Choi-Shadow Estimators

• Choi-shadow estimators are methods that represent quantum channels as Choi-type states and use shadow tomography to estimate linear functionals directly.
• They recast process inference as state inference on a bipartite system, reducing computational overhead by bypassing full process matrix reconstruction.
• Variants such as branch-resolved, optimized POVM, and pseudo-Choi extend their applicability to dynamic teleportation, robust measurement, and Hamiltonian learning.

Searching arXiv for papers on Choi-shadow estimators and related Choi/shadow process tomography. arxiv_search(query="Choi shadow estimators process tomography Choi shadows", max_results=10) arxiv_search(query="shadow process tomography Choi isomorphism classical Choi shadows", max_results=10) Choi-shadow estimators are estimator constructions that encode a quantum process into a Choi-state, Choi operator, branch Choi operator, or pseudo-Choi state, and then apply shadow-tomography machinery to estimate target functionals without full process reconstruction. In the canonical channel setting, a CPTP map $E$ is represented by the Choi operator

$$\eta = (\mathcal I_A\otimes E_B)\big[\ket{\omega}\bra{\omega}\big],$$

with channel action recovered through

$$E(\rho)=\operatorname{Tr}_A\!\big[(\rho^T\otimes \mathbb I_B)\,\eta\big].$$

Consequently, any quantity of the form $\operatorname{Tr}[E(\rho)O]$ becomes the linear functional $\operatorname{Tr}[\eta(\rho^T\otimes O)]$, which can be estimated from shadow data rather than from a fully reconstructed process matrix (Kunjummen et al., 2021). Recent work extends this template to branch-resolved dynamic teleportation via classical Choi shadows (Edwards et al., 30 Apr 2026), to generalized-measurement shadow process tomography with optimized POVMs (Wang et al., 30 Jun 2025), and to Hamiltonian learning via pseudo-Choi states that play a Choi-like role for generators rather than channels (Castaneda et al., 2023).

1. Core representation and estimator principle

The defining move is the Choi reduction: process inference is recast as state inference on a larger bipartite system. In one standard normalization, the Choi-state of a channel $\mathcal N_A$ acting on a $d$-dimensional system is

$J_{\mathcal N_{AB}} = (\mathcal N_A \otimes \mathcal I_B)\!\left(\ketbra{\Phi^+}_{AB}\right), \qquad \ket{\Phi^+}_{AB}=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}\ket{ii}_{AB},$

and the inverse relation is

$$\mathcal N_A(\rho) = d\,\mathrm{tr}_B\!\left[(\mathbb I_A\otimes \rho^t)\,J_{\mathcal N_{AB}}\right].$$

The Choi object therefore fully encodes the channel, and positivity plus the marginal condition

$$J_{\mathcal N_{AB}} \ge 0,\qquad \mathrm{tr}_A[J_{\mathcal N_{AB}}]=\frac{\mathbb I_B}{d}$$

characterize CPTP maps (Stratton et al., 2024).

Taken together, the quantum-shadow papers suggest a common estimator architecture: encode the unknown process into a Choi-type object; measure that object through randomized or generalized measurements; construct unbiased or least-squares single-shot reconstructions; and evaluate linear observables on the reconstructed Choi samples. This shifts the computational burden from full tomography to observable-specific estimation, while preserving access to many channel properties from a common data stream (Kunjummen et al., 2021).

Framework	Encoded object	Distinguishing feature
Shadow process tomography (Kunjummen et al., 2021)	Choi operator $$\eta = (\mathcal I_A\otimes E_B)\big[\ket{\omega}\bra{\omega}\big],$$0	Direct channel-function estimation
Classical branch Choi shadows (Edwards et al., 30 Apr 2026)	Branch Choi operator $$\eta = (\mathcal I_A\otimes E_B)\big[\ket{\omega}\bra{\omega}\big],$$1	Branch-resolved dynamic-circuit analysis
POVM-SPT (Wang et al., 30 Jun 2025)	Normalized Choi state $$\eta = (\mathcal I_A\otimes E_B)\big[\ket{\omega}\bra{\omega}\big],$$2	Optimized POVMs minimize shadow norm
Pseudo-Choi shadows (Castaneda et al., 2023)	Pseudo-Choi state $$\eta = (\mathcal I_A\otimes E_B)\big[\ket{\omega}\bra{\omega}\big],$$3	Hamiltonian-coefficient estimation

2. Standard Choi-shadow construction for channels

The standard shadow process tomography protocol samples the Choi operator indirectly by randomized preparation and measurement on the input and output sides. For each shot, one samples $$\eta = (\mathcal I_A\otimes E_B)\big[\ket{\omega}\bra{\omega}\big],$$4, prepares $$\eta = (\mathcal I_A\otimes E_B)\big[\ket{\omega}\bra{\omega}\big],$$5, applies the channel, applies $$\eta = (\mathcal I_A\otimes E_B)\big[\ket{\omega}\bra{\omega}\big],$$6, measures $$\eta = (\mathcal I_A\otimes E_B)\big[\ket{\omega}\bra{\omega}\big],$$7, and records

$$\eta = (\mathcal I_A\otimes E_B)\big[\ket{\omega}\bra{\omega}\big],$$8

The corresponding Choi-shadow basis vector is

$$\eta = (\mathcal I_A\otimes E_B)\big[\ket{\omega}\bra{\omega}\big],$$9

With $$E(\rho)=\operatorname{Tr}_A\!\big[(\rho^T\otimes \mathbb I_B)\,\eta\big].$$0 the induced shadow measurement map on the input-output system, the single-shot Choi shadow is

$$E(\rho)=\operatorname{Tr}_A\!\big[(\rho^T\otimes \mathbb I_B)\,\eta\big].$$1

and repeated sampling yields $$E(\rho)=\operatorname{Tr}_A\!\big[(\rho^T\otimes \mathbb I_B)\,\eta\big].$$2 with $$E(\rho)=\operatorname{Tr}_A\!\big[(\rho^T\otimes \mathbb I_B)\,\eta\big].$$3 (Kunjummen et al., 2021).

Target functionals are then estimated linearly. For an input state $$E(\rho)=\operatorname{Tr}_A\!\big[(\rho^T\otimes \mathbb I_B)\,\eta\big].$$4 and output observable $$E(\rho)=\operatorname{Tr}_A\!\big[(\rho^T\otimes \mathbb I_B)\,\eta\big].$$5,

$$E(\rho)=\operatorname{Tr}_A\!\big[(\rho^T\otimes \mathbb I_B)\,\eta\big].$$6

so one single shadow sample induces

$$E(\rho)=\operatorname{Tr}_A\!\big[(\rho^T\otimes \mathbb I_B)\,\eta\big].$$7

The recommended aggregation is median-of-means: split the $$E(\rho)=\operatorname{Tr}_A\!\big[(\rho^T\otimes \mathbb I_B)\,\eta\big].$$8 samples into $$E(\rho)=\operatorname{Tr}_A\!\big[(\rho^T\otimes \mathbb I_B)\,\eta\big].$$9 groups of size $\operatorname{Tr}[E(\rho)O]$0, average within each group, and take the median. In the local-Pauli specialization, the inverse map is explicit,

$\operatorname{Tr}[E(\rho)O]$1

and the single-qubit estimator building block becomes

$\operatorname{Tr}[E(\rho)O]$2

A recurring practical point is that these estimators are not, in the first instance, reconstructed CPTP maps. They are shadows of the Choi operator from which channel properties are extracted linearly (Kunjummen et al., 2021).

3. Major variants of the estimator family

A branch-resolved variant was introduced for dynamic-circuit teleportation on superconducting hardware. There the mid-circuit measurement defines a quantum instrument

$\operatorname{Tr}[E(\rho)O]$3

with post-correction branch channel

$\operatorname{Tr}[E(\rho)O]$4

Each branch has a subnormalized branch Choi operator

$\operatorname{Tr}[E(\rho)O]$5

whose trace gives the branch probability $\operatorname{Tr}[E(\rho)O]$6. Using a two-qubit classical shadow over the reference-output pair $\operatorname{Tr}[E(\rho)O]$7, the branch-specific estimator is

$\operatorname{Tr}[E(\rho)O]$8

with $\operatorname{Tr}[E(\rho)O]$9 for unmitigated physical correction and post-processing, and $\operatorname{Tr}[\eta(\rho^T\otimes O)]$0 for PROM mitigation (Edwards et al., 30 Apr 2026).

A second major variant replaces random-unitary measurements by optimized generalized measurements. In POVM-SPT, a tensor-product POVM $\operatorname{Tr}[\eta(\rho^T\otimes O)]$1 is applied to the normalized Choi state $\operatorname{Tr}[\eta(\rho^T\otimes O)]$2, the least-squares reconstruction map is

$\operatorname{Tr}[\eta(\rho^T\otimes O)]$3

and single-shot Choi reconstructions are built as $\operatorname{Tr}[\eta(\rho^T\otimes O)]$4, with the process estimator

$\operatorname{Tr}[\eta(\rho^T\otimes O)]$5

The paper emphasizes that projection measurements are a special case of POVMs, so the optimized POVM cannot do worse than the best projective measurement in shadow-norm terms (Wang et al., 30 Jun 2025).

A third variant uses pseudo-Choi states for Hamiltonian learning. The pseudo-Choi state is defined on $\operatorname{Tr}[\eta(\rho^T\otimes O)]$6 by

$\operatorname{Tr}[\eta(\rho^T\otimes O)]$7

with $\operatorname{Tr}[\eta(\rho^T\otimes O)]$8. Coefficients are decoded through operators $\operatorname{Tr}[\eta(\rho^T\otimes O)]$9 and $\mathcal N_A$0 satisfying

$\mathcal N_A$1

so that

$\mathcal N_A$2

This preserves the same estimator logic—encode, shadow, decode—but shifts the target from channels to Hamiltonian parameters (Castaneda et al., 2023).

4. Accuracy, shadow norms, and complexity

The finite-sample guarantees are organized around shadow norms. For channel shadows, if $\mathcal N_A$3 are output operators and $\mathcal N_A$4 are input states, then

$\mathcal N_A$5

and

$\mathcal N_A$6

suffices for simultaneous estimation at error $\mathcal N_A$7 with probability at least $\mathcal N_A$8. The paper attributes the $\mathcal N_A$9 factor to the normalization convention for the unnormalized Choi operator, making it a process-specific overhead absent from ordinary normalized-state shadow estimation (Kunjummen et al., 2021).

In POVM-SPT, variance is controlled by

$d$0

and the worst-case norm over a family $d$1 is $d$2. The median-of-means sample complexity is

$d$3

more explicitly with

$d$4

The reported numerical gains are an approximate 7-fold reduction in the squared shadow norm for single-qubit input states and a reported $d$5-fold sample-complexity enhancement for 64-qubit input states relative to conventional SPT (Wang et al., 30 Jun 2025).

For pseudo-Choi Hamiltonian learning, the classical-shadow route yields query complexity

$d$6

while replacing classical shadows with quantum mean estimation improves this to

$d$7

The paper attributes the constant-observable behavior to bounded shadow norms for the decoding operators, stating $d$8 for Clifford shadows (Castaneda et al., 2023).

5. Operational uses and empirical behavior

The original channel-shadow framework was designed for estimating many targeted properties of large quantum channels rather than reconstructing the full process tensor. The paper gives explicit uses for transition probabilities, multitime correlation functions, channel concatenation, and the application of channel shadows to shadows of quantum states. It also identifies a sign problem in both concatenation and channel-on-state composition, because the induced coefficients become signed or quasiprobabilistic rather than strictly positive (Kunjummen et al., 2021).

Branch-resolved classical Choi shadows provide a concrete experimental demonstration of the value of preserving Choi information at the branch level. In dynamic teleportation experiments, physical correction, post-processing adjustments, and PROM-mitigated physical application were compared against full tomography of the branch Choi operators. At a shadow budget of 73,728 shots on layout 1, the reported RMSE values for the perfect $d$9 resource were $J_{\mathcal N_{AB}} = (\mathcal N_A \otimes \mathcal I_B)\!\left(\ketbra{\Phi^+}_{AB}\right), \qquad \ket{\Phi^+}_{AB}=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}\ket{ii}_{AB},$0 for physical correction and $J_{\mathcal N_{AB}} = (\mathcal N_A \otimes \mathcal I_B)\!\left(\ketbra{\Phi^+}_{AB}\right), \qquad \ket{\Phi^+}_{AB}=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}\ket{ii}_{AB},$1 for post-processing on the primary observable set, and $J_{\mathcal N_{AB}} = (\mathcal N_A \otimes \mathcal I_B)\!\left(\ketbra{\Phi^+}_{AB}\right), \qquad \ket{\Phi^+}_{AB}=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}\ket{ii}_{AB},$2 and $J_{\mathcal N_{AB}} = (\mathcal N_A \otimes \mathcal I_B)\!\left(\ketbra{\Phi^+}_{AB}\right), \qquad \ket{\Phi^+}_{AB}=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}\ket{ii}_{AB},$3 on the full observable family. The feed-forward penalty

$J_{\mathcal N_{AB}} = (\mathcal N_A \otimes \mathcal I_B)\!\left(\ketbra{\Phi^+}_{AB}\right), \qquad \ket{\Phi^+}_{AB}=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}\ket{ii}_{AB},$4

was $J_{\mathcal N_{AB}} = (\mathcal N_A \otimes \mathcal I_B)\!\left(\ketbra{\Phi^+}_{AB}\right), \qquad \ket{\Phi^+}_{AB}=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}\ket{ii}_{AB},$5 and $J_{\mathcal N_{AB}} = (\mathcal N_A \otimes \mathcal I_B)\!\left(\ketbra{\Phi^+}_{AB}\right), \qquad \ket{\Phi^+}_{AB}=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}\ket{ii}_{AB},$6 on layout 1, but $J_{\mathcal N_{AB}} = (\mathcal N_A \otimes \mathcal I_B)\!\left(\ketbra{\Phi^+}_{AB}\right), \qquad \ket{\Phi^+}_{AB}=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}\ket{ii}_{AB},$7 and $J_{\mathcal N_{AB}} = (\mathcal N_A \otimes \mathcal I_B)\!\left(\ketbra{\Phi^+}_{AB}\right), \qquad \ket{\Phi^+}_{AB}=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}\ket{ii}_{AB},$8 on layout 2 for the perfect and symmetric $J_{\mathcal N_{AB}} = (\mathcal N_A \otimes \mathcal I_B)\!\left(\ketbra{\Phi^+}_{AB}\right), \qquad \ket{\Phi^+}_{AB}=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}\ket{ii}_{AB},$9 resources, respectively. The paper reports a reversal in branch-quality ordering: PROM is best on the noisier-readout layout 1, whereas post-processing exceeds PROM for every branch on layout 2 (Edwards et al., 30 Apr 2026).

Pseudo-Choi shadows also support robustness statements beyond ordinary tomography. When the true Hamiltonian contains additional orthogonal terms outside the modeled class, the known coefficients remain estimable and the residual normalization can reveal the missing component through

$$\mathcal N_A(\rho) = d\,\mathrm{tr}_B\!\left[(\mathbb I_A\otimes \rho^t)\,J_{\mathcal N_{AB}}\right].$$0

For noisy resource states $$\mathcal N_A(\rho) = d\,\mathrm{tr}_B\!\left[(\mathbb I_A\otimes \rho^t)\,J_{\mathcal N_{AB}}\right].$$1, the paper states

$$\mathcal N_A(\rho) = d\,\mathrm{tr}_B\!\left[(\mathbb I_A\otimes \rho^t)\,J_{\mathcal N_{AB}}\right].$$2

so preparation noise adds an explicit additive error term (Castaneda et al., 2023).

6. Boundaries of the term and adjacent literatures

A common source of confusion is terminological. Not every result involving a Choi object and not every use of the word “shadow” belongs to the same estimator family. The operational interpretation of Choi rank, for example, is not an estimator construction but a channel property: if the encoded states in an entanglement-assisted exclusion task all have rank equal to the Choi rank $$\mathcal N_A(\rho) = d\,\mathrm{tr}_B\!\left[(\mathbb I_A\otimes \rho^t)\,J_{\mathcal N_{AB}}\right].$$3, then

$$\mathcal N_A(\rho) = d\,\mathrm{tr}_B\!\left[(\mathbb I_A\otimes \rho^t)\,J_{\mathcal N_{AB}}\right].$$4

This gives a universal exclusion bound, not a shadow estimator (Stratton et al., 2024).

Likewise, infinite-dimensional analogues of Choi matrices generalize the representation theory of maps on von Neumann factors rather than shadow tomography. For suitable normal completely bounded maps $$\mathcal N_A(\rho) = d\,\mathrm{tr}_B\!\left[(\mathbb I_A\otimes \rho^t)\,J_{\mathcal N_{AB}}\right].$$5, the paper defines two Choi-like objects,

$$\mathcal N_A(\rho) = d\,\mathrm{tr}_B\!\left[(\mathbb I_A\otimes \rho^t)\,J_{\mathcal N_{AB}}\right].$$6

and proves that $$\mathcal N_A(\rho) = d\,\mathrm{tr}_B\!\left[(\mathbb I_A\otimes \rho^t)\,J_{\mathcal N_{AB}}\right].$$7 or $$\mathcal N_A(\rho) = d\,\mathrm{tr}_B\!\left[(\mathbb I_A\otimes \rho^t)\,J_{\mathcal N_{AB}}\right].$$8 is equivalent to complete positivity of $$\mathcal N_A(\rho) = d\,\mathrm{tr}_B\!\left[(\mathbb I_A\otimes \rho^t)\,J_{\mathcal N_{AB}}\right].$$9. It also shows that universal existence of these correspondences for all normal completely bounded maps holds if and only if the factor is of type I (Han et al., 2023). This suggests a broader “Choi-object as compressed witness” perspective, but not a direct shadow-tomography protocol.

Outside quantum information, the term “shadow” appears in unrelated estimator traditions. A semiparametric DID paper develops a shadow-variable-based estimator for the ATT under MNAR post-treatment outcome missingness, using a fully observed variable $$J_{\mathcal N_{AB}} \ge 0,\qquad \mathrm{tr}_A[J_{\mathcal N_{AB}}]=\frac{\mathbb I_B}{d}$$0, an odds-ratio model, and stacked estimating equations (Li et al., 7 Jun 2026). A separate model-uncertainty paper studies constrained M-estimation with Lagrangian shadow prices and individual shadow prices,

$$J_{\mathcal N_{AB}} \ge 0,\qquad \mathrm{tr}_A[J_{\mathcal N_{AB}}]=\frac{\mathbb I_B}{d}$$1

to quantify the empirical relevance of candidate restrictions (Lee et al., 16 Apr 2026). These are methodologically distinct from Choi-shadow estimators in the quantum-process sense.

In the narrow technical sense established by the process-tomography literature, Choi-shadow estimators are therefore best understood as estimators that combine a Choi-type encoding with shadow-style measurement and reconstruction. Their common pattern is: represent the process by a state-like object, estimate linear functionals of that object efficiently, and trade full process recovery for scalable access to targeted observables and operational diagnostics (Kunjummen et al., 2021).