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A large body of recent work has begun to explore the potential of parametrized quantum circuits (PQCs) as machine learning models, within the framework of hybrid quantum-classical optimization. In particular, theoretical guarantees on the out-of-sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC-based models that depend explicitly on the strategy used for data-encoding. These imply bounds on the performance of trained PQC-based models on unseen data. Moreover, our results facilitate the selection of optimal data-encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC-based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC-based models via trigonometric functions. Our generalization bounds emphasize the importance of well-considered data-encoding strategies for PQC-based models.
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To harness the potential of noisy intermediate-scale quantum devices, it is paramount to find the best type of circuits to run hybrid quantum-classical algorithms. Key candidates are parametrized quantum circuits that can be effectively implemented on current devices. Here, we evaluate the capacity and trainability of these circuits using the geometric structure of the parameter space via the effective quantum dimension, which reveals the expressive power of circuits in general as well as of particular initialization strategies. We assess the expressive power of various popular circuit types and find striking differences depending on the type of entangling gates used. Particular circuits are characterized by scaling laws in their expressiveness. We identify a transition in the quantum geometry of the parameter space, which leads to a decay of the quantum natural gradient for deep circuits. For shallow circuits, the quantum natural gradient can be orders of magnitude larger in value compared to the regular gradient; however, both of them can suffer from vanishing gradients. By tuning a fixed set of circuit parameters to randomized ones, we find a region where the circuit is expressive, but does not suffer from barren plateaus, hinting at a good way to initialize circuits. We show an algorithm that prunes redundant parameters of a circuit without affecting its effective dimension. Our results enhance the understanding of parametrized quantum circuits and can be immediately applied to improve variational quantum algorithms.
Extracting tomographic information about quantum states is a crucial task in the quest towards devising high-precision quantum devices. Current schemes typically require measurement devices for tomography that are a priori calibrated to a high precision. Ironically, the accuracy of the measurement calibration is fundamentally limited by the accuracy of state preparation, establishing a vicious cycle. Here, we prove that this cycle can be broken and the fundamental dependence on the measurement devices significantly relaxed. We show that exploiting the natural low-rank structure of quantum states of interest suffices to arrive at a highly scalable blind tomography scheme with a classically efficient post-processing algorithm. We further improve the efficiency of our scheme by making use of the sparse structure of the calibrations. This is achieved by relaxing the blind quantum tomography problem to the task of de-mixing a sparse sum of low-rank quantum states. Building on techniques from model-based compressed sensing, we prove that the proposed algorithm recovers a low-rank quantum state and the calibration provided that the measurement model exhibits a restricted isometry property. For generic measurements, we show that our algorithm requires a close-to-optimal number measurement settings for solving the blind tomography task. Complementing these conceptual and mathematical insights, we numerically demonstrate that blind quantum tomography is possible by exploiting low-rank assumptions in a practical setting inspired by an implementation of trapped ions using constrained alternating optimization.
We provide the first inner bounds for sending private classical information over a quantum multiple access channel. We do so by using three powerful information theoretic techniques: rate splitting, quantum simultaneous decoding for multiple access channels, and a novel smoothed distributed covering lemma for classical quantum channels. Our inner bounds are given in the one shot setting and accordingly the three techniques used are all very recent ones specifically designed to work in this setting. The last technique is new to this work and is our main technical advancement. For the asymptotic iid setting, our one shot inner bounds lead to the natural quantum analogue of the best classical inner bounds for this problem.
We prove the first non-trivial one-shot inner bounds for sending quantum information over an entanglement unassisted two-sender quantum multiple access channel (QMAC) and an unassisted two-sender two-receiver quantum interference channel (QIC). Previous works only studied the unassisted QMAC in the limit of many independent and identical uses of the channel also known as the asymptotic iid limit, and did not study the unassisted QIC at all. We employ two techniques, rate splitting and successive cancellation}, in order to obtain our inner bound. Rate splitting was earlier used to obtain inner bounds, avoiding time sharing, for classical channels in the asymptotic iid setting. Our main technical contribution is to extend rate splitting from the classical asymptotic iid setting to the quantum one-shot setting. In the asymptotic iid limit our one-shot inner bound for QMAC approaches the rate region of Yard, Devetak and Hayden. For the QIC we get novel non-trivial rate regions in the asymptotic iid setting. All our results also extend to the case where limited entanglement assistance is provided, in both one-shot and asymptotic iid settings. The limited entanglement results for one-setting for both QMAC and QIC are new. For the QIC the limited entanglement results are new even in the asymptotic iid setting.
In this paper, we develop a compositional vector-based semantics of positive transitive sentences in quantum natural language processing for a non-English language, i.e. Persian, to compare the parametrized quantum circuits of two synonymous sentences in two languages, English and Persian. By considering grammar+meaning of a transitive sentence, we translate DisCoCat diagram via ZX-calculus into quantum circuit form. Also, we use a bigraph method to rewrite DisCoCat diagram and turn into quantum circuit in the semantic side.
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