Sampling Frequency Thresholds for Quantum Advantage of Quantum Approximate Optimization Algorithm

TL;DR

The paper evaluates whether QAOA can outperform leading classical solvers for MaxCut on 3-regular graphs, focusing on fixed-angle, non-variational single- and multi-shot QAOA and leveraging tensor-network simulations to predict performance. It demonstrates that, for depths up to $p\leq11$, the typical QAOA performance concentrates near the tree subgraph value $f_{\text{p-tree}}$ and that single-shot solutions concentrate around this mean, with multi-shot gains limited by the shrinking $1/\sqrt{N}$ variance. When benchmarked against state-of-the-art classical solvers (Gurobi, MQLib with BURER2002, and FLIP), zero-time classical solutions already surpass fixed-angle QAOA at $p=11$, and time-to-solution scales favor classical methods; quantum advantage would require high sampling rates (on the order of kHz) and deeper circuits ($p\gtrsim 12$) or much larger qubit counts, especially as $N$ grows. The study also provides a rigorous framework for comparing quantum and classical runtimes, includes subgraph-based performance bounds, and discusses the prospect that quantum advantage with QAOA MaxCut on 3-regular graphs may not be achievable on near-term devices, motivating exploration of alternative problem classes or enhanced QAOA variants. Overall, the work clarifies the stringent conditions under which QAOA could surpass classical heuristics and highlights both the potential and the limitations of near-term quantum devices for combinatorial optimization.

Abstract

In this work, we compare the performance of the Quantum Approximate Optimization Algorithm (QAOA) with state-of-the-art classical solvers such as Gurobi and MQLib to solve the combinatorial optimization problem MaxCut on 3-regular graphs. The goal is to identify under which conditions QAOA can achieve "quantum advantage" over classical algorithms, in terms of both solution quality and time to solution. One might be able to achieve quantum advantage on hundreds of qubits and moderate depth $p$ by sampling the QAOA state at a frequency of order 10 kHz. We observe, however, that classical heuristic solvers are capable of producing high-quality approximate solutions in linear time complexity. In order to match this quality for $\textit{large}$ graph sizes $N$, a quantum device must support depth $p>11$. Otherwise, we demonstrate that the number of required samples grows exponentially with $N$, hindering the scalability of QAOA with $p\leq11$. These results put challenging bounds on achieving quantum advantage for QAOA MaxCut on 3-regular graphs. Other problems, such as different graphs, weighted MaxCut, maximum independent set, and 3-SAT, may be better suited for achieving quantum advantage on near-term quantum devices.

Figures (10)

• Figure 1: Locus of quantum advantage over classical algorithms. A particular classical algorithm may return some solution to some ensemble of problems in time $T_C$ (horizontal axis) with some quality $C_C$ (vertical axis). Similarly, a quantum algorithm may return a different solution sampled in time $T_Q$, which may be faster (right) or slower (left) than classical, with a better (top) or worse (bottom) quality than classical. If QAOA returns better solutions faster than the classical, then there is clear advantage (top right), and conversely no advantage for worse solutions slower than the classical (bottom left).
• Figure 1: Counting types of subgraphs on sparse cycles to find upper and lower limits on QAOA expectation values. The presence of a finite number of cycles in an infinitely large graph slightly modifies the value of the QAOA expectation value by modifying the local subgraphs. As shown above, the edges that are modified as a part of each cycle for $p\leq 3$ are shown in black; vertices which connect to the rest of the graph are shown in red. Edge labels refer to subgraph indexing in Wurtz_guarantee.
• Figure 2: Time required for a single-shot QAOA to match classical MaxCut algorithms. The blue line shows time for comparing with the Gurobi solver and using $p=11$; the yellow line shows comparison with the FLIP algorithm and $p=6$. Each quantum device that runs MaxCut QAOA can be represented on this plot as a point, where the x-axis is the number of qubits and the y-axis is the time to solution. For any QAOA depth $p$, the quantum device should return at least one bitstring faster than the Y-value on this plot.
• Figure 2: Dependence of standard deviation of MaxCut cut fraction on $N$ and $p$ for random 3-regular graphs. Circle markers represent approximate evaluations over 1000 samples per graph and 20 graphs for each size $N$. The dashed line shows a fit to the approximate data using the $\propto \frac{1}{\sqrt N}$ scaling. Cross markers show exact standard deviation values for larger $N$ and $p=3$ with one graph per each $N$. These values were obtained using tensor network contraction via QTensor. The bold cross marker at $N=256$ is also an exact value of QAOA standard deviation at size corresponding to $N$ used on Figure \ref{['fig:timebounds']}. Note that this plot has a log-log scale.
• Figure 3: Zero-time performance for graphs of different size $N$. The Y-value is the cut fraction obtained by running corresponding algorithms for minimum possible time. This corresponds to the Y-value of the star marker in Fig. \ref{['fig:timebounds']}. Dashed lines show the expected QAOA performance for $p=11$ (blue) and $p=6$ (yellow). QAOA can outperform the FLIP algorithm at depth $p>6$, while for Gurobi it needs $p>11$. Note that in order to claim advantage, QAOA has to provide the zero-time solutions in faster time than FLIP or Gurobi does. These times are shown on Fig. \ref{['fig:adv_freq']}.