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arXiv:2601.17686

Exponential Quantum Speedup on Structured Hard Instances of Maximum Independent Set

Vicky Choi

quant-phSubmitted Jan 25, 2026• Announced Jan 27, 2026• 0 citations
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Abstract

Establishing quantum speedup for computationally hard problems of practical relevance, particularly combinatorial optimization problems, remains a central challenge in quantum computation. In this work, we identify a structurally defined family of classically hard maximum independent set (MIS) instances, and design and analyze a non-stoquastic adiabatic quantum optimization algorithm that exploits this structure. The algorithm runs in polynomial time and achieves an exponential speedup over both transverse-field quantum annealing and state-of-the-art classical solvers on these instances, under assumptions supported by analytical and numerical evidence. We identify the essential quantum mechanism enabling the speedup as the use of a non-stoquastic XX-driver to access a larger sign-structured admissible subspace beyond the stoquastic regime, which allows sign-generating quantum interference to create smooth evolution paths that bypass tunneling. This identifies a distinctive quantum mechanism underlying the speedup and explains why no efficient classical analogue is likely to exist. In addition, our analysis produces scalable small-scale models, derived from our structural reduction, that capture the essential dynamics of the algorithm. These models provide a concrete opportunity for verification of the quantum advantage mechanism on currently available universal quantum computers.

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18 pages, 8 figures, 2 tables

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Establishing quantum speedup for computationally hard problems of practical relevance, particularly combinatorial optimization problems, remains a central challenge in quantum computation. In this work, we identify a structurally defined family of classically hard maximum independent set (MIS) instances, and design and analyze a non-stoquastic adiabatic quantum optimization algorithm that exploits this structure. The algorithm runs in polynomial time and achieves an exponential speedup over both transverse-field quantum annealing and state-of-the-art classical solvers on these instances, under assumptions supported by analytical and numerical evidence. We identify the essential quantum mechanism enabling the speedup as the use of a non-stoquastic XX-driver to access a larger sign-structured admissible subspace beyond the stoquastic regime, which allows sign-generating quantum interference to create smooth evolution paths that bypass tunneling. This identifies a distinctive quantum mechanism underlying the speedup and explains why no efficient classical analogue is likely to exist. In addition, our analysis produces scalable small-scale models, derived from our structural reduction, that capture the essential dynamics of the algorithm. These models provide a concrete opportunity for verification of the quantum advantage mechanism on currently available universal quantum computers.
We present a theoretical analysis of the DIC-DAC-DOA algorithm, a non-stoquastic quantum algorithm for solving the Maximum Independent Set (MIS) problem. The algorithm runs in polynomial time and achieves exponential speedup over both transverse-field quantum annealing (TFQA) and classical algorithms on a structured family of NP-hard MIS instances, under assumptions supported by analytical and numerical evidence. The core of this speedup lies in the ability of the evolving ground state to develop both positive and negative amplitudes, enabled by the non-stoquastic XX-driver. This sign structure permits quantum interference that produces negative amplitudes in the computational basis, allowing efficient evolution paths beyond the reach of stoquastic algorithms, whose ground states remain strictly non-negative. In our analysis, the efficiency of the algorithm is measured by the presence or absence of an anti-crossing, rather than by spectral gap estimation as in traditional approaches. The key idea is to infer it from the crossing behavior of bare energy levels of relevant subsystems associated with the degenerate local minima (LM) and the global minimum (GM). The cliques of the critical LM, responsible for the anti-crossing in TFQA, can be efficiently identified to form the XX-driver graph. The resulting speedup can be attributed to two mechanisms: in the first stage, energy-guided localization within the same-sign block steers the ground state smoothly into the GM-supporting region, while in the second stage, the opposite-sign blocks are invoked and sign-generating quantum interference drives the evolution along an opposite-sign path. Finally, we derive scalable reduced models that provide a concrete opportunity for verification of the quantum advantage mechanism on currently available universal quantum computers.
Quantum computation provides exponential speedup for solving certain mathematical problems against classical computers. Motivated by current rapid experimental progress on quantum computing devices, various models of quantum computation have been investigated to show quantum computational supremacy. At a commercial side, quantum annealing machine realizes the quantum Ising model with a transverse field and heuristically solves combinatorial optimization problems. The computational power of this machine is closely related to adiabatic quantum computation (AQC) with a restricted type of Hamiltonians, namely stoquastic Hamiltonians, and has been thought to be relatively less powerful compared to universal quantum computers. Little is known about computational quantum speedup nor advantage in AQC with stoquastic Hamiltonians. Here we characterize computational capability of AQC with stoquastic Hamiltonians, which we call stoqAQC. We construct a concrete stoqAQC model, whose lowest energy gap is lower bounded polynomially, and hence the final state can be obtained in polynomial time. Then we show that it can simulate universal quantum computation if adaptive single-qubit measurements in non-standard bases are allowed on the final state. Even if the measurements are restricted to non-adaptive measurements to respect the robustness of AQC, the proposed model exhibits quantum computational supremacy; classical simulation is impossible under complexity theoretical conjectures. Moreover, it is found that such a stoqAQC model can simulate Shor's algorithm and solve the factoring problem in polynomial time. We also propose how to overcome the measurement imperfections via quantum error correction within the stoqAQC model and also an experimentally feasible verification scheme to test whether or not stoqAQC is done faithfully.
Designing quantum algorithms with a speedup over their classical analogs is a central challenge in quantum information science. Motivated by recent experimental observations of a superlinear quantum speedup in solving the Maximum Independent Set problem on certain unit-disk graph instances [Ebadi et al., Science 376, 6598 (2022)], we develop a theoretical framework to analyze the relative performance of the optimized quantum adiabatic algorithm and a broad class of classical Markov chain Monte Carlo algorithms. We outline conditions for the quantum adiabatic algorithm to achieve a quadratic speedup on hard problem instances featuring flat low-energy landscapes and provide example instances with either a quantum speedup or slowdown. We then introduce an additional local Hamiltonian with no sign problem to the optimized adiabatic algorithm to achieve a quadratic speedup over a wide class of classical simulated annealing, parallel tempering, and quantum Monte Carlo algorithms in solving these hard problem instances. Finally, we use this framework to analyze the experimental observations.
The Maximum Independent Set (MIS) problem is a fundamental combinatorial optimization task that can be naturally mapped onto the Ising Hamiltonian of neutral atom quantum processors. Given its connection to NP-hard problems and real-world applications, there has been significant experimental interest in exploring quantum advantage for MIS. Pioneering experiments on King's Lattice graphs suggested a quadratic speed-up over simulated annealing, but recent benchmarks using state-of-the-art methods found no clear advantage, likely due to the structured nature of the tested instances. In this work, we generate hard instances of unit-disk graphs by leveraging complexity theory results and varying key hardness parameters such as density and treewidth. For a fixed graph size, we show that increasing these parameters can lead to prohibitive classical runtime increases of several orders of magnitude. We then compare classical and quantum approaches on small instances and find that, at this scale, quantum solutions are slower than classical ones for finding exact solutions. Based on extended classical benchmarks at larger problem sizes, we estimate that scaling up to a thousand atoms with a 1 kHz repetition rate is a necessary step toward demonstrating a computational advantage with quantum methods.
We analyze the behavior of stoquastic transverse-field quantum annealing (TFQA) on a structured class of Maximum Independent Set (MIS) instances, using the same decomposition framework developed in our companion work on the DIC-DAC-DOA algorithm (Beyond Stoquasticity). For these instances, we provide a structural explanation for the anti-crossing arising from the competition between the energies associated with a set of degenerate local minima (LM) and the global minimum (GM), and analytically derive the associated exponentially small gap. Our analysis proceeds in two steps. First, we reduce the dynamics to an effective two-block Hamiltonian HcoreH_{core}Hcore​, constructed from the bare (decoupled) subsystems associated with the LM and GM. This reduction is justified analytically using the structural decomposition. Second, we reformulate the eigenvalue problem as a generalized eigenvalue problem in a non-orthogonal basis constructed from the bare eigenstates of the subsystems. This transformation enables a clean perturbative treatment of the anti-crossing structure, independent of the transverse field, unlike standard perturbation theory approach, which requires treating the transverse field as a small parameter. This paper serves as a supplementary companion to our main work on the DIC-DAC-DOA algorithm, where we demonstrate how appropriately designed non-stoquastic drivers can bypass this tunneling-induced bottleneck.
Non-stoquastic Hamiltonians have both positive and negative signs in off-diagonal elements in their matrix representation in the standard computational basis and thus cannot be simulated efficiently by the standard quantum Monte Carlo method due to the sign problem. We describe our analytical studies of this type of Hamiltonians with infinite-range non-random as well as random interactions from the perspective of possible enhancement of the efficiency of quantum annealing or adiabatic quantum computing. It is shown that multi-body transverse interactions like XX and XXXXX with positive coefficients appended to a stoquastic transverse-field Ising model render the Hamiltonian non-stoquastic and reduce a first-order quantum phase transition in the simple transverse-field case to a second-order transition. This implies that the efficiency of quantum annealing is exponentially enhanced, because a first-order transition has an exponentially small energy gap (and therefore exponentially long computation time) whereas a second-order transition has a polynomially decaying gap (polynomial computation time). The examples presented here represent rare instances where strong quantum effects, in the sense that they cannot be efficiently simulated in the standard quantum Monte Carlo, have analytically been shown to exponentially enhance the efficiency of quantum annealing for combinatorial optimization problems.