Preparation-Space Diagnostics and Logical Information Loss in a Driven Kerr-Cat Qubit

Technical Summary: Preparation-Space Diagnostics and Logical Information Loss in a Driven Kerr-Cat Qubit

Problem Statement
Kerr-cat qubits encode logical information in the two wells of a parametrically driven nonlinear oscillator. While bit-flip errors are exponentially suppressed during idling, gate operations require transiently deforming the double-well potential, which can drive the state across the barrier between wells. The central challenge addressed is identifying which initial quantum states (preparations) are vulnerable to a specific gate pulse and determining how classical phase-space transport diagnostics relate to the resulting quantum logical information loss. Specifically, the paper investigates whether the "turnstile" mechanism of classical transport—where a pulse splits the separatrix and carries trajectories across the dividing surface—accurately predicts the quantum outcome, and how the temporal profile of the pulse (e.g., sudden quench vs. smooth ramp) affects logical bit preservation.

Methodology
The study employs a comparative framework on a common "preparation space," defined as a two-dimensional disk of coherent states $|\alpha_0\rangle$ centered on the right-hand well.

1. Models:
   • Classical: The mean-field dynamics are governed by a pair of dissipative ordinary differential equations in the phase plane $(x, y)$, derived from the expectation value of the annihilation operator. The system includes a Kerr nonlinearity, a two-photon parametric pump, and single-photon loss.
   • Quantum: The system is modeled via the Lindblad master equation for the density operator $\rho$, incorporating the same Hamiltonian and a single-photon dissipator. Engineered two-photon dissipation is also simulated to test stabilization.
2. Diagnostics:
   • Classical Fields: The finite-time sensitivity (spectral norm of the Jacobian of the trajectory map) and the Lagrangian descriptor (trajectory arc length) are computed over the preparation grid.
   • Quantum Fields: The left-half-plane occupation $P_{left}(T)$ (probability of finding the oscillator in the left well) and the Loschmidt echo (fidelity between pulsed and idle evolutions) are calculated.
   • Logical Bit Metrics: The trace distance $D_{tr}$ between the two logical inputs ($|\pm \alpha_{cat}\rangle$) is computed to measure logical distinguishability, distinct from spatial transport.
   • Exploratory Diagnostic: The gate-induced Out-of-Time-Order Correlator (OTOC) is tested as a potential bridge between classical sensitivity and quantum operator growth.
3. Protocol Variation: The study sweeps pulse amplitude, width, and temporal profile (comparing a "quench" where integration starts at the pulse peak vs. a "smooth" full Gaussian ramp). It also varies the cat size (mean photon number) and dissipation rates.

Key Contributions and Results

• Protocol Dependence of Logical Erasure: The most significant finding is that logical information loss depends critically on the full temporal protocol, not merely pulse strength. A sudden quench (starting integration at the pulse peak) causes the logical trace distance to collapse to near zero ($D_{tr} \approx 0.013$), indicating the erasure of the bit into an incoherent mixture. In contrast, a smooth ramp of the same peak amplitude largely preserves the bit ($D_{tr} \approx 0.32$, retaining ~46% of idle distinguishability).
• Classical-Quantum Correspondence:
  • Spatial Tracking: The quantum left-occupation field $P_{left}$ tracks the classical transport pattern (leak vs. safe) with a correlation of $r \approx +0.61$ under the quench. However, the classical model predicts deterministic bit-flips, whereas the quantum system exhibits delocalization (scrambling) without necessarily flipping the bit in a classical sense.
  • Sensitivity Ridge: The classical finite-time sensitivity field robustly locates the transport boundary (the "ridge" coincides with the leak/safe boundary at the 99th percentile). However, it is a poor predictor of which specific preparations leak when the leaked set is a large bulk region, as the bulk has low sensitivity.
  • OTOC Failure: The gate-induced OTOC aligns with the classical sensitivity field only transiently during the mid-pulse phase ($r \approx +0.97$). By the end of the pulse, this correspondence collapses ($r \approx 0.09$) due to the polynomial growth of the background OTOC overwhelming the gate's signal. The paper records this as a negative result for using OTOC as a robust bridge to classical sensitivity in this regime.
• Loschmidt Echo as Early Warning: The Loschmidt echo, evaluated near the end of the pulse, serves as a strong predictor of the final quantum outcome in the quench regime ($r \approx -0.94$ with $P_{left}$). It effectively anticipates the logical scrambling before the final readout time.
• Stabilization and Cat Size:
  • Engineered Dissipation: Introducing engineered two-photon dissipation drives the classical leak fraction to zero. However, the quantum system still loses up to 47% of its "which-well" population for the most vulnerable preparations, demonstrating that the classical model underpredicts logical error in the presence of pair loss.
  • Cat Size: As the cat size increases (making the system more semiclassical), the quench-induced erasure weakens, and the system shifts toward deterministic classical transport (bit-flips rather than erasure).

Significance and Claims
The paper claims to establish that the "turnstile" mechanism of classical phase-space transport provides a spatial map of vulnerability but fails to predict the magnitude of logical information loss or the survival of the bit. The central conclusion is that logical bit preservation is a protocol-dependent phenomenon: a smooth ramp can suppress logical erasure even when the classical transport geometry suggests significant vulnerability.

The authors emphasize that classical diagnostics (sensitivity, leak fraction) are insufficient for gate design in the strong-driving regime because they cannot capture the transition from coherent transport to incoherent scrambling. The Loschmidt echo is identified as a necessary quantum diagnostic for predicting logical outcomes in the quench regime. The study explicitly notes that the "quench-versus-smooth" distinction is multi-parametric (involving ramp rate, dwell time, and integrated area) and that no single scalar parameter (like amplitude or Melnikov threshold) organizes the phenomenology under strong driving. The work serves as a rigorous validation of the limitations of mean-field models for Kerr-cat gate design and highlights the necessity of full open-system quantum simulations to assess logical fidelity.