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 offers 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.
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 offers 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.
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.
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.
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.
Quantum annealing aims to exploit quantum mechanics to speed up the search for the solution to optimization problems. Most problems exhibit complete connectivity between the logical spin variables after they are mapped to the Ising spin Hamiltonian of quantum annealing. To account for hardware constraints of current and future physical quantum annealers, methods enabling the embedding of fully connected graphs of logical spins into a constant-degree graph of physical spins are therefore essential. Here, we compare the recently proposed embedding scheme for quantum annealing with all-to-all connectivity by Lechner, Hauke, and Zoller (LHZ) [Sci. Adv. 1, e1500838 (2015)] to the commonly used minor embedding (ME) scheme. Using both simulated quantum annealing and parallel tempering simulations, we find that for a set of instances randomly chosen from a class of fully connected, random Ising problems, the ME scheme outperforms the LHZ scheme when using identical simulation parameters, despite the fault tolerance of the latter to weakly correlated spin-flip noise. This result persists even after we introduce several decoding strategies for the LHZ scheme, including a minimum-weight decoding algorithm that results in substantially improved performance over the original LHZ scheme. We explain the better performance of the ME scheme in terms of more efficient spin updates, which allows it to better tolerate the correlated spin-flip errors that arise in our model of quantum annealing. Our results leave open the question of whether the performance of the two embedding schemes can be improved using scheme-specific parameters and new error correction approaches.
One of the current major challenges surrounding the use of quantum annealers for solving practical optimization problems is their inability to encode even moderately sized problems, the main reason for this being the rigid layout of their quantum bits as well as their sparse connectivity. In particular, the implementation of constraints has become a major bottleneck in the embedding of practical problems, because the latter is typically achieved by adding harmful penalty terms to the problem Hamiltonian, a technique that often requires an all-to-all connectivity between the qubits. Recently, a novel technique designed to obviate the need for penalty terms was suggested; it is based on the construction of driver Hamiltonians that commute with the constraints of the problem, rendering the latter constants of motion. In this work we propose general guidelines for the construction of such driver Hamiltonians given an arbitrary set of constraints. We illustrate the broad applicability of our method by analyzing several diverse examples, namely, graph isomorphism, not-all-equal three-satisfiability, and the so-called Lechner-Hauke-Zoller constraints. We also discuss the significance of our approach in the context of current and future experimental quantum annealers.
Lechner-Hauke-Zoller (LHZ) parity embedding is one of the front-running methods for implementing difficult-to-engineer long-range interactions in quantum optimization problems. Continuous-time quantum walks are a leading approach for solving quantum optimization problems. Because they populate excited states, quantum walks can avoid the exponential gap-closing problems seen in other continuous-time techniques such as quantum annealing and adiabatic quantum computation. An important question, therefore, is how continuous-time quantum walks perform in combination with the LHZ parity embedding. By numerically simulating continuous-time quantum walks on four-, five-, and six-logical-qubit Sherrington-Kirkpatrick Ising spin glass instances embedded onto the LHZ parity embedding, we are able to verify the continued efficacy of heuristics used to estimate the optimal hopping rate and the numerical agreement with the theory behind the location of the lower bound of the LHZ parity constraint strength. In addition, by comparing several different LHZ-based decoding methods, we identified postreadout error correction techniques which improve the success probability of the quantum walk.