Ben W. Reichardt

Any AND-OR Formula of Size

Consider the problem of evaluating an AND-OR formula on an

N

N

Can Be Evaluated in Time

-bit black-box input. We present a bounded-error quantum algorithm that solves this problem in time

N^{1/2+o(1)}

N^{1/2+o(1)}

on a Quantum Computer

. In particular, approximately balanced formulas can be evaluated in

ANY AND-OR FORMULA OF SIZE N CAN BE EVALUATED IN TIME N1/2+o(1) ON A QUANTUM COMPUTER

O(\sqrt{N})

Classical command of quantum systems

queries, which is optimal. The idea of the algorithm is to apply phase estimation to a discrete-time quantum walk on a weighted tree whose spectrum encodes the value of the formula.

Quantum Query Complexity of State Conversion

Consider the problem of evaluating an AND-OR formula on an N-bit black-box input. We present a bounded-error quantum algorithm that solves this problem in time N 1/2+o(1) . In particular, approximately balanced formulas can be evaluated in O(√N) queries, which is optimal. The idea of the algorithm is to apply phase estimation to a discrete-time quantum walk on a weighted tree whose spectrum encodes the value of the formula.

The quantum adiabatic optimization algorithm and local minima

Quantum computation and cryptography both involve scenarios in which a user interacts with an imperfectly modelled or ‘untrusted’ system. It is therefore of fundamental and practical interest to devise tests that reveal whether the system is behaving as instructed. In 1969, Clauser, Horne, Shimony and Holt proposed an experimental test that can be passed by a quantum-mechanical system but not by a system restricted to classical physics. Here we extend this test to enable the characterization of a large quantum system. We describe a scheme that can be used to determine the initial state and to classically command the system to evolve according to desired dynamics. The bipartite system is treated as two black boxes, with no assumptions about their inner workings except that they obey quantum physics. The scheme works even if the system is explicitly designed to undermine it; any misbehaviour is detected. Among its applications, our scheme makes it possible to test whether a claimed quantum computer is truly quantum. It also advances towards a goal of quantum cryptography: namely, the use of ‘untrusted’ devices to establish a shared random key, with security based on the validity of quantum physics.

Quantum Universality from Magic States Distillation Applied to CSS Codes

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.