Transmitting information about quantum states over classical noisy channels is an important problem with applications to science, computing, and sensing. This task, however, poses fundamental challenges due to the exponential scaling of state space with system size. We introduce shadow tomography-based transmission with unequal error protection (STT-UEP), a novel communication protocol that enables efficient transmission of properties of quantum states, allowing decoder-side estimation of arbitrary local Pauli observables. Unlike conventional approaches requiring the transmission of a number of bits that is exponential in the number of qubits, STT-UEP achieves communication complexity that scales logarithmically with the number of observables, depending on the observable weight. The protocol exploits classical shadow tomography for measurement efficiency, and applies unequal error protection by encoding measurement bases with stronger channel codes than measurement outcomes. We provide theoretical guarantees on estimation accuracy as a function of the bit error probability of the classical channel, and validate the approach against several benchmarks via numerical results.
The fragile nature of quantum information makes it practically impossible to completely isolate a quantum state from noise under quantum channel transmissions. Quantum networks are complex systems formed by the interconnection of quantum processing devices through quantum channels. In this context, characterizing how channels introduce noise in transmitted quantum states is of paramount importance. Precise descriptions of the error distributions introduced by non-unitary quantum channels can inform quantum error correction protocols to tailor operations for the particular error model. In addition, characterizing such errors by monitoring the network with end-to-end measurements enables end-nodes to infer the status of network links. In this work, we address the end-to-end characterization of quantum channels in a quantum network by introducing the problem of Quantum Network Tomography. The solution for this problem is an estimator for the probabilities that define a Kraus decomposition for all quantum channels in the network, using measurements performed exclusively in the end-nodes. We study this problem in detail for the case of arbitrary star quantum networks with quantum channels described by a single Pauli operator, like bit-flip quantum channels. We provide solutions for such networks with polynomial sample complexity. Our solutions provide evidence that pre-shared entanglement brings advantages for estimation in terms of the identifiability of parameters.
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.
Shadow tomography is a scalable technique to characterise the quantum state of a quantum computer or quantum simulator. The protocol is based on the transformation of the outcomes of random measurements into the so-called classical shadows, which can later be transformed into samples of expectation values of the observables of interest. By construction, classical shadows are intrinsically sensitive to readout noise. In fact, the complicated structure of the readout noise due to crosstalk appears to be detrimental to its scalability. We show that classical shadows accept much more flexible constructions beyond the standard ones, which can eventually be made more conformable with readout noise. With this construction, we show that readout errors in classical shadows can be efficiently mitigated by randomly flipping the qubit before, and the classical outcome bit after the measurement, referred to as
X-twirling. That a single
X-gate is sufficient for mitigating readout noise for classical shadows is in contrast to Clifford-twirling, where the implementation of random Clifford gates is required.
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.