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arXiv:2004.11066 · Physical Review Research

Mean field analysis of reverse annealing for code-division multiple-access multiuser detection

Shunta Arai, Masayuki Ohzeki, Kazuyuki Tanaka

Published Jul 2, 2021• 36 citations
arXivHTMLPDF

Abstract

Code-division multiple-access (CDMA) multiuser detection is a kind of signal recovery problem. The main problem of CDMA multiuser detection is to estimate the original signal from the degraded information. In CDMA multiuser detection, the first-order phase transition happens. The first-order phase transition degrades the estimation performance. To avoid or mitigate the first-order phase transition, we apply adiabatic reverse annealing (ARA) to CDMA multiuser detection. In ARA, we introduce the initial Hamiltonian, which corresponds to the prior information of the original signal into quantum annealing (QA) formulation. The ground state of the initial Hamiltonian is the initial candidate solution. By using the prior information of the original signal, ARA enhances the performance of QA for CDMA multiuser detection. We evaluate the typical ARA performance of CDMA multiuser detection by means of statistical mechanics using the replica method. At first, we consider the oracle cases where the initial candidate solution is randomly generated with a fixed fraction of the original signal in the initial state. In the oracle cases, the first-order phase transition can be avoided or mitigated by ARA if we prepare for the proper initial candidate solution. We validate our theoretical analysis with quantum Monte Carlo simulations. The theoretical results to avoid the first-order phase transition are consistent with the numerical results. Next, we consider the practical cases where we prepare for the initial candidate solution obtained by commonly used algorithms. We show that the practical algorithms can exceed the threshold to avoid the first-order phase transition. Finally, we test the performance of ARA with the initial candidate solution obtained by the practical algorithm. In this case, ARA cannot avoid the first-order phase transition even if the initial candidate solution exceeds the threshold to avoid the first-order phase transition.

DOI
10.1103/physrevresearch.3.033006

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Code-division multiple-access (CDMA) multiuser detection is a kind of signal recovery problem. The main problem of CDMA multiuser detection is to estimate the original signal from the degraded information. In CDMA multiuser detection, the first-order phase transition happens. The first-order phase transition degrades the estimation performance. To avoid or mitigate the first-order phase transition, we apply adiabatic reverse annealing (ARA) to CDMA multiuser detection. In ARA, we introduce the initial Hamiltonian, which corresponds to the prior information of the original signal into quantum annealing (QA) formulation. The ground state of the initial Hamiltonian is the initial candidate solution. By using the prior information of the original signal, ARA enhances the performance of QA for CDMA multiuser detection. We evaluate the typical ARA performance of CDMA multiuser detection by means of statistical mechanics using the replica method. At first, we consider the oracle cases where the initial candidate solution is randomly generated with a fixed fraction of the original signal in the initial state. In the oracle cases, the first-order phase transition can be avoided or mitigated by ARA if we prepare for the proper initial candidate solution. We validate our theoretical analysis with quantum Monte Carlo simulations. The theoretical results to avoid the first-order phase transition are consistent with the numerical results. Next, we consider the practical cases where we prepare for the initial candidate solution obtained by commonly used algorithms. We show that the practical algorithms can exceed the threshold to avoid the first-order phase transition. Finally, we test the performance of ARA with the initial candidate solution obtained by the practical algorithm. In this case, ARA cannot avoid the first-order phase transition even if the initial candidate solution exceeds the threshold to avoid the first-order phase transition.
Abstract In this chapter, we analyze the typical performance of adiabatic reverse annealing (ARA) for Sourlas codes. Sourlas codes are representative error-correcting codes related to p-body spin-glass models and have a first-order phase transition for p>2p>2p>2 p > 2 , which degrades the estimation performance. In the ARA formulation, we introduce the initial Hamiltonian which incorporates the prior information of the solution into a vanilla quantum annealing (QA) formulation. The ground state of the initial Hamiltonian represents the initial candidate solution. To avoid the first-order phase transition, we apply ARA to Sourlas codes. We evaluate the typical ARA performance for Sourlas codes using the replica method. We show that ARA can avoid the first-order phase transition if we prepare for the proper initial candidate solution.
Adiabatic reverse annealing (ARA) is an improvement to conventional quantum annealing (QA) that uses an initial guess at the desired ground state to circumvent problematic phase transitions. Despite encouraging results in the closed-system setting, Ref. [1] has suggested on the basis of numerical simulations that ARA may lose its advantage in the presence of decoherence. Here, we revisit this problem from a more analytical perspective. Using the ppp-spin model as a solvable example, together with the adiabatic master equation to describe the effects of the environment (valid at weak coupling), we show that ARA can in fact succeed in open systems but that the temperature of the environment plays a key role. We first demonstrate that, in the adiabatic limit, the system will follow the instantaneous equilibrium state as long as the protocol does not pass through any (finite-temperature) phase transitions. Given this, there are two distinct mechanisms by which ARA can break down at high temperature: either there are no paths that avoid transitions, or the equilibrium state itself is disordered. When the temperature is sufficiently low that neither of these occur, then ARA succeeds. Remarkably, there are even situations in which the environment benefits ARA: we find parameter values for which no transition-avoiding paths exist at zero temperature but such paths appear at non-zero temperature.
Adiabatic reverse annealing (ARA) has been proposed as an improvement to conventional quantum annealing for solving optimization problems, in which one takes advantage of an initial guess at the solution to suppress problematic phase transitions. Here we interpret the performance of ARA through its effects on the free energy landscape, and use the intuition gained to introduce a classical analogue to ARA termed ``simulated reverse annealing'' (SRA). This makes it more difficult to claim that ARA provides a quantum advantage in solving a given problem, as not only must ARA succeed but the corresponding SRA must fail. As a solvable example, we analyze how both protocols behave in the infinite-range (non-disordered) ppp-spin model. Through both the thermodynamic phase diagrams and explicit dynamical behavior, we establish that the quantum algorithm has no advantage over its classical counterpart: SRA succeeds not only in every case where ARA does but even in a narrow range of parameters where ARA fails.
We examine the average-case performance of a code-division multiple-access (CDMA) multiuser demodulator in which quantum fluctuations are utilized to demodulate the original message within the context of Bayesian inference. The quantum fluctuations are built into the system as a transverse field in the infinite-range Ising spin glass model. We evaluate the performance measurements by using statistical mechanics. We confirm that the CDMA multiuser modulator using quantum fluctuations achieve roughly the same performance as the conventional CDMA multiuser modulator through thermal fluctuations on average. We also find that the relationship between the quality of the original information retrieval and the amplitude of the transverse field is somehow a "universal feature" in typical probabilistic information processing, viz., in image restoration, error-correcting codes, and CDMA multiuser demodulation.
Detecting active users in a non-orthogonal multiple access (NOMA) network poses a significant challenge for 5G/6G applications. Traditional algorithms tackling this task, relying on classical processors, have to make a compromise between performance and complexity. However, a quantum computing based strategy called quantum annealing (QA) can mitigate this tradeoff. In this paper, we first propose a mapping between the AUD searching problem and the identification of the ground state of an Ising Hamiltonian. Then, we compare the execution times of our QA approach for several code domain multiple access (CDMA) scenarios. We evaluate the impact of the cross-correlation properties of the chosen codes in a NOMA network for detecting the active user's set.
We study adiabatic reverse annealing (ARA) in an open system. In the closed system (unitary) setting, this annealing protocol allows avoidance of first-order quantum phase transitions of selected models, resulting in an exponential speedup compared with standard quantum annealing, provided that the initial state of the algorithm is close in Hamming distance to the target one. Here, we show that decoherence can significantly modify this conclusion: by resorting to the adiabatic master equation approach, we simulate the dynamics of the ferromagnetic ppp-spin model with p=3p=3p=3 under independent and collective dephasing. For both models of decoherence, we show that the performance of open system ARA is far less sensitive to the choice of the initial state than its unitary counterpart, and, most significantly, that open system ARA by and large loses its time to solution advantage compared to standard quantum annealing. These results suggest that as a stand-alone strategy ARA is unlikely to experimentally outperform standard ``forward'' quantum annealing, and that error mitigation strategies will likely be required in order to realize the benefits of ARA in realistic, noisy settings.