Pushing the boundary of quantum advantage in hard combinatorial optimization with probabilistic computers
Shuvro Chowdhury, Navid Anjum Aadit, Andrea Grimaldi, Eleonora Raimondo, Atharva Raut, P. Aaron Lott, Johan H. Mentink, Marek M. Rams, Federico Ricci‐Tersenghi, M. Chiappini, Luke S. Theogarajan, Tathagata Srimani, Giovanni Finocchio, Masoud Mohseni, Kerem Yunus Camsari
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.
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.
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 of D-Wave's annealing-based quantum simulators have established new benchmarks for quantum computational advantage [arXiv:2403.00910]. However, the precise location of the classical-quantum computational frontier remains an open question, as classical simulation strategies continue to evolve. Here, we demonstrate that time-dependent variational Monte Carlo (t-VMC) with a physically motivated Jastrow-Feenberg wave function can efficiently simulate the quantum annealing of spin glasses up to system sizes previously thought to be intractable. Our approach achieves accuracy comparable to that of quantum processing units while requiring only polynomially scaling computational resources, in stark contrast to entangled-limited tensor network methods that scale exponentially. For systems up to 128 spins on a three-dimensional diamond lattice, we maintain correlation errors below 7%, which match or exceed the precision of existing quantum hardware. Rigorous assessments of residual energies and time-dependent variational principle errors establish clear performance benchmarks for classical simulations. These findings substantially shift the quantum advantage frontier and underscore that classical variational techniques, which are not fundamentally constrained by entanglement growth, remain competitive at larger system sizes than previously anticipated.
Critical decision-making issues in science, engineering, and industry are based on combinatorial optimization; however, its application is inherently limited by the NP-hard nature of the problem. A specialized paradigm of analogue quantum computing, quantum annealing (QA), has been proposed to solve these problems by encoding optimization problems into physical energy landscapes and solving them by quantum tunnelling systematically through exploration of solution space. This is a critical review that summarizes the current applications of quantum annealing to combinatorial optimization and includes a theoretical background, hardware designs, algorithm implementation strategies, encoding and embedding schemes, protocols to benchmark quantum annealing, areas of implementation, and links with the quantum algorithms implementation with gate-based hardware and classical solvers. We develop a unified framework, relating adiabatic quantum dynamics, Ising and QUBO models, stoquastic and non-stoquastic Hamiltonians, and diabatic transitions to modern flux-qubit annealers (Chimera, Pegasus, Zephyr topologies), and emergent architectures (Lechner-Hauke-Zoller systems, Rydberg atom platforms), and hybrids of quantum and classical computation. Through our analysis, we find that overhead in embedding and encoding is the largest determinant of the scalability and performance (this is not just the number of qubits). Minor embeddings also usually have a physical qubit count per logical variable of between 5 and 12 qubits, which limits effective problem capacity by 80-92% and, due to chain-breaking errors, compromises the quality of solutions.
Quantum computing, leveraging quantum mechanics, promises to enhance optimization processes in artificial intelligence (AI) by addressing challenges faced by classical computing in solving complex problems. This study investigates the role of quantum computing in optimizing AI tasks, focusing on algorithms such as Quantum Approximate Optimization Algorithm (QAOA) and Quantum Annealing (QA) for improved efficiency in optimization. A hybrid quantum-classical approach was used to evaluate the performance of quantum algorithms in optimization tasks, including combinatorial optimization and machine learning model tuning. QAOA and QA were applied to benchmark problems, such as Max-Cut and Traveling Salesman Problem (TSP). The study involved a comparative analysis between quantum algorithms and classical optimization techniques using performance metrics like solution quality, computational time, and scalability. Data were gathered through simulation on a quantum computing platform, and results were analyzed using statistical methods to assess improvements. The results showed that quantum computing significantly outperformed classical methods in solving large-scale optimization problems. QAOA demonstrated a 30% improvement in solution quality for Max-Cut problems, and QA showed a 25% faster convergence in TSP compared to classical algorithms. Quantum algorithms demonstrated scalability advantages as the problem size increased, with quantum methods achieving up to a 40% reduction in computational time for large instances. Furthermore, 70% of the tested cases showed significant improvements in solution accuracy, confirming the potential of quantum computing in real-world AI applications. Quantum computing, through QAOA and QA, offers substantial improvements in solving complex optimization problems in AI, presenting a transformative opportunity for the future of optimization algorithms.
Abstract Quantum Monte Carlo (QMC) techniques are widely used in a variety of scientific problems and much work has been dedicated to developing optimized algorithms that can accelerate QMC on standard processors (CPU). With the advent of various special purpose devices and domain specific hardware, it has become increasingly important to establish clear benchmarks of what improvements these technologies offer compared to existing technologies. In this paper, we demonstrate 2 to 3 orders of magnitude acceleration of a standard QMC algorithm using a specially designed digital processor, and a further 2 to 3 orders of magnitude by mapping it to a clockless analog processor. Our demonstration provides a roadmap for 5 to 6 orders of magnitude acceleration for a transverse field Ising model (TFIM) and could possibly be extended to other QMC models as well. The clockless analog hardware can be viewed as the classical counterpart of the quantum annealer and provides performance within a factor of < 10 of the latter. The convergence time for the clockless analog hardware scales with the number of qubits as ∼ N, improving the ∼ N2 scaling for CPU implementations, but appears worse than that reported for quantum annealers by D-Wave.