Exploring the trade-off between fidelity- and time-optimal control of quantum unitary transformations

TL;DR

This paper investigates the trade-off between achieving high fidelity and minimizing control time in quantum gate implementation, introducing a Pareto front tracking method to identify critical times and analyze control effort.

Contribution

It presents a numerical Pareto front tracking method to analyze the fidelity-time trade-off in quantum control, revealing the critical time below which target transformations are unreachable.

Findings

Unit fidelity can be achieved for all times above a critical threshold.

Optimization effort increases superexponentially as control time approaches the critical time.

Operating below the critical time results in significant fidelity loss.

Abstract

Generating a unitary transformation in the shortest possible time is of practical importance to quantum information processing because it helps to reduce decoherence effects and improve robustness to additive control field noise. Many analytical and numerical studies have identified the minimum time necessary to implement a variety of quantum gates on coupled-spin qubit systems. This work focuses on exploring the Pareto front that quantifies the trade-off between the competitive objectives of maximizing the gate fidelity $\mathcal{F}$ and minimizing the control time $T$. In order to identify the critical time $T^{\ast}$, below which the target transformation is not reachable, as well as to determine the associated Pareto front, we introduce a numerical method of Pareto front tracking (PFT). We consider closed two- and multi-qubit systems with constant inter-qubit coupling strengths and each individual qubit controlled by a separate time-dependent external field. Our analysis demonstrates that unit fidelity (to a desired numerical accuracy) can be achieved at any $T \geq T^{\ast}$ in most cases. However, the optimization search effort rises superexponentially as $T$ decreases and approaches $T^{\ast}$. Furthermore, a small decrease in control time incurs a significant penalty in fidelity for $T < T^{\ast}$, indicating that it is generally undesirable to operate below the critical time. We investigate the dependence of the critical time $T^{\ast}$ on the coupling strength between qubits and the target gate transformation. Practical consequences of these findings for laboratory implementation of quantum gates are discussed.