[Paper Review] Effects of XX-catalysts on quantum annealing spectra with perturbative crossings

Natasha Feinstein, Louis Fry-Bouriaux+2 more|arXiv (Cornell University)|Mar 13, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper investigates how XX-catalysts—non-stoquastic couplings introduced to enhance the minimum energy gap in quantum annealing—dramatically alter gap scaling at perturbative crossings depending on problem encoding. It shows that identical catalysts can either suppress exponential gap closing or induce new, detrimental gap minima, depending on subtle changes in the problem’s spectral structure, particularly the sign evolution of ground-state vector components.

ABSTRACT

In adiabatic quantum annealing the required run-time to reach a given ground-state fidelity is dictated by the size of the minimum gap that appears between the ground and first excited state in the annealing spectrum. In general the presence of avoided level crossings demands an exponential increase in the annealing time with the system size which has consequences both for the efficiency of the algorithm and the required qubit coherence times. One promising avenue being explored to produce more favourable gap scaling is the introduction of non-stoquastic XX-couplings in the form of a catalyst - of particular interest are catalysts which utilise accessible information about the optimisation problem in their construction. Here we show extreme sensitivity of the effect of an XX-catalyst to subtle changes in the encoding of the optimisation problem. In particular, we observe that a targeted catalyst containing a single coupling at constant strength can significantly reduce the gap closing with system size at an avoided level crossing. For slightly different encodings of the same problems however, these same catalysts result in closing gaps in the annealing spectrum. To understand the origin of these closing gaps, we study how the evolution of the ground-state vector is altered by the presence of the catalyst and find that the negative components of the ground-state vector are key to understanding the response of the gap spectrum. We also consider how and when these closing gaps could be utilised in diabatic quantum annealing protocols - a promising alternative to adiabatic quantum annealing in which transitions to higher energy levels are exploited to reduce the run time of the algorithm.

Motivation & Objective

• To understand how non-stoquastic XX-catalysts affect the energy gap scaling at perturbative crossings in quantum annealing.
• To investigate the sensitivity of catalyst effectiveness to minor changes in problem encoding, particularly in the structure of the low-energy spectrum.
• To determine the conditions under which catalysts either suppress or induce additional gap minima in the annealing spectrum.
• To explore the implications of these findings for diabatic quantum annealing protocols, where non-adiabatic transitions can reduce runtimes.
• To assess the broader applicability of these results beyond the maximum weight independent set (MWIS) problem to other optimization problems with similar spectral features.

Proposed method

• The study analyzes the spectral response of small-scale quantum annealing instances of the maximum weight independent set (MWIS) problem under different problem encodings.
• It introduces targeted XX-catalysts with constant coupling strength to modify the Hamiltonian and probe their effect on the minimum gap at perturbative crossings.
• The analysis focuses on how the evolution of the ground-state vector—particularly the sign changes in its components—correlates with gap scaling behavior.
• Numerical simulations are performed on graphs up to 70 vertices to confirm the existence of additional gap minima and their dependence on catalyst strength.
• The researchers examine the conditions under which catalysts induce new gap minima, identifying a critical relationship between the s-value at minimum gap and the sign change of key ground-state components.
• The study evaluates the potential of such modified spectra for diabatic quantum annealing by assessing achievable ground-state fidelity at shorter annealing times.

Experimental results

Research questions

• RQ1 How does the effectiveness of an XX-catalyst in enhancing the minimum gap depend on the spectral structure of the problem Hamiltonian?
• RQ2 What causes the catalyst to suppress the gap in some encodings while inducing new, detrimental gap minima in others?
• RQ3 Under what conditions does the catalyst lead to the formation of an additional gap minimum in the spectrum?
• RQ4 How is the emergence of new gap minima related to the sign evolution of components in the ground-state vector?
• RQ5 Can the presence of such gap minima enable higher ground-state fidelity in shorter diabatic annealing times?

Key findings

• For problem encodings with milder (but still exponential) gap scaling at perturbative crossings, the XX-catalyst successfully enhances the minimum gap, improving scaling when optimized for each system size.
• In encodings with stronger intrinsic gap scaling, the same catalyst fails to improve scaling and can instead induce a new, exponentially closing gap minimum.
• The catalyst induces a new gap minimum when its strength matches the s-value at which key components of the ground-state vector change sign, indicating a critical transition point.
• The presence of this additional gap minimum enables higher ground-state fidelity at shorter annealing times, suggesting potential utility in diabatic quantum annealing protocols.
• The study confirms numerically that these effects persist up to 70-qubit instances, indicating robustness across moderate system sizes.
• The results suggest that catalyst design must be highly sensitive to problem encoding, as small changes can lead to drastically different spectral outcomes.