arXiv subjects

Kaifeng Bu

Publications and source records attributed to Kaifeng Bu.

At least 37 records · Page 2Linked to original sources

arXiv2024

Resource Theory of Non-Revivals with Applications to Quantum Many-Body Scars

The study of state revivals has a long history in dynamical systems. We introduce a resource theory to understand the use of state revivals in quantum physics, especially in quantum many-body scarred systems. In this theory, a state is said to contain no amount of resource if it experiences perfect revivals under some unitary evolution. All other states are said to be resourceful. We show that this resource bounds information scrambling. Furthermore, we show that quantum many-body scarred dynamics can produce revivals in the Hayden-Preskill decoding protocol and can also be used to recover damaged quantum information. Our theory establishes a framework to study information retrieval and its applications in quantum many-body physics.

quant-ph↗

arXiv2023

Quantum Entropy and Central Limit Theorem

We introduce a framework to study discrete-variable (DV) quantum systems based on qudits. It relies on notions of a mean state (MS), a minimal stabilizer-projection state (MSPS), and a new convolution. Some interesting consequences are: The MS is the closest MSPS to a given state with respect to the relative entropy; the MS is extremal with respect to the von Neumann entropy, demonstrating a ''maximal entropy principle in DV systems.'' We obtain a series of inequalities for quantum entropies and for Fisher information based on convolution, giving a ''second law of thermodynamics for quantum convolutions.'' We show that the convolution of two stabilizer states is a stabilizer state. We establish a central limit theorem, based on iterating the convolution of a zero-mean quantum state, and show this converges to its MS. The rate of convergence is characterized by the ''magic gap,'' which we define in terms of the support of the characteristic function of the state. We elaborate on two examples: the DV beam splitter and the DV amplifier.

quant-ph↗

arXiv2022

Barren plateaus from learning scramblers with local cost functions

The existence of barren plateaus has recently revealed new training challenges in quantum machine learning (QML). Uncovering the mechanisms behind barren plateaus is essential in understanding the scope of problems that QML can efficiently tackle. Barren plateaus have recently been shown to exist when learning global properties of random unitaries, which is relevant when learning black hole dynamics. Establishing whether local cost functions can circumvent these barren plateaus is pertinent if we hope to apply QML to quantum many-body systems. We prove a no-go theorem showing that local cost functions encounter barren plateaus in learning random unitary properties.

quant-ph↗

arXiv2022

Resource theory of quantum scrambling

Quantum scrambling refers to the spread of local quantum information into the many degrees of freedom of a quantum system. In this work, we introduce a resource theory of scrambling which incorporates two mechanisms, "entanglement scrambling" and "magic scrambling". We introduce two resource monotones called the Pauli growth and the OTOC (out-of-time-ordered correlator) magic for these two mechanisms, respectively. We use our resource theory to explain recent experimental observations of magic. We also show that both resource monotones can be used to bound the decoding fidelity in Yoshida's black hole decoding protocol. These applications provide an operational interpretation of the resource monotones defined in this work.

quant-ph↗

arXiv2022

Wasserstein Complexity of Quantum Circuits

Given a unitary transformation, what is the size of the smallest quantum circuit that implements it? This quantity, known as the quantum circuit complexity, is a fundamental property of quantum evolutions that has widespread applications in many fields, including quantum computation, quantum field theory, and black hole physics. In this letter, we obtain a new lower bound for the quantum circuit complexity in terms of a novel complexity measure that we propose for quantum circuits, which we call the quantum Wasserstein complexity. Our proposed measure is based on the quantum Wasserstein distance of order one (also called the quantum earth mover's distance), a metric on the space of quantum states. We also prove several fundamental and important properties of our new complexity measure, which stand to be of independent interest. Finally, we show that our new measure also provides a lower bound for the experimental cost of implementing quantum circuits, which implies a quantum limit on converting quantum resources to computational resources. Our results provide novel applications of the quantum Wasserstein distance and pave the way for a deeper understanding of the resources needed to implement a quantum computation.

quant-ph↗

arXiv2022

Privacy-Utility Trade-Off

In this paper, we investigate the privacy-utility trade-off (PUT) problem, which considers the minimal privacy loss at a fixed expense of utility. Several different kinds of privacy in the PUT problem are studied, including differential privacy, approximate differential privacy, maximal information, maximal leakage, Renyi differential privacy, Sibson mutual information and mutual information. The average Hamming distance is used to measure the distortion caused by the privacy mechanism. We consider two scenarios: global privacy and local privacy. In the framework of global privacy framework, the privacy-distortion function is upper-bounded by the privacy loss of a special mechanism, and lower-bounded by the optimal privacy loss with any possible prior input distribution. In the framework of local privacy, we generalize a coloring method for the PUT problem.

cs.IT↗

arXiv2022

Complexity of quantum circuits via sensitivity, magic, and coherence

Quantum circuit complexity-a measure of the minimum number of gates needed to implement a given unitary transformation-is a fundamental concept in quantum computation, with widespread applications ranging from determining the running time of quantum algorithms to understanding the physics of black holes. In this work, we study the complexity of quantum circuits using the notions of sensitivity, average sensitivity (also called influence), magic, and coherence. We characterize the set of unitaries with vanishing sensitivity and show that it coincides with the family of matchgates. Since matchgates are tractable quantum circuits, we have proved that sensitivity is necessary for a quantum speedup. As magic is another measure to quantify quantum advantage, it is interesting to understand the relation between magic and sensitivity. We do this by introducing a quantum version of the Fourier entropy-influence relation. Our results are pivotal for understanding the role of sensitivity, magic, and coherence in quantum computation.

quant-ph↗

arXiv2022

Classical simulation of quantum circuits by half Gauss sums

We give an efficient algorithm to evaluate a certain class of exponential sums, namely the periodic, quadratic, multivariate half Gauss sums. We show that these exponential sums become $\#\mathsf{P}$-hard to compute when we omit either the periodicity or quadraticity condition. We apply our results about these exponential sums to the classical simulation of quantum circuits, and give an alternative proof of the Gottesman-Knill theorem. We also explore a connection between these exponential sums and the Holant framework. In particular, we generalize the existing definition of affine signatures to arbitrary dimensions, and use our results about half Gauss sums to show that the Holant problem for the set of affine signatures is tractable.

quant-ph↗

arXiv2022

Classical shadows with Pauli-invariant unitary ensembles

The classical shadow estimation protocol is a noise-resilient and sample-efficient quantum algorithm for learning the properties of quantum systems. Its performance depends on the choice of a unitary ensemble, which must be chosen by a user in advance. What is the weakest assumption that can be made on the chosen unitary ensemble that would still yield meaningful and interesting results? To address this question, we consider the class of Pauli-invariant unitary ensembles, i.e. unitary ensembles that are invariant under multiplication by a Pauli operator. This class includes many previously studied ensembles like the local and global Clifford ensembles as well as locally scrambled unitary ensembles. For this class of ensembles, we provide an explicit formula for the reconstruction map corresponding to the shadow channel and give explicit sample complexity bounds. In addition, we provide two applications of our results. Our first application is to locally scrambled unitary ensembles, where we give explicit formulas for the reconstruction map and sample complexity bounds that circumvent the need to solve an exponential-sized linear system. Our second application is to the classical shadow tomography of quantum channels with Pauli-invariant unitary ensembles. Our results pave the way for more efficient or robust protocols for predicting important properties of quantum states, such as their fidelity, entanglement entropy, and quantum Fisher information.

quant-ph↗

arXiv2022

Quantifying scrambling in quantum neural networks

We characterize a quantum neural network's error in terms of the network's scrambling properties via the out-of-time-ordered correlator. A network can be trained by optimizing either a loss function or a cost function. We show that, with some probability, both functions can be bounded by out-of-time-ordered correlators. The gradients of these functions can be bounded by the gradient of the out-of-time-ordered correlator, demonstrating that the network's scrambling ability governs its trainability. Our results pave the way for the exploration of quantum chaos in quantum neural networks.

quant-ph↗

arXiv2021

Rademacher complexity of noisy quantum circuits

Noise in quantum systems is a major obstacle to implementing many quantum algorithms on large quantum circuits. In this work, we study the effects of noise on the Rademacher complexity of quantum circuits, which is a measure of statistical complexity that quantifies the richness of classes of functions generated by these circuits. We consider noise models that are represented by convex combinations of unitary channels and provide both upper and lower bounds for the Rademacher complexities of quantum circuits characterized by these noise models. In particular, we find a lower bound for the Rademacher complexity of noisy quantum circuits that depends on the Rademacher complexity of the corresponding noiseless quantum circuit as well as the free robustness of the circuit. Our results show that the Rademacher complexity of quantum circuits decreases with the increase in noise.

quant-ph↗

arXiv2021

Effects of quantum resources on the statistical complexity of quantum circuits

We investigate how the addition of quantum resources changes the statistical complexity of quantum circuits by utilizing the framework of quantum resource theories. Measures of statistical complexity that we consider include the Rademacher complexity and the Gaussian complexity, which are well-known measures in computational learning theory that quantify the richness of classes of real-valued functions. We derive bounds for the statistical complexities of quantum circuits that have limited access to certain resources and apply our results to two special cases: (1) stabilizer circuits that are supplemented with a limited number of T gates and (2) instantaneous quantum polynomial-time Clifford circuits that are supplemented with a limited number of CCZ gates. We show that the increase in the statistical complexity of a quantum circuit when an additional quantum channel is added to it is upper bounded by the free robustness of the added channel. Finally, we derive bounds for the generalization error associated with learning from training data arising from quantum circuits.

quant-ph↗

arXiv2021

On the statistical complexity of quantum circuits

In theoretical machine learning, the statistical complexity is a notion that measures the richness of a hypothesis space. In this work, we apply a particular measure of statistical complexity, namely the Rademacher complexity, to the quantum circuit model in quantum computation and study how the statistical complexity depends on various quantum circuit parameters. In particular, we investigate the dependence of the statistical complexity on the resources, depth, width, and the number of input and output registers of a quantum circuit. To study how the statistical complexity scales with resources in the circuit, we introduce a resource measure of magic based on the $(p,q)$ group norm, which quantifies the amount of magic in the quantum channels associated with the circuit. These dependencies are investigated in the following two settings: (i) where the entire quantum circuit is treated as a single quantum channel, and (ii) where each layer of the quantum circuit is treated as a separate quantum channel. The bounds we obtain can be used to constrain the capacity of quantum neural networks in terms of their depths and widths as well as the resources in the network.

quant-ph↗

arXiv2020

Benchmarking one-shot distillation in general quantum resource theories

We study the one-shot distillation of general quantum resources, providing a unified quantitative description of the maximal fidelity achievable in this task, and revealing similarities shared by broad classes of resources. We establish fundamental quantitative and qualitative limitations on resource distillation applicable to all convex resource theories. We show that every convex quantum resource theory admits a meaningful notion of a pure maximally resourceful state which maximizes several monotones of operational relevance and finds use in distillation. We endow the generalized robustness measure with an operational meaning as an exact quantifier of performance in distilling such maximal states in many classes of resources including bi- and multipartite entanglement, multi-level coherence, as well as the whole family of affine resource theories, which encompasses important examples such as asymmetry, coherence, and thermodynamics.

quant-ph↗

arXiv2020

Depth-Width Trade-offs for Neural Networks via Topological Entropy

One of the central problems in the study of deep learning theory is to understand how the structure properties, such as depth, width and the number of nodes, affect the expressivity of deep neural networks. In this work, we show a new connection between the expressivity of deep neural networks and topological entropy from dynamical system, which can be used to characterize depth-width trade-offs of neural networks. We provide an upper bound on the topological entropy of neural networks with continuous semi-algebraic units by the structure parameters. Specifically, the topological entropy of ReLU network with $l$ layers and $m$ nodes per layer is upper bounded by $O(l\log m)$. Besides, if the neural network is a good approximation of some function $f$, then the size of the neural network has an exponential lower bound with respect to the topological entropy of $f$. Moreover, we discuss the relationship between topological entropy, the number of oscillations, periods and Lipschitz constant.

cs.LG↗

arXiv2020

Operational Advantage of Quantum Resources in Subchannel Discrimination

One of the central problems in the study of quantum resource theories is to provide a given resource with an operational meaning, characterizing physical tasks in which the resource can give an explicit advantage over all resourceless states. We show that this can always be accomplished for all convex resource theories. We establish in particular that any resource state enables an advantage in a channel discrimination task, allowing for a strictly greater success probability than any state without the given resource. Furthermore, we find that the generalized robustness measure serves as an exact quantifier for the maximal advantage enabled by the given resource state in a class of subchannel discrimination problems, providing a universal operational interpretation to this fundamental resource quantifier. We also consider a wider range of subchannel discrimination tasks and show that the generalized robustness still serves as the operational advantage quantifier for several well-known theories such as entanglement, coherence, and magic.

quant-ph↗

arXiv2019

Duality of Graph Invariants

We study a new set of duality relations between weighted, combinatoric invariants of a graph $G$. The dualities arise from a non-linear transform $\mathfrak{B}$, acting on the weight function $p$. We define $\mathfrak{B}$ on a space of real-valued functions $\mathcal{O}$ and investigate its properties. We show that three invariants (weighted independence number, weighted Lovász number, and weighted fractional packing number) are fixed points of $\mathfrak{B}^{2}$, but the weighted Shannon capacity is not. We interpret these invariants in the study of quantum non-locality.

math.FA↗

arXiv2019

De Finetti Theorems for Braided Parafermions

The classical de Finetti theorem in probability theory relates symmetry under the permutation group with the independence of random variables. This result has application in quantum information. Here we study states that are invariant with respect to a natural action of the braid group, and we emphasize the pictorial formulation and interpretation of our results. We prove a new type of de Finetti theorem for the four-string, double-braid group acting on the parafermion algebra to braid qudits, a natural symmetry in the quon language for quantum information. We prove that a braid-invariant state is extremal if and only if it is a product state. Furthermore, we provide an explicit characterization of braid-invariant states on the parafermion algebra, including finding a distinction that depends on whether the order of the parafermion algebra is square free. We characterize the extremal nature of product states (an inverse de Finetti theorem).

math-ph↗