Quantum speedups in solving near-symmetric optimization problems by low-depth QAOA

TL;DR

This paper demonstrates that exponential quantum speedups are achievable with low-depth QAOA (specifically $p=1$) on families of planted, symmetric, and near-symmetric CSPs. By deriving exact and asymptotic success probabilities using saddle-point analysis, it proves $Ω(1/√n)$ success in general $S_n$-symmetric cases and $Ω(1)$ in structured instances, with extensions to $S_{n_1}×S_{n_2}$ symmetry. It shows that sparsifying symmetric costs preserves high QAOA success, yielding near-symmetric problems that remain efficiently solvable by quantum means, and provides extensive classical benchmarks indicating regimes where classical solvers require exponential time. Collectively, the results indicate a robust exponential quantum speedup over general-purpose classical algorithms for these symmetry-rich optimization problems and point to practical avenues for near-term quantum advantage via low-depth QAOA. The work also lays out future directions for exploring other symmetry groups and more sophisticated symmetry-breaking schemes while maintaining quantum advantages on real hardware.

Abstract

We present new advances towards achieving exponential quantum speedups for solving optimization problems by low-depth quantum algorithms. Specifically, we focus on families of combinatorial optimization problems that exhibit symmetry and contain planted solutions. We rigorously prove that the 1-step Quantum Approximate Optimization Algorithm (QAOA) can achieve a success probability of $Ω(1/\sqrt{n})$, and sometimes $Ω(1)$, for finding the exact solution in many cases. This allows us to prove a separation of $O(1)$ quantum queries and $Ω(n/\log n)$ classical queries required to find the planted solution in the latter setting. Furthermore, we construct near-symmetric optimization problems by randomly sampling the individual clauses of symmetric problems, and prove that the QAOA maintains a strong success probability in this setting even when the symmetry is broken. Finally, we construct various families of near-symmetric Max-SAT problems and benchmark state-of-the-art classical solvers, discovering instances where all known general-purpose classical algorithms require exponential time. Therefore, our results indicate that low-depth QAOA may achieve an exponential quantum speedup for optimization problems.

Figures (4)

• Figure 1: Performance of various classical SAT and Max-SAT solvers on a family of $S_n$-symmetric 3-SAT problems. (a) Example cost function plotted versus normalized Hamming distance to the hidden string ${\boldsymbol{s}}$ at $n=200$. (b) Run times plotted as a function of number of bits $n$ on a log-log scale, where $n^3$ is plotted as a guide to the eyes. Individual dots correspond to runs with different choices of ${\boldsymbol{s}}$, and dashed line connects the averages at each $n$. (c) Run times divided by $n^3$, which is roughly the number of clauses in the formula, plotted on a log-linear scale. The flat lines of kissat, EvalMaxSAT and MaxCDCL indicate their run time scales linearly with the problem size $\Theta(n^3)$.
• Figure 2: kissat run time rescaled by problem size for different near-$S_n$-symmetric $\ell$-SAT problems, where we explore the dependence with respect to locality $\ell$ in (a) and the sparsification fraction $f$ in (b). Here $f$ is the fraction of clauses sampled from the corresponding symmetric $\ell$-SAT problems to construct the near-symmetric problem. Dashed lines connect averages over 3$\sim$5 instances with different choices of ${\boldsymbol{s}}$ and clause sampling realizations. Error bars are standard errors of the mean.
• Figure 3: Performance of classical Max-SAT solvers on two families of near-$S_n$-symmetric Max-$\ell$-SAT problems for $\ell=4$ (top row) and $\ell=5$ (bottom row), constructed with clause sampling. (a)(d) Plot of the energy landscape as a function of the normalized Hamming distance to ${\boldsymbol{s}}$. (b)(e) Run times plotted as a function of number of bits $n$ on a log-log scale. (c)(f) Run times divided by $n^\ell$ plotted as a function of $n$ on a log-linear scale. Individual dots correspond to 3 problem instances at each $n$ with different clause sampling and choices of ${\boldsymbol{s}}$, and dashed lines connect their averages. The results suggest that akmaxsat and MaxCDCL take exponential time on both problems, but EvalMaxSAT seems to run in $O(n^\ell)$ time.
• Figure 4: Performance of classical Max-SAT solvers on two families of near-$(S_{n/2})^2$-symmetric Max-$\ell$-SAT problems for $\ell = 5$ (top row) and $\ell=6$ (bottom row), constructed with clause sampling. (a)(d) Contour plots of the energy landscapes of the two families of Max-$\ell$-SAT problems, as a function of the two subset Hamming distances to ${\boldsymbol{s}}$. (b)(e) Run times of various classical algorithms plotted as a function of number of bits $n$ on a log-log scale. (c)(f) Run times divided by $n^\ell$ plotted on a log-linear scale. Individual dots correspond to 3 problem instances at each $n$ with different clause sampling and choices of ${\boldsymbol{s}}$, and dashed lines connect their averages. For both problems, all three algorithms appear to require $\exp(n)$ run time in the large $n$ limit.

Theorems & Definitions (21)

• Definition 2.1: Symmetric CSPs
• Theorem 3.3
• Theorem 3.4
• proof : Proof of Theorem \ref{['thm:distinct-values']}
• Lemma 3.5
• proof
• Theorem 3.7
• proof
• Theorem 4.1
• Lemma 4.2