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arXiv:2301.10699 · Communications in Mathematical Physics

Testing Quantum Satisfiability

Ashley Montanaro, Changpeng Shao, Dominic Verdon

Published Sep 1, 2025• 1 citation
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Abstract

Abstract Quantum k-SAT (the problem of determining whether a k-local Hamiltonian is frustration-free) is known to be QMA $$_1$$ 1 -complete for $$k\ge 3$$ k ≥ 3 , and hence likely hard for quantum computers to solve. Building on a classical result of Alon and Shapira, we show that quantum k-SAT can be solved in randomised polynomial time given the ‘property testing’ promise that the instance is either satisfiable (by any state) or far from satisfiable by a product state; by ‘far from satisfiable by a product state’ we mean that $$\epsilon n^k$$ ϵ n k constraints must be removed before a product state solution exists, for some fixed $$\epsilon >0$$ ϵ > 0 . The proof has two steps: we first show that for a satisfiable instance of quantum k-SAT, most subproblems on a constant number of qubits are satisfiable by a product state. We then show that for an instance of quantum k-SAT which is far from satisfiable by a product state, most subproblems are unsatisfiable by a product state. Given the promise, quantum k-SAT may therefore be solved by checking satisfiability by a product state on randomly chosen subsystems of constant size.

DOI
10.1007/s00220-025-05377-4

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Abstract Quantum k-SAT (the problem of determining whether a k-local Hamiltonian is frustration-free) is known to be QMA 1_11​ 1 -complete for k≥3k\ge 3k≥3 k ≥ 3 , and hence likely hard for quantum computers to solve. Building on a classical result of Alon and Shapira, we show that quantum k-SAT can be solved in randomised polynomial time given the ‘property testing’ promise that the instance is either satisfiable (by any state) or far from satisfiable by a product state; by ‘far from satisfiable by a product state’ we mean that ϵnk\epsilon n^kϵnk ϵ n k constraints must be removed before a product state solution exists, for some fixed ϵ>0\epsilon >0ϵ>0 ϵ > 0 . The proof has two steps: we first show that for a satisfiable instance of quantum k-SAT, most subproblems on a constant number of qubits are satisfiable by a product state. We then show that for an instance of quantum k-SAT which is far from satisfiable by a product state, most subproblems are unsatisfiable by a product state. Given the promise, quantum k-SAT may therefore be solved by checking satisfiability by a product state on randomly chosen subsystems of constant size.
Abstract Estimating ground state energies of local Hamiltonian models is a central problem in quantum physics. The question of whether a given local Hamiltonian is frustration-free, meaning the ground state is the simultaneous ground state of all local interaction terms, is known as the Quantum k-SAT (k-QSAT) problem. In analogy to its classical Boolean constraint satisfaction counterpart, the NP-complete problem k-SAT, Quantum k-SAT is QMA1\hbox {QMA}_1QMA1​ QMA 1 -complete (for k≥3k\ge 3k≥3 k ≥ 3 , and where QMA1\hbox {QMA}_1QMA1​ QMA 1 is a quantum generalization of NP with one-sided error), and thus likely intractable. But whereas k-SAT has been well-studied for special tractable cases, as well as from a “parameterized complexity” perspective, much less is known in similar settings for k-QSAT. Here, we study the open problem of computing satisfying assignments to k-QSAT instances which have a “dimer covering” or “matching”; such systems are known to be frustration-free, but it remains open whether one can efficiently compute a ground state. Our results fall into three directions, all of which relate to the “dimer covering” setting: (1) We give a polynomial-time classical algorithm for k-QSAT when all qubits occur in at most two interaction terms or clauses. (2) We give a “parameterized algorithm” for k-QSAT instances from a certain non-trivial class, which allows us to obtain exponential speedups over brute force methods in some cases. This is achieved by reducing the problem to solving for a single root of a single univariate polynomial. An explicit family of hypergraphs, denoted Crash, for which such a speedup is obtained is introduced. (3) We conduct a structural graph theoretic study of 3-QSAT interaction graphs which have a “dimer covering”. We remark that the results of (2), in particular, introduce a number of new tools to the study of Quantum SAT, including graph theoretic concepts such as transfer filtrations and blow-ups from algebraic geometry.
Random quantum satisfiability is a natural quantum analogue of random constraint satisfaction and a basic model for frustration-free local Hamiltonians. Despite extensive work on its satisfiable and unsatisfiable regimes, the quantitative location of the random quantum kkk-SAT threshold has remained poorly understood, with the best general upper bounds leaving a large gap to the known lower bounds. In this paper we prove a new upper bound on the satisfiability threshold of random quantum kkk-SAT. Our result improves the previously known asymptotic upper bound by a factor of order kkk, giving a bound of order 2k/k2^k/k2k/k. The improvement is also significant at small values of kkk; in particular, for random quantum 333-SAT we obtain a substantially smaller explicit upper bound than the one previously available. The proof combines the geometric formulation of generic quantum satisfiability with a dimension-decay analysis of the full satisfying subspace. The key input is a multiplicative Shearer-type inequality for tensor-product subspaces, which quantifies how global dimension forces nontrivial local dimension on typical sets of qubits.
The Quantum k-SAT problem is the quantum generalization of the k-SAT problem. It is the problem whether a given local Hamiltonian is frustration-free. Frustration-free means that the ground state of the k-local Hamiltonian minimizes the energy of every local interaction term simultaneously. This is a central question in quantum physics and a canonical QMA_1-complete problem. The Quantum k-SAT problem is not as well studied as the classical k-SAT problem in terms of special tractable cases, approximation algorithms and parameterized complexity. In this paper, we will give a graph-theoretic study of the Quantum k-SAT problem with the structures core and radius. These hypergraph structures are important to solve the Quantum k-SAT problem. We can solve a Quantum k-SAT instance in polynomial time if the derived hypergraph has a core of size n-m+a, where a is a constant, and the radius is at most logarithmic. If it exists, we can find a core of size n-m+a with the best possible radius in polynomial time, whereas finding a general minimum core with minimal radius is NP-hard.
The local Hamiltonian problem is the canonical QMA\mathsf{QMA}QMA-complete problem, and O(2n)O(2^n)O(2n) time classical algorithms and O(2n/2)O(2^{n/2})O(2n/2) time quantum algorithms are known to solve the problem in the worst case. It is not clear how to improve these brute force strategies for a broad class of the problem because ground states are highly entangled in general, and we cannot directly apply known strategies for classical CSPs. In this work, we present exponentially faster classical and quantum algorithms under two mild assumptions: (1) the Hamiltonian is frustration-free on YES instances, and (2) it is approximately regular, meaning that every qubit is acted upon by approximately the same number of constraints. We complement these upper bounds by showing that, assuming (Q)SETH, quantum 5-SAT admits no non-trivial worst-case speedup. Our lower bound further demonstrates that the dependence of our algorithms on regularity is in some sense nearly optimal. Specifically, quantum 5-SAT remains (Q)SETH-hard even for Hamiltonians in which all but O(n)O(\sqrt{n})O(n​) qubits participate in only constantly many constraints, while the remaining O(n)O(\sqrt{n})O(n​) qubits each participate in O(n)O(\sqrt{n})O(n​) constraints. By contrast, if either the size of this high-degree subset or the degrees of its qubits is reduced by a factor of nδn^δnδ, for any δ>0δ>0δ>0, our algorithm solves the problem in time O(2(1−ε)n)O(2^{(1- \varepsilon)n})O(2(1−ε)n) for some ε>0\varepsilon>0ε>0. Together, our upper and lower bounds establish a fine-grained complexity dichotomy for quantum satisfiability.
Satisfiability (SAT) is a central problem in computer science, and advances in SAT-solving algorithms have a far-reaching impact across many fields. Recent works have proposed quantum SAT solvers based on Grover's algorithm, a quantum search technique. However, Grover-based approaches face a key limitation: they typically require prior knowledge of the number of satisfying assignments of the target Boolean formula. This information is unavailable in most practical settings. Quantum counting can be used to estimate this quantity, but it incurs a computational overhead that is several orders of magnitude higher than Grover search. In this paper, we propose a novel quantum SAT solver based on entanglement and equivalence checking. Our method does not assume prior knowledge of the number of solutions and is computationally more efficient than quantum counting. Although the worst case time complexity is inevitably exponential, we prove that the expected time complexity of our approach is only constant time O(1) over random Boolean functions. Experimental results also support our theoretical claim.
Local Quantum Search Algorithm for Random kkk-SAT with Ω(n1+ε)Ω(n^{1+ε})Ω(n1+ε) Clauses
The random k-SAT instances undergo a "phase transition" from being generally satisfiable to unsatisfiable as the clause number m passes a critical threshold, rknr_k nrk​n. This causes a drastic reduction in the number of satisfying assignments, shifting the problem from being generally solvable on classical computers to typically insolvable. Beyond this threshold, it is challenging to comprehend the computational complexity of random k-SAT. In quantum computing, Grover's search still yields exponential time requirements due to the neglect of structural information. Leveraging the structure inherent in search problems, we propose the k-local quantum search algorithm, which extends quantum search to structured scenarios. Grover's search, by contrast, addresses the unstructured case where k=n. Given that the search algorithm necessitates the presence of a target, we specifically focus on the problem of searching the interpretation of satisfiable instances of k-SAT, denoted as max-k-SSAT. If this problem is solvable in polynomial time, then k-SAT can also be solved within the same complexity. We demonstrate that, for small k≥3k \ge 3k≥3, any small ε>0ε>0ε>0 and sufficiently large n: ⋅\cdot⋅ k-local quantum search achieves general efficiency on random instances of max-k-SSAT with m=Ω(n2+δ+ε)m=Ω(n^{2+δ+ε})m=Ω(n2+δ+ε) using O(n)\mathcal{O}(n)O(n) iterations, and ⋅\cdot⋅ k-local adiabatic quantum search enhances the bound to m=Ω(n1+δ+ε)m=Ω(n^{1+δ+ε})m=Ω(n1+δ+ε) within an evolution time of O(n2)\mathcal{O}(n^2)O(n2). In both cases, the circuit complexity of each iteration is O(nk)\mathcal{O}(n^k)O(nk), and the efficiency is assured with overwhelming probability 1−O(erfc(nδ/2))1 - \mathcal{O}(\mathrm{erfc}(n^{δ/2}))1−O(erfc(nδ/2)). By modifying this algorithm capable of solving all instances of max-k-SSAT, we further prove that max-k-SSAT is polynomial on average when m=Ω(n2+ε)m=Ω(n^{2+ε})m=Ω(n2+ε) based on the average-case complexity theory.