Minimal resources for exact simulation of quantum walks

Abstract

A general approach to obtain reduced models for a wide class of discrete-time quantum systems is proposed. The obtained models not only reproduce exactly the output of a given quantum model, but are also guaranteed to satisfy physical constraints, namely complete positivity and preservation of total probability. A fundamental framework for exact model reduction of quantum systems is constructed leveraging on algebraic methods, as well as novel results on quantum conditional expectations in finite-dimensions. The proposed reduction algorithm is illustrated and tested on prototypical examples, including the quantum walk realizing Grover's algorithm.

Exact Model Reduction for Discrete-Time Conditional Quantum Dynamics

“…Notice that Assumption (A1) implies Assumption (A2): by definition N is M k -invariant for all k, hence for X P N , we have ErXs " ř k λ k M k rXs P N . This generalizes the results presented in [11, equation 3.3] and [13,Lemma 4] to the case of stochastic output matrices (not only deterministic ones). Assumption (A2) however, does not imply (A1).…”

“…p¨q " E ˝Kk . This ordering choice has been made to keep consistency with previous works [11,13] but similar results can be derived if the evolution precedes the measurement.…”

“…The proof of this follows from [13,Theorem 4] on the reducibility of switching dynamics. This can be applied to Proposition 2 by vectorizing the state and obtaining the corresponding matrix form of the superoperators.…”

“…Our approach builds on the ideas of [11], (see [12] for a generalization to quantum walks), where stochastic analogues of reachable and non-observable spaces are used to characterize equivalent hidden-Markov processes. Exploiting the ideas developed in [13,14] for unconditional dynamics, the method leverages algebraic quantum probability tools, in particular conditional expectations, to allow for reduced conditional models that generate the same measurement distribution, reproduce the average of observables of interest, and do so while maintaining the same structure and physical constraints as the original model. In general, existing reduction approaches cannot guarantee that the reduced evolution is completely positive T. Grigoletto and F. Ticozzi are with the Department of Information Engineering, University of Padova, Via Gradenigo 6, 35131 Padova, Italy.…”