arXiv subjects

Iordanis Kerenidis

Publications and source records attributed to Iordanis Kerenidis.

At least 37 records · Page 2Linked to original sources

arXiv2022

Quantum City: simulation of a practical near-term metropolitan quantum network

We present the architecture and analyze the applications of a metropolitan-scale quantum network that requires only limited hardware resources for end users. Using NetSquid, a quantum network simulation tool based on discrete events, we assess the performance of several quantum network protocols involving two or more users in various configurations in terms of topology, hardware and trust choices. Our analysis takes losses and errors into account and considers realistic parameters corresponding to present or near-term technology. Our results show that practical quantum-enhanced network functionalities are within reach today and can prepare the ground for further applications when more advanced technology becomes available.

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arXiv2022

Quantum protocol for electronic voting without election authorities

Electronic voting is a very useful but challenging internet-based protocol that despite many theoretical approaches and various implementations with different degrees of success, remains a contentious topic due to issues in reliability and security. Here we present a quantum protocol that exploits an untrusted source of multipartite entanglement to carry out an election without relying on election authorities, simultaneous broadcasting or computational assumptions, and whose result is publicly verifiable. The level of security depends directly on the fidelity of the shared multipartite entangled quantum state, and the protocol can be readily implemented for a few voters with state-of-the-art photonic technology.

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arXiv2022

Low depth algorithms for quantum amplitude estimation

We design and analyze two new low depth algorithms for amplitude estimation (AE) achieving an optimal tradeoff between the quantum speedup and circuit depth. For $β\in (0,1]$, our algorithms require $N= \tilde{O}( \frac{1}{ ε^{1+β}})$ oracle calls and require the oracle to be called sequentially $D= O( \frac{1}{ ε^{1-β}})$ times to perform amplitude estimation within additive error $ε$. These algorithms interpolate between the classical algorithm $(β=1)$ and the standard quantum algorithm ($β=0$) and achieve a tradeoff $ND= O(1/ε^{2})$. These algorithms bring quantum speedups for Monte Carlo methods closer to realization, as they can provide speedups with shallower circuits. The first algorithm (Power law AE) uses power law schedules in the framework introduced by Suzuki et al \cite{S20}. The algorithm works for $β\in (0,1]$ and has provable correctness guarantees when the log-likelihood function satisfies regularity conditions required for the Bernstein Von-Mises theorem. The second algorithm (QoPrime AE) uses the Chinese remainder theorem for combining lower depth estimates to achieve higher accuracy. The algorithm works for discrete $β=q/k$ where $k \geq 2$ is the number of distinct coprime moduli used by the algorithm and $1 \leq q \leq k-1$, and has a fully rigorous correctness proof. We analyze both algorithms in the presence of depolarizing noise and provide numerical comparisons with the state of the art amplitude estimation algorithms.

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arXiv2022

Quantum Reinforcement Learning via Policy Iteration

Quantum computing has shown the potential to substantially speed up machine learning applications, in particular for supervised and unsupervised learning. Reinforcement learning, on the other hand, has become essential for solving many decision making problems and policy iteration methods remain the foundation of such approaches. In this paper, we provide a general framework for performing quantum reinforcement learning via policy iteration. We validate our framework by designing and analyzing: \emph{quantum policy evaluation} methods for infinite horizon discounted problems by building quantum states that approximately encode the value function of a policy $π$; and \emph{quantum policy improvement} methods by post-processing measurement outcomes on these quantum states. Last, we study the theoretical and experimental performance of our quantum algorithms on two environments from OpenAI's Gym.

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arXiv2022

Quantum machine learning with subspace states

We introduce a new approach for quantum linear algebra based on quantum subspace states and present three new quantum machine learning algorithms. The first is a quantum determinant sampling algorithm that samples from the distribution $\Pr[S]= det(X_{S}X_{S}^{T})$ for $|S|=d$ using $O(nd)$ gates and with circuit depth $O(d\log n)$. The state of art classical algorithm for the task requires $O(d^{3})$ operations \cite{derezinski2019minimax}. The second is a quantum singular value estimation algorithm for compound matrices $\mathcal{A}^{k}$, the speedup for this algorithm is potentially exponential. It decomposes a $\binom{n}{k}$ dimensional vector of order-$k$ correlations into a linear combination of subspace states corresponding to $k$-tuples of singular vectors of $A$. The third algorithm reduces exponentially the depth of circuits used in quantum topological data analysis from $O(n)$ to $O(\log n)$. Our basic tool are quantum subspace states, defined as $|Col(X)\rangle = \sum_{S\subset [n], |S|=d} det(X_{S}) |S\rangle$ for matrices $X \in \mathbb{R}^{n \times d}$ such that $X^{T} X = I_{d}$, that encode $d$-dimensional subspaces of $\mathbb{R}^{n}$. We develop two efficient state preparation techniques, the first using Givens circuits uses the representation of a subspace as a sequence of Givens rotations, while the second uses efficient implementations of unitaries $Γ(x) = \sum_{i} x_{i} Z^{\otimes (i-1)} \otimes X \otimes I^{n-i}$ with $O(\log n)$ depth circuits that we term Clifford loaders.

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arXiv2021

Low depth amplitude estimation on a trapped ion quantum computer

Amplitude estimation is a fundamental quantum algorithmic primitive that enables quantum computers to achieve quadratic speedups for a large class of statistical estimation problems, including Monte Carlo methods. The main drawback from the perspective of near term hardware implementations is that the amplitude estimation algorithm requires very deep quantum circuits. Recent works have succeeded in somewhat reducing the necessary resources for such algorithms, by trading off some of the speedup for lower depth circuits, but high quality qubits are still needed for demonstrating such algorithms. Here, we report the results of an experimental demonstration of amplitude estimation on a state-of-the-art trapped ion quantum computer. The amplitude estimation algorithms were used to estimate the inner product of randomly chosen four-dimensional unit vectors, and were based on the maximum likelihood estimation (MLE) and the Chinese remainder theorem (CRT) techniques. Significant improvements in accuracy were observed for the MLE based approach when deeper quantum circuits were taken into account, including circuits with more than ninety two-qubit gates and depth sixty, achieving a mean additive estimation error on the order of $10^{-2}$. The CRT based approach was found to provide accurate estimates for many of the data points but was less robust against noise on average. Last, we analyze two more amplitude estimation algorithms that take into account the specifics of the hardware noise to further improve the results.

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arXiv2021

Improving quantum linear system solvers via a gradient descent perspective

Solving systems of linear equations is one of the most important primitives in quantum computing that has the potential to provide a practical quantum advantage in many different areas, including in optimization, simulation, and machine learning. In this work, we revisit quantum linear system solvers from the perspective of convex optimization, and in particular gradient descent-type algorithms. This leads to a considerable constant-factor improvement in the runtime (or, conversely, a several orders of magnitude smaller error with the same runtime/circuit depth). More precisely, we first show how the asymptotically optimal quantum linear system solver of Childs, Kothari, and Somma is related to the gradient descent algorithm on the convex function $\|A\vec x - \vec b\|_2^2$: their linear system solver is based on a truncation in the Chebyshev basis of the degree-$(t-1)$ polynomial (in $A$) that maps the initial solution $\vec{x}_1 := \vec{b}$ to the $t$-th iterate $\vec{x}_t$ in the basic gradient descent algorithm. Then, instead of starting from the basic gradient descent algorithm, we use the optimal Chebyshev iteration method (which can be viewed as an accelerated gradient descent algorithm) and show that this leads to considerable improvements in the quantum solver.

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arXiv2021

Quantum Spectral Clustering

Spectral clustering is a powerful unsupervised machine learning algorithm for clustering data with non convex or nested structures. With roots in graph theory, it uses the spectral properties of the Laplacian matrix to project the data in a low-dimensional space where clustering is more efficient. Despite its success in clustering tasks, spectral clustering suffers in practice from a fast-growing running time of $O(n^3)$, where $n$ is the number of points in the dataset. In this work we propose an end-to-end quantum algorithm performing spectral clustering, extending a number of works in quantum machine learning. The quantum algorithm is composed of two parts: the first is the efficient creation of the quantum state corresponding to the projected Laplacian matrix, and the second consists of applying the existing quantum analogue of the $k$-means algorithm. Both steps depend polynomially on the number of clusters, as well as precision and data parameters arising from quantum procedures, and polylogarithmically on the dimension of the input vectors. Our numerical simulations show an asymptotic linear growth with $n$ when all terms are taken into account, significantly better than the classical cubic growth. This work opens the path to other graph-based quantum machine learning algorithms, as it provides routines for efficient computation and quantum access to the Incidence, Adjacency, and projected Laplacian matrices of a graph.

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arXiv2021

Quantum algorithms for Second-Order Cone Programming and Support Vector Machines

We present a quantum interior-point method (IPM) for second-order cone programming (SOCP) that runs in time $\widetilde{O} \left( n\sqrt{r} \frac{ζκ}{δ^2} \log \left(1/ε\right) \right)$ where $r$ is the rank and $n$ the dimension of the SOCP, $δ$ bounds the distance of intermediate solutions from the cone boundary, $ζ$ is a parameter upper bounded by $\sqrt{n}$, and $κ$ is an upper bound on the condition number of matrices arising in the classical IPM for SOCP. The algorithm takes as its input a suitable quantum description of an arbitrary SOCP and outputs a classical description of a $δ$-approximate $ε$-optimal solution of the given problem. Furthermore, we perform numerical simulations to determine the values of the aforementioned parameters when solving the SOCP up to a fixed precision $ε$. We present experimental evidence that in this case our quantum algorithm exhibits a polynomial speedup over the best classical algorithms for solving general SOCPs that run in time $O(n^{ω+0.5})$ (here, $ω$ is the matrix multiplication exponent, with a value of roughly $2.37$ in theory, and up to $3$ in practice). For the case of random SVM (support vector machine) instances of size $O(n)$, the quantum algorithm scales as $O(n^k)$, where the exponent $k$ is estimated to be $2.59$ using a least-squares power law. On the same family random instances, the estimated scaling exponent for an external SOCP solver is $3.31$ while that for a state-of-the-art SVM solver is $3.11$.

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arXiv2021

Quantum gradient descent for linear systems and least squares

Quantum machine learning and optimization are exciting new areas that have been brought forward by the breakthrough quantum algorithm of Harrow, Hassidim and Lloyd for solving systems of linear equations. The utility of {classical} linear system solvers extends beyond linear algebra as they can be leveraged to solve optimization problems using iterative methods like gradient descent. In this work, we provide the first quantum method for performing gradient descent when the gradient is an affine function. Performing $τ$ steps of the gradient descent requires time $O(τC_S)$ for weighted least squares problems, where $C_S$ is the cost of performing one step of the gradient descent quantumly, which at times can be considerably smaller than the classical cost. We illustrate our method by providing two applications: first, for solving positive semidefinite linear systems, and, second, for performing stochastic gradient descent for the weighted least squares problem with reduced quantum memory requirements. We also provide a quantum linear system solver in the QRAM data structure model that provides significant savings in cost for large families of matrices.

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arXiv2021

Experimental demonstration of quantum advantage for NP verification with limited information

In recent years, many computational tasks have been proposed as candidates for showing a quantum computational advantage, that is an advantage in the time needed to perform the task using a quantum instead of a classical machine. Nevertheless, practical demonstrations of such an advantage remain particularly challenging because of the difficulty in bringing together all necessary theoretical and experimental ingredients. Here, we show an experimental demonstration of a quantum computational advantage in a prover-verifier interactive setting, where the computational task consists in the verification of an NP-complete problem by a verifier who only gets limited information about the proof sent by an untrusted prover in the form of a series of unentangled quantum states. We provide a simple linear optical implementation that can perform this verification task efficiently (within a few seconds), while we also provide strong evidence that, fixing the size of the proof, a classical computer would take much longer time (assuming only that it takes exponential time to solve an NP-complete problem). While our computational advantage concerns a specific task in a scenario of mostly theoretical interest, it brings us a step closer to potential useful applications, such as server-client quantum computing.

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arXiv2020

Nearest Centroid Classification on a Trapped Ion Quantum Computer

Quantum machine learning has seen considerable theoretical and practical developments in recent years and has become a promising area for finding real world applications of quantum computers. In pursuit of this goal, here we combine state-of-the-art algorithms and quantum hardware to provide an experimental demonstration of a quantum machine learning application with provable guarantees for its performance and efficiency. In particular, we design a quantum Nearest Centroid classifier, using techniques for efficiently loading classical data into quantum states and performing distance estimations, and experimentally demonstrate it on a 11-qubit trapped-ion quantum machine, matching the accuracy of classical nearest centroid classifiers for the MNIST handwritten digits dataset and achieving up to 100% accuracy for 8-dimensional synthetic data.

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arXiv2020

Quantum classification of the MNIST dataset with Slow Feature Analysis

Quantum machine learning carries the promise to revolutionize information and communication technologies. While a number of quantum algorithms with potential exponential speedups have been proposed already, it is quite difficult to provide convincing evidence that quantum computers with quantum memories will be in fact useful to solve real-world problems. Our work makes considerable progress towards this goal. We design quantum techniques for Dimensionality Reduction and for Classification, and combine them to provide an efficient and high accuracy quantum classifier that we test on the MNIST dataset. More precisely, we propose a quantum version of Slow Feature Analysis (QSFA), a dimensionality reduction technique that maps the dataset in a lower dimensional space where we can apply a novel quantum classification procedure, the Quantum Frobenius Distance (QFD). We simulate the quantum classifier (including errors) and show that it can provide classification of the MNIST handwritten digit dataset, a widely used dataset for benchmarking classification algorithms, with $98.5\%$ accuracy, similar to the classical case. The running time of the quantum classifier is polylogarithmic in the dimension and number of data points. We also provide evidence that the other parameters on which the running time depends (condition number, Frobenius norm, error threshold, etc.) scale favorably in practice, thus ascertaining the efficiency of our algorithm.

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arXiv2020

Prospects and challenges of quantum finance

Quantum computers are expected to have substantial impact on the finance industry, as they will be able to solve certain problems considerably faster than the best known classical algorithms. In this article we describe such potential applications of quantum computing to finance, starting with the state-of-the-art and focusing in particular on recent works by the QC Ware team. We consider quantum speedups for Monte Carlo methods, portfolio optimization, and machine learning. For each application we describe the extent of quantum speedup possible and estimate the quantum resources required to achieve a practical speedup. The near-term relevance of these quantum finance algorithms varies widely across applications - some of them are heuristic algorithms designed to be amenable to near-term prototype quantum computers, while others are proven speedups which require larger-scale quantum computers to implement. We also describe powerful ways to bring these speedups closer to experimental feasibility - in particular describing lower depth algorithms for Monte Carlo methods and quantum machine learning, as well as quantum annealing heuristics for portfolio optimization. This article is targeted at financial professionals and no particular background in quantum computation is assumed.

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arXiv2020

Composable Security for Multipartite Entanglement Verification

We present a composably secure protocol allowing $n$ parties to test an entanglement generation resource controlled by a possibly dishonest party. The test consists only in local quantum operations and authenticated classical communication once a state is shared among them and provides composable security, namely it can be used as a secure subroutine by $n$ honest parties within larger communication protocols to test if a source is sharing quantum states that are at least $ε$-close to the GHZ state. This claim comes on top of previous results on multipartite entanglement verification where the security was studied in the usual game-based model. Here, we improve the protocol to make it more suitable for practical use in a quantum network and we study its security in the Abstract Cryptography framework to highlight composability issues and avoid hidden assumptions. This framework is a top-to-bottom theory that makes explicit any piece of information that each component (party or resource) gets at every time-step of the protocol. Moreover any security proof, which amounts to showing indistinguishability between an ideal resource having the desired security properties (up to local simulation) and the concrete resource representing the protocol, is composable for free in this setting. This allows us to readily compose our basic protocol in order to create a composably secure multi-round protocol enabling honest parties to obtain a state close to a GHZ state or an abort signal, even in the presence of a noisy or malicious source. Our protocol can typically be used as a subroutine in a Quantum Internet, to securely share a GHZ state among the network before performing a communication or computation protocol.

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arXiv2020

Quantum weak coin flipping with a single photon

Weak coin flipping is among the fundamental cryptographic primitives which ensure the security of modern communication networks. It allows two mistrustful parties to remotely agree on a random bit when they favor opposite outcomes. Unlike other two-party computations, one can achieve information-theoretic security using quantum mechanics only: both parties are prevented from biasing the flip with probability higher than $1/2+ε$, where $ε$ is arbitrarily low. Classically, the dishonest party can always cheat with probability $1$ unless computational assumptions are used. Despite its importance, no physical implementation has been proposed for quantum weak coin flipping. Here, we present a practical protocol that requires a single photon and linear optics only. We show that it is fair and balanced even when threshold single-photon detectors are used, and reaches a bias as low as $ε=1/\sqrt{2}-1/2\approx 0.207$. We further show that the protocol may display quantum advantage over a few hundred meters with state-of-the-art technology.

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arXiv2020

Quantum Expectation-Maximization for Gaussian Mixture Models

The Expectation-Maximization (EM) algorithm is a fundamental tool in unsupervised machine learning. It is often used as an efficient way to solve Maximum Likelihood (ML) estimation problems, especially for models with latent variables. It is also the algorithm of choice to fit mixture models: generative models that represent unlabelled points originating from $k$ different processes, as samples from $k$ multivariate distributions. In this work we define and use a quantum version of EM to fit a Gaussian Mixture Model. Given quantum access to a dataset of $n$ vectors of dimension $d$, our algorithm has convergence and precision guarantees similar to the classical algorithm, but the runtime is only polylogarithmic in the number of elements in the training set, and is polynomial in other parameters - as the dimension of the feature space, and the number of components in the mixture. We generalize further the algorithm in two directions. First, we show how to fit any mixture model of probability distributions in the exponential family. Then, we show how to use this algorithm to compute the Maximum a Posteriori (MAP) estimate of a mixture model: the Bayesian approach to likelihood estimation problems. We discuss the performance of the algorithm on a dataset that is expected to be classified successfully by this algorithm, arguing that on those cases we can give strong guarantees on the runtime.

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arXiv2019

Quantum Algorithms for Deep Convolutional Neural Networks

Quantum computing is a new computational paradigm that promises applications in several fields, including machine learning. In the last decade, deep learning, and in particular Convolutional neural networks (CNN), have become essential for applications in signal processing and image recognition. Quantum deep learning, however remains a challenging problem, as it is difficult to implement non linearities with quantum unitaries. In this paper we propose a quantum algorithm for applying and training deep convolutional neural networks with a potential speedup. The quantum CNN (QCNN) is a shallow circuit, reproducing completely the classical CNN, by allowing non linearities and pooling operations. The QCNN is particularly interesting for deep networks and could allow new frontiers in image recognition, by using more or larger convolution kernels, larger or deeper inputs. We introduce a new quantum tomography algorithm with $\ell_{\infty}$ norm guarantees, and new applications of probabilistic sampling in the context of information processing. We also present numerical simulations for the classification of the MNIST dataset to provide practical evidence for the efficiency of the QCNN.

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