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arXiv:2401.07184 · Physical Review Letters

Scaling Advantage in Approximate Optimization with Quantum Annealing

Humberto Munoz-Bauza, Daniel A. Lidar

Published Apr 23, 2025• 35 citations
arXivHTMLPDF

Abstract

Quantum annealing is a heuristic optimization algorithm that exploits quantum evolution to find low-energy states. Quantum annealers have scaled up in recent years to tackle increasingly larger and more highly connected discrete optimization and quantum simulation problems. Nevertheless, a computational quantum advantage in exact optimization using quantum annealing hardware has so far remained elusive. Here, we present evidence for a quantum annealing scaling advantage in approximate optimization. The advantage is relative to the top classical heuristic algorithm: parallel tempering with isoenergetic cluster moves (PT-ICM). The setting is a family of 2D spin-glass problems with high-precision spin-spin interactions. To achieve this advantage, we implement quantum annealing correction (QAC): an embedding of a bit-flip error-correcting code with energy penalties that leverages the properties of the D-Wave Advantage quantum annealer to yield over 1,300 error-suppressed logical qubits on a degree-5 interaction graph. We generate random spin-glass instances on this graph and benchmark their time-to-epsilon, a generalization of the time-to-solution metric for low-energy states. We demonstrate that, with QAC, quantum annealing exhibits a scaling advantage over PT-ICM at sampling low-energy states with an optimality gap of at least 1.0%. This amounts to the first demonstration of an algorithmic quantum speedup in approximate optimization.

DOI
10.1103/physrevlett.134.160601

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Quantum annealing is a heuristic optimization algorithm that exploits quantum evolution to find low-energy states. Quantum annealers have scaled up in recent years to tackle increasingly larger and more highly connected discrete optimization and quantum simulation problems. Nevertheless, a computational quantum advantage in exact optimization using quantum annealing hardware has so far remained elusive. Here, we present evidence for a quantum annealing scaling advantage in approximate optimization. The advantage is relative to the top classical heuristic algorithm: parallel tempering with isoenergetic cluster moves (PT-ICM). The setting is a family of 2D spin-glass problems with high-precision spin-spin interactions. To achieve this advantage, we implement quantum annealing correction (QAC): an embedding of a bit-flip error-correcting code with energy penalties that leverages the properties of the D-Wave Advantage quantum annealer to yield over 1,300 error-suppressed logical qubits on a degree-5 interaction graph. We generate random spin-glass instances on this graph and benchmark their time-to-epsilon, a generalization of the time-to-solution metric for low-energy states. We demonstrate that, with QAC, quantum annealing exhibits a scaling advantage over PT-ICM at sampling low-energy states with an optimality gap of at least 1.0%. This amounts to the first demonstration of an algorithmic quantum speedup in approximate optimization.
Abstract Quantum annealing is a heuristic optimization algorithm that exploits quantum evolution to approximately find lowest energy states [1, 2]. Quantum annealers have scaled up in recent years to tackle increasingly larger and more highly connected discrete optimization and quantum simulation problems [3–7]. Nevertheless, despite numerous attempts, a computational quantum advantage in exact optimization using quantum annealing hardware has so far remained elusive [8– 16]. Here, we present evidence for a quantum annealing scaling advantage in approximate optimization. The advantage is relative to the top classical heuristic algorithm: parallel tempering with isoenergetic cluster moves (PT-ICM) [17]. The setting is a family of 2D spin-glass problems with high-precision spin-spin interactions. To achieve this advantage, we implement quantum annealing correction (QAC) [18]: an embedding of a bit-flip error-correcting code with energy penalties that leverages the properties of the D-Wave Advantage quantum annealer to yield over 1,300 error-suppressed logical qubits on a degree-5 interaction graph. We generate random spin-glass instances on this graph and benchmark their time-to-epsilon, a generalization of the time-to-solution metric [8] for low-energy states. We demonstrate that with QAC, quantum annealing exhibits a scaling advantage over PT-ICM at sampling low energy states with an optimality gap [19] of at least 1.0%. This amounts to the first demonstration of an algorithmic quantum speedup in approximate optimization.
Recent demonstrations on specialized benchmarks have reignited excitement for quantum computers, yet their advantage for real-world problems remains an open question. Here, we show that probabilistic computers, co-designed with hardware to implement Monte Carlo algorithms, provide a scalable classical pathway for solving hard optimization problems. We focus on two algorithms applied to three-dimensional spin glasses: discrete-time simulated quantum annealing and adaptive parallel tempering. We benchmark these methods against a leading quantum annealer. For simulated quantum annealing, increasing replicas improves residual energy scaling, consistent with extreme value theory. Adaptive parallel tempering, supported by non-local isoenergetic cluster moves, scales more favorably and outperforms simulated quantum annealing. Field Programmable Gate Arrays or specialized chips can implement these algorithms in modern hardware, leveraging massive parallelism to accelerate them while improving energy efficiency. Our results establish a rigorous classical baseline for assessing practical quantum advantage and present probabilistic computers as a scalable platform for real-world optimization challenges.
There has been considerable progress in the design and construction of quantum annealing devices. However, a conclusive detection of quantum speedup over traditional silicon-based machines remains elusive, despite multiple careful studies. In this work we outline strategies to design hard tunable benchmark instances based on insights from the study of spin glasses - the archetypal random benchmark problem for novel algorithms and optimization devices. We propose to complement head-to-head scaling studies that compare quantum annealing machines to state-of-the-art classical codes with an approach that compares the performance of different algorithms and/or computing architectures on different classes of computationally hard tunable spin-glass instances. The advantage of such an approach lies in having to only compare the performance hit felt by a given algorithm and/or architecture when the instance complexity is increased. Furthermore, we propose a methodology that might not directly translate into the detection of quantum speedup, but might elucidate whether quantum annealing has a "`quantum advantage" over corresponding classical algorithms like simulated annealing. Our results on a 496 qubit D-Wave Two quantum annealing device are compared to recently-used state-of-the-art thermal simulated annealing codes.
Abstract Quantum annealing has the potential to provide a speedup over classical algorithms in solving optimization problems. Just as for any other quantum device, suppressing Hamiltonian control errors will be necessary before quantum annealers can achieve speedups. Such analog control errors are known to lead to JJJ J -chaos, wherein the probability of obtaining the optimal solution, encoded as the ground state of the intended Hamiltonian, varies widely depending on the control error. Here, we show that JJJ J -chaos causes a catastrophic failure of quantum annealing, in that the scaling of the time-to-solution metric becomes worse than that of a deterministic (exhaustive) classical solver. We demonstrate this empirically using random Ising spin glass problems run on the two latest generations of the D-Wave quantum annealers. We then proceed to show that this doomsday scenario can be mitigated using a simple error suppression and correction scheme known as quantum annealing correction (QAC). By using QAC, the time-to-solution scaling of the same D-Wave devices is improved to below that of the classical upper bound, thus restoring hope in the speedup prospects of quantum annealing.
The observation of an unequivocal quantum speedup remains an elusive objective for quantum computing. The D-Wave quantum annealing processors have been at the forefront of experimental attempts to address this goal, given their relatively large numbers of qubits and programmability. A complete determination of the optimal time-to-solution (TTS) using these processors has not been possible to date, preventing definitive conclusions about the presence of a scaling advantage. The main technical obstacle has been the inability to verify an optimal annealing time within the available range. Here we overcome this obstacle and present a class of problem instances for which we observe an optimal annealing time using a D-Wave 2000Q processor over a range spanning up to more than 200020002000 qubits. This allows us to perform an optimal TTS benchmarking analysis and perform a comparison to several classical algorithms, including simulated annealing, spin-vector Monte Carlo, and discrete-time simulated quantum annealing. We establish the first example of a scaling advantage for an experimental quantum annealer over classical simulated annealing: we find that the D-Wave device exhibits certifiably better scaling than simulated annealing, with 95%95\%95% confidence, over the range of problem sizes that we can test. However, we do not find evidence for a quantum speedup: simulated quantum annealing exhibits the best scaling by a significant margin. Our construction of instance classes with verifiably optimal annealing times opens up the possibility of generating many new such classes, paving the way for further definitive assessments of scaling advantages using current and future quantum annealing devices.
In a recent Letter [H. Munoz-Bauza and D. Lidar, Phys. Rev. Lett. 134, 160601 (2025)], quantum annealing was reported to exhibit a scaling advantage in approximately solving quadratic unconstrained binary optimization (QUBO) problems. Here, we revisit these findings by employing the simulated bifurcation machine (SBM), a nonlinear dynamical system that exploits chaotic behavior rather than thermal fluctuations. Our approach originates from quantum dynamics and shares key operational features with quantum annealing: (i) nearly parallel evolution and (ii) a well-defined relation between the energy gap, run-time, and solution quality. We obtain comparable or superior scaling, closing the reported quantum-classical gap. We further show that the small instances studied previously are insufficient to infer asymptotic behavior. Extending the analysis to larger problems reveals robust classical performance, indicating that current quantum annealers are unlikely to exhibit a clear scaling advantage over SBM-like solvers on quantum-annealing-correction-type QUBO problems under the run-time accounting studied here. Finally, we identify sparse problem classes where future quantum devices could achieve a genuine scaling advantage, once hardware overheads are mitigated.