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arXiv:2503.14491

Partial Quantum Shadow Tomography for Structured Operators and its Experimental Demonstration using NMR

Aniket Sengupta, Arijit Chatterjee, G. J. Sreejith, Mahesh, T. S.

Published Mar 18, 2025• 0 citations
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Abstract

Quantum shadow tomography based on the classical shadow representation provides an efficient way to estimate properties of an unknown quantum state without performing a full quantum state tomography. In scenarios where estimating the expectation values for only certain classes of observables is required, obtaining information about the entire density matrix is unnecessary. We propose a partial quantum shadow tomography protocol that estimates a subset of density matrix elements relevant to the expectation values of structured observables. Specifically, we identify specific subsets of the tomographically complete set ${\mathrm{Cl}}(2)^{\otimes n}$ and a simple pseudo-inverse of the associated channel, which can be used to estimate all elements of the density matrix with the same active order. By restricting the protocol to smaller subsets of single-qubit Pauli measurements, it becomes experimentally more efficient. We demonstrate the advantage over unitary designs, such as the Clifford, full Pauli basis, and methods utilizing mutually unbiased bases, by analytically deriving error bounds and numerically evaluating the protocol on structured operators. We experimentally demonstrate the partial shadow estimation scheme for a wide class of two-qubit states (pure, entangled, and mixed) in the nuclear magnetic resonance (NMR) platform. The full density matrix, reconstructed experimentally by combining different partial estimators, achieves fidelities around 99%.

DOI
10.48550/arxiv.2503.14491

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Quantum shadow tomography based on the classical shadow representation provides an efficient way to estimate properties of an unknown quantum state without performing a full quantum state tomography. In scenarios where estimating the expectation values for only certain classes of observables is required, obtaining information about the entire density matrix is unnecessary. We propose a partial quantum shadow tomography protocol that estimates a subset of density matrix elements relevant to the expectation values of structured observables. Specifically, we identify specific subsets of the tomographically complete set Cl(2)⊗n{\mathrm{Cl}}(2)^{\otimes n}Cl(2)⊗n and a simple pseudo-inverse of the associated channel, which can be used to estimate all elements of the density matrix with the same active order. By restricting the protocol to smaller subsets of single-qubit Pauli measurements, it becomes experimentally more efficient. We demonstrate the advantage over unitary designs, such as the Clifford, full Pauli basis, and methods utilizing mutually unbiased bases, by analytically deriving error bounds and numerically evaluating the protocol on structured operators. We experimentally demonstrate the partial shadow estimation scheme for a wide class of two-qubit states (pure, entangled, and mixed) in the nuclear magnetic resonance (NMR) platform. The full density matrix, reconstructed experimentally by combining different partial estimators, achieves fidelities around 99%.
We introduce a direct estimation framework for reconstructing multiple density matrix elements of an unknown quantum state using classical shadow tomography. Traditional direct measurement protocols (DMPs), while effective for individual elements, suffer from poor scalability due to post-selection losses and the need for element-specific measurement configurations. In contrast, our method, DMP-ST, leverages random Clifford or biased mutually unbiased basis measurements to enable global estimation: a single dataset suffices to estimate arbitrary off-diagonal entries with high accuracy. We prove that estimating KKK off-diagonal matrix elements up to additive error εεε requires only O(log⁡K/ε2)\mathcal{O}(\log K/ε^2)O(logK/ε2) samples, achieving exponential improvement over conventional DMPs. The number of required measurement configurations can also be exponentially reduced for large K. When extended to full state tomography, DMP-ST attains trace distance error ≤ε\le ε≤ε with sample complexity O(d3log⁡d/ε2)\mathcal{O}(d^3 \log d/ε^2)O(d3logd/ε2), which is closed to the optimal scaling for single-copy measurements. Moreover, biased MUB measurements reduce sample complexity by a constant factor than random Clifford measurements. This work provides both theoretical guarantees and explicit protocols for efficient, entrywise quantum state reconstruction. It significantly advances the practicality of direct tomography, especially for high-dimensional systems and near-term quantum platforms.
We investigate the problem of efficiently estimating expectation values of large sets of observables from copies of an unknown many-body quantum state. This task, known as shadow tomography, is crucial for quantum simulation and quantum chemistry, where the relevant observables often include k -body fermionic or k -local Pauli operators. A key goal is to achieve sample efficiency, computational efficiency, and few-copy measurements. We introduce the notion of triply efficient shadow tomography to formalize these requirements. Prior work has shown that single-copy measurements suffice for k -local Pauli operators with constant k . However, we prove that for k -body fermionic observables and the full set of n -qubit Pauli operators, single-copy protocols fail to achieve sample-efficient tomography. We then present new protocols based on measuring two copies of the state at a time that overcome these lower bounds, providing a triply efficient protocol.
Recently introduced shadow tomography protocols use ``classical shadows'' of quantum states to predict many target functions of an unknown quantum state. Unlike full quantum state tomography, shadow tomography does not insist on accurate recovery of the density matrix for high rank mixed states. Yet, such a protocol makes multiple accurate predictions with high confidence, based on a moderate number of quantum measurements. One particular influential algorithm, proposed by Huang et al. [Huang, Kueng, and Preskill, Nat. Phys. 16, 1050 (2020)], requires additional circuits for performing certain random unitary transformations. In this paper, we avoid these transformations but employ an arbitrary informationally complete positive operator-valued measure and show that such a procedure can compute kkk-bit correlation functions for quantum states reliably. We also show that, for this application, we do not need the median of means procedure of Huang et al. Finally, we discuss the contrast between the computation of correlation functions and fidelity of reconstruction of low rank density matrices.
Classical shadow tomography, harnessing randomized informationally complete (IC) measurements, provides an effective avenue for predicting many properties of unknown quantum states. In nnn-qubit systems, projections onto 2n+1{2}^{n}+12n+1 mutually unbiased bases (MUBs) are widely recognized as minimal and optimal IC measurements for full-state tomography. Mutually unbiased base circuits, structured as $\text{\ensuremath{-}}\mathrm{CZ}\text{\ensuremath{-}}S\text{\ensuremath{-}}H\text{\ensuremath{-}}$, form the minimal subset of the whole ${2}^{{n}^{2}+2n}{\ensuremath{\prod}}_{j=1}^{n}({4}^{j}\ensuremath{-}1)$ Clifford circuit ensemble. Each of them can be generated by nnn special MUB circuits. We study how to use MUB circuits as the ensemble in classical shadow tomography. For general observables, the variance to predict their expectation value is shown to be exponential to the number of qubits nnn. However, for a special class referred to as appropriate MUB-average (AMA) observables, the variance decreases to poly(n)\text{poly}(n)poly(n). Additionally, we find that through biased sampling of MUB circuits, the variance for non-AMA observables can again be reduced to poly(n)\text{poly}(n)poly(n) with the MUB-sparse condition. The performance and complexity of using the MUB and Clifford circuits as the ensemble in the classical shadow tomography are compared.
Classical shadow tomography provides an efficient method for predicting functions of an unknown quantum state from a few measurements of the state. It relies on a unitary channel that efficiently scrambles the quantum information of the state to the measurement basis. Facing the challenge of realizing deep unitary circuits on near-term quantum devices, we explore the scenario in which the unitary channel can be shallow and is generated by a quantum chaotic Hamiltonian via time evolution. We provide an unbiased estimator of the density matrix for all ranges of the evolution time. We analyze the sample complexity of the Hamiltonian-driven shadow tomography. For Pauli observables, we find that it can be more efficient than the unitary-2-design-based shadow tomography in a sequence of intermediate time windows that range from an order-1 scrambling time to a timescale of D1/6{D}^{1/6}D1/6, given the Hilbert space dimension DDD. In particular, the efficiency of predicting diagonal Pauli observables is improved by a factor of DDD without sacrificing the efficiency of predicting off-diagonal Pauli observables.
Quantum computers solve ever more complex tasks using steadily growing system sizes.Characterizing these quantum systems is vital, yet becoming increasingly challenging.The gold-standard method for this task is quantum state tomography (QST), capable of fully reconstructing a quantum state without prior knowledge.The measurement and classical computing costs, however, increase exponentially with the number of constituents (e.g., qubits)-a daunting bottleneck given the scale of existing and nearterm quantum devices.Here, we demonstrate a scalable and practical QST approach that only uses a single measurement setting, namely symmetric informationally complete (SIC) positive operator-valued measures (POVMs).We implement these nonorthogonal measurements on an ion trap quantum processor by utilizing additional energy levels within each ion-without requiring ancillary ions to assist in measurements.More precisely, we locally map the SIC POVM to orthogonal states embedded in a higher-dimensional system, which we read out using repeated in-sequence detections, thereby providing full tomographic information in every shot.Combining this SIC tomography with the recently developed randomized measurement toolbox ("classical shadows") proves to be a powerful combination.SIC tomography alleviates the need for choosing measurement settings at random ("derandomization"), while classical shadows enable the estimation of arbitrary polynomial functions of the density matrix orders of magnitudes faster than standard methods.The latter enables in-depth entanglement characterization, which we experimentally showcase on a five-qubit absolutely maximally entangled state.Moreover, the fact that the full tomography information is available in every shot enables online QST in real time (i.e., while the experiment is running).We demonstrate this on an eight-qubit entangled state (which has 2 8 • 2 8 -1 = 65 535 degrees of freedom), as well as for fast state identification.All in all, these features single out SIC-based classical shadow estimation as a highly scalable and convenient tool for quantum state characterization.