Quantum search algorithm tailored to clause satisfaction problems

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.