Monte Carlo approach for finding optimally controlled quantum gates with differential geometry

TL;DR

This paper addresses the problem of designing energetically efficient quantum gates in noisy environments by framing unitary evolution as geodesic curves on a unitary manifold and employing sub-Riemannian geometry. It introduces a Monte Carlo strategy to identify suitable initial co-states $\Lambda(0)$ that generate near-optimal control fields for single-qubit gates under dephasing and a two-qubit CNOT gate under crosstalk, comparing results to the Krotov method. The main contributions include a practical random-sampling approach to solving the geodesic equation, demonstrations of high gate fidelities (~0.99) with reduced energy cost, and insights into how energy-minimization can improve hardware stability and scalability. The results show that geodesic-based controls can match traditional optimal-control methods in fidelity while offering energy advantages, suggesting a valuable addition to optimal-control theory for quantum technologies.

Abstract

A unitary evolution in time may be treated as a curve in the manifold of the special unitary group. The length of such a curve can be related to the energetic cost of the associated computation, meaning a geodesic curve identifies an energetically optimal path. In this work, we employ sub-Riemannian geometry on the manifold of the unitary group to obtain optimally designed Hamiltonians for generating single-qubit gates in an environment with the presence of dephasing noise as well as a two-qubit gate under a time-constant crosstalk interaction. The resulting geodesic equation involves knowing the initial conditions of the parameters that cannot be obtained analytically. We then introduce a random sampling method combined with a minimization function and a cost function to find initial conditions that lead to optimal control fields. We also compare the optimized control fields obtained from the solutions of the geodesic equation with those extracted from the well-known Krotov method. Both approaches provide high fidelity values for the desired quantum gate implementation, but the geodesic method has the advantage of minimizing the required energy to execute the same task. These findings bring new insights for the design of more efficient fields in the arsenal of optimal control theory.

Figures (10)

• Figure 1: Gate fidelity as a function of time in units of gate time $\tau$ using the two optimized initial co-states calculated and shown in Eqs. (\ref{['Lambda0A']}) (continuous line) and (\ref{['Lambda0B']}) (dashed line). The fact that in both cases the gate is closely reached before $t=\tau$ shows that they do not correspond to a global minimum of energy.
• Figure 2: Gate fidelity as a function of time in units of gate time $\tau$ using the optimized initial co-state shown in Eq. (\ref{['Lambdag']}). After the minimum, the fidelity is monotonically increasing until reaching the unit at $t=\tau$. This suggests that the solution corresponds to the global minimum of energy. Moreover, this result coincides with the solution if we use the method described in Ref. morazotti2024optimized for this specific quantum gate.
• Figure 3: Energetic cost for the time evolution from $t=0$ to $t=\tau$ using Eqs. (\ref{['Lambda0A']}), (\ref{['Lambda0B']}), and (\ref{['Lambdag']}) as initial co-states. The energy functional is in units of $\hbar^2/\tau$, and according to Eq. (\ref{['energy_functional']}), the quantity proportional to the total energy spent is calculated as the area under the curves.
• Figure 4: Components of the control Hamiltonian for the three quantum gates: $\mathrm{H}$, $\mathrm{T}$, and $\mathrm{R}$, shown in Eqs. (\ref{['Hgate']}), (\ref{['Tgate']}), and (\ref{['Rgate']}) respectively. The components $x, y, z$ correspond directly to the indices $1,2,3$. The $y$-axis is in units of $\hbar/\tau$.
• Figure 5: Average gate fidelity as a function of time for the gates $\mathrm{H}$, $\mathrm{T}$, and $\mathrm{R}$, calculated using the numerically obtained control fields in the master equation shown in Eq. (\ref{['master_equation']}) in the interaction picture. The average fidelity values at the gate time $t=\tau$ are $0.987998$, $0.991376$, and $0.989268$.