Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2404.11850

Experimental Hybrid Shadow Tomography and Distillation

Peng, Xu-Jie, Liu, Qing, Lu Liu, Ting Zhang, Zhou, You, Lu He

Published Apr 18, 2024• 1 citation
arXivHTMLPDF

Abstract

Characterization of quantum states is a fundamental requirement in quantum science and technology. As a promising framework, shadow tomography shows significant efficiency in estimating linear functions, however, for the challenging nonlinear ones, it requires measurements at an exponential cost. Here, we implement an advanced shadow protocol, so-called hybrid shadow~(HS) tomography, to reduce the measurement cost in the estimation of nonlinear functions in an optical system. We design and realize a deterministic quantum Fredkin gate with single photon, achieving high process fidelity of $0.935\pm0.001$. Utilizing this novel Fredkin gate, we demonstrate HS in the estimations, like the higher-order moments up to 4, and reveal that the sample complexity of HS is significantly reduced compared with the original shadow protocol. Furthermore, we utilize these higher-degree functions to implement virtual distillation, which effectively extracts a high-purity quantum state from two noisy copies. The virtual distillation is also verified in a proof-of-principle demonstration of quantum metrology, further enhancing the accuracy of parameter estimation. Our results suggest that HS is efficient in state characterization and promising for quantum technologies.

DOI
10.48550/arxiv.2404.11850

Related papers

Experimental shadow tomography beyond single-copy measurements

Xu-Jie Peng, Qing Liu, Lu Liu +3 more

Shadow tomography is a promising way for quantum system characterization, enabling the extraction of multiple properties of unknown quantum states using only single-copy randomized measurements. While single-copy measurements are experimentally friendly and efficient for estimating linear functions, they show limited efficiency for nonlinear ones. In contrast, multicopy measurements enabled by joint entangling operations can remarkably reduce the sample complexity for nonlinear functions. Here, we design and realize a deterministic optical three-qubit Fredkin gate by comprehensive control over multiple degrees of freedom of a single photon, achieving controlled-swap operation with high process fidelity of $0.935\ifmmode\pm\else\textpm\fi{}0.001$. Utilizing this Fredkin gate, we demonstrate hybrid shadow tomography, in which the higher-order functions like the moments can be efficiently estimated from the two-copy swap operation in conjunction with the single-copy randomized measurement. Our demonstration indicates a significant reduction in sample complexity compared with the single-copy setting. Furthermore, we show the potential advantage of our scheme in a proof-of-principle quantum metrology experiment, wherein the accuracy of parameter estimation is enhanced with the assistance of virtual distillation. Our results suggest that hybrid shadow tomography is an efficient and powerful tool for shadow-estimation-enabled quantum information processing.

Physical Review Applied•Jan 29, 2025•7 citations

Logical shadow tomography: Efficient estimation of error-mitigated observables

Hong-Ye Hu, Ryan LaRose, Yi‐Zhuang You +2 more

We introduce a technique to estimate error-mitigated expectation values on noisy quantum computers. Our technique performs shadow tomography on a logical state to produce a memory-efficient classical reconstruction of the noisy density matrix. Using efficient classical post-processing, one can mitigate errors by projecting a general nonlinear function of the noisy density matrix into the codespace. The subspace expansion and virtual distillation can be viewed as special cases of the new framekwork. We show our method is favorable in the quantum and classical resources overhead. Relative to subspace expansion which requires $O\left(2^{N} \right)$ samples to estimate a logical Pauli observable with $[[N, k]]$ error correction code, our technique requires only $O\left(4^{k} \right)$ samples. Relative to virtual distillation, our technique can compute powers of the density matrix without additional copies of quantum states or quantum memory. We present numerical evidence using logical states encoded with up to sixty physical qubits and show fast convergence to error-free expectation values with only $10^5$ samples under 1% depolarizing noise.

arXiv:2203.07263•Mar 14, 2022•7 citations

Dimension-Free Polylogarithmic Quantum Shadow Tomography

Fernando Granha Jeronimo, Qizhao Huang, Lenny Liu

Shadow Tomography is a fundamental problem in quantum information theory. Given multiple copies of an unknown $d$-dimensional quantum state $ρ$ and a known collection of observables $E_1,\ldots,E_M$, the goal is to estimate all expectation values $\{\text{Tr}(ρE_i)\}_{i=1}^M$ to additive accuracy $\varepsilon$ with probability at least $1-δ$. An elusive open question from the seminal shadow tomography work of Aaronson is whether this task admits a dimension-independent sample complexity with only polylogarithmic dependence on $M$, as suggested by the best-known lower bounds. In this work, we propose two different quantum protocols for shadow tomography with the best sample complexity \[ O\left( \frac{\log(M)\log(M/δ)}{\varepsilon^2} \right), \] which is polylogarithmic in the number of observables and independent of the dimension of the unknown state, thereby answering Aaronson's original question while also providing an exponential improvement in the prior best dimension independent sample complexity of shadow tomography from Sinha (STOC 2025) and, more recently, Chen, O'Donnell, Pelecanos, and Wright. Our approach first reduces the general shadow tomography problem to a finite-ensemble estimation problem via a minimax argument. We then develop an observable-independent protocol that repeatedly applies the pretty-good measurement while updating the prior distribution over the finite ensemble according to the measurement outcomes. A tail analysis of the resulting estimation error yields simultaneous accuracy guarantees for all observables and a cubic-logarithmic upper bound. We also introduce a refined recovery-label measurement for the same finite ensemble, which yields the bound in our main theorem.

quant-pharXiv:2608.06345•Aug 7, 2026•0 citations

A computationally efficient approach to quantum state reconstruction using robust classical shadows

Sanjay Sharma, Shyam Akashe, Govind Murari Upadhyay +1 more

Quantum state tomography is a fundamental technique for characterizing quantum systems, but its scalability remains a significant challenge. This paper investigates an efficient alternative using classical shadows to reconstruct a Bell state with high fidelity. Classical shadows provide a compressed representation of a quantum state using randomized measurements, reducing the measurement complexity compared to full quantum tomography when full state reconstruction is not required. We employed a quantum circuit to generate a Bell state and utilize 1000 snapshots to construct its classical shadow. The reconstructed density matrix is evaluated using fidelity and norm difference metrics against the ideal Bell state. Our analysis demonstrates that as the number of snapshots increases, the fidelity of reconstruction stabilizes around 0.98-1.0, with the norm difference decreasing accordingly. The results also confirm the convergence of reconstructed states towards the ideal Bell state, highlighting the efficiency and accuracy of classical shadows in quantum state estimation. Shallow shadow tomography uses shallow random circuits to estimate global quantum state properties on noisy hardware. By combining low-depth entangling gates with randomized measurements and noise-aware postprocessing, it reduces sample complexity compared to Pauli shadows while remaining experimentally feasible. Recent experiments on superconducting processors confirmed up to fivefold measurement savings.

Scientific Reports•Feb 2, 2026•0 citations

Error Mitigation in Bosonic Systems via Virtual Distillation

Leonardo Finocchiaro, Marco Robbio, Diogo Gomes +5 more

Virtual distillation is a promising error-mitigation technique that exploits multiple copies of a noisy quantum state to estimate observables as if measured on a purified state. Although originally introduced in the context of bosonic many-body systems under the name of virtual cooling, its development and applications have largely focused on qubit-based quantum computation. Here, we establish a framework for virtual distillation in bosonic quantum information processing and continuous-variable quantum computing. Building on a diagonalization of cyclic shift operators implemented with passive linear-optical interferometers, we derive experimentally accessible protocols for estimating virtually distilled expectation values of observables relevant to bosonic architectures. In particular, we show how to recover noise-mitigated expectation values of number operators, phase-shift operators, and arbitrary quadratures from multi-copy measurements. For number operators, we further demonstrate the estimation of virtually distilled correlators of arbitrary order through the characteristic function of the photon-number distribution. We apply the framework to states affected by photon loss and dephasing, two of the dominant noise mechanisms in bosonic quantum computation, and quantify the resulting suppression of noise contributions. Our results extend virtual distillation beyond its original setting and provide a practical route toward error-mitigated measurements in bosonic quantum processors using experimentally available linear-optical resources.

quant-pharXiv:2607.04914•Jul 7, 2026•0 citations

Contractive unitary and classical shadow tomography

Yadong Wu, Ce Wang, Juan Yao +3 more

The rapid development of quantum technology demands efficient characterization of complex quantum many-body states. However, full quantum state tomography requires an exponential number of measurements in system size, preventing its practical use in large-scale quantum devices. A major recent breakthrough in this direction, called classical shadow tomography, significantly reduces the sample complexity, the number of samples needed to estimate properties of a state, by implementing random Clifford rotations before measurements. Despite many recent efforts, reducing the sample complexity below 2 k for extracting any non-successive local operators with a size ~ k remains a challenge. In this work, we achieve a significantly smaller sample complexity of ~ 1.8 k using a protocol that hybridizes locally random and globally deterministic unitary operations. The key insight is the discovery of a deterministic global unitary, termed as contractive unitary , which is more efficient in reducing the operator size to enhance tomography efficiency. The contractive unitary perfectly matches the advantages of the atom array quantum computation platform and is readily realized in the atom array quantum processor. More importantly, it highlights a new strategy in classical shadow tomography, demonstrating that a random-deterministic hybridized protocol can be more efficient than fully random measurements.

arXiv:2412.01850•npj Quantum Information•Apr 4, 2026•2 citations
Characterization of quantum states is a fundamental requirement in quantum science and technology. As a promising framework, shadow tomography shows significant efficiency in estimating linear functions, however, for the challenging nonlinear ones, it requires measurements at an exponential cost. Here, we implement an advanced shadow protocol, so-called hybrid shadow~(HS) tomography, to reduce the measurement cost in the estimation of nonlinear functions in an optical system. We design and realize a deterministic quantum Fredkin gate with single photon, achieving high process fidelity of 0.935±0.0010.935\pm0.0010.935±0.001. Utilizing this novel Fredkin gate, we demonstrate HS in the estimations, like the higher-order moments up to 4, and reveal that the sample complexity of HS is significantly reduced compared with the original shadow protocol. Furthermore, we utilize these higher-degree functions to implement virtual distillation, which effectively extracts a high-purity quantum state from two noisy copies. The virtual distillation is also verified in a proof-of-principle demonstration of quantum metrology, further enhancing the accuracy of parameter estimation. Our results suggest that HS is efficient in state characterization and promising for quantum technologies.
Shadow tomography is a promising way for quantum system characterization, enabling the extraction of multiple properties of unknown quantum states using only single-copy randomized measurements. While single-copy measurements are experimentally friendly and efficient for estimating linear functions, they show limited efficiency for nonlinear ones. In contrast, multicopy measurements enabled by joint entangling operations can remarkably reduce the sample complexity for nonlinear functions. Here, we design and realize a deterministic optical three-qubit Fredkin gate by comprehensive control over multiple degrees of freedom of a single photon, achieving controlled-swap operation with high process fidelity of $0.935\ifmmode\pm\else\textpm\fi{}0.001$. Utilizing this Fredkin gate, we demonstrate hybrid shadow tomography, in which the higher-order functions like the moments can be efficiently estimated from the two-copy swap operation in conjunction with the single-copy randomized measurement. Our demonstration indicates a significant reduction in sample complexity compared with the single-copy setting. Furthermore, we show the potential advantage of our scheme in a proof-of-principle quantum metrology experiment, wherein the accuracy of parameter estimation is enhanced with the assistance of virtual distillation. Our results suggest that hybrid shadow tomography is an efficient and powerful tool for shadow-estimation-enabled quantum information processing.
We introduce a technique to estimate error-mitigated expectation values on noisy quantum computers. Our technique performs shadow tomography on a logical state to produce a memory-efficient classical reconstruction of the noisy density matrix. Using efficient classical post-processing, one can mitigate errors by projecting a general nonlinear function of the noisy density matrix into the codespace. The subspace expansion and virtual distillation can be viewed as special cases of the new framekwork. We show our method is favorable in the quantum and classical resources overhead. Relative to subspace expansion which requires O(2N)O\left(2^{N} \right)O(2N) samples to estimate a logical Pauli observable with [[N,k]][[N, k]][[N,k]] error correction code, our technique requires only O(4k)O\left(4^{k} \right)O(4k) samples. Relative to virtual distillation, our technique can compute powers of the density matrix without additional copies of quantum states or quantum memory. We present numerical evidence using logical states encoded with up to sixty physical qubits and show fast convergence to error-free expectation values with only 10510^5105 samples under 1% depolarizing noise.
Shadow Tomography is a fundamental problem in quantum information theory. Given multiple copies of an unknown ddd-dimensional quantum state ρρρ and a known collection of observables E1,…,EME_1,\ldots,E_ME1​,…,EM​, the goal is to estimate all expectation values {Tr(ρEi)}i=1M\{\text{Tr}(ρE_i)\}_{i=1}^M{Tr(ρEi​)}i=1M​ to additive accuracy ε\varepsilonε with probability at least 1−δ1-δ1−δ. An elusive open question from the seminal shadow tomography work of Aaronson is whether this task admits a dimension-independent sample complexity with only polylogarithmic dependence on MMM, as suggested by the best-known lower bounds. In this work, we propose two different quantum protocols for shadow tomography with the best sample complexity O(log⁡(M)log⁡(M/δ)ε2), O\left( \frac{\log(M)\log(M/δ)}{\varepsilon^2} \right), O(ε2log(M)log(M/δ)​), which is polylogarithmic in the number of observables and independent of the dimension of the unknown state, thereby answering Aaronson's original question while also providing an exponential improvement in the prior best dimension independent sample complexity of shadow tomography from Sinha (STOC 2025) and, more recently, Chen, O'Donnell, Pelecanos, and Wright. Our approach first reduces the general shadow tomography problem to a finite-ensemble estimation problem via a minimax argument. We then develop an observable-independent protocol that repeatedly applies the pretty-good measurement while updating the prior distribution over the finite ensemble according to the measurement outcomes. A tail analysis of the resulting estimation error yields simultaneous accuracy guarantees for all observables and a cubic-logarithmic upper bound. We also introduce a refined recovery-label measurement for the same finite ensemble, which yields the bound in our main theorem.
Quantum state tomography is a fundamental technique for characterizing quantum systems, but its scalability remains a significant challenge. This paper investigates an efficient alternative using classical shadows to reconstruct a Bell state with high fidelity. Classical shadows provide a compressed representation of a quantum state using randomized measurements, reducing the measurement complexity compared to full quantum tomography when full state reconstruction is not required. We employed a quantum circuit to generate a Bell state and utilize 1000 snapshots to construct its classical shadow. The reconstructed density matrix is evaluated using fidelity and norm difference metrics against the ideal Bell state. Our analysis demonstrates that as the number of snapshots increases, the fidelity of reconstruction stabilizes around 0.98-1.0, with the norm difference decreasing accordingly. The results also confirm the convergence of reconstructed states towards the ideal Bell state, highlighting the efficiency and accuracy of classical shadows in quantum state estimation. Shallow shadow tomography uses shallow random circuits to estimate global quantum state properties on noisy hardware. By combining low-depth entangling gates with randomized measurements and noise-aware postprocessing, it reduces sample complexity compared to Pauli shadows while remaining experimentally feasible. Recent experiments on superconducting processors confirmed up to fivefold measurement savings.
Virtual distillation is a promising error-mitigation technique that exploits multiple copies of a noisy quantum state to estimate observables as if measured on a purified state. Although originally introduced in the context of bosonic many-body systems under the name of virtual cooling, its development and applications have largely focused on qubit-based quantum computation. Here, we establish a framework for virtual distillation in bosonic quantum information processing and continuous-variable quantum computing. Building on a diagonalization of cyclic shift operators implemented with passive linear-optical interferometers, we derive experimentally accessible protocols for estimating virtually distilled expectation values of observables relevant to bosonic architectures. In particular, we show how to recover noise-mitigated expectation values of number operators, phase-shift operators, and arbitrary quadratures from multi-copy measurements. For number operators, we further demonstrate the estimation of virtually distilled correlators of arbitrary order through the characteristic function of the photon-number distribution. We apply the framework to states affected by photon loss and dephasing, two of the dominant noise mechanisms in bosonic quantum computation, and quantify the resulting suppression of noise contributions. Our results extend virtual distillation beyond its original setting and provide a practical route toward error-mitigated measurements in bosonic quantum processors using experimentally available linear-optical resources.
The rapid development of quantum technology demands efficient characterization of complex quantum many-body states. However, full quantum state tomography requires an exponential number of measurements in system size, preventing its practical use in large-scale quantum devices. A major recent breakthrough in this direction, called classical shadow tomography, significantly reduces the sample complexity, the number of samples needed to estimate properties of a state, by implementing random Clifford rotations before measurements. Despite many recent efforts, reducing the sample complexity below 2 k for extracting any non-successive local operators with a size ~ k remains a challenge. In this work, we achieve a significantly smaller sample complexity of ~ 1.8 k using a protocol that hybridizes locally random and globally deterministic unitary operations. The key insight is the discovery of a deterministic global unitary, termed as contractive unitary , which is more efficient in reducing the operator size to enhance tomography efficiency. The contractive unitary perfectly matches the advantages of the atom array quantum computation platform and is readily realized in the atom array quantum processor. More importantly, it highlights a new strategy in classical shadow tomography, demonstrating that a random-deterministic hybridized protocol can be more efficient than fully random measurements.