arXiv subjects

Iordanis Kerenidis

Publications and source records attributed to Iordanis Kerenidis.

At least 55 records · Page 3Linked to original sources

arXiv2019

Quantum algorithms for feedforward neural networks

Quantum machine learning has the potential for broad industrial applications, and the development of quantum algorithms for improving the performance of neural networks is of particular interest given the central role they play in machine learning today. In this paper we present quantum algorithms for training and evaluating feedforward neural networks based on the canonical classical feedforward and backpropagation algorithms. Our algorithms rely on an efficient quantum subroutine for approximating the inner products between vectors in a robust way, and on implicitly storing large intermediate values in quantum random access memory for fast retrieval at later stages. The running times of our algorithms can be quadratically faster in the size of the network than their standard classical counterparts since they depend linearly on the number of neurons in the network, as opposed to the number of connections between neurons as in the classical case. This makes our algorithms suited for large-scale, highly-connected networks where the number of edges in the network dominates the classical algorithmic running time. Furthermore, networks trained by our quantum algorithm may have an intrinsic resilience to overfitting, as the algorithm naturally mimics the effects of classical techniques such as drop-out used to regularize networks. Our algorithms can also be used as the basis for new quantum-inspired classical algorithms which have the same dependence on the network dimensions as their quantum counterparts, but with quadratic overhead in other parameters that makes them relatively impractical.

quant-ph↗

arXiv2019

Quantum Algorithms for Portfolio Optimization

We develop the first quantum algorithm for the constrained portfolio optimization problem. The algorithm has running time $\widetilde{O} \left( n\sqrt{r} \frac{ζκ}{δ^2} \log \left(1/ε\right) \right)$, where $r$ is the number of positivity and budget constraints, $n$ is the number of assets in the portfolio, $ε$ the desired precision, and $δ, κ, ζ$ are problem-dependent parameters related to the well-conditioning of the intermediate solutions. If only a moderately accurate solution is required, our quantum algorithm can achieve a polynomial speedup over the best classical algorithms with complexity $\widetilde{O} \left( \sqrt{r}n^ω\log(1/ε) \right)$, where $ω$ is the matrix multiplication exponent that has a theoretical value of around $2.373$, but is closer to $3$ in practice. We also provide some experiments to bound the problem-dependent factors arising in the running time of the quantum algorithm, and these experiments suggest that for most instances the quantum algorithm can potentially achieve an $O(n)$ speedup over its classical counterpart.

math.OC↗

arXiv2019

Anonymity for practical quantum networks

Quantum communication networks have the potential to revolutionise information and communication technologies. Here we are interested in a fundamental property and formidable challenge for any communication network, that of guaranteeing the anonymity of a sender and a receiver when a message is transmitted through the network, even in the presence of malicious parties. We provide the first practical protocol for anonymous communication in realistic quantum networks.

quant-ph↗

arXiv2018

Multi-Party Protocols, Information Complexity and Privacy

We introduce a new information theoretic measure that we call Public Information Complexity (PIC), as a tool for the study of multi-party computation protocols, and of quantities such as their communication complexity, or the amount of randomness they require in the context of information-theoretic private computations. We are able to use this measure directly in the natural asynchronous message-passing peer-to-peer model and show a number of interesting properties and applications of our new notion: the Public Information Complexity is a lower bound on the Communication Complexity and an upper bound on the Information Complexity; the difference between the Public Information Complexity and the Information Complexity provides a lower bound on the amount of randomness used in a protocol; any communication protocol can be compressed to its Public Information Cost; an explicit calculation of the zero-error Public Information Complexity of the $k$-party, $n$-bit Parity function, where a player outputs the bit-wise parity of the inputs. The latter result also establishes that the amount of randomness needed by a private protocol that computes this function is $Ω(n)$.

cs.CC↗

arXiv2018

q-means: A quantum algorithm for unsupervised machine learning

Quantum machine learning is one of the most promising applications of a full-scale quantum computer. Over the past few years, many quantum machine learning algorithms have been proposed that can potentially offer considerable speedups over the corresponding classical algorithms. In this paper, we introduce q-means, a new quantum algorithm for clustering which is a canonical problem in unsupervised machine learning. The $q$-means algorithm has convergence and precision guarantees similar to $k$-means, and it outputs with high probability a good approximation of the $k$ cluster centroids like the classical algorithm. Given a dataset of $N$ $d$-dimensional vectors $v_i$ (seen as a matrix $V \in \mathbb{R}^{N \times d})$ stored in QRAM, the running time of q-means is $\widetilde{O}\left( k d \fracη{δ^2}κ(V)(μ(V) + k \fracηδ) + k^2 \frac{η^{1.5}}{δ^2} κ(V)μ(V) \right)$ per iteration, where $κ(V)$ is the condition number, $μ(V)$ is a parameter that appears in quantum linear algebra procedures and $η= \max_{i} ||v_{i}||^{2}$. For a natural notion of well-clusterable datasets, the running time becomes $\widetilde{O}\left( k^2 d \frac{η^{2.5}}{δ^3} + k^{2.5} \frac{η^2}{δ^3} \right)$ per iteration, which is linear in the number of features $d$, and polynomial in the rank $k$, the maximum square norm $η$ and the error parameter $δ$. Both running times are only polylogarithmic in the number of datapoints $N$. Our algorithm provides substantial savings compared to the classical $k$-means algorithm that runs in time $O(kdN)$ per iteration, particularly for the case of large datasets.

quant-ph↗

arXiv2018

Experimental demonstration of quantum advantage for one-way communication complexity

The goal of demonstrating a quantum advantage with currently available experimental systems is of utmost importance in quantum information science. While this remains elusive for quantum computation, the field of communication complexity offers the possibility to already explore and showcase this advantage for useful tasks. Here, we define such a task, the Sampling Matching problem, which is inspired by the Hidden Matching problem and features an exponential gap between quantum and classical protocols in the one-way communication model. Our problem allows by its conception a photonic implementation based on encoding in the phase of coherent states of light, the use of a fixed size linear optic circuit, and single-photon detection. This enables us to demonstrate experimentally an advantage in the transmitted information resource beyond a threshold input size, which would have been impossible to reach for the original Hidden Matching problem. Our demonstration has implications in various communication and cryptographic settings, for example for quantum retrieval games and quantum money.

quant-ph↗

arXiv2018

Practical and unconditionally secure spacetime-constrained oblivious transfer

Spacetime-constrained oblivious transfer (SCOT) extends the fundamental primitive of oblivious transfer to Minkowski space. SCOT and location oblivious data transfer (LODT) are the only known cryptographic tasks with classical inputs and outputs for which unconditional security needs both quantum theory and relativity. We give an unconditionally secure SCOT protocol that, contrasting previous SCOT and LODT protocols, is practical to implement with current technology, where distant agents need only communicate classical information, while quantum communication occurs at a single location. We also show that our SCOT protocol can be used to implement unconditionally secure quantum relativistic bit commitment.

quant-ph↗

arXiv2018

A Quantum Interior Point Method for LPs and SDPs

We present a quantum interior point method with worst case running time $\widetilde{O}(\frac{n^{2.5}}{ξ^{2}} μκ^3 \log (1/ε))$ for SDPs and $\widetilde{O}(\frac{n^{1.5}}{ξ^{2}} μκ^3 \log (1/ε))$ for LPs, where the output of our algorithm is a pair of matrices $(S,Y)$ that are $ε$-optimal $ξ$-approximate SDP solutions. The factor $μ$ is at most $\sqrt{2}n$ for SDPs and $\sqrt{2n}$ for LP's, and $κ$ is an upper bound on the condition number of the intermediate solution matrices. For the case where the intermediate matrices for the interior point method are well conditioned, our method provides a polynomial speedup over the best known classical SDP solvers and interior point based LP solvers, which have a worst case running time of $O(n^{6})$ and $O(n^{3.5})$ respectively. Our results build upon recently developed techniques for quantum linear algebra and pave the way for the development of quantum algorithms for a variety of applications in optimization and machine learning.

quant-ph↗

arXiv2018

Learning with Errors is easy with quantum samples

Learning with Errors is one of the fundamental problems in computational learning theory and has in the last years become the cornerstone of post-quantum cryptography. In this work, we study the quantum sample complexity of Learning with Errors and show that there exists an efficient quantum learning algorithm (with polynomial sample and time complexity) for the Learning with Errors problem where the error distribution is the one used in cryptography. While our quantum learning algorithm does not break the LWE-based encryption schemes proposed in the cryptography literature, it does have some interesting implications for cryptography: first, when building an LWE-based scheme, one needs to be careful about the access to the public-key generation algorithm that is given to the adversary; second, our algorithm shows a possible way for attacking LWE-based encryption by using classical samples to approximate the quantum sample state, since then using our quantum learning algorithm would solve LWE.

quant-ph↗

arXiv2017

Experimental investigation of practical unforgeable quantum money

Wiesner's unforgeable quantum money scheme is widely celebrated as the first quantum information application. Based on the no-cloning property of quantum mechanics, this scheme allows for the creation of credit cards used in authenticated transactions offering security guarantees impossible to achieve by classical means. However, despite its central role in quantum cryptography, its experimental implementation has remained elusive because of the lack of quantum memories and of practical verification techniques. Here, we experimentally implement a quantum money protocol relying on classical verification that rigorously satisfies the security condition for unforgeability. Our system exploits polarization encoding of weak coherent states of light and operates under conditions that ensure compatibility with state-of-the-art quantum memories. We derive working regimes for our system using a security analysis taking into account all practical imperfections. Our results constitute a major step towards a real-world realization of this milestone protocol.

quant-ph↗

arXiv2017

Quantum superiority for verifying NP-complete problems with linear optics

Demonstrating quantum superiority for some computational task will be a milestone for quantum technologies and would show that computational advantages are possible not only with a universal quantum computer but with simpler physical devices. Linear optics is such a simpler but powerful platform where classically-hard information processing tasks, such as Boson Sampling, can be in principle implemented. In this work, we study a fundamentally different type of computational task to achieve quantum superiority using linear optics, namely the task of verifying NP-complete problems. We focus on a protocol by Aaronson et al. (2008) that uses quantum proofs for verification. We show that the proof states can be implemented in terms of a single photon in an equal superposition over many optical modes. Similarly, the tests can be performed using linear-optical transformations consisting of a few operations: a global permutation of all modes, simple interferometers acting on at most four modes, and measurement using single-photon detectors. We also show that the protocol can tolerate experimental imperfections.

quant-ph↗

arXiv2017

The information cost of quantum memoryless protocols

We consider memoryless quantum communication protocols, where the two parties do not possess any memory besides their classical input and they take turns performing unitary operations on a pure quantum state that they exchange between them. Most known quantum protocols are of this type and recently a deep connection between memoryless protocols and Bell inequality violations has been explored recently by Buhrman et al. We study the information cost of memoryless quantum protocols by looking at a canonical problem: bounded-round quantum communication protocols for the one-bit AND function. We prove directly a tight lower bound of $Θ(\log(k) / k)$ for the information cost of AND for $k$-round memoryless quantum protocols and for the input distribution needed for the Disjointness function. It is not clear if memoryless protocols allow for a reduction between AND and Disjointness, due to the absence of private workspaces. We enhance the model by allowing the players to keep in their private classical workspace apart from their input also some classical private coins. Surprisingly, we show that every quantum protocol can be transformed into an equivalent quantum protocol with private coins that is perfectly private, i.e. the players only learn the value of the function and nothing more. Last, we consider the model where the players are allowed to use one-shot coins, i.e. private coins that can be used only once during the protocol. While in the classical case, private coins and one-shot coins are equivalent, in the quantum case, we prove that they are not. More precisely, we show that every quantum memoryless protocol with one-bit inputs that uses one-shot coins can be transformed into a memoryless quantum protocol without private coins and without increasing too much its information cost. Hence, while private coins always allow for private quantum protocols, one-shot coins do not.

quant-ph↗

arXiv2016

Shortcuts to quantum network routing

A quantum network promises to enable long distance quantum communication, and assemble small quantum devices into a large quantum computing cluster. Each network node can thereby be seen as a small few qubit quantum computer. Qubits can be sent over direct physical links connecting nearby quantum nodes, or by means of teleportation over pre-established entanglement amongst distant network nodes. Such pre-shared entanglement effectively forms a shortcut - a virtual quantum link - which can be used exactly once. Here, we present an abstraction of a quantum network that allows ideas from computer science to be applied to the problem of routing qubits, and manage entanglement in the network. Specifically, we consider a scenario in which each quantum network node can create EPR pairs with its immediate neighbours over a physical connection, and perform entanglement swapping operations in order to create long distance virtual quantum links. We proceed to discuss the features unique to quantum networks, which call for the development of new routing techniques. As an example, we present two simple hierarchical routing schemes for a quantum network of N nodes for a ring and sphere topology. For these topologies we present efficient routing algorithms requiring O(log N) qubits to be stored at each network node, O(polylog N) time and space to perform routing decisions, and O(log N) timesteps to replenish the virtual quantum links in a model of entanglement generation.

cs.NI↗

arXiv2016

Efficient quantum communications with multiplexed coherent state fingerprints

We provide the first example of a communication model and a distributed task, for which there exists a realistic quantum protocol which is asymptotically more efficient than any classical protocol, both in the communication and the information resources. For this, we extend a recently proposed coherent state mapping for quantum communication protocols, introduce the notion of multiplexed coherent state fingerprints and show how to use them to design an efficient quantum protocol for estimating the Euclidean distance of two real vectors within a constant factor.

quant-ph↗

arXiv2016

Quantum Recommendation Systems

A recommendation system uses the past purchases or ratings of $n$ products by a group of $m$ users, in order to provide personalized recommendations to individual users. The information is modeled as an $m \times n$ preference matrix which is assumed to have a good rank-$k$ approximation, for a small constant $k$. In this work, we present a quantum algorithm for recommendation systems that has running time $O(\text{poly}(k)\text{polylog}(mn))$. All known classical algorithms for recommendation systems that work through reconstructing an approximation of the preference matrix run in time polynomial in the matrix dimension. Our algorithm provides good recommendations by sampling efficiently from an approximation of the preference matrix, without reconstructing the entire matrix. For this, we design an efficient quantum procedure to project a given vector onto the row space of a given matrix. This is the first algorithm for recommendation systems that runs in time polylogarithmic in the dimensions of the matrix and provides an example of a quantum machine learning algorithm for a real world application.

quant-ph↗

arXiv2016

Streaming Communication Protocols

We define the Streaming Communication model that combines the main aspects of communication complexity and streaming. We consider two agents that want to compute some function that depends on inputs that are distributed to each agent. The inputs arrive as data streams and each agent has a bounded memory. Agents are allowed to communicate with each other and also update their memory based on the input bit they read and the previous message they received. We provide tight tradeoffs between the necessary resources, i.e. communication and memory, for some of the canonical problems from communication complexity by proving a strong general lower bound technique. Second, we analyze the Approximate Matching problem and show that the complexity of this problem (i.e. the achievable approximation ratio) in the one-way variant of our model is strictly different both from the streaming complexity and the one-way communication complexity thereof.

cs.CC↗

arXiv2016

Optimal bounds for parity-oblivious random access codes

Random access coding is an information task that has been extensively studied and found many applications in quantum information. In this scenario, Alice receives an $n$-bit string $x$, and wishes to encode $x$ into a quantum state $ρ_x$, such that Bob, when receiving the state $ρ_x$, can choose any bit $i \in [n]$ and recover the input bit $x_i$ with high probability. Here we study two variants: parity-oblivious random access codes, where we impose the cryptographic property that Bob cannot infer any information about the parity of any subset of bits of the input apart from the single bits $x_i$; and even-parity-oblivious random access codes, where Bob cannot infer any information about the parity of any even-size subset of bits of the input. In this paper, we provide the optimal bounds for parity-oblivious quantum random access codes and show that they are asymptotically better than the optimal classical ones. Our results provide a large non-contextuality inequality violation and resolve the main open problem in a work of Spekkens, Buzacott, Keehn, Toner, and Pryde (2009). Second, we provide the optimal bounds for even-parity-oblivious random access codes by proving their equivalence to a non-local game and by providing tight bounds for the success probability of the non-local game via semidefinite programming. In the case of even-parity-oblivious random access codes, the cryptographic property holds also in the device-independent model.

quant-ph↗

arXiv2016

QMA with subset state witnesses

The class QMA plays a fundamental role in quantum complexity theory and it has found surprising connections to condensed matter physics and in particular in the study of the minimum energy of quantum systems. In this paper, we further investigate the class QMA and its related class QCMA by asking what makes quantum witnesses potentially more powerful than classical ones. We provide a definition of a new class, SQMA, where we restrict the possible quantum witnesses to the "simpler" subset states, i.e. a uniform superposition over the elements of a subset of n-bit strings. Surprisingly, we prove that this class is equal to QMA, hence providing a new characterisation of the class QMA. We also prove the analogous result for QMA(2) and describe a new complete problem for QMA and a stronger lower bound for the class QMA$_1$.

quant-ph↗