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arXiv:1508.06340

Linear time algorithm for quantum 2SAT

Itai Arad, Miklós Sántha, Aarthi Sundaram, Shengyu Zhang

Published Aug 26, 2015• 3 citations
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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 $H=\sum Π_{ij}$ 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 $\frac{1}{2}$, 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.

DOI
10.48550/arxiv.1508.06340

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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 ΠijΠ_{ij}Πij​ on a system of nnn qubits, and the task is to decide whether the Hamiltonian H=∑ΠijH=\sum Π_{ij}H=∑Πij​ has a 0-eigenvalue, or it is larger than 1/nα1/n^α1/nα for some α=O(1)α=O(1)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 12\frac{1}{2}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)O(n^4)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\Pi_{ij}Πij​ on a system of nnn qubits, and the task is to decide whether the Hamiltonian H=∑ΠijH=\sum \Pi_{ij}H=∑Πij​ has a 0-eigenvalue, or all eigenvalues are greater than 1/nα1/n^\alpha1/nα for some α=O(1)\alpha=O(1)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 12\frac{1}{2}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)O(n^4)O(n4) in the algebraic model of computation where every arithmetic operation on complex numbers can be performed in unit time, and nnn is the number of variables. In this paper we give a deterministic algorithm in the algebraic model with running time O(n+m)O(n+m)O(n+m), where mmm 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))O((n+m) M(n))O((n+m)M(n)), where M(n)M(n)M(n) denotes the complexity of multiplying two nnn-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 Boolean constraint satisfaction problem 3-SAT is arguably the canonical NP-complete problem. In contrast, 2-SAT can not only be decided in polynomial time, but in fact in deterministic linear time. In 2006, Bravyi proposed a physically motivated generalization of k-SAT to the quantum setting, defining the problem "quantum k-SAT". He showed that quantum 2-SAT is also solvable in polynomial time on a classical computer, in particular in deterministic time O(n^4), assuming unit-cost arithmetic over a field extension of the rational numbers, where n is number of variables. In this paper, we present an algorithm for quantum 2-SAT which runs in linear time, i.e. deterministic time O(n+m) for n and m the number of variables and clauses, respectively. Our approach exploits the transfer matrix techniques of Laumann et al. [QIC, 2010] used in the study of phase transitions for random quantum 2-SAT, and bears similarities with both the linear time 2-SAT algorithms of Even, Itai, and Shamir (based on backtracking) [SICOMP, 1976] and Aspvall, Plass, and Tarjan (based on strongly connected components) [IPL, 1979].
Quantum 3-SAT Is QMA1_11​-Complete
Quantum satisfiability is a constraint satisfaction problem that generalizes classical boolean satisfiability. In the quantum kkk-SAT problem, each constraint is specified by a kkk-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 kkk-SAT with k≥4k\geq 4k≥4 is QMA1_11​-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 QMA1_11​ [Bravyi, 2006], but its computational hardness was unknown until now. We prove that quantum 3-SAT is QMA1_11​-hard, and therefore complete for this complexity class.
We present a quantum adiabatic algorithm for a set of quantum 2-satisfiability (Q2SAT) problem, which is a generalization of 2-satisfiability (2SAT) problem. For a Q2SAT problem, we construct the Hamiltonian which is similar to that of a Heisenberg chain. All the solutions of the given Q2SAT problem span the subspace of the degenerate ground states. The Hamiltonian is adiabatically evolved so that the system stays in the degenerate subspace. Our numerical results suggest that the time complexity of our algorithm is O(n 3.9) for yielding non-trivial solutions for problems with the number of clauses m = dn(n – 1)/2 (d ≲ 0.1). We discuss the advantages of our algorithm over the known quantum and classical algorithms.
The local Hamiltonian problem is the canonical QMA\mathsf{QMA}QMA-complete problem, and O(2n)O(2^n)O(2n) time classical algorithms and O(2n/2)O(2^{n/2})O(2n/2) time quantum algorithms are known to solve the problem in the worst case. It is not clear how to improve these brute force strategies for a broad class of the problem because ground states are highly entangled in general, and we cannot directly apply known strategies for classical CSPs. In this work, we present exponentially faster classical and quantum algorithms under two mild assumptions: (1) the Hamiltonian is frustration-free on YES instances, and (2) it is approximately regular, meaning that every qubit is acted upon by approximately the same number of constraints. We complement these upper bounds by showing that, assuming (Q)SETH, quantum 5-SAT admits no non-trivial worst-case speedup. Our lower bound further demonstrates that the dependence of our algorithms on regularity is in some sense nearly optimal. Specifically, quantum 5-SAT remains (Q)SETH-hard even for Hamiltonians in which all but O(n)O(\sqrt{n})O(n​) qubits participate in only constantly many constraints, while the remaining O(n)O(\sqrt{n})O(n​) qubits each participate in O(n)O(\sqrt{n})O(n​) constraints. By contrast, if either the size of this high-degree subset or the degrees of its qubits is reduced by a factor of nδn^δnδ, for any δ>0δ>0δ>0, our algorithm solves the problem in time O(2(1−ε)n)O(2^{(1- \varepsilon)n})O(2(1−ε)n) for some ε>0\varepsilon>0ε>0. Together, our upper and lower bounds establish a fine-grained complexity dichotomy for quantum satisfiability.