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Anti-crossings occurrence as exponentially closing gaps in Quantum Annealing

Arthur Braida, Simon Martiel, Ioan Todinca

TL;DR

The paper addresses when avoided level crossings (AC) cause exponentially closing gaps in quantum annealing (QA) by deriving a perturbative AC condition. It formulates QA with $H(s)=(1-s)H_0+sH_1$, develops initial and final energy perturbations, and identifies a criterion $\lambda_0(\text{loc})>n\alpha_T$ for AC via the local-excited graph $G_{loc}$. Applied to MaxCut on bipartite graphs, it proves no AC for $d$-regular bipartite graphs (for $d\notin\{2,4\}$), implying QA can efficiently solve these instances, while constructing irregular graphs that satisfy AC conditions and exhibit exponentially small gaps numerically. The results illuminate how graph structure influences QA performance and raise questions about whether AC is the definitive indicator of QA inefficiency, given observed constant end-state success probabilities despite small gaps. Overall, the work provides a rigorous perturbative framework linking spectral properties to QA scalability and guides design principles for annealing schedules in structured combinatorial problems.

Abstract

This paper explores the phenomenon of avoided level crossings in quantum annealing, a promising framework for quantum computing that may provide a quantum advantage for certain tasks. Quantum annealing involves letting a quantum system evolve according to the Schrödinger equation, with the goal of obtaining the optimal solution to an optimization problem through measurements of the final state. However, the continuous nature of quantum annealing makes analytical analysis challenging, particularly with regard to the instantaneous eigenenergies. The adiabatic theorem provides a theoretical result for the annealing time required to obtain the optimal solution with high probability, which is inversely proportional to the square of the minimum spectral gap. Avoided level crossings can create exponentially closing gaps, which can lead to exponentially long running times for optimization problems. In this paper, we use a perturbative expansion to derive a condition for the occurrence of an avoided level crossing during the annealing process. We then apply this condition to the MaxCut problem on bipartite graphs. We show that no exponentially small gaps arise for regular bipartite graphs, implying that QA can efficiently solve MaxCut in that case. On the other hand, we show that irregularities in the vertex degrees can lead to the satisfaction of the avoided level crossing occurrence condition. We provide numerical evidence to support this theoretical development, and discuss the relation between the presence of exponentially closing gaps and the failure of quantum annealing.

Anti-crossings occurrence as exponentially closing gaps in Quantum Annealing

TL;DR

The paper addresses when avoided level crossings (AC) cause exponentially closing gaps in quantum annealing (QA) by deriving a perturbative AC condition. It formulates QA with , develops initial and final energy perturbations, and identifies a criterion for AC via the local-excited graph . Applied to MaxCut on bipartite graphs, it proves no AC for -regular bipartite graphs (for ), implying QA can efficiently solve these instances, while constructing irregular graphs that satisfy AC conditions and exhibit exponentially small gaps numerically. The results illuminate how graph structure influences QA performance and raise questions about whether AC is the definitive indicator of QA inefficiency, given observed constant end-state success probabilities despite small gaps. Overall, the work provides a rigorous perturbative framework linking spectral properties to QA scalability and guides design principles for annealing schedules in structured combinatorial problems.

Abstract

This paper explores the phenomenon of avoided level crossings in quantum annealing, a promising framework for quantum computing that may provide a quantum advantage for certain tasks. Quantum annealing involves letting a quantum system evolve according to the Schrödinger equation, with the goal of obtaining the optimal solution to an optimization problem through measurements of the final state. However, the continuous nature of quantum annealing makes analytical analysis challenging, particularly with regard to the instantaneous eigenenergies. The adiabatic theorem provides a theoretical result for the annealing time required to obtain the optimal solution with high probability, which is inversely proportional to the square of the minimum spectral gap. Avoided level crossings can create exponentially closing gaps, which can lead to exponentially long running times for optimization problems. In this paper, we use a perturbative expansion to derive a condition for the occurrence of an avoided level crossing during the annealing process. We then apply this condition to the MaxCut problem on bipartite graphs. We show that no exponentially small gaps arise for regular bipartite graphs, implying that QA can efficiently solve MaxCut in that case. On the other hand, we show that irregularities in the vertex degrees can lead to the satisfaction of the avoided level crossing occurrence condition. We provide numerical evidence to support this theoretical development, and discuss the relation between the presence of exponentially closing gaps and the failure of quantum annealing.
Paper Structure (27 sections, 3 theorems, 28 equations, 13 figures)

This paper contains 27 sections, 3 theorems, 28 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper:
  3. Quantum annealing and perturbative analysis
  4. Computing with quantum annealing:
  5. Perturbative analysis:
  6. Perturbative Analysis to QA
  7. Initial perturbation
  8. Final perturbation
  9. Energy crossing
  10. Application to MaxCut
  11. d-regular bipartite graphs
  12. Case $l=d/2$:
  13. Case $l>d/2$:
  14. Case $d=2$ and $d=4$.
  15. General bipartite graphs
  16. ...and 12 more sections

Key Result

Theorem 3.1

Under perturbative expansion validity, if $\lambda_0(\text{loc})$ is the largest eigenvalue of the adjacency matrix of $G_{loc}$ and $H_1$ has a unique ground state and a degenerated first eigenspace, we use a linear interpolation between $H_0$ and $H_1$ as defined above, then by defining and where $\Delta H_1 =E_{fs} -E_{gs}$ and $\langle H_1 \rangle_0$ is the mean of $H_1$'s eigenvalues, we ca

Figures (13)

  • Figure 1: A 5-cube with $G_{loc}$ highlighted with red nodes and thick black edges. Lightblue node is the unique ground state and blue edges show the connection between $G_{loc}$ and the ground state. Green nodes are all the other possible states with higher energies. The labels, once converted in binary, represent the state configuration.
  • Figure 2: Schematic energy landscape of $H_1$ corresponding to Figure \ref{['fig:gh0_cycle']}. $G_{loc}$ has only one component and is strongly connected to the ground state $x_{opt}$.
  • Figure 3: Schematic behavior of the three energy expansions. (left) a case with no AC and (right) case with AC.
  • Figure 4: Construction of a specific first excited configuration. The L/R partition (left) is natural in MaxCut. The U/D partition (right) is relevant if 1 is a minimal separator.
  • Figure 5: Configurations of $G$ in its ground state (left) and first excited state. (middle) is a configuration far from GS, (right) a configuration neighboring GS.
  • ...and 8 more figures

Theorems & Definitions (7)

  • Theorem 3.1
  • Theorem 4.1: NO AC - d-regular bipartite graphs
  • Claim 1
  • Claim 2
  • Claim 3
  • Lemma 4.2
  • proof