Editors: Salman Beigi and Robert König
CC-BY 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015)
Quantum query complexity is known to be characterized by the so-called quantum adversary bound. While this result has been proved in the standard discrete-time model of quantum computation, it also holds for continuous-time (or Hamiltonian-based) quantum computation, due to a known equivalence between these two query complexity models. In this work, we revisit this result by providing a direct proof in the continuous-time model. One originality of our proof is that it draws new connections
... ew connections between the adversary bound, a modern technique of theoretical computer science, and early theorems of quantum mechanics. Indeed, the proof of the lower bound is based on Ehrenfest's theorem, while the upper bound relies on the adiabatic theorem, as it goes by constructing a universal adiabatic quantum query algorithm. Another originality is that we use for the first time in the context of quantum computation a version of the adiabatic theorem that does not require a spectral gap. 1998 ACM Subject Classification F.2 Analysis of Algorithms and Problem Complexity 1 Introduction The quantum adversary method was originally introduced by Ambainis  for lower-bounding the quantum query complexity Q(f) of a function f. It is based on optimizing a matrix Γ assigning weights to pairs of inputs. It was later shown by Høyer et al.  that using negative weights also provides a lower bound, which is stronger for some functions. A series of works [26, 27, 25] then led to the breakthrough result that this generalized adversary bound, which we will simply call adversary bound from now on, actually characterizes the quantum query complexity of any function f with boolean output and binary input alphabet. This is shown by constructing a tight algorithm based on the dual of the semidefinite program corresponding to the adversary bound 1. Finally, Lee et al.  have generalized this result to the quantum query complexity of state conversion, where instead of computing a function f (x), one needs to convert a quantum state |ρ x into another quantum state |σ x. All these results where obtained in the usual discrete-time query model, where each query corresponds to applying a unitary oracle O x. In this model, an algorithm then consists in a series of input-independent unitaries U 1 , U 2 ,. .. , U T , interleaved with oracle calls O x. Another natural model is the continuous-time (or Hamiltonian-based) model where the oracle corresponds to a Hamiltonian H x , and the algorithm consists in applying a possibly 1 Note that constructing a tight algorithm for a specific problem using this method requires to find an optimal feasible point for the semidefinite program, so that this method is not necessarily constructive. The same limitation will affect the universal adiabatic algorithm in the present article.