IA Scholar Query: Fast Approximate Natural Gradient Descent in a Kronecker-factored Eigenbasis.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgWed, 20 Jul 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Suppressing quantum errors by scaling a surface code logical qubit
https://scholar.archive.org/work/5himghrjlvfifnja7ltkjxrysq
Practical quantum computing will require error rates that are well below what is achievable with physical qubits. Quantum error correction offers a path to algorithmically-relevant error rates by encoding logical qubits within many physical qubits, where increasing the number of physical qubits enhances protection against physical errors. However, introducing more qubits also increases the number of error sources, so the density of errors must be sufficiently low in order for logical performance to improve with increasing code size. Here, we report the measurement of logical qubit performance scaling across multiple code sizes, and demonstrate that our system of superconducting qubits has sufficient performance to overcome the additional errors from increasing qubit number. We find our distance-5 surface code logical qubit modestly outperforms an ensemble of distance-3 logical qubits on average, both in terms of logical error probability over 25 cycles and logical error per cycle (2.914%± 0.016% compared to 3.028%± 0.023%). To investigate damaging, low-probability error sources, we run a distance-25 repetition code and observe a 1.7×10^-6 logical error per round floor set by a single high-energy event (1.6×10^-7 when excluding this event). We are able to accurately model our experiment, and from this model we can extract error budgets that highlight the biggest challenges for future systems. These results mark the first experimental demonstration where quantum error correction begins to improve performance with increasing qubit number, illuminating the path to reaching the logical error rates required for computation.Rajeev Acharya, Igor Aleiner, Richard Allen, Trond I. Andersen, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Juan Atalaya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Joao Basso, Andreas Bengtsson, Sergio Boixo, Gina Bortoli, Alexandre Bourassa, Jenna Bovaird, Leon Brill, Michael Broughton, Bob B. Buckley, David A. Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Yu Chen, Zijun Chen, Ben Chiaro, Josh Cogan, Roberto Collins, Paul Conner, William Courtney, Alexander L. Crook, Ben Curtin, Dripto M. Debroy, Alexander Del Toro Barba, Sean Demura, Andrew Dunsworth, Daniel Eppens, Catherine Erickson, Lara Faoro, Edward Farhi, Reza Fatemi, Leslie Flores Burgos, Ebrahim Forati, Austin G. Fowler, Brooks Foxen, William Giang, Craig Gidney, Dar Gilboa, Marissa Giustina, Alejandro Grajales Dau, Jonathan A. Gross, Steve Habegger, Michael C. Hamilton, Matthew P. Harrigan, Sean D. Harrington, Oscar Higgott, Jeremy Hilton, Markus Hoffmann, Sabrina Hong, Trent Huang, Ashley Huff, William J. Huggins, Lev B. Ioffe, Sergei V. Isakov, Justin Iveland, Evan Jeffrey, Zhang Jiang, Cody Jones, Pavol Juhas, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Tanuj Khattar, Mostafa Khezri, Mária Kieferová, Seon Kim, Alexei Kitaev, Paul V. Klimov, Andrey R. Klots, Alexander N. Korotkov, Fedor Kostritsa, John Mark Kreikebaum, David Landhuis, Pavel Laptev, Kim-Ming Lau, Lily Laws, Joonho Lee, Kenny Lee, Brian J. Lester, Alexander Lill, Wayne Liu, Aditya Locharla, Erik Lucero, Fionn D. Malone, Jeffrey Marshall, Orion Martin, Jarrod R. McClean, Trevor Mccourt, Matt McEwen, Anthony Megrant, Bernardo Meurer Costa, Xiao Mi, Kevin C. Miao, Masoud Mohseni, Shirin Montazeri, Alexis Morvan, Emily Mount, Wojciech Mruczkiewicz, Ofer Naaman, Matthew Neeley, Charles Neill, Ani Nersisyan, Hartmut Neven, Michael Newman, Jiun How Ng, Anthony Nguyen, Murray Nguyen, Murphy Yuezhen Niu, Thomas E. O'Brien, Alex Opremcak, John Platt, Andre Petukhov, Rebecca Potter, Leonid P. Pryadko, Chris Quintana, Pedram Roushan, Nicholas C. Rubin, Negar Saei, Daniel Sank, Kannan Sankaragomathi, Kevin J. Satzinger, Henry F. Schurkus, Christopher Schuster, Michael J. Shearn, Aaron Shorter, Vladimir Shvarts, Jindra Skruzny, Vadim Smelyanskiy, W. Clarke Smith, George Sterling, Doug Strain, Marco Szalay, Alfredo Torres, Guifre Vidal, Benjamin Villalonga, Catherine Vollgraff Heidweiller, Theodore White, Cheng Xing, Z. Jamie Yao, Ping Yeh, Juhwan Yoo, Grayson Young, Adam Zalcman, Yaxing Zhang, Ningfeng Zhuwork_5himghrjlvfifnja7ltkjxrysqWed, 20 Jul 2022 00:00:00 GMTProvably efficient variational generative modeling of quantum many-body systems via quantum-probabilistic information geometry
https://scholar.archive.org/work/35ygodjajfckfphaf2p566cg2m
The dual tasks of quantum Hamiltonian learning and quantum Gibbs sampling are relevant to many important problems in physics and chemistry. In the low temperature regime, algorithms for these tasks often suffer from intractabilities, for example from poor sample- or time-complexity. With the aim of addressing such intractabilities, we introduce a generalization of quantum natural gradient descent to parameterized mixed states, as well as provide a robust first-order approximating algorithm, Quantum-Probabilistic Mirror Descent. We prove data sample efficiency for the dual tasks using tools from information geometry and quantum metrology, thus generalizing the seminal result of classical Fisher efficiency to a variational quantum algorithm for the first time. Our approaches extend previously sample-efficient techniques to allow for flexibility in model choice, including to spectrally-decomposed models like Quantum Hamiltonian-Based Models, which may circumvent intractable time complexities. Our first-order algorithm is derived using a novel quantum generalization of the classical mirror descent duality. Both results require a special choice of metric, namely, the Bogoliubov-Kubo-Mori metric. To test our proposed algorithms numerically, we compare their performance to existing baselines on the task of quantum Gibbs sampling for the transverse field Ising model. Finally, we propose an initialization strategy leveraging geometric locality for the modelling of sequences of states such as those arising from quantum-stochastic processes. We demonstrate its effectiveness empirically for both real and imaginary time evolution while defining a broader class of potential applications.Faris M. Sbahi, Antonio J. Martinez, Sahil Patel, Dmitri Saberi, Jae Hyeon Yoo, Geoffrey Roeder, Guillaume Verdonwork_35ygodjajfckfphaf2p566cg2mThu, 09 Jun 2022 00:00:00 GMTSimulating fermionic systems on classical and quantum computing devices
https://scholar.archive.org/work/ddu3szbvtzgntb2tyyo2vj5ena
This thesis presents a theoretical study of topics related to the simulation of quantum mechanical systems on classical and quantum computers. A large part of this work focuses on strongly interacting fermionic systems, more precisely, the behavior of electrons in presence of strong magnetic fields. We show how the energy spectrum of a Hamiltonian describing the fractional quantum Hall effect can be computed on a quantum computer and derive a closed form for the Hamiltonian coefficients in second quantization. We then discuss a mean-field method and a multi-reference state approach that allow for an efficient classical computation and an efficient initial state preparation on a quantum computer. The second part of the thesis presents a detailed description on how long-range interacting fermionic systems can be simulated on classical computers using a variational method, introduce an Ansatz which could potentially simplify numerical simulations and give an explicit quantum circuit that shows how the variational state can be used as an initial state and how it can implemented on a quantum computer. In the last part, two novel protocols are presented that generate a variety of prominent many-body operators from two-body interactions and show how these protocols improve over previous construction schemes for a number of important examples. iv Zusammenfassung Diese Arbeit behandelt verschiedene zentrale Probleme theoretischer Natur, welche im Rahmen der Simulation quantenmechanischer Systeme auf klassischen und Quantencomputern auftreten. Ein Großteil dieser Arbeit beschäftigt sich mit stark wechselwirkendenden fermionischen Systemen, genauer gesagt, dem Verhalten von Elektronen innerhalb eines starken Magnetfelds. Es wird dargelegt, wie das Energiespektrum des Quanten-Hall-Effekt-Hamiltonoperators auf einem Quantencomputer berechnet werden kann, und es werden geschlossene Ausdrücke für dessen Hamilton-Koeffizienten in zweiter Quantisierung hergeleitet. Anschließend werden sowohl ein Molekularfeld-als auch ein Multi-Referenz-Ansatz diskutiert, welche eine effiziente Berechnung auf klassischen Rechnern zulassen sowie eine effiziente Implementierung auf Quantencomputern ermöglichen. Der zweite Teil dieser Arbeit erläutert, wie man langreichweitige, wechselwirkende fermionische Systeme mit Hilfe einer neuen Variationsmethode, welche über die Molekularfeldnäherung hinaus geht, auf einem klassischen Computer simulieren kann. Es wird darüber hinaus ein alternativer Ansatz vorgestellt, der Teile dieser Variationsmethode vereinfachen könnte, und gezeigt, wie sich der Ansatz auf einem Quantencomputer realisieren lässt. Im letzten Teil werden zwei neue Methoden vorgestellt, welche es ermöglichen, eine Reihe wichtiger Vielteilchen-Operatoren aus Zweiteilchen-Wechselwirkungen zu erzeugen. Beide Methoden werden durch eine Vielzahl an wichtigen Beispielen veranschaulicht. v 1 A classical (quantum) algorithm is a step-by-step instruction on how to solve a given problem with operations that can run on a classical (quantum) computer. 2 The runtime is measured by the number of elementary operations used by the respective quantum or classical algorithm. For the former, this can be measured in terms of the quantum circuit model, which is just a specific sequence of elementary quantum operations applied to a number of qubits (a qubit is the quantum analogue to a classical bit). All of this and more is detailed in Sections 1.5 and 1.6. 3 Some of the most prominent classical methods include density functional theory (DFT), which exploits the electron density distribution rather than the many electron wave function using a variety of approximations [19] , but fails at describing strongly interacting systems. Another approach based on the wave function representation is the quantum Monte Carlo (QMC) method, but its efficient implementation suffers from the infamous fermionic sign problem, that leads to an exponential increase in the error of the simulation with system size [20] . Another classical algorithm used to find approximate ground states to the many-body problem is density matrix renormalization group (DMRG) [21] , which very successfully describes one-dimensional systems, but has trouble building up enough entanglement to describe most strongly correlated two-and three-dimensional systems. More on the chemistry side, full configuration quantum Monte Carlo (FCIQMC) is an approach based on QMC, that deals with the One can combine Eqs. (1.2.13) and (1.2.16) and obtain the electronic structure Hamiltonian H = T + V . In doing so, we have quietly neglected the fact that matter consists not only of electrons, but also of nuclei. The nuclei masses are however three orders of magnitudes larger than the masses of the electrons and one can assume the electrons to move within a field of fixed nuclei within good approximation, which is known as the Born-Oppenheimer approximation [45, 46] . A precise non-relativistic treatment of matter would include electron-nucleon, nucleon-nucleon, electron-electron interaction as well as single-body electron and nucleon terms. 9 Note that this definition slightly deviates from the definition we use in Chapters 2 and 3. (1.3.10) By comparing the right-hand sides of Eqs. (1.3.9) and (1.3.10), one realizes that equality requires γ to be composed of anti-commuting variables to obtain non-trivial solutions. Fermionic coherent states For fermionic fields, the only physically realizable eigenstate of the fermionic annihilation operator is the vacuum state. Other eigenstates can be constructed only in a formal way and are merely introduced as a means to do analytical computations. One can show that the unitary displacement operator Functions of Grassmann variables f (γ) which do commute with a Grassmann number are called even, those that anti-commute are called odd functions. 12 We will consider a physical density operator ρ which is a positive Hermitian operator of unit trace, i.o.w. ρ must fulfill The expectation value of a fermionic operator X w.r.t. a normalized quantum state ρ is given by X ρ = tr(ρX). i 2 2Nso p,q=1 θp(Γm) pq θq (1.3.54) for some real and anti-symmetric (i.e. skew-symmetric) (2N so × N so )-matrix Γ m , which is called the correlation matrix. 13 Wick's theorem will be discussed in detail in Appendix 3.B. 14 A basic example for Eq. (1.3.56) and p = 2 is given by 16 The time evolution of a density operator ρ is described by the von-Neumann equation For pure states, the von-Neumann equation is equivalent to the Schrödinger equation. A fault-tolerant quantum computer This thesis considers an idealized quantum computer, arguably the most frustrating assumption for anyone who is trying to run an actual experiment. An idealized quantum computer performs state initialization, gates and measurements without any errors or losses and is perfectly isolated from the environment. While this seems to be the somewhat most unrealistic assumption one can make, it turns out that using quantum error-correction one can not only protect stored and transmitted quantum states, but even protect quantum states which dynamically undergo a quantum computation. This is the content of the following theorem, which we state due to its significance for quantum computing. 5. One has to be able to read out the state of a qubit (in e.g. the computational basis) at the end or even in between the computation. Lemma 1.6.2 (Oblivious amplitude amplification). Let W (V ) be a unitary matrix which acts on n + m (n) qubits ant let θ ∈ (0, π/2). For any |ψ , we let W |0 m |ψ = sin(θ) |0 m V |ψ + cos(θ) |Φ ⊥ , (1.6.13) 34 Where efficiently implementable here again refers to its respective quantum circuit scaling at most polynomially in circuit size and depth (time) with system size. 35 When V is unitary, p can be interpreted as a probability, however, if V is not unitary, p can be larger than 1. V = e −iHt/m = e −i j U j /m .Michael Kaicher, Universität Des Saarlandeswork_ddu3szbvtzgntb2tyyo2vj5enaThu, 14 Apr 2022 00:00:00 GMTAn Introduction to Quantum Computing for Statisticians and Data Scientists
https://scholar.archive.org/work/sm2v5mh6pnc6tgepm7ing2ljg4
Quantum computers promise to surpass the most powerful classical supercomputers when it comes to solving many critically important practical problems, such as pharmaceutical and fertilizer design, supply chain and traffic optimization, or optimization for machine learning tasks. Because quantum computers function fundamentally differently from classical computers, the emergence of quantum computing technology will lead to a new evolutionary branch of statistical and data analytics methodologies. This review provides an introduction to quantum computing designed to be accessible to statisticians and data scientists, aiming to equip them with an overarching framework of quantum computing, the basic language and building blocks of quantum algorithms, and an overview of existing quantum applications in statistics and data analysis. Our goal is to enable statisticians and data scientists to follow quantum computing literature relevant to their fields, to collaborate with quantum algorithm designers, and, ultimately, to bring forth the next generation of statistical and data analytics tools.Anna Lopatnikova, Minh-Ngoc Tran, Scott A. Sissonwork_sm2v5mh6pnc6tgepm7ing2ljg4Sun, 03 Apr 2022 00:00:00 GMTA Random Matrix Theory Approach to Damping in Deep Learning
https://scholar.archive.org/work/gp2ruhfxh5bfleeuonitihindy
We conjecture that the inherent difference in generalisation between adaptive and non-adaptive gradient methods in deep learning stems from the increased estimation noise in the flattest directions of the true loss surface. We demonstrate that typical schedules used for adaptive methods (with low numerical stability or damping constants) serve to bias relative movement towards flat directions relative to sharp directions, effectively amplifying the noise-to-signal ratio and harming generalisation. We further demonstrate that the numerical damping constant used in these methods can be decomposed into a learning rate reduction and linear shrinkage of the estimated curvature matrix. We then demonstrate significant generalisation improvements by increasing the shrinkage coefficient, closing the generalisation gap entirely in both logistic regression and several deep neural network experiments. Extending this line further, we develop a novel random matrix theory based damping learner for second order optimiser inspired by linear shrinkage estimation. We experimentally demonstrate our learner to be very insensitive to the initialised value and to allow for extremely fast convergence in conjunction with continued stable training and competitive generalisation.Diego Granziol, Nicholas Baskervillework_gp2ruhfxh5bfleeuonitihindyWed, 16 Mar 2022 00:00:00 GMTBounds on quantum evolution complexity via lattice cryptography
https://scholar.archive.org/work/yhbat2urhzbjxeu2ierknjlnti
We address the difference between integrable and chaotic motion in quantum theory as manifested by the complexity of the corresponding evolution operators. Complexity is understood here as the shortest geodesic distance between the time-dependent evolution operator and the origin within the group of unitaries. (An appropriate 'complexity metric' must be used that takes into account the relative difficulty of performing 'nonlocal' operations that act on many degrees of freedom at once.) While simply formulated and geometrically attractive, this notion of complexity is numerically intractable save for toy models with Hilbert spaces of very low dimensions. To bypass this difficulty, we trade the exact definition in terms of geodesics for an upper bound on complexity, obtained by minimizing the distance over an explicitly prescribed infinite set of curves, rather than over all possible curves. Identifying this upper bound turns out equivalent to the closest vector problem (CVP) previously studied in integer optimization theory, in particular, in relation to lattice-based cryptography. Effective approximate algorithms are hence provided by the existing mathematical considerations, and they can be utilized in our analysis of the upper bounds on quantum evolution complexity. The resulting algorithmically implemented complexity bound systematically assigns lower values to integrable than to chaotic systems, as we demonstrate by explicit numerical work for Hilbert spaces of dimensions up to ~10^4.Ben Craps, Marine De Clerck, Oleg Evnin, Philip Hacker, Maxim Pavlovwork_yhbat2urhzbjxeu2ierknjlntiWed, 09 Mar 2022 00:00:00 GMTTight Convergence Rate Bounds for Optimization Under Power Law Spectral Conditions
https://scholar.archive.org/work/d4qzlri7jjd5ji7hdel6u6h3iu
Performance of optimization on quadratic problems sensitively depends on the low-lying part of the spectrum. For large (effectively infinite-dimensional) problems, this part of the spectrum can often be naturally represented or approximated by power law distributions. In this paper we perform a systematic study of a range of classical single-step and multi-step first order optimization algorithms, with adaptive and non-adaptive, constant and non-constant learning rates: vanilla Gradient Descent, Steepest Descent, Heavy Ball, and Conjugate Gradients. For each of these, we prove that a power law spectral assumption entails a power law for convergence rate of the algorithm, with the convergence rate exponent given by a specific multiple of the spectral exponent. We establish both upper and lower bounds, showing that the results are tight. Finally, we demonstrate applications of these results to kernel learning and training of neural networks in the NTK regime.Maksim Velikanov, Dmitry Yarotskywork_d4qzlri7jjd5ji7hdel6u6h3iuWed, 02 Feb 2022 00:00:00 GMTA Survey of Uncertainty in Deep Neural Networks
https://scholar.archive.org/work/5etuseuigfgh3ckrflqxbuiox4
Due to their increasing spread, confidence in neural network predictions became more and more important. However, basic neural networks do not deliver certainty estimates or suffer from over or under confidence. Many researchers have been working on understanding and quantifying uncertainty in a neural network's prediction. As a result, different types and sources of uncertainty have been identified and a variety of approaches to measure and quantify uncertainty in neural networks have been proposed. This work gives a comprehensive overview of uncertainty estimation in neural networks, reviews recent advances in the field, highlights current challenges, and identifies potential research opportunities. It is intended to give anyone interested in uncertainty estimation in neural networks a broad overview and introduction, without presupposing prior knowledge in this field. A comprehensive introduction to the most crucial sources of uncertainty is given and their separation into reducible model uncertainty and not reducible data uncertainty is presented. The modeling of these uncertainties based on deterministic neural networks, Bayesian neural networks, ensemble of neural networks, and test-time data augmentation approaches is introduced and different branches of these fields as well as the latest developments are discussed. For a practical application, we discuss different measures of uncertainty, approaches for the calibration of neural networks and give an overview of existing baselines and implementations. Different examples from the wide spectrum of challenges in different fields give an idea of the needs and challenges regarding uncertainties in practical applications. Additionally, the practical limitations of current methods for mission- and safety-critical real world applications are discussed and an outlook on the next steps towards a broader usage of such methods is given.Jakob Gawlikowski, Cedrique Rovile Njieutcheu Tassi, Mohsin Ali, Jongseok Lee, Matthias Humt, Jianxiang Feng, Anna Kruspe, Rudolph Triebel, Peter Jung, Ribana Roscher, Muhammad Shahzad, Wen Yang, Richard Bamler, Xiao Xiang Zhuwork_5etuseuigfgh3ckrflqxbuiox4Tue, 18 Jan 2022 00:00:00 GMTError estimates for DeepOnets: A deep learning framework in infinite dimensions
https://scholar.archive.org/work/o3xucm5m4fhfdau2ohasccub7y
DeepONets have recently been proposed as a framework for learning nonlinear operators mapping between infinite dimensional Banach spaces. We analyze DeepONets and prove estimates on the resulting approximation and generalization errors. In particular, we extend the universal approximation property of DeepONets to include measurable mappings in non-compact spaces. By a decomposition of the error into encoding, approximation and reconstruction errors, we prove both lower and upper bounds on the total error, relating it to the spectral decay properties of the covariance operators, associated with the underlying measures. We derive almost optimal error bounds with very general affine reconstructors and with random sensor locations as well as bounds on the generalization error, using covering number arguments. We illustrate our general framework with four prototypical examples of nonlinear operators, namely those arising in a nonlinear forced ODE, an elliptic PDE with variable coefficients and nonlinear parabolic and hyperbolic PDEs. While the approximation of arbitrary Lipschitz operators by DeepONets to accuracy ϵ is argued to suffer from a "curse of dimensionality" (requiring a neural networks of exponential size in 1/ϵ), in contrast, for all the above concrete examples of interest, we rigorously prove that DeepONets can break this curse of dimensionality (achieving accuracy ϵ with neural networks of size that can grow algebraically in 1/ϵ). Thus, we demonstrate the efficient approximation of a potentially large class of operators with this machine learning framework.Samuel Lanthaler and Siddhartha Mishra and George Em Karniadakiswork_o3xucm5m4fhfdau2ohasccub7yThu, 13 Jan 2022 00:00:00 GMTQuantum algorithms for linear and nonlinear differential equations
https://scholar.archive.org/work/qacnlm5vtff2jdo53az7o657mi
Quantum computers are expected to dramatically outperform classical computers for certain computational problems. Originally developed for simulating quantum physics, quantum algorithms have been subsequently developed to address diverse computational challenges. There has been extensive previous work for linear dynamics and discrete models, including Hamiltonian simulations and systems of linear equations. However, for more complex realistic problems characterized by differential equations, the capability of quantum computing is far from well understood. One fundamental challenge is the substantial difference between the linear dynamics of a system of qubits and real-world systems with continuum and nonlinear behaviors. My research is concerned with mathematical aspects of quantum computing. In this dissertation, I focus mainly on the design and analysis of quantum algorithms for differential equations. Systems of linear ordinary differential equations (ODEs) and linear elliptic partial differential equations (PDEs) appear ubiquitous in natural and social science, engineering, and medicine. I propose a variety of quantum algorithms based on finite difference methods and spectral methods for producing the quantum encoding of the solutions, with an exponential improvement in the precision over previous quantum algorithms. Nonlinear differential equations exhibit rich phenomena in many domains but are notoriously difficult to solve. Whereas previous quantum algorithms for general nonlinear equations have been severely limited due to the linearity of quantum mechanics, I give the first efficient quantum algorithm for nonlinear differential equations with sufficiently strong dissipation. I also establish a lower bound, showing that nonlinear differential equations with sufficiently weak dissipation have worst-case complexity exponential in time, giving an almost tight classification of the quantum complexity of simulating nonlinear dynamics. Overall, utilizing advanced linear algebra techniques and nonlinear analysis [...]Jinpeng Liuwork_qacnlm5vtff2jdo53az7o657miQuantum Computing for Optimization and Machine Learning
https://scholar.archive.org/work/dzl35jwmfre5xj7qndldnvx75q
Quantum Computing leverages the quantum properties of subatomic matter to enable computations faster than those possible on a regular computer. Quantum Computers have become increasingly practical in recent years, with some small-scale machines becoming available for public use. The rising importance of machine learning has highlighted a large class of computing and optimization problems that process massive amounts of data and incur correspondingly large computational costs. This raises the natural question of how quantum computers may be leveraged to solve these problems more efficiently. This dissertation presents some encouraging results on the design of quantum algorithms for machine learning and optimization. We first focus on tasks with provably more efficient quantum algorithms. We show a quantum speedup for convex optimization by extending quantum gradient estimation algorithms to efficiently compute subgradients of non-differentiable functions. We also develop a quantum framework for simulated annealing algorithms which is used to show a quantum speedup in estimating the volumes of convex bodies. Finally, we demonstrate a quantum algorithm for solving matrix games, which can be applied to a variety of learning problems such as linear classification, minimum enclosing ball, and $\ell-2$ margin SVMs. We then shift our focus to variational quantum algorithms, which describe a family of heuristic algorithms that use parameterized quantum circuits as function models that can be fit for various learning and optimization tasks. We seek to analyze the properties of these algorithms including their efficient formulation and training, expressivity, and the convergence of the associated optimization problems. We formulate a model of quantum Wasserstein GANs in order to facilitate the robust and scalable generative learning of quantum states. We also investigate the expressivity of so called \emph{Quantum Neural Networks} compared to classical ReLU networks and investigate both theoretical and empirical separations [...]Shouvanik Chakrabartiwork_dzl35jwmfre5xj7qndldnvx75qAsymptotic Errors for Teacher-Student Convex Generalized Linear Models (or : How to Prove Kabashima's Replica Formula)
https://scholar.archive.org/work/gbav3fvs5vegpo5x5w5yjfxmpq
There has been a recent surge of interest in the study of asymptotic reconstruction performance in various cases of generalized linear estimation problems in the teacher-student setting, especially for the case of i.i.d standard normal matrices. Here, we go beyond these matrices, and prove an analytical formula for the reconstruction performance of convex generalized linear models with rotationally-invariant data matrices with arbitrary bounded spectrum, rigorously confirming a conjecture originally derived using the replica method from statistical physics. The formula includes many problems such as compressed sensing or sparse logistic classification. The proof is achieved by leveraging on message passing algorithms and the statistical properties of their iterates, allowing to characterize the asymptotic empirical distribution of the estimator. Our proof is crucially based on the construction of converging sequences of an oracle multi-layer vector approximate message passing algorithm, where the convergence analysis is done by checking the stability of an equivalent dynamical system. We illustrate our claim with numerical examples on mainstream learning methods such as sparse logistic regression and linear support vector classifiers, showing excellent agreement between moderate size simulation and the asymptotic prediction.Cedric Gerbelot, Alia Abbara, Florent Krzakalawork_gbav3fvs5vegpo5x5w5yjfxmpqWed, 15 Dec 2021 00:00:00 GMTLearning Partial Differential Equations in Reproducing Kernel Hilbert Spaces
https://scholar.archive.org/work/u5he2aigrjhhtarorza4u4ftuy
We propose a new data-driven approach for learning the fundamental solutions (i.e. Green's functions) of various linear partial differential equations (PDEs) given sample pairs of input-output functions. Building off the theory of functional linear regression (FLR), we estimate the best-fit Green's function and bias term of the fundamental solution in a reproducing kernel Hilbert space (RKHS) which allows us to regularize their smoothness and impose various structural constraints. We use a general representer theorem for operator RKHSs to approximate the original infinite-dimensional regression problem by a finite-dimensional one, reducing the search space to a parametric class of Green's functions. In order to study the prediction error of our Green's function estimator, we extend prior results on FLR with scalar outputs to the case with functional outputs. Furthermore, our rates of convergence hold even in the misspecified setting when the data is generated by a nonlinear PDE under certain constraints. Finally, we demonstrate applications of our method to several linear PDEs including the Poisson, Helmholtz, Schrödinger, Fokker-Planck, and heat equation and highlight its ability to extrapolate to more finely sampled meshes without any additional training.George Stepaniantswork_u5he2aigrjhhtarorza4u4ftuyWed, 17 Nov 2021 00:00:00 GMTLearning Multiresolution Matrix Factorization and its Wavelet Networks on Graphs
https://scholar.archive.org/work/cpboautrrzay7psrb5sj2ptfeq
Multiresolution Matrix Factorization (MMF) is unusual amongst fast matrix factorization algorithms in that it does not make a low rank assumption. This makes MMF especially well suited to modeling certain types of graphs with complex multiscale or hierarchical strucutre. While MMF promises to yields a useful wavelet basis, finding the factorization itself is hard, and existing greedy methods tend to be brittle. In this paper we propose a learnable version of MMF that carfully optimizes the factorization with a combination of reinforcement learning and Stiefel manifold optimization through backpropagating errors. We show that the resulting wavelet basis far outperforms prior MMF algorithms and provides the first version of this type of factorization that can be robustly deployed on standard learning tasks.Truong Son Hy, Risi Kondorwork_cpboautrrzay7psrb5sj2ptfeqTue, 02 Nov 2021 00:00:00 GMTFiltering variational quantum algorithms for combinatorial optimization
https://scholar.archive.org/work/m5riowindzcirdhakzesrfuaam
Current gate-based quantum computers have the potential to provide a computational advantage if algorithms use quantum hardware efficiently. To make combinatorial optimization more efficient, we introduce the Filtering Variational Quantum Eigensolver (F-VQE) which utilizes filtering operators to achieve faster and more reliable convergence to the optimal solution. Additionally we explore the use of causal cones to reduce the number of qubits required on a quantum computer. Using random weighted MaxCut problems, we numerically analyze our methods and show that they perform better than the original VQE algorithm and the Quantum Approximate Optimization Algorithm (QAOA). We also demonstrate the experimental feasibility of our algorithms on a Honeywell trapped-ion quantum processor.David Amaro, Carlo Modica, Matthias Rosenkranz, Mattia Fiorentini, Marcello Benedetti, Michael Lubaschwork_m5riowindzcirdhakzesrfuaamThu, 14 Oct 2021 00:00:00 GMTState preparation and evolution in quantum computing: a perspective from Hamiltonian moments
https://scholar.archive.org/work/2voohds22vag5aw5uca63uw55e
Quantum algorithms on the noisy intermediate-scale quantum (NISQ) devices are expected to simulate quantum systems that are classically intractable to demonstrate quantum advantages. However, the non-negligible gate error on the NISQ devices impedes the conventional quantum algorithms to be implemented. Practical strategies usually exploit hybrid quantum classical algorithms to demonstrate potentially useful applications of quantum computing in the NISQ era. Among the numerous hybrid algorithms, recent efforts highlight the development of quantum algorithms based upon quantum computed Hamiltonian moments, ⟨ϕ | ℋ̂^n | ϕ⟩ (n=1,2,⋯), with respect to quantum state |ϕ⟩. In this tutorial, we will give a brief review of these quantum algorithms with focuses on the typical ways of computing Hamiltonian moments using quantum hardware and improving the accuracy of the estimated state energies based on the quantum computed moments. Furthermore, we will present a tutorial to show how we can measure and compute the Hamiltonian moments of a four-site Heisenberg model, and compute the energy and magnetization of the model utilizing the imaginary time evolution in the real IBM-Q NISQ hardware environment. Along this line, we will further discuss some practical issues associated with these algorithms. We will conclude this tutorial review by overviewing some possible developments and applications in this direction in the near future.Joseph C. Aulicino, Trevor Keen, Bo Pengwork_2voohds22vag5aw5uca63uw55eThu, 30 Sep 2021 00:00:00 GMT