[Paper Review] A Polynomial Time Bounded-error Quantum Algorithm for Boolean Satisfiability

Ahmed Younes, Jonathan E. Rowe|arXiv (Cornell University)|Jul 16, 2015

Quantum Computing Algorithms and Architecture3 references3 citations

TL;DR

This paper proposes a polynomial-time bounded-error quantum algorithm in BQP for the MAX-E3-SAT problem, a fundamental NP-hard problem, using iterative partial negation and partial measurement to amplify the amplitude of variable assignments that maximize satisfied clauses. The algorithm achieves arbitrarily high success probability $1 - \epsilon$ in $O(m^2)$ time for $m$ clauses, or $O(n^6)$ in the worst case for $n$ variables, demonstrating that NP is contained in BQP in a non-relativized setting.

ABSTRACT

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 $O(m^2)$ for an E3-CNF Boolean formula with $m$ clauses and in the worst case runs in $O(n^6)$ for an E3-CNF Boolean formula with $n$ 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 $1-ε$ for small $ε>0$ 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.

Motivation & Objective

• To resolve the longstanding open question of whether NP is contained in BQP.
• To provide a concrete quantum algorithm that solves MAX-E3-SAT with bounded error and polynomial resources.
• To demonstrate that NP-complete problems like E3-SAT can be decided with high probability using quantum computation.
• To overcome limitations of prior quantum approaches by using partial measurement in amplitude amplification instead of standard unitary iteration.

Proposed method

• The algorithm prepares a superposition of all $2^n$ variable assignments using Hadamard gates on $n$ qubits.
• It encodes each E3-CNF clause as a GT4 quantum gate, entangling variable assignments with a truth vector indicating clause satisfaction.
• A novel amplitude amplification technique applies iterative partial negation to an auxiliary qubit based on the number of satisfied clauses, increasing amplitude for better solutions.
• Partial measurement is used to collapse the state toward the best solution, with the process repeated to amplify success probability.
• Dummy clauses are added to increase the density of 1s in the truth vector, improving the initial success probability of measuring the optimal state.
• The algorithm runs in $O(m^2)$ time for $m$ clauses and $O(n^6)$ in the worst case, achieving success probability $1 - \epsilon$ for any $\epsilon > 0$.

Experimental results

Research questions

• RQ1 Can a quantum algorithm solve MAX-E3-SAT in polynomial time with bounded error, thereby placing NP in BQP?
• RQ2 Does the use of partial measurement in amplitude amplification enable a more efficient solution than standard unitary amplitude amplification?
• RQ3 Can the algorithm achieve arbitrarily high success probability $1 - \epsilon$ using only polynomial resources?
• RQ4 Is it possible to use quantum amplitude amplification with partial measurements to maximize the number of satisfied clauses in an E3-CNF formula?
• RQ5 Can this approach be extended to decide the satisfiability of E3-CNF formulas by checking if all $m$ clauses are satisfied?

Key findings

• The proposed quantum algorithm solves MAX-E3-SAT in $O(m^2)$ time for $m$ clauses, with a worst-case runtime of $O(n^6)$ for $n$ variables.
• The algorithm achieves a success probability of at least $1 - \epsilon$ for any $\epsilon > 0$ using polynomially many qubits and operations.
• The use of dummy clauses increases the 1-density in the truth vector, enabling the success probability of the first iteration to exceed 99% when $Pr_{\max} = 0.99$.
• The algorithm can determine whether an E3-CNF formula is satisfiable by checking if all $m$ clauses are satisfied in the output state.
• The method generalizes to E$k$-CNF formulas, requiring $O(n^{2k})$ iterations, showing scalability beyond 3-CNF.
• The result implies that NP is contained in BQP in the non-relativized world, challenging the common belief that BQP does not contain NP.