arXiv:2606.13638

Optimal classical shadow estimation of unitary channels at Heisenberg limit

Entong He, Zihao Li, Noam Scully, Sisi Zhou, Yuxiang Yang

quant-phSubmitted Jun 11, 2026• Announced Jun 12, 2026• 0 citations

Abstract

Full tomography of an unknown quantum evolution is resource-intensive and often unnecessary when the goal is only to predict selected properties. This motivates the study of classical shadow estimation of unitary channels (CSEU), a task in which one queries an unknown $d$-dimensional unitary $U$ and stores classical data that can later be used to predict expectation values $\mathrm{tr}[O \cdot UρU^\dagger]$ up to additive error $\varepsilon$ for arbitrary input states $ρ$ and observables $O$. We propose a parallel, non-adaptive CSEU protocol using $\mathcal{O}(d\varepsilon^{-1})$ queries when the input states or observables have constant rank. This achieves Heisenberg scaling with respect to $\varepsilon$ and is query-optimal, as we prove a matching $Ω(d\varepsilon^{-1})$ lower bound that remains valid even with stronger access to the unknown unitary. Our query-optimal CSEU protocol provides a versatile and powerful tool for quantum learning theory, pushing the performance limits of several fundamental learning tasks, including unitary channel tomography, Hamiltonian learning, boundary-regime quantum channel tomography, Pauli transfer matrix learning, inverse-free amplitude estimation, pure-state property estimation, and shallow-circuit learning. Remarkably, we show that optimal unitary channel tomography can be achieved using only parallel queries, closing the gap between the best achievable efficiency of parallel and sequential tomography protocols. Together, these applications establish our framework as a fundamental tool for learning properties of quantum processes, particularly for certain key tasks that require high precision.

Comments: 27+59 pages, 10 figures, 2 tables

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arXiv:2606.13638

Optimal classical shadow estimation of unitary channels at Heisenberg limit

Entong He, Zihao Li, Noam Scully, Sisi Zhou, Yuxiang Yang

quant-phSubmitted Jun 11, 2026• Announced Jun 12, 2026• 0 citations

arXiv HTML PDF

Abstract

d

-dimensional unitary

U

and stores classical data that can later be used to predict expectation values

\mathrm{tr}[O \cdot UρU^\dagger]

up to additive error

\varepsilon

for arbitrary input states

ρ

and observables

O

. We propose a parallel, non-adaptive CSEU protocol using

\mathcal{O}(d\varepsilon^{-1})

queries when the input states or observables have constant rank. This achieves Heisenberg scaling with respect to

\varepsilon

and is query-optimal, as we prove a matching

Ω(d\varepsilon^{-1})

lower bound that remains valid even with stronger access to the unknown unitary. Our query-optimal CSEU protocol provides a versatile and powerful tool for quantum learning theory, pushing the performance limits of several fundamental learning tasks, including unitary channel tomography, Hamiltonian learning, boundary-regime quantum channel tomography, Pauli transfer matrix learning, inverse-free amplitude estimation, pure-state property estimation, and shallow-circuit learning. Remarkably, we show that optimal unitary channel tomography can be achieved using only parallel queries, closing the gap between the best achievable efficiency of parallel and sequential tomography protocols. Together, these applications establish our framework as a fundamental tool for learning properties of quantum processes, particularly for certain key tasks that require high precision.

Comments: 27+59 pages, 10 figures, 2 tables

Related papers

Nearly Query-Optimal Classical Shadow Estimation of Unitary Channels

Zihao Li, Changhao Yi, You Zhou +1 more

Classical shadow estimation (CSE) is a powerful tool for learning the properties of quantum states and quantum processes. Here we consider the CSE task for quantum unitary channels. By querying an unknown unitary channel U multiple times in quantum experiments, the goal is to learn a classical description from which one can accurately predict many different linear properties of the channel, i.e., the expectation values of arbitrary observables measured on the output of U upon arbitrary input states. Based on collective measurements on multiple systems, we propose a query-efficient protocol for this task, whose query complexity has a quadratic advantage over the previous best approach for this problem, and almost saturates the information-theoretic lower bound. To further enhance practicality, we also present a variant protocol using only single-copy measurements, which still offers much better query performance than previous protocols that do not use quantum memory, and can serve as a key subroutine for learning an arbitrary unknown Hamiltonian from dynamics. In addition to linear properties of unitary channels, our protocol can also be applied to simultaneously predict many nonlinear properties, such as out-of-time-ordered correlators.Classical shadow estimation (CSE) is a powerful tool for learning the properties of quantum states and quantum processes. Here we consider the CSE task for quantum unitary channels. By querying an unknown unitary channel U multiple times in quantum experiments, the goal is to learn a classical description from which one can accurately predict many different linear properties of the channel, i.e., the expectation values of arbitrary observables measured on the output of U upon arbitrary input states. Based on collective measurements on multiple systems, we propose a query-efficient protocol for this task, whose query complexity has a quadratic advantage over the previous best approach for this problem, and almost saturates the information-theoretic lower bound. To further enhance practicality, we also present a variant protocol using only single-copy measurements, which still offers much better query performance than previous protocols that do not use quantum memory, and can serve as a key subroutine for learning an arbitrary unknown Hamiltonian from dynamics. In addition to linear properties of unitary channels, our protocol can also be applied to simultaneously predict many nonlinear properties, such as out-of-time-ordered correlators.

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Query-optimal unitary channel tomography in diamond distance with parallel access

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In the study of quantum process tomography, it has remained open how to learn a unitary channel in diamond distance with strictly parallel queries as efficiently as with sequential queries. Sequential queries allow one to exploit adaptivity to refine the estimate and adjust the learning strategy accordingly, whereas such adjustments are impossible for parallel queries. From the perspective of higher-order quantum operations, any parallel learning strategy can be simulated by a sequential one, whereas the converse does not hold in general, suggesting that sequential learning strategies are potentially more powerful. Contrary to this intuition, we present a new protocol for learning an unknown $d$ -dimensional unitary channel to within $\varepsilon$ in diamond distance using $\mathcal{O}(d^2/\varepsilon)$ parallel queries, matching the lower bound presented in [Haah, Kothari, O'Donnell, and Tang, FOCS '23]. Our protocol thus closes the gap between parallel and sequential strategies for unitary channel tomography. As applications, it achieves optimal query complexity for boundary-regime quantum channel tomography and improves the query efficiency of the best-known protocol for tomography of fermionic linear optics.In the study of quantum process tomography, it has remained open how to learn a unitary channel in diamond distance with strictly parallel queries as efficiently as with sequential queries. Sequential queries allow one to exploit adaptivity to refine the estimate and adjust the learning strategy accordingly, whereas such adjustments are impossible for parallel queries. From the perspective of higher-order quantum operations, any parallel learning strategy can be simulated by a sequential one, whereas the converse does not hold in general, suggesting that sequential learning strategies are potentially more powerful. Contrary to this intuition, we present a new protocol for learning an unknown $d$-dimensional unitary channel to within $\varepsilon$ in diamond distance using $\mathcal{O}(d^2/\varepsilon)$ parallel queries, matching the lower bound presented in [Haah, Kothari, O'Donnell, and Tang, FOCS '23]. Our protocol thus closes the gap between parallel and sequential strategies for unitary channel tomography. As applications, it achieves optimal query complexity for boundary-regime quantum channel tomography and improves the query efficiency of the best-known protocol for tomography of fermionic linear optics.

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Structure, optimality, and symmetry in shadow unitary inversion

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Reversing unknown quantum dynamics is vital for quantum control and learning, yet full unitary inversion requires a resource-intensive O(d2) queries. Because many applications only require the reversed evolution for a given observable, a critical gap remains in understanding the minimal resources for such targeted reversal. Here, we address this by introducing shadow unitary inversion. We establish a lower bound showing the query complexity must scale at least linearly with system dimension for spectrally biased observables, with the constant determined by the observable’s spectral properties. For qubit case, we construct an explicit, deterministic three-query sequential protocol achieving exact shadow inversion and completely characterize all admissible channels, with numerical evidence suggesting optimality. For higher dimensions, we develop a semidefinite-programming formulation and introduce a representation-theoretic symmetry reduction that decomposes the optimization into invariant blocks, substantially reducing the problem size. Shadow unitary inversion thus offers a resource-efficient path to inverse-dynamics estimation for future quantum control, diagnostics, and learning tasks. Reversing unitary operations is crucial in quantum computing and control, yet remains a challenging task. The authors introduce shadow unitary inversion, achieving efficient inversion at the expectation value level with reduced query complexity, offering a promising approach for quantum error correction and information recovery.Reversing unknown quantum dynamics is vital for quantum control and learning, yet full unitary inversion requires a resource-intensive O(d2) queries. Because many applications only require the reversed evolution for a given observable, a critical gap remains in understanding the minimal resources for such targeted reversal. Here, we address this by introducing shadow unitary inversion. We establish a lower bound showing the query complexity must scale at least linearly with system dimension for spectrally biased observables, with the constant determined by the observable’s spectral properties. For qubit case, we construct an explicit, deterministic three-query sequential protocol achieving exact shadow inversion and completely characterize all admissible channels, with numerical evidence suggesting optimality. For higher dimensions, we develop a semidefinite-programming formulation and introduce a representation-theoretic symmetry reduction that decomposes the optimization into invariant blocks, substantially reducing the problem size. Shadow unitary inversion thus offers a resource-efficient path to inverse-dynamics estimation for future quantum control, diagnostics, and learning tasks. Reversing unitary operations is crucial in quantum computing and control, yet remains a challenging task. The authors introduce shadow unitary inversion, achieving efficient inversion at the expectation value level with reduced query complexity, offering a promising approach for quantum error correction and information recovery.

arXiv:2510.24880Communications PhysicsMay 20, 20260 citations

Quantum channel tomography: optimal bounds and a Heisenberg-to-classical phase transition

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How many black-box queries to a quantum channel are needed to learn its full classical description? This question lies at the heart of quantum channel tomography (also known as quantum process tomography), a fundamental task in the characterization and validation of quantum hardware. Despite extensive prior work, the optimal query complexity for quantum channel tomography is far from fully understood. In this paper, we study tomography of an unknown quantum channel with input dimension $d_1$ , output dimension $d_2$ , and Kraus rank at most $r$ , to within error $\varepsilon$ . We identify the dilation rate $τ= r d_2 / d_1$ (which always satisfies $τ\geq 1$ due to the trace preservation of quantum channels) as a key parameter, and establish that the optimal query complexity of channel tomography exhibits distinct scaling laws across three regimes of $τ$ . - In the boundary regime ( $τ= 1$ ): we show that the query complexity is $Θ(r d_1 d_2/\varepsilon)$ for Choi trace norm error $\varepsilon$ , and is upper bounded by $O(\min\{r d_1^{1.5} d_2/\varepsilon, r d_1 d_2/\varepsilon^2\})$ and lower bounded by $Ω(r d_1 d_2/\varepsilon)$ for diamond norm error $\varepsilon$ . - In the away-from-boundary regime ( $τ\geq 1+Ω(1)$ ): we show that the query complexity is $Θ(r d_1 d_2/\varepsilon^2)$ for both Choi trace norm and diamond norm errors $\varepsilon$ . Our results uncover a sharp Heisenberg-to-classical phase transition in the query complexity of quantum channel tomography: at $τ=1$ , the optimal query complexity exhibits Heisenberg scaling $1/\varepsilon$ , whereas for $τ\geq 1+Ω(1)$ , it exhibits classical scaling $1/\varepsilon^2$ . In addition, we show that in the near-boundary regime ( $1< τ< 1+o(1)$ ), the query complexity exhibits a mixture of Heisenberg and classical scaling behaviors.How many black-box queries to a quantum channel are needed to learn its full classical description? This question lies at the heart of quantum channel tomography (also known as quantum process tomography), a fundamental task in the characterization and validation of quantum hardware. Despite extensive prior work, the optimal query complexity for quantum channel tomography is far from fully understood. In this paper, we study tomography of an unknown quantum channel with input dimension $d_1$, output dimension $d_2$, and Kraus rank at most $r$, to within error $\varepsilon$. We identify the dilation rate $τ= r d_2 / d_1$ (which always satisfies $τ\geq 1$ due to the trace preservation of quantum channels) as a key parameter, and establish that the optimal query complexity of channel tomography exhibits distinct scaling laws across three regimes of $τ$. - In the boundary regime ($τ= 1$): we show that the query complexity is $Θ(r d_1 d_2/\varepsilon)$ for Choi trace norm error $\varepsilon$, and is upper bounded by $O(\min\{r d_1^{1.5} d_2/\varepsilon, r d_1 d_2/\varepsilon^2\})$ and lower bounded by $Ω(r d_1 d_2/\varepsilon)$ for diamond norm error $\varepsilon$. - In the away-from-boundary regime ($τ\geq 1+Ω(1)$): we show that the query complexity is $Θ(r d_1 d_2/\varepsilon^2)$ for both Choi trace norm and diamond norm errors $\varepsilon$. Our results uncover a sharp Heisenberg-to-classical phase transition in the query complexity of quantum channel tomography: at $τ=1$, the optimal query complexity exhibits Heisenberg scaling $1/\varepsilon$, whereas for $τ\geq 1+Ω(1)$, it exhibits classical scaling $1/\varepsilon^2$. In addition, we show that in the near-boundary regime ($1< τ< 1+o(1)$), the query complexity exhibits a mixture of Heisenberg and classical scaling behaviors.

quant-pharXiv:2604.17369Apr 21, 20260 citations

Instance-optimal high-precision shadow tomography with few-copy measurements: A metrological approach

Senrui Chen, Weiyuan Gong, Sisi Zhou

We study the sample complexity of shadow tomography in the high-precision regime under realistic measurement constraints. Given an unknown $d$ -dimensional quantum state $ρ$ and a known set of observables $\{O_i\}_{i=1}^m$ , the goal is to estimate expectation values $\{\mathrm{tr}(O_iρ)\}_{i=1}^m$ to accuracy $ε$ in $L_p$ -norm, using possibly adaptive measurements that act on $O(\mathrm{polylog}(d))$ number of copies of $ρ$ at a time. We focus on the regime where $ε$ is below an instance-dependent threshold. Our main contribution is an instance-optimal characterization of the sample complexity as $\tildeΘ(Γ_p/ε^2)$ , where $Γ_p$ is a function of $\{O_i\}_{i=1}^m$ defined via an optimization formula involving the inverse Fisher information matrix. Previously, tight bounds were known only in special cases, e.g. Pauli shadow tomography with $L_\infty$ -norm error. Concretely, we first analyze a simpler oblivious variant where the goal is to estimate an observable of the form $\sum_{i=1}^m α_i O_i$ with $\|α\|_q = 1$ (where $q$ is dual to $p$ ) revealed after the measurement. For single-copy measurements, we obtain a sample complexity of $Θ(Γ^{\mathrm{ob}}_p/ε^2)$ . We then show $\tildeΘ(Γ_p/ε^2)$ is necessary and sufficient for the original problem, with the lower bound applying to unbiased, bounded estimators. Our upper bounds rely on a two-step algorithm combining coarse tomography with local estimation. Notably, $Γ^{\mathrm{ob}}_\infty = Γ_\infty$ . In both cases, allowing $c$ -copy measurements improves the sample complexity by at most $Ω(1/c)$ . Our results establish a quantitative correspondence between quantum learning and metrology, unifying asymptotic metrological limits with finite-sample learning guarantees.We study the sample complexity of shadow tomography in the high-precision regime under realistic measurement constraints. Given an unknown $d$-dimensional quantum state $ρ$ and a known set of observables $\{O_i\}_{i=1}^m$, the goal is to estimate expectation values $\{\mathrm{tr}(O_iρ)\}_{i=1}^m$ to accuracy $ε$ in $L_p$-norm, using possibly adaptive measurements that act on $O(\mathrm{polylog}(d))$ number of copies of $ρ$ at a time. We focus on the regime where $ε$ is below an instance-dependent threshold. Our main contribution is an instance-optimal characterization of the sample complexity as $\tildeΘ(Γ_p/ε^2)$, where $Γ_p$ is a function of $\{O_i\}_{i=1}^m$ defined via an optimization formula involving the inverse Fisher information matrix. Previously, tight bounds were known only in special cases, e.g. Pauli shadow tomography with $L_\infty$-norm error. Concretely, we first analyze a simpler oblivious variant where the goal is to estimate an observable of the form $\sum_{i=1}^m α_i O_i$ with $\|α\|_q = 1$ (where $q$ is dual to $p$) revealed after the measurement. For single-copy measurements, we obtain a sample complexity of $Θ(Γ^{\mathrm{ob}}_p/ε^2)$. We then show $\tildeΘ(Γ_p/ε^2)$ is necessary and sufficient for the original problem, with the lower bound applying to unbiased, bounded estimators. Our upper bounds rely on a two-step algorithm combining coarse tomography with local estimation. Notably, $Γ^{\mathrm{ob}}_\infty = Γ_\infty$. In both cases, allowing $c$-copy measurements improves the sample complexity by at most $Ω(1/c)$. Our results establish a quantitative correspondence between quantum learning and metrology, unifying asymptotic metrological limits with finite-sample learning guarantees.

quant-pharXiv:2602.04952Feb 6, 20260 citations

Query-optimal estimation of unitary channels in diamond distance

Jeongwan Haah, Robin Kothari, Ryan O’Donnell +1 more

We consider process tomography for unitary quantum channels. Given access to an unknown unitary channel acting on a d-dimensional qudit, we aim to output a classical description of a unitary that is $\varepsilon$ -close to the unknown unitary in diamond norm. We design an algorithm achieving error $\varepsilon$ using $O\left(\mathrm{~d}^{2} / \varepsilon\right)$ applications of the unknown channel and only one qudit. This improves over prior results, which use $O\left(\mathrm{~d}^{3} / \varepsilon^{2}\right)$ [via standard process tomography] or $O\left(\mathrm{~d}^{2.5} / \varepsilon\right)$ [Yang, Renner, and Chiribella, PRL 2020] applications. To show this result, we introduce a simple technique to “bootstrap” an algorithm that can produce constant-error estimates to one that can produce $\varepsilon$ -error estimates with the Heisenberg scaling. Finally, we prove a complementary lower bound showing that estimation requires $\Omega\left(\mathrm{d}^{2} / \varepsilon\right)$ applications, even with access to the inverse or controlled versions of the unknown unitary. This shows that our algorithm has both optimal query complexity and optimal space complexity.We consider process tomography for unitary quantum channels. Given access to an unknown unitary channel acting on a d-dimensional qudit, we aim to output a classical description of a unitary that is $\varepsilon$-close to the unknown unitary in diamond norm. We design an algorithm achieving error $\varepsilon$ using $O\left(\mathrm{~d}^{2} / \varepsilon\right)$ applications of the unknown channel and only one qudit. This improves over prior results, which use $O\left(\mathrm{~d}^{3} / \varepsilon^{2}\right)$ [via standard process tomography] or $O\left(\mathrm{~d}^{2.5} / \varepsilon\right)$ [Yang, Renner, and Chiribella, PRL 2020] applications. To show this result, we introduce a simple technique to “bootstrap” an algorithm that can produce constant-error estimates to one that can produce $\varepsilon$-error estimates with the Heisenberg scaling. Finally, we prove a complementary lower bound showing that estimation requires $\Omega\left(\mathrm{d}^{2} / \varepsilon\right)$ applications, even with access to the inverse or controlled versions of the unknown unitary. This shows that our algorithm has both optimal query complexity and optimal space complexity.

arXiv:2302.14066Annual Symposium on Foundations of Computer ScienceNov 6, 202323 citations

d

-dimensional unitary

U

and stores classical data that can later be used to predict expectation values

\mathrm{tr}[O \cdot UρU^\dagger]

up to additive error

\varepsilon

for arbitrary input states

ρ

and observables

O

. We propose a parallel, non-adaptive CSEU protocol using

\mathcal{O}(d\varepsilon^{-1})

queries when the input states or observables have constant rank. This achieves Heisenberg scaling with respect to

\varepsilon

and is query-optimal, as we prove a matching

Ω(d\varepsilon^{-1})

d

-dimensional unitary channel to within

\varepsilon

in diamond distance using

\mathcal{O}(d^2/\varepsilon)

parallel queries, matching the lower bound presented in [Haah, Kothari, O'Donnell, and Tang, FOCS '23]. Our protocol thus closes the gap between parallel and sequential strategies for unitary channel tomography. As applications, it achieves optimal query complexity for boundary-regime quantum channel tomography and improves the query efficiency of the best-known protocol for tomography of fermionic linear optics.

d_1

, output dimension

d_2

, and Kraus rank at most

r

, to within error

\varepsilon

. We identify the dilation rate

τ= r d_2 / d_1

(which always satisfies

τ\geq 1

due to the trace preservation of quantum channels) as a key parameter, and establish that the optimal query complexity of channel tomography exhibits distinct scaling laws across three regimes of

τ

. - In the boundary regime (

τ= 1

): we show that the query complexity is

Θ(r d_1 d_2/\varepsilon)

for Choi trace norm error

\varepsilon

, and is upper bounded by

O(\min\{r d_1^{1.5} d_2/\varepsilon, r d_1 d_2/\varepsilon^2\})

and lower bounded by

Ω(r d_1 d_2/\varepsilon)

for diamond norm error

\varepsilon

. - In the away-from-boundary regime (

τ\geq 1+Ω(1)

): we show that the query complexity is

Θ(r d_1 d_2/\varepsilon^2)

for both Choi trace norm and diamond norm errors

\varepsilon

. Our results uncover a sharp Heisenberg-to-classical phase transition in the query complexity of quantum channel tomography: at

τ=1

, the optimal query complexity exhibits Heisenberg scaling

1/\varepsilon

, whereas for

τ\geq 1+Ω(1)

, it exhibits classical scaling

1/\varepsilon^2

. In addition, we show that in the near-boundary regime (

1< τ< 1+o(1)

), the query complexity exhibits a mixture of Heisenberg and classical scaling behaviors.

We study the sample complexity of shadow tomography in the high-precision regime under realistic measurement constraints. Given an unknown

d

-dimensional quantum state

ρ

and a known set of observables

\{O_i\}_{i=1}^m

, the goal is to estimate expectation values

\{\mathrm{tr}(O_iρ)\}_{i=1}^m

to accuracy

ε

L_p

-norm, using possibly adaptive measurements that act on

O(\mathrm{polylog}(d))

number of copies of

ρ

at a time. We focus on the regime where

ε

is below an instance-dependent threshold. Our main contribution is an instance-optimal characterization of the sample complexity as

\tildeΘ(Γ_p/ε^2)

, where

Γ_p

is a function of

\{O_i\}_{i=1}^m

defined via an optimization formula involving the inverse Fisher information matrix. Previously, tight bounds were known only in special cases, e.g. Pauli shadow tomography with

L_\infty

-norm error. Concretely, we first analyze a simpler oblivious variant where the goal is to estimate an observable of the form

\sum_{i=1}^m α_i O_i

with

\|α\|_q = 1

(where

q

is dual to

p

) revealed after the measurement. For single-copy measurements, we obtain a sample complexity of

Θ(Γ^{\mathrm{ob}}_p/ε^2)

. We then show

\tildeΘ(Γ_p/ε^2)

is necessary and sufficient for the original problem, with the lower bound applying to unbiased, bounded estimators. Our upper bounds rely on a two-step algorithm combining coarse tomography with local estimation. Notably,

Γ^{\mathrm{ob}}_\infty = Γ_\infty

. In both cases, allowing

c

-copy measurements improves the sample complexity by at most

Ω(1/c)

. Our results establish a quantitative correspondence between quantum learning and metrology, unifying asymptotic metrological limits with finite-sample learning guarantees.

We consider process tomography for unitary quantum channels. Given access to an unknown unitary channel acting on a d-dimensional qudit, we aim to output a classical description of a unitary that is

\varepsilon

-close to the unknown unitary in diamond norm. We design an algorithm achieving error

\varepsilon

using

O\left(\mathrm{~d}^{2} / \varepsilon\right)

applications of the unknown channel and only one qudit. This improves over prior results, which use

O\left(\mathrm{~d}^{3} / \varepsilon^{2}\right)

[via standard process tomography] or

O\left(\mathrm{~d}^{2.5} / \varepsilon\right)

[Yang, Renner, and Chiribella, PRL 2020] applications. To show this result, we introduce a simple technique to “bootstrap” an algorithm that can produce constant-error estimates to one that can produce

\varepsilon

-error estimates with the Heisenberg scaling. Finally, we prove a complementary lower bound showing that estimation requires

\Omega\left(\mathrm{d}^{2} / \varepsilon\right)

applications, even with access to the inverse or controlled versions of the unknown unitary. This shows that our algorithm has both optimal query complexity and optimal space complexity.