arXiv subjects

András Gilyén

Publications and source records attributed to András Gilyén.

At least 37 records · Page 2Linked to original sources

arXiv2022

Quantum tomography using state-preparation unitaries

We describe algorithms to obtain an approximate classical description of a $d$-dimensional quantum state when given access to a unitary (and its inverse) that prepares it. For pure states we characterize the query complexity for $\ell_q$-norm error up to logarithmic factors. As a special case, we show that it takes $\widetildeΘ(d/\varepsilon)$ applications of the unitaries to obtain an $\varepsilon$-$\ell_2$-approximation of the state. For mixed states we consider a similar model, where the unitary prepares a purification of the state. In this model we give an efficient algorithm for obtaining Schatten $q$-norm estimates of a rank-$r$ mixed state, giving query upper bounds that are close to optimal. In particular, we show that a trace-norm ($q=1$) estimate can be obtained with $\widetilde{\mathcal{O}}(dr/\varepsilon)$ queries. This improves (assuming our stronger input model) the $\varepsilon$-dependence over the algorithm of Haah et al.\ (2017) that uses a joint measurement on $\widetilde{\mathcal{O}}(dr/\varepsilon^2)$ copies of the state. To our knowledge, the most sample-efficient results for pure-state tomography come from setting the rank to $1$ in generic mixed-state tomography algorithms, which can be computationally demanding. We describe sample-optimal algorithms for pure states that are easy and fast to implement. Along the way we show that an $\ell_\infty$-norm estimate of a normalized vector induces a (slightly worse) $\ell_q$-norm estimate for that vector, without losing a dimension-dependent factor in the precision. We also develop an unbiased and symmetric version of phase estimation, where the probability distribution of the estimate is centered around the true value. Finally, we give an efficient method for estimating multiple expectation values, improving over the recent result by Huggins et al.\ (2021) when the measurement operators do not fully overlap.

quant-ph↗

arXiv2022

An improved quantum-inspired algorithm for linear regression

We give a classical algorithm for linear regression analogous to the quantum matrix inversion algorithm [Harrow, Hassidim, and Lloyd, Physical Review Letters'09, arXiv:0811.3171] for low-rank matrices [Wossnig, Zhao, and Prakash, Physical Review Letters'18, arXiv:1704.06174], when the input matrix $A$ is stored in a data structure applicable for QRAM-based state preparation. Namely, suppose we are given an $A \in \mathbb{C}^{m\times n}$ with minimum non-zero singular value $σ$ which supports certain efficient $\ell_2$-norm importance sampling queries, along with a $b \in \mathbb{C}^m$. Then, for some $x \in \mathbb{C}^n$ satisfying $\|x - A^+b\| \leq \varepsilon\|A^+b\|$, we can output a measurement of $|x\rangle$ in the computational basis and output an entry of $x$ with classical algorithms that run in $\tilde{\mathcal{O}}\big(\frac{\|A\|_{\mathrm{F}}^6\|A\|^6}{σ^{12}\varepsilon^4}\big)$ and $\tilde{\mathcal{O}}\big(\frac{\|A\|_{\mathrm{F}}^6\|A\|^2}{σ^8\varepsilon^4}\big)$ time, respectively. This improves on previous "quantum-inspired" algorithms in this line of research by at least a factor of $\frac{\|A\|^{16}}{σ^{16}\varepsilon^2}$ [Chia, Gilyén, Li, Lin, Tang, and Wang, STOC'20, arXiv:1910.06151]. As a consequence, we show that quantum computers can achieve at most a factor-of-12 speedup for linear regression in this QRAM data structure setting and related settings. Our work applies techniques from sketching algorithms and optimization to the quantum-inspired literature. Unlike earlier works, this is a promising avenue that could lead to feasible implementations of classical regression in a quantum-inspired settings, for comparison against future quantum computers.

cs.DS↗

arXiv2022

Quantum algorithm for Petz recovery channels and pretty good measurements

The Petz recovery channel plays an important role in quantum information science as an operation that approximately reverses the effect of a quantum channel. The pretty good measurement is a special case of the Petz recovery channel, and it allows for near-optimal state discrimination. A hurdle to the experimental realization of these vaunted theoretical tools is the lack of a systematic and efficient method to implement them. This paper sets out to rectify this lack: using the recently developed tools of quantum singular value transformation and oblivious amplitude amplification, we provide a quantum algorithm to implement the Petz recovery channel when given the ability to perform the channel that one wishes to reverse. Moreover, we prove that, in some sense, our quantum algorithm's usage of the channel implementation cannot be improved by more than a quadratic factor. Our quantum algorithm also provides a procedure to perform pretty good measurements when given multiple copies of the states that one is trying to distinguish.

quant-ph↗

arXiv2022

Improved Quantum Algorithms for Fidelity Estimation

Fidelity is a fundamental measure for the closeness of two quantum states, which is important both from a theoretical and a practical point of view. Yet, in general, it is difficult to give good estimates of fidelity, especially when one works with mixed states over Hilbert spaces of very high dimension. Although, there has been some progress on fidelity estimation, all prior work either requires a large number of identical copies of the relevant states, or relies on unproven heuristics. In this work, we improve on both of these aspects by developing new and efficient quantum algorithms for fidelity estimation with provable performance guarantees in case at least one of the states is approximately low-rank. Our algorithms use advanced quantum linear algebra techniques, such as the quantum singular value transformation, as well as density matrix exponentiation and quantum spectral sampling. As a complementary result, we prove that fidelity estimation to any non-trivial constant additive accuracy is hard in general, by giving a sample complexity lower bound that depends polynomially on the dimension. Moreover, if circuit descriptions for the relevant states are provided, we show that the task is hard for the complexity class called (honest verifier) quantum statistical zero knowledge via a reduction to a closely related result by Watrous.

quant-ph↗

arXiv2021

Scalable Benchmarks for Gate-Based Quantum Computers

In the near-term "NISQ"-era of noisy, intermediate-scale, quantum hardware and beyond, reliably determining the quality of quantum devices becomes increasingly important: users need to be able to compare them with one another, and make an estimate whether they are capable of performing a given task ahead of time. In this work, we develop and release an advanced quantum benchmarking framework in order to help assess the state of the art of current quantum devices. Our testing framework measures the performance of universal quantum devices in a hardware-agnostic way, with metrics that are aimed to facilitate an intuitive understanding of which device is likely to outperform others on a given task. This is achieved through six structured tests that allow for an immediate, visual assessment of how devices compare. Each test is designed with scalability in mind, making this framework not only suitable for testing the performance of present-day quantum devices, but also of those released in the foreseeable future. The series of tests are motivated by real-life scenarios, and therefore emphasise the interplay between various relevant characteristics of quantum devices, such as qubit count, connectivity, and gate and measurement fidelity. We present the benchmark results of twenty-one different quantum devices from IBM, Rigetti and IonQ.

quant-ph↗

arXiv2020

(Sub)Exponential advantage of adiabatic quantum computation with no sign problem

We demonstrate the possibility of (sub)exponential quantum speedup via a quantum algorithm that follows an adiabatic path of a gapped Hamiltonian with no sign problem. This strengthens the superpolynomial separation recently proved by Hastings. The Hamiltonian that exhibits this speed-up comes from the adjacency matrix of an undirected graph, and we can view the adiabatic evolution as an efficient $\mathcal{O}(\mathrm{poly}(n))$-time quantum algorithm for finding a specific "EXIT" vertex in the graph given the "ENTRANCE" vertex. On the other hand we show that if the graph is given via an adjacency-list oracle, there is no classical algorithm that finds the "EXIT" with probability greater than $\exp(-n^δ)$ using at most $\exp(n^δ)$ queries for $δ= \frac15 - o(1)$. Our construction of the graph is somewhat similar to the "welded-trees" construction of Childs et al., but uses additional ideas of Hastings for achieving a spectral gap and a short adiabatic path.

quant-ph↗

arXiv2020

Complete classification of trapping coins for quantum walks on the 2D square lattice

One of the unique features of discrete-time quantum walks is called trapping, meaning the inability of the quantum walker to completely escape from its initial position, albeit the system is translationally invariant. The effect is dependent on the dimension and the explicit form of the local coin. A four state discrete-time quantum walk on a square lattice is defined by its unitary coin operator, acting on the four dimensional coin Hilbert space. The well known example of the Grover coin leads to a partial trapping, i.e., there exists some escaping initial state for which the probability of staying at the initial position vanishes. On the other hand, some other coins are known to exhibit strong trapping, where such escaping state does not exist. We present a systematic study of coins leading to trapping, explicitly construct all such coins for discrete-time quantum walks on the 2D square lattice, and classify them according to the structure of the operator and the manifestation of the trapping effect. We distinguish three types of trapping coins exhibiting distinct dynamical properties, as exemplified by the existence or non-existence of the escaping state and the area covered by the spreading wave-packet.

quant-ph↗

arXiv2020

Symmetries, graph properties, and quantum speedups

Aaronson and Ambainis (2009) and Chailloux (2018) showed that fully symmetric (partial) functions do not admit exponential quantum query speedups. This raises a natural question: how symmetric must a function be before it cannot exhibit a large quantum speedup? In this work, we prove that hypergraph symmetries in the adjacency matrix model allow at most a polynomial separation between randomized and quantum query complexities. We also show that, remarkably, permutation groups constructed out of these symmetries are essentially the only permutation groups that prevent super-polynomial quantum speedups. We prove this by fully characterizing the primitive permutation groups that allow super-polynomial quantum speedups. In contrast, in the adjacency list model for bounded-degree graphs (where graph symmetry is manifested differently), we exhibit a property testing problem that shows an exponential quantum speedup. These results resolve open questions posed by Ambainis, Childs, and Liu (2010) and Montanaro and de Wolf (2013).

quant-ph↗

arXiv2020

Quantum SDP-Solvers: Better upper and lower bounds

Brandão and Svore very recently gave quantum algorithms for approximately solving semidefinite programs, which in some regimes are faster than the best-possible classical algorithms in terms of the dimension $n$ of the problem and the number $m$ of constraints, but worse in terms of various other parameters. In this paper we improve their algorithms in several ways, getting better dependence on those other parameters. To this end we develop new techniques for quantum algorithms, for instance a general way to efficiently implement smooth functions of sparse Hamiltonians, and a generalized minimum-finding procedure. We also show limits on this approach to quantum SDP-solvers, for instance for combinatorial optimizations problems that have a lot of symmetry. Finally, we prove some general lower bounds showing that in the worst case, the complexity of every quantum LP-solver (and hence also SDP-solver) has to scale linearly with $mn$ when $m\approx n$, which is the same as classical.

quant-ph↗

arXiv2019

Convex optimization using quantum oracles

We study to what extent quantum algorithms can speed up solving convex optimization problems. Following the classical literature we assume access to a convex set via various oracles, and we examine the efficiency of reductions between the different oracles. In particular, we show how a separation oracle can be implemented using $\tilde{O}(1)$ quantum queries to a membership oracle, which is an exponential quantum speed-up over the $Ω(n)$ membership queries that are needed classically. We show that a quantum computer can very efficiently compute an approximate subgradient of a convex Lipschitz function. Combining this with a simplification of recent classical work of Lee, Sidford, and Vempala gives our efficient separation oracle. This in turn implies, via a known algorithm, that $\tilde{O}(n)$ quantum queries to a membership oracle suffice to implement an optimization oracle (the best known classical upper bound on the number of membership queries is quadratic). We also prove several lower bounds: $Ω(\sqrt{n})$ quantum separation (or membership) queries are needed for optimization if the algorithm knows an interior point of the convex set, and $Ω(n)$ quantum separation queries are needed if it does not.

quant-ph↗

arXiv2019

A Unified Framework of Quantum Walk Search

The main results on quantum walk search are scattered over different, incomparable frameworks, most notably the hitting time framework, originally by Szegedy, the electric network framework by Belovs, and the MNRS framework by Magniez, Nayak, Roland and Santha. As a result, a number of pieces are currently missing. For instance, the electric network framework allows quantum walks to start from an arbitrary initial state, but it only detects marked elements. In recent work by Ambainis et al., this problem was resolved for the more restricted hitting time framework, in which quantum walks must start from the stationary distribution. We present a new quantum walk search framework that unifies and strengthens these frameworks. This leads to a number of new results. For instance, the new framework not only detects, but finds marked elements in the electric network setting. The new framework also allows one to interpolate between the hitting time framework, which minimizes the number of walk steps, and the MNRS framework, which minimizes the number of times elements are checked for being marked. This allows for a more natural tradeoff between resources. Whereas the original frameworks only rely on quantum walks and phase estimation, our new algorithm makes use of a technique called quantum fast-forwarding, similar to the recent results by Ambainis et al. As a final result we show how in certain cases we can simplify this more involved algorithm to merely applying the quantum walk operator some number of times. This answers an open question of Ambainis et al.

quant-ph↗

arXiv2019

Quantum algorithms for zero-sum games

We derive sublinear-time quantum algorithms for computing the Nash equilibrium of two-player zero-sum games, based on efficient Gibbs sampling methods. We are able to achieve speed-ups for both dense and sparse payoff matrices at the cost of a mildly increased dependence on the additive error compared to classical algorithms. In particular we can find $\varepsilon$-approximate Nash equilibrium strategies in complexity $\tilde{O}(\sqrt{n+m}/\varepsilon^3)$ and $\tilde{O}(\sqrt{s}/\varepsilon^{3.5})$ respectively, where $n\times m$ is the size of the matrix describing the game and $s$ is its sparsity. Our algorithms use the LP formulation of the problem and apply techniques developed in recent works on quantum SDP-solvers. We also show how to reduce general LP-solving to zero-sum games, resulting in quantum LP-solvers that have complexities $\tilde{O}(\sqrt{n+m}γ^3)$ and $\tilde{O}(\sqrt{s}γ^{3.5})$ for the dense and sparse access models respectively, where $γ$ is the relevant "scale-invariant" precision parameter

quant-ph↗

arXiv2019

The Power Light Cone of the Discrete Bak-Sneppen, Contact and other local processes

We consider a class of random processes on graphs that include the discrete Bak-Sneppen (DBS) process and the several versions of the contact process (CP), with a focus on the former. These processes are parametrized by a probability $0\leq p \leq 1$ that controls a local update rule. Numerical simulations reveal a phase transition when $p$ goes from 0 to 1. Analytically little is known about the phase transition threshold, even for one-dimensional chains. In this article we consider a power-series approach based on representing certain quantities, such as the survival probability or the expected number of steps per site to reach the steady state, as a power-series in $p$. We prove that the coefficients of those power series stabilize as the length $n$ of the chain grows. This is a phenomenon that has been used in the physics community but was not yet proven. We show that for local events $A,B$ of which the support is a distance $d$ apart we have $\mathrm{cor}(A,B) = \mathcal{O}(p^d)$. The stabilization allows for the (exact) computation of coefficients for arbitrary large systems which can then be analyzed using the wide range of existing methods of power series analysis.

math.PR↗

arXiv2019

Quadratic speedup for finding marked vertices by quantum walks

A quantum walk algorithm can detect the presence of a marked vertex on a graph quadratically faster than the corresponding random walk algorithm (Szegedy, FOCS 2004). However, quantum algorithms that actually find a marked element quadratically faster than a classical random walk were only known for the special case when the marked set consists of just a single vertex, or in the case of some specific graphs. We present a new quantum algorithm for finding a marked vertex in any graph, with any set of marked vertices, that is (up to a log factor) quadratically faster than the corresponding classical random walk.

quant-ph↗

arXiv2019

Distributional property testing in a quantum world

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.

quant-ph↗

arXiv2018

Quantum-inspired low-rank stochastic regression with logarithmic dependence on the dimension

We construct an efficient classical analogue of the quantum matrix inversion algorithm (HHL) for low-rank matrices. Inspired by recent work of Tang, assuming length-square sampling access to input data, we implement the pseudoinverse of a low-rank matrix and sample from the solution to the problem $Ax=b$ using fast sampling techniques. We implement the pseudo-inverse by finding an approximate singular value decomposition of $A$ via subsampling, then inverting the singular values. In principle, the approach can also be used to apply any desired "smooth" function to the singular values. Since many quantum algorithms can be expressed as a singular value transformation problem, our result suggests that more low-rank quantum algorithms can be effectively "dequantised" into classical length-square sampling algorithms.

cs.DS↗

arXiv2018

The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation

We apply the framework of block-encodings, introduced by Low and Chuang (under the name standard-form), to the study of quantum machine learning algorithms and derive general results that are applicable to a variety of input models, including sparse matrix oracles and matrices stored in a data structure. We develop several tools within the block-encoding framework, such as singular value estimation of a block-encoded matrix, and quantum linear system solvers using block-encodings. The presented results give new techniques for Hamiltonian simulation of non-sparse matrices, which could be relevant for certain quantum chemistry applications, and which in turn imply an exponential improvement in the dependence on precision in quantum linear systems solvers for non-sparse matrices. In addition, we develop a technique of variable-time amplitude estimation, based on Ambainis' variable-time amplitude amplification technique, which we are also able to apply within the framework. As applications, we design the following algorithms: (1) a quantum algorithm for the quantum weighted least squares problem, exhibiting a 6-th power improvement in the dependence on the condition number and an exponential improvement in the dependence on the precision over the previous best algorithm of Kerenidis and Prakash; (2) the first quantum algorithm for the quantum generalized least squares problem; and (3) quantum algorithms for estimating electrical-network quantities, including effective resistance and dissipated power, improving upon previous work.

quant-ph↗

arXiv2018

Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics

Quantum computing is powerful because unitary operators describing the time-evolution of a quantum system have exponential size in terms of the number of qubits present in the system. We develop a new "Singular value transformation" algorithm capable of harnessing this exponential advantage, that can apply polynomial transformations to the singular values of a block of a unitary, generalizing the optimal Hamiltonian simulation results of Low and Chuang. The proposed quantum circuits have a very simple structure, often give rise to optimal algorithms and have appealing constant factors, while usually only use a constant number of ancilla qubits. We show that singular value transformation leads to novel algorithms. We give an efficient solution to a certain "non-commutative" measurement problem and propose a new method for singular value estimation. We also show how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum. Finally, as a quantum machine learning application we show how to efficiently implement principal component regression. "Singular value transformation" is conceptually simple and efficient, and leads to a unified framework of quantum algorithms incorporating a variety of quantum speed-ups. We illustrate this by showing how it generalizes a number of prominent quantum algorithms, including: optimal Hamiltonian simulation, implementing the Moore-Penrose pseudoinverse with exponential precision, fixed-point amplitude amplification, robust oblivious amplitude amplification, fast QMA amplification, fast quantum OR lemma, certain quantum walk results and several quantum machine learning algorithms. In order to exploit the strengths of the presented method it is useful to know its limitations too, therefore we also prove a lower bound on the efficiency of singular value transformation, which often gives optimal bounds.

quant-ph↗