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  • Review Article
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Prospects for quantum enhancement with diabatic quantum annealing

Nature Reviews Physics volume 3, pages 466–489 (2021)Cite this article

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Abstract

Optimization, sampling and machine learning are topics of broad interest that have inspired significant developments and new approaches in quantum computing. One such approach is quantum annealing (QA). In this Review, we assess the prospects for algorithms within the general framework of QA to achieve a quantum speedup relative to classical state-of-the-art methods. We argue for continued exploration in the QA framework on the basis that improved coherence times and control capabilities will enable the near-term exploration of several heuristic quantum optimization algorithms. These continuous-time Hamiltonian computation algorithms rely on control protocols that are more advanced than those in traditional ground-state QA, while still being considerably simpler than those used in gate-model implementations. The inclusion of coherent diabatic transitions to excited states results in a generalization we refer to collectively as diabatic quantum annealing, which we believe is the most promising route to quantum enhancement within this framework. Other promising variants of traditional QA include reverse annealing, continuous-time quantum walks and analogues of parameterized quantum circuit ansatzes for machine learning. Most of these algorithms have no known efficient classical simulations, making them worthy of further investigation with quantum hardware in the intermediate-scale regime.

Key points

  • Quantum annealing (QA) is an optimization and sampling heuristic that has been implemented in the first commercial quantum computing devices featuring the largest number of qubits to date.

  • Despite a decade of effort, evidence of a quantum speedup is lacking, so researchers have turned to alternative, new QA protocols that deviate from the traditional forward-annealing, ground-state paradigm.

  • New QA protocols exhibit promising early signs for possible quantum speedups.

  • Diabatic QA appears particularly promising: it is unlikely to be efficiently classically simulatable, yet, it retains most of the simplicity of the original QA paradigm, while being less demanding than the gate model of quantum computing.

  • Alternative QA protocols can be explored in a state-of-the-art manner by embracing the full range of new out-of-equilibrium quantum dynamics generated by time-dependent effective transverse-field Ising Hamiltonians that can be natively implemented by inductively coupled flux qubits, both existing and projected at application scale.

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Fig. 1: An illustration of the lowest four energy levels of a parameterized Hamiltonian H(s).
Fig. 2: A numerical study of the MAX-2-SAT problem for n = 20 qubits.
Fig. 3: Coherent adiabatic reverse annealing for the p-spin model: \({{\boldsymbol{H}}}_{{\boldsymbol{Z}}}{\boldsymbol{=}}{\boldsymbol{-}}\,\frac{{\bf{1}}}{{\boldsymbol{n}}}{{\boldsymbol{(}}{{\boldsymbol{\sum }}}_{{\boldsymbol{i}}{\boldsymbol{=}}{\bf{1}}}^{{\boldsymbol{n}}}{{\boldsymbol{Z}}}_{{\boldsymbol{i}}}{\boldsymbol{)}}}^{{\boldsymbol{p}}}\).
Fig. 4: The D-Wave reverse annealing protocol.
Fig. 5: Optimal Quantum Approximate Optimization Algorithm approaches coherent diabatic quantum annealing.

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Acknowledgements

We are grateful to many of our colleagues in the IARPA-QEO and DARPA-QAFS programmes, in particular, A. Kerman, E. Rieffel, F. Wilhelm and K. Zick, for their comments and insights. We also acknowledge helpful discussions with Dr. Richard Harris about D-Wave’s reverse annealing protocol. The research is based upon work (partially) supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA) and the Defense Advanced Research Projects Agency (DARPA), via the U.S. Army Research Office contract W911NF-17-C-0050. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, DARPA or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes, notwithstanding any copyright annotation thereon. This material is based upon work supported in part by the National Science Foundation the Quantum Leap Big Idea under Grant No. OMA-1936388.

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Authors and Affiliations

  1. Center for Quantum Information and Control (CQuIC), Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM, USA

    E. J. Crosson

  2. Departments of Electrical and Computer Engineering, Chemistry, Physics & Astronomy, and Center for Quantum Information Science & Technology (CQIST), University of Southern California, Los Angeles, CA, USA

    D. A. Lidar

Authors
  1. E. J. Crosson
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  2. D. A. Lidar
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The authors contributed equally to all aspects of the article.

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Correspondence to D. A. Lidar.

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Nature Reviews Physics thanks Mohammad Amin, Shu Tanaka and the other anonymous reviewer for their contribution to the peer review of this work.

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Glossary

Gate model

Or gate-based quantum computing. The ‘standard’ universal model of quantum computing. Qubits are initialized in a product state, then driven through single-qubit and two-qubit gates. Qubits are measured at the end, typically in the computational basis.

Stoquastic Hamiltonians

Hamiltonians whose off-diagonal elements are all nonpositive in a given basis, usually the computational basis.

Nonstoquastic Hamiltonians

Hamiltonians with at least one positive off-diagonal element in a given basis, usually the computational basis.

J-chaos

Extreme sensitivity of the output distribution of analogue computation to the values of the coupling parameters in the Hamiltonian.

Fault-tolerance

The ability to simulate a noise-free quantum computation with faulty operations. This usually involves concatenated quantum error correction with operations whose noise is below a threshold level.

Quantum Monte Carlo

A family of classical algorithms that sample from the thermal equilibrium distribution of quantum systems. These algorithms are usually efficient in the absence of a ‘sign problem’ and for stoquastic Hamiltonians.

MAX-2-SAT

An NP-hard constraint satisfaction problem consisting of clauses of ORs of two binary variables or their negations, with ANDs between clauses. The problem is to find the variable assignment that maximizes the number of satisfied clauses.

Oracle

A black box for which the cost of computing some function (even on a superposition) is one unit.

Graph Laplacian

L = D − A, where Dii is the degree of each vertex and Di≠j = 0 (degree matrix), and Aij = 1 when there is an edge between vertices i and j, and zero otherwise (adjacency matrix).

Unstructured search

Essentially the ‘needle in a haystack’ problem. Given a function f(x): {0,…, N} ↦ {0, 1} such that \(f(x)={\delta }_{x,{x}_{0}}\) with x0 unknown, find x0 in the smallest number of queries to f.

Boolean hypercube

The Boolean hypercube is the graph whose vertices are the set of all length n bit strings ({0, 1}n). Two vertices are adjacent if and only if they differ on exactly one bit.

MAX-K-LIN-2

An NP-hard problem consisting of a system of linear equations (modulo 2) in ≤K variables. The problem is to find the variable assignment that maximizes the number of satisfied equations.

Trotterized

Trotterization is breaking up a time evolution over some finite duration into a very large number of tiny steps that add up to the original duration. The term comes from the Lie–Trotter formula.

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Crosson, E.J., Lidar, D.A. Prospects for quantum enhancement with diabatic quantum annealing. Nat Rev Phys 3, 466–489 (2021). https://doi.org/10.1038/s42254-021-00313-6

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