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arXiv:1903.09452 · Physical Review A

Robust controllability of two-qubit Hamiltonian dynamics

Ryosuke Sakai, Akihito Soeda, Mio Murao, Daniel Klaus Burgarth

Published Oct 8, 2019• 5 citations

Abstract

Physically, quantum gates (unitary gates) for quantum computation are implemented by controlling the Hamiltonian dynamics of quantum systems. When full descriptions of the Hamiltonians are given, the set of implementable quantum gates is easily characterized by quantum control theory. In many real systems, however, the Hamiltonians may include unknown parameters due to the difficulty of performing precise measurements or instability of the system. In this paper, we consider the situation that some parameters of the Hamiltonian are unknown, but we still want to perform a robust control of the Hamiltonian dynamics to implement a quantum gate irrespectively to the unknown parameters. The existence of the robust control was previously shown for single-qubit systems, and a constructive method was developed for two-qubit systems if a full control of each qubit is available. We analytically investigate the robust controllability of two-qubit systems, and apply Lie-algebraic approaches to handle the cases where only one of the two qubits is controllable. We also numerically analyze the robust controllability of the two-qubit systems where the analytical approach is not necessarily applicable and investigate the relationship between the robust controllability of systems with a discrete and continuous unknown parameter.

DOI: 10.1103/physreva.100.042305

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Robust control of quantum systems against parametric uncertainties is crucial for the practical implementation of quantum technologies. However, whether all gates are robustly realizable, i.e., can be precisely realized against errors, remain not fully understood. This paper investigates the identification of robustly realizable quantum gates from a Lie algebraic perspective. For single-parameter uncertainties, we derive an algebraic criterion that is straightforward to test. Numerical simulations involving single-qubit control systems demonstrate the validity of the proposed criterion.

In practical scenarios, quantum systems often encounter significant uncertainties-such as unknown initial states, imprecise physical parameters, and unmodeled noise that challenge robust quantum control. We propose a feedback approach leveraging a reduced quantum filter capable of handling both Hamiltonian perturbations and additional unmodeled dissipative dynamics, without requiring prior knowledge of initial states or precise system parameters. Notably, the reduced filter tracks only

\mathcal{O}(N)

real parameters instead of

\mathcal{O}\left(N^{2}\right)

, where

N

is the Hilbert space dimension. By substantially lowering the dimensionality of the filter, this method significantly simplifies real-time implementation and offers a scalable path to robust quantum control in uncertain environments.

Quantum-information processing systems are often operated through time-dependent controls; choosing these controls in a way that makes the resulting operation insensitive to variations in unknown or uncontrollable system parameters is an important prerequisite for obtaining high-fidelity gate operations. In this article we present a numerical method for constructing such robust control sequences for a quite general class of quantum-information processing systems. As an application of the method we have designed a robust implementation of a phase-shift operation central to rare-earth-metal quantum computing, an ensemble quantum computing system proposed by Ohlsson et al. [Opt. Commun. 201, 71 (2002)]. In this case the method has been used to obtain a high degree of insensitivity with respect to differences between ensemble members, but it is equally well suited for quantum computing with a single physical system.

This technical note presents a sampled-data approach to the robust control of a single qubit (quantum bit). The required robustness is defined using a sliding mode domain and the control law is designed offline and then utilized online with a single qubit having bounded uncertainties. Two classes of uncertainties are considered involving the system Hamiltonian and the coupling strength of the system-environment interaction. Four cases are analyzed in detail including without decoherence, with amplitude damping decoherence, phase damping decoherence and depolarizing decoherence. Sampling periods are specifically designed for these cases to guarantee the required robustness. Two sufficient conditions are presented for the design of a unitary control for the cases without decoherence and with amplitude damping decoherence. The proposed approach has potential applications in quantum error-correction and in constructing robust quantum gates.

An important step in building a quantum computer is calibrating experimentally implemented quantum gates to produce operations that are close to ideal unitaries. The calibration step involves estimating the error in gates and then using controls to correct the implementation. Quantum process tomography is a standard technique for estimating these errors, but is both time consuming, (when one only wants to learn a few key parameters), and requires resources, like perfect state preparation and measurement, that might not be available. With the goal of efficiently estimating specific errors using minimal resources, we develop a parameter estimation technique, which can gauge two key parameters (amplitude and off-resonance errors) in a single-qubit gate with provable robustness and efficiency. In particular, our estimates achieve the optimal efficiency, Heisenberg scaling. Our main theorem making this possible is a robust version of the phase estimation procedure of Higgins et al. [B. L. Higgins, New J. Phys. 11, 073023 (2009)].

Dynamically corrected gates were recently introduced [K. Khodjasteh and L. Viola, Phys. Rev. Lett. 102, 080501 (2009)] as a tool to achieve decoherence-protected quantum gates based on open-loop Hamiltonian engineering. Here, we further expand the framework of dynamical quantum error correction, with emphasis on elucidating under what conditions decoherence suppression can be ensured while performing a generic target quantum gate, using only available bounded-strength control resources. Explicit constructions for physically relevant error models are detailed, including arbitrary linear decoherence and pure dephasing on qubits. The effectiveness of dynamically corrected gates in an illustrative non-Markovian spin-bath setting is investigated numerically, confirming the expected fidelity performance in a wide parameter range. Robustness against a class of systematic control errors is automatically incorporated in the perturbative error regime.