Optimizing stimulated Raman adiabatic passage for leakage suppression via Pontryagin's maximum principle

1. Problem Statement

Context: Stimulated Raman Adiabatic Passage (STIRAP) is a robust protocol for high-fidelity quantum state transfer in ideal three-level systems, relying on the adiabatic following of a "dark state."
Challenge: In realistic physical systems, particularly superconducting circuits (e.g., transmons), the energy spectrum is not an isolated three-level manifold but a multilevel structure with finite anharmonicity.

• Leakage: Finite-amplitude control pulses inevitably couple the target three-level subspace to higher-lying energy levels (leakage states).
• Limitations: Standard STIRAP protocols, designed for ideal three-level models, suffer from significant fidelity loss and transient population in leakage states when applied to these multilevel systems.
• Gap: Existing optimization methods (like GRAPE or Krotov) often treat control waveforms as arbitrary functions, which can lead to experimentally unfeasible pulses or fail to explicitly incorporate leakage penalties and physical constraints (like bandwidth and amplitude limits) simultaneously. There is a need for a framework that optimizes STIRAP specifically for leakage suppression while maintaining experimental feasibility.

2. Methodology

The authors propose a Pontryagin's Maximum Principle (PMP) based optimal control framework tailored for multilevel STIRAP.

A. System Modeling

• Multilevel Chain Model: The system is modeled as an $N$-state nearest-neighbor chain under the Rotating Wave Approximation (RWA).
• Hamiltonian: The effective Hamiltonian includes diagonal detunings and off-diagonal couplings ($\Omega_{j,j+1}$) between adjacent levels.
• Leakage Definition: The Hilbert space is partitioned into a target manifold (e.g., $|0\rangle, |1\rangle, |2\rangle$) and a leakage manifold (e.g., $|3\rangle, |4\rangle, \dots$).
• Dissipation: Energy relaxation is incorporated via an effective non-Hermitian Hamiltonian ($H_{nh} = \tilde{H} - i/2 \sum C_\mu^\dagger C_\mu$), allowing for efficient simulation of decay during optimization.

B. Optimization Framework (PMP)

• Objective Functional: A Bolza-type cost function $J[u]$ is defined to minimize:
  1. Terminal Cost: Infidelity of the target state ($1 - |\langle 2|\psi(T)\rangle|^2$).
  2. Running Cost: Transient occupation of the intermediate state ($|1\rangle$) and leakage states ($|3\rangle, |4\rangle$), plus control energy.
• Control Parametrization: Instead of optimizing arbitrary waveforms, the control fields (Pump and Stokes pulses) are restricted to Gaussian pulse families. This reduces the infinite-dimensional functional optimization to a low-dimensional parameter search:
  $$u = (A_p, A_s, t_{0p}, t_{0s}, \sigma_p, \sigma_s)$$
  (Amplitudes, center times, and widths).
• Gradient Derivation:
  • The authors derive the PMP state-costate equations in a complex Hilbert space.
  • They explicitly calculate the gradients of the objective functional with respect to the Gaussian pulse parameters ($\partial J / \partial u_k$) using the chain rule and the derived control gradients ($\delta J / \delta \Omega$).
  • This enables efficient gradient-based optimization (e.g., using BFGS or trust-region methods) without needing to discretize the entire time-domain waveform.

C. Algorithm

The optimization follows an iterative loop:

1. Forward Propagation: Evolve the state $|\psi(t)\rangle$ from $t=0$ to $T$ using current parameters.
2. Backward Propagation: Evolve the costate $\langle \lambda(t)|$ from $t=T$ to $0$ using the adjoint equation.
3. Gradient Calculation: Compute the gradient of the cost function with respect to the 6 Gaussian parameters.
4. Update: Update parameters to minimize the cost function.

3. Key Contributions

1. Unified Multilevel Framework: The paper establishes a rigorous theoretical framework that explicitly treats STIRAP in a multilevel chain, accounting for leakage pathways induced by finite anharmonicity within a single effective Hamiltonian.
2. PMP for Constrained Gaussian Pulses: Unlike standard gradient ascent methods that optimize discrete time-points, this work adapts PMP to optimize a small set of physically meaningful Gaussian parameters. This ensures the resulting pulses are experimentally feasible (smooth, bandwidth-limited) and preserves the counterintuitive ordering essential for STIRAP.
3. Explicit Leakage Penalty: The cost function explicitly penalizes population in leakage states, directly addressing the primary failure mode of STIRAP in superconducting circuits.
4. Complex Hilbert Space Formulation: The derivation of PMP gradients in a complex Hilbert space avoids the computational overhead of decomposing states into real and imaginary parts, maintaining mathematical elegance and efficiency.

4. Results

The method was applied to a five-level superconducting transmon model (targeting $|0\rangle \to |2\rangle$ with $|3\rangle, |4\rangle$ as leakage).

• Fidelity Improvement:
  • Initial Protocol: Standard counterintuitive Gaussian STIRAP achieved a fidelity of 0.911.
  • Optimized Protocol: PMP-optimized pulses achieved a fidelity of 0.998.
• Leakage Suppression:
  • Maximum leakage population ($P_3 + P_4$) was reduced from 0.0499 to 0.0221.
  • Transient occupation of the intermediate state $|1\rangle$ was also significantly reduced.
• Duration Reduction: The optimized protocol shortened the total pulse duration from 80 ns to 47.8 ns while maintaining high fidelity, demonstrating that the optimization found a more efficient adiabatic path rather than just a slower one.
• Robustness Analysis:
  • Amplitude Miscalibration: The optimized pulses showed a broader high-fidelity region against drive amplitude errors compared to the initial protocol.
  • Detuning Drifts: The method exhibited superior robustness against Raman detuning bias ($\delta$) and transition frequency drifts.
  • Anharmonicity Variations: The optimized solution remained effective even when the anharmonicity scale ($\eta_\alpha$) deviated from the nominal value, indicating it compensates for structural distortions of the dark state.

5. Significance

• Practical Implementation: The work provides a practical route to implementing high-fidelity STIRAP in real-world superconducting quantum processors, where leakage is a dominant error source.
• Experimental Feasibility: By restricting optimization to Gaussian pulses, the results are immediately applicable to experimental setups without requiring complex, hard-to-generate waveforms.
• Generalizability: While demonstrated on transmons, the framework is general. It can be applied to other multilevel systems (cold atoms, molecules) and other adiabatic control tasks (subspace transfer, gate design) where leakage suppression and robustness are critical.
• Theoretical Advancement: It bridges the gap between idealized adiabatic theory and realistic optimal control by integrating multilevel dynamics, dissipation, and experimental constraints into a unified PMP-based variational framework.

In summary, the paper demonstrates that by using Pontryagin's Maximum Principle to optimize the parameters of simple Gaussian pulses within a multilevel model, one can simultaneously achieve high-fidelity state transfer, significant leakage suppression, and enhanced robustness against experimental imperfections.