[Theory] UC Theory Seminar

[Alexander Razborov via Theory]

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Departments of Mathematics & Computer Science
Combinatorics & Theory Seminar

Thursday, February 28
Ry 251@ 3:30pm

Scott Aaronson
University of Texas at Austin

Title:  Gentle Measurement of Quantum States and Differential Privacy

Abstract: In differential privacy (DP), we want to query a database about n
users, in a way that "leaks at most eps about any individual user,"
conditioned on any outcome of the query.  Meanwhile, in gentle
measurement, we want to measure n quantum states, in a way that
"damages the states by at most alpha," conditioned on any outcome of
the measurement.  In both cases, we can achieve the goal by techniques
like deliberately adding noise to the outcome before returning it.  We
prove a new and general connection between the two subjects.
Specifically, on products of n quantum states, any measurement that is
alpha-gentle for small alpha is also O(alpha)-DP, and any product
measurement that is eps-DP is also O(eps*sqrt(n)) -gentle.

Illustrating the power of this connection, we apply it to the recently
studied problem of shadow tomography.  Given an unknown d-dimensional
quantum state rho, as well as known two-outcome measurements
E_1,...,E_m, shadow tomography asks us to estimate Pr[E_i accepts
rho], for every i in [m], by measuring few copies of rho.  Using our
connection theorem, together with a quantum analog of the so-called
private multiplicative weights algorithm of Hardt and Rothblum, we
give a protocol to solve this problem using O((log m)^2 (log d)^2)
copies of rho, compared to Aaronson's previous bound of ~O((log m)^4
(log d)).  Our protocol has the advantages of being online (that is,
the E_i's are processed one at a time), gentle, and conceptually
simple.

Joint work with Guy Rothblum (to appear in STOC'2019).

(Refreshments will be served prior to the talk in Ry. 255 @ 3:15pm)