arXiv subjects

Min-Hsiu Hsieh

Publications and source records attributed to Min-Hsiu Hsieh.

At least 73 records · Page 4Linked to original sources

arXiv2024

Geometrically Local Quantum and Classical Codes from Subdivision

A geometrically local quantum code is an error correcting code situated within $\mathbb{R}^D$, where the checks only act on qubits within a fixed spatial distance. The main question is: What is the optimal dimension and distance for a geometrically local code? Recently, Portnoy made a significant breakthrough with codes achieving optimal dimension and distance up to polylogs. However, the construction invokes a somewhat advanced mathematical result that involves lifting a chain complex to a manifold. This paper bypasses this step and streamlines the construction by noticing that a family of good quantum low-density parity-check codes, balanced product codes, naturally carries a two-dimensional structure. Together with a new embedding result that will be shown elsewhere, this quantum code achieves the optimal dimension and distance in all dimensions. In addition, we show that the code has an optimal energy barrier. We also discuss similar results for classical codes.

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arXiv2024

Quantum-Train: Rethinking Hybrid Quantum-Classical Machine Learning in the Model Compression Perspective

We introduces the Quantum-Train(QT) framework, a novel approach that integrates quantum computing with classical machine learning algorithms to address significant challenges in data encoding, model compression, and inference hardware requirements. Even with a slight decrease in accuracy, QT achieves remarkable results by employing a quantum neural network alongside a classical mapping model, which significantly reduces the parameter count from $M$ to $O(\text{polylog} (M))$ during training. Our experiments demonstrate QT's effectiveness in classification tasks, offering insights into its potential to revolutionize machine learning by leveraging quantum computational advantages. This approach not only improves model efficiency but also reduces generalization errors, showcasing QT's potential across various machine learning applications.

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arXiv2024

Oracle Separation between Noisy Quantum Polynomial Time and the Polynomial Hierarchy

This work investigates the oracle separation between the physically motivated complexity class of noisy quantum circuits, inspired by definitions such as those presented by Chen, Cotler, Huang, and Li (2022). We establish that with a constant error rate, separation can be achieved in terms of NP. When the error rate is $Ω(\log n/n)$, we can extend this result to the separation of PH. Notably, our oracles, in all separations, do not necessitate error correction schemes or fault tolerance, as all quantum circuits are of constant depth. This indicates that even quantum computers with minor errors, without error correction, may surpass classical complexity classes under various scenarios and assumptions. We also explore various common noise settings and present new classical hardness results, generalizing those found in studies by Raz and Tal (2022) and Bassirian, Bouland, Fefferman, Gunn, and Tal (2021), which are of independent interest.

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arXiv2024

Operational Markovianization in Randomized Benchmarking

A crucial task to obtain optimal and reliable quantum devices is to quantify their overall performance. The average fidelity of quantum gates is a particular figure of merit that can be estimated efficiently by Randomized Benchmarking (RB). However, the concept of gate-fidelity itself relies on the crucial assumption that noise behaves in a predictable, time-local, or so-called Markovian manner, whose breakdown can naturally become the leading source of errors as quantum devices scale in size and depth. We analytically show that error suppression techniques such as Dynamical Decoupling (DD) and Randomized Compiling (RC) can operationally Markovianize RB: i) fast DD reduces non-Markovian RB to an exponential decay plus longer-time corrections, while on the other hand, ii) RC generally does not affect the average, but iii) it always suppresses the variance of such RB outputs. We demonstrate these effects numerically with a qubit noise model. Our results show that simple and efficient error suppression methods can simultaneously tame non-Markovian noise and allow for standard and reliable gate quality estimation, a fundamentally important task in the path toward fully functional quantum devices.

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arXiv2024

Equivalence Checking of Parameterised Quantum Circuits

Parameterised quantum circuits (PQCs) hold great promise for demonstrating quantum advantages in practical applications of quantum computation. Examples of successful applications include the variational quantum eigensolver, the quantum approximate optimisation algorithm, and quantum machine learning. However, before executing PQCs on real quantum devices, they undergo compilation and optimisation procedures. Given the inherent error-proneness of these processes, it becomes crucial to verify the equivalence between the original PQC and its compiled or optimised version. Unfortunately, most existing quantum circuit verifiers cannot directly handle parameterised quantum circuits; instead, they require parameter substitution to perform verification. In this paper, we address the critical challenge of equivalence checking for PQCs. We propose a novel compact representation for PQCs based on tensor decision diagrams. Leveraging this representation, we present an algorithm for verifying PQC equivalence without the need for instantiation. Our approach ensures both effectiveness and efficiency, as confirmed by experimental evaluations. The decision-diagram representations offer a powerful tool for analysing and verifying parameterised quantum circuits, bridging the gap between theoretical models and practical implementations.

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arXiv2024

Training Classical Neural Networks by Quantum Machine Learning

In recent years, advanced deep neural networks have required a large number of parameters for training. Therefore, finding a method to reduce the number of parameters has become crucial for achieving efficient training. This work proposes a training scheme for classical neural networks (NNs) that utilizes the exponentially large Hilbert space of a quantum system. By mapping a classical NN with $M$ parameters to a quantum neural network (QNN) with $O(\text{polylog} (M))$ rotational gate angles, we can significantly reduce the number of parameters. These gate angles can be updated to train the classical NN. Unlike existing quantum machine learning (QML) methods, the results obtained from quantum computers using our approach can be directly used on classical computers. Numerical results on the MNIST and Iris datasets are presented to demonstrate the effectiveness of our approach. Additionally, we investigate the effects of deeper QNNs and the number of measurement shots for the QNN, followed by the theoretical perspective of the proposed method. This work opens a new branch of QML and offers a practical tool that can greatly enhance the influence of QML, as the trained QML results can benefit classical computing in our daily lives.

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arXiv2024

Characterizing First Arrival Position Channels: Noise Distribution and Capacity Analysis

This paper introduces a novel mathematical model for Molecular Communication (MC) systems, utilizing First Arrival Position (FAP) as a fundamental mode of information transmission. We address two critical challenges: the characterization of FAP density and the establishment of capacity bounds for channels with vertically-drifted FAP. Our method relate macroscopic Partial Differential Equation (PDE) models to microscopic Stochastic Differential Equation (SDE) models, resulting in a precise expression that links FAP density with elliptic-type Green's function. This formula is distinguished by its wide applicability across any spatial dimensions, any drift directions, and various receiver geometries. We demonstrate the practicality of our model through case studies: 2D and 3D planar receivers. The accuracy of our formula is also validated by particle-based simulations. Advancing further, the explicit FAP density forms enable us to establish closed-form upper and lower bounds for the capacity of vertically-drifted FAP channels under a second-moment constraint, significantly advancing the understanding of FAP channels in MC systems.

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arXiv2024

Efficient Algorithms for All Port-Based Teleportation Protocols

Port-based teleportation (PBT) is a form of quantum teleportation in which no corrective unitary is required on the part of the receiver. Two primary regimes exist - deterministic PBT in which teleportation is always successful, but is imperfect, and probabilistic PBT, in which teleportation succeeds with probability less than one, but teleportation is perfect upon a success. Two further regimes exist within each of these in which the resource state used for the teleportation is fixed to a maximally entangled state, or free to be optimised. Recently, works resolved the long-standing problem of efficiently implementing port-based teleportation, tackling the two deterministic cases for qudits. Here, we provide algorithms in all four regimes for qubits. Emphasis is placed on the practicality of these algorithms, where we give polynomial improvements in the known gate complexity for PBT, as well as an exponential improvement in the required number of ancillas (albeit in separate protocols). Our approach to the implementation of the square-root measurement in PBT can be directly generalised to other highly symmetric state ensembles. For certain families of states, such a framework yields efficient algorithms in the case that the Petz recovery algorithm for the square-root measurement runs in exponential time.

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arXiv2024

Tradeoff Constructions for Quantum Locally Testable Codes

In this work, we continue the search for quantum locally testable codes (qLTCs) of new parameters by presenting three constructions that can make new qLTCs from old. The first analyses the soundness of a quantum code under Hastings' weight reduction construction for qLDPC codes arXiv:2102.10030 to give a weight reduction procedure for qLTCs. Secondly, we describe a novel `soundness amplification' procedure for qLTCs which can increase the soundness of any qLTC to a constant while preserving its distance and dimension, with an impact only felt on its locality. Finally, we apply the AEL distance amplification construction to the case of qLTCs for the first time which can turn a high-distance qLTC into one with linear distance, at the expense of other parameters. These constructions can be used on as-yet undiscovered qLTCs to obtain new parameters, but we also find a number of present applications to prove the existence of codes in previously unknown parameter regimes. In particular, applications of these operations to the hypersphere product code arXiv:1608.05089 and the hemicubic code arXiv:1911.03069 yield many previously unknown parameters. Additionally, soundness amplification can be used to produce the first asymptotically good testable quantum code (rather than locally testable) - that being one with linear distance and dimension, as well as constant soundness. Lastly, applications of all three results are described to an upcoming work.

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arXiv2024

Incorporating Zero-Probability Constraints to Device-Independent Randomness Expansion

One of the distinguishing features of quantum theory is that its measurement outcomes are usually unpredictable or, equivalently, random. Moreover, this randomness is certifiable with minimal assumptions in the so-called device-independent (DI) paradigm, where a device's behavior does not need to be presupposed but can be verified through the statistics it produces. In this work, we explore various forms of randomness that are certifiable in this setting, where two users can perform two binary-outcome measurements on their shared entangled state. In this case, even though the Clauser-Horne-Shimony-Holt (CHSH) Bell-inequality violation is a pre-requisite for the generation of DI certifiable randomness, the CHSH value alone does not generally give a tight bound on the certifiable randomness. Here, we determine the certifiable randomness when zero-probability constraints are incorporated into the task of DI randomness expansion for the standard local and global randomness and the so-called "blind" randomness. Asymptotically, we observe consistent improvements in the amount of DI certifiable randomness (of all kinds) as we increase the number zero constraints for a wide range of given CHSH Bell violations. However, if we further optimize over the allowed CHSH values, then benefits of these additional constraints over the standard CHSH-based protocol are only found in the case of global and blind randomness. In contrast, in the regimes of finite data, these zero constraints only give a slight improvement in the local randomness rate when compared with all existing protocols.

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arXiv2023

A Gell-Mann & Low Theorem Perspective on Quantum Computing: New Paradigm for Designing Quantum Algorithm

The Gell-Mann & Low theorem is a cornerstone of Quantum Field Theory (QFT) and condensed matter physics, and many-body perturbation theory is a foundational tool for treating interactions. However, their integration into quantum algorithms remains a largely unexplored area of research, with current quantum simulation algorithms predominantly operating in the Schrödinger picture, leaving the potential of the interaction picture largely untapped. Our Variational Interaction-Picture S-matrix Ansatz (VIPSA) now fills this gap, specifically in the context of the Fermi-Hubbard model -- a canonical paradigm in condensed matter physics which is intricately connected to phenomena such as high-temperature superconductivity and Mott insulator transitions. This work offers a new conceptual perspective for variational quantum computing based upon the Gell-Mann & Low theorem. We achieve this by employing an innovative mathematical technique to explicitly unfold the normalized S-matrix, thereby enabling the systematic reconstruction of the Dyson series on a quantum computer, order by order. This method stands in contrast to the conventional reliance on Trotter expansion for adiabatic time evolution, marking a conceptual shift towards more sophisticated quantum algorithmic design. We leverage the strengths of the recently developed ADAPT-VQE algorithm, tailoring it to reconstruct perturbative terms effectively. Our simulations indicate that this method not only successfully recovers the Dyson series but also exhibits robust and stable convergence. We believe that our approach shows great promise in generalizing to more complex scenarios without increasing algorithmic complexity.

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arXiv2023

(Quantum) complexity of testing signed graph clusterability

This study examines clusterability testing for a signed graph in the bounded-degree model. Our contributions are two-fold. First, we provide a quantum algorithm with query complexity $\tilde{O}(N^{1/3})$ for testing clusterability, which yields a polynomial speedup over the best classical clusterability tester known [arXiv:2102.07587]. Second, we prove an $\tildeΩ(\sqrt{N})$ classical query lower bound for testing clusterability, which nearly matches the upper bound from [arXiv:2102.07587]. This settles the classical query complexity of clusterability testing, and it shows that our quantum algorithm has an advantage over any classical algorithm.

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arXiv2023

Implementation of Trained Factorization Machine Recommendation System on Quantum Annealer

Factorization Machine (FM) is the most commonly used model to build a recommendation system since it can incorporate side information to improve performance. However, producing item suggestions for a given user with a trained FM is time-consuming. It requires a run-time of $O((N_m \log N_m)^2)$, where $N_m$ is the number of items in the dataset. To address this problem, we propose a quadratic unconstrained binary optimization (QUBO) scheme to combine with FM and apply quantum annealing (QA) computation. Compared to classical methods, this hybrid algorithm provides a faster than quadratic speedup in finding good user suggestions. We then demonstrate the aforementioned computational advantage on current NISQ hardware by experimenting with a real example on a D-Wave annealer.

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arXiv2023

Problem-Dependent Power of Quantum Neural Networks on Multi-Class Classification

Quantum neural networks (QNNs) have become an important tool for understanding the physical world, but their advantages and limitations are not fully understood. Some QNNs with specific encoding methods can be efficiently simulated by classical surrogates, while others with quantum memory may perform better than classical classifiers. Here we systematically investigate the problem-dependent power of quantum neural classifiers (QCs) on multi-class classification tasks. Through the analysis of expected risk, a measure that weighs the training loss and the generalization error of a classifier jointly, we identify two key findings: first, the training loss dominates the power rather than the generalization ability; second, QCs undergo a U-shaped risk curve, in contrast to the double-descent risk curve of deep neural classifiers. We also reveal the intrinsic connection between optimal QCs and the Helstrom bound and the equiangular tight frame. Using these findings, we propose a method that uses loss dynamics to probe whether a QC may be more effective than a classical classifier on a particular learning task. Numerical results demonstrate the effectiveness of our approach to explain the superiority of QCs over multilayer Perceptron on parity datasets and their limitations over convolutional neural networks on image datasets. Our work sheds light on the problem-dependent power of QNNs and offers a practical tool for evaluating their potential merit.

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arXiv2023

On the Capacity of Zero-Drift First Arrival Position Channels in Diffusive Molecular Communication

Recent advancements in understanding the impulse response of the first arrival position (FAP) channel in molecular communication (MC) have illuminated its Shannon capacity. While Lee et al. shed light on FAP channel capacities with vertical drifts, the zero-drift scenario remains a conundrum, primarily due to the challenges associated with the heavy-tailed Cauchy distributions whose first and second moments do not exist, rendering traditional mutual information constraints ineffective. This paper unveils a novel characterization of the zero drift FAP channel capacity for both 2D and 3D. Interestingly, our results reveal a 3D FAP channel capacity that is double its 2D counterpart, hinting at a capacity increase with spatial dimension growth. Furthermore, our approach, which incorporates a modified logarithmic constraint and an output signal constraint, offers a simplified and more intuitive formula (similar to the well-known Gaussian case) for estimating FAP channel capacity.

cs.IT↗

arXiv2023

Approximate reconstructability of quantum states and noisy quantum secret sharing schemes

We introduce and analyse approximate quantum secret sharing in a formal cryptographic setting, wherein a dealer encodes and distributes a quantum secret to players such that authorized structures (sets of subsets of players) can approximately reconstruct the quantum secret and omnipotent adversarial agents controlling non-authorized subsets of players are approximately denied the quantum secret. In particular, viewing the map encoding the quantum secret to shares for players in an authorized structure as a quantum channel, we show that approximate reconstructability of the quantum secret by these players is possible if and only if the information leakage, given in terms of a certain entanglement-assisted capacity of the complementary quantum channel to the players outside the structure and the environment, is small.

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arXiv2023

Decision Diagrams for Symbolic Verification of Quantum Circuits

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.

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arXiv2023

Counting stabiliser codes for arbitrary dimension

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 (Ref. [23]) 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., $d=p^m$, but when the qudits are Galois-qudits). The proof in Ref. Ref. [23] 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 $[[n,k]]_d$ codes. Our work overlaps with Ref. Ref. [23] when $d$ is a prime and in this case our results match exactly, but the results differ for the more generic case. Despite that, the overall order of magnitude of the number of stabilizer codes scales agnostic of whether the dimension is prime or non-prime. This is surprising since the method employed to count the number of stabilizer states (or more generally stabilizer codes) depends on whether $d$ is prime or not. The cardinality of stabilizer states, which was so far known only for the prime-dimensional case (and the Galois qudit prime-power dimensional case) plays an important role as a quantifier in many topics in quantum computing. Salient among these are the resource theory of magic, design theory, de Finetti theorem for stabilizer states, the study and optimisation of the classical simulability of Clifford circuits, the study of quantum contextuality of small-dimensional systems and the study of Wigner-functions. Our work makes available this quantifier for the generic case, and thus is an important step needed to place results for quantum computing with non-prime dimensional quantum systems on the same pedestal as prime-dimensional systems.

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