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arXiv:2111.08795 · Physical Review A

Projection-operator-based Newton method for the trajectory optimization of closed quantum systems

Jieqiu Shao, Joshua Combes, John Hauser, Marco M. Nicotra

Published Mar 9, 2022• 12 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.105.032605

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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.
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.
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.
Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton's method takes into account curvature information and thereby often improves convergence. Here, we develop quantum versions of these iterative optimization algorithms and apply them to polynomial optimization with a unit norm constraint. In each step, multiple copies of the current candidate are used to improve the candidate using quantum phase estimation, an adapted quantum state exponentiation scheme, as well as quantum matrix multiplications and inversions. The required operations perform polylogarithmically in the dimension of the solution vector and exponentially in the number of iterations. Therefore, the quantum algorithm can be useful for high-dimensional problems where a small number of iterations is sufficient.
We develop quantum algorithms for optimal control over a prescribed finite time interval (finite horizon), focusing on continuous-time linear-quadratic regulation (LQR). This foundational control problem often requires solving a nonlinear matrix Riccati equation, which can be costly for large systems and long control durations on conventional classical solvers. We instead work directly with the state-costate optimality equations, reformulating the LQR problem as a multiple-shooting linear system. Importantly, we prove a condition-number bound for the system that is independent of the horizon, under stabilizability and detectability assumptions. By incorporating the initial and final conditions into a single linear system, we can solve the subsequent linear system more efficiently with a quantum linear systems subroutine. For a given initial state, our quantum algorithms return classical estimates of the optimal cost and the optimal control inputs at a selected time, without constructing a Riccati solution operator. Given sparse-matrix block-encodings, an initial-state preparation oracle and a fixed number of control inputs, the query and gate complexities scale polylogarithmically with system size and control duration, provided that sparsity, relevant norms, and stability parameters are bounded. The dependence on the target error is inverse-linear up to logarithmic factors. We establish the required bounds uniformly in system size and duration for a family of damped mass-spring chains with two endpoint controls. Numerical experiments support the analysis.
Abstract Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton’s method takes into account curvature information and thereby often improves convergence. Here, we develop quantum versions of these iterative optimization algorithms and apply them to polynomial optimization with a unit norm constraint. In each step, multiple copies of the current candidate are used to improve the candidate using quantum phase estimation, an adapted quantum state exponentiation scheme, as well as quantum matrix multiplications and inversions. The required operations perform polylogarithmically in the dimension of the solution vector and exponentially in the number of iterations. Therefore, the quantum algorithm can be useful for high-dimensional problems where a small number of iterations is sufficient.