Quantum neural networks (QNNs) have emerged as a leading strategy to establish applications in machine learning, chemistry, and optimization. While the applications of QNN have been widely investigated, its theoretical foundation remains less understood. In this paper, we formulate a theoretical framework for the expressive ability of data re-uploading quantum neural networks that consist of interleaved encoding circuit blocks and trainable circuit blocks. First, we prove that single-qubit quantum neural networks can approximate any univariate function by mapping the model to a partial Fourier series. We in particular establish the exact correlations between the parameters of the trainable gates and the Fourier coefficients, resolving an open problem on the universal approximation property of QNN. Second, we discuss the limitations of single-qubit native QNNs on approximating multivariate functions by analyzing the frequency spectrum and the flexibility of Fourier coefficients. We further demonstrate the expressivity and limitations of single-qubit native QNNs via numerical experiments. We believe these results would improve our understanding of QNNs and provide a helpful guideline for designing powerful QNNs for machine learning tasks.

Distributed quantum information processing is essential for building quantum networks and enabling more extensive quantum computations. In this regime, several spatially separated parties share a multipartite quantum system, and the most natural set of operations are Local Operations and Classical Communication (LOCC). As a pivotal part in quantum information theory and practice, LOCC has led to many vital protocols such as quantum teleportation. However, designing practical LOCC protocols is challenging due to LOCC’s intractable structure and limitations set by near-term quantum devices. Here we introduce LOCCNet, a machine learning framework facilitating protocol design and optimization for distributed quantum information processing tasks. As applications, we explore various quantum information tasks such as entanglement distillation, quantum state discrimination, and quantum channel simulation. We discover novel protocols with evident improvements, in particular, for entanglement distillation with quantum states of interest in quantum information. Our approach opens up new opportunities for exploring entanglement and its applications with machine learning, which will potentially sharpen our understanding of the power and limitations of LOCC.

Magic state manipulation is a crucial component in the leading approaches to realizing scalable, fault-tolerant, and universal quantum computation. Related to magic state manipulation is the resource theory of magic states, for which one of the goals is to characterize and quantify quantum “magic.” In this paper, we introduce the family of thauma measures to quantify the amount of magic in a quantum state, and we exploit this family of measures to address several open questions in the resource theory of magic states. As a first application, we use the min-thauma to bound the regularized relative entropy of magic. As a consequence of this bound, we find that two classes of states with maximal mana, a previously established magic measure, cannot be interconverted in the asymptotic regime at a rate equal to one. This result resolves a basic question in the resource theory of magic states and reveals a fundamental difference between the resource theory of magic states and other resource theories such as entanglement and coherence. As a second application, we establish the hypothesis testing thauma as an efficiently computable benchmark for the one-shot distillable magic, which in turn leads to a variety of bounds on the rate at which magic can be distilled, as well as on the overhead of magic state distillation. Finally, we prove that the max-thauma can outperform mana in benchmarking the efficiency of magic state distillation.

To achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we develop a resource theory for magic quantum channels to characterize and quantify the quantum “magic” or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimension $d$, it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely positive-Wigner-preserving quantum operations as free operations, and we show that they can be efficiently simulated via a classical algorithm. Second, we introduce two efficiently computable magic measures for quantum channels, called the mana and thauma of a quantum channel. As applications, we show that these measures not only provide fundamental limits on the distillable magic of quantum channels, but they also lead to lower bounds for the task of synthesizing non-Clifford gates. Third, we propose a classical algorithm for simulating noisy quantum circuits, whose sample complexity can be quantified by the mana of a quantum channel. We further show that this algorithm can outperform another approach for simulating noisy quantum circuits, based on channel robustness. Finally, we explore the threshold of non-stabilizerness for basic quantum circuits under depolarizing noise.

We study the one-shot and asymptotic quantum communication assisted with the positive-partial-transpose-preserving (PPT) and no-signalling (NS) codes. We first show improved general semidefinite programming (SDP) finite blocklength converse bounds for quantum communication with a given infidelity tolerance and utilize them to study the depolarizing channel and amplitude damping channel in a small blocklength. Based on the one-shot bounds, we then derive a general SDP strong converse bound for the quantum capacity of an arbitrary quantum channel. In particular, we prove that the SDP strong converse bound is always smaller than or equal to the partial transposition bound introduced by Holevo and Werner, and the inequality could be strict. Furthermore, we show that the SDP strong converse bound can be refined as the max-Rains information, which is an analog to the Rains information introduced in [Tomamichel/Wilde/Winter, IEEE Trans. Inf. Theory 63:715, 2017]. This also implies that it is always no smaller than the Rains information. Finally, we establish an inequality relationship among some of these known strong converse bounds on quantum capacity.

We show a meaningful theory of classical communication over quantum channels when assisted by no-signalling (NS) and PPT-preserving (PPT) codes, for which both the optimal success probability of a given transmission rate and one-shot $ε$-error capacity are formalized as semidefinite programs (SDPs). Based on this, we obtain improved SDP finite blocklength converse bounds for general quantum channels, which also reduce to the converse bound of Polyanskiy, Poor, and Verdú for classical channels. Furthermore, we derive two SDP strong converse bounds for the classical capacity of a general quantum channel: for any code with a rate exceeding either of the two bounds of the channel, the success probability vanishes exponentially fast as the number of channel uses increases. In particular, applying our efficiently computable bounds, we derive improved upper bounds to the classical capacity of the class of amplitude damping channels. Remarkably, we also establish the strong converse property for the classical and private capacities of a new class of quantum channels. We finally study the zero-error setting and provide efficiently computable upper bounds on the one-shot zero-error capacity of a general quantum channel.

Quantum Lovász number, as a natural generalization of the celebrated Lovśz number in graph theory, is the best known efficiently computable upper bound of the entanglement-assisted zero-error classical capacity of a quantum channel. However, it remains an intriguing open problem whether this number always coincides with the entanglement-assisted zero-error capacity. Here we show that there is a strict gap between these two quantities for a class of explicitly constructed qutrit-to-qutrit channels. Our key approach is to show that for this class of channels both the one-shot zero-error communication capacity and the one-shot zero-error simulation cost in the presence of quantum no-signalling correlations (QNSC), a more broader class of resources than entanglement, are exactly four, while the quantum Lov’ź number is strictly larger than that. Interestingly, for this class of channels, the quantum fractional packing number is strictly larger than the QNSC-assisted or quantum feedback-assisted zero-error classical capacity. Note that these quantities are all equal for classical channels.

We demonstrate the irreversibility of asymptotic entanglement manipulation under quantum operations that completely preserve positivity of partial transpose (PPT), which resolves a major open problem in quantum information theory. To be more specific, we show that for any rank-two mixed state supporting on the $3øtimes3$ antisymmetric subspace, the amount of distillable entanglement by PPT operations is strictly smaller than one entanglement bit (ebit) while its entanglement cost under PPT operations is exactly one ebit. As a byproduct, we find that for this class of quantum states, both the Rains' bound and its regularization, are strictly less than the asymptotic relative entropy of entanglement with respect to PPT states. So in general there is no unique entanglement measure for the manipulation of entanglement by PPT operations. We further present a feasible sufficient condition for the irreversibility of entanglement manipulation under PPT operations.