Quantum statistical zero-knowledge

In this paper we propose a definition for (honest verifier) quantum statistical zero-knowledge interactive proof systems and study the resulting complexity class, which we denote QSZK. We prove several facts regarding this class: • The following natural problem is a complete promise problem for QSZK: given instructions for preparing two mixed quantum states, are the states close together or far apart in the trace norm metric? By instructions for preparing a mixed quantum state we mean the description of a quantum circuit that produces the mixed state on some specified subset of its qubits, assuming all qubits are initially in the |0〉 state. This problem is a quantum generalization of the complete promise problem of Sahai and Vadhan [33] for (classical) statistical zero-knowledge. • QSZK is closed under complement. • QSZK ⊆ PSPACE. (At present it is not known if arbitrary quantum interactive proof systems can be simulated in PSPACE, even for one-round proof systems.) • Any honest verifier quantum statistical zero-knowledge proof system can be parallelized to a two-message (i.e., one-round) honest verifier quantum statistical zero-knowledge proof system. (For arbitrary quantum interactive proof systems it is known how to parallelize to three messages, but not two.) Moreover, the one-round proof system can be taken to be such that the prover sends only one qubit to the verifier in order to achieve completeness and soundness error exponentially close to 0 and 1/2, respectively. These facts establish close connections between classical statistical zero-knowledge and our definition for quantum statistical zero-knowledge, and give some insight regarding the effect of this zero-knowledge restriction on quantum interactive proof systems.