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arXiv:1809.06725

Precise quantum control via unsharp measurements and feedback operations

Du Ran, Ye‐Hong Chen, Zhi‐Cheng Shi, Yang, Zhen-Biao, Jie Song, Yan Xia

Published Sep 18, 2018• 0 citations
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Abstract

In this paper, we propose a scheme to eliminate the influence of noises on system dynamics, by means of a sequential unsharp measurements and unitary feedback operations. The unsharp measurements are carried out periodically during system evolution, while the feedback operations are well designed based on the eigenstates of the density matrices of the exact (noiseless) dynamical states and its corresponding post-measurement states. For illustrative examples, we show that the dynamical trajectory errors caused by both static and non-static noises are successfully eliminated in typical two-level and multi-level systems, i.e., the high-fidelity quantum dynamics can be maintained. Furthermore, we discuss the influence of noise strength and measurement strength on the degree of precise quantum control. Crucially, the measurement-feedback scheme is quite universal in that it can be applied to precise quantum control for any dimension systems. Thus, it naturally finds extensive applications in quantum information processing.

DOI
10.48550/arxiv.1809.06725

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In this paper, we propose a scheme to eliminate the influence of noises on system dynamics, by means of a sequential unsharp measurements and unitary feedback operations. The unsharp measurements are carried out periodically during system evolution, while the feedback operations are well designed based on the eigenstates of the density matrices of the exact (noiseless) dynamical states and its corresponding post-measurement states. For illustrative examples, we show that the dynamical trajectory errors caused by both static and non-static noises are successfully eliminated in typical two-level and multi-level systems, i.e., the high-fidelity quantum dynamics can be maintained. Furthermore, we discuss the influence of noise strength and measurement strength on the degree of precise quantum control. Crucially, the measurement-feedback scheme is quite universal in that it can be applied to precise quantum control for any dimension systems. Thus, it naturally finds extensive applications in quantum information processing.
Abstract In this paper, we generalize the application of a recently proposed quantum control technique called self-fulfilling prophesy (Uys et al 2018 Phys. Rev. A 97 060102) to achieve high-fidelity target quantum dynamics which are disturbed by unwanted external couplings, based on sequential unsharp measurements and tailored unitary feedback operations. The unsharp measurements are carried out periodically during system evolution, while the feedback operations are well-designed based on the eigenstates of the density matrices of the expected (without external couplings) dynamical states and their corresponding post-measurement states. By means of quantum state estimation, the mechanism of the measurement–feedback scheme is explained generally. For illustrative examples, we show that the dynamical trajectory errors caused by both constant couplings and time-varying couplings are substantially reduced in typical two-level and multi-level systems, i.e. the high-fidelity quantum dynamics are achieved. Furthermore, we discuss the influence of external coupling strength and measurement strength on system dynamics. Crucially, the scheme is universal and can be applied to any dimension systems. Thus, it may find applications in quantum information processing.
Measurement combined with feedback that aims to restore a presumed premeasurement quantum state will yield this state after a few measurement-feedback cycles even if the actual state of the system initially had no resemblance to the presumed state. Here we introduce this mechanism of self-fulfilling prophecy and show that it can be used to prepare finite-dimensional quantum systems in target states or force them into target dynamics. Using two-level systems as an example, we demonstrate that self-fulfilling prophecy protects the system against noise and tolerates imprecision of feedback up to the level of the measurement strength. By means of unsharp measurements the system can be driven deterministically into arbitrary, smooth quantum trajectories.
Abstract The recently proposed quantum control method called self‐fulfilling prophesy is investigated in multi‐level cases, based on a sequence of measurement‐feedback operations. The feedback operation is elaborately designed with respect to the eigenstates of the density matrix of the target state and its post‐measurement state. This design procedure is suitable for the generation of any predefined coherent superpositions of multiple quantum states and elicitation of desired quantum dynamics. For the sake of clearness, arbitrarily prescribed superposition states in three‐ and four‐level systems are prepared, and quantum dynamics achieved as desired. The simulation results indicate that the scheme tolerates modest imprecisions of feedback operation and is robust against sudden perturbations. Thus, the scheme enables new insight on quantum manipulations in a variety of multi‐level systems to be achieved.
The advance of experimental techniques in circuit quantum electrodynamics over last decadehas made it possible to monitor the evolution of qubits under continuous measurements in\nreal time . This kind of measurements, as compared to the discrete ones, provides a stream\nof information for feedback to alter the system's quantum dynamics continuously in a desired\nmanner, thus allowing broader possibilities. We obtain the information at a cost, the backaction on the states. However, the back-action turns out not to be just something disturbing \nand harmful to the control but instead can be harnessed, together with the control, to steer the system to our targets. \nThis thesis discusses one such application, namely to use measurements to create entanglement from remote qubits under locally optimal control. It is illustrated that by carefully designing symmetric measurement and feedback operators, we can construct efficient Markovian or non-Markovian protocols which generate N-qubit W,\ngeneral Dicke and Greenberger-Horne-Zeilinger states with high fidelity above 94% up to\nN = 100. In some cases, the fidelity even approaches unity. Following that, we describe how\nto take advantages of the continuous stream of information to protect the unitary dynamics of\na quantum system subject to noises. We develop a practical protocol for continuous operation\nof a quantum error correcting code for protection of coherent evolution due to an encoded\nHamiltonian against environmental errors, using the three qubit bit-flip code and bit-flip\nerrors as a canonical example. We demonstrate, for both quantum memory and adiabatic\nHamiltonians, that the protocol, once optimized, can significantly reduce logical errors with\na performance which is close to the theoretical limit of the bit-flip code's capability. In the\nadiabatic Hamiltonian case, we also propose some novel correction operators instead of the\nconventional ones to further reduce the logical errors.
Unsharp measurements allow the estimation and tracking of quantum wave functions in real time with minimal disruption of the dynamics. Here we demonstrate that high-fidelity state monitoring, and hence quantum control, is possible, even in the presence of classical dephasing and amplitude noise, by simulating such measurements on a two-level system undergoing Rabi oscillations. Finite estimation fidelity is found to persist indefinitely after the decoherence times set by the noise fields in the absence of measurement.
This paper investigates the dynamic suppression of quantum noise within the framework of quantum control theory. The core challenge in implementing precise quantum operations lies in the significant influence of environmental noise, which introduces uncertainties and degrades control fidelity. We propose a novel approach utilizing real-time feedback control, leveraging quantum sensors to monitor the noise levels within a quantum system. Based on this continuous assessment, control parameters are dynamically adjusted to mitigate the impact of noise. The effectiveness of this method is demonstrated through theoretical analysis and mathematical modeling. We aim to enhance the accuracy and stability of quantum control processes, ultimately contributing to the reliable execution of complex quantum operations. The system is described by the following Hamiltonian: H=H0+HIH = H_0 + H_IH=H0​+HI​, where H0H_0H0​ is the system Hamiltonian and HIH_IHI​ represents the interaction Hamiltonian. The time evolution of the system is governed by the equation: d∣ψ(t)>dt=iℏ[H,∣ψ(t)]\frac{d|ψ(t)>}{dt} = \frac{i}{\hbar} [H, |ψ(t)]dtd∣ψ(t)>​=ℏi​[H,∣ψ(t)]. Noise is modeled as a stochastic term, adding to the Hamiltonian: H=H0+HI+HNH = H_0 + H_I + H_NH=H0​+HI​+HN​, where HNH_NHN​ represents the noise Hamiltonian. The control pulse can be described as: U(t)=∑i=1Nαi e−iωitU(t) = \sum_{i=1}^N \alpha_i \, e^{-i\omega_i t}U(t)=∑i=1N​αi​e−iωi​t, where αi\alpha_iαi​ are the amplitudes and ωi\omega_iωi​ are the carrier frequencies. The fidelity of the quantum operation is evaluated using the overlap between the initial and final states: Fidelity=∣∣2Fidelity = ||^2Fidelity=∣∣2. The control parameters are optimized to maximize the fidelity. The quantum state evolution is described by the equation: ∣ψ(t+δt)>=U(t)∣ψ(t)>|ψ(t+\delta t)> = U(t) |ψ(t)>∣ψ(t+δt)>=U(t)∣ψ(t)>, where U(t) is the control pulse. The time-dependent Schrodinger equation is: i/ħ ∂/∂t |ψ(t)> = H|ψ(t)>. The noise introduces uncertainties which can be represented as: σ2=−2\sigma^2 = - ^2σ2=−2, where N is the noise operator.