De-Signing Hamiltonians for Quantum Adiabatic Optimization

Elizabeth Crosson1, Tameem Albash1,2, Itay Hen3,4, and A. P. Young5

1Center for Quantum Information and Control (CQuIC), Department of Physics and Astronomy , University of New Mexico, Albuquerque, NM 87131, USA
2Electrical and Computer Engineering, University of New Mexico, Albuquerque, NM 87131, USA
3Information Sciences Institute, University of Southern California, Marina del Rey, California 90292, USA
4Department of Physics and Astronomy and Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA
5Department of Physics, University of California, Santa Cruz, California 95064, USA

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Abstract

Quantum fluctuations driven by non-stoquastic Hamiltonians have been conjectured to be an important and perhaps essential missing ingredient for achieving a quantum advantage with adiabatic optimization. We introduce a transformation that maps every non-stoquastic adiabatic path ending in a classical Hamiltonian to a corresponding stoquastic adiabatic path by appropriately adjusting the phase of each matrix entry in the computational basis. We compare the spectral gaps of these adiabatic paths and find both theoretically and numerically that the paths based on non-stoquastic Hamiltonians have generically smaller spectral gaps between the ground and first excited states, suggesting they are less useful than stoquastic Hamiltonians for quantum adiabatic optimization. These results apply to any adiabatic algorithm which interpolates to a final Hamiltonian that is diagonal in the computational basis.

Non-stoquastic Hamiltonians, defined as Hamiltonians with positive or non-real off-diagonal matrix elements in every choice of local basis, have been conjectured to be an important and perhaps essential missing ingredient for achieving a quantum advantage with quantum adiabatic optimization (QAO). This conjecture stems primarily from the fact that adiabatic computation based on non-stoquastic Hamiltonians can be universal for quantum computing and is not efficiently simulable by quantum Monte Carlo. However, in order for non-stoquastic Hamiltonians to provide a quantum advantage, they must enhance the minimum spectral gap along the Hamiltonian interpolation of the adiabatic algorithm relative to their stoquastic counterpart. In this work we define a locality-preserving mapping which takes every non-stoquastic QAO protocol to a corresponding stoquastic QAO protocol. Considering various ensembles (dense matrices, signed graphs, and local Hamiltonians) we find that the non-stoquastic adiabatic paths have smaller spectral gaps than the corresponding stoquastic adiabatic paths, with high probability. Our results call into question the promise attributed to non-stoquastic drivers to serve as generic catalysts of quantum speedups in QAO.

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Cited by

[1] E. J. Crosson and D. A. Lidar, "Prospects for Quantum Enhancement with Diabatic Quantum Annealing", arXiv:2008.09913.

[2] Jarrod R. McClean, Matthew P. Harrigan, Masoud Mohseni, Nicholas C. Rubin, Zhang Jiang, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush, and Hartmut Neven, "Low depth mechanisms for quantum optimization", arXiv:2008.08615.

[3] Marios Ioannou, Stephen Piddock, Milad Marvian, Joel Klassen, and Barbara M. Terhal, "Sign-curing local Hamiltonians: termwise versus global stoquasticity and the use of Clifford transformations", arXiv:2007.11964.

[4] Eleni Marina Lykiardopoulou, Alex Zucca, Sam A. Scivier, and Mohammad H. Amin, "Improving nonstoquastic quantum annealing with spin-reversal transformations", arXiv:2010.00065.

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