Quantum computing and hidden variables

This paper initiates the study of hidden variables from a quantum computing perspective. For us, a hidden-variable theory is simply a way to convert a unitary matrix that maps one quantum state to another into a stochastic matrix that maps the initial probability distribution to the final one in some fixed basis. We list five axioms that we might want such a theory to satisfy and then investigate which of the axioms can be satisfied simultaneously. Toward this end, we propose a new hidden-variable theory based on network flows. In a second part of the paper, we show that if we could examine the entire history of a hidden variable, then we could efficiently solve problems that are believed to be intractable even for quantum computers. In particular, under any hidden-variable theory satisfying a reasonable axiom, we could solve the graph isomorphism problem in polynomial time, and could search an $N$-item database using $O\left(N^{1∕3}\right)$ queries, as opposed to $O\left(N^{1∕2}\right)$ queries with Grover’s search algorithm. On the other hand, the $N^{1∕3}$ bound is optimal, meaning that we could probably not solve $\mathrm{NP}$-complete problems in polynomial time. We thus obtain the first good example of a model of computation that appears slightly more powerful than the quantum computing model.

Figure 1

A network (weighted directed graph with source and sink) corresponding to the unitary $U$ and state $\mid \psi ⟩$.