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arXiv:2211.14862 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Control limit for the quantum state preparation under stochastic control errors

Kohei Kobayashi

Published Sep 1, 2023• 0 citations
arXivHTMLPDF

Abstract

We investigate the effect of stochastic control errors on the Hamiltonian that controls a closed quantum system. Quantum information technologies require careful control for preparing a desired state used as an information resource. However, because the stochastic control errors inevitably appear in realistic situations, it is difficult to completely implement the control Hamiltonian. Under this error, the actual performance of quantum control is far away from the ideal one, and thus it is of great importance to evaluate the effect of the control errors. In this paper, we derive a lower bound of the fidelity between two closed quantum systems obeying the dynamics with and without errors. This bound reveals a reachable and unreachable set of the controlled quantum system under stochastic noises. Also, it is easily computable without considering the stochastic process and needing the full dynamics of the states. We demonstrate the actual performance of this bound via a simple control example. Furthermore, based on this result, we quantitatively evaluate the probability of obtaining the target state in the presence of control errors.

DOI
10.1098/rspa.2023.0215

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We investigate the effect of stochastic control errors on the Hamiltonian that controls a closed quantum system. Quantum information technologies require careful control for preparing a desired state used as an information resource. However, because the stochastic control errors inevitably appear in realistic situations, it is difficult to completely implement the control Hamiltonian. Under this error, the actual performance of quantum control is far away from the ideal one, and thus it is of great importance to evaluate the effect of the control errors. In this paper, we derive a lower bound of the fidelity between two closed quantum systems obeying the dynamics with and without errors. This bound reveals a reachable and unreachable set of the controlled quantum system under stochastic noises. Also, it is easily computable without considering the stochastic process and needing the full dynamics of the states. We demonstrate the actual performance of this bound via a simple control example. Furthermore, based on this result, we quantitatively evaluate the probability of obtaining the target state in the presence of control errors.
We investigate the effect of analog control errors which deterministically occurs on isolated quantum dynamics. Quantum information technologies require careful control for preparing a desired quantum state used as an information resource. However, in realistic experiment systems, it is difficult to implement the driving Hamiltonian without analog errors and the actual performance of quantum control is far away from the ideal one. Towards this problem, we derive a lower bound of the overlap between two isolated quantum systems obeying time evolution in the absence and presence of deterministic control errors. We demonstrate the effectiveness of the bound through some examples. Furthermore, by using this bound, we give an analytical estimate on the probability of obtaining the target state under any control errors.
Quantum information technologies require careful control for generating and preserving a desired target quantum state. The biggest practical obstacle is of course decoherence. Therefore, the reachability analysis, which in our scenario aims to estimate the distance between the controlled state under decoherence and the target state, is of great importance to evaluate the realistic performance of those technologies. This paper presents a lower bound of the fidelity-based distance for a general open Markovian quantum system driven by the decoherence process and several types of control including feedback. The lower bound is straightforward to calculate and can be used as a guide for choosing the target state, as demonstrated in some examples. Moreover, the lower bound is applied to derive a theoretical limit in some quantum metrology problems based on a large-size atomic ensemble under control and decoherence.
Abstract To implement quantum information technologies, carefully designed control for preparing a desired state plays a key role. However, in realistic situation, the actual performance of those methodologies is severely limited by decoherence. Therefore, it is important to evaluate how close we can steer the controlled state to a desired target state under decoherence. In this paper, we provide an upper bound of the distance between the two controlled quantum systems in the presence and absence of decoherence. The bound quantifies the degree of achievement of the control for a given target state under decoherence, and can be straightforwardly calculated without solving any equation. Moreover, the upper bound is applied to derive a theoretical limit of the probability for obtaining the target state under decoherence.
Precisely characterizing and controlling realistic quantum systems under noises is a challenging frontier in quantum sciences and technologies. In developing reliable controls for open quantum systems, one is often confronted with the problem of the lack of knowledge on the system controllability. The purpose of this paper is to give a numerical approach to this problem, that is, to approximately compute the reachable set of states for coherently controlled quantum Markovian systems. The approximation consists of setting both upper and lower bounds for system's reachable region of states. Furthermore, we apply our reachability analysis to the control of the relaxation dynamics of a two-qubit nuclear magnetic resonance spin system. We implement some experimental tasks of quantum state engineering in this open system at a near optimal performance in view of purity: e.g., increasing polarization and preparing pseudopure states. These results demonstrate the usefulness of our theory and show interesting and promising applications of environment-assisted quantum dynamics.
Analog quantum simulation is a promising path towards solving classically intractable problems in many-body physics on near-term quantum devices. However, the presence of noise limits the size of the system and the length of time that can be simulated. In our work, we consider an error model in which the actual Hamiltonian of the simulator differs from the target Hamiltonian we want to simulate by small local perturbations, which are assumed to be random and unbiased. We analyze the error accumulated in observables in this setting and show that, due to stochastic error cancellation, with high probability the error scales as the square root of the number of qubits instead of linearly. We explore the concentration phenomenon of this error as well as its implications for local observables in the thermodynamic limit. Moreover, we show that stochastic error cancellation also manifests in the fidelity between the target state at the end of time-evolution and the actual state we obtain in the presence of noise. This indicates that, to reach a certain fidelity, more noise can be tolerated than implied by the worst-case bound if the noise comes from many statistically independent sources.
We investigate the effect of stochastic control errors in the time-dependent Hamiltonian on isolated quantum dynamics. The control errors are formulated as time-dependent stochastic noise in the Schrödinger equation. For a class of stochastic control errors, we establish a threshold theorem that provides a sufficient condition to obtain the target state, which should be determined in noiseless isolated quantum dynamics, as a relation between the number of measurements and noise strength. The theorem guarantees that if the sum of the noise strengths is less than the inverse of computational time, the target state can be obtained through a constant-order number of measurements. If the opposite is true, the number of measurements to guarantee obtaining the target state increases exponentially with computational time. Our threshold theorem can be applied to any isolated quantum dynamics such as quantum annealing and adiabatic quantum computation. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.