Belief propagation with quantum messages for quantum-enhanced classical communications

Abstract

For space-based laser communications, when the mean photon number per received optical pulse is much smaller than one, there is a large gap between communications capacity achievable with a receiver that performs individual pulse-by-pulse detection, and the quantum-optimal “joint-detection receiver” that acts collectively on long codeword-blocks of modulated pulses; an effect often termed “superadditive capacity”. In this paper, we consider the simplest scenario where a large superadditive capacity is known: a pure-loss channel with a coherent-state binary phase-shift keyed (BPSK) modulation. The two BPSK states can be mapped conceptually to two non-orthogonal states of a qubit, described by an inner product that is a function of the mean photon number per pulse. Using this map, we derive an explicit construction of the quantum circuit of a joint-detection receiver based on a recent idea of “belief-propagation with quantum messages” (BPQM). We quantify its performance improvement over the Dolinar receiver that performs optimal pulse-by-pulse detection, which represents the best “classical” approach. We analyze the scheme rigorously and show that it achieves the quantum limit of minimum average error probability in discriminating 8 (BPSK) codewords of a length-5 binary linear code with a tree factor graph. Our result suggests that a BPQM receiver might attain the Holevo capacity of this BPSK-modulated pure-loss channel. Moreover, our receiver circuit provides an alternative proposal for a quantum supremacy experiment, targeted at a specific application that can potentially be implemented on a small, special-purpose, photonic quantum computer capable of performing cat-basis universal qubit logic.

Quantum message-passing algorithm for optimal and efficient decoding

“…To illustrate why BPQM achieves optimal bit-wise decoding (as also described in [1,2]), we employ a formalism where classical and quantum states and channels can be represented by a variant of Forney factor graphs. This formalism is based on [4,5] and is introduced in detail in Section 2.3.…”

“…It has been observed empirically in [2] that the circuits for decoding subsequent bits can be simplified by constructing them from an updated factor graph where the information of previously decoded codeword bits is incoroporated. We do not address this optimization here and leave its analysis as an open question for further research.…”

“…, X k . Since the Factor graph only contains equality and check nodes, the random variable in question must be of the form X g e, 1 1 `X g e, 2 2 `¨¨¨`X g e,k k…”

“…Like BP, this decoder is bitwise optimal on codes with tree Tanner graphs. In [2] Rengaswamy et al proposed several simplifications of the algorithm and coined the name belief propagation with quantum messages (BPQM). Moreover, quite surprisingly, they also observed numerically that BPQM is blockwise optimal for a simple example code, i.e.…”

“…The algorithm presents a genuine quantum counterpart to decoding based on the classical belief propagation algorithm, which has found wide success in classical coding theory when used in conjunction with LDPC or Turbo codes. More recently Rengaswamy et al [2] numerically observed that BPQM implements the optimal decoder on a small example code, in that it implements the optimal measurement for distinguishing the quantum output states for the set of input codewords. Here we significantly expand the understanding, formalism, and applicability of the BPQM algorithm with the following contributions.…”