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In this work, we compute the number of [[n,k]]d stabilizer codes made up of d-dimensional qudits, for arbitrary positive integers d. In a seminal work by Gross \cite{Gross2006} the number of [[n,k]]d stabilizer codes was computed for the case when d is a prime (or the power of a prime, i.e., d=pm, but when the qudits are Galois-qudits). The proof in \cite{Gross2006} is inapplicable to the non-prime case. For our proof, we introduce a group structure to [[n,k]]d codes, and use this in conjunction with the Chinese remainder theorem to count the number of [[n

Quantum computers are next-generation devices that hold promise to perform calculations beyond the reach of classical computers. A leading method towards achieving this goal is through quantum machine learning, especially quantum generative learning. Due to the intrinsic probabilistic nature of quantum mechanics, it is reasonable to postulate that quantum generative learning models (QGLMs) may surpass their classical counterparts. As such, QGLMs are receiving growing attention from the quantum physics and computer science communities, where various QGLMs that can be efficiently implemented on near-term quantum machines with potential computational advantages are proposed. In this paper, we review the current progress of QGLMs from the perspective of machine learning. Particularly, we interpret these QGLMs, covering quantum circuit Born machines, quantum generative adversarial networks, quantum Boltzmann machines, and quantum variational autoencoders, as the quantum extension of classical generative learning models. In this context, we explore their intrinsic relations and their fundamental differences. We further summarize the potential applications of QGLMs in both conventional machine learning tasks and quantum physics. Last, we discuss the challenges and further research directions for QGLMs.

With the rapid development of quantum computing automatic verification of quantum circuits becomes more and more important While several decision diagrams DDs have been introduced in quantum circuit simulation and verification none of them supports symbolic computation Algorithmic manipulations of symbolic objects however have been identified as crucial if not indispensable for several verification tasks This paper proposes the first decision diagram approach for operating symbolic objects and verifying quantum circuits with symbolic terms As a notable example our symbolic tensor decision diagrams symbolic TDD could verify the functionality of the 160 qubit quantum Fourier transform circuit within three minutes Moreover as demonstrated on Bernstein Vazirani algorithm Grover s algorithm and the bit flip error correction code the symbolic TDD enables efficient verification of quantum circuits with user supplied oracles and or classical controls

The recognition of entanglement states is a notoriously difficult problem when no prior information is available. Here, we propose an efficient quantum adversarial bipartite entanglement detection scheme to address this issue. Our proposal reformulates the bipartite entanglement detection as a two-player zero-sum game completed by parameterized quantum circuits, where a two-outcome measurement can be used to query a classical binary result about whether the input state is bipartite entangled or not. In principle, for an N-qubit quantum state, the runtime complexity of our proposal is O(poly(N)T) with T being the number of iterations. We experimentally implement our protocol on a linear optical network and exhibit its effectiveness to accomplish the bipartite entanglement detection for 5-qubit quantum pure states and 2-qubit quantum mixed states. Our work paves the way for using near-term quantum machines to tackle entanglement detection on multipartite entangled quantum systems.

Entanglement-assisted concatenated quantum codes (EACQCs), constructed by concatenating two quantum codes, are proposed. These EACQCs show significant advantages over standard concatenated quantum codes (CQCs). First, we prove that, unlike standard CQCs, EACQCs can beat the nondegenerate Hamming bound for entanglement-assisted quantum error-correction codes (EAQECCs). Second, we construct families of EACQCs with parameters better than the best-known standard quantum error-correction codes (QECCs) and EAQECCs. Moreover, these EACQCs require very few Einstein-Podolsky-Rosen (EPR) pairs to begin with. Finally, it is shown that EACQCs make entanglement-assisted quantum communication possible, even if the ebits are noisy. Furthermore, EACQCs can outperform CQCs in entanglement fidelity over depolarizing channels if the ebits are less noisy than the qubits. We show that the error-probability threshold of EACQCs is larger than that of CQCs when the error rate of ebits is sufficiently lower than that of qubits. Specifically, we derive a high threshold of 47% when the error probability of the preshared entanglement is 1% to that of qubits.

BACKGROUND & AIMS: The outcome of HBV infection, including the dynamics of HBsAg and HBV virological reactivation, among patients coinfected with HCV receiving direct-acting antivirals (DAAs) remains unclear. Thus, we aimed to analyze HBV-related outcomes in these patients. METHODS: Serial HBsAg and HBV DNA levels were measured in 79 HBV/HCV-coinfected patients receiving DAAs (13 receiving anti-HBV nucleot(s)ide analog [NUC] therapy simultaneously). The endpoints included HBsAg dynamics and seroclearance, HBV reactivation (HBV DNA >1 log increase or >100 IU/ml if undetectable at baseline) and HBV-related clinical reactivation. RESULTS: HBsAg levels declined from a median of 73.3 IU/ml at baseline to 16.2 IU/ml at the end-of-DAA treatment and increased to 94.1 IU/ml at 12 months post-treatment. During a mean 11.1-months of follow-up, 8 (10.1%) patients experienced HBsAg seroclearance and 30 (38.0%) HBV reactivation (12-month cumulative incidence, 10.3% and 40.4%, respectively). Patients with pre-treatment HBsAg ≤10 IU/ml had a significantly higher rate of HBsAg seroclearance (hazard ratio [HR] 8.52; 95% CI 1.048-69.312) and lower risk of HBV reactivation than those with pre-treatment HBsAg >10 IU/ml (HR 2.88; 95% CI 1.057-7.844) in multivariate analyses. Six patients (4 cirrhotics) not receiving NUC therapy experienced HBV-related clinical reactivation; 3 of the 4 cirrhotics developed liver failure and 2 died despite immediate NUC therapy. Compared to untreated HBV-monoinfected patients, HBV/HCV-coinfected patients without NUC prophylaxis had a similar rate of HBsAg seroclearance, but a significantly higher risk of HBV reactivation following DAA therapy (HR 6.59; 95% CI 2.488-17.432). CONCLUSIONS: DAA-treated HBV/HCV-coinfected patients had significantly higher rates of HBV seroclearance, particularly among those with low pre-treatment HBsAg titer, but were at higher risk of HBV reactivation, particularly among those with higher pre-treatment HBsAg titer. Prophylacti

Abstract Quantum neural network (QNN), or equivalently, the variational quantum circuits with a gradient-based classical optimizer, has been broadly applied to many experimental proposals for noisy intermediate scale quantum (NISQ) devices. However, the learning capability of QNN remains largely unknown due to the non-convex optimization landscape, the measurement error, and the unavoidable gate noise introduced by NISQ machines. In this study, we theoretically explore the learnability of QNN from the perspective of the trainability and generalization. Particularly, we derive the convergence performance of QNN under the NISQ setting, and identify classes of computationally hard concepts that can be efficiently learned by QNN. Our results demonstrate that large gate noise, few quantum measurements, and deep circuit depth will lead to poor convergence rates of QNN towards the empirical risk minimization. Moreover, we prove that any concept class, which is efficiently learnable by a restricted quantum statistical query (QSQ) learning model, can also be efficiently learned by QNN. Since the restricted QSQ learning model can tackle certain problems such as parity learning with a runtime speedup, our result suggests that QNN established on NISQ devices will retain the quantum advantage. Our work provides the theoretical guidance for developing advanced QNNs and opens up avenues for exploring quantum advantages using NISQ devices.

Many quantum algorithms that claim speed-up over their classical counterparts only generate quantum states as solutions instead of their final classical description. The additional step to decode quantum states into classical vectors normally will destroy the quantum advantage in most scenarios because all existing tomographic methods require runtime that is polynomial with respect to the state dimension. In this work, we present an efficient read-out protocol that yields the classical vector form of the generated state, so it will achieve the end-to-end advantage for those quantum algorithms. Our protocol suits the case that the output state lies in the row space of the input matrix, of rank $r$, that is stored in the quantum random access memory. The quantum resources for decoding the state in $\ell^2$ norm with $\epsilon$ error require $\poly(r,1/\epsilon)$ copies of the output state and $\poly(r, \kappa^r,1/\epsilon)$ queries to the input oracles, where $\kappa$ is the condition number of the input matrix. With our read-out protocol, we completely characterise the end-to-end resources for quantum linear equation solvers and quantum singular value decomposition. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure, which we believe, will be of independent interest.

Quantum Neural Networks (QNNs) have been recently proposed as generalizations of classical neural networks to achieve the quantum speed-up. Despite the potential to outperform classical models, serious bottlenecks exist for training QNNs; namely, QNNs with random structures have poor trainability due to the vanishing gradient with rate exponential to the input qubit number. The vanishing gradient could seriously influence the applications of large-size QNNs. In this work, we provide a viable solution with theoretical guarantees. Specifically, we prove that QNNs with tree tensor and step controlled architectures have gradients that vanish at most polynomially with the qubit number. We numerically demonstrate QNNs with tree tensor and step controlled structures for the application of binary classification. Simulations show faster convergent rates and better accuracy compared to QNNs with random structures.