Quantum algorithm to solve function inversion with time–space trade-off

Abstract

In general, it is a difficult problem to solve the inverse of any function. With the inverse implication operation, we present a quantum algorithm for solving the inversion of function via using time---space trade-off in this paper. The details are as follows. Let function $$f(x)=y$$f(x)=y have k solutions, where $$x\in \{0, 1\}^{n}, y\in \{0, 1\}^{m}$$x?{0,1}n,y?{0,1}m for any integers n, m. We show that an iterative algorithm can be used to solve the inverse of function f(x) with successful probability $$1-\left( 1-\frac{k}{2^{n}}\right) ^{L}$$1-1-k2nL for $$L\in Z^{+}$$L?Z+. The space complexity of proposed quantum iterative algorithm is O(Ln), where L is the number of iterations. The paper concludes that, via using time---space trade-off strategy, we improve the successful probability of algorithm.