Download presentation

Presentation is loading. Please wait.

Published byKatherine O'Keefe Modified over 5 years ago

1
BQP/qpoly EXP/poly Scott Aaronson UC Berkeley

2
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?

3
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

4
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

5
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

6
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

7
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)?

Similar presentations

OK

1cs542g-term1-2006 High Dimensional Data So far we’ve considered scalar data values f i (or interpolated/approximated each component of vector values.

1cs542g-term1-2006 High Dimensional Data So far we’ve considered scalar data values f i (or interpolated/approximated each component of vector values.

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google