Kraus Constrained Sequence Learning For Quantum Trajectories from Continuous Measurement

Priyanshi Singh; Krishna Bhatia

cs.LGMar 5, 2026

Kraus Constrained Sequence Learning For Quantum Trajectories from Continuous Measurement

Authors: Priyanshi Singh, Krishna Bhatia

Organizations: Department of Computer Science and Engineering SRM Institute of Science and Technology, Chennai, India · QuantumAI Lab, Fractal Analytics Mumbai, India

Abstract

Real-time reconstruction of conditional quantum states from continuous measurement records is a fundamental requirement for quantum feedback control, yet standard stochastic master equation (SME) solvers require exact model specification, known system parameters, and are sensitive to parameter mismatch. While neural sequence models can fit these stochastic dynamics, the unconstrained predictors can violate physicality such as positivity or trace constraints, leading to unstable rollouts and unphysical estimates. We propose a Kraus-structured output layer that converts the hidden representation of a generic sequence backbone into a completely positive trace preserving (CPTP) quantum operation, yielding physically valid state updates by construction. We instantiate this layer across diverse backbones, RNN, GRU, LSTM, TCN, ESN and Mamba; including Neural ODE as a comparative baseline, on stochastic trajectories characterized by parameter drift. Our evaluation reveals distinct trade-offs between gating mechanisms, linear recurrence, and global attention. Across all models, Kraus-LSTM achieves the strongest results, improving state estimation quality by 7% over its unconstrained counterpart while guaranteeing physically valid predictions in non-stationary regimes.

Figures & tables

Item	Symbol	Value
System dimension	$d$	2 (Single Qubit)
Time step	$\Delta t$	$0.005\,\mathrm{s}$
Sequence length	$T$	2000 steps ( $10\,\mathrm{s}$ total)
# trajectories	train/test	$2000/300$
Measurement efficiency	$\eta$	1.0 (Ideal)
Measured observable	$L$	$\sigma_{z}$

Table 1: Dataset Specification for Non-Stationary Switching Dynamics.

Figure 1: Comparative performance of Baseline (unconstrained) vs. Kraus-constrained models. Error bars represent one standard deviation. The Kraus Head provides a consistent fidelity lift for recurrent backbones, effectively acting as a physics-informed regularizer

Model	Baseline Fid. $\uparrow$	Kraus Fid. $\uparrow$	$\Delta$ $\uparrow$
RNN (Vanilla)	0.6985	0.6949	$-0.0036$
GRU	0.7462	0.7655	$+0.0193$
LSTM	0.6997	0.7683	$+0.0686$
ESN	0.5125	0.6117	$+0.0992$
TCN (Standard)	0.6863	0.7122	$+0.0259$
TCN (Gated)	0.6872	0.7055	$+0.0183$

Table 2: Baseline vs. Kraus-constrained performance on the switching dataset (test set).

\Delta

denotes Kraus minus baseline.

Model	Avg. Fidelity $\uparrow$	Bures Dist. $\downarrow$	Phase 1 Fid.	Phase 2 Fid.
Kraus-Mamba	0.7388	0.2918	0.7469	0.7360
Neural-ODE	0.7387	0.2918	0.7546	0.7294
Kraus-RNN (Vanilla)	0.6949	0.3430	0.6904	0.6971
Kraus-ESN	0.6117	0.4440	0.6119	0.6098
Kraus-GRU	0.7655	0.2619	0.7690	0.7630
Kraus-LSTM	0.7683	0.2621	0.7707	0.7659

Table 3: Performance Metrics across Dynamical Regimes. Phase 1 denotes steady-state tracking; Phase 2 denotes tracking following the orthogonal Hamiltonian switch. Bures distance serves as the natural geodesic metric for quantum state consistency.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Model	Type	Hidden Dim	LR	Batch	Epochs
Kraus-GRU	Kraus	256	$5\times 10^{-4}$	128	100
Kraus-LSTM	Kraus	256	$10^{-3}$	32	100
Kraus-RNN	Kraus	256	$5\times 10^{-4}$	32	100
Kraus-ESN	Kraus	256	$5\times 10^{-4}$	128	100
Kraus-TCN (Std)	Kraus	256	$5\times 10^{-4}$	32	100
Kraus-TCN (Gated)	Kraus	256	$5\times 10^{-4}$	32	100

Appendix

Table 4: Hyperparameter configurations for all architectural families.

Kraus Models (256 Hidden Dim)	Params	Baseline Models (256 Hidden Dim)	Params
Kraus-Transformer (3L)	5.23M	Baseline-Transformer	2.42M
Kraus-TCN Gated	3.68M*	Baseline-TCN Gated	3.68M
Kraus-TCN Standard	1.84M	Baseline-TCN Standard	1.84M
Kraus-Mamba	1.21M	Baseline-Mamba	1.02M
Kraus-LSTM (2L)	1.06M	Baseline-LSTM	0.79M
Kraus-GRU (2L)	0.79M	Baseline-GRU	0.60M

Appendix

Table 5: Trainable Parameter Counts for Kraus-Constrained and Baseline Architectures (All models at 256 hidden dimensions).

Figure 2: Visualization of the Switching Dataset. (Top) The normalized homodyne measurement record $y_{t}$ provided as input to the models. (Bottom) Ground-truth Bloch vector components showing the Phase 1 ( $\sigma_{x}$ -rotation) to Phase 2 ( $\sigma_{y}$ -rotation) transition at the vertical dashed line. The stochastic “jitter” in the Bloch components illustrates the measurement back-action (kicks) modeled by the SME.

Figure 3: Layer-wise gradient norms for Kraus-RNN (left) and Kraus-GRU (right) computed from trained checkpoints. The Kraus-RNN exhibits severe gradient imbalance, with input_proj.weight nearly vanishing ( $\sim\!10^{-2.5}$ ) while recurrent weights in later layers receive substantially larger gradients. In contrast, Kraus-GRU maintains comparatively uniform gradient flow across all layers.

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Aug 16, 2026J/K move · Enter open · S save

Continuous Quantum Feedback Control via Kraus-Parameterized Belief Reinforcement Learning

Priyanshi Singh, Krishna Bhatia

Department of Computer Science and Engineering SRM Institute of Science and Technology Chennai, India · QuantumAI Lab Fractal AI Research Mumbai, India

Feedback Control Quantum RL Physics-Informed RL Reinforcement Learning

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Quantum feedback control requires acting on noisy continuous measurement records without direct access to the underlying quantum state. We propose Kraus-Parameterized Belief Reinforcement Learning, a pipeline in which a recurrent encoder, constrained to the Stiefel manifold, produces density-matrix estimates that are guaranteed positive-semidefinite and trace-normalized by construction, embedding quantum state geometry directly into the learning loop. A Proximal Policy Optimization (PPO) actor then maps these physically valid belief states to continuous control actions. On a simulated continuously monitored qubit, the resulting policy achieves stable feedback control, maintaining a measurement-conditioned belief fidelity of approximately 0.77-0.80 and exhibiting substantially lower return variance than a parameter-matched LSTM-history baseline across both nominal and out-of-distribution conditions. Although gains in raw target fidelity are modest, the geometric constraint guarantees a physically valid, interpretable belief representation and yields markedly more stable control under measurement inefficiency and abrupt dynamics switches. These results indicate that physics-informed neural memory is a practical inductive bias for reliable quantum feedback control.

Abstract

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