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arXiv:2310.09644 · Physical Review A

Classical shadow tomography with mutually unbiased bases

Yu Wang, Wei Cui

Published Jun 3, 2024• 10 citations
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Abstract

Classical shadow tomography, harnessing randomized informationally complete (IC) measurements, provides an effective avenue for predicting many properties of unknown quantum states. In $n$-qubit systems, projections onto ${2}^{n}+1$ 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 $n$ 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 $n$. However, for a special class referred to as appropriate MUB-average (AMA) observables, the variance decreases to $\text{poly}(n)$. Additionally, we find that through biased sampling of MUB circuits, the variance for non-AMA observables can again be reduced to $\text{poly}(n)$ 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.

DOI
10.1103/physreva.109.062406

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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 \ensuremath-CZ\ensuremath-S\ensuremath-H\ensuremath-\text{\ensuremath{-}}\mathrm{CZ}\text{\ensuremath{-}}S\text{\ensuremath{-}}H\text{\ensuremath{-}}\ensuremath-CZ\ensuremath-S\ensuremath-H\ensuremath-, form the minimal subset of the whole 2n2+2n\ensuremath∏j=1n(4j\ensuremath−1){2}^{{n}^{2}+2n}{\ensuremath{\prod}}_{j=1}^{n}({4}^{j}\ensuremath{-}1)2n2+2n\ensuremath∏j=1n​(4j\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.
Predicting properties of large-scale quantum systems is crucial for the development of quantum science and technology. Shadow estimation is an efficient method for this task based on randomized measurements, where many-qubit random Clifford circuits are used for estimating global properties like quantum fidelity. Here we introduce the minimal Clifford measurement (MCM) to reduce the number of possible random circuits to the minimum, while keeping the effective postprocessing channel in shadow estimation. In particular, we show that MCM requires 2n+1{2}^{n}+12n+1 distinct Clifford circuits, and it can be realized by mutually unbiased bases, with nnn as the total qubit number. By applying the ZZZ-tableau formalism, this ensemble of circuits can be synthesized to the $\ensuremath{-}\mathrm{S}\ensuremath{-}\mathrm{CZ}\ensuremath{-}\mathrm{H}\ensuremath{-}$ structure, which can be decomposed to $2n\ensuremath{-}1$ fixed circuit modules, and the total circuit depth is at most n+1n+1n+1. Compared to the original Clifford measurements, our MCM reduces the circuit complexity and the compilation costs. In addition, we find the sampling advantage of MCM on estimating off-diagonal operators, and extend this observation to the biased-MCM scheme to enhance the sampling improvement further.
An Efficient Quantum Circuit Construction Method for Mutually Unbiased Bases in nnn-Qubit Systems
Mutually unbiased bases (MUBs) play a crucial role in numerous applications within quantum information science, such as quantum state tomography, error correction, entanglement detection, and quantum cryptography. Utilizing 2n+12^n + 12n+1 MUB circuits provides a minimal and optimal measurement strategy for reconstructing all nnn-qubit unknown states. It significantly reduces the number of measurements compared to the traditional 4n4^n4n Pauli observables, also enhancing the robustness of quantum key distribution (QKD) protocols. Previous circuit designs that rely on a single generator can result in exponential gate costs for some MUB circuits. In this work, we present an efficient algorithm to generate each of the 2n+12^n + 12n+1 quantum MUB circuits on nnn-qubit systems within O(n3)O(n^3)O(n3) time. The algorithm features a three-stage structure, and we have calculated the average number of different gates for random sampling. Additionally, we have identified two linear properties: the entanglement part can be directly defined into 2n−32n - 32n−3 fixed sub-parts, and the knowledge of nnn special MUB circuits is sufficient to construct all 2n+12^n + 12n+1 MUB circuits. This new efficient and simple circuit construction paves the way for the implementation of a complete set of MUBs in diverse quantum information processing tasks on high-dimensional quantum systems.
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 generalize the classical shadow tomography scheme to a broad class of finite-depth or finite-time local unitary ensembles, known as locally scrambled quantum dynamics, where the unitary ensemble is invariant under local-basis transformations. In this case, the reconstruction map for the classical shadow tomography depends only on the average entanglement feature of classical snapshots. We provide an unbiased estimator of the quantum state as a linear combination of reduced classical snapshots in all subsystems, where the combination coefficients are solely determined by the entanglement feature. We also bound the number of experimental measurements required for the tomography scheme, so-called sample complexity, by formulating the operator shadow norm in the entanglement feature formalism. We numerically demonstrate our approach for finite-depth local unitary circuits and finite-time local-Hamiltonian generated evolutions. The shallow-circuit measurement can achieve a lower tomography complexity compared to the existing method based on Pauli or Clifford measurements. Our approach is also applicable to approximately locally scrambled unitary ensembles with a controllable bias that vanishes quickly. Surprisingly, we find a single instance of time-dependent local Hamiltonian evolution is sufficient to perform an approximate tomography as we numerically demonstrate it using a paradigmatic spin chain Hamiltonian modeled after trapped ion or Rydberg atom quantum simulators. Our approach significantly broadens the application of classical shadow tomography on near-term quantum devices.
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.
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.