Quantum search algorithm tailored to clause satisfaction problems

TL;DR

This paper introduces a new quantum search algorithm for clause satisfaction problems that reduces the need for coupled computation circuits, potentially simplifying implementation in quantum computing.

Contribution

A novel quantum search method that relaxes the coupling requirement among computation circuits for clause satisfaction problems.

Findings

Reduces circuit coupling complexity

Offers implementation advantages in quantum algorithms

Maintains effectiveness in finding satisfying assignments

Abstract

Many important computer science problems can be reduced to clause satisfaction problem. We are given $n$ Boolean variables $x_{k}$ and $m$ clauses $c_{j}$ where each clause is a function of values of some of the variables. We want to find an assignment $i$ of variables for which all $m$ clauses are satisfied. Let $f_{j}(i)$ be a binary function which is $1$ if $j^{\rm th}$ clause is satisfied by the assignment $i$ else $f_{j}(i) = 0$. Then the solution is $r$ for which $f(i=r) = 1$, where $f(i)$ is the AND function of all $f_{j}(i)$. In quantum computing, Grover`s algorithm can be used to find $r$. A crucial component of this algorithm is the selective phase inversion $I_{r}$ of the solution state encoding $r$. $I_{r}$ is implemented by computing $f(i)$ for all $i$ in superposition which requires computing AND of all $m$ binary functions $f_{j}(i)$. Hence there must be coupling between the computation circuits for each $f_{j}(i)$. In this paper, we present an alternative quantum search algorithm which relaxes the requirement of such couplings. Hence it offers implementation advantages for clause satisfaction problems.