventional soft-decision decoding cannot be applied directly to concatenated high-rate quantum codes.

[0028] The decoding method according to the present disclosure will be described below. The present invention is based on minimum distance decoding. Minimum distance decoding is equivalent to maximum likelihood decoding and is known as high-performance decoding. In minimum distance decoding, the codeword that is closest to the bit string of the measurement result of the physical quantum bit, that is, the codeword with the smallest Hamming distance from the bit string of the measurement result, is estimated, and the corresponding encoded bit string is returned. However, for example, s ZZ stabilizer operators and s X [[n,k=ns with X stabilizer operators Z -s X ,d]]For CSS code, the codeword is 2 k+sX For high-rate quantum codes with large k, it is not possible to efficiently find the minimum distance codeword, so some approximation is required.

[0029] Therefore, in the present invention, only the coded bit sequence with the smallest distance in each code block at each level is left as a candidate. Figure 3 is a flowchart showing the decoding method of the present invention for a level 3 concatenated quantum code. Instead of the smallest distance, coded bit sequences with the second smallest distance or the third smallest distance may be left as candidates. This makes it possible to efficiently search for coded bit sequences corresponding to codewords with a small Hamming distance from the bit sequence of the measurement result. Figure 4 is a flowchart showing the decoding method of the present invention for a general level L concatenated quantum code.

[0030] However, for example, in the subroutine S3 of Fig. 3, when an attempt is made to construct a level 3 code word by taking one coded bit sequence of each minimum distance candidate from all level 2 code blocks, the parity check equation corresponding to the eigenvalue of the Z stabilizer operator being 1 is not necessarily satisfied, and there may be no corresponding code word. Z -s X ,d]]In CSS code, out of n bits of the codeword, (ns Z ) are determined, the remaining s Z Bit is s Z s corresponding to the eigenvalues ​​of the Z stabilizer operators being all 1 Z We focus on the fact that the parity check equations are uniquely determined. By using this, we can determine the number of parity checks (ns Z ) code blocks are selected one by one from the minimum distance candidates, and the remaining s Z The coded bit strings of the code blocks are ZBy determining the parity check formulas (without using the minimum distance candidates), it is possible to ensure that the parity check formula is always satisfied. For example, the stabilizers of a level 3 concatenated [[6,4,2]] quantum code ([[6,4,2]] are Z1Z2Z3Z4Z5Z6 and X1X2X3X4X5X6) are s Z =s X In S3 of Figure 3 for (k = 1), using the above tensor notation, i =n i -s Zi -s Xi , i=1,2,3), and the bit value corresponding to each Q is represented by b. Then, for example, the first (n3-s Z3 ) = 5 code blocks of coded bit strings {b (2) i,j,l |i=1,…,5, j=1,…,4, l=1,…4} are selected one by one from the minimum distance candidates, and the remaining s Z3 = 1 (6th) code block coded bit string {b (2) 6,j,l | j=1,…,4, l=1,…4} is determined to satisfy the following parity check equation, which corresponds to the eigenvalue of Z1Z2Z3Z4Z5Z6 being 1: This can be easily solved, and the coded bit string for the sixth code block is given by: s Z Even in the case of a general code where s is 2 or more, Z Similarly, by solving the mod 2 simultaneous equations, we can obtain the remaining s Z The coded bit sequences of the code blocks can be determined.

[0031] In the subroutine S3 of FIG. 3, the distance of the coded bit string of the level 3 code block is calculated by selecting one by one from the minimum distance candidates (n3-s Z3 ) level-2 code blocks and the minimum distances determined from their coded bit sequences and parity check formulas. Z3The distance is calculated as the sum of the distances of the coded bit sequences of the level 2 code blocks. However, as in the latter case, the distance between the coded bit sequences of other code blocks and the coded bit sequence determined by the parity check formula is generally different from the minimum distance of that code block. Therefore, the distance must be calculated again. In this way, in order to calculate the distance when the values ​​of the coded bit sequences are given, it is generally Z -s X ,d]]In CSS code, when a k-bit encoded bit string is given, s of the n bits of the codeword X Once the number is determined, the remaining (ns X )=(k+s Z ) bits are k conditions that a k-bit encoded bit string has a given value, and the above s Z For example, in the example of S3 in FIG. 3 for the level 3 concatenated [[6,4,2]] quantum code, the coded bit string {b (2) 6,j,l | j=1,...,4, l=1,...4} is determined as above by the coded bit sequence of the minimum distance candidate of other code blocks and the parity check formula. The distance of this sixth level-2 code block is calculated as follows. For example, the first s X = coded bit sequence of one code block {b (1) 6,1,l |l=1,…4} is selected one by one from the minimum distance candidates, and the remaining (ns X )=(k+s Z ) = 5 level 1 code blocks coded bit strings {b (1) 6,j,l | j=2,…,6, l=1,…4} is a string in which the four encoded Z operators in [[6,4,2]] are Z L1 =Z1Z2, Z L2 =Z2Z3, Z L3 =Z4Z5, Z L4=Z5Z6 (there can be other definitions, but we define it this way here) and the eigenvalue of Z1Z2Z3Z4Z5Z6 is 1. Z =1 parity check equation, the following five conditional expressions are determined to be satisfied. This can be easily solved, and the coded bit strings of the second to sixth level 1 code blocks are given by the following equations:

[0032] The distances for the second to sixth level-1 code blocks for which coded bit sequences have been determined in this way generally differ from the minimum distance for each coded block, so distance calculations are required again, but level-1 distance calculations can be easily performed. For example, if the level-1 code is [[6,4,2]] as in the example above, in each level-1 code block, if the parity of the bit sequence resulting from the measurement of the six physical quantum bits that make up it is even, the bit sequence is considered a level-1 codeword. If the corresponding level-1 coded bit sequence matches the given coded bit sequence, the distance is 0; if they do not match, the distance is 2. On the other hand, if the parity is odd, six bit sequences obtained by inverting one bit of the bit sequence are considered level-1 codewords. If one of the six corresponding level-1 coded bit sequences matches the given coded bit sequence, the distance is 1; if none of them match, the distance is 3. For other codes, if the number of codewords for the code used in level 1 is sufficiently small, the desired distance can be calculated by calculating the distances for all codewords and finding the minimum value. The distance of the sixth level-2 code block is calculated as the sum of the distances of the level-1 code blocks, but the distance thus obtained depends on the combination of how the level-1 code block for the coded bit sequence is selected from the minimum distance candidates and how one code block is selected from the minimum distance candidates. Therefore, the distance is calculated for all combinations of the selection methods, and the minimum value is taken as the distance of the sixth level-2 code block.

[0033] Here, when there are many combinations of selection methods, the number of combinations of selection methods can be kept below a desired threshold and the calculation time can be shortened by narrowing down the candidates (for example, by randomly selecting only one candidate) starting from the level 1 code block with the most minimum distance candidates (or the largest minimum distance). In this way, the distance calculation for the code block given an encoded bit string is completed. The above example is for the case where an encoded bit string of a level 2 code block is given, but similar distance calculations can be performed for levels 3 and above. As can be seen from the above example, distance calculations for the encoded bit strings given at the next lower level are required, and finally, distance calculations for the encoded bit string given at level 1 are required, but this is feasible.

[0034] In this way, for example, in the subroutine S3 of FIG. 3, a level 3 codeword can be constructed from the coded bit strings of the minimum distance candidates of level 2, and the distance can be calculated. This distance is determined by selecting one coded bit string from the minimum distance candidates (n3-s Z3 ) level-2 code blocks and the combination of how one coded bit sequence is selected from these minimum distance candidates. In subroutine S3, the distance is calculated for all of these combinations, the minimum distance among them is taken, and the corresponding level-3 coded bit sequence is enumerated. When there are an extremely large number of combinations, the candidates are narrowed down to a small number (for example, by randomly selecting only one candidate) starting from the level-2 code block with the most minimum distance candidates (or the largest minimum distance), thereby making it possible to keep the number of combinations below a desired threshold and shorten the calculation time.

[0035] 5 to 8 are flowcharts showing the processing steps of the above-mentioned subroutine S3 for level 3 concatenated [[6,4,2]] quantum codes. Since each loop process is independent, they can all be executed in parallel. The decoding device of the present invention enables high-speed decoding by executing these loop processes in parallel using multiple processors such as a multi-core CPU (central processing unit), GPU (graphics processing unit), or FPGA (field-programmable gate array).

[0036] (Example 1) An example of a concatenated [[6,4,2]] quantum code will be described in detail. [[6,4,2]] has a distance of 2 and cannot be corrected but can only be detected, but by concatenating them, correction becomes possible. A concatenated quantum code made by concatenating L [[6,4,2]]s is [[6 L ,4 L ,2 L ]], and in principle, (2 L-1 -1) independent physical qubit errors can be corrected. As mentioned above, the stabilizers for [[6,4,2]] are Z1Z2Z3Z4Z5Z6 and X1X2X3X4X5X6 (hence a type of CSS code). The four coded Z operators are Z L1 =Z1Z2, Z L2 =Z2Z3, Z L3 =Z4Z5, Z L4 =Z5Z6, and the four coded X operators are X L1 =X2X3, X L2 =X1X2, X L3 =X5X6, X L4 =X4X5.

[0037] The reasons for focusing on [[6,4,2]] are as follows. First, it is the n=6 case of the well-known [[n,n-2,2]] quantum error-detecting code. Compared to quantum error-correcting codes, this code has a simpler structure and a higher code rate, making it suitable for use in error-resilient quantum computing. However, as is, it can only detect errors, not correct them. Therefore, by concatenating these codes to increase the code distance, error correction becomes possible. High-rate concatenated quantum error-detecting codes that concatenate such high-rate quantum error-detecting codes have not yet been studied. Furthermore, the reason for choosing n=6 is that n=4 results in a relatively low code rate, while n=8 is expected to make error-resilient coding difficult. Table 1 shows the coding rate r, number of physical qubits n, number of coding qubits k, and code distance d for the concatenation level L when n=6. Table 1: Characteristics of concatenated [[6,4,2]] codes

[0038] Figure 9 shows a quantum circuit for an encoder that can generate any coded state of [[6,4,2]]. Figure 10 shows a schematic diagram of a quantum circuit for an encoder of the zero state, a special case of [[6,4,2]]. Because [[6,4,2]] is a CSS code, the coded CNOT gate (encoded controlled-NOT gate) can be performed transversally (i.e., by performing a physical CNOT gate between corresponding physical qubits). Figure 11 shows this. As shown in Figure 12, the coded Hadamard gate can be performed by performing a physical Hadamard gate on all physical qubits and then swapping the physical bits. By using the above encoder, coded CNOT gate, and coded Hadamard gate, we can generate the coded zero state of the concatenated [[6,4,2]] code.

[0039] In this example, as the most basic performance evaluation of the decoding method, an independent bit flip error was made in each physical quantum bit in the ideal logical zero state (logical zero state, where all logical quantum bits are 0) of the concatenated [[6,4,2]] code, and the decoding failure probability (the probability that the decoded result does not become an encoded bit string of all 0s, decoding failure probability) when the Z-basis measurement result (measurement is also assumed to be ideal) was decoded using the decoding method of the present invention was evaluated by numerical simulation. The results are shown in Figure 13(a). In this example, the N ｔｈ３ 100000, and its counterpart at level 4, N ｔｈ４ 100000, M in Fig. 7 ｔｈ２ 6, its level 3 counterpart M ｔｈ３ was set to 12. The same applies to the other embodiments below. For comparison, Figures 13(b) and 13(c) show simulation results using hard-decision decoding and soft-decision decoding, respectively. As mentioned above, since strict soft-decision decoding cannot be performed, symbol MAP decoding was performed, which calculates the posterior probability for each coded bit at each level. As is clear from the figure, the high threshold and low decoding failure probability show that the method disclosed herein has higher performance than hard-decision decoding and soft-decision decoding.

[0040] Example 2 The decoding method of the present disclosure can also be used for error detection. The fewer the number of minimum distance candidates at each level, the higher the reliability. Furthermore, the acceptance probability can be increased by appropriately relaxing the condition for the number of minimum distance candidates. Therefore, for example, the highest error detection performance can be achieved by accepting only when there is only one minimum distance candidate at all levels and rejecting otherwise. Figure 14 is a flowchart showing the processing procedure for level 2 of this embodiment. Figure 15 shows the results of a numerical simulation when the decoding method of the present invention is used for error detection. Here, the same simulation as shown in Figure 13(a) was performed, and the error correction results of Figure 13 were compared with the decoding failure probability when no errors were detected by error detection. Figure 15(a) shows the case of level 2, and Figure 15(b) shows the case of level 3. It can be seen that error detection can significantly reduce the decoding failure probability when no errors are detected.

[0041] (Example 3) The decoding method of the present disclosure can also be used when erasure errors are present. An erasure error is an error in which the information of a physical quantum bit is completely lost, but its position can be identified. To use the decoding method of the present disclosure when erasure errors are present, it is only necessary to exclude the physical quantum bit in which the erasure error occurred from the Hamming distance calculation in the distance calculation at level 1, and the other procedures are as described above. Figure 16 is a flowchart showing the processing procedure of level 1 of this embodiment. Figure 17 shows the results of a numerical simulation when the decoding method of the present disclosure is used for error detection. It can be seen that the threshold value (the erasure error probability and bit flip error probability at which the decoding failure probability at level 3 and level 4 is equal) is much higher than that of conventional methods.

[0042] (Example 4) An example using a circuit error model in which errors occur in the physical CNOT gate, the initialization of the physical qubit, and the measurement of the physical qubit is shown. Here, it is assumed that the memory error of the physical qubit and the single-qubit gate error can be ignored. For example, such an error model is appropriate for cold atoms and cold ions. The circuit error model requires the use of an error-resilient encoder. Below, an example of an error-resilient zero-state encoder for the concatenated [[6,4,2]] code is described.

[0043] 18 shows an example of a level 1 coded zero-state error-resilient encoder. If the result of the Z-basis measurement of the seventh physical qubit is 0, it is accepted; if it is 1, it is rejected and the encoder starts over.

[0044] Figure 19 shows an example of an error-resilient encoder for level 2 coding in the zero state. Error detection decoding is performed on the measurement results of the seventh level 1 code block. If an error is detected or if any of the decoded results is 1, the process is restarted. Ｌ１ and EDZ Ｌ１ and represent level 1 X error detection and Z error detection, respectively, which are performed by a conventional method called FLAG (see, for example, Non-Patent Document 4).

[0045] 20 is a diagram showing an example of X error detection (EDX) and Z error detection (EDZ) at level 1. In this case, if the results of the Z-basis measurements of the two physical qubits are both 0, they are accepted, and if not, they are restarted from the beginning.

[0046] 21 is a diagram showing an example of X error detection (EDX) and Z error detection (EDZ) at level 2. The level 3 coded zero state can be generated in a similar manner to the level 2 coded zero state of FIG. 19, but the level 2 X error detection EDX Ｌ２ and Z error detection EDZ Ｌ２ are executed as shown in Fig. 21. Here, the results of the Z-basis measurement of the level 2 code block are subjected to decoding for error detection, and if an error is detected or the decoded result contains at least one 1, the process is restarted from the beginning.

[0047] Figures 22 and 23 show examples of error-resilient encoders for the level 4 coding zero state of the concatenated [[6,4,2]] code. The level 4 coding zero state can also be generated in the same way as the level 3 coding zero state of Figures 19 and 21. However, in this case, the acceptance probability is low and the number of iterations may increase. Therefore, the encoder shown in Figure 22 or Figure 23 is useful as a method to increase the acceptance probability. Here, EDT Ｌ３ is error-detecting teleportation for level 3 coded qubits, and is the error-correcting teleportation of Figure 2, where decoding is changed to error detection. Error detection (or error correction) decoding is performed on the results of the Z-basis measurement of the final level 3 code block. If no error is detected and the parity of the decoded result is all 0, it is accepted; otherwise, it is rejected and the process is started over. At level 4, a technique for increasing the acceptance probability in the error-detecting decoding method of the present invention (a technique for accepting when the number of shortest distance candidates is 1 only at high levels) is particularly useful.

[0048] 24 shows the results of a numerical simulation evaluating the performance of error correction teleportation in a circuit error model using the decoding method of the present invention. In this simulation, an ideal (error-free) logical Bell state is prepared, and error correction resistant teleportation is performed 10 times on one side of it using a circuit error model. The obtained logical Bell state is returned to the zero state using an ideal logical CNOT gate and a logical Hadamard gate, and the results of this ideal measurement are decoded. The error correction failure probability p １０ Calculate the probability of one error correction failure p １ o p １ = 1 - (1 - p １０ ) ０．１The calculations are shown in Table 1. In this simulation, encoding was performed using the method shown in Figure 18 for level 1, Figures 19 and 20 for level 2, Figures 19 and 21 for level 3, and Figure 22 for level 4. Error detection was performed as much as possible up to level 3, and at level 4, EDT accepted a case where the number of minimum distance candidates for both level 3 and level 2 was 1, and the decoding of the final measurement result accepted a case where the minimum distance candidate for level 3 was 1. It can be seen that the probability of error correction failure decreases with the level, and the threshold is relatively high at approximately 0.8%.

[0049] (Example 5) A decoding device and a quantum computer using a concatenated [[6,4,2]] code according to Example 5 of the present invention will be described. FIG. 25 is a schematic functional block diagram of a quantum computer 30 according to this example. The quantum computer 30 includes any of the decoding devices 31 described above, a control device 32, a physical system 33 having a physical quantum bit, and a measurement device 34. Preparation of an error-tolerant logical zero state, Z-basis measurement of the logical quantum bit, and quantum error correction using error-correction teleportation can be performed as described above. Logical gate operations will be described below.

[0050] All necessary logic gate operations can be realized in principle by the method described in Non-Patent Document 1. That is, Clifford gates (Hadamard gates and CNOT gates) can be realized by quantum teleportation using the auxiliary states obtained by executing them on logical Bell states, which are prepared in advance. These auxiliary states are so-called stabilizer states, and can be prepared in an error-tolerant manner. In other words, the desired state is first prepared in a non-error-tolerant manner, followed by error correction, and finally state verification using stabilizer measurements is performed. If the desired measurement result is obtained, it is accepted; if not, it is rejected and the process is repeated from the beginning. Non-Clifford gates (T gates or R gates) can be realized by quantum teleportation using the auxiliary states obtained by executing them on logical Bell states, which are prepared in advance as auxiliary states. This auxiliary state is a so-called stabilizer state, and can be prepared in an error-tolerant manner. In other words, if the desired measurement result is obtained, it is accepted; if not, it is rejected and the process is repeated from the beginning. YThe (π / 4) gate) can also be implemented by quantum teleportation using an error-resilient auxiliary state prepared by a method called magic state distillation (an efficient method for magic state distillation for concatenated codes is also known. For example, see Non-Patent Document 5). However, the above general gate implementation methods are not very efficient. Below, we will explain a more efficient method.

[0051] First, we will explain the logical Hadamard gate. When performing a logical Hadamard gate on all logical qubits in a code block, it can be performed using a physical Hadamard gate and a physical swap gate as shown in Figure 12. When performing a logical Hadamard gate on some logical qubits in a code block, it can be performed as shown in the example of Figure 26, using another code block (called the coded +0 state) with a plus state |+> = H|0> = (|0> + |1>) / √2 at the position corresponding to the logical qubit on which the gate is to be performed and a zero state |0> at the other position. Note that the logical CNOT gate and logical Hadamard gate in Figure 26 can be performed as shown in Figures 11 and 12, respectively, because they are performed on all logical qubits in the code block. Therefore, it is important to prepare the coded +0 state in an error-resilient manner. Figure 26 shows how to perform a logical Hadamard gate on some logical qubits in a code block using the coded +0 state. The arrow indicates that the gate is executed when the decoded result of the measurement is 1. ECT stands for error-correcting teleportation.

[0052] Figures 27, 28, and 29 show error-resilient methods for generating three level-1 coding +0 states of a concatenated [[6,4,2]] code. Figure 30 shows a swap gate for the level-1 coding qubit of a concatenated [[6,4,2]] code. In Figures 28 and 29, the measurement results of the physical qubits are accepted if they are all 0; otherwise, they are rejected and the process is restarted. By combining the three level-1 coding +0 states of Figures 27 to 29, the coding Hadamard gate of Figure 12, and the swap gate of Figure 30, any level-1 coding +0 state can be prepared in an error-resilient manner.

[0053] Figure 31 shows state verification for generating an error-resilient level 2 coding + 0 state. Any level 2 coding + 0 state can be prepared using the level 1 coding + 0 state obtained as described above and the encoder of Figure 9. However, the encoder of Figure 9 is not error-resilient. Therefore, four level 2 coding + 0 states obtained in this way are prepared, and one error-resilient level 2 coding + 0 state is generated by the state verification of Figure 31 (if the decoded results of all measurement results are 0, it is accepted; if not, start over).

[0054] Next, the logical phase gate S will be described. Fig. 32 shows a method for executing the logical phase gate S. A logical phase gate for any logical quantum bit in a code block can be executed as shown in Fig. 32 by using an auxiliary state in which all logical quantum bits in the code block are eigenstates |Y> of the logical Y operator with an eigenvalue of 1, a logical Hadamard gate for the logical quantum bit that executes the gate, and a logical CNOT gate that can be executed as shown in Fig. 11.

[0055] FIG. 33 shows an error-resilient |YYYY> Ｌ The state verification for preparation of |YYYY> in FIG. Ｌ can be prepared in an error-resilient manner by preparing two non-error-resilient encoders in the encoder of FIG. 9 and then performing the state verification in FIG. 33 (accept if the decoded results of all measurement results are 0, otherwise start over from the beginning).

[0056] Logic R that is a non-Clifford gate Ｙ The (π / 4) gate can be implemented in an error-tolerant manner using conventional methods (see, for example, Non-Patent Document 5) and the logical Hadamard gate of FIG.

[0057] Finally, the logical CNOT gate will be described. Fig. 34 shows a method for error-resilient execution of the logical CNOT gate between different code blocks. To execute the logical CNOT gate on a specific logical qubit pair between different code blocks, the coding +0 state can be used to execute the gate in an error-resilient manner as shown in Fig. 34.

[0058] Figure 35 shows a method for error-resilient implementation of a logical CNOT gate within the same code block. To implement a logical CNOT gate for a specific logical qubit pair within the same code block, another code block (called the coded B0 state) having the Bell state |B>=(|00>+|11>) / √2 in a position corresponding to the logical qubit pair on which the gate is to be implemented and the zero state |0> in another position can be used, as in the example of Figure 35.

[0059] In Figure 36, the level 1 encoding B0 state |B １２ 00> Ｌ１ Here, if the measurement result of the physical qubit is 0, it is accepted, otherwise it is rejected and we start over. Any level 1 coded B0 state can also be prepared in an error-resilient manner in a similar way.

[0060] Figure 37 shows how to generate one level-1 coded BB state (all Bell states in the code block) of a concatenated [[6,4,2]] code in an error-resilient manner. Any level-1 coded BB state can be prepared in an error-resilient manner in a similar manner.

[0061] Fig. 38 shows a method for generating an error-resilient coded B0 state using a coded BB state. In general, a coded B0 state can be prepared in an error-resilient manner as shown in Fig. 38 using a coded BB state and a coded +0 state.

[0062] 39 and 40 show how to generate two level-2 encoded BB states of a concatenated [[6,4,2]] code in an error-resilient manner. (2) 1,j and Q (2) 2,j , Q (2) 3,j and Q (2) 4,j (j=1,...,4) is a Bell state. (2) i,1 and Q (2) i,2 , Q (2) i,3 and Q (2) i,4(i = 1, ..., 4) form a Bell state. The coded CNOT gates in Figures 39 and 40 can be implemented using the coded CNOT gates in the level 1 code block described above. In this way, a level 2 coded BB state consisting of coded qubit pairs in the same row and column can be prepared in an error-resilient manner in a similar manner. Coded BB states consisting of pairs in the same axis direction at levels 3 and above of the concatenated [[6,4,2]] code can also be prepared in an error-resilient manner in a similar manner. From the above, it is possible to implement a logical CNOT gate for pairs in the same axis direction at any level of the concatenated [[6,4,2]] code in an error-resilient manner.

[0063] [First Embodiment] Fig. 41 is a functional block diagram of a decoding device 1 according to a first embodiment. The decoding device 1 decodes the measurement results of a concatenated quantum code formed by concatenating multiple quantum codes. The decoding device 1 includes a level 1 minimum distance candidate calculation unit 11, a level i minimum distance candidate calculation unit 14, and an output unit 20, where i takes an integer value of i = 2, ..., L. In Fig. 41, the configurations for i = 2, 3, i, and L are shown as representative examples.

[0064] The level 1 minimum distance candidate calculation unit 11 calculates the minimum distance, which is the smallest value of the Hamming distance between the code word and the measurement result, for each code block in level 1, and selects the coded bit sequence corresponding to the minimum distance for each code block in level 1 as the minimum distance candidate for that code block. The level 2 minimum distance candidate calculation unit 12 calculates the minimum distance for each code block in level 2 using the minimum distance and minimum distance candidate for level 1, and selects the coded bit sequence corresponding to the minimum distance for each code block in level 2 as the minimum distance candidate for that code block. The level 3 minimum distance candidate calculation unit 13 calculates the minimum distance for each code block in level 3 using the minimum distance and minimum distance candidate for level 2, and selects the coded bit sequence corresponding to the minimum distance for each code block in level 3 as the minimum distance candidate for that code block. The level i minimum distance candidate calculation unit 14 calculates the minimum distance for each code block in level i using the minimum distance and minimum distance candidate for level (i-1), and selects the coded bit sequence corresponding to the minimum distance for each code block in level i as the minimum distance candidate for that code block. A level L minimum distance candidate calculation unit 19 calculates the minimum distance of the code block of level L using the minimum distance and minimum distance candidate of level (L-1), and selects an encoded bit string corresponding to the minimum distance of the code block of level L as a minimum distance candidate of the code block of level L. An output unit 20 selects and outputs one of the minimum distance candidates of level L.

[0065] According to this embodiment, concatenated high-rate quantum codes can be decoded efficiently with high performance.

[0066] [Second embodiment] The second embodiment is a modification of the decoding device of the first embodiment. ｉ Let s be the number of qubits in the quantum code at level i. ｉ When the number of parity checks corresponding to the Z stabilizer of the quantum code at level i is set as (n ｉ -s ｉ ) minimum distance candidates at level (i-1) and s ｉ Using the parity check conditions, the remaining s ｉThe coded bit strings of the level (i-1) code blocks are calculated.

[0067] According to this embodiment, decoding can be achieved so that the parity check equation is always satisfied.

[0068] [Third Embodiment] The third embodiment is a modification of the decoding device of the second embodiment. In this embodiment, for i=3, . . . , L, k ｉ is the number of encoded quantum bits of the quantum code at level i, the level i minimum distance candidate calculation unit is ｉ In calculating the minimum distance of the coded bit strings of the code blocks of level (i-1), n ｉ－１ From the minimum distance candidates of the code blocks of level (i-2), ｉ－１ -k ｉ－１ -s ｉ－１ ) minimum distance candidates are selected, and the remaining (k ｉ－１ +s ｉ－１ ) coded bit sequences of level (i-2) are coded bit sequences of level (i-1) code blocks and s ｉ－１ The minimum distance of the coded bit strings of each code block at level (i-1) is calculated based on the parity check conditions.

[0069] According to this embodiment, the parity check conditions can be taken into consideration when calculating the minimum distance.

[0070] [Fourth Embodiment] The fourth embodiment is a modification of the decoding device of the second or third embodiment. In this embodiment, when the total number of minimum distance candidates at level (i-1) is greater than a predetermined threshold, the level i minimum distance candidate calculation unit narrows down and selects candidates in order from the code block at level (i-1) with the greatest number of minimum distance candidates, so that the total number of candidates is equal to or less than the threshold.

[0071] According to this embodiment, when the number of candidates at level (i-1) is large, the total number of minimum distance candidates can be kept below a predetermined threshold value.

[0072] Fifth Embodiment The fifth embodiment is a modification of the decoding device of the third embodiment. In this embodiment, when the total number of minimum distance candidates at level (i-2) is greater than a predetermined threshold, the level i minimum distance candidate calculation unit narrows down and selects candidates in order from the code block at level (i-2) with the greatest number of minimum distance candidates, so that the total number of candidates is equal to or less than the threshold.

[0073] According to this embodiment, when the number of candidates at level (i-2) is large, the total number of minimum distance candidates can be kept below a predetermined threshold value.

[0074] Sixth Embodiment The sixth embodiment is a modification of the decoding device of the first embodiment. In this embodiment, for a natural number L' between 1 and L, the level 1 minimum distance candidate calculation unit outputs a failure if any one of the minimum distance candidates at (L - L' + 1) levels between L' and L is not 1.

[0075] According to this embodiment, it is possible to perform exceptional processing when the measurement result includes a loss error.

[0076] [Seventh Embodiment] The seventh embodiment is a modification of the decoding device of the first embodiment. In this embodiment, if the measurement result includes an erasure error, the level 1 minimum distance candidate calculation unit calculates the Hamming distance after removing the bit corresponding to the erasure error.

[0077] According to this embodiment, it is possible to perform exceptional processing when the measurement result includes a loss error.

[0078] Eighth Embodiment Fig. 42 is a flowchart showing the processing steps of a decoding method according to an eighth embodiment. This method includes step S11 of calculating a level 1 minimum distance candidate, step S14 of calculating a level i minimum distance candidate, and output step S20, where i takes an integer value of i=2, ..., L. Fig. 42 shows steps i=2, i, and L as representative steps.

[0079] In step S11, when the number of concatenations of the concatenated quantum code is L, the minimum value of the Hamming distance between the codeword and the measurement result, i.e., the minimum distance, is found for each code block of level 1, and an encoded bit sequence corresponding to the minimum distance for each code block of level 1 is selected as a minimum distance candidate for each code block of level 1. In step S12, for each code block of level 2 consisting of a plurality of code blocks of level 1, the minimum distance and minimum distance candidate for level 1 are used to find the minimum distance for each code block of level 2, and an encoded bit sequence corresponding to the minimum distance for each code block of level 2 is selected as a minimum distance candidate for each code block of level 2. In step S13, for each code block of level 3 consisting of a plurality of code blocks of level 2, the minimum distance and minimum distance candidate for level 2 are used to find the minimum distance for each code block of level 3, and an encoded bit sequence corresponding to the minimum distance for each code block of level 3 is selected as a minimum distance candidate for each code block of level 3. In step S14, for each code block of level i consisting of a plurality of code blocks of level (i-1), the minimum distance of each code block of level i is calculated using the minimum distance and minimum distance candidate of level (i-1), and an encoded bit sequence corresponding to the minimum distance of each code block of level i is selected as a minimum distance candidate of each code block of level i. In step S19, for a code block of level L consisting of a plurality of code blocks of level (L-1), the minimum distance of the code block of level L is calculated using the minimum distance and minimum distance candidate of level (L-1), and an encoded bit sequence corresponding to the minimum distance of each code block of level L is selected as a minimum distance candidate of the code block of level L. In output step S20, one of the minimum distance candidates of level L is selected and output.

[0080] According to this embodiment, concatenated high-rate quantum codes can be decoded efficiently with high performance.

[0081] [Ninth Embodiment] The ninth embodiment is a program. Fig. 42 shows the procedure of processing executed by this program. This program causes a computer to execute the following steps: a level 1 minimum distance candidate calculation step of calculating, when the number of concatenations of a concatenated quantum code is L, the minimum value of the Hamming distance between a code word and a measurement result, i.e., the minimum distance, in each code block of level 1, and selecting an encoded bit sequence corresponding to the minimum distance of each code block of level 1 as a minimum distance candidate for each code block of level 1; a level i minimum distance candidate calculation step of calculating, for i = 2, ..., L, the minimum distance of each code block of level i consisting of multiple code blocks of level (i-1), using the minimum distance and minimum distance candidate of level (i-1), and selecting an encoded bit sequence corresponding to the minimum distance of each code block of level i as a minimum distance candidate for each code block of level i; and a selection step of selecting and outputting one of the minimum distance candidates of level L.

[0082] According to this embodiment, a function for efficiently decoding concatenated high-rate quantum codes with high performance up to at least level 2 can be implemented as a computer program.

[0083] Tenth Embodiment A tenth embodiment is a quantum computer, which includes the decryption device of the first embodiment.

[0084] According to this embodiment, it is possible to realize a quantum computer having a function of decoding concatenated high-rate quantum codes efficiently with high performance.

[0085] The present invention has been described above based on the embodiments. These embodiments are merely examples, and it will be understood by those skilled in the art that various modifications are possible in the combination of each component and each treatment process, and that such modifications are also within the scope of the present invention.

[0086] Any combination of the above-described embodiments and modifications is also useful as an embodiment of the present disclosure. A new embodiment resulting from the combination has the combined effects of the respective embodiments and modifications.

[0087] When understanding the abstract technical ideas of the embodiments and modifications, the technical ideas should not be interpreted as being limited to the contents of the embodiments and modifications. The above-described embodiments and modifications are merely illustrative examples, and many design modifications, such as changes, additions, and deletions of components, are possible. In the embodiments, the contents in which such design modifications are possible are emphasized by adding the notation "embodiment." However, design modifications are also permitted even in contents without such notation.

[0088] The present invention relates to a decoding device, a decoding method, a program, and a quantum computer.

[0089] 1. Decoding device. 11. Level 1 minimum distance candidate calculation unit. 12. Level 2 minimum distance candidate calculation unit. 13. Level 3 minimum distance candidate calculation unit. 14. Level i minimum distance candidate calculation unit. 19. Level L minimum distance candidate calculation unit. 20. Output unit. S11. Level 1 minimum distance candidate calculation step. S12. Level 2 minimum distance candidate calculation step. S13. Level 3 minimum distance candidate calculation step. S14. Level i minimum distance candidate calculation step. S19. Level L minimum distance candidate calculation step. S20. Output step.

Claims

1. A decoding device that decodes the measurement results of a concatenated quantum code formed by concatenating a plurality of quantum codes, wherein the number of concatenations of the concatenated quantum code is L, comprising: a level 1 minimum distance candidate calculation unit that calculates, for each code block of level 1, a minimum distance that is the smallest value of the Hamming distance between a code word and a measurement result, and selects an encoded bit sequence corresponding to the minimum distance of each code block of level 1 as a minimum distance candidate for each code block of level 1; a level i minimum distance candidate calculation unit that, for i=2, ..., L, for each code block of level i consisting of code blocks of a plurality of levels (i-1), calculates the minimum distance of each code block of level i using the minimum distance and minimum distance candidate of level (i-1), and selects an encoded bit sequence corresponding to the minimum distance of each code block of level i as a minimum distance candidate for each code block of level i; and an output unit that selects and outputs one of the minimum distance candidates of level L.

2. n ｉ Let s be the number of qubits in the quantum code at level i. ｉ is the number of parity checks corresponding to the Z stabilizer of the quantum code of level i, the level i minimum distance candidate calculation unit is ｉ -s ｉ ) minimum distance candidates at level (i-1) and s ｉ Using the parity check conditions, the remaining s ｉ 2. The decoding device according to claim 1, wherein the decoding device calculates coded bit strings of level (i-1) code blocks.

3. For i = 3, ..., L, k ｉ is the number of encoded quantum bits of the quantum code at level i, ｉ In calculating the minimum distance of the coded bit strings of the level (i-1) code blocks, n ｉ－１ From the minimum distance candidates of the code blocks of level (i-2), ｉ－１ -k ｉ－１ -s ｉ－１ ) minimum distance candidates are selected, and the remaining (k ｉ－１ +s ｉ－１ ) the level (i-2) coded bit sequences are combined with the level (i-1) coded bit sequences of the level (i-1) code blocks and the level (i-1) coded bit sequences of the level (i-2) code blocks. ｉ－１ 3. The decoding device according to claim 2, wherein the minimum distance of the coded bit strings of each code block of said level (i-1) is calculated by determining the minimum distance of the coded bit strings of each code block of said level (i-1) from among said parity check conditions.

4. A decoding device as described in claim 2 or 3, characterized in that when the total number of minimum distance candidates at level (i-1) is greater than a predetermined threshold, the level i minimum distance candidate calculation unit narrows down and selects candidates in order from the code block at level (i-1) with the largest number of minimum distance candidates so that the total number of candidates is less than or equal to the threshold.

5. The decoding device described in claim 3, characterized in that when the total number of minimum distance candidates at level (i-2) is greater than a predetermined threshold, the level i minimum distance candidate calculation unit narrows down and selects candidates in order from the code block at level (i-2) with the largest number of minimum distance candidates so that the total number of candidates is less than or equal to the threshold.

6. The decoding device described in claim 1, characterized in that for a natural number L' between 1 and L, the level 1 minimum distance candidate calculation unit outputs a failure if any one of the number of minimum distance candidates at levels between L' and L (L-L'+1) is not 1.

7. The decoding device according to claim 1, characterized in that, when the measurement result also includes an erasure error, the level 1 minimum distance candidate calculation unit calculates the Hamming distance after removing the bit corresponding to the erasure error.

8. A decoding method for decoding a measurement result of a concatenated quantum code formed by concatenating a plurality of quantum codes, comprising: a level 1 minimum distance candidate calculation step of calculating, where L is the number of concatenations of the concatenated quantum code, the minimum value of the Hamming distance between the codeword and the measurement result, i.e., the minimum distance, for each code block of level 1, and selecting an encoded bit sequence corresponding to the minimum distance for each code block of level 1 as a minimum distance candidate for each code block of level 1; a level i minimum distance candidate calculation step of calculating, for i=2, ..., L, for each code block of level i consisting of a plurality of code blocks of level (i-1), using the minimum distance and minimum distance candidate for level (i-1) to calculate the minimum distance for each code block of level i, and selecting an encoded bit sequence corresponding to the minimum distance for each code block of level i as a minimum distance candidate for level i; and a selection step of selecting and outputting one of the minimum distance candidates for level L.

9. A program for decoding measurement results of a concatenated quantum code formed by concatenating a plurality of quantum codes, the program causing a computer to execute the following steps: a level 1 minimum distance candidate calculation step, where L is the number of concatenations of the concatenated quantum code, for each code block of level 1, calculating the minimum value of the Hamming distance between the code word and the measurement result, i.e., the minimum distance, and selecting an encoded bit string corresponding to the minimum distance of each code block of level 1 as a minimum distance candidate for each code block of level 1; a level i minimum distance candidate calculation step, for i=2, ..., L, for each code block of level i consisting of a plurality of code blocks of level (i-1), calculating the minimum distance of each code block of level i using the minimum distance and minimum distance candidate of level (i-1), and selecting an encoded bit string corresponding to the minimum distance of each code block of level i as a minimum distance candidate for level i; and a selection step, selecting and outputting one of the minimum distance candidates of level L.

10. A quantum computer equipped with the decoding device according to claim 1.

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