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A linear time quantum algorithm for 3SAT

Zachary B. Walters

Published Sep 29, 2015• 0 citations
OpenAlex

Abstract

A quantum algorithm for the NP complete problem of satisfying boolean formulas with three variables per clause (3SAT) is presented. Departing from traditional models of quantum computation, this algorithm makes extensive use of irreversible operations to incoherently transfer population from states which do not solve some problem of interest to states which do. Provided that a solution exists, the algorithm yields exponential decay of nonsolution probability, at a rate controlled by the user.

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Finding solutions to MaxSAT in a quantum computer

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Abstract We introduce a quantum algorithm designed to encode a 3-MaxSAT instance f(z) comprising n Boolean variables and m clauses. In the initial stage, we construct an unitary operator F ∈ SU(2n) whose corresponding eigenvalues are phase-encoded to reflect the total count of non-satisfiable clauses observed when evaluating f(z) under a truth assignment (a potential solution) ϕj . Our second stage consists in combining F along Quantum Phase Estimation to map the induced phase shifts into measurable quantum states in only O(mlog(m)) time due to commuting properties of F. Our objective is to identify a state |ϕj⟩ with the smallest phase shift induced by F. Such eigenstate |ϕj⟩ directly corresponds to the truth assignment minimizing the count of non-satisfiable clauses, thereby constituting our solution for f(z). We demonstrate how we can use this encoding along with Amplitude Amplification to find such solutions to MaxSAT in O(mlog(m)1.414n) time. In perspective, this search algorithm may represent an exponential speedup over MaxSAT algorithms, whose complexity grows exponentially as a function of the number of clauses.

•Jul 10, 2024•0 citations
A quantum algorithm for the NP complete problem of satisfying boolean formulas with three variables per clause (3SAT) is presented. Departing from traditional models of quantum computation, this algorithm makes extensive use of irreversible operations to incoherently transfer population from states which do not solve some problem of interest to states which do. Provided that a solution exists, the algorithm yields exponential decay of nonsolution probability, at a rate controlled by the user.
The Deutsch model of quantum computation is extended to allow for thermodynamically irreversible operations by allowing the system of interest to interact with an outside reservoir. A set of irreversible logical error correction superoperators are constructed which allow the rapid concentration of probability from an exponentially large search space into a small number of logically defined states. These capabilities are used to construct a linear time solution algorithm for the NP complete problem 3SAT.
We present a multi-step quantum algorithm for solving the 333-bit exact cover problem, which is one of the NP-complete problems. Unlike the brute force methods have been tried before, in this algorithm, we showed that by applying the clauses of the Boolean formula sequentially and introducing non-unitary operations, the state that satisfies all of the clauses can be projected out from an equal superposition of all computational basis states step by step, and the search space is reduced exponentially. The runtime of the algorithm is proportional to the number of clauses, therefore scales polynomial to the size of the problem. Our results indicate that quantum computers may be able to outperform classical computers in solving NP-complete problems.
This paper proposes a quantum algorithm for solving the tautology and the satisfiability problems for a Boolean formula. Let’s say we are given a Boolean formula. The variables of the Boolean formula can take only two values—TRUE or FALSE. The tautology problem asks whether the Boolean formula always evaluates to TRUE for all values of its Boolean variables. The satisfiability problem asks whether the Boolean formula can evaluate to TRUE for at least one set of values of its Boolean variables. This paper proposes that both problems can be solved using the proposed quantum algorithm. The tautology problem is known to be co-NP-complete. The satisfiability problem is known to be NP-complete.
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].
The aim of the paper is to answer a long-standing open problem on the relationship between NP and BQP. The paper shows that BQP contains NP by proposing a BQP quantum algorithm for the MAX-E3-SAT problem which is a fundamental NP-hard problem. Given an E3-CNF Boolean formula, the aim of the MAX-E3-SAT problem is to find the variable assignment that maximizes the number of satisfied clauses. The proposed algorithm runs in O(m2)O(m^2)O(m2) for an E3-CNF Boolean formula with mmm clauses and in the worst case runs in O(n6)O(n^6)O(n6) for an E3-CNF Boolean formula with nnn inputs. The proposed algorithm maximizes the set of satisfied clauses using a novel iterative partial negation and partial measurement technique. The algorithm is shown to achieve an arbitrary high probability of success of 1−ε1-ε1−ε for small ε>0ε>0ε>0 using a polynomial resources. In addition to solving the MAX-E3-SAT problem, the proposed algorithm can also be used to decide if an E3-CNF Boolean formula is satisfiable or not, which is an NP-complete problem, based on the maximum number of satisfied clauses.
Abstract We introduce a quantum algorithm designed to encode a 3-MaxSAT instance f(z) comprising n Boolean variables and m clauses. In the initial stage, we construct an unitary operator F ∈ SU(2n) whose corresponding eigenvalues are phase-encoded to reflect the total count of non-satisfiable clauses observed when evaluating f(z) under a truth assignment (a potential solution) ϕj . Our second stage consists in combining F along Quantum Phase Estimation to map the induced phase shifts into measurable quantum states in only O(mlog(m)) time due to commuting properties of F. Our objective is to identify a state |ϕj⟩ with the smallest phase shift induced by F. Such eigenstate |ϕj⟩ directly corresponds to the truth assignment minimizing the count of non-satisfiable clauses, thereby constituting our solution for f(z). We demonstrate how we can use this encoding along with Amplitude Amplification to find such solutions to MaxSAT in O(mlog(m)1.414n) time. In perspective, this search algorithm may represent an exponential speedup over MaxSAT algorithms, whose complexity grows exponentially as a function of the number of clauses.