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arXiv:quant-ph/0610044 · Physical Review B

Rapid-state purification protocols for a Cooper pair box

Elias J. Griffith, Charles D. Hill, Jason F. Ralph, Howard M. Wiseman, Kurt A. Jacobs

Published Jan 9, 2007• 24 citations
arXivHTMLPDF

Abstract

We propose techniques for implementing two different rapid-state purification schemes, within the constraints present in a superconducting charge qubit system. Both schemes use a continuous measurement of charge $(z)$ measurements and seek to minimize the time required to purify the conditional state. Our methods are designed to make the purification process relatively insensitive to rotations about the $x$-axis, due to the Josephson tunneling Hamiltonian. The first proposed method, based on the scheme of Jacobs [Phys. Rev. A 67, 030301(R) (2003)] uses the measurement results to control bias $(z)$ pulses so as to rotate the Bloch vector onto the $x$-axis of the Bloch sphere. The second proposed method, based on the scheme of Wiseman and Ralph [New J. Phys. 8, 90 (2006)] uses a simple feedback protocol which tightly rotates the Bloch vector about an axis almost parallel with the measurement axis. We compare the performance of these and other techniques by a number of different measures.

DOI
10.1103/physrevb.75.014511

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We propose techniques for implementing two different rapid-state purification schemes, within the constraints present in a superconducting charge qubit system. Both schemes use a continuous measurement of charge (z)(z)(z) measurements and seek to minimize the time required to purify the conditional state. Our methods are designed to make the purification process relatively insensitive to rotations about the xxx-axis, due to the Josephson tunneling Hamiltonian. The first proposed method, based on the scheme of Jacobs [Phys. Rev. A 67, 030301(R) (2003)] uses the measurement results to control bias (z)(z)(z) pulses so as to rotate the Bloch vector onto the xxx-axis of the Bloch sphere. The second proposed method, based on the scheme of Wiseman and Ralph [New J. Phys. 8, 90 (2006)] uses a simple feedback protocol which tightly rotates the Bloch vector about an axis almost parallel with the measurement axis. We compare the performance of these and other techniques by a number of different measures.
This paper discusses the use of continuous weak measurement and quantum feedback for the rapid purification of the quantum state of a model solid state qubit: a superconducting Cooper pair box. The feedback algorithm uses Jacobs' rapid purification protocol, which starts with a completely mixed state and applies controls to rotate the qubit Bloch vector onto the plane orthogonal to the measurement axis. This rotation maximises the rate of increase of the average purity of the state but can require large changes in the control fields to produce the required rotation. Since solid state qubits have finite controls and feedback channels have limited bandwidth, such rotations may not be practical. This paper applies Jacobs' protocol to the Cooper pair box with realistic control fields.
Abstract. This paper reconsiders the properties of a scheme for the rapid purification of the quantum state of a qubit, proposed recently in Jacobs 2003 Phys. Rev. A 67 030301(R). The qubit starts in a completely mixed state, and information is obtained by a continuous measurement. Jacobs ’ rapid purification protocol uses Hamiltonian feedback control to maximise the average purity of the qubit for a given time, with a factor of two increase in the purification rate over the no-feedback protocol. However, by re-examining the latter approach, we show that it mininises the average time taken for a qubit to reach a given purity. In fact, the average time taken for the no-feedback protocol beats that for Jacobs ’ protocol by a factor of two. We discuss how this is compatible with Jacobs ’ result, and the usefulness of the different approaches. Reconsidering Rapid Qubit Purification by Feedback 2 1.
The work of this thesis focuses on stochastic methods for the simulation of superconducting charge qubits, also known as Cooper pair boxes, which are promising candidates for large scale quantum information processing. To aid non-physicists, a brief outline of the structure of quantum mechanics is provided using the· Dirac formalism. Using the so called 'Backreaction effect', we consider if any information can be obtained regarding the qubit behaviour, through detecting the changes observed in the frequency spectrum of the coupled biasing circuitry, modelled as a dissipative oscillator circuit. The process of modelling dissipative quantum systems is described, however an alternative approach called 'quantum trajectories' is used rather than the traditional Calderia-Leggett model, as the time evolution of a single 'trajectory' represents the evolution of an individual system coupled to a noisy environment. Through the noise generated by an excited qubit, the energy level structure of the qubit can be probed with a microwave drive field, by observing the noise power within the biasing circuit. We consider a biasing circuit of unusually high resonant frequency which can drive the qubit, this creates frequency splitting features that would not normally be observed. Weak measurement is also examined as this is closely related to the stochastic 'quantum trajectories', where the measurement is recorded by the observer rather than lost to an environment. Weakly measuring a qubit does not completely collapse it and therefore 'quantum feedback' may be employed to alter the qubit controls favourably. In particular we consider the problem of purifying a weakly measured system rapidly; given a qubit in the completely mixed state what is the best feedback to become confident in the actual qubit state quickly. There are two optimal feedback protocols proposed by Jacobs [1] and by Wiseman and Ralph [2] for purifying qubits that have ideal controls. However, we adapt these protocols for the charge qubit, whose finite Hamiltonian resources and non-zero ax tunnelling term means the Bloch vector can not be easily held in the optimal location.
This paper reconsiders the claimed rapidity of a scheme for the purification of the quantum state of a qubit, proposed recently in Jacobs (2003 Phys. Rev. A 67 030301(R)). The qubit starts in a completely mixed state, and information is obtained by a continuous measurement. Jacobs' rapid purification protocol uses Hamiltonian feedback control to maximize the average purity of the qubit for a given time , with a factor of two increase in the purification rate over the no-feedback protocol. However, by re-examining the latter approach, we show that it minimizes the average time taken for a qubit to reach a given purity . In fact, the average time taken for the no-feedback protocol beats that for Jacobs' protocol by a factor of two. We discuss how this is compatible with Jacobs' result and the usefulness of the different approaches.
Rapid purification by feedback---specifically, reducing the mean impurity faster than by measurement alone---can be achieved by choosing the eigenbasis of the density matrix to be unbiased relative to the measurement basis. Here we further examine the protocol introduced by Combes and Jacobs [Phys. Rev. Lett. 96, 010504 (2006)] involving continuous measurement of the observable Jz{J}_{z}Jz​ for a DDD-dimensional system. We rigorously rederive the lower bound (2/3)(D+1)(2/3)(D+1)(2/3)(D+1) on the achievable speedup factor and also an upper bound, namely D2/2{D}^{2}/2D2/2, for all feedback protocols that use measurements in unbiased bases. Finally, we extend our results to nnn independent measurements on a register of nnn qubits and derive an upper bound on the achievable speedup factor that scales linearly with nnn.
We present a number of rapid-purification feedback protocols for optical homodyne detection of a single optical qubit. We derive first a protocol that speeds up the rate of increase of the average purity of the system, and find that as in the equivalent protocol for a nondissipative measurement, this generates a deterministic evolution for the purity in the limit of strong feedback. We also consider two analogs of the Wiseman-Ralph rapid-purification protocol in this setting, and show that as in that protocol they speed up the average time taken to reach a fixed level of purity. We also examine how the performance of these algorithms changes with detection efficiency, being an important practical consideration.