arXiv subjects

Song Cheng

Publications and source records attributed to Song Cheng.

At least 37 records · Page 2Linked to original sources

arXiv2024

Simulating Noisy Variational Quantum Algorithms: A Polynomial Approach

Large-scale variational quantum algorithms are widely recognized as a potential pathway to achieve practical quantum advantages. However, the presence of quantum noise might suppress and undermine these advantages, which blurs the boundaries of classical simulability. To gain further clarity on this matter, we present a novel polynomial-scale method based on the path integral of observable's back-propagation on Pauli paths (OBPPP). This method efficiently approximates expectation values of operators in variational quantum algorithms with bounded truncation error in the presence of single-qubit Pauli noise. Theoretically, we rigorously prove: 1) For a constant minimal non-zero noise rate $γ$, OBPPP's time and space complexity exhibit a polynomial relationship with the number of qubits $n$, the circuit depth $L$. 2) For variable $γ$, in scenarios where more than two non-zero noise factors exist, the complexity remains $\mathrm{Poly}\left(n,L\right)$ if $γ$ exceeds $1/\log{L}$, but grows exponential with $L$ when $γ$ falls below $1/L$. Numerically, we conduct classical simulations of IBM's zero-noise extrapolated experimental results on the 127-qubit Eagle processor [Nature \textbf{618}, 500 (2023)]. Our method attains higher accuracy and faster runtime compared to the quantum device. Furthermore, our approach allows us to simulate noisy outcomes, enabling accurate reproduction of IBM's unmitigated results that directly correspond to raw experimental observations. Our research reveals the vital role of noise in classical simulations and the derived method is general in computing the expected value for a broad class of quantum circuits and can be applied in the verification of quantum computers.

quant-ph↗

arXiv2024

Ab initio intermolecular interactions mediate thermochemically real-fluid effects that affect system reactivity

The properties of supercritical fluids are dictated by intermolecular interactions that involve two or more molecules. Such intermolecular interactions were described via intermolecular potentials in historical supercritical combustion modeling studies, but have been treated empirically and with no consideration of radical interactions or multi-body interactions involving more than two molecules. This approach has been adopted long ago, assuming sufficient characterization of real-fluid effects during supercritical combustion. Here, with data from ab initio multi-body intermolecular potentials, non-empirical Virial Equation of State (EoS), and real-fluid thermochemical and kinetic simulations, we reveal that empirical intermolecular potentials can lead to significant errors in representing supercritical fluids under common combustion situations, which can be impressively described by ab initio intermolecular potentials. These interactions are also found to greatly influence autoignition delay times, a common measure of global reactivity, with significant contributions from radical interactions and multi-body interactions. It is therefore of necessity to incorporate ab initio intermolecular interactions in studying supercritical combustion and various dynamic systems involving supercritical fluids, which has now been enabled through the new framework developed in the present study.

physics.chem-ph↗

arXiv2024

A ZX-Calculus Approach for the Construction of Graph Codes

Quantum Error-Correcting Codes (QECCs) play a crucial role in enhancing the robustness of quantum computing and communication systems against errors. Within the realm of QECCs, stabilizer codes, and specifically graph codes, stand out for their distinct attributes and promising utility in quantum technologies. This study underscores the significance of devising expansive QECCs and adopts the ZX-calculus a graphical language adept at quantum computational reasoning-to depict the encoders of graph codes effectively. Through the integration of ZX-calculus with established encoder frameworks, we present a nuanced approach that leverages this graphical representation to facilitate the construction of large-scale QECCs. Our methodology is rigorously applied to examine the intricacies of concatenated graph codes and the development of holographic codes, thus demonstrating the practicality of our graphical approach in addressing complex quantum error correction challenges. This research contributes to the theoretical understanding of quantum error correction and offers practical tools for its application, providing objective advancements in the field of quantum computing.

quant-ph↗

arXiv2024

Two-dimensional polarized superfluids under the prism of the fermion sign problem

Understanding if attractive fermions in an unbalanced occupation of its flavors can give rise to a superfluid state in two dimensions (2D), realizing the Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) state, presents a long-standing question. A limitation on its solution by numerics is posed by the sign problem, which constrains the applicability of quantum Monte Carlo techniques at sufficiently low temperatures and large lattice sizes, where a potential signature of polarized superfluidity would be unambiguous. By using a recently explored argument that the sign problem may be used instead to infer quantum critical behavior, we explore the regime where partial polarization occurs in the phase diagram, further showing that the average sign $\langle {\cal S}\rangle$ of quantum Monte Carlo weights tracks the criticality between balanced (or fully polarized) and polarized phases. Using the attractive Hubbard model with an unbalanced population, our investigation expands the scope of problems in which $\langle {\cal S}\rangle$ can be used for monitoring critical behavior, providing compelling albeit indirect evidence for the robustness of an FFLO phase in 2D.

cond-mat.str-el↗

arXiv2023

Exact results of dynamical structure factor of Lieb-Liniger model

The dynamical structure factor (DSF) represents a measure of dynamical density-density correlations in a quantum many-body system. Due to the complexity of many-body correlations and quantum fluctuations in a system of an infinitely large Hilbert space, such kind of dynamical correlations often impose a big theoretical challenge. For one dimensional (1D) quantum many-body systems, qualitative predictions of dynamical response functions are usually carried out by using the Tomonaga-Luttinger liquid (TLL) theory. In this scenario, a precise evaluation of the DSF for a 1D quantum system with arbitrary interaction strength remains a formidable task. In this paper, we use the form factor approach based on algebraic Bethe ansatz theory to calculate precisely the DSF of Lieb-Liniger model with an arbitrary interaction strength at a large scale of particle number. We find that the DSF for a system as large as 2000 particles enables us to depict precisely its line-shape from which the power-law singularity with corresponding exponents in the vicinities of spectral thresholds naturally emerge. It should be noted that, the advantage of our algorithm promises an access to the threshold behavior of dynamical correlation functions, further confirming the validity of nonlinear TLL theory besides Kitanine et. al. 2012 J. Stat. Mech. P09001. Finally we discuss a comparison of results with the results from the ABACUS method by J.-S. Caux 2009 J. Math. Phys. 50 095214 as well as from the strongly coupling expansion by Brand and Cherny 2005 Phys. Rev. A 72 033619.

cond-mat.quant-gas↗

arXiv2020

Simulating Noisy Quantum Circuits with Matrix Product Density Operators

Simulating quantum circuits with classical computers requires resources growing exponentially in terms of system size. Real quantum computer with noise, however, may be simulated polynomially with various methods considering different noise models. In this work, we simulate random quantum circuits in 1D with Matrix Product Density Operators (MPDO), for different noise models such as dephasing, depolarizing, and amplitude damping. We show that the method based on Matrix Product States (MPS) fails to approximate the noisy output quantum states for any of the noise models considered, while the MPDO method approximates them well. Compared with the method of Matrix Product Operators (MPO), the MPDO method reflects a clear physical picture of noise (with inner indices taking care of the noise simulation) and quantum entanglement (with bond indices taking care of two-qubit gate simulation). Consequently, in case of weak system noise, the resource cost of MPDO will be significantly less than that of the MPO due to a relatively small inner dimension needed for the simulation. In case of strong system noise, a relatively small bond dimension may be sufficient to simulate the noisy circuits, indicating a regime that the noise is large enough for an `easy' classical simulation. Moreover, we propose a more effective tensor updates scheme with optimal truncations for both the inner and the bond dimensions, performed after each layer of the circuit, which enjoys a canonical form of the MPDO for improving simulation accuracy. With truncated inner dimension to a maximum value $κ$ and bond dimension to a maximum value $χ$, the cost of our simulation scales as $\sim NDκ^3χ^3$, for an $N$-qubit circuit with depth $D$.

quant-ph↗

arXiv2020

Supervised Learning with Projected Entangled Pair States

Tensor networks, a model that originated from quantum physics, has been gradually generalized as efficient models in machine learning in recent years. However, in order to achieve exact contraction, only tree-like tensor networks such as the matrix product states and tree tensor networks have been considered, even for modeling two-dimensional data such as images. In this work, we construct supervised learning models for images using the projected entangled pair states (PEPS), a two-dimensional tensor network having a similar structure prior to natural images. Our approach first performs a feature map, which transforms the image data to a product state on a grid, then contracts the product state to a PEPS with trainable parameters to predict image labels. The tensor elements of PEPS are trained by minimizing differences between training labels and predicted labels. The proposed model is evaluated on image classifications using the MNIST and the Fashion-MNIST datasets. We show that our model is significantly superior to existing models using tree-like tensor networks. Moreover, using the same input features, our method performs as well as the multilayer perceptron classifier, but with much fewer parameters and is more stable. Our results shed light on potential applications of two-dimensional tensor network models in machine learning.

cs.CV↗

arXiv2020

Compressing deep neural networks by matrix product operators

A deep neural network is a parametrization of a multilayer mapping of signals in terms of many alternatively arranged linear and nonlinear transformations. The linear transformations, which are generally used in the fully connected as well as convolutional layers, contain most of the variational parameters that are trained and stored. Compressing a deep neural network to reduce its number of variational parameters but not its prediction power is an important but challenging problem toward the establishment of an optimized scheme in training efficiently these parameters and in lowering the risk of overfitting. Here we show that this problem can be effectively solved by representing linear transformations with matrix product operators (MPOs), which is a tensor network originally proposed in physics to characterize the short-range entanglement in one-dimensional quantum states. We have tested this approach in five typical neural networks, including FC2, LeNet-5, VGG, ResNet, and DenseNet on two widely used data sets, namely, MNIST and CIFAR-10, and found that this MPO representation indeed sets up a faithful and efficient mapping between input and output signals, which can keep or even improve the prediction accuracy with a dramatically reduced number of parameters. Our method greatly simplifies the representations in deep learning, and opens a possible route toward establishing a framework of modern neural networks which might be simpler and cheaper, but more efficient.

cs.LG↗

arXiv2020

Domain Adaption for Knowledge Tracing

With the rapid development of online education system, knowledge tracing which aims at predicting students' knowledge state is becoming a critical and fundamental task in personalized education. Traditionally, existing methods are domain-specified. However, there are a larger number of domains (e.g., subjects, schools) in the real world and the lacking of data in some domains, how to utilize the knowledge and information in other domains to help train a knowledge tracing model for target domains is increasingly important. We refer to this problem as domain adaptation for knowledge tracing (DAKT) which contains two aspects: (1) how to achieve great knowledge tracing performance in each domain. (2) how to transfer good performed knowledge tracing model between domains. To this end, in this paper, we propose a novel adaptable framework, namely adaptable knowledge tracing (AKT) to address the DAKT problem. Specifically, for the first aspect, we incorporate the educational characteristics (e.g., slip, guess, question texts) based on the deep knowledge tracing (DKT) to obtain a good performed knowledge tracing model. For the second aspect, we propose and adopt three domain adaptation processes. First, we pre-train an auto-encoder to select useful source instances for target model training. Second, we minimize the domain-specific knowledge state distribution discrepancy under maximum mean discrepancy (MMD) measurement to achieve domain adaptation. Third, we adopt fine-tuning to deal with the problem that the output dimension of source and target domain are different to make the model suitable for target domains. Extensive experimental results on two private datasets and seven public datasets clearly prove the effectiveness of AKT for great knowledge tracing performance and its superior transferable ability.

cs.LG↗

arXiv2019

Enhancing Item Response Theory for Cognitive Diagnosis

Cognitive diagnosis is a fundamental and crucial task in many educational applications, e.g., computer adaptive test and cognitive assignments. Item Response Theory (IRT) is a classical cognitive diagnosis method which can provide interpretable parameters (i.e., student latent trait, question discrimination, and difficulty) for analyzing student performance. However, traditional IRT ignores the rich information in question texts, cannot diagnose knowledge concept proficiency, and it is inaccurate to diagnose the parameters for the questions which only appear several times. To this end, in this paper, we propose a general Deep Item Response Theory (DIRT) framework to enhance traditional IRT for cognitive diagnosis by exploiting semantic representation from question texts with deep learning. In DIRT, we first use a proficiency vector to represent students' proficiency in knowledge concepts and embed question texts and knowledge concepts to dense vectors by Word2Vec. Then, we design a deep diagnosis module to diagnose parameters in traditional IRT by deep learning techniques. Finally, with the diagnosed parameters, we input them into the logistic-like formula of IRT to predict student performance. Extensive experimental results on real-world data clearly demonstrate the effectiveness and interpretation power of DIRT framework.

cs.AI↗

arXiv2019

Tree Tensor Networks for Generative Modeling

Matrix product states (MPS), a tensor network designed for one-dimensional quantum systems, has been recently proposed for generative modeling of natural data (such as images) in terms of `Born machine'. However, the exponential decay of correlation in MPS restricts its representation power heavily for modeling complex data such as natural images. In this work, we push forward the effort of applying tensor networks to machine learning by employing the Tree Tensor Network (TTN) which exhibits balanced performance in expressibility and efficient training and sampling. We design the tree tensor network to utilize the 2-dimensional prior of the natural images and develop sweeping learning and sampling algorithms which can be efficiently implemented utilizing Graphical Processing Units (GPU). We apply our model to random binary patterns and the binary MNIST datasets of handwritten digits. We show that TTN is superior to MPS for generative modeling in keeping correlation of pixels in natural images, as well as giving better log-likelihood scores in standard datasets of handwritten digits. We also compare its performance with state-of-the-art generative models such as the Variational AutoEncoders, Restricted Boltzmann machines, and PixelCNN. Finally, we discuss the future development of Tensor Network States in machine learning problems.

stat.ML↗

arXiv2018

FFLO correlation and free fluids in the one-dimensional attractive Hubbard model

In this Rapid Communication we show that low energy macroscopic properties of the one-dimensional (1D) attractive Hubbard model exhibit two fluids of bound pairs and of unpaired fermions. Using the thermodynamic Bethe ansatz equations of the model, we first determine the low temperature phase diagram and analytically calculate the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing correlation function for the partially-polarized phase. We then show that for such a FFLO-like state in the low density regime the effective chemical potentials of bound pairs and unpaired fermions behave like two free fluids. Consequently, the susceptibility, compressibility and specific heat obey simple additivity rules, indicating the `free' particle nature of interacting fermions on a 1D lattice. In contrast to the continuum Fermi gases, the correlation critical exponents and thermodynamics of the attractive Hubbard model essentially depend on two lattice interacting parameters. Finally, we study scaling functions, the Wilson ratio and susceptibility which provide universal macroscopic properties/dimensionless constants of interacting fermions at low energy.

cond-mat.str-el↗

arXiv2018

Universal thermodynamics of the one-dimensional attractive Hubbard model

The one-dimensional (1D) Hubbard model, describing electrons on a lattice with an on-site repulsive interaction, provides a paradigm for the physics of quantum many-body phenomena. Here by solving the thermodynamic Bethe ansatz equations we study the universal thermodynamics, quantum criticality and magnetism of the 1D attractive Hubbard model. We show that the compressibility and the susceptibility of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-like state obey simple additivity rules at low temperatures, indicating an existence of two free quantum fluids. The magnetic properties, such as magnetization and susceptibility, reveal three physical regions: quantum fluids at low temperatures, a non-Fermi liquid at high temperatures and the quantum fluid to non-Fermi liquid crossover in between. The lattice interaction is seen to significantly influence the nature of the FFLO-like state in 1D. Furthermore, we show that the dimensionless Wilson ratio provides an ideal parameter to map out the various phase boundaries and to characterize the two free fluids of the FLLO-like state. The quantum scaling functions for the thermal and magnetic properties yield the same dynamic critical exponent $z=2$ and correlation critical exponent $ν=1/2$ in the quantum critical region whenever a phase transition occurs. Our results provide a rigorous understanding of quantum criticality and free fluids of many-body systems on a 1D lattice.

cond-mat.str-el↗

arXiv2018

Asymptotic correlation functions and FFLO signature for the one-dimensional attractive Hubbard model

We study the long-distance asymptotic behavior of various correlation functions for the one-dimensional (1D) attractive Hubbard model in a partially polarized phase through the Bethe ansatz and conformal field theory approaches. We particularly find the oscillating behavior of these correlation functions with spatial power-law decay, of which the pair (spin) correlation function oscillates with a frequency $Δk_F$ ($2Δk_F$). Here $Δk_F=π(n_\uparrow-n_\downarrow)$ is the mismatch in the Fermi surfaces of spin-up and spin-down particles. Consequently, the pair correlation function in momentum space has peaks at the mismatch $k=Δk_F$, which has been observed in recent numerical work on this model. These singular peaks in momentum space together with the spatial oscillation suggest an analog of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state in the 1D Hubbard model. The parameter $β$ representing the lattice effect becomes prominent in critical exponents which determine the power-law decay of all correlation functions. We point out that the backscattering of unpaired fermions and bound pairs within their own Fermi points gives a microscopic origin of the FFLO pairing in 1D.

cond-mat.str-el↗

arXiv2018

Equivalence of restricted Boltzmann machines and tensor network states

The restricted Boltzmann machine (RBM) is one of the fundamental building blocks of deep learning. RBM finds wide applications in dimensional reduction, feature extraction, and recommender systems via modeling the probability distributions of a variety of input data including natural images, speech signals, and customer ratings, etc. We build a bridge between RBM and tensor network states (TNS) widely used in quantum many-body physics research. We devise efficient algorithms to translate an RBM into the commonly used TNS. Conversely, we give sufficient and necessary conditions to determine whether a TNS can be transformed into an RBM of given architectures. Revealing these general and constructive connections can cross-fertilize both deep learning and quantum many-body physics. Notably, by exploiting the entanglement entropy bound of TNS, we can rigorously quantify the expressive power of RBM on complex data sets. Insights into TNS and its entanglement capacity can guide the design of more powerful deep learning architectures. On the other hand, RBM can represent quantum many-body states with fewer parameters compared to TNS, which may allow more efficient classical simulations.

cond-mat.str-el↗

arXiv2017

Information Perspective to Probabilistic Modeling: Boltzmann Machines versus Born Machines

We compare and contrast the statistical physics and quantum physics inspired approaches for unsupervised generative modeling of classical data. The two approaches represent probabilities of observed data using energy-based models and quantum states respectively.Classical and quantum information patterns of the target datasets therefore provide principled guidelines for structural design and learning in these two approaches. Taking the restricted Boltzmann machines (RBM) as an example, we analyze the information theoretical bounds of the two approaches. We verify our reasonings by comparing the performance of RBMs of various architectures on the standard MNIST datasets.

physics.data-an↗

arXiv2017

Phase transition of the q-state clock model: duality and tensor renormalization

We investigate the critical behavior and the duality property of the ferromagnetic $q$-state clock model on the square lattice based on the tensor-network formalism. From the entanglement spectra of local tensors defined in the original and dual lattices, we obtain the exact self-dual points for the model with $q \leq 5 $ and approximate self-dual points for $q \geq 6$. We calculate accurately the lower and upper critical temperatures for the six-state clock model from the fixed-point tensors determined using the higher-order tensor renormalization group method and compare with other numerical results.

cond-mat.stat-mech↗

arXiv2015

Lifetimes and Oscillator Strengths for Ultraviolet Transitions in Singly-Ionized Lead

We present the results of lifetime measurements made using beam-foil techniques on levels of astrophysical interest in Pb II producing lines at 1203.6 A ($6s6p^{2}$ $^{2}D_{3/2}$) and 1433.9 A ($6s^{2}6d$ $^{2}D_{3/2}$). We also report the first detection of the Pb II $\lambda1203$ line in the interstellar medium (ISM) from an analysis of archival spectra acquired by the Space Telescope Imaging Spectrograph (STIS) onboard the Hubble Space Telescope (HST). The oscillator strengths derived from our experimental lifetimes for Pb II $λ\lambda1203$, $1433$ are generally consistent with recent theoretical results, including our own relativistic calculations. Our analysis of high-resolution HST/STIS spectra helps to confirm the relative strengths of the Pb II $λλ1203$, $1433$ lines. However, the oscillator strength that we obtain for Pb II $\lambda1433$ ($0.321\pm0.034$) is significantly smaller than earlier theoretical values, which have been used to study the abundance of Pb in the ISM. Our revised oscillator strength for $\lambda1433$ yields an increase in the interstellar abundance of Pb of 0.43 dex over determinations based on the value given by Morton, indicating that the depletion of Pb onto interstellar dust grains is less severe than previously thought.

astro-ph.IM↗