Quantum annealing (QA) is a continuous-time heuristic quantum algorithm for solving or approximately solving classical optimization problems. The algorithm uses a schedule to interpolate between a driver Hamiltonian with an easy-to-prepare ground state and a problem Hamiltonian whose ground state encodes solutions to an optimization problem. The standard implementation relies on the evolution being adiabatic: keeping the system in the instantaneous ground state with high probability and requiring a time scale inversely related to the minimum energy gap between the instantaneous ground and excited states. However, adiabatic evolution can lead to evolution times that scale exponentially with the system size, even for computationally simple problems. Here, we study whether non-adiabatic evolutions with optimized annealing schedules can bypass this exponential slowdown for one such class of problems called the frustrated ring model. For sufficiently optimized annealing schedules and system sizes of up to 39 qubits, we provide numerical evidence that we can avoid the exponential slowdown. Our work highlights the potential of highly-controllable QA to circumvent bottlenecks associated with the standard implementation of QA.
Due to intense interest in the potential applications of quantum computing, it is critical to understand the basis for potential exponential quantum advantage in quantum chemistry. Here we gather the evidence for this case in the most common task in quantum chemistry, namely, ground-state energy estimation, for generic chemical problems where heuristic quantum state preparation might be assumed to be efficient. The availability of exponential quantum advantage then centers on whether features of the physical problem that enable efficient heuristic quantum state preparation also enable efficient solution by classical heuristics. Through numerical studies of quantum state preparation and empirical complexity analysis (including the error scaling) of classical heuristics, in both ab initio and model Hamiltonian settings, we conclude that evidence for such an exponential advantage across chemical space has yet to be found. While quantum computers may still prove useful for ground-state quantum chemistry through polynomial speedups, it may be prudent to assume exponential speedups are not generically available for this problem.
more » « less- Award ID(s):
- 2102505
- NSF-PAR ID:
- 10405822
- Publisher / Repository:
- Nature Publishing Group
- Date Published:
- Journal Name:
- Nature Communications
- Volume:
- 14
- Issue:
- 1
- ISSN:
- 2041-1723
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract Interacting many-electron problems pose some of the greatest computational challenges in science, with essential applications across many fields. The solutions to these problems will offer accurate predictions of chemical reactivity and kinetics, and other properties of quantum systems 1–4 . Fermionic quantum Monte Carlo (QMC) methods 5,6 , which use a statistical sampling of the ground state, are among the most powerful approaches to these problems. Controlling the fermionic sign problem with constraints ensures the efficiency of QMC at the expense of potentially significant biases owing to the limited flexibility of classical computation. Here we propose an approach that combines constrained QMC with quantum computation to reduce such biases. We implement our scheme experimentally using up to 16 qubits to unbias constrained QMC calculations performed on chemical systems with as many as 120 orbitals. These experiments represent the largest chemistry simulations performed with the help of quantum computers, while achieving accuracy that is competitive with state-of-the-art classical methods without burdensome error mitigation. Compared with the popular variational quantum eigensolver 7,8 , our hybrid quantum-classical computational model offers an alternative path towards achieving a practical quantum advantage for the electronic structure problem without demanding exceedingly accurate preparation and measurement of the ground-state wavefunction.more » « less
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Abstract Quantum annealing is a powerful alternative model of quantum computing, which can succeed in the presence of environmental noise even without error correction. However, despite great effort, no conclusive demonstration of a quantum speedup (relative to state of the art classical algorithms) has been shown for these systems, and rigorous theoretical proofs of a quantum advantage (such as the adiabatic formulation of Grover’s search problem) generally rely on exponential precision in at least some aspects of the system, an unphysical resource guaranteed to be scrambled by experimental uncertainties and random noise. In this work, we propose a new variant of quantum annealing, called RFQA, which can maintain a scalable quantum speedup in the face of noise and modest control precision. Specifically, we consider a modification of flux qubit-based quantum annealing which includes low-frequency oscillations in the directions of the transverse field terms as the system evolves. We show that this method produces a quantum speedup for finding ground states in the Grover problem and quantum random energy model, and thus should be widely applicable to other hard optimization problems which can be formulated as quantum spin glasses. Further, we explore three realistic noise channels and show that the speedup from RFQA is resilient to 1/
f -like local potential fluctuations and local heating from interaction with a sufficiently low temperature bath. Another noise channel, bath-assisted quantum cooling transitions, actually accelerates the algorithm and may outweigh the negative effects of the others. We also detail how RFQA may be implemented experimentally with current technology. -
Abstract Variational quantum algorithms have the potential for significant impact on high-dimensional optimization, with applications in classical combinatorics, quantum chemistry, and condensed matter. Nevertheless, the optimization landscape of these algorithms is generally nonconvex, leading the algorithms to converge to local, rather than global, minima and the production of suboptimal solutions. In this work, we introduce a variational quantum algorithm that couples classical Markov chain Monte Carlo techniques with variational quantum algorithms, allowing the former to provably converge to global minima and thus assure solution quality. Due to the generality of our approach, it is suitable for a myriad of quantum minimization problems, including optimization and quantum state preparation. Specifically, we devise a Metropolis–Hastings method that is suitable for variational quantum devices and use it, in conjunction with quantum optimization, to construct quantum ensembles that converge to Gibbs states. These performance guarantees are derived from the ergodicity of our algorithm’s state space and enable us to place analytic bounds on its time-complexity. We demonstrate both the effectiveness of our technique and the validity of our analysis through quantum circuit simulations for MaxCut instances, solving these problems deterministically and with perfect accuracy, as well as large-scale quantum Ising and transverse field spin models of up to 50 qubits. Our technique stands to broadly enrich the field of variational quantum algorithms, improving and guaranteeing the performance of these promising, yet often heuristic, methods.more » « less
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INTRODUCTION Solving quantum many-body problems, such as finding ground states of quantum systems, has far-reaching consequences for physics, materials science, and chemistry. Classical computers have facilitated many profound advances in science and technology, but they often struggle to solve such problems. Scalable, fault-tolerant quantum computers will be able to solve a broad array of quantum problems but are unlikely to be available for years to come. Meanwhile, how can we best exploit our powerful classical computers to advance our understanding of complex quantum systems? Recently, classical machine learning (ML) techniques have been adapted to investigate problems in quantum many-body physics. So far, these approaches are mostly heuristic, reflecting the general paucity of rigorous theory in ML. Although they have been shown to be effective in some intermediate-size experiments, these methods are generally not backed by convincing theoretical arguments to ensure good performance. RATIONALE A central question is whether classical ML algorithms can provably outperform non-ML algorithms in challenging quantum many-body problems. We provide a concrete answer by devising and analyzing classical ML algorithms for predicting the properties of ground states of quantum systems. We prove that these ML algorithms can efficiently and accurately predict ground-state properties of gapped local Hamiltonians, after learning from data obtained by measuring other ground states in the same quantum phase of matter. Furthermore, under a widely accepted complexity-theoretic conjecture, we prove that no efficient classical algorithm that does not learn from data can achieve the same prediction guarantee. By generalizing from experimental data, ML algorithms can solve quantum many-body problems that could not be solved efficiently without access to experimental data. RESULTS We consider a family of gapped local quantum Hamiltonians, where the Hamiltonian H ( x ) depends smoothly on m parameters (denoted by x ). The ML algorithm learns from a set of training data consisting of sampled values of x , each accompanied by a classical representation of the ground state of H ( x ). These training data could be obtained from either classical simulations or quantum experiments. During the prediction phase, the ML algorithm predicts a classical representation of ground states for Hamiltonians different from those in the training data; ground-state properties can then be estimated using the predicted classical representation. Specifically, our classical ML algorithm predicts expectation values of products of local observables in the ground state, with a small error when averaged over the value of x . The run time of the algorithm and the amount of training data required both scale polynomially in m and linearly in the size of the quantum system. Our proof of this result builds on recent developments in quantum information theory, computational learning theory, and condensed matter theory. Furthermore, under the widely accepted conjecture that nondeterministic polynomial-time (NP)–complete problems cannot be solved in randomized polynomial time, we prove that no polynomial-time classical algorithm that does not learn from data can match the prediction performance achieved by the ML algorithm. In a related contribution using similar proof techniques, we show that classical ML algorithms can efficiently learn how to classify quantum phases of matter. In this scenario, the training data consist of classical representations of quantum states, where each state carries a label indicating whether it belongs to phase A or phase B . The ML algorithm then predicts the phase label for quantum states that were not encountered during training. The classical ML algorithm not only classifies phases accurately, but also constructs an explicit classifying function. Numerical experiments verify that our proposed ML algorithms work well in a variety of scenarios, including Rydberg atom systems, two-dimensional random Heisenberg models, symmetry-protected topological phases, and topologically ordered phases. CONCLUSION We have rigorously established that classical ML algorithms, informed by data collected in physical experiments, can effectively address some quantum many-body problems. These rigorous results boost our hopes that classical ML trained on experimental data can solve practical problems in chemistry and materials science that would be too hard to solve using classical processing alone. Our arguments build on the concept of a succinct classical representation of quantum states derived from randomized Pauli measurements. Although some quantum devices lack the local control needed to perform such measurements, we expect that other classical representations could be exploited by classical ML with similarly powerful results. How can we make use of accessible measurement data to predict properties reliably? Answering such questions will expand the reach of near-term quantum platforms. Classical algorithms for quantum many-body problems. Classical ML algorithms learn from training data, obtained from either classical simulations or quantum experiments. Then, the ML algorithm produces a classical representation for the ground state of a physical system that was not encountered during training. Classical algorithms that do not learn from data may require substantially longer computation time to achieve the same task.more » « less
