Quantum tomography is a widely applicable tool for complete characterization of quantum states and processes. We develop a method for precision-guaranteed quantum process tomography. With the use of the Choi–Jamiołkowski isomorphism, we generalize the recently suggested extended norm minimization estimator for the case of quantum processes. Our estimator is based on the Hilbert–Schmidt distance for quantum processes. Specifically, we discuss the application of our method for characterizing quantum gates of a superconducting quantum processor in the framework of the IBM Q Experience.Quantum tomography is a widely applicable tool for complete characterization of quantum states and processes. We develop a method for precision-guaranteed quantum process tomography. With the use of the Choi–Jamiołkowski isomorphism, we generalize the recently suggested extended norm minimization estimator for the case of quantum processes. Our estimator is based on the Hilbert–Schmidt distance for quantum processes. Specifically, we discuss the application of our method for characterizing quantum gates of a superconducting quantum processor in the framework of the IBM Q Experience.

arXiv:1911.00277Optical EngineeringJan 10, 20207 citations

Quantum Tomography for Imaging

Giacomo Mauro D’Ariano, Lorenzo Maccone

Electronic Notes in Discrete MathematicsJun 19, 20052 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

Energy-independent tomography of Gaussian states

Lennart Bittel, Mele, Francesco Anna, Jens Eisert +1 more

The exploration of tomography of bosonic Gaussian states is presumably as old as quantum optics, but only recently, their precise and rigorous study have been moving into the focus of attention, motivated by technological developments. In this work, we present an efficient and experimentally feasible Gaussian state tomography algorithm with provable recovery trace-distance guarantees, whose sample complexity depends only on the number of modes, and - remarkably - is independent of the state's photon number or energy, up to doubly logarithmic factors. Our algorithm yields a doubly-exponential improvement over existing methods, and it employs operations that are readily accessible in experimental settings: the preparation of an auxiliary squeezed vacuum, passive Gaussian unitaries, and homodyne detection. At its core lies an adaptive strategy that systematically reduces the total squeezing of the system, enabling efficient tomography. Quite surprisingly, this proves that estimating a Gaussian state in trace distance is generally more efficient than directly estimating its covariance matrix. Our algorithm is particularly well-suited for applications in quantum metrology and sensing, where highly squeezed - and hence high-energy - states are commonly employed. As a further contribution, we establish improved sample complexity bounds for standard heterodyne tomography, equipping this widely used protocol with rigorous trace-norm guarantees.The exploration of tomography of bosonic Gaussian states is presumably as old as quantum optics, but only recently, their precise and rigorous study have been moving into the focus of attention, motivated by technological developments. In this work, we present an efficient and experimentally feasible Gaussian state tomography algorithm with provable recovery trace-distance guarantees, whose sample complexity depends only on the number of modes, and - remarkably - is independent of the state's photon number or energy, up to doubly logarithmic factors. Our algorithm yields a doubly-exponential improvement over existing methods, and it employs operations that are readily accessible in experimental settings: the preparation of an auxiliary squeezed vacuum, passive Gaussian unitaries, and homodyne detection. At its core lies an adaptive strategy that systematically reduces the total squeezing of the system, enabling efficient tomography. Quite surprisingly, this proves that estimating a Gaussian state in trace distance is generally more efficient than directly estimating its covariance matrix. Our algorithm is particularly well-suited for applications in quantum metrology and sensing, where highly squeezed - and hence high-energy - states are commonly employed. As a further contribution, we establish improved sample complexity bounds for standard heterodyne tomography, equipping this widely used protocol with rigorous trace-norm guarantees.

arXiv:2508.14979Aug 20, 20250 citations

Pretty good shadow tomography

Sabee Grewal, Angelos Pelecanos, Jack Spilecki +2 more

We give a shadow tomography algorithm that uses $O(\log(m) \log(m/δ) / \varepsilon^2)$ copies of the input state. Specifically, given a quantum state $ρ$ and observables $O_1,\ldots,O_m$ , the algorithm outputs estimates of $\mathrm{tr}(O_iρ)$ for all $i\in[m]$ that are all accurate to within $\pm\varepsilon$ , except with probability at most $δ$ . The key idea is to work in the average-case setting: for a random state drawn from a known prior, a pretty good measurement on $O(\log(m)/\varepsilon^2)$ copies gives a constant-error guarantee on all but a small fraction of the prior. Boosting this guarantee gives an average-case shadow tomography algorithm, and the minimax principle then (non-constructively) implies the existence of a measurement with the same worst-case guarantee. We also identify an obstruction to further improvement with this approach. While pretty good measurements amplify success over the randomness of the prior, they do not amplify success over the randomness of measurement. This issue appears to be inherent, and gives rise to the remaining $\log(m)$ factor separating the complexity of the quantum problem from its classical counterpart.We give a shadow tomography algorithm that uses $O(\log(m) \log(m/δ) / \varepsilon^2)$ copies of the input state. Specifically, given a quantum state $ρ$ and observables $O_1,\ldots,O_m$, the algorithm outputs estimates of $\mathrm{tr}(O_iρ)$ for all $i\in[m]$ that are all accurate to within $\pm\varepsilon$, except with probability at most $δ$. The key idea is to work in the average-case setting: for a random state drawn from a known prior, a pretty good measurement on $O(\log(m)/\varepsilon^2)$ copies gives a constant-error guarantee on all but a small fraction of the prior. Boosting this guarantee gives an average-case shadow tomography algorithm, and the minimax principle then (non-constructively) implies the existence of a measurement with the same worst-case guarantee. We also identify an obstruction to further improvement with this approach. While pretty good measurements amplify success over the randomness of the prior, they do not amplify success over the randomness of measurement. This issue appears to be inherent, and gives rise to the remaining $\log(m)$ factor separating the complexity of the quantum problem from its classical counterpart.

quant-pharXiv:2610.06781Oct 6, 20260 citations

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 give a shadow tomography algorithm that uses

O(\log(m) \log(m/δ) / \varepsilon^2)

copies of the input state. Specifically, given a quantum state

ρ

and observables

O_1,\ldots,O_m

, the algorithm outputs estimates of

\mathrm{tr}(O_iρ)

for all

i\in[m]

that are all accurate to within

\pm\varepsilon

, except with probability at most

δ

. The key idea is to work in the average-case setting: for a random state drawn from a known prior, a pretty good measurement on

O(\log(m)/\varepsilon^2)

copies gives a constant-error guarantee on all but a small fraction of the prior. Boosting this guarantee gives an average-case shadow tomography algorithm, and the minimax principle then (non-constructively) implies the existence of a measurement with the same worst-case guarantee. We also identify an obstruction to further improvement with this approach. While pretty good measurements amplify success over the randomness of the prior, they do not amplify success over the randomness of measurement. This issue appears to be inherent, and gives rise to the remaining

\log(m)

factor separating the complexity of the quantum problem from its classical counterpart.