Geodesic Pulse Engineering (GEOPE)

• Geodesic Pulse Engineering (GEOPE) is a quantum control paradigm that employs geodesics on the SU(2^n) manifold to design efficient quantum gate pulses.
• It leverages a convex update algorithm that minimizes geodesic length, resulting in faster convergence and higher fidelity than conventional gradient methods.
• GEOPE addresses experimental constraints by optimizing pulse sequences over accessible Hamiltonian terms, making it viable for complex multi-qubit gate synthesis.

Geodesic Pulse Engineering (GEOPE) is a quantum optimal control paradigm that uses the differential geometry of unitary groups—specifically, geodesics on the Riemannian manifold of $\mathrm{SU}(2^n)$—to design control pulses that implement desired quantum gates with maximal efficiency, minimal path length, and strong adherence to experimental constraints. Unlike standard gradient-based pulse optimization algorithms, GEOPE frames the evolution as a minimal-distance trajectory in unitary space, producing control solutions that typically converge faster, yield higher fidelities, and possess greater interpretability in terms of system geometry and physical limitations (Lewis et al., 22 Aug 2025).

1. Geodesic Principles in Quantum Control

GEOPE is built on the concept of a geodesic—a shortest path connecting two points—in the context of the group manifold $\mathrm{SU}(2^n)$ for $n$-qubit quantum systems. If $U_G$ is the current implemented unitary evolution and $V$ is the target gate, the geodesic connecting them is given by

$X(t) = U_G \cdot \exp(i t \Gamma)$

where

$\Gamma = \log(U_G^\dagger V)$

with $t \in [0, 1]$. The generator $\Gamma$ provides the tangent direction of the ideal path; its principal branch guarantees minimal geodesic length.

The quantum control problem is recast: at each algorithmic step, the update to the control pulse parameters (typically, discretized over $L$ piecewise-constant intervals) seeks to best follow this geodesic, even when system actuation is limited to a subset of possible Hamiltonian terms (due to experimental or hardware constraints).

2. Algorithmic Construction

The control pulse protocol is broken into $L$ intervals, with each interval $\ell$ parameterized by a vector $\phi_\ell$ describing the weights of experimentally accessible Hamiltonian generators (e.g., specific Pauli strings). The evolution operator for each interval is $U(\phi_\ell) = \exp[i H(\phi_\ell)]$; the total operation is ordered as

$U_G(\Phi) = U(\phi_L) \cdots U(\phi_1)$

where $\Phi$ collects all control vectors.

At each optimization step:

• The algorithm computes the current $U_G$ and the principal geodesic direction $\Gamma$ toward $V$.
• The update $\delta\Phi = \{\delta\phi_{\ell,k}\}$ is chosen such that the tangent vector produced by the available Hamiltonian terms matches, as closely as possible, the ideal geodesic direction. This involves minimizing the convex cost function:

$$L(\delta\Phi) = \|\sum_{\ell,k\in\mathcal{H}} J_{\ell,k}(\Phi) \delta\phi_{\ell,k} - i U_G(\Phi)\Gamma\|^2$$

where $J_{\ell,k}$ is the Jacobian—the derivative of $U_G$ with respect to the $k$th control in the $\ell$th interval; $\mathcal{H}$ denotes the accessible set of Hamiltonian terms.

After the update direction is found, a line search (e.g., golden-section search) selects the optimal step size $\eta$ along this direction. If no fidelity improvement is achieved, a corrective “kick” is applied using a Gram–Schmidt procedure to avoid local minima.

3. Comparison to Gradient-Based Methods

Traditional methods such as GRAPE (Gradient Ascent Pulse Engineering) maximize the fidelity

$$F(\Phi, V) = \frac{1}{N}|\operatorname{Tr}(U_G^\dagger V)|$$

by taking local gradients in the control parameters. This does not generally align with the geodesic direction toward the target gate; indeed, as shown by first-order expansions involving the Fréchet derivative, the local fidelity gradient and the optimal geodesic direction need not coincide.

GEOPE’s primary advantage is the geometric consistency of its updates:

• Each update explicitly minimizes geodesic length, subject to control constraints, resulting in faster convergence.
• The least-squares update guarantees maximal decrease of geodesic distance, up to the limitations imposed by $\mathcal{H}$.
• Numerical results indicate that GEOPE achieves high-fidelity solutions ($\epsilon < 10^{-9}$) in $10$–$13$ iterations for complex three-qubit gates, contrasted with hundreds of iterations required by local-gradient methods. In more challenging five- or six-qubit gates, GEOPE reached solutions not accessible to GRAPE in reasonable runtime (Lewis et al., 22 Aug 2025).

4. Hamiltonian Constraints and Practical Implementability

GEOPE operates under experimental restrictions—such as limited control Hamiltonian terms. For example, in a 2D neutral Rydberg atom array, the accessible Hamiltonian is

$$H(t) = \sum_{i<j} J_{ij} \sigma_i^z \sigma_j^z + \sum_i \left[\Omega_i(t) \sigma_i^x + \Delta_i(t) \sigma_i^z\right]$$

with long-range $J_{ij}$ (decaying as $r_{ij}^{-6}$), controllable local amplitudes ($\Omega_i$), and detunings ($\Delta_i$). GEOPE efficiently finds pulse sequences for complex gates (Toffoli, CCZ, QFT) that adhere to these hardware constraints. Notably, the convex update step allows integration with piecewise or time-dependent constraints as well as norm or bandwidth limitations.

5. Mathematical Foundations

The following equations summarize the mathematical structure:

Component	Formula	Description
Geodesic	$X(t) = U_G \exp(it\Gamma)$	Minimal path in $\mathrm{SU}(2^n)$
Geodesic Generator	$\Gamma = \log(U_G^\dagger V)$	Direction to target unitary
Control Update	$$U_G(\Phi+\delta\Phi) \approx U_G(\Phi) + \sum_{\ell,k} J_{\ell,k}\delta\phi_{\ell,k}$$	Linearization
Least-Squares Cost	$$L(\delta\Phi) = \|\sum_{\ell,k} J_{\ell,k}\delta\phi_{\ell,k} - i U_G\Gamma\|^2$$	Minimization criterion

The update $\delta\Phi$ is selected to best approximate $iU_G\Gamma$ given the physical limits of $\mathcal{H}$.

6. Numerical Performance and Solution Accessibility

Empirical benchmarks cover:

• Three-qubit gate synthesis (Toffoli, controlled-Z): GEOPE success probability reaches unity in at least $10\times$ fewer iterations than GRAPE.
• Five-qubit QFT construction: GEOPE finds solutions reliably; GRAPE trials fail to converge within $300$ iterations.
• Six-qubit gates: GEOPE is able to reach solutions not found by GRAPE.

These results demonstrate that GEOPE is particularly advantageous when the solution landscape is complex or strongly nonlinear, and control limitations preclude direct gradient-following approaches.

7. Implications, Extensions, and Outlook

GEOPE enables quantum gate design protocols that inherently maximize efficiency with respect to both path length and resource requirements on the underlying manifold of unitaries. Its convex update construction is highly compatible with custom constraints, piecewise or smooth pulse parameterizations, and integration with advanced hardware. The geodesic-centric formalism can be generalized to other manifold-based control settings, including open quantum systems with additional Lindbladian structure, or to quantum control in continuous variable systems.

A plausible implication is that GEOPE will be increasingly relevant as quantum control moves to higher qubit numbers and experimental platforms become more complex—a domain where local-gradient methods often stall or converge slowly. By connecting quantum gate synthesis to geometrically grounded optimization in unitary group space, GEOPE provides a foundation for scalable, high-fidelity, constraint-aware quantum control.