[Paper Review] Time-optimal control with finite bandwidth

Masashi Hirose, Paola Cappellaro|arXiv (Cornell University)|Oct 23, 2015

Quantum Information and Cryptography1 references3 citations

TL;DR

This paper introduces the Fourier-Approximated Time-Optimal (FATO) control method to achieve time-optimal quantum gate operations in qubits under finite control bandwidth, using Fourier series approximations of ideal bang-bang control protocols. It analytically links control bandwidth limitations to gate fidelity, showing that FATO enables high-fidelity, time-optimal control even beyond the rotating wave approximation, with fidelity losses scaling predictably with Fourier truncation error.

ABSTRACT

Optimal control theory provides recipes to achieve quantum operations with high fidelity and speed, as required in quantum technologies such as quantum sensing and computation. While technical advances have achieved the ultrastrong driving regime in many physical systems, these capabilities have yet to be fully exploited for the precise control of quantum systems, as other limitations, such as the generation of higher harmonics or the finite bandwidth of the control fields, prevent the implementation of theoretical time-optimal control. Here we present a method to achieve time-optimal control of qubit systems that can take advantage of fast driving beyond the rotating wave approximation. We exploit results from optimal control theory to design driving protocols that can be implemented with realistic, finite-bandwidth control and we find a relationship between bandwidth limitations and achievable control fidelity.

Motivation & Objective

• To address the gap between theoretical time-optimal control and practical implementation due to finite control bandwidth in quantum systems.
• To develop a practical control protocol that achieves time-optimal gate operations in qubits when the control field is constrained in bandwidth and amplitude.
• To quantify the trade-off between gate fidelity and control bandwidth in ultrastrong driving regimes where the rotating wave approximation fails.
• To provide an analytical framework linking Fourier series truncation to control infidelity, enabling design of high-fidelity control sequences.

Proposed method

• The FATO method approximates ideal bang-bang (BB) control protocols using truncated Fourier series, ensuring the control field remains real and bounded in amplitude.
• It models the control field as a sum of sinusoidal components within a finite bandwidth $[0, riangle heta]$, with coefficients derived from the Fourier expansion of the ideal BB function.
• The method uses a toggling frame transformation to analyze the error propagator due to truncation, enabling analytical treatment of the effective Hamiltonian.
• A first-order Magnus expansion is applied to the error Hamiltonian to derive an analytical expression for gate infidelity in terms of the mean square error of the Fourier approximation.
• The fidelity is computed via the trace of the error propagator, leading to closed-form expressions for infidelity as a function of the Fourier truncation error and system parameters.
• The approach is extended to two-qubit control by generalizing the error Hamiltonian and fidelity calculation to multi-qubit dynamics.

Experimental results

Research questions

• RQ1 How can time-optimal control be achieved in qubit systems when the control field is limited to finite bandwidth?
• RQ2 What is the quantitative relationship between control bandwidth and achievable gate fidelity in time-optimal protocols?
• RQ3 Can Fourier series approximation of ideal bang-bang control preserve high fidelity under realistic hardware constraints?
• RQ4 How does the fidelity of time-optimal control degrade with increasing truncation of the Fourier series in the control field?
• RQ5 To what extent can the FATO method be generalized to multi-qubit systems under finite-bandwidth control?

Key findings

• The FATO method achieves time-optimal control with gate fidelity that depends inversely on the mean square error of the Fourier series truncation, with fidelity loss quantified analytically.
• In the weak-driving regime, the fidelity is given by $ F = \cos\left(\frac{\pi}{2}\tan(\theta)\mathcal{E}_{K}\right) $, where $ \mathcal{E}_{K} $ is the mean square error of the truncated Fourier series.
• For ultrastrong driving, the fidelity for X and Y rotations is approximated as $ F_{s}^{x} \approx \cos\left[\frac{2}{\pi}\sin(\theta)\mathcal{E}_{K}\right] $ and $ F_{s}^{y} \approx \cos\left[\frac{2}{\pi}\tan(\theta)\mathcal{E}_{K}\right] $, respectively.
• The method enables gate times equal to the theoretical time-optimal limit even under finite-bandwidth constraints, outperforming conventional on-resonance driving in speed.
• The FATO protocol is robust against control field errors and can be extended to multi-qubit systems with minimal modification.
• The study establishes control bandwidth as a fundamental limiting resource in the trade-off between speed and fidelity in quantum control.