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Chinese Physics B

Encoding entanglement-assisted quantum stabilizer codes

王云江, 白宝明, Li Zhuo, 彭进业, 肖鹤玲

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

We address the problem of encoding entanglement-assisted (EA) quantum error-correcting codes (QECCs) and of the corresponding complexity.We present an iterative algorithm from which a quantum circuit composed of CNOT,H,and S gates can be derived directly with complexity O(n～2) to encode the qubits being sent.Moreover,we derive the number of each gate consumed in our algorithm according to which we can design EA QECCs with low encoding complexity.Another advantage brought by our algorithm is the easiness and efficiency of programming on classical computers.

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We address the problem of encoding entanglement-assisted (EA) quantum error-correcting codes (QECCs) and of the corresponding complexity.We present an iterative algorithm from which a quantum circuit composed of CNOT,H,and S gates can be derived directly with complexity O(n～2) to encode the qubits being sent.Moreover,we derive the number of each gate consumed in our algorithm according to which we can design EA QECCs with low encoding complexity.Another advantage brought by our algorithm is the easiness and efficiency of programming on classical computers.
We address the problem of encoding entanglement-assisted (EA) quantum error-correcting codes (QECCs) and of the corresponding complexity. We present an iterative algorithm from which a quantum circuit composed of CNOT , H , and S gates can be derived directly with complexity O ( n 2 ) to encode the qubits being sent. Moreover, we derive the number of each gate consumed in our algorithm according to which we can design EA QECCs with low encoding complexity. Another advantage brought by our algorithm is the easiness and efficiency of programming on classical computers.
In encoder circuits built on the stabilizer formalism, the dominant contribution to circuit complexity comes from the use of controlled (CNOT) gates, making CNOT-count reduction a central circuit-design objective. Entanglement-assisted (EA) quantum QC-LDPC codes offer strong error-correction capabilities with structured parity-check matrices, but their practical use depends on efficient encoder circuits and the availability of pre-shared Bell pairs (ebits). In this paper, we adopt a prior entanglement-assisted QC-LDPC (EAQC) encoder construction. We formulate the encoder optimization as a search over GF(2) row operations that decompose the binary matrix derived from its CNOT sub-sequence. We solve this problem using a beam search algorithm guided by a Hamming-distance heuristic. For the tested EA quantum QC-LDPC code families, the proposed method achieves CNOT-count reductions of 7.3-34.0% relative to the baseline EAQC encoder. The optimized circuits also outperform the Patel-Markov-Hayes and greedy cost-minimization baselines, and are verified by stabilizer-tableau simulation. These results show that substantial encoder simplification is possible for structured EA QC-LDPC codes.
Quantum error-correcting codes are used to protect quantum information from decoherence. A raw state is mapped, by an encoding circuit, to a codeword so that the most likely quantum errors from a noisy quantum channel can be removed after a decoding process.A good encoding circuit should have some desired features, such as low depth, few gates, and so on. In this paper, we show how to practically implement an encoding circuit of gate complexity O(n(n - k + c)/ log n) for an [[n, k; c]] quantum stabilizer code with the help of c pairs of maximally-entangled states. For the special case of an [[n, k]] stabilizer code with c = 0, the encoding complexity is O(n(n-k)/ log n), which is previously known to be O(n2/ log n). For c > 0, this suggests that the benefits from shared entanglement come at an additional cost of encoding complexity.Finally we discuss decoding of entanglement-assisted quantum stabilizer codes and extend previously known computational hardness results on decoding quantum stabilizer codes.
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 need to be protected by quantum error-correcting codes against decoherence. One of the most interesting and useful classes of quantum codes is the class of quantum stabilizer codes. Entanglement-assisted (EA) quantum codes are a class of stabilizer codes that make use of preshared entanglement between the sender and the receiver. We provide several code constructions for entanglement-assisted quantum codes. The MacWilliams identity for quantum codes leads to linear programming bounds on the minimum distance. We find new constraints on the simplified stabilizer group and the logical group, which help improve the linear programming bounds on entanglement-assisted quantum codes. The results also can be applied to standard stabilizer codes. In the real world, quantum gates are faulty. To implement quantum computation fault-tolerantly, quantum codes with certain properties are needed. We first analyze Knill's postselection scheme in a two-dimensional architecture. The error performance of this scheme is better than other known concatenated codes. Then we propose several methods to protect syndrome extraction against measurement errors.
Entanglement-assisted (EA) stabilizer codes enhance the rate of error correction in relation to codes with no pre-shared entanglement. Meanwhile, bosonic error-correcting codes, such as the Gottesman-Kitaev-Preskill (GKP) code, can be concatenated with qubit stabilizer codes to significantly reduce the logical failure probability of those stabilizer codes. First, we combine the above two concepts to propose an EA version of the qubit-into-oscillators concatenated code that chains an EA-stabilizer (outer) code with a GKP (inner) code. As an example we present a three-qubit EA-repetition concatenated with a GKP code. Second, we propose an EA version of the non-Gaussian oscillator-into-oscillators concatenated code that chains a GKP (outer) code with an EA-stabilizer (inner) code. As an example we present a GKP code concatenated with a three-qubit EA repetition code that uses two maximally entangled modes (emodes) and suppresses the variances of both position and momentum quadrature errors of a data mode. Furthermore, we generalize the latter example to a family of GKP code concatenated with a nnn-qubit EA repetition code that uses n−1{n-1}n−1 emodes and suppresses the variances of both position and momentum quadrature errors of a data mode by a factor 1/n{1/n}1/n.