Technical Summary: Resource-Bounded Controllability Benchmarking of Open Quantum Systems

Problem Statement

The practical realization of quantum technologies requires implementing high-fidelity operations under finite hardware resources and open-system dynamics. While idealized closed-system controllability is well-defined by a binary Lie-algebraic criterion (whether the drift and control Hamiltonians span $\mathfrak{su}(d)$), this theoretical condition fails to predict achievable performance in real devices. Real-world systems operate in open regimes where decoherence, drift, bandwidth limits, and non-ideal pulses induce non-unitary dynamics. Furthermore, noise is often partially characterized, control-dependent, and potentially non-Markovian.

Existing literature on open-system controllability typically relies on specific models (e.g., Markovian Lindblad dynamics) or abstract reachability analyses that do not directly address the engineering question: Given a fixed, hardware-realisable family of control pulses and explicit resource constraints (e.g., amplitude and bandwidth), which target operations are achievable to high fidelity, and how does this capability scale with available resources? Pure "whitebox" physics models often lack accuracy in high-fidelity regimes due to unmodeled dissipation, while "blackbox" data-driven models lack diagnostic structure and require excessive data.

Methodology

The authors propose a resource-bounded framework that treats practical controllability as a device-engineering problem. The approach quantifies the distribution of best-achievable fidelities over an ensemble of target gates rather than seeking a binary reachability condition. The methodology consists of four main stages:

1. Graybox Modeling:
   The core of the framework is a trained "graybox" surrogate model. This model retains the known coherent dynamics generated by the drift and control Hamiltonians ($H_0(t)$) but learns a compact, control-dependent description of open-system distortions ($V_O(T)$) from probing data.

   • The system dynamics are expressed in the interaction picture relative to $H_0(t)$.
   • The model is trained on pairs of control parameters $\vec{\theta}$ and measured expectation vectors $\vec{E}$ over a tomographically complete basis.
   • Once trained, the model acts as a fast lookup engine, predicting the effective channel for any candidate waveform within the trained control space without requiring full physical simulation.

2. Channel Reconstruction:
   For a given candidate control waveform, the graybox model predicts the coherent propagator and the set of effective observable-space correction operators. These predictions are used to reconstruct the process matrix (Choi matrix) of the implemented quantum channel in a fixed tomography basis.

3. Fidelity Scoring and Optimization:

   • A set of $N_G$ target gates is drawn independently from the Haar measure on $SU(d)$.
   • For each target gate, the framework evaluates the process fidelity against all waveforms in a constrained "admissible" pulse family.
   • The admissible family is defined by explicit resource budgets (e.g., maximum amplitude $A_{cap}$ and bandwidth $B_{cap}$).
   • The best-achievable fidelity $F^*_g(A_0)$ is determined by maximizing fidelity over all feasible controls for a specific target $g$ and budget level $A_0$.

4. Controllability Metrics:
   Practical controllability is characterized statistically by the distribution of best-achievable infidelities ($I_g = 1 - F^*_g$) over the Haar ensemble.

   • The primary metric is an area-based summary ($CM$), defined as the integral of the survival function of the infidelity distribution.
   • Mathematically, this is equivalent to the 1-Wasserstein distance ($W_1$) between the empirical distribution of best-achievable infidelities and the ideal point mass at zero infidelity.
   • This metric is computed across a grid of resource budgets to generate "controllability surfaces."

Key Results

The framework was demonstrated numerically for a driven qubit ($d=2$) coupled to a finite-dimensional bath, under three regimes:

1. Closed-system reference: Ideal unitary dynamics.
2. Classical-noise model: Stochastic dephasing.
3. Combined quantum-plus-classical noise: Dissipative auxiliary modes plus classical dephasing.

One-Dimensional Studies:

• Distributions of best-achievable infidelity were analyzed for amplitude scaling and bandwidth (Gaussian width) scaling.
• The closed-system case collapsed to a step function at zero infidelity ($CM \approx 0$), confirming the framework's validity.
• In noisy regimes, the infidelity distributions shifted right, with the combined noise model showing the poorest performance ($CM \approx 0.021$ for amplitude scaling).
• The results showed a consistent ordering: Closed-system > Classical-noise > Combined noise.

Two-Dimensional Studies:

• Joint amplitude-bandwidth controllability surfaces ($CM(A_{cap}, B_{cap})$) revealed a distinct low-resource–transition–saturation structure.
• Low-resource: Small caps restrict the pulse family, resulting in low controllability.
• Transition: As caps relax, controllability improves rapidly.
• Saturation: Further resource increases yield diminishing returns.
• Noise Impact: Noise not only lowers the overall controllability level but deforms the useful region. In the combined noise case, the surface is reshaped such that bandwidth dependence becomes more concentrated in low-bandwidth regions, even though absolute performance is lower than in the classical-noise case.
• Resource Hierarchy: The study suggests amplitude acts as the primary resource governing the onset of controllability, while bandwidth acts as a secondary resource that refines the regime once sufficient drive strength is available.

Significance and Claims

The paper claims to provide a device-level notion of controllability for fixed, hardware-realisable control families in the presence of partially unknown open-system noise. Its significance lies in:

1. Shifting the Paradigm: Moving from binary theoretical reachability to a quantitative, resource-bounded assessment of achievable performance distributions.
2. Operational Mapping: Converting practical controllability into an interpretable "resource landscape" (the controllability surface) that links specific hardware constraints (amplitude and bandwidth) to achievable gate performance.
3. Benchmarking Utility: Providing a pipeline to compare different control families, noise regimes, or hardware configurations by quantifying the change in the distribution of achievable gate fidelities (via the 1-Wasserstein distance).
4. Practical Relevance: Demonstrating that finite control resources and open-system noise jointly constrain gate performance, offering a method to identify the specific resource levels required to enter a high-performance regime.

The authors note limitations, including that the graybox model is valid only within the training region (no extrapolation), bandwidth is modeled via Gaussian width rather than explicit hardware filters, and the metrics are numerical summaries over finite samples rather than exact analytic criteria. The framework is presented as a tool for characterizing and benchmarking control capabilities in realistic, resource-constrained settings.