BQP/qpoly EXP/poly Scott Aaronson UC Berkeley

BQP/qpoly Class of languages recognized by a bounded-error polytime quantum algorithm, with a polysize quantum advice state | n that depends only on the input size Buhrman: Is BQP/qpoly anything/poly?

Our Result BQP/qpoly EXP/poly Means we shouldnt hope for an unrelativized separation between BQP/poly and BQP/qpolysince it would imply P/poly EXP/poly, which is equivalent to EXP P/poly

Proof Sketch Given a BQP/qpoly algorithm, make error prob. exponentially small by taking | n p(n) as advice On input x {0,1} n, loop through all y x in lexicographic order For i {0,1}, let S i be set of advice states that cause algorithm to output i with prob. 1-c -n. Then there exist orthogonal subspaces H 0,H 1 s.t. all states in S i are exponentially close to H i To see this: acceptance probability on advice | can be written | x |, for some Hermitian p.s.d. x with eigenvalues in [0,1]. Let H 0,H 1 be subspaces spanned by eigenvectors of x corresponding to eigenvalues in [0,1/3], [2/3,1] respectively

The Subspaces Let T y be subspace of | s compatible with inputs 1,…,y (initially T 0 = whole Hilbert space) Let T y = whichever has larger dimension: projection of T y-1 onto H 0, or projection of T y-1 onto H 1 Unless classical advice says to pick the subspace of smaller dimension! Each time we pick smaller subspace, dim(T y ) is at least halved. So advice needs to intervene only polynomially many times H1H1 H0H0

The Subspaces Can do everything in EXP (diagonalize exponentially large matrix y, loop over all inputs, etc.) Main technical fact: Error (distance from T y to | n p(n) ) stays bounded over all iterations

Open Problems Oracle separation between BQP/poly and BQP/qpoly Is BQP/qpoly PSPACE/poly? Is BQP/qpoly PP/poly relative to an oracle? Any natural problems in BQP/qpoly (besides cousins of QMA problems)?