Quantum interactive proofs with short messages

TL;DR

This paper shows that quantum interactive proof systems with short messages are equivalent in power to well-known complexity classes, using quantum state tomography and the quantum de Finetti theorem.

Contribution

It demonstrates that short-message quantum interactive proofs can be simulated by standard models, establishing their equivalence in computational power.

Findings

Short messages can be eliminated without changing the proof system's power.

Variants with short messages are equivalent to QMA and BQP classes.

Uses quantum state tomography and de Finetti theorem for proofs.

Abstract

This paper considers three variants of quantum interactive proof systems in which short (meaning logarithmic-length) messages are exchanged between the prover and verifier. The first variant is one in which the verifier sends a short message to the prover, and the prover responds with an ordinary, or polynomial-length, message; the second variant is one in which any number of messages can be exchanged, but where the combined length of all the messages is logarithmic; and the third variant is one in which the verifier sends polynomially many random bits to the prover, who responds with a short quantum message. We prove that in all of these cases the short messages can be eliminated without changing the power of the model, so the first variant has the expressive power of QMA and the second and third variants have the expressive power of BQP. These facts are proved through the use of quantum state tomography, along with the finite quantum de Finetti theorem for the first variant.