A quantum algorithm for syndrome decoding of classical error-correcting linear block codes

Abstract

We construct a quantum-classical Viterbi decoder for the classical errorcorrecting codes. Viterbi decoding is a trellis-based procedure for maximum likelihood decoding of classical error-correcting codes. In this article, we show that any number of paths with the minimum Hamming distance with respect to the received erroneous vector present in the trellis can be found using the quantum approximate optimization algorithm. We construct a generalized method to map the Viterbi decoding problem into a parameterized quantum circuit for any classical linear block codes. We propose a uniform parameter optimization strategy to optimize the parameterized quantum circuit. We observe that the proposed method is efficient for generating low-depth trainable parameterized quantum circuits. This renders the hybrid decoder more efficient than previous attempts at making quantum Viterbi algorithm. We show that using uniform parameter optimization, we obtain parameters more efficiently for the parameterized quantum circuit than many previous attempts made through random sampling and fixing the parameters.

Short Block-length Channel Coded Modulation with Random Linear Codes and QAOA Decoding

“…We conjecture that near-optimum ML performance is achievable for message lengths k ≤ 250 with p ∈ [1,10] where R = 0.5 and SN R = 10 dB as a proof-of-concept example for short block-lengths. The remainder of the paper is organized as follows.…”

“…In [9], channel decoding utilizing QAOA is introduced for binary linear codes in binary-symmetric channels (BSCs) with theoretical studies for layer-1 circuits and simulations including systematic Reed-Muller codes and Hamming codes with small code dimensions. QAOA is exploited to improve Viterbi decoding in [10] and for turbo detection in coded MIMO systems with recurrent neural networks in [12]. Syndrome decoding of binary linear codes including Hamming code and quantum codes is realized by exploiting QAOA in [11].…”

“…Besides that, (γ, β) values for p ≤ 6 are optimized with initialization of angles and then utilizing nonlinear optimization tools based on quasi-Newton method as described in [8]. Heuristic extrapolation methods based on curve fitting are utilized for p ∈ [7,10].…”

“…We also include r ≡ E J,h [ C QAOA / C(s)] and r * ≡ E J,h [min zm {C QAOA (z m )} / C(s)] denoting the ratios of C QAOA and min zm {C QAOA (z m )}, respectively, to C(s). We exploit simulations for k = 25 to accurately observe BER for varying SN R while simulating k ∈ [10,26] for R = 0.5 allows to analyze performances for increasing k. BER obtained with QAOA and ML solutions for varying M L is very close to unity for p ≥ 2. Therefore, the angles for p ∈ [7,10] obtained with heuristic extrapolation methods introduced in [8] provide high performance.…”

“…QAOA decoder is practically pre-optimized based on only R, SN R and the depth p. We simulate a total of 532000 QAOA circuits for varying R, k, p and SN R while results are shared in [15]. Simulations for BPSK based generator matrices and binary symbols show near-optimum BER and BLER for varying R in [0.3, 1], SN R ∈ [0, 10] dB and k ∈ [10,26] with depth p ∈ [1,4]. We extrapolate performance for large k and p ∈ [1,10] by obtaining near optimum performance reaching k ≤ 250 for R = 0.5 and SN R = 10 dB as a proof-of-concept example.…”