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Quantum Optimal Control and its Applications to Shaken Lattice Interferometry and Superconducting Qubits

Shao, Jieqiu

Published Jun 25, 2024• 0 citations
OpenAlex

Abstract

Quantum optimal control is an important technology that enables fast state preparationand gate design. In the absence of an analytic solution, most quantum optimal control methods rely on an iterative scheme to update the solution estimate. At present, the convergence rate of existing solvers is, at most, superlinear. In this thesis, we develop a new general purpose solver for quantum optimal control, Quantum PRojection Operator-based Newton method for Trajectory Optimization (Q-PRONTO). Specifically, the proposed approach uses a projection operator to incorporate the Schr¨odinger equation directly into the cost function, which is then minimized using a Newton descent method. The resulting method guarantees monotonic convergence at every iteration and quadratic convergence in proximity of the solution. We then implement the method to different quantum applications: the first application is shaken lattice interferometry, which is a novel atom-based interferometry for quantum sensing; the second application is fluxonium qubit, which is a novel superconducting qubit for quantum computing. The potential of Q-PRONTO is showcased by computing an interferometry sequence for shaken lattice interferometry and designing a quantum gate for fluxonium qubit. The encouraging simulation results represent a first step in the development of a new paradigm for both quantum sensing and quantum computing.

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•Jan 1, 2011•0 citations
Quantum optimal control is an important technology that enables fast state preparationand gate design. In the absence of an analytic solution, most quantum optimal control methods rely on an iterative scheme to update the solution estimate. At present, the convergence rate of existing solvers is, at most, superlinear. In this thesis, we develop a new general purpose solver for quantum optimal control, Quantum PRojection Operator-based Newton method for Trajectory Optimization (Q-PRONTO). Specifically, the proposed approach uses a projection operator to incorporate the Schr¨odinger equation directly into the cost function, which is then minimized using a Newton descent method. The resulting method guarantees monotonic convergence at every iteration and quadratic convergence in proximity of the solution. We then implement the method to different quantum applications: the first application is shaken lattice interferometry, which is a novel atom-based interferometry for quantum sensing; the second application is fluxonium qubit, which is a novel superconducting qubit for quantum computing. The potential of Q-PRONTO is showcased by computing an interferometry sequence for shaken lattice interferometry and designing a quantum gate for fluxonium qubit. The encouraging simulation results represent a first step in the development of a new paradigm for both quantum sensing and quantum computing.
Quantum optimal control is an important technology that enables fast state preparation and gate design. In the absence of an analytic solution, most quantum optimal control methods rely on an iterative scheme to update the solution estimate. At present, the convergence rate of existing solvers is, at most, superlinear. This paper develops a general-purpose solver for quantum optimal control based on the projection operator Newton method for trajectory optimization, or PRONTO. Specifically, the proposed approach uses a projection operator to incorporate the Schr\"odinger equation directly into the cost function, which is then minimized using Newton's descent method. At each iteration, the descent direction is obtained by computing the analytic solution to a linear-quadratic trajectory-optimization problem. The resulting method guarantees monotonic convergence at every iteration and quadratic convergence in the proximity of the solution. The potential of PRONTO is showcased by solving the optimal state-to-state mapping problem for a qubit and providing comparisons to a state-of-the-art quantum optimal control method.
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.
The Quantum Projection Operator-Based Newton Method for Trajectory Optimization (q-pronto) is a numerical method for solving quantum optimal control problems. This paper significantly improves prior versions of the quantum projection operator by introducing a regulator that stabilizes the solution estimate at every iteration. This modification is shown to not only improve the convergence rate of the algorithm, but also steer the solver towards better local minima compared to the unregulated case. Numerical examples showcase how q-pronto can be used to solve multi-input quantum optimal control problems featuring time-varying costs and undesirable populations that ought to be avoided during the transient.
This paper demonstrates how quantum optimal control can be used to perform shaken lattice interferometry. The first objective is to translate the five fundamental stages of interferometry (splitting, propagation, reflection, counter propagation and recombination) into quantum optimal control problems parametrized by the time horizon of each stage. The timing of each stage is then studied in relationship to its overall influence on the interferometer performance. This is done by comparing the population distributions obtained for a range of different accelerations and using Fisher information to estimate the sensitivity of the resulting accelerometer. These encouraging results highlight the effectiveness of quantum optimal control for the the design of next-generation atom-based interferometers.
Achieving scalable quantum computing demands high-fidelity operations capable of mitigating population leakage into non-computational states. Physics-Informed Neural Networks (PINNs) have recently emerged as a powerful paradigm to unify quantum hardware characterization (inverse problems) and pulse engineering (direct problems), laying the foundational architecture for autonomous quantum processors. However, standard PINN frameworks face severe numerical bottlenecks, such as spectral bias, when attempting to simultaneously solve highly oscillatory multi-level dynamics and optimize continuous control fields under strict global phase constraints. In this work, we propose an enhanced PINN scheme for quantum optimal control (PINNQOC) that circumvents these limitations by incorporating Fourier feature embeddings, dynamic epoch normalization, and an informed pre-training routine. To rigorously evaluate its performance, we systematically benchmark our framework against two premier continuous control solvers: the first-order Krotov method and the second-order Projection Operator Newton Method for Trajectory Optimization (PRONTO). These techniques are applied to implement multiple quantum gates on a truncated three-level fluxonium qubit and a four-level Nitrogen-Vacancy center coupled to a Carbon-13 nuclear spin. Our advanced PINNQOC approach successfully suppresses population leakage while achieving gate fidelities exceeding 99.9%\%%, matching the efficacy of traditional solvers. Finally, we provide a comprehensive analysis of computational times, iteration efficiency, and mean leakage, highlighting the distinct trade-offs and avenues for embedding physics-guided machine learning into automated quantum hardware pipelines.
The control of matter and energy at a fundamental level will be a cornerstone of new technologies for years to come. This idea is exemplified in a distilled form by controlling the dynamics of quantum mechanical systems via a time-dependent potential. The contributions detailed within this work focus on the computational aspects of formulating and solving quantum control problems efficiently. The accurate numerical computation of optimal controls of infinite dimensional quantum control problems is a very difficult task that requires to take into account the features of the original infinite dimensional problem. An important issue is the choice of the functional space where the minimization process is defined. A systematic comparison of L2 versus H1 based minimization shows that the choice of the appropriate functional space matters and has many consequences in the implementation of some optimization techniques. A matrix-free cascadic BFGS algorithm is introduced in the L 2 and H1 settings and it is demonstrated that the choice of H1 over L2 results in a substantial performance and robustness increase. A comparison between optimal control resulting from function space minimization and the control obtained by minimization over Chebyshev and POD basis function coefficients is presented. A theoretical and computational framework is presented to obtain accurate controls for fast quantum state transitions that are needed in a host of applications such as nano electronic devices and quantum computing. This method is based on a reduced Hessian Krylov-Newton scheme applied to a norm-preserving discrete model of a dipole quantum control problem. The use of second-order numerical methods for solving the control problem is justified proving existence of optimal solutions and analyzing first- and second-order optimality conditions. Criteria for the discretization of the non-convex optimization problem and for the formulation of the Hessian are given to ensure accurate gradients and a symmetric Hessian. Robustness of the Newton approach is obtained using a globalization strategy with a robust linesearch procedure. Results of numerical experiments demonstrate that the Newton approach presented in this dissertation is able to provide fast and accurate controls for high-energy state transitions. Control of bound-to-bound and bound-to-continuum transitions in open quantum systems and vector field control of two-dimensional systems is presented. An efficient space-time spectral discretization of the time-dependent Schrodinger equation and preconditioning strategy for a fast approximate solution with Krylov methods is outlined.