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Performance guarantees of light-cone variational quantum algorithms for the maximum cut problem

Xiaoyang Wang, Yuexin Su, Tongyang Li

TL;DR

This work tackles barren-plateau limitations in variational quantum algorithms for MaxCut by introducing a light-cone VQA built from directed $ZY$ gates that induces a DAG-based circuit with finite local depth. Through local analyses, it proves optimality of the light-cone orientation for 3-regular graphs, achieving a strong worst-case bound $\alpha_0\ge 0.7926$ that improves to $0.8333$ with angle-relaxation, and introduces the bipolar-$ZY$ construction with scalable BP-free performance. Numerical simulations show a favorable scaling of median time-to-solution compared with the best classical solver, and hardware demonstrations on 72- and 148-qubit IBM devices surpass a classical hardness threshold, while QAOA remains limited in the tested regime. The results establish a practical pathway for solving classically hard MaxCut instances on near-term quantum hardware and motivate extensions to other models via the same light-cone, BP-free framework.

Abstract

Variational quantum algorithms (VQAs) are promising to demonstrate the advantage of near-term quantum computing over classical computing in practical applications, such as the maximum cut (MaxCut) problem. However, current VQAs such as the quantum approximate optimization algorithm (QAOA) have lower performance guarantees compared to the best-known classical algorithm, and suffer from hard optimization processes due to the barren plateau problem. We propose a light-cone VQA by choosing an optimal gate sequence of the standard VQAs, which enables a significant improvement in solution accuracy while avoiding the barren plateau problem. Specifically, we prove that the light-cone VQA with one round achieves an approximation ratio of 0.7926 for the MaxCut problem in the worst case of $3$-regular graphs, which is higher than that of the 3-round QAOA, and can be further improved to 0.8333 by an angle-relaxation procedure. Finally, our numerical results indicate an exponential speed-up in finding the exact solution using the light-cone VQA compared with the classical algorithm. Using IBM's quantum devices, we demonstrate that the single-round light-cone VQA exceeds the known classical hardness threshold in both 72- and 148-qubit demonstrations, whereas $p$-round $\text{QAOA}$ with $p=1,2,3$ does not in the latter one. Our work highlights a promising route towards solving classically hard problems on practical quantum devices.

Performance guarantees of light-cone variational quantum algorithms for the maximum cut problem

TL;DR

This work tackles barren-plateau limitations in variational quantum algorithms for MaxCut by introducing a light-cone VQA built from directed gates that induces a DAG-based circuit with finite local depth. Through local analyses, it proves optimality of the light-cone orientation for 3-regular graphs, achieving a strong worst-case bound that improves to with angle-relaxation, and introduces the bipolar- construction with scalable BP-free performance. Numerical simulations show a favorable scaling of median time-to-solution compared with the best classical solver, and hardware demonstrations on 72- and 148-qubit IBM devices surpass a classical hardness threshold, while QAOA remains limited in the tested regime. The results establish a practical pathway for solving classically hard MaxCut instances on near-term quantum hardware and motivate extensions to other models via the same light-cone, BP-free framework.

Abstract

Variational quantum algorithms (VQAs) are promising to demonstrate the advantage of near-term quantum computing over classical computing in practical applications, such as the maximum cut (MaxCut) problem. However, current VQAs such as the quantum approximate optimization algorithm (QAOA) have lower performance guarantees compared to the best-known classical algorithm, and suffer from hard optimization processes due to the barren plateau problem. We propose a light-cone VQA by choosing an optimal gate sequence of the standard VQAs, which enables a significant improvement in solution accuracy while avoiding the barren plateau problem. Specifically, we prove that the light-cone VQA with one round achieves an approximation ratio of 0.7926 for the MaxCut problem in the worst case of -regular graphs, which is higher than that of the 3-round QAOA, and can be further improved to 0.8333 by an angle-relaxation procedure. Finally, our numerical results indicate an exponential speed-up in finding the exact solution using the light-cone VQA compared with the classical algorithm. Using IBM's quantum devices, we demonstrate that the single-round light-cone VQA exceeds the known classical hardness threshold in both 72- and 148-qubit demonstrations, whereas -round with does not in the latter one. Our work highlights a promising route towards solving classically hard problems on practical quantum devices.

Paper Structure

This paper contains 29 sections, 11 theorems, 131 equations, 30 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Light-cone variational ansatz
  3. Performance guarantees by local analysis
  4. Bipolar-$ZY$ ansatz
  5. Multi-angle bipolar-$ZY$ ansatz
  6. Numerical and hardware results
  7. Numerical simulation
  8. Hardware implementation
  9. Classical simulability
  10. Discussion
  11. Construction of the lightcone-$ZY$ ansatz
  12. Truncation error of the $k$-local analysis
  13. Proof of Theorem 1
  14. Averaged heads in-degree of directed $D$-regular graphs
  15. Biconnected decomposition for the MaxCut problem
  16. ...and 14 more sections

Key Result

Corollary 1

For the lightcone-$ZY_p$ ansatz solving MaxCut on $D$-regular graphs with $N$ nodes, the variance of the energy expectation is lower bounded by

Figures (30)

  • Figure 1: Main ideas and results of this work. (a) and (b) illustrate the standard VQA with $\mathcal{O}(N)$ rounds and the light-cone VQA with one round, respectively. The light-cone VQA has two-qubit gates expanded by the backward light-cone of the local observable $\hat{O}$. (c) The single-round light-cone VQA for the MaxCut problem. The light-cone VQA can be represented by a directed acyclic graph. The directed edge denotes a $ZY$ gate $e^{-i\theta ZY/2}$, and can be decomposed into a single-qubit gate $R_Y(\theta)\equiv e^{-i\theta Y/2}$ and two CNOT gates. (d) Performance guarantees of the bipolar-$ZY_p$ ansatz, a specific light-cone ansatz with $p$-round, and its comparison with QAOA and classical algorithms. $0.9412$ is the hardness threshold 10.1145/502090.502098.
  • Figure 2: The lightcone-$ZY_1$ ansatz for a graph $G$ using the breadth-first search (BFS) traversal. BFS gives a directed acyclic graph (DAG, left panel) of $G$. The topological order of the DAG and the edge direction give the gate sequence and the $ZY$ orientation of the lightcone-$ZY$ ansatz (right panel), respectively.
  • Figure 3: Schematic illustration of $k$-local analysis. $k$-local analysis calculates the expectation of $Z_iZ_i$ by truncating the quantum circuit $k$ steps away from the center edge $(i,j)$. The truncation error of $k$-local analysis can be well controlled since the distribution of $Z_iZ_j$ expectation decays exponentially with $k$, as shown schematically in the left panel.
  • Figure 4: Two kinds of directed edges $(i\to j)$ in $3$-regular graphs with $j$'s in-degree $\deg^-(j)=1$ (left panel) and $\deg^-(j)=2$ (right panel) and their lightcone-$ZY_1$ circuits. In both circuits, $ZY$ gates entering $j$ are arranged to the left of gates leaving $j$. Directed edges with $\deg^-(j)=1$ and $\deg^-(j)=2$ contribute to the terms of $\sin\theta$ and $\cos\theta\sin\theta$ in Eq. \ref{['eq:alpha_0_lower-bound']}, respectively.
  • Figure 5: Bipolar-$ZY$ and the angle-relaxed ansatz. (a) Construction procedures of the bipolar-$ZY_p$ ansatz for a biconnected graph described in Algorithm \ref{['alg:dag-ansatz']}. The graph is biconnected because it remains connected if any one node were to be removed. $s$ and $t$ are chosen as the source and sink nodes in the bipolar orientation. (b) Illustration of the angle-relaxed bipolar-$ZY_1$ ansatz for a $2$-regular graph of length $8$ with $3$ free angles labeled for each $ZY$ gate.
  • ...and 25 more figures

Theorems & Definitions (20)

  • Definition 2.1: $ZY$ orientation
  • Definition 2.2: gate sequence
  • Corollary 1
  • Definition 2.3: $k$-local analysis
  • Proposition 1
  • Theorem 1
  • Definition 2.4: Bipolar orientation
  • Theorem 2
  • Corollary 2
  • Definition B.1: Pauli path of the $ZY$ ansatz
  • ...and 10 more