# Quantum query complexity of state conversion

### Venue

Proceeding of 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS'11) (2011), pp. 344-353

### Publication Year

2011

### Authors

Troy Lee, Rajat Mittal, Ben Reichardt, Robert Spalek, Mario Szegedy

### BibTeX

## Abstract

State conversion generalizes query complexity to the problem of converting between
two input-dependent quantum states by making queries to the input. We characterize
the complexity of this problem by introducing a natural information-theoretic norm
that extends the Schur product operator norm. The complexity of converting between
two systems of states is given by the distance between them, as measured by this
norm. In the special case of function evaluation, the norm is closely related to
the general adversary bound, a semi-definite program that lower-bounds the number
of input queries needed by a quantum algorithm to evaluate a function. We thus
obtain that the general adversary bound characterizes the quantum query complexity
of any function whatsoever. This generalizes and simplifies the proof of the same
result in the case of boolean input and output. Also in the case of function
evaluation, we show that our norm satisfies a remarkable composition property,
implying that the quantum query complexity of the composition of two functions is
at most the product of the query complexities of the functions, up to a constant.
Finally, our result implies that discrete and continuous-time query models are
equivalent in the bounded-error setting, even for the general state-conversion
problem.