arXiv subjects

András Gilyén

Publications and source records attributed to András Gilyén.

40 records · Page 3Linked to original sources

arXiv2018

Optimizing quantum optimization algorithms via faster quantum gradient computation

We consider a generic framework of optimization algorithms based on gradient descent. We develop a quantum algorithm that computes the gradient of a multi-variate real-valued function $f:\mathbb{R}^d\rightarrow \mathbb{R}$ by evaluating it at only a logarithmic number of points in superposition. Our algorithm is an improved version of Stephen Jordan's gradient computation algorithm, providing an approximation of the gradient $\nabla f$ with quadratically better dependence on the evaluation accuracy of $f$, for an important class of smooth functions. Furthermore, we show that most objective functions arising from quantum optimization procedures satisfy the necessary smoothness conditions, hence our algorithm provides a quadratic improvement in the complexity of computing their gradient. We also show that in a continuous phase-query model, our gradient computation algorithm has optimal query complexity up to poly-logarithmic factors, for a particular class of smooth functions. Moreover, we show that for low-degree multivariate polynomials our algorithm can provide exponential speedups compared to Jordan's algorithm in terms of the dimension $d$. One of the technical challenges in applying our gradient computation procedure for quantum optimization problems is the need to convert between a probability oracle (which is common in quantum optimization procedures) and a phase oracle (which is common in quantum algorithms) of the objective function $f$. We provide efficient subroutines to perform this delicate interconversion between the two types of oracles incurring only a logarithmic overhead, which might be of independent interest. Finally, using these tools we improve the runtime of prior approaches for training quantum auto-encoders, variational quantum eigensolvers (VQE), and quantum approximate optimization algorithms (QAOA).

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arXiv2018

Improvements in Quantum SDP-Solving with Applications

Following the first paper on quantum algorithms for SDP-solving by Brandão and Svore in 2016, rapid developments has been made on quantum optimization algorithms. Recently Brandão et al. improved the quantum SDP-solver in the so-called quantum state input model, where the input matrices of the SDP are given as purified mixed states. They also gave the first non-trivial application of quantum SDP-solving by obtaining a more efficient algorithm for the problem of shadow tomography (proposed by Aaronson in 2017). In this paper we improve on all previous quantum SDP-solvers. Mainly we construct better Gibbs-samplers for both input models, which directly gives better bounds for SDP-solving. For an SDP with $m$ constraints involving $n\times n$ matrices, our improvements yield an $\widetilde{\mathcal O}\left( \left( \sqrt{m} + \sqrt{n}γ\right)s γ^4\right)$ upper bound on SDP-solving in the sparse matrix input model and an $\widetilde{\mathcal O}\left( \left(\sqrt{m}+B^{2.5}γ^{3.5} \right)Bγ^4 \right)$ upper bound in the quantum state input model. We then apply these results to the problem of shadow tomography to simultaneously improve the best known upper bounds on sample complexity due to Aaronson and complexity due Brandao et al. Furthermore, we apply our quantum SDP-solvers to the problems of quantum state discrimination and E-optimal design. In both cases we beat the classical lower bound in terms of some parameters, at the expense of heavy dependence on some other parameters. Finally we prove two lowers bounds for solving SDPs using quantum algorithms: (1) $\tildeΩ(\sqrt{m}B/\eps)$ in the quantum state input model, and (2) $\tildeΩ(\sqrt{m}α/\eps)$ in the quantum operator input model. These lower bounds show that the $\sqrt{m}$ factor and the polynomial dependence on the parameters $B,α$, and $1/\eps$ are necessary.

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arXiv2016

On preparing ground states of gapped Hamiltonians: An efficient Quantum Lovász Local Lemma

A frustration-free local Hamiltonian has the property that its ground state minimises the energy of all local terms simultaneously. In general, even deciding whether a Hamiltonian is frustration-free is a hard task, as it is closely related to the QMA1-complete quantum satisfiability problem (QSAT) -- the quantum analogue of SAT, which is the archetypal NP-complete problem in classical computer science. This connection shows that the frustration-free property is not only relevant to physics but also to computer science. The Quantum Lovász Local Lemma (QLLL) provides a sufficient condition for frustration-freeness. A natural question is whether there is an efficient way to prepare a frustration-free state under the conditions of the QLLL. Previous results showed that the answer is positive if all local terms commute. In this work we improve on the previous constructive results by designing an algorithm that works efficiently for non-commuting terms as well, assuming that the system is "uniformly" gapped, by which we mean that the system and all its subsystems have an inverse polynomial energy gap. Also, our analysis works under the most general condition for the QLLL, known as Shearer's bound. Similarly to the previous results, our algorithm has the charming feature that it uses only local measurement operations corresponding to the local Hamiltonian terms.

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arXiv2015

Exponential Sensitivity and its Cost in Quantum Physics

State selective protocols, like entanglement purification, lead to an essentially non-linear quantum evolution, unusual in naturally occurring quantum processes. Sensitivity to initial states in quantum systems, stemming from such non-linear dynamics, is a promising perspective for applications. Here we demonstrate that chaotic behaviour is a rather generic feature in state selective protocols: exponential sensitivity can exist for all initial states in an experimentally realisable optical scheme. Moreover, any complex rational polynomial map, including the example of the Mandelbrot set, can be directly realised. In state selective protocols, one needs an ensemble of initial states, the size of which decreases with each iteration. We prove that exponential sensitivity to initial states in any quantum system have to be related to downsizing the initial ensemble also exponentially. Our results show that magnifying initial differences of quantum states (a Schrödinger microscope) is possible, however, there is a strict bound on the number of copies needed.

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