Advances in quantum technology require scalable techniques to efficiently extract information from a quantum system. Traditional tomography is limited to a handful of qubits, and shadow tomography has been suggested as a scalable replacement for larger systems. Shadow tomography is conventionally analyzed based on outcomes of ideal projective measurements on the system upon application of randomized unitaries. Here, we suggest that shadow tomography can be much more straightforwardly formulated for generalized measurements, or positive operator valued measures. Based on the idea of the least-square estimator shadow tomography with generalized measurements is both more general and simpler than the traditional formulation with randomization of unitaries. In particular, this formulation allows us to analyze theoretical aspects of shadow tomography in detail. For example, we provide a detailed study of the implication of symmetries in shadow tomography. Moreover, with this generalization we also demonstrate how the optimization of measurements for shadow tomography tailored toward a particular set of observables can be carried out.
Advances in quantum technology require scalable techniques to efficiently extract information from a quantum system. Traditional tomography is limited to a handful of qubits, and shadow tomography has been suggested as a scalable replacement for larger systems. Shadow tomography is conventionally analyzed based on outcomes of ideal projective measurements on the system upon application of randomized unitaries. Here, we suggest that shadow tomography can be much more straightforwardly formulated for generalized measurements, or positive operator valued measures. Based on the idea of the least-square estimator shadow tomography with generalized measurements is both more general and simpler than the traditional formulation with randomization of unitaries. In particular, this formulation allows us to analyze theoretical aspects of shadow tomography in detail. For example, we provide a detailed study of the implication of symmetries in shadow tomography. Moreover, with this generalization we also demonstrate how the optimization of measurements for shadow tomography tailored toward a particular set of observables can be carried out.
Quantum process tomography (QPT) is crucial for advancing quantum technologies, including quantum computers, quantum networks and quantum sensors. Shadow process tomography (SPT) utilizes the Choi isomorphism to map QPT to shadow state tomography (SST), significantly reducing the sample complexity for extracting information from quantum processes. However, SPT relies on random unitary operators and complicates the determination of the optimal unitary operator that minimizes the shadow norm, which is the key factor influencing the sample complexity. In this work, we propose a generalized SPT framework that minimizes the shadow norm by replacing unitary operators with generalized measurements (POVMs). This approach, termed shadow process tomography with POVMs (POVM-SPT), uses convex optimization to identify the optimal POVM for minimizing the shadow norm, thereby further reducing sample complexity. We demonstrate the identification of the optimal POVM through numerical simulations and provide the corresponding optimization algorithms. Our numerical experiments demonstrate that POVM-SPT achieves a substantial reduction in shadow norm compared to conventional SPT, with an approximate 7-fold improvement for single-qubit input states and a remarkable 2180-fold enhancement for 64-qubit input states. These results reveal that POVM-SPT offers significant advantages in simplifying SPT tasks, particularly for large-scale quantum systems.
Randomized measurement protocols such as classical shadows represent powerful resources for quantum technologies, with applications ranging from quantum state characterization and process tomography to machine learning and error mitigation. Recently, the notion of measurement dual frames, in which classical shadows are generalized to dual operators of positive operator-valued measure (POVM) effects, resurfaced in the literature. This brought attention to additional degrees of freedom in the postprocessing stage of randomized measurements that are often neglected by established techniques. In this work, we leverage dual frames to construct improved observable estimators from informationally complete measurement samples. We introduce novel classes of parametrized frame superoperators and optimization-free dual frames based on empirical frequencies, which offer advantages over their canonical counterparts while retaining computational efficiency. Remarkably, this comes at almost no quantum or classical cost, thus rendering dual frame optimization a valuable addition to the randomized measurement toolbox. Published by the American Physical Society 2024
Shadow tomography for quantum states provides a sample efficient approach for predicting the properties of quantum systems when the properties are restricted to expectation values of 2-outcome POVMs. However, these shadow tomography procedures yield poor bounds if there are more than 2 outcomes per measurement. In this paper, we consider a general problem of learning properties from unknown quantum states: given an unknown d-dimensional quantum state ρ and M unknown quantum measurements M1,...,MM with K≥2 outcomes, estimating the probability distribution for applying Mi on ρ to within total variation distance ε. Compared to the special case when K=2, we need to learn unknown distributions instead of values. We develop an online shadow tomography procedure that solves this problem with high success probability requiring O~(Klog2Mlogd/ε4) copies of ρ. We further prove an information-theoretic lower bound that at least Ω(min{d2,K+logM}/ε2) copies of ρ are required to solve this problem with high success probability. Our shadow tomography procedure requires sample complexity with only logarithmic dependence on M and d and is sample-optimal for the dependence on K.
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 E1,…,EM, the goal is to estimate all expectation values {Tr(ρEi)}i=1M to additive accuracy ε 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(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.
Classical shadow tomography is a powerful randomized measurement protocol for predicting many properties of a quantum state with few measurements. Two classical shadow protocols have been extensively studied in the literature: the single-qubit (local) Pauli measurement, which is well suited for predicting local operators but inefficient for large operators; and the global Clifford measurement, which is efficient for low-rank operators but infeasible on near-term quantum devices due to the extensive gate overhead. In this work, we demonstrate a scalable classical shadow tomography approach for generic randomized measurements implemented with finite-depth local Clifford random unitary circuits, which interpolates between the limits of Pauli and Clifford measurements. The method combines the recently proposed locally-scrambled classical shadow tomography framework with tensor network techniques to achieve scalability for computing the classical shadow reconstruction map and evaluating various physical properties. The method enables classical shadow tomography to be performed on shallow quantum circuits with superior sample efficiency and minimal gate overhead and is friendly to noisy intermediate-scale quantum (NISQ) devices. We show that the shallow-circuit measurement protocol provides immediate, exponential advantages over the Pauli measurement protocol for predicting quasi-local operators. It also enables a more efficient fidelity estimation compared to the Pauli measurement.
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.