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

Burhan Gülbahar

Published Nov 27, 2023• 1 citation
Publisher

Abstract

Quantum approximate optimization algorithm (QAOA) is used for NP-hard problems on noisy intermediate-scale quantum (NISQ) devices. We demonstrate QAOA’s near-optimum maximum likelihood (ML) decoding for short block-lengths in Gaussian channels using random linear codes and channel coded modulation. Simulations with a p-layer QAOA decoder for  p ∈ [1, 4], coding rates R = k/n ∈ [0.3, 1], signal-to-noise ratio (SNR)  ∈  [0, 10] dB and k ∈ [10, 26] show near-optimum bit and block error rates. We conjecture near-optimum performance for p ∈ [1, 10], R = 0.5, SNR = 10 dB and k ≦ 250 indicating QAOA’s potential in short block-length decoding.

DOI
10.36227/techrxiv.24587403.v1

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Quantum approximate optimization algorithm (QAOA) is used for NP-hard problems on noisy intermediate-scale quantum (NISQ) devices. We demonstrate QAOA’s near-optimum maximum likelihood (ML) decoding for short block-lengths in Gaussian channels using random linear codes and channel coded modulation. Simulations with a p-layer QAOA decoder for  p ∈ [1, 4], coding rates R = k/n ∈ [0.3, 1], signal-to-noise ratio (SNR)  ∈  [0, 10] dB and k ∈ [10, 26] show near-optimum bit and block error rates. We conjecture near-optimum performance for p ∈ [1, 10], R = 0.5, SNR = 10 dB and k ≦ 250 indicating QAOA’s potential in short block-length decoding.
Quantum approximate optimization algorithm (QAOA) is used for NP-hard problems on noisy intermediate-scale quantum (NISQ) devices. We demonstrate QAOA's near-optimum maximum likelihood (ML) decoding for short block-lengths in Gaussian channels using random linear codes and channel coded modulation. Simulations with a p-layer QAOA decoder for p ∈ [1, 4], coding rates R = k/n ∈ [0.3, 1], signal-to-noise ratio (SNR) ∈ [0, 10] dB and k ∈ [10, 26] show near-optimum bit and block error rates. We conjecture near-optimum performance for p ∈ [1, 10], R = 0.5, SNR = 10 dB and k ≦ 250 indicating QAOA's potential in short block-length decoding.
Motivated by the recent advancement of quantum processors, we investigate quantum approximate optimization algorithm (QAOA) to employ quasi-maximum-likelihood (ML) decoding of classical channel codes. QAOA is a hybrid quantum-classical variational algorithm, which is advantageous for the near-term noisy intermediate-scale quantum (NISQ) devices, where the fidelity of quantum gates is limited by noise and de-coherence. We first describe how to construct Ising Hamiltonian model to realize quasi-ML decoding with QAOA. For level-1 QAOA, we derive the systematic way to generate theoretical expressions of cost expectation for arbitrary binary linear codes. Focusing on [7], [4] Hamming code as an example, we analyze the impact of the degree distribution in associated generator matrix on the quantum decoding performance. The excellent performance of higher-level QAOA decoding is verified when Pauli rotation angles are optimized through meta-heuristic variational quantum eigensolver (VQE). Furthermore, we demonstrate the QAOA decoding performance in a real quantum device.
Recent advances in quantum technologies pave the way for noisy intermediate-scale quantum (NISQ) devices, where the quantum approximation optimization algorithm (QAOA) constitutes a promising candidate for demonstrating tangible quantum advantages based on NISQ devices. In this paper, we consider the maximum likelihood (ML) detection problem of binary symbols transmitted over a multiple-input and multiple-output (MIMO) channel, where finding the optimal solution is exponentially hard using classical computers. Here, we apply the QAOA for the ML detection by encoding the problem of interest into a level-pppQAOA circuit having2p2p2pvariational parameters, which can be optimized by classical optimizers. This level-pppQAOA circuit is constructed by applying the prepared Hamiltonian to our problem and the initial Hamiltonian alternately inpppconsecutive rounds. More explicitly, we first encode the optimal solution of the ML detection problem into the ground state of a problem Hamiltonian. Using the quantum adiabatic evolution technique, we provide both analytical and numerical results for characterizing the evolution of the eigenvalues of the quantum system used for ML detection. Then, for level-1 QAOA circuits, we derive the analytical expressions of the expectation values of the QAOA and discuss the complexity of the QAOA based ML detector. Explicitly, we evaluate the computational complexity of the classical optimizer used and the storage requirement of simulating the QAOA. Finally, we evaluate the bit error rate (BER) of the QAOA based ML detector and compare it both to the classical ML detector and to the classical minimum mean squared error (MMSE) detector, demonstrating that the QAOA based ML detector is capable of approaching the performance of the classical ML detector.
Quantum-approximate optimization algorithm (QAOA) is promising in Noisy Intermediate-Scale Quantum (NISQ) computers with applications for NP-hard combinatorial optimization problems. It is recently utilized for NP-hard maximum-likelihood (ML) detection problem with challenges of optimization, simulation and performance analysis forn×nmultiple-input multiple output (MIMO) systems with largen. QAOA is recently applied by Farhi et al. on infinite size limit of Sherrington-Kirkpatrick (SK) model with a cost model including only quadratic terms. In this article, we extend the model by including also linear terms and then realize SK modeling of massive MIMO ML detection. The proposed design targets near ML performance while with complexity includingO(16p) initial operations independent from problem instance and sizenfor optimizing QAOA angles andO(n2p) quantum operations for each instance. We provide both optimized and extrapolated angles forp∈ [1, 14] and signal-to-noise (SNR)p≥ 4 for 25×25 and 12 × 12 MIMO systems modulated with BPSK and QPSK, respectively. We present two conjectures about concentration properties of QAOA and near-optimum performance for next generation massive MIMO systems coveringn< 300.
Quantum-approximate optimization algorithm (QAOA) is promising in Noisy Intermediate-Scale Quantum (NISQ) computers with applications for NP-hard combinatorial optimization problems. It is recently utilized for NP-hard maximum-likelihood (ML) detection problem with fundamental challenges of optimization, simulation and performance analysis for n x n multiple-input multiple output (MIMO) systems with large n. QAOA is recently applied by Farhi et al. on infinite size limit of Sherrington-Kirkpatrick (SK) model with a cost model including only quadratic terms. In this article, we extend application of QAOA on SK model by including also linear terms and then realize SK modeling of massive MIMO ML detection by ensuring independence from specific problem instance and size n while preserving computational complexity of O(16p) designed by Farhi et al. We provide both optimized and extrapolated angles for p in the set [1, 14] and signal-to-noise (SNR) < 12 dB achieving near-optimum ML performance for 25 x 25 MIMO system with p >= 4 in extensive simulations where 236500 different QAOA circuits are simulated. We present two conjectures about the concentration properties of QAOA and its near-optimum performance for massive MIMO systems with large sizes covering n < 300 promising significant performance for next generation massive MIMO systems.
In the massive multiple-input and multiple-output (Massive MIMO) systems, the maximum likelihood (ML) detection problem is NP-hard and becoming classically intricate with the number of transmitting antennas and symbols increasing. The quantum approximate optimization algorithm (QAOA), a leading candidate algorithm running in the noisy intermediate-scale quantum (NISQ) devices, can show quantum advantage for approximately solving combinatorial optimization problems. In this paper, we propose the QAOA based on the maximum likelihood detection solver of binary symbols. In the proposed scheme, we first conduct a universal and compact analytical expression for the expectation value of the 1-level QAOA. Second, a Bayesian optimization based parameters initialization is presented, which can speedup the convergence of the QAOA to a lower local minimum and improve the probability of measuring the exact solution. Compared to the state-of-the-art QAOA based ML detection algorithm, our scheme has the more universal and compact expectation value expression of the 1-level QAOA, and requires few quantum resources and has the higher probability to obtain the exact solution.