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arXiv:1008.3291 · Physical Review A

Unifying parameter estimation and the Deutsch-Jozsa algorithm for continuous variables

Marcin Zwierz, Carlos A. Pérez-Delgado, Pieter Kok

Published Oct 20, 2010• 9 citations
arXivHTMLPDF

Abstract

We reveal a close relationship between quantum metrology and the Deutsch-Jozsa algorithm on continuous-variable quantum systems. We develop a general procedure, characterized by two parameters, that unifies parameter estimation and the Deutsch-Jozsa algorithm. Depending on which parameter we keep constant, the procedure implements either the parameter-estimation protocol or the Deutsch-Jozsa algorithm. The parameter-estimation part of the procedure attains the Heisenberg limit and is therefore optimal. Due to the use of approximate normalizable continuous-variable eigenstates, the Deutsch-Jozsa algorithm is probabilistic. The procedure estimates a value of an unknown parameter and solves the Deutsch-Jozsa problem without the use of any entanglement.

DOI
10.1103/physreva.82.042320

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We reveal a close relationship between quantum metrology and the Deutsch-Jozsa algorithm on continuous-variable quantum systems. We develop a general procedure, characterized by two parameters, that unifies parameter estimation and the Deutsch-Jozsa algorithm. Depending on which parameter we keep constant, the procedure implements either the parameter-estimation protocol or the Deutsch-Jozsa algorithm. The parameter-estimation part of the procedure attains the Heisenberg limit and is therefore optimal. Due to the use of approximate normalizable continuous-variable eigenstates, the Deutsch-Jozsa algorithm is probabilistic. The procedure estimates a value of an unknown parameter and solves the Deutsch-Jozsa problem without the use of any entanglement.
This thesis presents three different results in quantum information theory. The first result addresses the theoretical foundations of quantum metrology. The Heisenberg limit considered as the ultimate limit in quantum metrology sets a lower bound on how precisely a physical quantity can be measured given a certain amount of resources in any possible measurement. Recently, however, several measurement procedures have been proposed in which the Heisenberg limit seemed to be surpassed. This led to an extensive debate over the question how the sensitivity scales with the physical resources and the computational resources that are used in estimation procedures. Here, we reconcile the physical definition of the relevant resources with the information-theoretical scaling in terms of the query complexity of a quantum network. This leads to a novel and ultimate Heisenberg limit that applies to all conceivable measurement procedures. The second result reveals a close relationship between quantum metrology and the Deutsch-Jozsa algorithm over continuous-variable quantum systems. Here, we develop a general procedure, characterized by two parameters, that unifies parameter estimation and the Deutsch-Jozsa algorithm. The procedure estimates a value of an unknown parameter with Heisenberg-limited precision or solves the Deutsch-Jozsa problem in a single run without the use of any entanglement. The third result illustrates how physical principles that govern interaction of light and matter can be efficiently employed to create a computational resource for a (one-way) quantum computer. More specifically, we demonstrate theoretically a scheme based on atomic ensembles and the dipole blockade mechanism for generation of the so-called cluster states in a single step. This procedure is significantly more efficient than any known robust probabilistic entangling operation.
We examine the conditions under which quantum parameter estimation is compatible with the concept of a ``true'' quantity value: a unique value of the estimated quantity that exists prior to and independently of measurement. While quantum parameter estimation typically assumes that the quantity to be estimated is a parameter, we argue that this assumption can be lifted, opening the way to a more general formulation of estimation that may apply even in situations where the quantity to be estimated is no longer a parameter, but instead a quantum variable susceptible to quantum fluctuations. We identify three assumptions that characterize the way estimation is often presented in the quantum metrology literature, namely, as a procedure for inferring an unknown ``true value", which is assumed to be unique and to exist prior to and independently of measurement. Using our more general framework, we prove a no-go theorem showing that the preceding assumptions are jointly incompatible with two further assumptions: the completeness of the quantum state (no `hidden variables'), and the presence of entanglement between the probe systems and the system under measurement. We note that the presence of such entanglement is a generic prediction of standard quantum theory. Consequently, if one is not willing to accept hidden variables, nor willing to modify standard quantum theory by introducing a new physical mechanism that destroys entanglement at some scale or regime, then a realist account of estimation is ultimately impossible.
This paper is a study of the continuous-variable Deutsch–Jozsa algorithm. First, we review an existing version of the algorithm for qunat states (Pati and Braunstein 2002 arXiv:0207108v1), and then, we present a realistic version of the Deutsch–Jozsa algorithm for continuous variables, which can be implemented in a physical quantum system given the appropriate oracle. Under these conditions, we have a probabilistic algorithm for deciding the function with a very high success rate with a single call to the oracle. Finally, we look at the effects of errors in both of these continuous-variable algorithms and how they affect the chances of success. We find that the algorithm is generally robust for errors in initialization and the oracle, but less so for errors in the measurement apparatus and the Fourier transform. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.
In this paper, we propose a novel algorithm that solves a generalized version of the Deutsch-Jozsa problem. The proposed algorithm has the potential to classify an oracle U F , that represents an unknown Boolean function on n Boolean variables, to one of 2 n different classes instead of only two classes which are constant and balanced classes in the case of Deutsch-Jozsa algorithm. The proposed algorithm is based on the use of entanglement measure to explore 2 n - 2 additional classes compared to the standard Deutsch-Jozsa algorithm. In addition, the comparison between the proposed quantum algorithm and the classical one is investigated in details. The comparison shows that the proposed algorithm is faster when the number of Boolean variables exceed 14 variables.
A pivotal task in quantum metrology, and quantum parameter estimation in general, is to design schemes that achieve the highest precision with the given resources. Standard models of quantum metrology usually assume that the dynamics is fixed and that the highest precision is achieved by preparing the optimal probe states and performing optimal measurements. However, in many practical experimental settings, additional controls are usually available to alter the dynamics. Here we propose to use optimal control methods for further improvement of the precision limit of quantum parameter estimation. We show that, by exploring the additional degree of freedom offered by the controls, a higher-precision limit can be achieved. In particular, we show that the precision limit under the controlled schemes can go beyond the constraints put by the coherent time, which is in contrast with the standard scheme where the precision limit is always bounded by the coherent time.