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Complexity of Basis-Restricted Local Hamiltonians

Ma, Henry

Published May 1, 2025• 0 citations
OpenAlex

Abstract

A major goal of quantum complexity theory is to understand which computational problems can be solved with access to certain quantum resources. The subfield of Hamiltonian complexity specifically considers computational problems that ask about properties of local Hamiltonians, which are of critical importance in quantum complexity because they can be viewed as quantum generalizations of classical constraint satisfaction problems. In this work, we study the complexity of certain restricted variants of the Quantum-k-Sat problem, a quantum analog of the NP-complete k-Sat problem. We introduce new variants of Quantum-k-Sat which place a basis restriction on the input Hamiltonian H = Σᵢ hᵢ . Each variant is defined by a fixed collection of bases B₁, . . . , Bᵣ of n-qubit space. We require that each Hamiltonian term hi must be diagonal in one of these bases. Our results resolve the complexity of certaim basis-restricted variants of Quantum-k-Sat. First we show the Quantum-6-Sat problem with Hamiltonian terms restricted to be diagonal in an X/Z mixed basis is QMA₁-complete. Second, we combine basis restriction with the restriction of commutativity, and show the following easiness result, which applies generally to higher-level quantum systems (qudits) and bases Q and R (which are real-valued and satisfy an overlap condition): The commmuting Quantum-Sat problem on qudits, where Hamiltonian terms are either diagonal in the Q basis, the R basis, or a single mixed Q/R basis, is in NP.

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A major goal of quantum complexity theory is to understand which computational problems can be solved with access to certain quantum resources. The subfield of Hamiltonian complexity specifically considers computational problems that ask about properties of local Hamiltonians, which are of critical importance in quantum complexity because they can be viewed as quantum generalizations of classical constraint satisfaction problems. In this work, we study the complexity of certain restricted variants of the Quantum-k-Sat problem, a quantum analog of the NP-complete k-Sat problem. We introduce new variants of Quantum-k-Sat which place a basis restriction on the input Hamiltonian H = Σᵢ hᵢ . Each variant is defined by a fixed collection of bases B₁, . . . , Bᵣ of n-qubit space. We require that each Hamiltonian term hi must be diagonal in one of these bases. Our results resolve the complexity of certaim basis-restricted variants of Quantum-k-Sat. First we show the Quantum-6-Sat problem with Hamiltonian terms restricted to be diagonal in an X/Z mixed basis is QMA₁-complete. Second, we combine basis restriction with the restriction of commutativity, and show the following easiness result, which applies generally to higher-level quantum systems (qudits) and bases Q and R (which are real-valued and satisfy an overlap condition): The commmuting Quantum-Sat problem on qudits, where Hamiltonian terms are either diagonal in the Q basis, the R basis, or a single mixed Q/R basis, is in NP.
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.
We study several problems related to properties of nonnegative matrices that arise at the boundary between quantum and classical probabilistic computation. Our results are twofold. First, we identify a large class of quantum Hamiltonians describing systems of qubits for which the adiabatic evolution can be efficiently simulated on a classical probabilistic computer. These are stoquastic local Hamiltonians with a “frustration-free” ground-state. A Hamiltonian belongs to this class iff it can be represented as H=∑aHaH=\sum_{a}H_{a}H=∑a​Ha​ where (1) every term HaH_{a}Ha​ acts nontrivially on a constant number of qubits, (2) every term HaH_{a}Ha​ has real nonpositive off-diagonal matrix elements in the standard basis, and (3) the ground-state of H is a ground-state of every term HaH_{a}Ha​. Second, we generalize the Cook–Levin theorem proving NP-completeness of the satisfiability problem to the complexity class MA (Merlin–Arthur games)—a probabilistic analogue of NP. Specifically, we construct a quantum version of the k-SAT problem which we call “stoquastic k-SAT” such that stoquastic k-SAT is contained in MA for any constant k, and any promise problem in MA is Karp-reducible to stoquastic 6-SAT. This result provides the first nontrivial example of a MA-complete promise problem.
We study several problems related to properties of non-negative matrices that arise at the boundary between quantum and classical probabilistic computation. Our results are twofold. First, we identify a large class of quantum Hamiltonians describing systems of qubits for which the adiabatic evolution can be efficiently simulated on a classical probabilistic computer. These are stoquastic local Hamiltonians with a "frustration free" ground-state. A Hamiltonian belongs to this class iff it can be represented as H=∑aHaH=\sum_a H_aH=∑a​Ha​ where (1) every term HaH_aHa​ acts non-trivially on a constant number of qubits, (2) every term HaH_aHa​ has real non-positive off-diagonal matrix elements in the standard basis, and (3) the ground-state of HHH is a ground-state of every term HaH_aHa​. Secondly, we generalize the Cook-Levin theorem proving NP-completeness of the satisfiability problem to the complexity class MA -- a probabilistic analogue of NP. Specifically, we construct a quantum version of the k-SAT problem which we call "stoquastic k-SAT" such that stoquastic k-SAT is contained in MA for any constant kkk, and any promise problem in MA is Karp-reducible to stoquastic 6-SAT. This result provides the first non-trivial example of a MA-complete promise problem.
The local Hamiltonian problem plays the equivalent role of SAT in quantum complexity theory. Understanding the complexity of the intermediate case in which the constraints are quantum but all local terms in the Hamiltonian commute, is of importance for conceptual, physical and computational complexity reasons. Bravyi and Vyalyi showed in 2003 [10], using a clever application of the representation theory of C*-algebras, that if the terms in the Hamiltonian are all two-local, the problem is in NP, and the entanglement in the ground states is local. The general case remained open since then. In this paper we extend this result beyond the two-local case, to the case of three-qubit interactions. We then extend our results even further, and show that NP verification is possible for three-wise interaction between qutrits as well, as long as the interaction graph is planar and also "nearly Euclidean" in some well-defined sense. The proofs imply that in all such systems, the entanglement in the ground states is local. These extensions imply an intriguing sharp transition phenomenon in commuting Hamiltonian systems: the ground spaces of 3-local "physical" systems based on qubits and qutrits are diagonalizable by a basis whose entanglement is highly local, while even slightly more involved interactions (the particle dimensionality or the locality of the interaction is larger) already exhibit an important long-range entanglement property called Topological Order. Our results thus imply that Kitaev's celebrated Toric code construction is, in a well defined sense, optimal as a construction of Topological Order based on commuting Hamiltonians.
Quantum satisfiability is a constraint satisfaction problem that generalizes classical boolean satisfiability. In the quantum k-SAT problem, each constraint is specified by a k-local projector and is satisfied by any state in its nullspace. Bravyi showed that quantum 2-SAT can be solved efficiently on a classical computer and that quantum k-SAT with k ≥ 4 is QMA1-complete [4]. Quantum 3-SAT was known to be contained in QMA1[4], but its computational hardness was unknown until now. We prove that quantum 3-SAT is QMA1-hard, and therefore complete for this complexity class.
The calculation of ground-state energies of physical systems can be formalised as the k-local Hamiltonian problem, which is the natural quantum analogue of classical constraint satisfaction problems. One way of making the problem more physically meaningful is to restrict the Hamiltonian in question by picking its terms from a fixed set S. Examples of such special cases are the Heisenberg and Ising models from condensed-matter physics. In this work we characterise the complexity of this problem for all 2-local qubit Hamiltonians. Depending on the subset S, the problem falls into one of the following categories: in P, NP-complete, polynomial-time equivalent to the Ising model with transverse magnetic fields, or QMA-complete. The third of these classes contains NP and is contained within StoqMA. The characterisation holds even if S does not contain any 1-local terms, for example, we prove for the first time QMA-completeness of the Heisenberg and XY interactions in this setting. If S is assumed to contain all 1-local terms, which is the setting considered by previous work, we have a characterisation that goes beyond 2-local interactions: for any constant k, all k-local qubit Hamiltonians whose terms are picked from a fixed set S correspond to problems either in P, polynomial-time equivalent to the Ising model with transverse magnetic fields, or QMA-complete. These results are a quantum analogue of Schaefer's dichotomy theorem for boolean constraint satisfaction problems.