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
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n dimensional convex body within multiplicative error ε usingÕ(n^{3}+ n^{2.5}/ε ) queries to a membership oracle andÕ(n^{5}+n^{4.5}/ε) additional arithmetic operations. For comparison, the best known classical algorithm usesÕ(n^{3.5}+n^{3}/ε^{2}) queries andÕ(n^{5.5}+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 continuousspace quantum walks with rigorous bounds on discretization error. To complement our quantum algorithms, we also prove that volume estimation requiresΩ (√ n+1/ε) quantum membership queries, which rules out the possibility of exponential quantum speedup inn and shows optimality of our algorithm in 1/ε up to polylogarithmic factors. 
A quantum neural network (QNN) is a parameterized mapping efficiently implementable on nearterm Noisy IntermediateScale Quantum (NISQ) computers. It can be used for supervised learning when combined with classical gradientbased optimizers. Despite the existing empirical and theoretical investigations, the convergence of QNN training is not fully understood. Inspired by the success of the neural tangent kernels (NTKs) in probing into the dynamics of classical neural networks, a recent line of works proposes to study overparameterized QNNs by examining a quantum version of tangent kernels. In this work, we study the dynamics of QNNs and show that contrary to popular belief it is qualitatively different from that of any kernel regression: due to the unitarity of quantum operations, there is a nonnegligible deviation from the tangent kernel regression derived at the random initialization. As a result of the deviation, we prove the atmost sublinear convergence for QNNs with Pauli measurements, which is beyond the explanatory power of any kernel regression dynamics. We then present the actual dynamics of QNNs in the limit of overparameterization. The new dynamics capture the change of convergence rate during training, and implies that the range of measurements is crucial to the fast QNN convergence.more » « less

null (Ed.)We initiate the study of quantum algorithms for escaping from saddle points with provable guarantee. Given a function f : R n → R , our quantum algorithm outputs an ϵ approximate secondorder stationary point using O ~ ( log 2 ( n ) / ϵ 1.75 ) queries to the quantum evaluation oracle (i.e., the zerothorder oracle). Compared to the classical stateoftheart algorithm by Jin et al. with O ~ ( log 6 ( n ) / ϵ 1.75 ) queries to the gradient oracle (i.e., the firstorder 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.more » « less

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 bestknown 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.more » « less

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 ndimensional convex body within multiplicative error ϵ using Õ (n3+n2.5/ϵ) queries to a membership oracle and Õ (n5+n4.5/ϵ) additional arithmetic operations. For comparison, the best known classical algorithm uses Õ (n4+n3/ϵ2) queries and Õ (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 continuousspace quantum walks with rigorous bounds on discretization error.more » « less

The study of quantum generative models is wellmotivated, not only because of its importance in quantum machine learning and quantum chemistry but also because of the perspective of its implementation on nearterm 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 3qubit quantum circuit of ~50 gates that well approximates a 3qubit 1d Hamiltonian simulation circuit that requires over 10k gates using standard techniques.more » « less

We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speedups. We consider SDP instances with m constraint matrices, each of dimension n, rank at most r, and sparsity s. The first algorithm assumes an input model where one is given access to an oracle to the entries of the matrices at unit cost. We show that it has run time O~(s^2 (sqrt{m} epsilon^{10} + sqrt{n} epsilon^{12})), with epsilon the error of the solution. 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 input matrices are given as quantum states. We show that its run time is O~(sqrt{m}+poly(r))*poly(log m,log n,B,epsilon^{1}), with B an upper bound on the tracenorm of all input matrices. In particular the complexity depends only polylogarithmically 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 rho with rank at most r, we show we can find in time sqrt{m}*poly(log m,log n,r,epsilon^{1}) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as rho on the m measurements, up to error epsilon. 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 update method. One of our main technical contributions is a quantum Gibbs state sampler for lowrank Hamiltonians, given quantum states encoding these Hamiltonians, with a polylogarithmic dependence on its dimension, which is based on ideas developed in quantum principal component analysis. We also develop a "fast" quantum OR lemma with a quadratic improvement in gate complexity over the construction of Harrow et al. [Harrow et al., 2017]. We believe both techniques might be of independent interest.more » « less

We investigate quantum algorithms for classification, a fundamental problem in machine learning, with provable guarantees. Given n ddimensional data points, the stateoftheart (and optimal) classical algorithm for training classifiers with constant margin by Clarkson et al. runs in Õ (n+d), which is also optimal in its input/output model. We design sublinear quantum algorithms for the same task running in Õ (\sqrt{n}+\sqrt{d}), 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 endtoend applications. We also demonstrate a tight lower bound (up to polylog factors) and discuss the possibility of implementation on nearterm quantum machines.more » « less