Simon's problem

In computational complexity theory and quantum computing, Simon's problem is a computational problem that is proven to be solved exponentially faster on a quantum computer than on a classical (that is, traditional) computer. The quantum algorithm solving Simon's problem, usually called Simon's algorithm, served as the inspiration for Shor's algorithm. Both problems are special cases of the abelian hidden subgroup problem, which is now known to have efficient quantum algorithms. The problem is set in the model of decision tree complexity or query complexity and was conceived by Daniel Simon in 1994. Simon exhibited a quantum algorithm that solves Simon's problem exponentially faster and with exponentially fewer queries than the best probabilistic (or deterministic) classical algorithm. In particular, Simon's algorithm uses a linear number of queries and any classical probabilistic algorithm must use an exponential number of queries. This problem yields an oracle separation between the complexity classes BPP (bounded-error classical query complexity) and BQP (bounded-error quantum query complexity). This is the same separation that the Bernstein–Vazirani algorithm achieves, and different from the separation provided by the Deutsch–Jozsa algorithm, which separates P and EQP. Unlike the Bernstein–Vazirani algorithm, Simon's algorithm's separation is exponential. Because this problem assumes the existence of a highly-structured "black box" oracle to achieve its speedup, this problem has little practical value. However, without such an oracle, exponential speedups cannot easily be proven, since this would prove that P is different from PSPACE. Given a function (implemented by a black box or oracle) with the promise that, for some unknown , for all , if and only if , where denotes bitwise XOR. The goal is to identify by making as few queries to as possible. Note that if and only if Furthermore, for some and in , is unique (not equal to ) if and only if . This means that is two-to-one when , and one-to-one when .