arXiv subjects

Min-Hsiu Hsieh

Publications and source records attributed to Min-Hsiu Hsieh.

At least 127 records · Page 7Linked to original sources

arXiv2021

Quantum Optimization for Training Quantum Neural Networks

Training quantum neural networks (QNNs) using gradient-based or gradient-free classical optimisation approaches is severely impacted by the presence of barren plateaus in the cost landscapes. In this paper, we devise a framework for leveraging quantum optimisation algorithms to find optimal parameters of QNNs for certain tasks. To achieve this, we coherently encode the cost function of QNNs onto relative phases of a superposition state in the Hilbert space of the network parameters. The parameters are tuned with an iterative quantum optimisation structure using adaptively selected Hamiltonians. The quantum mechanism of this framework exploits hidden structure in the QNN optimisation problem and hence is expected to provide beyond-Grover speed up, mitigating the barren plateau issue.

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arXiv2021

Asymmetric Quantum Concatenated and Tensor Product Codes with Large Z-Distances

In this paper, we present a new construction of asymmetric quantum codes (AQCs) by combining classical concatenated codes (CCs) with tensor product codes (TPCs), called asymmetric quantum concatenated and tensor product codes (AQCTPCs) which have the following three advantages. First, only the outer codes in AQCTPCs need to satisfy the orthogonal constraint in quantum codes, and any classical linear code can be used for the inner, which makes AQCTPCs very easy to construct. Second, most AQCTPCs are highly degenerate, which means they can correct many more errors than their classical TPC counterparts. Consequently, we construct several families of AQCs with better parameters than known results in the literature. Third, AQCTPCs can be efficiently decoded although they are degenerate, provided that the inner and outer codes are efficiently decodable. In particular, we significantly reduce the inner decoding complexity of TPCs from $Ω(n_2a^{n_1})(a>1)$ to $O(n_2)$ by considering error degeneracy, where $n_1$ and $n_2$ are the block length of the inner code and the outer code, respectively. Furthermore, we generalize our concatenation scheme by using the generalized CCs and TPCs correspondingly.

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arXiv2021

Quantum algorithm for estimating Renyi entropies of quantum states

We describe a quantum algorithm to estimate the $α$-Renyi entropy of an unknown density matrix $ρ\in\mathcal{C}^{d\times d}$ for $α\neq 1$ by combining the recent technique of quantum singular value transformations with the method of estimating normalised traces in the one clean qubit model. We consider an oracular input model where the input state is prepared via a quantum oracle that outputs a purified version of the state, assumed to be non-singular. Our method outputs an estimate of the $α$-Renyi entropy to additive precision $ε$, using an expected total number $O\left(\frac{1}{(xε)^2}\right)$ of independent applications of a quantum circuit which coherently queries the input unitary $O\left(\frac{1}δ\log \frac{d}ε\right)$ times, in each case measuring a single output qubit. Here $δ$ is a lower cutoff on the smallest eigenvalue of $ρ$ and $x=\frac{1}{d}\!\mathop{Tr}{ρ^α}$. The expected number of measurements made in this method can be compared to results in the sample complexity model that generally require $Θ(d^2/ε^2)$ samples. Furthermore, we also show that multiplicative approximations can be obtained by iteratively using additive approximations, with an overhead logarithmic in the dimension $d$.

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arXiv2020

Toward Trainability of Quantum Neural Networks

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.

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arXiv2020

Simple Bounds for One-shot Pure-State Distillation in General Resource Theories

We present bounds for distilling many copies of a pure state from an arbitrary initial state in a general quantum resource theory. Our bounds apply to operations that are able to generate no more than a $δ$ amount of resource, where $δ\geq 0$ is a given parameter. To maximize applicability of our upper bound, we assume little structure on the set of free states under consideration besides a weak form of superadditivity of the function $G_{min}(ρ)$, which measures the overlap between $ρ$ and the set of free states. Our bounds are given in terms of this function and the robustness of resource. Known results in coherence and entanglement theory are reproduced in this more general framework.

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arXiv2020

One-shot Capacity bounds on the Simultaneous Transmission of Classical and Quantum Information

We study the communication capabilities of a quantum channel under the most general channel model known as the one-shot model. Unlike classical channels that can only be used to transmit classical information (bits), a quantum channel can be used for transmission of classical information, quantum information (qubits) and simultaneous transmission of classical and quantum information. In this work, we investigate the one-shot capabilities of a quantum channel for simultaneously transmitting of bits and qubits. This problem was studied in the asymptotic regime for a memoryless channel and a regularized characterization of the capacity region was reported. It is known that the transmission of private classical information is closely related to the problem of quantum information transmission. We resort to this idea and find achievable and converse bounds on the simultaneous transmission of the public and private classical information. then by shifting the classical private rate to the quantum information rate, the obtained rate regions will be translated into rate regions of thThis in turn, leads to a rate region for simulttaneous transmission of classical and quantum information. In the case of asymptotic i.i.d. setting, our one-shot result is evaluated to the known results in the literature. Our main tools used in the achievability proofs are position-based decoding and convex-split lemma.

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arXiv2020

On the learnability of quantum neural networks

We consider the learnability of the quantum neural network (QNN) built on the variational hybrid quantum-classical scheme, which remains largely unknown due to the non-convex optimization landscape, the measurement error, and the unavoidable gate errors introduced by noisy intermediate-scale quantum (NISQ) machines. Our contributions in this paper are multi-fold. First, we derive the utility bounds of QNN towards empirical risk minimization, and show that large gate noise, few quantum measurements, and deep circuit depth will lead to the poor utility bounds. This result also applies to the variational quantum circuits with gradient-based classical optimization, and can be of independent interest. We then prove that QNN can be treated as a differentially private (DP) model. Thirdly, we show that if a concept class can be efficiently learned by QNN, then it can also be effectively learned by QNN even with gate noise. This result implies the same learnability of QNN whether it is implemented on noiseless or noisy quantum machines. We last exhibit that the quantum statistical query (QSQ) model can be effectively simulated by noisy QNN. Since the QSQ model can tackle certain tasks with runtime speedup, our result suggests that the modified QNN implemented on NISQ devices will retain the quantum advantage. Numerical simulations support the theoretical results.

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arXiv2020

Quantum noise protects quantum classifiers against adversaries

Noise in quantum information processing is often viewed as a disruptive and difficult-to-avoid feature, especially in near-term quantum technologies. However, noise has often played beneficial roles, from enhancing weak signals in stochastic resonance to protecting the privacy of data in differential privacy. It is then natural to ask, can we harness the power of quantum noise that is beneficial to quantum computing? An important current direction for quantum computing is its application to machine learning, such as classification problems. One outstanding problem in machine learning for classification is its sensitivity to adversarial examples. These are small, undetectable perturbations from the original data where the perturbed data is completely misclassified in otherwise extremely accurate classifiers. They can also be considered as `worst-case' perturbations by unknown noise sources. We show that by taking advantage of depolarisation noise in quantum circuits for classification, a robustness bound against adversaries can be derived where the robustness improves with increasing noise. This robustness property is intimately connected with an important security concept called differential privacy which can be extended to quantum differential privacy. For the protection of quantum data, this is the first quantum protocol that can be used against the most general adversaries. Furthermore, we show how the robustness in the classical case can be sensitive to the details of the classification model, but in the quantum case the details of classification model are absent, thus also providing a potential quantum advantage for classical data that is independent of quantum speedups. This opens the opportunity to explore other ways in which quantum noise can be used in our favour, as well as identifying other ways quantum algorithms can be helpful that is independent of quantum speedups.

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arXiv2020

Quantum transfer component analysis for domain adaptation

Domain adaptation, a crucial sub-field of transfer learning, aims to utilize known knowledge of one data set to accomplish tasks on another data set. In this paper, we perform one of the most representative domain adaptation algorithms, transfer component analysis (TCA), on quantum devices. Two different quantum implementations of this transfer learning algorithm; namely, the linear-algebra-based quantum TCA algorithm and the variational quantum TCA algorithm, are presented. The algorithmic complexity of the linear-algebra-based quantum TCA algorithm is $O(\mathrm{poly}(\log (n_{s} + n_{t})))$, where $n_{s}$ and $n_{t}$ are input sample size. Compared with the corresponding classical algorithm, the linear-algebra-based quantum TCA can be performed on a universal quantum computer with exponential speedup in the number of given samples. Finally, the variational quantum TCA algorithm based on a quantum-classical hybrid procedure, that can be implemented on the near term quantum devices, is proposed.

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arXiv2019

On Dimension-free Tail Inequalities for Sums of Random Matrices and Applications

In this paper, we present a new framework to obtain tail inequalities for sums of random matrices. Compared with existing works, our tail inequalities have the following characteristics: 1) high feasibility--they can be used to study the tail behavior of various matrix functions, e.g., arbitrary matrix norms, the absolute value of the sum of the sum of the $j$ largest singular values (resp. eigenvalues) of complex matrices (resp. Hermitian matrices); and 2) independence of matrix dimension --- they do not have the matrix-dimension term as a product factor, and thus are suitable to the scenario of high-dimensional or infinite-dimensional random matrices. The price we pay to obtain these advantages is that the convergence rate of the resulting inequalities will become slow when the number of summand random matrices is large. We also develop the tail inequalities for matrix random series and matrix martingale difference sequence. We also demonstrate usefulness of our tail bounds in several fields. In compressed sensing, we employ the resulted tail inequalities to achieve a proof of the restricted isometry property when the measurement matrix is the sum of random matrices without any assumption on the distributions of matrix entries. In probability theory, we derive a new upper bound to the supreme of stochastic processes. In machine learning, we prove new expectation bounds of sums of random matrices matrix and obtain matrix approximation schemes via random sampling. In quantum information, we show a new analysis relating to the fractional cover number of quantum hypergraphs. In theoretical computer science, we obtain randomness-efficient samplers using matrix expander graphs that can be efficiently implemented in time without dependence on matrix dimensions.

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arXiv2019

Quantum algorithm for finding the negative curvature direction in non-convex optimization

We present an efficient quantum algorithm aiming to find the negative curvature direction for escaping the saddle point, which is the critical subroutine for many second-order non-convex optimization algorithms. We prove that our algorithm could produce the target state corresponding to the negative curvature direction with query complexity O(polylog(d) /ε), where d is the dimension of the optimization function. The quantum negative curvature finding algorithm is exponentially faster than any known classical method which takes time at least O(d /\sqrtε). Moreover, we propose an efficient quantum algorithm to achieve the classical read-out of the target state. Our classical read-out algorithm runs exponentially faster on the degree of d than existing counterparts.

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arXiv2019

A Quantum-inspired Algorithm for General Minimum Conical Hull Problems

A wide range of fundamental machine learning tasks that are addressed by the maximum a posteriori estimation can be reduced to a general minimum conical hull problem. The best-known solution to tackle general minimum conical hull problems is the divide-and-conquer anchoring learning scheme (DCA), whose runtime complexity is polynomial in size. However, big data is pushing these polynomial algorithms to their performance limits. In this paper, we propose a sublinear classical algorithm to tackle general minimum conical hull problems when the input has stored in a sample-based low-overhead data structure. The algorithm's runtime complexity is polynomial in the rank and polylogarithmic in size. The proposed algorithm achieves the exponential speedup over DCA and, therefore, provides advantages for high dimensional problems.

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arXiv2019

Matrix Poincaré, Φ-Sobolev inequalities, and quantum ensembles

Sobolev-type inequalities have been extensively studied in the frameworks of real-valued functions and non-commutative $\mathbb{L}_p$ spaces, and have proven useful in bounding the time evolution of classical/quantum Markov processes, among many other applications. In this paper, we consider yet another fundamental setting - matrix-valued functions - and prove new Sobolev-type inequalities for them. Our technical contributions are two-fold: (i) we establish a series of matrix Poincaré inequalities for separably convex functions and general functions with Gaussian unitary ensembles inputs; and (ii) we derive $Φ$-Sobolev inequalities for matrix-valued functions defined on Boolean hypercubes and for those with Gaussian distributions. Our results recover the corresponding classical inequalities (i.e.~real-valued functions) when the matrix has one dimension. Finally, as an application of our technical outcomes, we derive the upper bounds for a fundamental entropic quantity - the Holevo quantity - in quantum information science since classical-quantum channels are a special instance of matrix-valued functions. This is obtained through the equivalence between the constants in the strong data processing inequality and the $Φ$-Sobolev inequality.

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arXiv2019

Quantum Sphere-Packing Bounds with Polynomial Prefactors

We study lower bounds on the optimal error probability in classical coding over classical-quantum channels at rates below the capacity, commonly termed quantum sphere-packing bounds. Winter and Dalai have derived such bounds for classical-quantum channels; however, the exponents in their bounds only coincide when the channel is classical. In this paper, we show that these two exponents admit a variational representation and are related by the Golden-Thompson inequality, reaffirming that Dalai's expression is stronger in general classical-quantum channels. Second, we establish a sphere-packing bound for classical-quantum channels, which significantly improves Dalai's prefactor from the order of subexponential to polynomial. Furthermore, the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes in the order of $o(\log n / n)$, indicating our sphere-packing bound is almost exact in the high rate regime. Finally, for a special class of symmetric classical-quantum channels, we can completely characterize its optimal error probability without the constant composition code assumption. The main technical contributions are two converse Hoeffding bounds for quantum hypothesis testing and the saddle-point properties of error exponent functions.

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arXiv2019

Efficient Online Quantum Generative Adversarial Learning Algorithms with Applications

The exploration of quantum algorithms that possess quantum advantages is a central topic in quantum computation and quantum information processing. One potential candidate in this area is quantum generative adversarial learning (QuGAL), which conceptually has exponential advantages over classical adversarial networks. However, the corresponding learning algorithm remains obscured. In this paper, we propose the first quantum generative adversarial learning algorithm-- the quantum multiplicative matrix weight algorithm (QMMW)-- which enables the efficient processing of fundamental tasks. The computational complexity of QMMW is polynomially proportional to the number of training rounds and logarithmically proportional to the input size. The core concept of the proposed algorithm combines QuGAL with online learning. We exploit the implementation of QuGAL with parameterized quantum circuits, and numerical experiments for the task of entanglement test for pure state are provided to support our claims.

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arXiv2019

Publicness, Privacy and Confidentiality in the Single-Serving Quantum Broadcast Channel

The 2-receiver broadcast channel is studied: a network with three parties where the transmitter and one of the receivers are the primarily involved parties and the other receiver considered as third party. The messages that are determined to be communicated are classified into public, private and confidential based on the information they convey. The public message contains information intended for both parties and is required to be decoded correctly by both of them, the private message is intended for the primary party only, however, there is no secrecy requirement imposed upon it meaning that it can possibly be exposed to the third party and finally the confidential message containing information intended exclusively for the primary party such that this information must be kept completely secret from the other receiver. A trade-off arises between the rates of the three messages, when one of the rates is high, the other rates may need to be reduced to guarantee the reliable transmission of all three messages. The encoder performs the necessary equivocation by virtue of dummy random numbers whose rate is assumed to be limited and should be considered in the trade-off as well. We study this trade-off in the one-shot regime of a quantum broadcast channel by providing achievability and (weak) converse regions. In the achievability, we prove and use a conditional version of the convex-split lemma as well as position-based decoding. By studying the asymptotic behaviour of our bounds, we will recover several well-known asymptotic results in the literature.

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arXiv2019

Channel Simulation and Coded Source Compression

Coded source compression, also known as source compression with helpers, has been a major variant of distributed source compression, but has hitherto received little attention in the quantum regime. This work treats and solves the corresponding quantum coded source compression through an observation that connects coded source compression with channel simulation. First, we consider classical source coding with quantum side information where the quantum side information is observed by a helper and sent to the decoder via a classical channel. We derive a single-letter characterization of the achievable rate region for this problem. The direct coding theorem of our result is proved via the measurement compression theory of Winter, a quantum-to-classical channel simulation. Our result reveals that a helper's scheme which separately conducts a measurement and a compression is suboptimal, and measurement compression seems necessary to achieve the optimal rate region. We then study coded source compression in the fully quantum regime, where two different scenarios are considered depending on the types of communication channels between the legitimate source and the receiver. We further allow entanglement assistance from the quantum helper in both scenarios. We characterize the involved quantum resources, and derive single-letter expressions of the achievable rate region. The direct coding proofs are based on well-known quantum protocols, the quantum state merging protocol and the fully quantum Slepian-Wolf protocol, together with the quantum reverse Shannon theorem.

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arXiv2019

Quantum Speedup in Adaptive Boosting of Binary Classification

In classical machine learning, a set of weak classifiers can be adaptively combined to form a strong classifier for improving the overall performance, a technique called adaptive boosting (or AdaBoost). However, constructing the strong classifier for a large data set is typically resource consuming. Here we propose a quantum extension of AdaBoost, demonstrating a quantum algorithm that can output the optimal strong classifier with a quadratic speedup in the number of queries of the weak classifiers. Our results also include a generalization of the standard AdaBoost to the cases where the output of each classifier may be probabilistic even for the same input. We prove that the update rules and the query complexity of the non-deterministic classifiers are the same as those of deterministic classifiers, which may be of independent interest to the classical machine-learning community. Furthermore, the AdaBoost algorithm can also be applied to data encoded in the form of quantum states; we show how the training set can be simplified by using the tools of t-design. Our approach describes a model of quantum machine learning where quantum speedup is achieved in finding the optimal classifier, which can then be applied for classical machine-learning applications.

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