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Recent years have witnessed the remarkable success of applying Graph machine learning (GML) to node/graph classification and link prediction. However, edge classification task that enjoys numerous real-world applications such as social network analysis and cybersecurity, has not seen significant advancement. To address this gap, our study pioneers a comprehensive approach to edge classification. We identify a novel `Topological Imbalance Issue', which arises from the skewed distribution of edges across different classes, affecting the local subgraph of each edge and harming the performance of edge classifications. Inspired by the recent studies in node classification that the performance discrepancy exists with varying local structural patterns, we aim to investigate if the performance discrepancy in topological imbalanced edge classification can also be mitigated by characterizing the local class distribution variance. To overcome this challenge, we introduce Topological Entropy (TE), a novel topological-based metric that measures the topological imbalance for each edge. Our empirical studies confirm that TE effectively measures local class distribution variance, and indicate that prioritizing edges with high TE values can help address the issue of topological imbalance. Based on this, we develop two strategies - Topological Reweighting and TE Wedge-based Mixup - to focus training on (synthetic) edges based on their TEs. While topological reweighting directly manipulates training edge weights according to TE, our wedge-based mixup interpolates synthetic edges between high TE wedges. Ultimately, we integrate these strategies into a novel topological imbalance strategy for edge classification: TopoEdge. Through extensive experiments, we demonstrate the efficacy of our proposed strategies on newly curated datasets and thus establish a new benchmark for (imbalanced) edge classification.

Random reversible and quantum circuits form random walks on the alternating group $\mathrm{Alt}(2^n)$ and unitary group $\mathrm{SU}(2^n)$, respectively. Known bounds on the spectral gap for the $t$-th moment of these random walks have inverse-polynomial dependence in both $n$ and $t$. We prove that the gap for random reversible circuits is $\Omega(n^{-3})$ for all $t\geq 1$, and the gap for random quantum circuits is $\Omega(n^{-3})$ for $t \leq \Theta(2^{n/2})$. These gaps are independent of $t$ in the respective regimes. We can further improve both gaps to $n^{-1}/\mathrm{polylog}(n, t)$ for $t\leq 2^{\Theta(n)}$, which is tight up to polylog factors. Our spectral gap results have a number of consequences: 1) Random reversible circuits with $\mathcal{O}(n^4 t)$ gates form multiplicative-error $t$-wise independent (even) permutations for all $t\geq 1$; for $t \leq \Theta(2^{n/6.1})$, we show that $\tilde{\mathcal{O}}(n^2 t)$ gates suffice. 2) Random quantum circuits with $\mathcal{O}(n^4 t)$ gates form multiplicative-error unitary $t$-designs for $t \leq \Theta(2^{n/2})$; for $t\leq \Theta(2^{2n/5})$, we show that $\tilde{\mathcal{O}}(n^2t)$ gates suffice. 3) The robust quantum circuit complexity of random circuits grows linearly for an exponentially long time, proving the robust Brown--Susskind conjecture [BS18,BCHJ+21]. Our spectral gap bounds are proven by reducing random quantum circuits to a more structured walk: a modification of the ``$\mathrm{PFC}$ ensemble'' from [MPSY24] together with an expander on the alternating group due to Kassabov [Kas07a], for which we give an efficient implementation using reversible circuits. In our reduction, we approximate the structured walk with local random circuits without losing the gap, which uses tools from the study of frustration-free Hamiltonians., Comment: 80 pages, 5 figures, v2: added references and minor changes in the presentation

Mid-circuit measurement (MCM) is a crucial ingredient in the development of fault-tolerant quantum computation. While there have been rapid experimental progresses in realizing MCM, a systematic method for characterizing noisy MCM is still under exploration. In this work we develop an algorithm to characterize noisy MCM, via a generalization of cycle benchmarking -- a standard approach for characterizing the Pauli noise channel of Clifford gates. The key idea is to use a joint Fourier transform on the classical and quantum registers and then estimate parameters in the Fourier space, analogous to Pauli fidelities used in cycle benchmarking. Furthermore, we develop a theory of the noise learnability of MCM, which determines what information can be learned about the noise model (in the presence of state preparation and measurement noise) and what cannot, which shows that all learnable information can be learned using our algorithm. As an application, we show how to use the learned information to test the independence between measurement noise and state preparation noise in a MCM. Finally, we conduct numerical simulations to illustrate the practical applicability of the algorithm. Similar to cycle benchmarking, we expect the algorithm to provide a useful toolkit that is of experimental interest., Comment: 27 pages, 9 figures

A quantum system coupled to a bath at some fixed, finite temperature converges to its Gibbs state. This thermalization process defines a natural, physically-motivated model of quantum computation. However, whether quantum computational advantage can be achieved within this realistic physical setup has remained open, due to the challenge of finding systems that thermalize quickly, but are classically intractable. Here we consider sampling from the measurement outcome distribution of quantum Gibbs states at constant temperatures, and prove that this task demonstrates quantum computational advantage. We design a family of commuting local Hamiltonians (parent Hamiltonians of shallow quantum circuits) and prove that they rapidly converge to their Gibbs states under the standard physical model of thermalization (as a continuous-time quantum Markov chain). On the other hand, we show that no polynomial time classical algorithm can sample from the measurement outcome distribution by reducing to the classical hardness of sampling from noiseless shallow quantum circuits. The key step in the reduction is constructing a fault-tolerance scheme for shallow IQP circuits against input noise., Comment: To appear in FOCS 2024. v2: Improved main result to constant locality

We construct random walks on simple Lie groups that quickly converge to the Haar measure for all moments up to order $t$. Specifically, a step of the walk on the unitary or orthognoal group of dimension $2^{\mathsf n}$ is a random Pauli rotation $e^{\mathrm i \theta P /2}$. The spectral gap of this random walk is shown to be $\Omega(1/t)$, which coincides with the best previously known bound for a random walk on the permutation group on $\{0,1\}^{\mathsf n}$. This implies that the walk gives an $\varepsilon$-approximate unitary $t$-design in depth $O(\mathsf n t^2 + t \log 1/\varepsilon)d$ where $d=O(\log \mathsf n)$ is the circuit depth to implement $e^{\mathrm i \theta P /2}$. Our simple proof uses quadratic Casimir operators of Lie algebras., Comment: 21 pages, 1 figure

Despite fundamental interests in learning quantum circuits, the existence of a computationally efficient algorithm for learning shallow quantum circuits remains an open question. Because shallow quantum circuits can generate distributions that are classically hard to sample from, existing learning algorithms do not apply. In this work, we present a polynomial-time classical algorithm for learning the description of any unknown $n$-qubit shallow quantum circuit $U$ (with arbitrary unknown architecture) within a small diamond distance using single-qubit measurement data on the output states of $U$. We also provide a polynomial-time classical algorithm for learning the description of any unknown $n$-qubit state $\lvert \psi \rangle = U \lvert 0^n \rangle$ prepared by a shallow quantum circuit $U$ (on a 2D lattice) within a small trace distance using single-qubit measurements on copies of $\lvert \psi \rangle$. Our approach uses a quantum circuit representation based on local inversions and a technique to combine these inversions. This circuit representation yields an optimization landscape that can be efficiently navigated and enables efficient learning of quantum circuits that are classically hard to simulate., Comment: 10 pages, 14 figures (7 inline; 7 floating) + 76-page appendix

Graph Neural Networks (GNNs) have shown great promise in learning node embeddings for link prediction (LP). While numerous studies aim to improve the overall LP performance of GNNs, none have explored its varying performance across different nodes and its underlying reasons. To this end, we aim to demystify which nodes will perform better from the perspective of their local topology. Despite the widespread belief that low-degree nodes exhibit poorer LP performance, our empirical findings provide nuances to this viewpoint and prompt us to propose a better metric, Topological Concentration (TC), based on the intersection of the local subgraph of each node with the ones of its neighbors. We empirically demonstrate that TC has a higher correlation with LP performance than other node-level topological metrics like degree and subgraph density, offering a better way to identify low-performing nodes than using cold-start. With TC, we discover a novel topological distribution shift issue in which newly joined neighbors of a node tend to become less interactive with that node's existing neighbors, compromising the generalizability of node embeddings for LP at testing time. To make the computation of TC scalable, We further propose Approximated Topological Concentration (ATC) and theoretically/empirically justify its efficacy in approximating TC and reducing the computation complexity. Given the positive correlation between node TC and its LP performance, we explore the potential of boosting LP performance via enhancing TC by re-weighting edges in the message-passing and discuss its effectiveness with limitations. Our code is publicly available at https://github.com/YuWVandy/Topo_LP_GNN.

Recommender systems are effective tools for mitigating information overload and have seen extensive applications across various domains. However, the single focus on utility goals proves to be inadequate in addressing real-world concerns, leading to increasing attention to fairness-aware and diversity-aware recommender systems. While most existing studies explore fairness and diversity independently, we identify strong connections between these two domains. In this survey, we first discuss each of them individually and then dive into their connections. Additionally, motivated by the concepts of user-level and item-level fairness, we broaden the understanding of diversity to encompass not only the item level but also the user level. With this expanded perspective on user and item-level diversity, we re-interpret fairness studies from the viewpoint of diversity. This fresh perspective enhances our understanding of fairness-related work and paves the way for potential future research directions. Papers discussed in this survey along with public code links are available at https://github.com/YuyingZhao/Awesome-Fairness-and-Diversity-Papers-in-Recommender-Systems .

We give a polynomial time classical algorithm for sampling from the output distribution of a noisy random quantum circuit in the regime of anti-concentration to within inverse polynomial total variation distance. This gives strong evidence that, in the presence of a constant rate of noise per gate, random circuit sampling (RCS) cannot be the basis of a scalable experimental violation of the extended Church-Turing thesis. Our algorithm is not practical in its current form, and does not address finite-size RCS based quantum supremacy experiments., Comment: 27 pages, 2 figures

Recently, several quantum benchmarking algorithms have been developed to characterize noisy quantum gates on today's quantum devices. A well-known issue in benchmarking is that not everything about quantum noise is learnable due to the existence of gauge freedom, leaving open the question of what information about noise is learnable and what is not, which has been unclear even for a single CNOT gate. Here we give a precise characterization of the learnability of Pauli noise channels attached to Clifford gates, showing that learnable information corresponds to the cycle space of the pattern transfer graph of the gate set, while unlearnable information corresponds to the cut space. This implies the optimality of cycle benchmarking, in the sense that it can learn all learnable information about Pauli noise. We experimentally demonstrate noise characterization of IBM's CNOT gate up to 2 unlearnable degrees of freedom, for which we obtain bounds using physical constraints. In addition, we give an attempt to characterize the unlearnable information by assuming perfect initial state preparation. However, based on the experimental data, we conclude that this assumption is inaccurate as it yields unphysical estimates, and we obtain a lower bound on state preparation noise., Comment: 33 pages, 11 figures. Close to accepted version

As small quantum computers are becoming available on different physical platforms, a benchmarking task known as cross-platform verification has been proposed that aims to estimate the fidelity of states prepared on two quantum computers. This task is fundamentally distributed, as no quantum communication can be performed between the two physical platforms due to hardware constraints, which prohibits a joint SWAP test. In this paper we settle the sample complexity of this task across all measurement and communication settings. The essence of the task, which we call distributed quantum inner product estimation, involves two players Alice and Bob who have $k$ copies of unknown states $\rho,\sigma$ (acting on $\mathbb{C}^{d}$) respectively. Their goal is to estimate $\mathrm{Tr}(\rho\sigma)$ up to additive error $\varepsilon\in(0,1)$, using local quantum operations and classical communication. In the weakest setting where only non-adaptive single-copy measurements and simultaneous message passing are allowed, we show that $k=O(\max\{1/\varepsilon^2,\sqrt{d}/\varepsilon\})$ copies suffice. This achieves a savings compared to full tomography which takes $\Omega(d^3)$ copies with single-copy measurements. Surprisingly, we also show that the sample complexity must be at least $\Omega(\max\{1/\varepsilon^2,\sqrt{d}/\varepsilon\})$, even in the strongest setting where adaptive multi-copy measurements and arbitrary rounds of communication are allowed. This shows that the success achieved by shadow tomography, for sample-efficiently learning the properties of a single system, cannot be generalized to the distributed setting. Furthermore, the fact that the sample complexity remains the same with single and multi-copy measurements contrasts with single system quantum property testing, which often demonstrate exponential separations in sample complexity with single and multi-copy measurements., Comment: 47 pages, 3 figures