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Lobachevskii Journal of Mathematics

Exploration of Quantum Control Landscapes for Single-qubit Gate Generation in Few-Level Fluxonium Approximations

I. M. Korolev, Alexander Nikolaevich Pechen

Published Jun 1, 2026• 0 citations
Publisher

Abstract

Abstract In this work, we explore quantum control landscapes for the single qubit optimal gate generation problem for fluxonium models of superconducting systems in the vicinity of the null control. Ultimately, the fluxonium qubit is described by an infinite-dimensional Hamiltonian, for which using the two lowest levels provides only a minimal approximation. A more detailed analysis requires considering more energy levels for a single qubit. We consider two-, three-, four-, and five-level approximations for a single qubit, and as an example, the generation of a Hadamard gate. We investigate how taking into account different numbers of energy levels for a single qubit affects the efficiency of optimization and control landscape properties. As a result, we find an unexpected sharp difference in the behavior for various values of the parameters and for various numbers of energy levels used for the approximation, including non-monotonic behavior of the optimization efficiency in some cases.

DOI
10.1134/s1995080226619612

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Abstract In this work, we explore quantum control landscapes for the single qubit optimal gate generation problem for fluxonium models of superconducting systems in the vicinity of the null control. Ultimately, the fluxonium qubit is described by an infinite-dimensional Hamiltonian, for which using the two lowest levels provides only a minimal approximation. A more detailed analysis requires considering more energy levels for a single qubit. We consider two-, three-, four-, and five-level approximations for a single qubit, and as an example, the generation of a Hadamard gate. We investigate how taking into account different numbers of energy levels for a single qubit affects the efficiency of optimization and control landscape properties. As a result, we find an unexpected sharp difference in the behavior for various values of the parameters and for various numbers of energy levels used for the approximation, including non-monotonic behavior of the optimization efficiency in some cases.
In this work, we consider a formulation of the optimal gate generation problem for a fluxonium model of superconducting systems and apply it to the numerical analysis of the corresponding quantum control landscape in a vicinity of the null control. The model includes Schrödinger equation with control and maximization of the objective functionals for gate generation and observable mean value. For a qutrit fluxonium model, the third level can be considered as one of the three computational states. We exploit gradient of the control objective and GRAPE algorithm to perform numerical investigation of the quantum control landscapes for generation of the Hadamard, T\mathrm{T}T , and S\mathrm{S}S gates and for maximizing mean value of a quantum observable in a vicinity of the null control. The obtained results show high efficiency of gradient-based optimization when it starts from the null control and decrease of the efficiency with increase of the distance of the initial control from the null control for the gate generation problems, and an opposite behaviour for the mean value maximization. For gate generation, the obtained behavior differs (is opposite) from the behavior found previously for maximizing mean value of an observable for Λ\LambdaΛ -type systems, that may indicate either the absence or low influence of trap at the null control for a fluxonium.
An important problem in quantum computation is the generation of single-qubit quantum gates such as Hadamard (H) and π/8 (T) gates, which are components of a universal set of gates. Qubits in experimental realizations of quantum computing devices are interacting with their environment. While the environment is often considered as an obstacle leading to a decrease in the gate fidelity, in some cases, it can be used as a resource. Here, we consider the problem of the optimal generation of H and T gates using coherent control and the environment as a resource acting on the qubit via incoherent control. For this problem, we studied the quantum control landscape, which represents the behavior of the infidelity as a functional of the controls. We considered three landscapes, with infidelities defined by steering between two, three (via Goerz–Reich–Koch approach), and four matrices in the qubit Hilbert space. We observed that, for the H gate, which is a Clifford gate, for all three infidelities, the distributions of minimal values obtained with a gradient search have a simple form with just one peak. However, for the T gate, which is a non-Clifford gate, the situation is surprisingly different—this distribution for the infidelity defined by two matrices also has one peak, whereas distributions for the infidelities defined by three and four matrices have two peaks, which might indicate the possible existence of two isolated minima in the control landscape. It is important that, among these three infidelities, only those defined with three and four matrices guarantee the closeness of the generated gate to a target and can be used as a good measure of closeness. We studied sets of optimized solutions for the most general and previously unexplored case of coherent and incoherent controls acting together and discovered that they form sub-manifolds in the control space, and unexpectedly, in some cases, two isolated sub-manifolds.
In this work, we study the detailed structure of quantum control landscape for the problem of single-qubit phase shift gate generation on the fast time scale. In previous works, the absence of traps for this problem was proved on various time scales. A special critical point which was known to exist in quantum control landscapes was shown to be either a saddle or a global extremum, depending on the parameters of the control system. However, in case of a saddle, the numbers of negative and positive eigenvalues of the Hessian at this point and their magnitudes have not been studied. At the same time, these numbers and magnitudes determine the relative ease or difficulty for practical optimization in a vicinity of the critical point. In this work, we compute the numbers of negative and positive eigenvalues of the Hessian at this saddle point and, moreover, give estimates on magnitude of these eigenvalues. We also significantly simplify our previous proof of the theorem about this saddle point of the Hessian (Theorem 3 in [22]).
We present a gradient-based method to construct memory-efficient, high-fidelity, single-qubit gates for fluxonium qubits. These gates are constructed using a sequence of single-flux quantum (SFQ) pulses that are sent to the qubit through either capacitive or inductive coupling. The schedule of SFQ pulses is constructed with an on-ramp and an off-ramp applied prior to and after a pulse train, where the pulses are spaced at intervals equal to the qubit period. We reduce the optimization problem to the scheduling of a fixed number of SFQ pulses in the on-ramp and solve it by relaxing the discretization constraint of the SFQ clock as an intermediate step, allowing the use of the Broyden-Fletcher-Goldfarb-Shanno optimizer. Using this approach, gate fidelities of 99.99 % can be achieved for inductive coupling and 99.9 % for capacitive coupling, with leakage being the main source of coherent errors for both approaches.
Scaling up to a full-scale multi-qubit processor adds complexity to the control signals input requirements which can be addressed by the use of specially designed cryogenic interface circuits. In this paper, we discuss the use of cryogenic classical electronics for control of a fluxonium artificial atom. In recent years, fluxonium has gained notoriety among the superconducting qubit community for exceptionally long coherence times and large anharmonicity - both highly desirable properties when designing the next generation of qubit systems. The feasibility of controlling a fluxonium based quantum processors with a compact control system ultimately enables a highly scalable system; however, the cryogenic environment poses very strict requirements on electronics. Many trade-offs arise in terms of noise, power consumption, and circuit complexity. Addressing these requires comprehensive simulations that account for the physics of fluxonium qubits and the potential impact of control circuits on their performance. This paper focuses on the models and specification derivation for qubit control and explores constraints in fluxonium-based system design. The models and specifications derived enable the design of optimal control circuits targeted for scalable fluxonium quantum processors.
Optimizing the control of large quantum systems is a resource-intensive task, where the convergence rate of the underlying iterative schemes is crucial. The recently developed Quantum Projection Operator-Based Newton Method for Trajectory Optimization (Q-PRONTO) offers a unique quadratic convergence for controlling quantum states, while most model-based solvers guarantee only an approximately linear convergence rate. As a proof of concept, here, we design and benchmark single-qubit gates optimized by Q-PRONTO based on the circuit Hamiltonian of a fluxonium superconducting qubit. To ensure that we can apply a long string of arbitrary pulses to the circuit, we modify the firmware associated with the Quantum Instrumentation Control Kit (QICK) by altering its memory allocation. We characterize the performance of the optimal solutions using standard randomized benchmarking and achieve a total error per gate of 6.6×10−46.6\times10^{-4}6.6×10−4.