Fingerprints of classical memory in quantum hysteresis

Technical Summary: Fingerprints of Classical Memory in Quantum Hysteresis

Problem Statement
High-fidelity coherent control is fundamental to quantum information science, yet standard theoretical models often assume an idealized scenario where a time-dependent Hamiltonian is prescribed and followed exactly. In practice, the Hamiltonian experienced by a quantum device is generated by a classical control stack (room-temperature electronics, wiring, attenuators, bias tees, and filters) with finite bandwidth and internal dynamics. This control chain introduces "classical memory," causing the realized field $\Phi(t)$ to lag behind the commanded waveform $u(t)$. This lag can generate hysteresis loops in experimental data (e.g., plotting an observable against the command) that mimic signatures of dissipation or non-Markovian quantum memory (information backflow), even when the quantum system itself is perfectly isolated and evolves unitarily. Distinguishing between genuine quantum memory and artifacts arising from classical filtering is a critical challenge for interpreting data and designing robust control protocols.

Methodology
The authors propose a minimal framework where the Hamiltonian at time $t$ depends on past values of a control command through a causal kernel $K$. The realized field is modeled as a convolution:
$$\Phi(t) = \int_{-\infty}^{t} K(t-s) u(s) ds$$
The total Hamiltonian is $\hat{H}(t) = \hat{H}_A + \Phi(t)\hat{M}$, where $\hat{H}_A$ is a drift term and $\hat{M}$ is a fixed operator. The paper assumes the kernel $K$ is real and causal, ensuring the instantaneous Hamiltonian remains Hermitian and the device evolution remains unitary.

Key methodological components include:

1. Time-Local Embedding: For kernels representable as sums of decaying exponentials (motivated by passive RC network physics), the non-local convolution is converted into a set of coupled ordinary differential equations (ODEs) for auxiliary classical variables $\Phi_k(t)$. This allows the control memory to be treated as a finite-dimensional classical dynamical system coupled to the quantum state.
2. Geometric Loop Measures: The authors define oriented loop areas in three planes to quantify hysteresis:
   • $A_{u\Phi} = \oint \Phi du$: Measures memory intrinsic to the control channel (lag between command and realized field).
   • $A_{\Phi O} = \oint O d\Phi$: Measures history dependence of the observable relative to the realized field (diagnostic of non-adiabatic quantum dynamics).
   • $A_{uO} = \oint O du$: The experimentally accessible loop between command and observable.
3. Physical Derivations: The paper provides two justifications for the kernel model:
   • Classical: Based on network synthesis theory, where passive RC ladders yield transfer functions with poles on the negative real axis, resulting in impulse responses that are sums of decaying exponentials.
   • Quantum: A microscopic derivation using the Kubo linear response formalism. The control hardware is modeled as a quantum channel driven by a classical source. In the weak-coupling limit, the realized field is the retarded susceptibility of the channel, which reduces to the causal kernel form when the channel exhibits dissipative (overdamped) dynamics.

Key Contributions and Results

• Separation of Mechanisms: The framework demonstrates that a non-zero $A_{uO}$ (hysteresis in the command-observable plane) does not necessarily imply non-Markovian quantum memory. In the adiabatic regime, $A_{\Phi O} \approx 0$, yet $A_{uO}$ can be non-zero solely due to the classical lag $u \to \Phi$. The paper derives a linear relationship in the weak-drive limit: $A_{uO} \approx f'(0) A_{u\Phi}$, where $f(\Phi)$ is the adiabatic response function.
• Diagnostic Criteria: The authors establish that $A_{\Phi O}$ serves as a clean diagnostic for genuine state-history dependence. If $A_{\Phi O} \neq 0$ while the control channel is accounted for, it indicates coherent non-adiabatic effects (e.g., transitions between energy levels) rather than classical filtering artifacts. Conversely, if $A_{\Phi O} = 0$ but $A_{uO} \neq 0$, the hysteresis is purely a fingerprint of classical control memory.
• Single-Qubit Case Studies: Analytical solutions are provided for a transverse-driven qubit ($\hat{H} = \sigma_z + \Phi(t)\sigma_x$) and a longitudinal commuting model. These illustrate how the loop areas behave under different drive frequencies and filter time constants.
• Numerical Validation: Simulations of a transverse-driven qubit with a single-pole (RC) filter confirm the theoretical predictions. The results show a crossover from a regime dominated by classical memory (where $A_{uO}$ tracks $A_{u\Phi}$) to a regime where non-adiabatic quantum effects become prominent as the drive frequency increases relative to the gap.
• Physical Realism: The paper connects the phenomenological kernel to realistic cryogenic control lines, showing that the memory time scale $\tau_c$ corresponds to the inverse of the control stack's bandwidth (e.g., $\tau_c \sim 1$ ns for GHz bandwidths).

Significance
The paper claims to provide a "clean way to distinguish between memory in the control and genuine non-Markovian dynamics of the state." Its primary significance lies in offering a pragmatic, experimentally grounded vocabulary (the loop areas) to interpret hysteretic features in quantum control data without misattributing classical filtering effects to open-system quantum memory.

The authors emphasize that their framework is aimed at the interpretation and separation of mechanisms rather than the inversion of distortion maps for pulse design, though they note the framework naturally supports distortion-aware optimal control. By showing that classical control memory can generate deterministic "fingerprint" loops even in closed, unitary systems, the work cautions against over-interpreting hysteresis as evidence of information backflow. Instead, it suggests that reporting control-channel diagnostics (such as $A_{u\Phi}$ or transfer function estimates) alongside observable hysteresis plots is essential for accurate characterization of quantum devices. The work bridges classical circuit theory and quantum control, validating the use of exponential-mode kernels derived from both RC network synthesis and microscopic Kubo linear response.