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 $\Pi_{ij}$ on a system of $n$ qubits, and the task is to decide whether the Hamiltonian $H=\sum \Pi_{ij}$ has a 0-eigenvalue, or it is larger than $1/n^\alpha$ for some $\alpha=O(1)$. 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 $\frac{1}{2}$, 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.
Highlights
Various formulations of the satisfiability problem of Boolean formulae arguably constitute the centerpiece of classical complexity theory
A great amount of attention has been paid to the SAT problem, in which we are given a formula in the form of a conjunction of clauses, where each clause is a disjunction of literals, and the task is to find a satisfying assignment if there is one, or determine that none exists when the formula is unsatisfiable
The algorithm of Krom [19] based on the resolution principle and on transitive closure computation decides if the formula is satisfiable in time O(n3) and finds a satisfying assignment in time O(n4)
Summary
Various formulations of the satisfiability problem of Boolean formulae arguably constitute the centerpiece of classical complexity theory. Notice that in the YES case, the energy of each projector at a ground state is necessarily 0 because, by definition, projectors are non-negative operators This corresponds to a perfectly satisfiable formula. To maintain a linear running time, we propagate these two choices simultaneously until one of the propagations stops without contradiction, in which case the corresponding qubit assignment is made final. One possibility would be to consider complex numbers with bounded precision in which case exact computation is no more possible and an error analysis should be Figure 1: Handling a contradicting cycle: (a) we slide edges that touch i along the two paths to j until (b) we get a double edge with two “tails.” (c) we use a structure lemma to deduce that at least one of these edges can be written as a product projector (a dashed edge). To us, [3] handles these instances by using transfer matrix techniques to find discretizing cycles [20]
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