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arXiv:2510.26189 · Physical Review Applied

Practical hybrid decoding scheme for parity-encoded spin systems

Yoshihiro Nambu

Published Dec 17, 2025• 0 citations
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Abstract

We propose a practical hybrid decoding scheme for the parity-encoding architecture. This architecture was first introduced by N. Sourlas [Phys. Rev. Lett. , 070601 (2005)] 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 [Sci. Adv. , e1500838 (2015)] 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 (LDPC) 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; or (2) using deterministic decoding techniques for the classical LDPC code, such as belief propagation and a 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 the soft-annealing concept proposed by Sourlas.

DOI
10.1103/drry-blqs

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We propose a practical hybrid decoding scheme for the parity-encoding architecture. This architecture was first introduced by N. Sourlas [Phys. Rev. Lett. , 070601 (2005)] 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 [Sci. Adv. , e1500838 (2015)] 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 (LDPC) 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; or (2) using deterministic decoding techniques for the classical LDPC code, such as belief propagation and a 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 the soft-annealing concept proposed by Sourlas.
This paper presents classical benchmark simulations of a practical hybrid decoding scheme for parity-encoded spin systems, which is well-suited to the development of quantum annealing devices based on on-chip superconducting technology. We compared the performance of finding the optimal solution using two embedding schemes for emulating all-to-all connectivity from local interactions: the SLHZ scheme, proposed by Sourlas, Lechner, Hauke, and Zoller, and the commonly used minor embedding (ME) scheme. We found that the SLHZ scheme is more efficient than the ME scheme when combined with postreadout classical decoding based on the classical bit-flipping algorithm, although the SLHZ scheme itself is substantially less efficient than the ME scheme.
Quantum annealing (QA) has emerged as a promising candidate for fast solvers for combinatorial optimization problems (COPs) and has attracted the interest of many researchers. Since COP is logically encoded in the Ising interaction among spins, its realization necessitates a spin system with all-to-all connectivity, presenting technical challenges in the physical implementation of large-scale QA devices. W. Lechner, P. Hauke, and P. Zoller proposed a parity-encoding (PE) architecture consisting of an expanded spin system with only local connectivity among them to circumvent this difficulty in developing near-future QA devices. They suggested that this architecture not only alleviates implementation challenges and enhances scalability but also possesses intrinsic fault tolerance. This paper proposes a practical decoding method tailored to correlated spin-flip errors in spin readout of PE architecture. Our work is based on the close connection between PE architecture and classical low-density parity-check (LDPC) codes. We show that the bit-flip (BF) decoding algorithm can correct independent and identically distributed errors in the readout of the SLHZ system with comparable performance to the belief propagation (BP) decoding algorithm. Then, we show evidence that the proposed BF decoding algorithm can efficiently correct correlated spinflip errors by simulation. The result suggests that introducing post-readout BF decoding reduces the computational cost of QA using the PE architecture and improves the performance of global optimal solution search. Our results emphasize the importance of the proper selection of decoding algorithms to exploit the inherent fault tolerance potential of the PE architecture.
Lechner, Hauke, and Zoller proposed a parity-encoded spin-embedding scheme for quantum annealing (QA) with all-to-all connectivity to avoid the issue of limited connectivity in near-term QA hardware and to enable the implementation thereof using only geometrically local interactions between spins fabricated on the planar substrate. Nevertheless, the redundant encoding of logical information, i.e., using a large number of spins to embed the logical information, increases the computational cost and reduces the efficiency. In this study, we show through Monte Carlo simulation that this redundant encoding may be exploited to solve the problems of the inefficiency and computational cost of the parity-encoded scheme by incorporating appropriate decoding, namely classical post-processing, of the spins to retrieve the logical information. Our findings open up the possibility of parity-encoded schemes for realizing the QA with near-term quantum technologies.
The Sourlas-Lechner-Hauke-Zoller (SLHZ) scheme for quantum annealing uses the parity to encode logical variables and has several advantages, but it has not been implemented for large-scale quantum annealers. If the SLHZ-based approach can be implemented on currently available quantum annealers, we can evaluate its performance. An efficient method to embed the parity-encoded model into the hardware graphs of available quantum annealers is one of the key elements for this approach. We propose a qubit-efficient embedding scheme for parity-encoded Hamiltonians on quantum annealers with the Zephyr connectivity. We give an explicit constructive embedding of the interaction graph of an intermediate Hamiltonian, which contains only one- and two-body interactions, into the Zephyr graph. Our embedding maps each spin to a two-qubit chain using systematic chain-assignment rules. Its validity is verified via the resulting chain-to-chain connectivity. Our embedding also provides practical flexibility. Chains assigned to ancillary spins allow reduction to a single physical qubit, leading to options to avoid inactive qubits. The number of required qubits per spin in the parity Hamiltonian is three, which is fewer than that for a known embedding scheme for the Pegasus graph.
Recently, W. Lechner, P. Hauke, and P. Zoller [Sci. Adv. 1, e1500838 (2015)] have proposed a quantum annealing architecture, in which a classical spin glass with all-to-all pairwise connectivity is simulated by a spin glass with geometrically local interactions. We interpret this architecture as a classical error-correcting code, which is highly robust against weakly correlated bit-flip noise, and we analyze the code's performance using a belief-propagation decoding algorithm. Our observations may also apply to more general encoding schemes and noise models.
In this paper, the LDPC (Low Density Parity Check Codes) decoding algorithm has been investigated using statistical mechanics properties. The main advantage of LDPC codes is their performance that works in near Shannon limit, and the ability to use iterative decoding algorithms. In 1989 N. Sourlas showed that error-correcting codes can be considered as Spin-Glass systems thus making it possible to model LDPC codes as an Ising model, opening the way for information theorists to solve coding problems with the power of statistical mechanics. According to N. Sourlas, the decoding problem can be solved by finding the ground state of the corresponding spin-system Hamiltonian. The main goal of this paper is to review as simple as possible the statistical properties of LDPC codes, and how it is used especially facing the decoding problem.