TIME: 1:30-2:20 pm,  April 25, 2006

PLACE: CSE 403

TITLE: Quantum Information and the PCP Theorem 

SPEAKER:  Ran Raz
          Weizmann Insitutte & Microsoft Research

ABSTRACT:
I will discuss the following two results. I will assume no prior 
knowledge of quantum information or the PCP theorem.

1) The membership of x in SAT (for x of length n) can be 
proved by a logarithmic-size quantum state Psi, together with 
a polynomial-size classical proof consisting of blocks of length 
polylog(n) bits each, such that after measuring the state Psi
a verifier only needs to read ONE of the blocks of the classical 
proof.  This shows that if a short quantum witness is available then
a (classical) PCP with only one query is possible.

2) The class QIP/qpoly contains ALL languages.
That is: for any language L (even non-recursive), the membership 
of x in L (for x of length n) can be proved by a 
polynomial-size quantum interactive proof, where the verifier is a 
polynomial-size quantum circuit with working space initiated with 
some quantum state Psi(L,n)$ (depending only on L and n).

The interactive proof that we give is of only one round, and the 
messages communicated are classical. The advice Psi(L,n) given 
to the verifier can also be replaced by a classical probabilistic 
advice, as long as this advice is kept as a secret from the prover.
The second result can hence be interpreted as:
The class IP/rpoly contains all languages.

For the proof of the second result, we introduce the 
"quantum low-degree-extension" of a string of bits.
The first result requires an additional machinery of 
"quantum low-degree-test".