Quantum Channel Interpolation

• Quantum channel interpolation is a framework for constructing CPTP maps from partial specifications, convex mixtures, and dynamical synthesis.
• Advanced methods use Choi matrix constraints, Kraus representations, and semidefinite programming to optimize interpolation accuracy and enforce structural properties.
• Experimental implementations on platforms like neutral atoms and superconducting circuits demonstrate high fidelities and low errors, underscoring its practical significance.

Quantum channel interpolation denotes a family of constructions for producing a controlled set of quantum channels from partial specifications, component channels, or discrete-time observations. In the literature, the term is used in several technically distinct senses: exact or approximate interpolation constraints of the form $\Phi(X_i)=Y_i$ in the Choi representation; convex interpolation of channels through ensembles such as $\mathcal{E}=\sum_i p_i\mathcal{E}_i$; dynamical interpolation and extrapolation obtained by learning or synthesizing a short-time CPTP step and composing it in time; and parameterized interpolation paths used to generate non-Hermitian degeneracies of channel superoperators. A broader, non-equivalent usage blends solutions of the von Neumann equation and a classical master equation by intervention on the evolved solutions; that construction preserves purity but is generally nonlinear in $\rho$ and therefore does not define a linear CPTP quantum channel in the usual sense (Visser et al., 2023, Roy et al., 2021, Pirandola et al., 2018, Hu et al., 2018, Wong et al., 21 Jul 2025, Kadowaki, 2017).

1. Mathematical setting and core representations

A quantum channel is a linear, completely positive and trace-preserving map $$\Phi:B(\mathcal{H}_{\mathrm{in}})\to B(\mathcal{H}_{\mathrm{out}})$$, equivalently representable by Kraus operators, a Choi matrix, a Stinespring dilation, or a superoperator/PTM. The Kraus form is

$$\Phi(X)=\sum_j V_j X V_j^\dagger,\qquad \sum_j V_j^\dagger V_j=I_{\mathrm{in}},$$

while the Choi matrix $J_\Phi$ satisfies $J_\Phi\succeq 0$ and $$\operatorname{Tr}_{\mathrm{out}}J_\Phi=I_{\mathrm{in}}$$. In the Choi picture, the interpolation constraints are linear: $$\operatorname{Tr}_{\mathrm{in}}\!\big[(I_{\mathrm{out}}\otimes X_i^{\mathsf T})J_\Phi\big]=Y_i.$$ This is the basis for exact and approximate interpolation as a feasibility or optimization problem over the convex set of CPTP maps (Roy et al., 2021).

For open-system dynamics, interpolation is also tied to continuous-time evolution. In the Markovian, time-homogeneous case, the GKSL equation

$\frac{d\rho}{dt}=\mathcal{L}(\rho)$

generates a semigroup $\mathcal{E}=\sum_i p_i\mathcal{E}_i$0 with $\mathcal{E}=\sum_i p_i\mathcal{E}_i$1. The condition of CP-divisibility, namely the existence of a CPTP map $\mathcal{E}=\sum_i p_i\mathcal{E}_i$2 such that $\mathcal{E}=\sum_i p_i\mathcal{E}_i$3, is the structural condition that underpins interpolation to unseen $\mathcal{E}=\sum_i p_i\mathcal{E}_i$4 and extrapolation beyond a training window; it is exactly satisfied for time-homogeneous Lindblad dynamics (Visser et al., 2023).

For single-qubit channels, the PTM or affine Bloch representation is especially useful. In that representation, the nontrivial spectral data are carried by a real $\mathcal{E}=\sum_i p_i\mathcal{E}_i$5 distortion matrix $\mathcal{E}=\sum_i p_i\mathcal{E}_i$6, while the TP condition fixes the first row of the full $\mathcal{E}=\sum_i p_i\mathcal{E}_i$7 superoperator. This representation is central both to generator extraction for continuous interpolation and to spectral phase classifications used in exceptional-point constructions (Wong et al., 21 Jul 2025).

2. Interpolation as a conic or semidefinite program

In the operator interpolation problem, one is given Hermitian pairs $\mathcal{E}=\sum_i p_i\mathcal{E}_i$8 and seeks a CPTP map $\mathcal{E}=\sum_i p_i\mathcal{E}_i$9 satisfying $\rho$0 for all $\rho$1. The Choi formulation turns this into a finite-dimensional conic feasibility problem: $\rho$2, $\rho$3, and the linear constraints above. Approximate interpolation introduces residual-slack matrices $\rho$4 with

$\rho$5

and minimizes $\rho$6, optionally together with a bounded trace-preservation slack. In the notation of the source, feasibility with zero objective recovers exact interpolation by a CPTP map; otherwise the objective quantifies the interpolation error via the slack decomposition (Roy et al., 2021).

The same framework supports structural constraints on the channel class by restricting the feasible Choi variable to a cone or affine slice. The paper develops explicit conic programs for entanglement-breaking, random unitary, and degradable channels. For entanglement-breaking channels, the Choi matrix must lie in a separable PSD cone; for random unitary channels, it must lie in the convex hull of unitary Choi matrices; for degradable channels, approximate degradability is encoded by coupling the interpolation constraints to Watrous’ SDP for the diamond norm. The general abstraction is a “semi-SDP” over a cone $\rho$7 associated with a channel property $\rho$8, and Theorem 6.3 states the corresponding interpolation program for arbitrary convex channel classes (Roy et al., 2021).

A particularly explicit existence result concerns entanglement-breaking interpolation for orthogonal data. If $\rho$9 is an orthogonal set of positive matrices with the identity matrix in $$\Phi:B(\mathcal{H}_{\mathrm{in}})\to B(\mathcal{H}_{\mathrm{out}})$$0, and $$\Phi:B(\mathcal{H}_{\mathrm{in}})\to B(\mathcal{H}_{\mathrm{out}})$$1 is another set of positive matrices, then the condition $$\Phi:B(\mathcal{H}_{\mathrm{in}})\to B(\mathcal{H}_{\mathrm{out}})$$2 for each $$\Phi:B(\mathcal{H}_{\mathrm{in}})\to B(\mathcal{H}_{\mathrm{out}})$$3 is equivalent to the existence of an entanglement-breaking trace-preserving map $$\Phi:B(\mathcal{H}_{\mathrm{in}})\to B(\mathcal{H}_{\mathrm{out}})$$4 such that $$\Phi:B(\mathcal{H}_{\mathrm{in}})\to B(\mathcal{H}_{\mathrm{out}})$$5 for all $$\Phi:B(\mathcal{H}_{\mathrm{in}})\to B(\mathcal{H}_{\mathrm{out}})$$6. The construction proceeds through rank-one Kraus operators and therefore yields a measure-and-prepare channel (Roy et al., 2021).

3. Convex interpolation by conditional simulation

A second major meaning of quantum channel interpolation is convex mixing of channels, implemented operationally through a control system. For a discrete ensemble $$\Phi:B(\mathcal{H}_{\mathrm{in}})\to B(\mathcal{H}_{\mathrm{out}})$$7, the average channel

$$\Phi:B(\mathcal{H}_{\mathrm{in}})\to B(\mathcal{H}_{\mathrm{out}})$$8

has Choi mixture $$\Phi:B(\mathcal{H}_{\mathrm{in}})\to B(\mathcal{H}_{\mathrm{out}})$$9. If each component admits an LOCC simulation

$$\Phi(X)=\sum_j V_j X V_j^\dagger,\qquad \sum_j V_j^\dagger V_j=I_{\mathrm{in}},$$0

then one introduces a classical control register

$$\Phi(X)=\sum_j V_j X V_j^\dagger,\qquad \sum_j V_j^\dagger V_j=I_{\mathrm{in}},$$1

a control–program state

$$\Phi(X)=\sum_j V_j X V_j^\dagger,\qquad \sum_j V_j^\dagger V_j=I_{\mathrm{in}},$$2

and a controlled LOCC $$\Phi(X)=\sum_j V_j X V_j^\dagger,\qquad \sum_j V_j^\dagger V_j=I_{\mathrm{in}},$$3 such that

$$\Phi(X)=\sum_j V_j X V_j^\dagger,\qquad \sum_j V_j^\dagger V_j=I_{\mathrm{in}},$$4

This conditional simulation identity holds without requiring joint teleportation covariance of the component ensemble; joint teleportation covariance is sufficient but not necessary (Pirandola et al., 2018).

The construction extends to asymptotic simulations, continuous ensembles, and memory channels. In the continuous case, $$\Phi(X)=\sum_j V_j X V_j^\dagger,\qquad \sum_j V_j^\dagger V_j=I_{\mathrm{in}},$$5 is replaced by $$\Phi(X)=\sum_j V_j X V_j^\dagger,\qquad \sum_j V_j^\dagger V_j=I_{\mathrm{in}},$$6 throughout, producing an operational interpolation for $$\Phi(X)=\sum_j V_j X V_j^\dagger,\qquad \sum_j V_j^\dagger V_j=I_{\mathrm{in}},$$7. In memory settings, a multi-index control state $$\Phi(X)=\sum_j V_j X V_j^\dagger,\qquad \sum_j V_j^\dagger V_j=I_{\mathrm{in}},$$8 tracks classically correlated component choices across channel uses. The same control-based architecture therefore interpolates not only between finitely many channels but also over continuous parameter families and multi-use blocks with classical memory (Pirandola et al., 2018).

The significance of conditional simulation is not limited to implementation. Because it supports teleportation stretching of adaptive protocols, it leads to converse bounds for two-way assisted capacities. For the average channel, the stretched form $$\Phi(X)=\sum_j V_j X V_j^\dagger,\qquad \sum_j V_j^\dagger V_j=I_{\mathrm{in}},$$9 implies

$J_\Phi$0

and hence

$J_\Phi$1

with continuous-ensemble analogues obtained by replacing sums by integrals (Pirandola et al., 2018).

4. Dynamical interpolation and extrapolation from repeated CPTP steps

A third setting concerns time interpolation and extrapolation from a learned or synthesized short-time channel. In the variational Stinespring framework, a single-step map is realized by preparing an ancilla in $J_\Phi$2, applying a parameterized joint unitary $J_\Phi$3 on system and ancilla, and tracing out the ancilla: $J_\Phi$4 Because the map is implemented through a Stinespring unitary, it is CPTP by construction for all $J_\Phi$5. Training uses observable data at discrete times $J_\Phi$6 and minimizes either a single-step or multistep loss on expectation values, typically for Pauli-string observables. When the dynamics are time-homogeneous, extrapolation is performed by channel powers,

$J_\Phi$7

implemented physically by reapplying the same learned dilation with fresh ancillas at each step. Continuous interpolation is optional: from a reconstructed superoperator $J_\Phi$8 one defines $J_\Phi$9 and then $J_\Phi\succeq 0$0, with branch-cut stabilization or a GKSL-constrained fit when needed. Theoretical guarantees include CPTP preservation under composition and the extrapolation bound $J_\Phi\succeq 0$1 for step error $J_\Phi\succeq 0$2 (Visser et al., 2023).

Neutral-atom hardware is singled out because entangled ancillas can be spatially transported. The implementation exploits long-lived ground-manifold storage for ancillas, coherent tweezer transport, and the fact that only the active system plus one ancilla set are actuated per step. By Stinespring, $J_\Phi\succeq 0$3, and in the experiments a one-qubit channel used two ancillas while a two-qubit channel used three ancillas. Reported predictive errors were small: for one-qubit decay with Rabi drive, training on two steps with $J_\Phi\succeq 0$4 and two ancillas gave average Bures error at $J_\Phi\succeq 0$5, growing to $J_\Phi\succeq 0$6 after $J_\Phi\succeq 0$7 reapplications; for a two-qubit decay task with $J_\Phi\succeq 0$8 and three ancillas, the average Bures error at $J_\Phi\succeq 0$9 was $$\operatorname{Tr}_{\mathrm{out}}J_\Phi=I_{\mathrm{in}}$$0; and for a two-qubit TFIM with decay it was $$\operatorname{Tr}_{\mathrm{out}}J_\Phi=I_{\mathrm{in}}$$1. Under equalized “equivalent evolutions,” pulse-based and stochastic-gate training outperformed plain gate-based training (Visser et al., 2023).

A related experimental route synthesizes arbitrary single-qubit channels with one ancilla qubit and measurement-based adaptive control, then obtains a continuous CPTP path by repetition. In that platform, any single-qubit channel is implemented as a deterministic convex combination of two quasiextreme channels, and repeated application of a small-step channel realizes

$$\operatorname{Tr}_{\mathrm{out}}J_\Phi=I_{\mathrm{in}}$$2

For dephasing,

$$\operatorname{Tr}_{\mathrm{out}}J_\Phi=I_{\mathrm{in}}$$3

and for amplitude damping,

$$\operatorname{Tr}_{\mathrm{out}}J_\Phi=I_{\mathrm{in}}$$4

The experiment reported $$\operatorname{Tr}_{\mathrm{out}}J_\Phi=I_{\mathrm{in}}$$5, $$\operatorname{Tr}_{\mathrm{out}}J_\Phi=I_{\mathrm{in}}$$6 for the storage cavity, ancilla coherence times $$\operatorname{Tr}_{\mathrm{out}}J_\Phi=I_{\mathrm{in}}$$7 and $$\operatorname{Tr}_{\mathrm{out}}J_\Phi=I_{\mathrm{in}}$$8, and arbitrary-channel tests with worst-case state-generation fidelity averaged $$\operatorname{Tr}_{\mathrm{out}}J_\Phi=I_{\mathrm{in}}$$9 at $$\operatorname{Tr}_{\mathrm{in}}\!\big[(I_{\mathrm{out}}\otimes X_i^{\mathsf T})J_\Phi\big]=Y_i.$$0 across six random target channels; the average diamond distance at $$\operatorname{Tr}_{\mathrm{in}}\!\big[(I_{\mathrm{out}}\otimes X_i^{\mathsf T})J_\Phi\big]=Y_i.$$1 was approximately $$\operatorname{Tr}_{\mathrm{in}}\!\big[(I_{\mathrm{out}}\otimes X_i^{\mathsf T})J_\Phi\big]=Y_i.$$2 (Hu et al., 2018).

Platform and task	Construction	Reported result
Neutral atom, 1-qubit decay	Learned Stinespring step composed with fresh ancillas	Average Bures error $$\operatorname{Tr}_{\mathrm{in}}\!\big[(I_{\mathrm{out}}\otimes X_i^{\mathsf T})J_\Phi\big]=Y_i.$$3 at $$\operatorname{Tr}_{\mathrm{in}}\!\big[(I_{\mathrm{out}}\otimes X_i^{\mathsf T})J_\Phi\big]=Y_i.$$4, $$\operatorname{Tr}_{\mathrm{in}}\!\big[(I_{\mathrm{out}}\otimes X_i^{\mathsf T})J_\Phi\big]=Y_i.$$5 after 9 reapplications
Neutral atom, 2-qubit decay	Same variational dilation framework	Average Bures error at $$\operatorname{Tr}_{\mathrm{in}}\!\big[(I_{\mathrm{out}}\otimes X_i^{\mathsf T})J_\Phi\big]=Y_i.$$6
Superconducting circuit, arbitrary single-qubit channels	One ancilla plus adaptive control, repeated channel simulation	Worst-case state-generation fidelity averaged $$\operatorname{Tr}_{\mathrm{in}}\!\big[(I_{\mathrm{out}}\otimes X_i^{\mathsf T})J_\Phi\big]=Y_i.$$7 at $$\operatorname{Tr}_{\mathrm{in}}\!\big[(I_{\mathrm{out}}\otimes X_i^{\mathsf T})J_\Phi\big]=Y_i.$$8; average diamond distance $$\operatorname{Tr}_{\mathrm{in}}\!\big[(I_{\mathrm{out}}\otimes X_i^{\mathsf T})J_\Phi\big]=Y_i.$$9 at $\frac{d\rho}{dt}=\mathcal{L}(\rho)$0

5. Interpolated channels and exceptional points

Quantum channel interpolation has also been used to generate exceptional points directly at the level of CPTP maps. For a single qubit in PTM form, the nontrivial spectrum is that of the real $\frac{d\rho}{dt}=\mathcal{L}(\rho)$1 distortion matrix $\frac{d\rho}{dt}=\mathcal{L}(\rho)$2. Because $\frac{d\rho}{dt}=\mathcal{L}(\rho)$3 is real, its eigenvalues are either all real or consist of one real eigenvalue plus a complex conjugate pair. This yields a phase classification: a $\frac{d\rho}{dt}=\mathcal{L}(\rho)$4-exact phase, in which all eigenvalues and eigenvectors are real, and a $\frac{d\rho}{dt}=\mathcal{L}(\rho)$5-broken phase, in which a complex conjugate pair appears. The transition between these phases occurs where eigenvalues coalesce, and when the corresponding eigenvectors also coalesce the transition point is an exceptional point (Wong et al., 21 Jul 2025).

The interpolation itself is the convex channel mixture

$\frac{d\rho}{dt}=\mathcal{L}(\rho)$6

which is CPTP for all $\frac{d\rho}{dt}=\mathcal{L}(\rho)$7 because convex combinations of CPTP maps remain CPTP. In the explicit two-channel construction, the spectrum of the interpolated distortion matrix contains a pair $\frac{d\rho}{dt}=\mathcal{L}(\rho)$8 that coalesce at

$\frac{d\rho}{dt}=\mathcal{L}(\rho)$9

and the corresponding eigenvectors also coalesce there, establishing a second-order exceptional point. The construction extends to three channels,

$\mathcal{E}=\sum_i p_i\mathcal{E}_i$00

where the reported convergence of EP2 lines produces an EP3 at

$\mathcal{E}=\sum_i p_i\mathcal{E}_i$01

The paper interprets these as non-Markovian channel exceptional points because the interpolated maps need not admit a time-local GKSL generator with nonnegative rates for all intermediate parameter values (Wong et al., 21 Jul 2025).

The experimental implementation used a two-qubit NMR quantum computer. Rather than a full Stinespring dilation for arbitrary single-qubit channels, the channel was decomposed as

$\mathcal{E}=\sum_i p_i\mathcal{E}_i$02

with each component implemented using one ancilla qubit through a circuit containing two $\mathcal{E}=\sum_i p_i\mathcal{E}_i$03 gates, two $\mathcal{E}=\sum_i p_i\mathcal{E}_i$04 gates, and two CNOTs. Quantum process tomography over the Pauli eigenstates $\mathcal{E}=\sum_i p_i\mathcal{E}_i$05 reconstructed the channel by a maximum-likelihood CPTP fit. The reconstructed channel at $\mathcal{E}=\sum_i p_i\mathcal{E}_i$06 achieved process fidelity $\mathcal{E}=\sum_i p_i\mathcal{E}_i$07, the fidelity stayed above $\mathcal{E}=\sum_i p_i\mathcal{E}_i$08 across the interpolation range $\mathcal{E}=\sum_i p_i\mathcal{E}_i$09, and the nontrivial eigenvalues of the reconstructed superoperator coalesced at $\mathcal{E}=\sum_i p_i\mathcal{E}_i$10 in agreement with theory (Wong et al., 21 Jul 2025).

6. Distinct usages, limitations, and common points of confusion

The cited literature does not use “quantum channel interpolation” for a single canonical procedure. One line of work means exact or approximate operator matching under CPTP constraints; another means convex interpolation of channels through control-assisted simulation; another means interpolation in physical time by learning or synthesizing a small-step map and composing it; and another means convexly interpolating channels to traverse spectral phase boundaries of the superoperator. A related but technically different usage mixes coherent and classical dynamics by intervening on their infinitesimal solutions. In that scheme, one evolves $\mathcal{E}=\sum_i p_i\mathcal{E}_i$11 by the von Neumann equation and a population vector $\mathcal{E}=\sum_i p_i\mathcal{E}_i$12 by a classical master equation, then replaces amplitudes by

$\mathcal{E}=\sum_i p_i\mathcal{E}_i$13

retaining phases from the quantum evolution. The source states explicitly that the resulting evolution map on $\mathcal{E}=\sum_i p_i\mathcal{E}_i$14 is nonlinear and therefore does not define a linear CPTP quantum channel in general, even though it always produces a valid rank-one density matrix and reduces, in a weak-coupling/asymptotic regime for a two-level system, to equations equivalent to the optical Bloch equations (Kadowaki, 2017).

Several limitations recur across the channel-based formulations. Matrix-logarithm interpolation can be ill-conditioned near the negative real axis or for nearly defective superoperators, and even a stable branch of $\mathcal{E}=\sum_i p_i\mathcal{E}_i$15 may fail to produce a Lindblad generator when the true dynamics are not CP-divisible; the recommended alternatives are GKSL-constrained fitting, piecewise generators, or direct discrete-time powers (Visser et al., 2023). Exact enforcement of entanglement-breaking structure through separable-cone membership is NP-hard, so PPT conditions are used as tractable relaxations in practice (Roy et al., 2021). Conditional simulation removes the need for joint teleportation covariance, but resource overhead grows with the number of control flags, and continuous ensembles require idealized orthogonal flags (Pirandola et al., 2018). Convex interpolation preserves CPTP only for $\mathcal{E}=\sum_i p_i\mathcal{E}_i$16 with $\mathcal{E}=\sum_i p_i\mathcal{E}_i$17; nonconvex interpolations can break complete positivity or trace preservation (Wong et al., 21 Jul 2025). In repeated-step learning, faithful long-horizon prediction becomes challenging when the underlying process is non-Markovian or data are sparse, and ancilla and parameter counts grow quickly with system size (Visser et al., 2023).

Despite these differences, the constructions share a common aim: to parameterize admissible open-system transformations while retaining physically meaningful constraints. In the exact-feasibility setting the constraint is membership in a prescribed convex channel class; in conditional simulation it is LOCC realizability of channel mixtures; in repeated-step dynamics it is CPTP preservation under composition; and in exceptional-point constructions it is a CPTP path through superoperator phase space. This suggests that “quantum channel interpolation” is best understood as a family of CPTP-preserving parameterization strategies rather than a single formalism.