Approaching the Quantum Speed Limit in Quantum Gates with Geometric Control

TL;DR

This paper extends the Mandelstam–Tamm quantum speed limit to evolution operators in arbitrary dimensions and derives the associated geometric constraints for saturating the bound. It analyzes saturation possibilities across low-dimensional unitary families, highlighting that MT-QSL saturation is typical only in constrained cases and may be unattainable in higher dimensions. A geometric optimal-control framework based on Pontryagin's Maximum Principle is developed to approach the MT-QSL by minimizing plane- and phase-deviation costs, and this approach is tested on four two-qubit gates, showing significant gains in speed efficiency with minimal fidelity loss, especially when combining plane and phase costs. The authors also compare PMP with CRAB optimization, finding PMP generally faster and often more effective, though CRAB can offer competitive results under certain constraints. Overall, the work provides a principled route to faster quantum gate implementations by leveraging geometric QSL principles, with potential extensions to shortcut-to-adiabaticity and open-system dynamics.

Abstract

We present a geometric optimization method for implementing quantum gates by optimally controlling the Hamiltonian parameters, with the goal of approaching the Mandelstam-Tamm Quantum Speed Limit (MT-QSL). Achieving this bound requires satisfying precise geometric conditions that govern the evolution of quantum states. We extend this geometric framework to quantum unitary operators in arbitrary dimensions and analyze the conditions necessary for saturation of the bound. Additionally, we show that the quantum brachistochrone, when generalized to operators, does not, in general, saturate the MT-QSL bound. Finally, we propose a systematic optimal control strategy based on geometric principles to approach the quantum speed limit for unitary driving. We illustrate this optimization method on a set of four well-known two-qubit quantum gates. Our procedure significantly reduces the deviation from the optimal quantum speed limit while preserving high quantum fidelity.

Figures (2)

• Figure 1: Implementation of the gate $U_{\rm QFT}$ after geometric PMP optimization with a full Hamiltonian control [$(p_1=p_2=0.5)$, 1st line of Table \ref{['tab:OptPlanePhase']}]. (a):Instantaneous QSL efficiency $\eta(t)$ as a function of the rescaled time $t/t_f$. (b):Fidelity $\mathcal{F}=|\langle U_{\rm target},U(t) \rangle|^2,$ cost functionals $f_{\rm Plane}(U(t),H(t))$ (dash-dotted blue line) and $f_{\rm Phase}(U(t),H(t))$ (dotted red line) as a function of the rescaled time $t/t_f$. (c):Dynamical couplings $\Omega_{12}^R(t)$ (dash-dotted blue line), $\Omega_{12}^I(t)$ (dotted blue line), $\Omega_{13}^R(t)$ (dash-dotted red line), $\Omega_{13}^I(t)$ (dotted red line), $\Omega_{23}^R(t)$ (dash-dotted green line), $\Omega_{23}^I(t)$ (dotted green line). Note that the lines of $\Omega_{12}^R(t)$, $\Omega_{13}^R(t)$ and $\Omega_{14}^R(t)$ are superimposed, as well as the lines of $\Omega_{12}^I(t)$, $\Omega_{13}^I(t)$ and $\Omega_{14}^I(t)$. (d):Dynamical couplings $\Omega_{23}^R(t)$ (dash-dotted blue line), $\Omega_{23}^I(t)$ (dotted blue line), $\Omega_{24}^R(t)$ (dash-dotted red line), $\Omega_{24}^I(t)$ (dotted red line), $\Omega_{34}^R(t)$ (dash-dotted green line), $\Omega_{34}^I(t)$ (dotted green line). The lines of $\Omega_{23}^R(t)$ and $\Omega_{34}^R(t)$ are superimposed. (e):Dynamical couplings $\Delta_{11}(t)$ (dash-dotted blue line), $\Delta_{22}(t)$ (dash-dotted red line), $\Delta_{33}(t)$ (dotted green line), $\Delta_{44}(t)$ (dotted purple line). The lines of $\Delta_{11}(t)$ and $\Delta_{33}(t)$ are superimposed, as well as $\Delta_{22}(t)$ and $\Delta_{44}(t)$. All couplings in (c,d,e) are given in units of $1/t_f$. (f):Matrix elements of the operator $\tilde{U}(t_f)=U(t_f) e^{- i \varphi}$ (real and imaginary parts), with $\varphi= {\rm Arg}[\langle U_{\rm target}, U(t_f) \rangle]$.
• Figure 2: Implementation of the gate $U_{\rm "QFT"}$ after geometric CRAB optimization with a full Hamiltonian control [$(p_1=0.5,p_2=0.1)$, 2nd line of Table \ref{['tab:CRABQSLOptimization']}]. (a):Instantaneous QSL efficiency $\eta(t)$ as a function of the rescaled time $t/t_f$. (b):Fidelity $\mathcal{F}=|\langle U_{\rm target},U(t) \rangle|^2,$ running cost functions $f_{\rm Plane}(U(t),H(t))$ (dash-dotted blue line) and $f_{\rm Phase}(U(t),H(t))$ (dotted red line) as a function of the rescaled time $t/t_f$. (c):Dynamical couplings $\Omega_{12}^R(t)$ (dash-dotted blue line), $\Omega_{12}^I(t)$ (dotted blue line), $\Omega_{13}^R(t)$ (dash-dotted red line), $\Omega_{13}^I(t)$ (dotted red line), $\Omega_{23}^R(t)$ (dash-dotted green line), $\Omega_{23}^I(t)$ (dotted green line). (d):Dynamical couplings $\Omega_{23}^R(t)$ (dash-dotted blue line), $\Omega_{23}^I(t)$ (dotted blue line), $\Omega_{24}^R(t)$ (dash-dotted red line), $\Omega_{24}^I(t)$ (dotted red line), $\Omega_{34}^R(t)$ (dash-dotted green line), $\Omega_{34}^I(t)$ (dotted green line). (e): Dynamical couplings $\Delta_{11}(t)$ (dash-dotted blue line), $\Delta_{22}(t)$ (dash-dotted red line), $\Delta_{33}(t)$ (dotted green line), $\Delta_{44}(t)$ (dotted purple line). All couplings in (c,d,e) are given in units of $1/t_f$.(f):Matrix elements of the operator $\tilde{U}(t_f)=U(t_f) e^{- i \varphi}$ (Real and Imaginary parts), with $\varphi= {\rm Arg}[\langle U_{\rm target}, U(t_f) \rangle]$.