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Generate Seed Generators for Quantum Stabilizer Codes

Xiao Fang-ying, Chen Hanwu

Published Jan 1, 2009• 0 citations
Publisher

Abstract

Stabilizer is a very important description method of quantum error-correcting code. Before to generate quantum codes based on the stabilizer it needs to generate seed generator. The problem of finding seed generators is transformed into the problems of determining linear independence or linear dependence of vectors and finding the solutions of equations; we also present quick determinant method, sort and find method and solve equations method to verify the validity of the operator; we present the total number of seed generators of the stabilizer code. The method introduced in this paper can be applied to the quantum stabilizer code contains both type 1 and type 2 generator; and also can be applied to the quantum stabilizer code contains only type 1 generators. Through standardize the generator of the stabilizer and the seed generators at the same time; it can quickly get the quantum encoding circuit. It also presented the number of 2-qubit gates and 1-qubit gates needed for the encoding circuit.

DOI
10.1109/csie.2009.168

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Stabilizer is a very important description method of quantum error-correcting code. Before to generate quantum codes based on the stabilizer it needs to generate seed generator. The problem of finding seed generators is transformed into the problems of determining linear independence or linear dependence of vectors and finding the solutions of equations; we also present quick determinant method, sort and find method and solve equations method to verify the validity of the operator; we present the total number of seed generators of the stabilizer code. The method introduced in this paper can be applied to the quantum stabilizer code contains both type 1 and type 2 generator; and also can be applied to the quantum stabilizer code contains only type 1 generators. Through standardize the generator of the stabilizer and the seed generators at the same time; it can quickly get the quantum encoding circuit. It also presented the number of 2-qubit gates and 1-qubit gates needed for the encoding circuit.
Quantum computing is an emerging technology that has the potential to achieve exponential speedups over their classical counterparts. To achieve quantum advantage, quantum principles are being applied to fields such as communications, information processing, and artificial intelligence. However, quantum computers face a fundamental issue since quantum bits are extremely noisy and prone to decoherence. Keeping qubits error free is one of the most important steps towards reliable quantum computing. Different stabilizer codes for quantum error correction have been proposed in past decades and several methods have been proposed to import classical error correcting codes to the quantum domain. Design of encoding and decoding circuits for the stabilizer codes have also been proposed. Optimization of these circuits in terms of the number of gates is critical for reliability of these circuits. In this paper, we propose a procedure for optimization of encoder circuits for stabilizer codes. Using the proposed method, we optimize the encoder circuit in terms of the number of 2-qubit gates used. The proposed optimized eight-qubit encoder uses 18 CNOT gates and 4 Hadamard gates, as compared to 14 single qubit gates, 33 2-qubit gates, and 6 CCNOT gates in a prior work. The encoder and decoder circuits are verified using IBM Qiskit. We also present encoder circuits for the Steane code and a 13-qubit code, that are optimized with respect to the number of gates used, leading to a reduction in number of CNOT gates by 1 and 8, respectively.
Quantum computers have the potential to provide exponential speedups over their classical counterparts. Quantum principles are being applied to fields such as communications, information processing, and artificial intelligence to achieve quantum advantage. However, quantum bits are extremely noisy and prone to decoherence. Thus, keeping the qubits error free is extremely important toward reliable quantum computing. Quantum error correcting codes have been studied for several decades and methods have been proposed to import classical error correcting codes to the quantum domain. Along with the exploration into novel and more efficient quantum error correction codes, it is also essential to design circuits for practical realization of these codes. This article serves as a tutorial on designing and simulating quantum encoder and decoder circuits for stabilizer codes. We first describe Shor’s 9-qubit code which was the first quantum error correcting code. We discuss the stabilizer formalism along with the design of encoding and decoding circuits for stabilizer codes such as the five-qubit code and Steane code. We also design nearest neighbor compliant circuits for the above codes. The circuits were simulated and verified using IBM Qiskit.
In this paper, a new method for the construction of a quantum stabilizer code from circulant permutation matrices is discussed. First, we choose a finite-length vector randomly, and we can construct circulant permutation matrices from the vectors. Then, the parity-check matrix can be produce from the circulant permutation matrices. Hence, the generators of stabilizer code are determined according to the parity-check matrix and quantum stabilizer group are defined from the generators. From the stabilizer group, codewords of the proposed quantum codes can also be generated. Finally, a complete efficient encoding and decoding quantum circuit of [[7,1,3]] is proposed. [[7,1,3]] is stabilizer code that construction based on our method is an seven-qubit code that protects a one-qubit state with up to one error, which is very important for quantum information processing.
To implement a quantum error-correction protocol, we first need a scheme to prepare our state in the correct subspace of the code, and this can be done using a unitary encoding circuit. Majorana codes are special since any gates that transform such codes must preserve fermionic parity. In this paper, we present an algorithm that uses the stabilizer matrix to compute unitary encoding circuits for Majorana codes. We present two approaches, both of which use a version of Gaussian elimination with row operations replaced with elementary fermionic Clifford operations. One approach uses an additional ancilla mode and works for all Majorana stabilizer codes, while the second approach does not use an ancilla but does not work if the total parity is inside the stabilizer group.
The quantum block stabilizer codes have developed to overcome noise of communication channels. Although a lot of efforts have investigated their error correction properties, just Gottesman has designed an algorithm for their encoding. Number of gates required for the encoding circuit account for most of the overhead needed for practical implementation of quantum communication protocols. In this paper a different approach for encoding quantum block codes is proposed which is conceptually easier than Gottesman's encoding algorithm. Moreover, the simulation results show that the number of gates decreases by a factor 43% on average.