Linear time algorithm for quantum 2SAT

Abstract

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 Π ij on a system of n qubits, and the task is to decide whether the Hamiltonian H = Π ij has a 0-eigenvalue, or it is larger than 1/n α 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 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.

The Complexity of Translationally Invariant Spin Chains with Low Local Dimension

“…On the other hand, just as in classical computer science 2-SAT is solvable in polynomial time, its quantum analogue-the Quantum 2-SAT, a special case of the 2-Local Hamiltonian problem 3 -can also be solved deterministically in polynomial time: [20] proved an O(n 4 ) runtime bound, and later a linear-time algorithm was discovered independently by [21] and [22]. Yet the resemblance with classical results goes further: Quantum 4-SAT and later Quantum 3-SAT were shown to be QMA 1 -complete [20,23].…”

Quantum independent-set problem and non-Abelian adiabatic mixing

“…Applications and perspective -The key of our algorithm, adiabatic non-abelian mixing, can be applied to other problems that have multiple solutions with one or more solutions easy to find or already found. For example, a class of quantum 2-SAT problems have multiple solutions and one of their trivial solutions is precisely |−1, −1, −1, • • • , −1 [22][23][24][25][26].…”

A Quantum Version of Schöning's Algorithm Applied to Quantum 2-SAT

“…The Quantum Algorithm may solve arbitrary Quantum 2-SAT instances in polynomial time, but our analysis can only show that it succeeds in polynomial time on Restricted Quantum 2-SAT. On the other hand, Bravyi's algorithm and recent linear-time quantum algorithms [9,10] give procedures for deciding all Quantum 2-SAT instances in polynomial time, but are classical algorithms. Our algorithm is a quantum algorithm, so our analysis techniques may be of broader interest.…”

“…Analogously to classical k-SAT, Quantum 3-SAT is QMA 1 -complete [6], while Quantum 2-SAT can be solved in polynomial time [7]. Interestingly, existing algorithms for Quantum 2-SAT have paralleled algorithms for classical 2-SAT: Bravyi's original algorithm for Quantum 2-SAT is similar to Krom's algorithm for classical 2-SAT [8] and uses inference rules; and two recent linear-time algorithms for Quantum 2-SAT [9,10] use ideas from linear-time classical 2-SAT algorithms [4,5].…”