[Paper Review] On the relevance of avoided crossings away from quantum critical point to the complexity of quantum adiabatic algorithm

Sergey Knysh, Vadim Smelyanskiy|arXiv (Cornell University)|May 17, 2010

Quantum Computing Algorithms and Architecture5 references22 citations

TL;DR

This paper challenges the claim that avoided crossings near zero transverse field cause exponentially small gaps in the quantum adiabatic algorithm for random NP-complete problems. Using corrected perturbation theory that accounts for exponential ground-state degeneracy at zero field, the authors show that such avoided crossings require large transverse fields ($O(1)$), where perturbation theory breaks down, implying the proposed failure mechanism is unlikely. The analysis suggests the minimum gap does not scale exponentially with $N$, though the ultimate complexity remains open.

ABSTRACT

Two recent preprints [B. Altshuler, H. Krovi, and J. Roland, "Quantum adiabatic optimization fails for random instances of NP-complete problems", arXiv:0908.2782 and "Anderson localization casts clouds over adiabatic quantum optimization", arXiv:0912.0746] argue that random 4th order perturbative corrections to the energies of local minima of random instances of NP-complete problem lead to avoided crossings that cause the failure of quantum adiabatic algorithm (due to exponentially small gap) close to the end, for very small transverse field that scales as an inverse power of instance size N. The theoretical portion of this work does not to take into account the exponential degeneracy of the ground and excited states at zero field. A corrected analysis shows that unlike those in the middle of the spectrum, avoided crossings at the edge would require high [O(1)] transverse fields, at which point the perturbation theory may become divergent due to quantum phase transition. This effect manifests itself only in large instances [exp(0.02 N) >> 1], which might be the reason it had not been observed in the authors' numerical work. While we dispute the proposed mechanism of failure of quantum adiabatic algorithm, we cannot draw any conclusions on its ultimate complexity.

Motivation & Objective

• To challenge the claim that avoided crossings near zero transverse field cause exponentially small gaps in quantum adiabatic optimization for random NP-complete problems.
• To correct the perturbative analysis in prior work by accounting for exponential degeneracy of ground and excited states at zero field.
• To assess whether avoided crossings at the spectrum edge can lead to exponentially small gaps under realistic conditions.
• To clarify the role of perturbation theory in estimating energy splittings and gap scaling in random combinatorial optimization problems.
• To evaluate the validity of the proposed mechanism for algorithmic failure in the quantum adiabatic model.

Proposed method

• Re-analyzes the energy splitting between localized states using degenerate and non-degenerate perturbation theory, correcting for the exponential degeneracy at zero field.
• Identifies that $O(\lambda^3)$ corrections arise from degenerate perturbation theory in certain clause configurations, not just $O(\lambda^4)$ from standard perturbation theory.
• Demonstrates that avoided crossings at the spectrum edge require transverse fields of order $O(1)$, beyond the regime where perturbation theory is valid.
• Uses numerical results to test scaling of corrections, showing they may grow logarithmically rather than as a power of $N$, contradicting prior assumptions.
• Considers the impact of hypergraph structure and trimming on degeneracy, showing that exponential degeneracy persists in most models.
• Argues that non-perturbative methods may be required to assess the true probability of small gaps for finite $\lambda < \lambda_c$.

Experimental results

Research questions

• RQ1 Can avoided crossings at the edge of the spectrum lead to exponentially small gaps in the quantum adiabatic algorithm for random instances of NP-complete problems?
• RQ2 Does the perturbative analysis in prior work correctly account for the exponential degeneracy of states at zero transverse field?
• RQ3 What is the required transverse field strength for avoided crossings to occur at the spectrum edge, and does this invalidate perturbation theory?
• RQ4 How do perturbative corrections scale with system size $N$ in the presence of degenerate ground states?
• RQ5 Is the proposed mechanism of failure via small gaps near $\lambda \to 0$ physically plausible given the breakdown of perturbation theory at large $\lambda$?

Key findings

• The perturbative analysis in Ref. Alt09 fails to account for exponential degeneracy of ground and excited states at zero field, leading to incorrect conclusions about avoided crossings.
• Avoided crossings at the spectrum edge require transverse fields of order $O(1)$, where perturbation theory becomes unreliable due to quantum phase transitions.
• The energy splitting from degenerate perturbation theory can be $O(\lambda^3)$, not just $O(\lambda^4)$, and such corrections are not easily removable by hypergraph trimming.
• Numerical results suggest corrections grow slowly with $N$, possibly logarithmically, contradicting the assumed power-law scaling in prior work.
• The minimum gap in the quantum adiabatic algorithm does not scale exponentially with $N$ under the corrected analysis, though the ultimate complexity remains undetermined.
• Non-perturbative methods may be necessary to assess the true likelihood of exponentially small gaps for finite $\lambda < \lambda_c$.