### DIMACS Theoretical Computer Science Seminar

Title: Quantum Adversary Method with a Twist

Speaker: **Troy Lee**

Date: Wednesday, September 12, 2007 11:00-12:00pm

Location: DIMACS Center, CoRE Bldg, Room 431, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

The quantum adversary method is one of the most successful techniques for proving lower bounds on quantum query complexity. It gives optimal lower bounds for many problems, has application to classical complexity in formula size lower bounds, and is versatile with equivalent formulations in terms of weight schemes, eigenvalues, and Kolmogorov complexity. All these formulations rely on the principle that if an algorithm successfully computes a function then, in particular, it is able to distinguish between inputs which map to different values.

We present a stronger version of the adversary method which goes beyond this principle to make explicit use of the stronger condition that the algorithm actually computes the function. This new method, which we call ADV+-, has all the advantages of the old: it is a lower bound on bounded-error quantum query complexity, its square is a lower bound on formula size, and it behaves well with respect to function composition. Moreover ADV+- is always at least as large as the standard adversary method ADV, and we show an example of a monotone function for which
ADV+-(f)=\Omega(ADV(f)^{1.098}). We also give examples showing that
ADV+- does not face limitations of ADV like the certificate complexity
barrier and the property testing barrier.

Joint work with Peter Hoyer and Robert Spalek.