Optimal Control of thermally noisy quantum gates in a multilevel system

TL;DR

The paper develops a thermodynamically consistent optimal-control framework for implementing quantum gates in open systems experiencing Markovian thermal noise. By employing a NAME-based master equation and an invariant-based construction of time-dependent jump operators, it achieves high-fidelity gates while harnessing environmental interactions. Through a Liouville-space OCT that optimizes over ancilla-assisted architectures and direct qubit control, it demonstrates both robustness to thermal effects and a dissipation-assisted mechanism that concentrates dynamics within a decoherence-resilient subspace. Key findings show that adding ancillas generally improves noise resilience, while small direct control on the logical subspace is crucial for substantial mitigation; two-qubit C-iX gates benefit from OCT within a finite temperature–dissipation window, with energy exchange to the bath providing an energetic lens on the process. Overall, the work provides a concrete, thermodynamically grounded route toward high-fidelity quantum gates in realistic, dissipative settings.

Abstract

Quantum systems are inherently sensitive to environmental noise and imperfections in external control fields, posing a significant challenge for the practical implementation of quantum technologies. These noise sources degrade the fidelity of quantum gates, making their mitigation a key requirement for realizing reliable quantum computing. In this study, we apply Optimal Control Theory (OCT) within a thermodynamically consistent framework to design and stabilize high-fidelity quantum gates under Markovian noise. Our approach focuses on thermal relaxation and incorporates these effects into the control protocol, wherein external driving fields not only govern the system's unitary evolution but also modulate its interaction with the environment. By leveraging this interplay, we demonstrate that OCT can enable entropy-modifying processes, such as targeted cooling or heating, while maintaining high-fidelity gate performance in noisy environments. To validate our approach, we employ high-precision numerical methods for an open quantum system implementing one- or two-qubit gates embedded in a larger Hilbert space. The results showcase robust gate operation even under significant dissipative influences, offering a concrete path toward fault-tolerant quantum computation under realistic conditions.

Figures (9)

• Figure 1: Instantaneous transition (Bohr) frequencies $\omega_{ij}(t)$ (a.u.) for all two-level sub-manifolds of the driven three-level system, plotted versus time (a.u.). We show here two different cases with the same drift Hamiltonian but different control protocols (dashed and solid). these are the frequencies used in fig. (\ref{['fig:2_case_Control_1Qbit']}) for the single qubit Hadamard gate.
• Figure 2: Schematic illustration of the system architecture. The primary quantum system consists of isolated qubits $Q_1, Q_2, \ldots, Q_N$. The control field acts indirectly via a set of ancilla modes $a_1, a_2, \ldots, a_N$, resembling Raman transitions. All components, including both qubits and ancillas, are coupled to a shared thermal environment $H_E$, which introduces decoherence and dissipation. This structure underlies the control framework analyzed in this study, where noise is mitigated and coherence is preserved through optimal control strategies tailored to this topology.
• Figure 3: Normalized infidelity and the purity loss of the map vs the relaxation rate $\gamma$ at fixed temperature $T = 5$. Left axis (solid lines): logarithmic ratio of infidelities $\log_{10}({IF}_{Noise}/{IF}_{U})$ plotted as a function of the system–bath coupling rate $\gamma$. Right axis (dashed lines): Generalized purity Eq. (\ref{['eq:purity']}) of the final reduced qubit map versus $\gamma$. Each color in the graph corresponds to a different number of ancillas coupled to the single qubit.
• Figure 4: Normalized infidelity and purity loss of the map with respect to temperature performance at a fixed system-bath coupling rate of $\gamma = 0.01$. Left axis: The logarithmic ratio of infidelities (infidelity loss) $\log_{10}({IF}_{Noise}/{IF}_{U})$ represented by a solid line against the dimensionless temperature $T$ (in units of the characteristic transition frequency $\omega_0$). Right Axis: the purity of the target map as a function of temperature, indicated by a dashed line. In this context, ${IF} = 1 - {F}$, where ${IF}_U$ is the infidelity of the closed quantum system (as a reference) and ${IF}_{Noise}$ refers to the uncontrolled or naively driven reference state. Each color in the graph corresponds to a different number of ancillas coupled with a single qubit. This representation effectively highlights the effect of temperature on the fidelity and purity of the qubit states.
• Figure 5: Mitigating thermal noise at fixed reduced temperature $T \approx 10^{-4}\,\omega_{ij}$. (a) Logarithmic of the ratio of the control gain as a function of the relaxation rate $\gamma$: $G(\gamma)=\log_{10}\!({IF}_{C}/{IF}_{Noise})$ vs. $\log_{10}\gamma$. (b) additional fidelity loss of the unitary reference when coupled to the bath, quantified by $\log_{10}\!(\mathrm{IF}_{\mathrm{noise}}/\mathrm{IF}_{U})$; this metric rises monotonically with the relaxation rate $\gamma$. (c) Population projections $P_i(t)$ (a.u.) under the optimized protocols.Two distinct optimal controls (G1, G2) with different waveform shapes and amplitudes.