We study single-copy shadow tomography in the adversarial robust setting, where the goal is to learn the expectation values of M observables O1,…,OM with ε accuracy, but γ-fraction of the outcomes can be arbitrarily corrupted by an adversary. We show that all non-adaptive shadow tomography algorithms must incur an error of ε=Ω~(γmin{M,d}) for some choice of observables, even with unlimited copies. Unfortunately, the classical shadows algorithm by [HKP20] and naive algorithms that directly measure each observable suffer even more. We design an algorithm that achieves an error of ε=O~(γmaxi∈[M]∥Oi∥HS), which nearly matches our worst-case error lower bound for M≥d and guarantees better accuracy when the observables have stronger structure. Remarkably, the algorithm only needs n=γ21log(M/δ) copies to achieve that error with probability at least 1−δ, matching the sample complexity of the classical shadows algorithm that achieves the same error without corrupted measurement outcomes. Our algorithm is conceptually simple and easy to implement. Classical simulation for fidelity estimation shows that our algorithm enjoys much stronger robustness than [HKP20] under adversarial noise. Finally, based on a reduction from full-state tomography to shadow tomography, we prove that for rank r states, both the near-optimal asymptotic error of ε=O~(γr) and copy complexity O~(dr2/ε2)=O~(dr/γ2) can be achieved for adversarially robust state tomography, closing the large gap in [ABCL25] where optimal error can only be achieved using pseudo-polynomial number of copies in d.
Accurate methods of assessing the performance of quantum gates are extremely important. Quantum process tomography and randomized benchmarking are the current favored methods. Quantum process tomography gives detailed information, but significant approximations must be made to reduce this information to a form quantum error correction simulations can use. Randomized benchmarking typically outputs just a single number, the fidelity, giving no information on the structure of errors during the gate. Neither method is optimized to assess gate performance within an error detection circuit, where gates will be actually used in a large-scale quantum computer. Specifically, the important issues of error composition and error propagation lie outside the scope of both methods. We present a fast, simple, and scalable method of obtaining exactly the information required to perform effective quantum error correction from the output of continuously running error detection circuits, enabling accurate prediction of large-scale behavior.
We propose an iterative algorithm that computes the maximum-likelihood estimate in quantum state tomography. The optimization error of the algorithm converges to zero at an O((1/k)logD) rate, where k denotes the number of iterations and D denotes the dimension of the quantum state. The per-iteration computational complexity of the algorithm is O(D3+ND2), where N denotes the number of measurement outcomes. The algorithm can be considered as a parameter-free correction of the RρR method [A. I. Lvovsky. Iterative maximum-likelihood reconstruction in quantum homodyne tomography. \textit{J. Opt. B: Quantum Semiclass. Opt.} 2004] [G. Molina-Terriza et al. Triggered qutrits for quantum communication protocols. \textit{Phys. Rev. Lett.} 2004.].
At the intersection of quantum computing and machine learning, quantum machine learning (QML) is poised to revolutionize artificial intelligence. However, the vulnerability of the current generation of quantum computers to noise and computational error poses a significant barrier to this vision. Whilst quantum error correction (QEC) offers a promising solution for almost any type of hardware noise, its application requires millions of qubits to encode even a simple logical algorithm, rendering it impractical in the near term. In this chapter, we examine strategies for integrating QEC and quantum error detection (QED) into QML under realistic resource constraints. We first quantify the resource demands of fully error-corrected QML and propose a partial QEC approach that reduces overhead while enabling error correction. We then demonstrate the application of a simple QED method, evaluating its impact on QML performance and highlighting challenges we have yet to overcome before we achieve fully fault-tolerant QML.