Evidence of Scaling Advantage for the Quantum Approximate Optimization Algorithm on a Classically Intractable Problem

TL;DR

This work investigates whether QAOA can surpass classical solvers on a classically intractable problem by studying LABS with fixed QAOA schedules. Through extensive noiseless simulations, the authors show that QAOA with depth p=12 scales as ~1.46^N and, when augmented with quantum minimum finding, as ~1.21^N, outperforming the best classical heuristics ~1.34^N. They develop a fixed-parameter strategy based on Fourier reparameterization and parameter averaging across small N, and prove a theorem showing how QAOA-derived initial states can accelerate minimum finding. On hardware, they implement QAOA with an algorithm-specific parity-check error-detection scheme on trapped-ion devices, achieving up to 65% gap-closure relative to noiseless results via post-selection. Collectively, the results provide evidence that QAOA can serve as a building block for quantum speedups on hard optimization problems, motivating further advances in hardware and fault-tolerant execution.

Abstract

The quantum approximate optimization algorithm (QAOA) is a leading candidate algorithm for solving optimization problems on quantum computers. However, the potential of QAOA to tackle classically intractable problems remains unclear. Here, we perform an extensive numerical investigation of QAOA on the low autocorrelation binary sequences (LABS) problem, which is classically intractable even for moderately sized instances. We perform noiseless simulations with up to 40 qubits and observe that the runtime of QAOA with fixed parameters scales better than branch-and-bound solvers, which are the state-of-the-art exact solvers for LABS. The combination of QAOA with quantum minimum finding gives the best empirical scaling of any algorithm for the LABS problem. We demonstrate experimental progress in executing QAOA for the LABS problem using an algorithm-specific error detection scheme on Quantinuum trapped-ion processors. Our results provide evidence for the utility of QAOA as an algorithmic component that enables quantum speedups.

Figures (24)

• Figure 1: Classical and quantum algorithms applied to the LABS problem.A, Diagram of the LABS problem (with example of $N=5$). The problem involves non-local two-body (black lines) and four-body (blue lines) interactions. B, Time-to-solution (TTS) of classical solvers. For the sizes considered, we observe clear exponential scaling with exponents matching their asymptotic values reported in the literature (see Table \ref{['tab:summary']}). C, Diagram of QAOA circuit for a 5-qubit example. Starting from a uniform superposition of the computational basis states, we apply $p$ layers of phase and mixing operators, followed by measurement in the computational basis. D, The distribution over $21\leq N \leq 31$ (for LABS) and $34$ random instances (for MaxCut on random 3-regular graphs with 20 nodes) of Pearson product-moment correlation coefficients relating the Hamming distance of bitstrings from the optimal solution with the objective value of the bitstring. LABS has a much lower correlation between the Hamming distance and objective, indicating that it is much harder than the commonly considered MaxCut problem.
• Figure 2: QAOA runtime scaling.A, The quality of the exponential fit for different choices of minimum $N$ to include in the fit. $N\geq 28$ results in a robust fit, the quality of which does not deteriorate with $p$. $N=40$ is omitted as it was only simulated up to $p=22$. B, TTS of QAOA at $p=12$. Clear exponential scaling is observed. C, The scaling exponent of QAOA runtime for different QAOA depths $p$. Shaded area shows $95\%$ confidence interval. Increasing $p$ beyond $p\approx 12$ does not lead to better scaling.
• Figure 3: QAOA dynamics under different parameter schedules.A, The gain in success probability $p^{\text{opt}}$ from applying step $p$ of QAOA and amplitude amplification (AA). The gain is defined as $p^{\text{opt}}_{\text{at step }p} / p^{\text{opt}}_{\text{at step }(p-1)}$. The gain at $p=1$ is over the random guess. Only one line is plotted for amplitude amplification since the lines for the values of $N$ considered are visually indistinguishable. For small $p$, a QAOA layer gives orders of magnitude larger gain than a step of AA. B, Fixed QAOA parameters for $p=30$ chosen with respect to the QAOA energy $\langle C\rangle_{\text{MF}}$ ("MF") and probability of sampling the optimal solution ("$p^{\text{opt}}$"). Different choice of optimization objective gives different resulting parameters. C, Probability of obtaining a binary string corresponding to a given energy level of the LABS problem (the zeroth energy level is the ground state or optimal solution; lower is better). When parameters are optimized with respect to the expected merit factor (labeled "MF"), the QAOA output state is concentrated around the mean and fails to obtain a high overlap with the ground state. On the other hand, when parameters are optimized with respect to $p^{\text{opt}}$ (labeled "$p^{\text{opt}}$"), the QAOA state has a high overlap with both the ground state and higher energy states. The probability of obtaining the ground state is $27.3$ times greater for QAOA with parameters optimized with respect to $p^{\text{opt}}$ at $p=40$.
• Figure 4: Experimental results on trapped-ion system.A, Decomposition of four-body interaction terms into a two-body $R_{\textsc{z}\textsc{z}}$ gate and four two-body $\textsc{cnot}$ gates, which can be realized via native $R_{\textsc{z}\textsc{z}}$ gates. B, Energy density plot from experimental measured bitstrings for $N$=13. Energy index is arranged in energy ascending order. As a comparison, the distributions for noiseless $p=1$ QAOA simulation and random guess (assuming uniform distribution of all possible bitstrings) are shown. C, Experimental results up to 18 qubits on a trapped-ion quantum device (H1-1) with QAOA layer $p=1$ with optimized QAOA parameters. The error bars are calculated with 99% confidence intervals hereafter. D, Illustration of parity check circuit. The $\textsc{z}$ and $\textsc{x}$ parities of states are mapped to ancillary qubits after implementation of full (or part of) phase operators via $\textsc{cz}$ and $\textsc{cnot}$ gates, respectively, followed by mid-circuit measurement on the ancillary qubits to extract the parity syndrome result. E, Experimental results for circuit with parity check. Three mid-circuit $\textsc{z}$-parity and $\textsc{x}$-parity checks were performed using six ancillary qubits. The ancillae can also be reused after appropriate reset during the circuit. The red data points were run on the Quantinuum H2 hardware while the blue data were from the H1-1 device. Data run on the H1-1 device without any ancillary qubits are shown in grey. Circles (diamonds) are the data without (with) post-selection. The abbreviation ED refers to the error detection via the parity checks. Number of mid-circuit parity checks is fixed to be two for $N=10, 11$ and three for all other $N$. Improvement in expected merit factor after post-selection according to parity syndrome measurement is observed.
• Figure 5: Visualization of how the fixed parameters are obtained.A, Optimized QAOA parameters $\bm\beta$ (top lines) and $\bm\gamma$ (bottom lines) for $p=21$. $\bm\gamma$ is multiplied by $N/24$ (constant factor of $\frac{1}{24}$ added for figure readability in both subfigures). B, Fixed parameters obtained by taking the arithmetic mean over the optimized parameters.

Theorems & Definitions (6)

• Lemma 1: Exponential Quantum Search, Ref. Brassard_2002
• Theorem 1
• proof
• Lemma 1: Exponential Quantum Search, Ref. Brassard_2002
• Theorem 1
• proof