 
Summary: Quantum Computing and Hidden Variables
Scott Aaronson
Institute for Advanced Study, Princeton
This paper initiates the study of hidden variables from a quantum computing perspective. For
us, a hiddenvariable 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
hiddenvariable 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 hiddenvariable
theory satisfying a reasonable axiom, we could solve the Graph Isomorphism problem in polynomial
time, and could search an Nitem database using O N1/3
queries, as opposed to O N1/2
queries
with Grover's search algorithm. On the other hand, the N1/3
bound is optimal, meaning that we
could probably not solve NPcomplete 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.
