@article {2448, title = {Quantum algorithm for estimating volumes of convex bodies}, journal = {ACM Transactions on Quantum Computing}, volume = {4}, year = {2023}, month = {4/2023}, chapter = {20}, abstract = {

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 O~(n3.5+n2.5/ε) queries to a membership oracle and O~(n5.5+n4.5/ε) additional arithmetic operations. For comparison, the best known classical algorithm uses O~(n4+n3/ε2) queries and O~(n6+n5/ε2) 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.

}, url = {https://arxiv.org/abs/1908.03903}, author = {Shouvanik Chakrabarti and Andrew M. Childs and Shih-Han Hung and Tongyang Li and Chunhao Wang and Xiaodi Wu} } @article {2913, title = {Hamiltonian simulation with random inputs}, journal = {Phys. Rev. Lett. 129, 270502}, volume = {129}, year = {2022}, month = {12/30/2022}, abstract = {

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 O(n) in the worst case to O(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.

}, doi = {https://doi.org/10.1103/PhysRevLett.129.270502}, url = {https://arxiv.org/abs/2111.04773}, author = {Qi Zhao and You Zhou and Alexander F. Shaw and Tongyang Li and Andrew M. Childs} } @article {3187, title = {Quantum simulation of real-space dynamics}, journal = {Quantum}, volume = {6}, year = {2022}, month = {11/8/2022}, pages = {860}, abstract = {

Quantum simulation is a prominent application of quantum computers. While there is extensive previous work on simulating finite-dimensional systems, less is known about quantum algorithms for real-space dynamics. We conduct a systematic study of such algorithms. In particular, we show that the dynamics of a d-dimensional Schr{\"o}dinger equation with η particles can be simulated with gate complexity O~(ηdFpoly(log(g\′/ϵ))), where ϵ is the discretization error, g\′ controls the higher-order derivatives of the wave function, and F measures the time-integrated strength of the potential. Compared to the best previous results, this exponentially improves the dependence on ϵ and g\′ from poly(g\′/ϵ) to poly(log(g\′/ϵ)) and polynomially improves the dependence on T and d, while maintaining best known performance with respect to η. For the case of Coulomb interactions, we give an algorithm using η3(d+η)Tpoly(log(ηdTg\′/(Δϵ)))/Δ one- and two-qubit gates, and another using η3(4d)d/2Tpoly(log(ηdTg\′/(Δϵ)))/Δ one- and two-qubit gates and QRAM operations, where T is the evolution time and the parameter Δ regulates the unbounded Coulomb interaction. We give applications to several computational problems, including faster real-space simulation of quantum chemistry, rigorous analysis of discretization error for simulation of a uniform electron gas, and a quadratic improvement to a quantum algorithm for escaping saddle points in nonconvex optimization.

}, doi = {https://doi.org/10.22331/q-2022-11-17-860}, url = {https://doi.org/10.22331\%2Fq-2022-11-17-860}, author = {Andrew M. Childs and Jiaqi Leng and Tongyang Li and Jin-Peng Liu and Chenyi Zhang} } @article {2645, title = {Quantum Algorithms for Escaping from Saddle Points}, journal = {Quantum}, volume = {5}, year = {2021}, month = {8/19/2021}, abstract = {

We initiate the study of quantum algorithms for escaping from saddle points with provable guarantee. Given a function f:Rn\→R, our quantum algorithm outputs an ϵ-approximate second-order stationary point using O~(log2n/ϵ1.75) 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 O~(log6n/ϵ1.75) queries to the gradient oracle (i.e., the first-order oracle), our quantum algorithm is polynomially better in terms of n and matches its complexity in terms of 1/ϵ. Our quantum algorithm is built upon two techniques: First, we replace the classical perturbations in gradient descent methods by simulating quantum wave equations, which constitutes the polynomial speedup in n for escaping from saddle points. Second, we 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 quantum speedup.

}, doi = {https://doi.org/10.22331/q-2021-08-20-529}, url = {https://arxiv.org/abs/2007.10253}, author = {Chenyi Zhang and Jiaqi Leng and Tongyang Li} } @article {2633, title = {Quantum exploration algorithms for multi-armed bandits}, journal = {Proceedings of the 35th Conference on Artificial Intelligence (AAAI 2021)}, volume = {35}, year = {2021}, month = {2021}, pages = {10102-10110}, abstract = {

\ 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 O~(\∑ni=2Δ\−2i\−\−\−\−\−\−\−\−\√) quantum queries, where Δi represents the difference between the mean reward of the best arm and the ith-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).

}, url = {https://ojs.aaai.org/index.php/AAAI/article/view/17212}, author = {Daochen Wang and Xuchen You and Tongyang Li and Andrew M. Childs} } @article {2886, title = {Quantum query complexity with matrix-vector products}, journal = {Proceedings of the 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021), Leibniz International Proceedings in Informatics}, volume = {198}, year = {2021}, month = {3/14/2021}, pages = {55:1-55:19}, abstract = {

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.

}, doi = {https://arxiv.org/ct?url=https\%3A\%2F\%2Fdx.doi.org\%2F10.4230\%2FLIPIcs.ICALP.2021.55\&v=2de93347}, url = {https://arxiv.org/abs/2102.11349}, author = {Andrew M. Childs and Shih-Han Hung and Tongyang Li} } @article {2388, title = {Distributional property testing in a quantum world}, journal = {Proceedings of ITCS 2020}, volume = {25}, year = {2020}, month = {02/02/2019}, pages = {1-25}, abstract = {

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.

}, doi = {http://dx.doi.org/10.4230/LIPIcs.ITCS.2020.25}, url = {https://arxiv.org/abs/1902.00814}, author = {Andras Gilyen and Tongyang Li} } @article {2207, title = {Quantum algorithms and lower bounds for convex optimization}, journal = {Quantum}, volume = {4}, year = {2020}, month = {12/18/2019}, abstract = {

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 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 Ω~(n\−\−\√) evaluation queries and Ω(n\−\−\√) membership queries.

}, doi = {https://doi.org/10.22331/q-2020-01-13-221}, url = {https://arxiv.org/abs/1809.01731}, author = {Shouvanik Chakrabarti and Andrew M. Childs and Tongyang Li and Xiaodi Wu} } @article {2501, title = {Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning}, journal = {to appear in Proceedings of STOC 2020}, year = {2020}, month = {6/18/2020}, abstract = {

We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang\&$\#$39;s breakthrough quantum-inspired algorithm for recommendation systems [STOC\&$\#$39;19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gily{\'e}n et al. [STOC\&$\#$39;19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffice to generalize all recent results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: l2-norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.

}, doi = {https://doi.org/10.1145/3357713.3384314}, url = {https://arxiv.org/abs/1910.06151}, author = {Nai-Hui Chia and Andras Gilyen and Tongyang Li and Han-Hsuan Lin and Ewin Tang and Chunhao Wang} } @article {2722, title = {Sublinear classical and quantum algorithms for general matrix games}, journal = {To appear in the Thirty-Fifth AAAI Conference on Artificial Intelligence (AAAI 2021)}, year = {2020}, month = {12/11/2020}, abstract = {

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\∈Rn\×d, sublinear algorithms for the matrix game minx\∈Xmaxy\∈Yy⊤Ax were previously known only for two special cases: (1) Y being the l1-norm unit ball, and (2) X being either the l1- or the l2-norm unit ball. We give a sublinear classical algorithm that can interpolate smoothly between these two cases: for any fixed q\∈(1,2], we solve the matrix game where X is a lq-norm unit ball within additive error ε in time O~((n+d)/ε2). We also provide a corresponding sublinear quantum algorithm that solves the same task in time O~((n\−\−\√+d\−\−\√)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 Carath{\'e}odory problem and the lq-margin support vector machines as applications.

}, url = {https://arxiv.org/abs/2012.06519}, author = {Tongyang Li and Chunhao Wang and Shouvanik Chakrabarti and Xiaodi Wu} } @article {2105, title = {Quantum query complexity of entropy estimation}, journal = {IEEE Transactions on Information Theory}, volume = {65}, year = {2019}, month = {10/16/2017}, pages = {2899-2921}, abstract = {

Estimation of Shannon and R\´enyi 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- R\´enyi 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-R\´enyi entropy estimation for all α \≥ 0, including a tight bound for the collision-entropy (2-R\´enyi 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 = +\∞), as well as the Kullback-Leibler divergence between two distributions. Moreover, we complement our results with quantum lower bounds on α-R\´enyi entropy estimation for all α \≥ 0. Our approach is inspired by the pioneering work of Bravyi, Harrow, and Hassidim (BHH) [13] on quantum algorithms for distributional property testing, however, with many new technical ingredients. For Shannon entropy and 0-R\´enyi entropy estimation, we improve the performance of the BHH framework, especially its error dependence, using Montanaro\’s approach to estimating the expected output value of a quantum subroutine with bounded variance [41] and giving a fine-tuned error analysis. For general α-R\´enyi entropy estimation, we further develop a procedure that recursively approximates α-R\´enyi entropy for a sequence of αs, which is in spirit similar to a cooling schedule in simulated annealing. For special cases such as integer α \≥ 2 and α = +\∞ (i.e., the min-entropy), we reduce the entropy estimation problem to the α-distinctness and the dlog ne-distinctness problems, respectively. We exploit various techniques to obtain our lower bounds for different ranges of α, including reductions to (variants of) existing lower bounds in quantum query complexity as well as the polynomial method inspired by the celebrated quantum lower bound for the collision problem.

}, doi = {https://doi.org/10.1109/TIT.2018.2883306}, url = {https://arxiv.org/abs/1710.06025}, author = {Tongyang Li and Xiaodi Wu} } @article {2500, title = {Quantum Wasserstein Generative Adversarial Networks}, journal = {Advances in Neural Information Processing Systems (NIPS)}, volume = {32}, year = {2019}, month = {2019/10/31}, abstract = {

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.

}, doi = {https://papers.nips.cc/paper/8903-quantum-wasserstein-generative-adversarial-networks.pdf}, url = {https://arxiv.org/abs/1911.00111}, author = {Shouvanik Chakrabarti and Yiming Huang and Tongyang Li and Soheil Feizi and Xiaodi Wu} } @article {2338, title = {Quantum-inspired classical sublinear-time algorithm for solving low-rank semidefinite programming via sampling approaches}, year = {2019}, abstract = {

Semidefinite programming (SDP) is a central topic in mathematical optimization with extensive studies on its efficient solvers. Recently, quantum algorithms with superpolynomial speedups for solving SDPs have been proposed assuming access to its constraint matrices in quantum superposition. Mutually inspired by both classical and quantum SDP solvers, in this paper we present a sublinear classical algorithm for solving low-rank SDPs which is asymptotically as good as existing quantum algorithms. Specifically, given an SDP with m constraint matrices, each of dimension n and rank poly(logn), our algorithm gives a succinct description and any entry of the solution matrix in time O(m\⋅poly(logn,1/ε)) given access to a sample-based low-overhead data structure of the constraint matrices, where ε 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 both the SDP solvers based on the matrix multiplicative weight (MMW) framework and the recent studies of quantum-inspired machine learning algorithms. The cost of solving SDPs by MMW mainly comes from the exponentiation of Hermitian matrices, and we propose two new technical ingredients (compared to previous sample-based algorithms) for this task that may be of independent interest: . Weighted sampling: assuming sampling access to each individual constraint matrix A1,\…,Aτ, we propose a procedure that gives a good approximation of A=A1+...+Aτ. . Symmetric approximation: we propose a sampling procedure that gives low-rank spectral decomposition of a Hermitian matrix A. This improves upon previous sampling procedures that only give low-rank singular value decompositions, losing the signs of eigenvalues.

}, url = {https://arxiv.org/abs/1901.03254}, author = {Nai-Hui Chia and Tongyang Li and Han-Hsuan Lin and Chunhao Wang} } @article {2376, title = {Sublinear quantum algorithms for training linear and kernel-based classifiers}, journal = {Proceedings of the 36th International Conference on Machine Learning (ICML 2019) PMLR }, volume = {97}, year = {2019}, month = {04/03/2019}, pages = {3815-3824}, abstract = {

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 O~(n+d) time. We design sublinear quantum algorithms for the same task running in O~(n\−\−\√+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 Θ~(n\−\−\√).\ 

}, url = {https://arxiv.org/abs/1904.02276}, author = {Tongyang Li and Shouvanik Chakrabarti and Xiaodi Wu} } @article {2309, title = {Quantum SDP Solvers: Large Speed-ups, Optimality, and Applications to Quantum Learning}, journal = {To appear at the 46th International Colloquium on Automata, Languages and Programming (ICALP 2019)}, year = {2018}, abstract = {

We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with m constraint matrices, each of dimension n, rank r, and sparsity s. The first algorithm assumes an input model where one is given access to entries of the matrices at unit cost. We show that it has run time O~(s2(m\−\−\√ε\−10+n\−\−\√ε\−12)), where ε is the error. This gives an optimal dependence in terms of m,n and quadratic improvement over previous quantum algorithms when m\≈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 O~(m\−\−\√+poly(r))\⋅poly(logm,logn,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 the problem of learning a good description of a quantum state with respect to a set of measurements: Given m measurements and copies of an unknown state ρ, we show we can find in time m\−\−\√\⋅poly(logm,logn,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\&$\#$39; 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.

}, url = {https://arxiv.org/abs/1710.02581}, author = {Fernando G. S. L. Brand{\~a}o and Amir Kalev and Tongyang Li and Cedric Yen-Yu Lin and Krysta M. Svore and Xiaodi Wu} } @article {1902, title = {Efficient simulation of sparse Markovian quantum dynamics}, journal = {Quantum Information and Computation}, volume = {17}, year = {2017}, month = {2017/09/01}, pages = {901-947}, abstract = {

Quantum algorithms for simulating Hamiltonian dynamics have been extensively developed, but there has been much less work on quantum algorithms for simulating the dynamics of open quantum systems. We give the first efficient quantum algorithms for simulating Markovian quantum dynamics generated by Lindbladians that are not necessarily local. We introduce two approaches to simulating sparse Lindbladians. First, we show how to simulate Lindbladians that act within small invariant subspaces using a quantum algorithm to implement sparse Stinespring isometries. Second, we develop a method for simulating sparse Lindblad operators by concatenating a sequence of short-time evolutions. We also show limitations on Lindbladian simulation by proving a no-fast-forwarding theorem for simulating sparse Lindbladians in a black-box model.

}, doi = {10.26421/QIC17.11-12}, url = {https://arxiv.org/abs/1611.05543}, author = {Andrew M. Childs and Tongyang Li} } @article {2106, title = {Exponential Quantum Speed-ups for Semidefinite Programming with Applications to Quantum Learning}, year = {2017}, month = {2017/10/06}, abstract = {

We give semidefinite program (SDP) quantum solvers with an exponential speed-up over classical ones. Specifically, we consider SDP instances with m constraint matrices of dimension n, each of rank at most r, and assume that the input matrices of the SDP are given as quantum states (after a suitable normalization). Then we show there is a quantum algorithm that solves the SDP feasibility problem with accuracy ǫ by using \√ m log m \· poly(log n,r, ǫ \−1 ) quantum gates. The dependence on n provides an exponential improvement over the work of Brand \˜ao and Svore [6] and the work of van Apeldoorn et al. [23], and demonstrates an exponential quantum speed-up when m and r are small. We apply the SDP solver to the problem of learning 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, we show we can find in time \√ m log m \· poly(log n,r, ǫ \−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 ǫ. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes\’ principle. As in previous work, we obtain our algorithm by \“quantizing\” classical SDP solvers based on the matrix multiplicative weight update 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 based on the techniques developed in quantum principal component analysis, which could be of independent interest. Our quantum SDP solver is different from previous ones in the following two aspects: (1) it follows from a zero-sum game approach of Hazan [11] of solving SDPs rather than the primal-dual approach by Arora and Kale [5]; and (2) it does not rely on any sparsity assumption of the input matrices.

}, url = {https://arxiv.org/abs/1710.02581}, author = {Fernando G. S. L. Brand{\~a}o and Amir Kalev and Tongyang Li and Cedric Yen-Yu Lin and Krysta M. Svore and Xiaodi Wu} }