arXiv subjects

Tongyang Li

Publications and source records attributed to Tongyang Li.

71 records · Page 4Linked to original sources

arXiv2022

Quantum Multi-Armed Bandits and Stochastic Linear Bandits Enjoy Logarithmic Regrets

Multi-arm bandit (MAB) and stochastic linear bandit (SLB) are important models in reinforcement learning, and it is well-known that classical algorithms for bandits with time horizon $T$ suffer $Ω(\sqrt{T})$ regret. In this paper, we study MAB and SLB with quantum reward oracles and propose quantum algorithms for both models with $O(\mbox{poly}(\log T))$ regrets, exponentially improving the dependence in terms of $T$. To the best of our knowledge, this is the first provable quantum speedup for regrets of bandit problems and in general exploitation in reinforcement learning. Compared to previous literature on quantum exploration algorithms for MAB and reinforcement learning, our quantum input model is simpler and only assumes quantum oracles for each individual arm.

cs.LG↗

arXiv2021

Quantum autoencoders for communication-efficient quantum cloud computing

In the model of quantum cloud computing, the server executes a computation on the quantum data provided by the client. In this scenario, it is important to reduce the amount of quantum communication between the client and the server. A possible approach is to transform the desired computation into a compressed version that acts on a smaller number of qubits, thereby reducing the amount of data exchanged between the client and the server. Here we propose quantum autoencoders for quantum gates (QAEGate) as a method for compressing quantum computations. We illustrate it in concrete scenarios of single-round and multi-round communication and validate it through numerical experiments. A bonus of our method is it does not reveal any information about the server's computation other than the information present in the output.

quant-ph↗

arXiv2021

Escape saddle points by a simple gradient-descent based algorithm

Escaping saddle points is a central research topic in nonconvex optimization. In this paper, we propose a simple gradient-based algorithm such that for a smooth function $f\colon\mathbb{R}^n\to\mathbb{R}$, it outputs an $ε$-approximate second-order stationary point in $\tilde{O}(\log n/ε^{1.75})$ iterations. Compared to the previous state-of-the-art algorithms by Jin et al. with $\tilde{O}((\log n)^{4}/ε^{2})$ or $\tilde{O}((\log n)^{6}/ε^{1.75})$ iterations, our algorithm is polynomially better in terms of $\log n$ and matches their complexities in terms of $1/ε$. For the stochastic setting, our algorithm outputs an $ε$-approximate second-order stationary point in $\tilde{O}((\log n)^{2}/ε^{4})$ iterations. Technically, our main contribution is an idea of implementing a robust Hessian power method using only gradients, which can find negative curvature near saddle points and achieve the polynomial speedup in $\log n$ compared to the perturbed gradient descent methods. Finally, we also perform numerical experiments that support our results.

math.OC↗

arXiv2021

Hamiltonian simulation with random inputs

The algorithmic error of digital quantum simulations is usually explored in terms of the spectral norm distance between the actual and ideal evolution operators. In practice, this worst-case error analysis may be unnecessarily pessimistic. To address this, we develop a theory of average-case performance of Hamiltonian simulation with random initial states. We relate the average-case error to the Frobenius norm of the multiplicative error and give upper bounds for the product formula (PF) and truncated Taylor series methods. As applications, we estimate average-case error for digital Hamiltonian simulation of general lattice Hamiltonians and $k$-local Hamiltonians. In particular, for the nearest-neighbor Heisenberg chain with $n$ spins, the error is quadratically reduced from $\mathcal O(n)$ in the worst case to $\mathcal O(\sqrt{n})$ on average for both the PF method and the Taylor series method. Numerical evidence suggests that this theory accurately characterizes the average error for concrete models. We also apply our results to error analysis in the simulation of quantum scrambling.

quant-ph↗

arXiv2021

Quantum algorithm for estimating volumes of convex bodies

Estimating the volume of a convex body is a central problem in convex geometry and can be viewed as a continuous version of counting. We present a quantum algorithm that estimates the volume of an $n$-dimensional convex body within multiplicative error $ε$ using $\tilde{O}(n^{3}+n^{2.5}/ε)$ queries to a membership oracle and $\tilde{O}(n^{5}+n^{4.5}/ε)$ additional arithmetic operations. For comparison, the best known classical algorithm uses $\tilde{O}(n^{4}+n^{3}/ε^{2})$ queries and $\tilde{O}(n^{6}+n^{5}/ε^{2})$ additional arithmetic operations. To the best of our knowledge, this is the first quantum speedup for volume estimation. Our algorithm is based on a refined framework for speeding up simulated annealing algorithms that might be of independent interest. This framework applies in the setting of "Chebyshev cooling", where the solution is expressed as a telescoping product of ratios, each having bounded variance. We develop several novel techniques when implementing our framework, including a theory of continuous-space quantum walks with rigorous bounds on discretization error. To complement our quantum algorithms, we also prove that volume estimation requires $Ω(\sqrt n+1/ε)$ quantum membership queries, which rules out the possibility of exponential quantum speedup in $n$ and shows optimality of our algorithm in $1/ε$ up to poly-logarithmic factors.

quant-ph↗

arXiv2021

Quantum algorithms for escaping from saddle points

We initiate the study of quantum algorithms for escaping from saddle points with provable guarantee. Given a function $f\colon\mathbb{R}^{n}\to\mathbb{R}$, our quantum algorithm outputs an $ε$-approximate second-order stationary point using $\tilde{O}(\log^{2} (n)/ε^{1.75})$ queries to the quantum evaluation oracle (i.e., the zeroth-order oracle). Compared to the classical state-of-the-art algorithm by Jin et al. with $\tilde{O}(\log^{6} (n)/ε^{1.75})$ queries to the gradient oracle (i.e., the first-order oracle), our quantum algorithm is polynomially better in terms of $\log n$ and matches its complexity in terms of $1/ε$. Technically, our main contribution is the idea of replacing the classical perturbations in gradient descent methods by simulating quantum wave equations, which constitutes the improvement in the quantum query complexity with $\log n$ factors for escaping from saddle points. We also show how to use a quantum gradient computation algorithm due to Jordan to replace the classical gradient queries by quantum evaluation queries with the same complexity. Finally, we also perform numerical experiments that support our theoretical findings.

quant-ph↗

arXiv2021

Quantum query complexity with matrix-vector products

We study quantum algorithms that learn properties of a matrix using queries that return its action on an input vector. We show that for various problems, including computing the trace, determinant, or rank of a matrix or solving a linear system that it specifies, quantum computers do not provide an asymptotic speedup over classical computation. On the other hand, we show that for some problems, such as computing the parities of rows or columns or deciding if there are two identical rows or columns, quantum computers provide exponential speedup. We demonstrate this by showing equivalence between models that provide matrix-vector products, vector-matrix products, and vector-matrix-vector products, whereas the power of these models can vary significantly for classical computation.

quant-ph↗

arXiv2020

Quantum exploration algorithms for multi-armed bandits

Identifying the best arm of a multi-armed bandit is a central problem in bandit optimization. We study a quantum computational version of this problem with coherent oracle access to states encoding the reward probabilities of each arm as quantum amplitudes. Specifically, we show that we can find the best arm with fixed confidence using $\tilde{O}\bigl(\sqrt{\sum_{i=2}^nΔ^{\smash{-2}}_i}\bigr)$ quantum queries, where $Δ_{i}$ represents the difference between the mean reward of the best arm and the $i^\text{th}$-best arm. This algorithm, based on variable-time amplitude amplification and estimation, gives a quadratic speedup compared to the best possible classical result. We also prove a matching quantum lower bound (up to poly-logarithmic factors).

quant-ph↗

arXiv2020

Sublinear classical and quantum algorithms for general matrix games

We investigate sublinear classical and quantum algorithms for matrix games, a fundamental problem in optimization and machine learning, with provable guarantees. Given a matrix $A\in\mathbb{R}^{n\times d}$, sublinear algorithms for the matrix game $\min_{x\in\mathcal{X}}\max_{y\in\mathcal{Y}} y^{\top} Ax$ were previously known only for two special cases: (1) $\mathcal{Y}$ being the $\ell_{1}$-norm unit ball, and (2) $\mathcal{X}$ being either the $\ell_{1}$- or the $\ell_{2}$-norm unit ball. We give a sublinear classical algorithm that can interpolate smoothly between these two cases: for any fixed $q\in (1,2]$, we solve the matrix game where $\mathcal{X}$ is a $\ell_{q}$-norm unit ball within additive error $ε$ in time $\tilde{O}((n+d)/{ε^{2}})$. We also provide a corresponding sublinear quantum algorithm that solves the same task in time $\tilde{O}((\sqrt{n}+\sqrt{d})\textrm{poly}(1/ε))$ with a quadratic improvement in both $n$ and $d$. Both our classical and quantum algorithms are optimal in the dimension parameters $n$ and $d$ up to poly-logarithmic factors. Finally, we propose sublinear classical and quantum algorithms for the approximate Carathéodory problem and the $\ell_{q}$-margin support vector machines as applications.

quant-ph↗

arXiv2020

On the cut dimension of a graph

Let $G = (V,w)$ be a weighted undirected graph with $m$ edges. The cut dimension of $G$ is the dimension of the span of the characteristic vectors of the minimum cuts of $G$, viewed as vectors in $\{0,1\}^m$. For every $n \ge 2$ we show that the cut dimension of an $n$-vertex graph is at most $2n-3$, and construct graphs realizing this bound. The cut dimension was recently defined by Graur et al.\ \cite{GPRW20}, who show that the maximum cut dimension of an $n$-vertex graph is a lower bound on the number of cut queries needed by a deterministic algorithm to solve the minimum cut problem on $n$-vertex graphs. For every $n\ge 2$, Graur et al.\ exhibit a graph on $n$ vertices with cut dimension at least $3n/2 -2$, giving the first lower bound larger than $n$ on the deterministic cut query complexity of computing mincut. We observe that the cut dimension is even a lower bound on the number of \emph{linear} queries needed by a deterministic algorithm to solve mincut, where a linear query can ask any vector $x \in \mathbb{R}^{\binom{n}{2}}$ and receives the answer $w^T x$. Our results thus show a lower bound of $2n-3$ on the number of linear queries needed by a deterministic algorithm to solve minimum cut on $n$-vertex graphs, and imply that one cannot show a lower bound larger than this via the cut dimension. We further introduce a generalization of the cut dimension which we call the $\ell_1$-approximate cut dimension. The $\ell_1$-approximate cut dimension is also a lower bound on the number of linear queries needed by a deterministic algorithm to compute minimum cut. It is always at least as large as the cut dimension, and we construct an infinite family of graphs on $n=3k+1$ vertices with $\ell_1$-approximate cut dimension $2n-2$, showing that it can be strictly larger than the cut dimension.

cs.CC↗

arXiv2020

Quantum-inspired sublinear algorithm for solving low-rank semidefinite programming

Semidefinite programming (SDP) is a central topic in mathematical optimization with extensive studies on its efficient solvers. In this paper, we present a proof-of-principle sublinear-time algorithm for solving SDPs with low-rank constraints; specifically, given an SDP with $m$ constraint matrices, each of dimension $n$ and rank $r$, our algorithm can compute any entry and efficient descriptions of the spectral decomposition of the solution matrix. The algorithm runs in time $O(m\cdot\mathrm{poly}(\log n,r,1/\varepsilon))$ given access to a sampling-based low-overhead data structure for the constraint matrices, where $\varepsilon$ is the precision of the solution. In addition, we apply our algorithm to a quantum state learning task as an application. Technically, our approach aligns with 1) SDP solvers based on the matrix multiplicative weight (MMW) framework by Arora and Kale [TOC '12]; 2) sampling-based dequantizing framework pioneered by Tang [STOC '19]. In order to compute the matrix exponential required in the MMW framework, we introduce two new techniques that may be of independent interest: $\bullet$ Weighted sampling: assuming sampling access to each individual constraint matrix $A_{1},\ldots,A_τ$, we propose a procedure that gives a good approximation of $A=A_{1}+\cdots+A_τ$. $\bullet$ Symmetric approximation: we propose a sampling procedure that gives the \emph{spectral decomposition} of a low-rank Hermitian matrix $A$. To the best of our knowledge, this is the first sampling-based algorithm for spectral decomposition, as previous works only give singular values and vectors.

cs.DS↗

arXiv2019

Quantum algorithms and lower bounds for convex optimization

While recent work suggests that quantum computers can speed up the solution of semidefinite programs, little is known about the quantum complexity of more general convex optimization. We present a quantum algorithm that can optimize a convex function over an $n$-dimensional convex body using $\tilde{O}(n)$ queries to oracles that evaluate the objective function and determine membership in the convex body. This represents a quadratic improvement over the best-known classical algorithm. We also study limitations on the power of quantum computers for general convex optimization, showing that it requires $\tildeΩ(\sqrt n)$ evaluation queries and $Ω(\sqrt{n})$ membership queries.

quant-ph↗

arXiv2019

Quantum Wasserstein Generative Adversarial Networks

The study of quantum generative models is well-motivated, not only because of its importance in quantum machine learning and quantum chemistry but also because of the perspective of its implementation on near-term quantum machines. Inspired by previous studies on the adversarial training of classical and quantum generative models, we propose the first design of quantum Wasserstein Generative Adversarial Networks (WGANs), which has been shown to improve the robustness and the scalability of the adversarial training of quantum generative models even on noisy quantum hardware. Specifically, we propose a definition of the Wasserstein semimetric between quantum data, which inherits a few key theoretical merits of its classical counterpart. We also demonstrate how to turn the quantum Wasserstein semimetric into a concrete design of quantum WGANs that can be efficiently implemented on quantum machines. Our numerical study, via classical simulation of quantum systems, shows the more robust and scalable numerical performance of our quantum WGANs over other quantum GAN proposals. As a surprising application, our quantum WGAN has been used to generate a 3-qubit quantum circuit of ~50 gates that well approximates a 3-qubit 1-d Hamiltonian simulation circuit that requires over 10k gates using standard techniques.

quant-ph↗

arXiv2019

Quantum SDP Solvers: Large Speed-ups, Optimality, and Applications to Quantum Learning

We give two quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with $m$ constraint matrices, each of dimension $n$, rank at most $r$, and sparsity $s$. The first algorithm assumes access to an oracle to the matrices at unit cost. We show that it has run time $\tilde{O}(s^2(\sqrt{m}ε^{-10}+\sqrt{n}ε^{-12}))$, with $ε$ the error of the solution. This gives an optimal dependence in terms of $m, n$ and quadratic improvement over previous quantum algorithms when $m\approx n$. The second algorithm assumes a fully quantum input model in which the matrices are given as quantum states. We show that its run time is $\tilde{O}(\sqrt{m}+\text{poly}(r))\cdot\text{poly}(\log m,\log n,B,ε^{-1})$, with $B$ an upper bound on the trace-norm of all input matrices. In particular the complexity depends only poly-logarithmically in $n$ and polynomially in $r$. We apply the second SDP solver to learn a good description of a quantum state with respect to a set of measurements: Given $m$ measurements and a supply of copies of an unknown state $ρ$ with rank at most $r$, we show we can find in time $\sqrt{m}\cdot\text{poly}(\log m,\log n,r,ε^{-1})$ a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as $ρ$ on the $m$ measurements, up to error $ε$. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics. As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians with a poly-logarithmic dependence on its dimension, which could be of independent interest.

quant-ph↗

arXiv2019

Sublinear quantum algorithms for training linear and kernel-based classifiers

We investigate quantum algorithms for classification, a fundamental problem in machine learning, with provable guarantees. Given $n$ $d$-dimensional data points, the state-of-the-art (and optimal) classical algorithm for training classifiers with constant margin runs in $\tilde{O}(n+d)$ time. We design sublinear quantum algorithms for the same task running in $\tilde{O}(\sqrt{n} +\sqrt{d})$ time, a quadratic improvement in both $n$ and $d$. Moreover, our algorithms use the standard quantization of the classical input and generate the same classical output, suggesting minimal overheads when used as subroutines for end-to-end applications. We also demonstrate a tight lower bound (up to poly-log factors) and discuss the possibility of implementation on near-term quantum machines. As a side result, we also give sublinear quantum algorithms for approximating the equilibria of $n$-dimensional matrix zero-sum games with optimal complexity $\tildeΘ(\sqrt{n})$.

quant-ph↗

arXiv2019

Distributional property testing in a quantum world

A fundamental problem in statistics and learning theory is to test properties of distributions. We show that quantum computers can solve such problems with significant speed-ups. In particular, we give fast quantum algorithms for testing closeness between unknown distributions, testing independence between two distributions, and estimating the Shannon / von Neumann entropy of distributions. The distributions can be either classical or quantum, however our quantum algorithms require coherent quantum access to a process preparing the samples. Our results build on the recent technique of quantum singular value transformation, combined with more standard tricks such as divide-and-conquer. The presented approach is a natural fit for distributional property testing both in the classical and the quantum case, demonstrating the first speed-ups for testing properties of density operators that can be accessed coherently rather than only via sampling; for classical distributions our algorithms significantly improve the precision dependence of some earlier results.

quant-ph↗

arXiv2017

Quantum query complexity of entropy estimation

Estimation of Shannon and Rényi entropies of unknown discrete distributions is a fundamental problem in statistical property testing and an active research topic in both theoretical computer science and information theory. Tight bounds on the number of samples to estimate these entropies have been established in the classical setting, while little is known about their quantum counterparts. In this paper, we give the first quantum algorithms for estimating $α$-Rényi entropies (Shannon entropy being 1-Renyi entropy). In particular, we demonstrate a quadratic quantum speedup for Shannon entropy estimation and a generic quantum speedup for $α$-Rényi entropy estimation for all $α\geq 0$, including a tight bound for the collision-entropy (2-Rényi entropy). We also provide quantum upper bounds for extreme cases such as the Hartley entropy (i.e., the logarithm of the support size of a distribution, corresponding to $α=0$) and the min-entropy case (i.e., $α=+\infty$), as well as the Kullback-Leibler divergence between two distributions. Moreover, we complement our results with quantum lower bounds on $α$-Rényi entropy estimation for all $α\geq 0$.

quant-ph↗