Globally optimal control of quantum dynamics

TL;DR

The paper tackles the difficulty of constrained quantum-control problems with dense local extrema by introducing QCPOP, a framework that recasts control optimization as a global polynomial problem. By combining a truncated Magnus expansion, Chebyshev interpolation, and a finite control expansion, QCPOP converts the objective into a polynomial in a small set of coefficients, enabling global solutions via SOS hierarchies or homotopy continuation. Empirical results show QCPOP finds true global optima in single runs and outperforms gradient-based methods on challenging tasks, while also enabling Hamiltonian identification and offering a computability perspective on quantum control within analog computation models. The approach substantially expands the tractable regime for quantum-control design and suggests broad utility for quantum technologies and beyond.

Abstract

Optimization of constrained quantum control problems powers quantum technologies. This task becomes very difficult when these control problems are nonconvex and plagued with dense local extrema. For such problems current optimization methods must be repeated many times to find good solutions, each time requiring many simulations of the system. Here, we present quantum control via polynomial optimization (QCPOP), a method that eliminates this problem by directly finding globally optimal solutions. The resulting increase in speed, which can be a thousandfold or more, makes it possible to solve problems that were previously intractable. This remarkable advance is due to global optimization methods recently developed for polynomial functions. We demonstrate the power of this method by showing that it obtains an optimal solution in a single run for a problem in which local extrema are so dense that gradient methods require thousands of runs to reach a similar fidelity. Since QCPOP is able to find the global optimum for quantum control, we expect that it will not only enhance the utility of quantum control by making it much easier to find the necessary protocols, but also provide a key tool for understanding the precise limits of quantum technologies. Finally, we note that the ability to cast quantum control as polynomial optimization resolves an open question regarding the computability of exact solutions to quantum control problems.

Figures (6)

• Figure 1: The control problem of Pechen and Tannor pechen_are_2011 which has a trap at the point where the control variable, $\lambda$, vanishes. Using QCPOP the trap poses no problem for the homotopy continuation method which returns all the extrema (shown by the red crosses). a) The solid blue line is the value of the objective function (the expectation value of $O$), being a symmetric function of the control variable, $\lambda$. The green (orange) dashed line is the objective function resulting from using the Taylor (Chebyshev) approximation in QCPOP (to tenth order). b) A diagram of the control problem. The system has three levels, each with a different value of the observable, $O$. The control Hamiltonian, $V$, couples $\ket{1}$ to $\ket{2}$ and $\ket{2}$ to $\ket{3}$.
• Figure 2: Violin plots of the infidelity between the target unitary, $U^\star$, and the realized unitary, $\widehat{U}$, defined in Sec. \ref{['SecEvaluatingQCPOP']}, for each of 1000 target unitaries and for three quantum control methods: GRAPE, CRAB, and QCPOP. Note that $f_{PSU}$ denotes the fidelity that is defined as $f_{PSU} = \left| \Tr(\widehat{U}^\dagger U^\star ) \right| / 3$; whereas, $|1 - f_{PSU}|$ is the infidelity.
• Figure 3: Histogram plots for Magnus series convergence tests [see the main text just after Eq. \ref{['eq:Omega3']}] (A) when exact solutions $\bm x^\star$ for quantum control problem \ref{['eq:QuantumCoherentControl']} are used; (B) when global minimizers $\widehat{\bm x}$ for \ref{['Eq:UminusUtarget']}, obtained by the TSSOS Julia package, are employed.
• Figure 4: Comparing the global minima for \ref{['Eq:UminusUtarget']}, obtained by the TSSOS Julia package, with the value of the polynomial objective function \ref{['Eq:UminusUtarget']} when an exact solution $\bm x^\star$ for quantum control problem \ref{['eq:QuantumCoherentControl']} is used.
• Figure 5: A relation between the quantum control problem \ref{['eq:QuantumCoherentControl']} and its approximate polynomial formulation \ref{['Eq:UminusUtarget']}. $\widehat{U}$ is the evolution operator obtained via polynomial optimization \ref{['Eq:UminusUtarget']}.