Trainable Quantum Channels

• Trainable quantum channels are parameterized CPTP maps that adjust their dynamics through learnable parameters, unifying methods like Kraus, Stinespring, and mixed-unitary representations.
• They are implemented via diverse strategies such as static hardware couplings, adaptive ancilla-assisted schemes, and direct state-space techniques to simulate both unitary and non-unitary dynamics.
• These channels transform non-unitary noise into computational resources, enhancing optimization, error mitigation, and quantum communication in modern quantum devices.

Trainable quantum channels are parameterized completely positive trace-preserving (CPTP) maps whose parameters are optimized from data, from task losses, or from hardware design constraints so that the resulting open-system dynamics realizes a desired transformation. In this setting, the trainable object may be a Kraus map, a Stinespring dilation, a mixture of unitaries, a Gaussian or generalized Gaussian photonic channel, or an effective reduced channel induced by a larger system–environment evolution. The notion therefore unifies several lines of work that had often been treated separately: quantum process learning, hardware-native gate synthesis, variational open-system modeling, and non-unitary quantum machine learning. A recurring theme is that unitary variational models arise as a special case, while non-unitary degrees of freedom are promoted from noise sources to computational primitives (Wen et al., 14 Jun 2026, Cemin et al., 2024, Shen et al., 2016).

1. Formal definition and mathematical representations

The common mathematical object is a CPTP map

$$\mathcal{E}_{\theta}(\rho)=\sum_k K_k(\theta)\,\rho\,K_k^\dagger(\theta),\qquad \sum_k K_k^\dagger(\theta)K_k(\theta)=I,$$

with learnable parameters $\theta$. This representation appears throughout the literature, from programmable superconducting implementations and stochastic channel simulators to neural quasi-inverse models and communication-oriented variational circuits (Shen et al., 2016, Hu et al., 2018, Aziz et al., 13 Jun 2025, Rathi et al., 2023).

An equivalent parameterization is given by Stinespring dilation,

$$\mathcal{E}_{\theta}(\rho)=\operatorname{Tr}_{E}\!\big[U(\theta)\,(\rho\otimes \rho_E)\,U^\dagger(\theta)\big],$$

where the channel is realized by a unitary on system plus environment followed by a partial trace. This construction is central in hardware-native channel synthesis, in neural-network learning of reduced NISQ dynamics, and in variational extrapolation of Lindblad evolution on neutral-atom systems (Cemin et al., 2024, Visser et al., 2023, Shen et al., 2016). It is also the mechanism by which a static Hamiltonian acting on a larger qubit network induces an effective subsystem channel,

$$\mathcal{E}_{w}(\rho_S)=\operatorname{Tr}_{E}\!\big[U_{\mathrm{global}}(\rho_S\otimes \sigma_E)U_{\mathrm{global}}^\dagger\big],$$

with $U_{\mathrm{global}}=e^{-iH(w)}$ and trainable couplings $w$ (Banchi et al., 2016).

The Choi representation provides a third standard description. For a $d$-dimensional input space,

$$J_{\mathcal{E}}=(\mathcal{E}\otimes \mathbb{I})(|\Phi^+\rangle\langle \Phi^+|),$$

with CPTP constraints $J_{\mathcal{E}}\succeq 0$ and $$\operatorname{Tr}_{\mathrm{out}}J_{\mathcal{E}}=I_{\mathrm{in}}$$. Choi states, process matrices $\theta$0, and Pauli transfer matrices recur as optimization targets and validation objects in channel tomography, arbitrary-channel simulation, and structured random-unitary learning (Hu et al., 2018, Smart et al., 28 Jan 2025, Shen et al., 2016).

A further formal shift appears when channels are treated as model primitives rather than merely dynamical maps. For an observable $\theta$1, trainable-channel models induce an effective observable

$\theta$2

so that the prediction is $\theta$3. In unitary models, $\theta$4 is a similarity transform and preserves spectrum; in channel models, each Kraus branch can deform the spectrum, which the recent non-unitary learning literature identifies as a source of added expressivity (Wen et al., 14 Jun 2026).

2. Principal parameterization strategies

One major family learns channels through static physical couplings. In supervised quantum gate teaching, a time-independent Hamiltonian $\theta$5 with pairwise interactions is trained so that the natural evolution $\theta$6 implements a target unitary on a subsystem after tracing out ancillas. The trainable parameters are static couplings $\theta$7 and, optionally, local fields $\theta$8; no pulse sequence or time-dependent control is required, and execution reduces to state preparation, passive evolution, and measurement (Banchi et al., 2016). This makes the learned map an induced subsystem channel even when the target is unitary.

A second family uses adaptive ancilla-assisted constructions. Universal channel construction with a single ancilla qubit, quantum non-demolition readout, and adaptive control realizes arbitrary CPTP maps by traversing a binary tree of ancilla outcomes in $\theta$9 rounds for Kraus rank $$\mathcal{E}_{\theta}(\rho)=\operatorname{Tr}_{E}\!\big[U(\theta)\,(\rho\otimes \rho_E)\,U^\dagger(\theta)\big],$$0 (Shen et al., 2016). In superconducting cQED, this viewpoint becomes experimentally explicit: arbitrary single-qubit channels are implemented by measurement-based adaptive control, deterministic convex mixing of quasiextreme channels, ancilla reset, and repetition to synthesize continuous-time open-system dynamics (Hu et al., 2018).

A third family learns channels by parameterizing the dilation unitary itself. In NISQ device learning, the joint system–environment unitary is represented by a neural network, unitarity is enforced through torch.nn.utils.parametrizations.orthogonal, and the environment is initialized in a fixed pure state, which guarantees CPTP reduced dynamics by construction (Cemin et al., 2024). A closely related variational strategy on neutral-atom systems learns a step channel $$\mathcal{E}_{\theta}(\rho)=\operatorname{Tr}_{E}\!\big[U(\theta)\,(\rho\otimes \rho_E)\,U^\dagger(\theta)\big],$$1 from short-time measurement data and extrapolates long-time evolution by repeated application with fresh ancillas (Visser et al., 2023).

A fourth class uses mixed-unitary or structured stochastic channels. Density quantum neural networks parameterize

$$\mathcal{E}_{\theta}(\rho)=\operatorname{Tr}_{E}\!\big[U(\theta)\,(\rho\otimes \rho_E)\,U^\dagger(\theta)\big],$$2

with $$\mathcal{E}_{\theta}(\rho)=\operatorname{Tr}_{E}\!\big[U(\theta)\,(\rho\otimes \rho_E)\,U^\dagger(\theta)\big],$$3 in the probability simplex, so the Kraus operators are $$\mathcal{E}_{\theta}(\rho)=\operatorname{Tr}_{E}\!\big[U(\theta)\,(\rho\otimes \rho_E)\,U^\dagger(\theta)\big],$$4 and CPTP is automatic (Coyle et al., 2024). Orthogonal random unitary channels similarly decompose a channel into pre- and post-unitaries around a Pauli stochastic core,

$$\mathcal{E}_{\theta}(\rho)=\operatorname{Tr}_{E}\!\big[U(\theta)\,(\rho\otimes \rho_E)\,U^\dagger(\theta)\big],$$5

with simplex-constrained Pauli probabilities and contracted quantum learning for the coherent parts (Smart et al., 28 Jan 2025).

A fifth family is native to continuous-variable photonics. There the trainable object is a Gaussian, Gaussian-plus-photodetection, or generalized Gaussian channel, with classical inputs encoded into channel parameters $$\mathcal{E}_{\theta}(\rho)=\operatorname{Tr}_{E}\!\big[U(\theta)\,(\rho\otimes \rho_E)\,U^\dagger(\theta)\big],$$6 or their generalized Gaussian analogues by polynomial maps from $$\mathcal{E}_{\theta}(\rho)=\operatorname{Tr}_{E}\!\big[U(\theta)\,(\rho\otimes \rho_E)\,U^\dagger(\theta)\big],$$7 (Rosati, 2022). A related but discrete-variable photonic construction uses a programmable interferometer acting on path-encoded system and environment qubits, with polarization as an auxiliary/program qubit; the resulting channel is

$$\mathcal{E}_{\theta}(\rho)=\operatorname{Tr}_{E}\!\big[U(\theta)\,(\rho\otimes \rho_E)\,U^\dagger(\theta)\big],$$8

and tunable interferometric angles realize dephasing, amplitude damping, generalized amplitude damping, bit-flip, and squeezed generalized amplitude damping channels (Araujo et al., 25 Feb 2025).

Finally, there are direct state-space parameterizations. A neural quasi-inverse for qubit noise learns a map $$\mathcal{E}_{\theta}(\rho)=\operatorname{Tr}_{E}\!\big[U(\theta)\,(\rho\otimes \rho_E)\,U^\dagger(\theta)\big],$$9 on Bloch vectors, uses a physics-inspired loss to discourage norm expansion, and validates CPTP physicality only after training by quantum process tomography and Kraus reconstruction (Aziz et al., 13 Jun 2025). A different classical-to-quantum channel arises in trainable discrete feature embeddings, where a discrete input $$\mathcal{E}_{w}(\rho_S)=\operatorname{Tr}_{E}\!\big[U_{\mathrm{global}}(\rho_S\otimes \sigma_E)U_{\mathrm{global}}^\dagger\big],$$0 selects a codeword-dependent unitary $$\mathcal{E}_{w}(\rho_S)=\operatorname{Tr}_{E}\!\big[U_{\mathrm{global}}(\rho_S\otimes \sigma_E)U_{\mathrm{global}}^\dagger\big],$$1 and prepares

$$\mathcal{E}_{w}(\rho_S)=\operatorname{Tr}_{E}\!\big[U_{\mathrm{global}}(\rho_S\otimes \sigma_E)U_{\mathrm{global}}^\dagger\big],$$2

yielding a trainable embedding channel for variational classifiers (Thumwanit et al., 2021).

3. Objectives, losses, and optimization procedures

The choice of loss function depends on whether the target is a known channel, an unknown effective process, or a task-specific predictor. In supervised gate teaching, the target unitary is known, so a teaching set

$$\mathcal{E}_{w}(\rho_S)=\operatorname{Tr}_{E}\!\big[U_{\mathrm{global}}(\rho_S\otimes \sigma_E)U_{\mathrm{global}}^\dagger\big],$$3

is sampled from Haar-random inputs, and the average fidelity objective is

$$\mathcal{E}_{w}(\rho_S)=\operatorname{Tr}_{E}\!\big[U_{\mathrm{global}}(\rho_S\otimes \sigma_E)U_{\mathrm{global}}^\dagger\big],$$4

with loss $$\mathcal{E}_{w}(\rho_S)=\operatorname{Tr}_{E}\!\big[U_{\mathrm{global}}(\rho_S\otimes \sigma_E)U_{\mathrm{global}}^\dagger\big],$$5 and online stochastic gradient descent updates

$$\mathcal{E}_{w}(\rho_S)=\operatorname{Tr}_{E}\!\big[U_{\mathrm{global}}(\rho_S\otimes \sigma_E)U_{\mathrm{global}}^\dagger\big],$$6

under Haar sampling and decaying learning rate (Banchi et al., 2016).

When the task is to infer a channel from trajectories, objectives are usually state-space discrepancies. For reduced-channel learning on NISQ devices, the loss is mean squared error in coherence vectors,

$$\mathcal{E}_{w}(\rho_S)=\operatorname{Tr}_{E}\!\big[U_{\mathrm{global}}(\rho_S\otimes \sigma_E)U_{\mathrm{global}}^\dagger\big],$$7

optimized with Adam on time-series data reconstructed from Pauli measurements (Cemin et al., 2024). For neutral-atom extrapolation, the variational channel is trained from measured observables by

$$\mathcal{E}_{w}(\rho_S)=\operatorname{Tr}_{E}\!\big[U_{\mathrm{global}}(\rho_S\otimes \sigma_E)U_{\mathrm{global}}^\dagger\big],$$8

or its multi-step generalization, with finite differences, SPSA, parameter-shift, or pulse-based adjoint optimal control (Visser et al., 2023).

In trainable open-system learning models, task losses resemble standard supervised learning objectives. Channel-enhanced classifiers use binary cross-entropy with L2 regularization,

$$\mathcal{E}_{w}(\rho_S)=\operatorname{Tr}_{E}\!\big[U_{\mathrm{global}}(\rho_S\otimes \sigma_E)U_{\mathrm{global}}^\dagger\big],$$9

with Adam and cosine-decay schedules (Wen et al., 14 Jun 2026). Trainable discrete embeddings combine classification loss with a geometric regularizer

$U_{\mathrm{global}}=e^{-iH(w)}$0

where $U_{\mathrm{global}}=e^{-iH(w)}$1 is the covariance of codeword Bloch vectors, so that the embedding retains QRAC-like spread while adapting to the task (Thumwanit et al., 2021). Communication autoencoders optimize cross-entropy for classical and entanglement-assisted tasks, and trace distance for quantum communication, with PennyLane–JAX autodiff and a variant of Adam (Rathi et al., 2023).

Several works optimize directly over process structure. The quasi-inverse model minimizes the mean of the square of the modified trace distance,

$U_{\mathrm{global}}=e^{-iH(w)}$2

where the target is scaled by $U_{\mathrm{global}}=e^{-iH(w)}$3 to bias the output toward the Bloch ball (Aziz et al., 13 Jun 2025). Orthogonal random unitary learning uses a multi-objective loss

$U_{\mathrm{global}}=e^{-iH(w)}$4

alternating simplex-constrained Pauli updates with contracted or resolution-of-the-identity unitary updates (Smart et al., 28 Jan 2025). Across these settings, validation commonly relies on process fidelity, state fidelity, Bures distance, trace distance, $U_{\mathrm{global}}=e^{-iH(w)}$5-matrix discrepancies, or diamond-distance surrogates (Hu et al., 2018, Visser et al., 2023).

4. Learnability, gradient structure, and trainability

A central theoretical result concerns sample-efficient learning in continuous-variable photonics. For Gaussian circuits, the probability function class has pseudo-dimension

$U_{\mathrm{global}}=e^{-iH(w)}$6

yielding sample complexity

$U_{\mathrm{global}}=e^{-iH(w)}$7

For Gaussian plus photodetection channels,

$U_{\mathrm{global}}=e^{-iH(w)}$8

with analogous $U_{\mathrm{global}}=e^{-iH(w)}$9 sample scaling, while generalized Gaussian channels satisfy covering-number bounds leading to

$w$0

The notable feature is that these bounds depend on the number of modes $w$1 and resource-dependent constants, but not on circuit depth (Rosati, 2022).

A different aspect of trainability is gradient extraction cost. For density quantum neural networks with commuting-generator blocks, the gradient of

$w$2

decomposes linearly over the sub-unitaries, and the total number of circuits needed for unbiased gradients scales as

$w$3

for commuting-block sub-unitaries with block counts $w$4 (Coyle et al., 2024). Orthogonal random unitary learning goes further by replacing parameter-shift scaling with contracted quantum learning; in the resolution-of-the-identity variant, only two quantum function evaluations per iteration are needed for unitary updates, while Pauli probabilities are updated by Riemannian gradient descent on the simplex (Smart et al., 28 Jan 2025).

Recent work has also formalized finite-sample trainability guarantees. For clipped gradient samples $w$5 with range $w$6, the sample variance obeys the dimension-independent concentration bound

$w$7

and, for $w$8, a sufficient sample size is

$w$9

This framework is used to define an operational gradient-to-noise trainability criterion and to document an empirical anticorrelation between expressibility and trainability across common PQC ansätze (Röseler et al., 15 Mar 2026). Within explicitly non-unitary models, trainable channels modify optimization geometry in two ways: unitary-direction gradients become ensemble averages over channel branches, and channel parameters add extra optimization directions through Kraus derivatives. For amplitude-damping and phase-damping channels, this has been tied to faster and smoother optimization in supervised learning experiments (Wen et al., 14 Jun 2026).

5. Hardware realizations and application domains

Superconducting platforms provide several canonical realizations. Static gate teaching on pairwise-interaction qubit networks demonstrates that few-qubit devices can be trained to implement nontrivial gates such as Toffoli and Fredkin with no time-dependent control, by embedding the target in hardware couplings and ancillary qubits (Banchi et al., 2016). A separate universal cQED protocol constructs arbitrary channels with a single ancilla qubit, QND measurement, and adaptive control in depth $d$0 for Kraus rank $d$1 (Shen et al., 2016). Experimentally, repetitive channel simulation in cQED achieved arbitrary single-qubit channels, deterministic convex mixing with one ancilla, and continuous-time Liouvillian synthesis; the reported metrics include $d$2 for dephasing round-trip, average worst-case state-generation fidelity $d$3 at $d$4 across six target channels, and average diamond distance $d$5 at $d$6 (Hu et al., 2018).

Neutral-atom and photonic implementations emphasize programmability and extrapolation. The neutral-atom Stinespring method learned a fixed-step channel from short-time data and extrapolated by repeated application with fresh ancillas, reporting averaged Bures errors of approximately $d$7 on the first step for a single-qubit decay benchmark, approximately $d$8 for a two-qubit interacting decay benchmark, and approximately $d$9 for a two-spin TFIM-with-decay benchmark (Visser et al., 2023). In photonics, a programmable interferometer realizes phase-damping, amplitude-damping, generalized amplitude damping, bit-flip, and squeezed generalized amplitude damping by tuning beam-splitter angles, phases, and polarization rotations; for generalized amplitude damping, the reconstructed system–environment density matrix agreed with theory at approximately $$J_{\mathcal{E}}=(\mathcal{E}\otimes \mathbb{I})(|\Phi^+\rangle\langle \Phi^+|),$$0 fidelity (Araujo et al., 25 Feb 2025). At the continuous-variable level, the same platform class admits provably trainable Gaussian and generalized Gaussian channel families with depth-independent sample complexity (Rosati, 2022).

Another application class is device characterization and noise identification. Learning reduced channels on NISQ devices from Pauli-measurement trajectories can recover effective stroboscopic maps even when a time-independent Floquet Lindbladian does not exist, and on ibmq_ehningen the learned channel identified cross-talk consistent with a weak effective $$J_{\mathcal{E}}=(\mathcal{E}\otimes \mathbb{I})(|\Phi^+\rangle\langle \Phi^+|),$$1 coupling

$$J_{\mathcal{E}}=(\mathcal{E}\otimes \mathbb{I})(|\Phi^+\rangle\langle \Phi^+|),$$2

between two simultaneously driven qubits (Cemin et al., 2024). Steady-state learning of local non-unital channels uses only local expectation values on a single steady state and scales linearly with system size; on ibm_lagos v1.0.32, the learned model produced 2-qubit reduced-density-matrix trace distances $$J_{\mathcal{E}}=(\mathcal{E}\otimes \mathbb{I})(|\Phi^+\rangle\langle \Phi^+|),$$3 and $$J_{\mathcal{E}}=(\mathcal{E}\otimes \mathbb{I})(|\Phi^+\rangle\langle \Phi^+|),$$4 for two deterministic maps, versus $$J_{\mathcal{E}}=(\mathcal{E}\otimes \mathbb{I})(|\Phi^+\rangle\langle \Phi^+|),$$5 and $$J_{\mathcal{E}}=(\mathcal{E}\otimes \mathbb{I})(|\Phi^+\rangle\langle \Phi^+|),$$6 for the ideal model (Ilin et al., 2023).

Trainable channels also appear in quantum learning and communication. Quantum autoencoders for channel coding train encoder and decoder circuits around fixed CPTP noise models and recover or closely approach known classical, entanglement-assisted, and quantum communication benchmarks, while also exposing superadditivity effects for depolarizing noise under larger GHZ inputs (Rathi et al., 2023). Trainable discrete feature embeddings define classical-to-quantum channels that retain QRAC-level qubit efficiency while improving classification of parity, Breast Cancer, Titanic, and reduced MNIST datasets; for example, the 3-bit parity benchmark reached classified ratio $$J_{\mathcal{E}}=(\mathcal{E}\otimes \mathbb{I})(|\Phi^+\rangle\langle \Phi^+|),$$7 with a single embedding qubit, and the 4-by-4 MNIST experiment achieved test accuracy $$J_{\mathcal{E}}=(\mathcal{E}\otimes \mathbb{I})(|\Phi^+\rangle\langle \Phi^+|),$$8 with a 9-qubit $$J_{\mathcal{E}}=(\mathcal{E}\otimes \mathbb{I})(|\Phi^+\rangle\langle \Phi^+|),$$9 trainable embedding (Thumwanit et al., 2021). Channel-enhanced classifiers that insert trainable amplitude-damping or phase-damping blocks into variational circuits report markedly faster and smoother optimization; on 4-qubit MNIST, phase-damping reduced the required optimization steps by about $J_{\mathcal{E}}\succeq 0$0 relative to the unitary baseline, and on the Electrical Grid Stability Simulated Dataset the accuracy gains were $J_{\mathcal{E}}\succeq 0$1 for phase damping and $J_{\mathcal{E}}\succeq 0$2 for amplitude damping over a 10-qubit unitary model (Wen et al., 14 Jun 2026). A complementary direction uses neural networks to learn quasi-inverse CPTP recovery maps for Pauli and amplitude-damping noise, reconstructing Kraus operators by process tomography and verifying completeness up to numerical tolerance $J_{\mathcal{E}}\succeq 0$3 (Aziz et al., 13 Jun 2025).

6. Conceptual clarifications, limitations, and open problems

A common misconception is that a trainable quantum channel must always mean that the physical noise process itself is optimized. The literature uses the term more broadly. In some frameworks, the channel is the learned object; this is the case for reduced-channel identification, quasi-inverse recovery, ORUC learning, and non-unitary variational layers (Cemin et al., 2024, Aziz et al., 13 Jun 2025, Smart et al., 28 Jan 2025, Wen et al., 14 Jun 2026). In others, the physical channel is fixed and the trainable objects are the encoder, decoder, or embedding around it, as in communication autoencoders and trainable discrete feature maps (Rathi et al., 2023, Thumwanit et al., 2021). A second misconception is that channels are merely nuisances. Several papers explicitly argue the opposite: non-unitary dynamics can enlarge the hypothesis space, modulate effective-observable spectra, and improve optimization (Wen et al., 14 Jun 2026).

The main technical limitations are structural. There are no known analytic criteria guaranteeing that a target unitary can be embedded with pairwise couplings and a specified ancilla budget in static-hardware teaching (Banchi et al., 2016). Learnability results for generalized Gaussian channels require fixed non-Gaussian combination coefficients and boundedness parameters $J_{\mathcal{E}}\succeq 0$4; learning arbitrary photodetection parameters remains hard because the pseudo-dimension scales as $J_{\mathcal{E}}\succeq 0$5 (Rosati, 2022). Steady-state recovery depends on non-unitality, locality, unique mixing, and negligible cross-talk; for unital channels, steady states tend toward the maximally mixed state and become weakly informative (Ilin et al., 2023). Stinespring-based NISQ and neutral-atom models are naturally tailored to Markovian or CP-divisible dynamics, and performance can degrade when memory effects become strong (Cemin et al., 2024, Visser et al., 2023).

A further divide concerns how physicality is enforced. Stinespring parameterizations, simplex-constrained mixed-unitary models, and interferometric system–environment constructions are CPTP by design (Cemin et al., 2024, Coyle et al., 2024, Araujo et al., 25 Feb 2025). Direct Bloch-vector neural maps instead rely on a physics-inspired loss during training and CPTP validation only after training through tomography and Kraus reconstruction (Aziz et al., 13 Jun 2025). This suggests an open methodological fault line between flexible surrogate parameterizations and hard physical constraints. Across the field, the principal unresolved questions are scalability to larger systems, robust learning under realistic decoherence and detector imperfections, extension beyond Markovianity, and the systematic design of channel ansätze that are simultaneously expressive, hardware-feasible, and trainable (Visser et al., 2023, Cemin et al., 2024, Röseler et al., 15 Mar 2026).