Practical hybrid decoding scheme for parity-encoded spin systems

TL;DR

This work presents a practical hybrid decoding strategy for the SLHZ parity-encoded spin architecture, linking LDPC-type parity constraints to spin-glass decoding and leveraging Gallager’s bit-flipping (BF) as a scalable post-readout algorithm. By framing post-readout decoding in Bayesian terms and exploiting weight-3 syndromes, the authors show that BF can match belief-propagation performance under i.i.d. noise while remaining computationally efficient. They further demonstrate a two-stage MCMC-BF hybrid that tolerates non-i.i.d. leakage and dramatically reduces sampling needs, suggesting a viable two-stage approach to soft-annealing in near-term quantum annealers. The results indicate that carefully tuned decoding, especially the Lagrange weights and hybridization with stochastic sampling, can unlock the latent potential of SLHZ for practical quantum optimization.

Abstract

We propose a practical hybrid decoding scheme for the parity-encoding architecture. This architecture was first introduced by N. Sourlas as a computational technique for tackling hard optimization problems, especially those modeled by spin systems such as the Ising model and spin glasses, and reinvented by W. Lechner, P. Hauke, and P. Zoller to develop quantum annealing devices. We study the specific model, called the SLHZ model, aiming to achieve a near-term quantum annealing device implemented solely through geometrically local spin interactions. Taking account of the close connection between the SLHZ model and a classical low-density-parity-check code, two approaches can be chosen for the decoding: (1) finding the ground state of a spin Hamiltonian derived from the SLHZ model, which can be achieved via stochastic decoders such as a quantum annealer or a classical Monte Carlo sampler; (2) using deterministic decoding techniques for the classical LDPC code, such as belief propagation and bit-flip decoder. The proposed hybrid approach combines the two approaches by applying bit-flip decoding to the readout of the stochastic decoder based on the SLHZ model. We present simulations demonstrating that this approach can reveal the latent potential of the SLHZ model, realizing soft-annealing concept proposed by Sourlas.

Figures (12)

• Figure 1: A considered model for a communication system.
• Figure 2: Two examples of a bipartite graph for $K=4$ logical spins. The dark blue circle with a label $\{k,l\}$ represents the spin variable $x_{kl}$, while the red circles with a label $\{k,l,m,n\}$ or $\{k,l,m\}$ represent the weight-4 syndrome $s_{klmn}^{(4)}$ and weight-3 syndrome $s_{klm}^{(3)}$, respectively. Let us relabel the variables with blue letters. An element of the code-word vector $\boldsymbol{x}=(x_{1},\ldots,x_{N_{v}})\in\{ \pm1\} ^{N_{v}}$ is called a variable node (VN). The $i$-th syndrome of $\boldsymbol{x}$ is defined by $s_{i}(\boldsymbol{x})=\prod_{k\in N(i)}x_{k}\in\{ \pm1\}$ and an element of vector $\boldsymbol{s}(\boldsymbol{x})=(s_{1}(\boldsymbol{x}),\ldots,s_{N_{c}}(\boldsymbol{x}))\in\{\pm1\} ^{N_{c}}$ is called a check node (CN), where $N(i)=\{ j:H_{ij}(H_{ij}^{'})=1\}$ is the VNs adjacent to a CN $i$$(1\leq i\leq N_{c})$ and $M(j)=\{i:H_{ij}(H_{ij}^{'})=1\}$ is the CNs adjacent to a VN $j$$(1\leq j\leq N_{v})$. The column and row weights of the parity-check matrix are defined by $d_{v}(i)=|M(j)|$ and $d_{c}(i)=|N(i)|$, respectively.
• Figure 3: Bipartite graph topologically equivalent to Fig.\ref{['fig:2']}(a). This graph avoids any edge crossings.
• Figure 4: Binary symmetric channel associated with the parameters $\gamma_{ij}$ and $p_{k}$.
• Figure 5: Bipartite graph for $K=5$ logical spins. Solid blue lines show an example of the shortest loop of edges connecting VNs and CNs, which has a length of 6.