This paper considers the design and decoding of polar codes for general classical-quantum (CQ) channels. It focuses on decoding via belief-propagation with quantum messages (BPQM) and, in particular, the idea of paired-measurement BPQM (PM-BPQM) decoding. Since the PM-BPQM decoder admits a classical density evolution (DE) analysis, one can use DE to design a polar code for any CQ channel and then efficiently compute the trade-off between code rate and error probability. We have also implemented and tested a classical simulation of our PM-BPQM decoder for polar codes. While the decoder can be implemented efficiently on a quantum computer, simulating the decoder on a classical computer actually has exponential complexity. Thus, simulation results for the decoder are somewhat limited and are included primarily to validate our theoretical results.
This paper considers the design and decoding of polar codes for general classical-quantum (CQ) channels. It focuses on decoding via belief-propagation with quantum messages (BPQM) and, in particular, the idea of paired-measurement BPQM (PM-BPQM) decoding. Since the PM-BPQM decoder admits a classical density evolution (DE) analysis, one can use DE to design a polar code for any CQ channel and then efficiently compute the trade-off between code rate and error probability. We have also implemented and tested a classical simulation of our PM-BPQM decoder for polar codes. While the decoder can be implemented efficiently on a quantum computer, simulating the decoder on a classical computer actually has exponential complexity. Thus, simulation results for the decoder are somewhat limited and are included primarily to validate our theoretical results.
Belief propagation with quantum messages (BPQM) provides a low-complexity alternative to collective measurements for communication over classical--quantum channels. Prior BPQM constructions and density-evolution (DE) analyses have focused on binary alphabets. Here, we generalize BPQM to symmetric q-ary pure-state channels (PSCs) whose output Gram matrix is circulant. For this class, we show that bit-node and check-node combining can be tracked efficiently via closed-form recursions on the Gram-matrix eigenvalues, independent of the particular physical realization of the output states. These recursions yield explicit BPQM unitaries and analytic bounds on the fidelities of the combined channels in terms of the input-channel fidelities. This provides a DE framework for symmetric q-ary PSCs that allows one to estimate BPQM decoding thresholds for LDPC codes and to construct polar codes on these channels.
Belief propagation (BP) is a classical algorithm that approximates the marginal distribution associated with a factor graph by passing messages between adjacent nodes in the graph. It gained popularity in the 1990’s as a powerful decoding algorithm for LDPC codes. In 2016, Renes introduced a belief propagation with quantum messages (BPQM) and described how it could be used to decode classical codes defined by tree factor graphs that are sent over the classical-quantum pure-state channel. In this work, we propose an extension of BPQM to general binary-input symmetric classical-quantum (BSCQ) channels based on the implementation of a symmetric "paired measurement". While this new paired-measurement BPQM (PMBPQM) approach is suboptimal in general, it provides a concrete BPQM decoder that can be implemented with local operations. Finally, we demonstrate that density evolution can be used to analyze the performance of PMBPQM on tree factor graphs. As an application, we compute noise thresholds of some LDPC codes with BPQM decoding for a class of BSCQ channels.
While the ultimate limits on the rate of quantum communication over noisy channels are well-established, very few efficient coding schemes achieving these limits are known. This contrasts sharply with transmitting classical data over classical channels, where polar codes and low-density parity-check codes both efficiently achieve the classical capacity of any noisy channel. In this work we construct an efficient entanglement generation scheme which achieves the coherent information for a broad class of channels. For degradable channels in the class, which includes truncated versions of the pure loss Bosonic channel, the resulting scheme achieves the quantum capacity. The construction makes use of polar codes for two kinds of classical-input, quantum-output (CQ) channels: those with commuting outputs and those whose outputs are heralded mixtures of pure states. The former case is amenable to decoding by classical polar decoding algorithms, the latter by the belief propagation with quantum messages (BPQM) algorithm. To our knowledge, this construction is the first explicit and efficiently decodable protocol that achieves the energy-constrained quantum capacity of the pure-loss Bosonic channel.
Belief propagation with quantum messages (BPQM) is a quantum algorithm that decodes classical codes transmitted over classical--quantum channels. It realizes optimal decoding on tree factor graphs over pure-state classical-quantum channels. However, this tree-based analysis does not ensure vanishing block-error probability for LDPC Tanner graphs with cycles. In this work, we construct a two-stage BPQM decoder for random
q-ary LDPC codes over symmetric
q-ary pure-state channels, where
q is prime, and prove that its ensemble-average block-error probability vanishes as the blocklength
N tends to infinity. For regular ensembles with
dv≥3, fidelity bounds yield double-exponential decay of the average symbol-error probability throughout the BPQM success region. We apply depth-
ℓ BPQM to coordinates with tree neighbourhoods and treat the remaining coordinates as erasures. With a suitable
ℓ=Θ(loglogN), a noncommutative union bound controls the BPQM decoding errors, while the minimum-distance property guarantees erasure recovery. We also extend the analysis to finite-support irregular ensembles. These results are relevant to quantum algorithms based on Regev's reduction, where coherent decoding uncomputes a codeword register. Decoded quantum interferometry (DQI) uses a closely related Fourier-based framework that reduces sparse max-LINSAT optimization problems to LDPC decoding problems on pure-state channels. Our results justify the use of BPQM in the decoding step of DQI and of coding-theoretic algorithms based on Regev's reduction whenever the code is drawn from one of the random LDPC ensembles analyzed here and the induced memoryless symmetric pure-state channel lies in the BPQM success region. Curiously, our numerical results indicate that DQI+BPQM achieves a satisfaction ratio that closely matches that of simulated annealing.