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arXiv:2310.18749 · Physical Review Applied

Minimal-Clifford shadow estimation by mutually unbiased bases

Qingyue Zhang, Qing Liu, You Zhou

Published Jun 3, 2024• 12 citations
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Abstract

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 ${2}^{n}+1$ distinct Clifford circuits, and it can be realized by mutually unbiased bases, with $n$ as the total qubit number. By applying the $Z$-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 $n+1$. 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.

DOI
10.1103/physrevapplied.21.064001

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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−S\ensuremath−CZ\ensuremath−H\ensuremath−\ensuremath{-}\mathrm{S}\ensuremath{-}\mathrm{CZ}\ensuremath{-}\mathrm{H}\ensuremath{-}\ensuremath−S\ensuremath−CZ\ensuremath−H\ensuremath− structure, which can be decomposed to 2n\ensuremath−12n\ensuremath{-}12n\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.
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.
Efficient fidelity estimation of multiqubit quantum states is crucial to many applications in quantum information processing. However, to estimate the infidelity εεε with multiplicative precision, conventional estimation protocols require (order) 1/ε21/ε^21/ε2 different circuits in addition to 1/ε21/ε^21/ε2 samples, which is quite resource-intensive for high-precision fidelity estimation. Here we introduce an efficient estimation protocol by virtue of common randomized measurements (CRM) integrated with shadow estimation based on the Clifford group, which only requires 1/ε1/ε1/ε circuits. Moreover, in many scenarios of practical interest, in the presence of depolarizing or Pauli noise for example, our protocol only requires a constant number of circuits, irrespective of the infidelity εεε and the qubit number. For large and intermediate quantum systems, quite often one circuit is already sufficient. In the course of study, we clarify the performance of CRM shadow estimation based on the Clifford group and 4-designs and highlight its advantages over standard and thrifty shadow estimation.
We provide practical and powerful schemes for learning properties of a quantum state using a small number of measurements. Specifically, we present a randomized measurement scheme modulated by the depth of a random quantum circuit in one spatial dimension. This scheme interpolates between two known classical shadows schemes based on random Pauli measurements and random Clifford measurements. We focus on the regime where depth scales logarithmically in the system size and provide evidence that this retains the desirable sample complexity properties of both extremal schemes while also being experimentally feasible. We present methods for two key tasks; estimating expectation values of certain observables from generated classical shadows and, computing upper bounds on the depth-modulated shadow norm, thus providing rigorous guarantees on the accuracy of the output estimates. We achieve our findings by bringing together tools from shadow estimation, random circuits, and tensor networks.
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.
Shadow estimation is a powerful framework for predicting quantum properties from randomized measurements. While 333-design protocols achieve optimal worst-case performance, the minimal number of measurement bases required for such optimality has remained open. Here we prove that Θ(d2)Θ(d^2)Θ(d2) measurement bases are both necessary and sufficient for worst-case optimal shadow estimation and construct an explicit basis family. In stark contrast, any state 222-design already suffices for average-case optimality: the mean squared shadow norm of normalized observables is bounded by a universal constant, and we prove strong concentration for Haar-random states, yielding constant sample complexity for generic pure-state fidelity estimation. Easily implementable 222-designs -- from mutually unbiased bases, cyclic measurements, or shallow O(log⁡n)\mathcal{O}(\log n)O(logn)-depth circuits -- enable optimal average-case protocols with remarkably simple measurement strategies. Our results establish a fundamental complexity separation: worst-case estimation requires Θ(d2)Θ(d^2)Θ(d2) bases, whereas average-case performance requires only Θ(d)Θ(d)Θ(d) bases, with broad implications for quantum information theory and near-term experiments.
Reliable and scalable readout strategies are essential for quantum technologies. As quantum processors grow, extracting useful information must remain feasible without measurement circuits becoming a dominant bottleneck. Randomized measurements and classical shadows provide a powerful route, but global estimation is conventionally associated with highly random ensembles that require increasing circuit depth and hence substantial experimental overhead. In this work, we show that substantially less randomness suffices when the readout is meaningfully adapted to the quantities being estimated. We introduce shallow phase shadows, based on a sparse Clifford-IQP ensemble, and prove efficient global estimation of stabilizer-state fidelities despite the ensemble not forming an approximate relative-error design. On all-to-all architectures, the protocol admits a constant-depth implementation using mid-circuit measurements and classical feedforward, or logarithmic depth without auxiliary systems. The protocol requires only controlled-phase entangling gates and offers a tunable trade-off between circuit resources and estimation accuracy, making it particularly amenable to experimentally relevant architectures with long-range connectivity. Our results show that scalable quantum readout need not reproduce generic randomness: task-adapted randomization can enable substantially shallower global characterization protocols.