We continue the study of the Stoquastic Local Hamiltonian problem, a physically motivated restriction of the QMA-complete Local Hamiltonian problem (Kitaev, Shen, and Vyalyi, 2002). For the $β$-gapped, frustration-free case, Bravyi, Bessen, and Terhal (2006) showed that the problem is MA-complete when $β= 1/\mathrm{poly}(n)$. Aharonov and Grilo (2019) derandomized this algorithm and proved membership in NP for constant gap $β= Ω(1)$. We present an improved algorithm and analysis, establishing membership in NP even when $β= Ω(1/(\log\log n))$. We complement our result with an explicit example demonstrating why the analysis does not extend directly to $β= o(1/\log\log n)$.
We continue the study of the Stoquastic Local Hamiltonian problem, a physically motivated restriction of the QMA-complete Local Hamiltonian problem (Kitaev, Shen, and Vyalyi, 2002). For the β-gapped, frustration-free case, Bravyi, Bessen, and Terhal (2006) showed that the problem is MA-complete when β=1/poly(n). Aharonov and Grilo (2019) derandomized this algorithm and proved membership in NP for constant gap β=Ω(1). We present an improved algorithm and analysis, establishing membership in NP even when β=Ω(1/(loglogn)). We complement our result with an explicit example demonstrating why the analysis does not extend directly to β=o(1/loglogn).
The QMA-completeness of the local Hamiltonian problem is a landmark result of the field of Hamiltonian complexity that studies the computational complexity of problems in quantum many-body physics. Since its proposal, substantial effort has been invested in better understanding the problem for physically motivated important families of Hamiltonians. In particular, the QMA-completeness of approximating the ground state energy of local Hamiltonians has been extended to the case where the Hamiltonians are geometrically local in one and two spatial dimensions. Among those physically motivated Hamiltonians, stoquastic Hamiltonians play a particularly crucial role, as they constitute the manifestly sign-free Hamiltonians in Monte Carlo approaches. Interestingly, for such Hamiltonians, the problem at hand becomes more ''classical'', being hard for the class MA (the randomized version of NP) and its complexity has tight connections with derandomization. In this work, we prove that both the two- and one-dimensional geometrically local analogues remain MA-hard with high enough qudit dimension. Moreover, we show that related problems are StoqMA-complete.
We study the complexity of Local Hamiltonian problems under the restriction that every local term is an unweighted occurrence of the same fixed interaction. In the resulting Single-Interaction Hamiltonian (SIH) model, once the interaction is fixed, the Hamiltonian is specified entirely by its ordered interaction hypergraph.
Our main technical tool is an exact compiler that converts any fixed finite alphabet of local interactions into a single interaction. On a designated auxiliary subspace, the compiled Hamiltonian reproduces the source Hamiltonian exactly, while the complete spectrum below a chosen separation scale is preserved with multiplicity. Combined with a finite-alphabet normalization, this gives a universal polynomial-time reduction from k-Local Hamiltonian to SIH3k+3 up to a known global energy scale and inverse-polynomial approximation.
Sharper constructions give QMA-completeness of SIH3 and of geometrically local SIH8 with bounded degree and interaction range. In the stoquastic setting, we obtain StoqMA-completeness of Stoq-SIH3 and QMA-completeness of 1-Pinned Stoq-SIH4. We also establish hardness results for uniformly system-size-dependent interactions and derive single-interaction formulations of the Hamiltonian quantum PCP conjecture, frustration-free, exponentially precise, and guided Local Hamiltonian problems. These results show that independently chosen local matrices and independently tunable coupling strengths are not necessary for a broad range of Hamiltonian-complexity phenomena.
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 QMA 1 -complete (for k≥3 k ≥ 3 , and where QMA1 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.
Stoquasticity, originating in sign-problem-free physical systems, gives rise to StoqMA, introduced by Bravyi, Bessen, and Terhal (2006), a quantum-inspired intermediate class between MA and AM. Unentanglement similarly gives rise to QMA(2), introduced by Kobayashi, Matsumoto, and Yamakami (CJTCS 2009), which generalizes QMA to two unentangled proofs and still has only the trivial NEXP upper bound. In this work, we initiate a systematic study of the power of unentanglement without destructive interference via StoqMA(2), the class of unentangled stoquastic Merlin--Arthur proof systems. Beyond its complexity-theoretic interest, StoqMA(2) is connected to the optimality of non-negative tensor optimization algorithms. We highlight:
1. NP⊆StoqMA(2) with O(n)-qubit proofs and completenes 1−2−polylog(n). Conversely, the Sum-of-Squares algorithm of Barak, Kelner, and Steurer (STOC 2014) gives an exponential-time upper bound for StoqMA(2). Our tightened analysis shows the optimality of our protocol and the BKS algorithm under ETH.
2. For logarithmic-size proofs, NP⊆StoqMA(2)log with completeness 1−O(n−2) and vanishing gap, while StoqMA(2)log⊆MA. Consequently, quantum-inspired randomness enables exponentially shorter unentangled proofs even under the assumption MA=NP.
3. For StoqMA(2)1, the parameter dependence in the general ETH-optimal time bound can be exponentially improved, or the bound achieved simultaneously with polynomial space.
Our lower bounds are obtained by stoquastizing the short-proof QMA(2) protocols using distribution testing techniques. Our upper bounds for the nearly perfect completeness case are proved via our rectangular closure testing framework.
The local Hamiltonian (LH) problem is the canonical QMA-complete problem introduced by Kitaev. In this paper, we show its hardness in a very strong sense: we show that the 3-local Hamiltonian problem on n qubits cannot be solved classically in time O(2(1−ε)n) for any ε>0 under the Strong Exponential-Time Hypothesis (SETH), and cannot be solved quantumly in time O(2(1−ε)n/2) for any ε>0 under the Quantum Strong Exponential-Time Hypothesis (QSETH). These lower bounds give evidence that the currently known classical and quantum algorithms for LH cannot be significantly improved.
Furthermore, we are able to demonstrate fine-grained complexity lower bounds for approximating the quantum partition function (QPF) with an arbitrary constant relative error. Approximating QPF with relative error is known to be equivalent to approximately counting the dimension of the solution subspace of QMA problems. We show the SETH and QSETH hardness to estimate QPF with constant relative error. We then provide a quantum algorithm that runs in O(2n) time for an arbitrary 1/poly(n) relative error, matching our lower bounds and improving the state-of-the-art algorithm by Bravyi, Chowdhury, Gosset, and Wocjan (Nature Physics 2022) in the low-temperature regime.
To prove our fine-grained lower bounds, we introduce the first size-preserving circuit-to-Hamiltonian construction that encodes the computation of a T-time quantum circuit acting on N qubits into a (d+1)-local Hamiltonian acting on N+O(T1/d) qubits. This improves the standard construction based on the unary clock, which uses N+O(T) qubits.