We introduce the fermionic satisfiability problem, Fermionic k -SAT: this is the problem of deciding whether there is a fermionic state in the null-space of a collection of fermionic, parity-conserving, projectors on n fermionic modes, where each fermionic projector involves at most k fermionic modes. We prove that this problem can be solved efficiently classically for k = 2 . In addition, we show that deciding whether there exists a satisfying assignment with a given fixed particle number parity can also be done efficiently classically for Fermionic 2-SAT: this problem is a quantum-fermionic extension of asking whether a classical 2-SAT problem has a solution with a given Hamming weight parity. We also prove that deciding whether there exists a satisfying assignment for particle-number-conserving Fermionic 2-SAT for some given particle number is NP-complete. Complementary to this, we show that Fermionic 9-SAT is QMA 1 -hard.
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 QMA
1-complete [S. Bravyi, Efficient Algorithm for a Quantum Analogue of 2-SAT, eprint arXiv:quant-ph/0602108, 2006]. Quantum 3-SAT was known to be contained in QMA
1 [Bravyi, 2006], but its computational hardness was unknown until now. We prove that quantum 3-SAT is QMA
1-hard, and therefore complete for this complexity class.
A canonical result about satisfiability theory is that the 2-SAT problem can be solved in linear time, despite the NP-hardness of the 3-SAT problem. In the quantum 2-SAT problem, we are given a family of 2-qubit projectors
Πij on a system of
n qubits, and the task is to decide whether the Hamiltonian
H=∑Πij has a 0-eigenvalue, or it is larger than
1/nα for some
α=O(1). The problem is not only a natural extension of the classical 2-SAT problem to the quantum case, but is also equivalent to the problem of finding the ground state of 2-local frustration-free Hamiltonians of spin
21, a well-studied model believed to capture certain key properties in modern condensed matter physics. While Bravyi has shown that the quantum 2-SAT problem has a classical polynomial-time algorithm, the running time of his algorithm is
O(n4). In this paper we give a classical algorithm with linear running time in the number of local projectors, therefore achieving the best possible complexity.
A canonical result about satisfiability theory is that the 2-SAT problem can be solved in linear time, despite the NP-hardness of the 3-SAT problem. In the quantum 2-SAT problem, we are given a family of 2-qubit projectors Q_{ij} on a system of n qubits, and the task is to decide whether the Hamiltonian H = sum Q_{ij} has a 0-eigenvalue, or it is larger than 1/n^c for some c = O(1). The problem is not only a natural extension of the classical 2-SAT problem to the quantum case, but is also equivalent to the problem of finding the ground state of 2-local frustration-free Hamiltonians of spin 1/2, a well-studied model believed to capture certain key properties in modern condensed matter physics. While Bravyi has shown that the quantum 2-SAT problem has a classical polynomial-time algorithm, the running time of his algorithm is O(n^4). In this paper we give a classical algorithm with linear running time in the number of local projectors, therefore achieving the best possible complexity.
A well-known result about satisfiability theory is that the 2-SAT problem can be solved in linear time, despite the NP-hardness of the 3-SAT problem. In the quantum 2-SAT problem, we are given a family of 2-qubit projectors
Πij on a system of
n qubits, and the task is to decide whether the Hamiltonian
H=∑Πij has a 0-eigenvalue, or all eigenvalues are greater than
1/nα for some
α=O(1). The problem is not only a natural extension of the classical 2-SAT problem to the quantum case, but is also equivalent to the problem of finding a ground state of 2-local frustration-free Hamiltonians of spin
21, a well-studied model believed to capture certain key properties in modern condensed matter physics. Bravyi has shown that the quantum 2-SAT problem has a deterministic algorithm of complexity
O(n4) in the algebraic model of computation where every arithmetic operation on complex numbers can be performed in unit time, and
n is the number of variables. In this paper we give a deterministic algorithm in the algebraic model with running time
O(n+m), where
m is the number of local projectors, therefore achieving the best possible complexity in that model. We also show that if in the input every number has a constant size representation then the bit complexity of our algorithm is
O((n+m)M(n)), where
M(n) denotes the complexity of multiplying two
n-bit integers. A conference version of this paper appeared in the Proceedings of the 43rd International Colloquium on Automata, Languages and Programming (ICALP'16), 2016.
The Kochen-Specker no-go theorem established that hidden-variable theories in quantum mechanics necessarily admit contextuality. This theorem is formally stated in terms of the partial Boolean algebra structure of projectors on a Hilbert space. Each partial Boolean algebra provides a semantics for interpreting propositional logic. In this paper, we examine the complexity of propositional satisfiablity for various classes of partial Boolean algebras. We first show that the satisfiability problem for the class of non-trivial partial Boolean algebras is NP-complete. Next, we consider the satisfiability problem for the class of partial Boolean algebras arising from projectors on finite dimensional Hilbert spaces. For real Hilbert spaces of dimension greater 2 and any complex Hilbert spaces of dimension greater than 3, we demonstrate that the satisfiablity problem is complete for the existential theory of the reals. Interestingly, the proofs of these results make use of Kochen-Specker sets as gadgets. As a corollary, we conclude that deciding quantum homomorphism in these fixed dimensions are also complete for the existential theory of the reals. Finally, we show that the satisfiability problems for the class of all Hilbert spaces and all finite-dimensional Hilbert spaces is undecidable.
The constraint satisfaction problems k-SAT and Quantum k-SAT (k-QSAT) are canonical NP-complete and QMA_1-complete problems (for k >= 3), respectively, where QMA_1 is a quantum generalization of NP with one-sided error. 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 "matching" or "dimer covering"; this is an NP problem whose decision variant is trivial, but whose search complexity remains open. Our results fall into three directions, all of which relate to the "matching" setting: (1) We give a polynomial-time classical algorithm for k-QSAT when all qubits occur in at most two 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 by reducing the problem to solving for a single root of a single univariate polynomial. (3) We conduct a structural graph theoretic study of 3-QSAT interaction graphs which have a "matching". 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 hope these prove useful elsewhere.