Rapid Measurement of Quantum Systems Using Feedback Control

Abstract

We introduce a feedback control algorithm that increases the speed at which a measurement extracts information about a d-dimensional system by a factor that scales as d 2 . Generalizing this algorithm, we apply it to a register of n qubits and show an improvement O(n). We derive analytical bounds on the benefit provided by the feedback and perform simulations that confirm that this speedup is achieved.PACS numbers: 02.30Yy, Recently it has been shown that it is possible to increase the speed at which a measurement purifies the state of a quantum system by using real-time feedback control as the measurement proceeds [1,2,3,4,5,6,7,8,9]. Specifically, by using feedback to keep the state of the system diagonal in a basis that is unbiased with respect to the measured observable, one can make the system purity increase deterministically at a rate faster than the increase of the average purity by measurement alone [1,2]. Since this protocol requires an unbiased basis, it exploits a purely quantum mechanical effect. However, as a consequence this quantum feedback protocol prevents the observer from obtaining full information about the initial preparation of the system, and is therefore not appropriate for use in applications such as communication channels [10]. It is for this reason that the effect of the protocol is termed rapid-purification and not rapid-measurement.Here we present a new protocol that can be applied to both quantum and classical systems, and that, in contrast to quantum rapid-purification, increases the rate at which the observer gains information about the initial preparation, as well as the rate at which the state is purified. This protocol therefore achieves not merely rapid purification, but also rapid measurement, and can be used in communication channels, state stabilisation, read out and error correction in quantum computers. It is also distinct from the rapid state-discrimination protocol introduced recently in [11], which is only applicable to non-orthogonal states and thus to quantum systems. Our protocol can be applied to all systems with dimension d > 2. Asumming the measured observable scales with d, we show that our protocol increases the speed of a measurement by a factor O(d 2 ) over an unaided measurement on a qudit. This is in contrast to the O(d) speed-up acheived by previous protocols [2]. We generalise our protocol for a register of n qubits, each being measured independently and continuously, and obtain an improvement O(n).The evolution of the state ρ of a system subject to a continuous measurement of an observable X is given by the stochastic master equation (SME) [13,14] whereThe measurement strength, γ, determines the rate at which information is extracted, and thus the rate at which the system is projected onto a single eigenstate of X [12]. We denote the continuous measurement record obtained by the observer as r(t), and dr = √ 4γ X dt + dW . We assume that the measurement strength is much smaller than the strength of the Hamiltonians that can be controlled by feedback, so ...

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