arXiv subjects

Adam Smith

Publications and source records attributed to Adam Smith.

At least 73 records · Page 4Linked to original sources

arXiv2022

Strong Memory Lower Bounds for Learning Natural Models

We give lower bounds on the amount of memory required by one-pass streaming algorithms for solving several natural learning problems. In a setting where examples lie in $\{0,1\}^d$ and the optimal classifier can be encoded using $κ$ bits, we show that algorithms which learn using a near-minimal number of examples, $\tilde O(κ)$, must use $\tilde Ω( dκ)$ bits of space. Our space bounds match the dimension of the ambient space of the problem's natural parametrization, even when it is quadratic in the size of examples and the final classifier. For instance, in the setting of $d$-sparse linear classifiers over degree-2 polynomial features, for which $κ=Θ(d\log d)$, our space lower bound is $\tildeΩ(d^2)$. Our bounds degrade gracefully with the stream length $N$, generally having the form $\tildeΩ\left(dκ\cdot \fracκ{N}\right)$. Bounds of the form $Ω(dκ)$ were known for learning parity and other problems defined over finite fields. Bounds that apply in a narrow range of sample sizes are also known for linear regression. Ours are the first such bounds for problems of the type commonly seen in recent learning applications that apply for a large range of input sizes.

cs.LG↗

arXiv2022

Data compression for quantum machine learning

The advent of noisy-intermediate scale quantum computers has introduced the exciting possibility of achieving quantum speedups in machine learning tasks. These devices, however, are composed of a small number of qubits, and can faithfully run only short circuits. This puts many proposed approaches for quantum machine learning beyond currently available devices. We address the problem of efficiently compressing and loading classical data for use on a quantum computer. Our proposed methods allow both the required number of qubits and depth of the quantum circuit to be tuned. We achieve this by using a correspondence between matrix-product states and quantum circuits, and further propose a hardware-efficient quantum circuit approach, which we benchmark on the Fashion-MNIST dataset. Finally, we demonstrate that a quantum circuit based classifier can achieve competitive accuracy with current tensor learning methods using only 11 qubits.

quant-ph↗

arXiv2022

Crossing a topological phase transition with a quantum computer

Quantum computers promise to perform computations beyond the reach of modern computers with profound implications for scientific research. Due to remarkable technological advances, small scale devices are now becoming available for use. One of the most apparent applications for such a device is the study of complex many-body quantum systems, where classical computers are unable to deal with the generic exponential complexity of quantum states. Even zero-temperature equilibrium phases of matter and the transitions between them have yet to be fully classified, with topologically protected phases presenting major difficulties. We construct and measure a continuously parametrized family of states crossing a symmetry protected topological phase transition on the IBM Q quantum computers. We present two complementary methods for measuring string order parameters that reveal the transition, and additionally analyse the effects of noise in the device using simple error models. The simulation that we perform is easily scalable and is a practical demonstration of the utility of near-term quantum computers for the study of quantum phases of matter and their transitions.

cond-mat.str-el↗

arXiv2022

The Price of Differential Privacy under Continual Observation

We study the accuracy of differentially private mechanisms in the continual release model. A continual release mechanism receives a sensitive dataset as a stream of $T$ inputs and produces, after receiving each input, an accurate output on the obtained inputs. In contrast, a batch algorithm receives the data as one batch and produces a single output. We provide the first strong lower bounds on the error of continual release mechanisms. In particular, for two fundamental problems that are widely studied and used in the batch model, we show that the worst case error of every continual release algorithm is $\tilde Ω(T^{1/3})$ times larger than that of the best batch algorithm. Previous work shows only a polylogarithimic (in $T$) gap between the worst case error achievable in these two models; further, for many problems, including the summation of binary attributes, the polylogarithmic gap is tight (Dwork et al., 2010; Chan et al., 2010). Our results show that problems closely related to summation -- specifically, those that require selecting the largest of a set of sums -- are fundamentally harder in the continual release model than in the batch model. Our lower bounds assume only that privacy holds for streams fixed in advance (the "nonadaptive" setting). However, we provide matching upper bounds that hold in a model where privacy is required even for adaptively selected streams. This model may be of independent interest.

cs.DS↗

arXiv2021

Skeleton of Matrix-Product-State-Solvable Models Connecting Topological Phases of Matter

Models whose ground states can be written as an exact matrix product state (MPS) provide valuable insights into phases of matter. While MPS-solvable models are typically studied as isolated points in a phase diagram, they can belong to a connected network of MPS-solvable models, which we call the MPS skeleton. As a case study where we can completely unearth this skeleton, we focus on the one-dimensional BDI class -- non-interacting spinless fermions with time-reversal symmetry. This class, labelled by a topological winding number, contains the Kitaev chain and is Jordan-Wigner-dual to various symmetry-breaking and symmetry-protected topological (SPT) spin chains. We show that one can read off from the Hamiltonian whether its ground state is an MPS: defining a polynomial whose coefficients are the Hamiltonian parameters, MPS-solvability corresponds to this polynomial being a perfect square. We provide an explicit construction of the ground state MPS, its bond dimension growing exponentially with the range of the Hamiltonian. This complete characterization of the MPS skeleton in parameter space has three significant consequences: (i) any two topologically distinct phases in this class admit a path of MPS-solvable models between them, including the phase transition which obeys an area law for its entanglement entropy; (ii) we illustrate that the subset of MPS-solvable models is dense in this class by constructing a sequence of MPS-solvable models which converge to the Kitaev chain (equivalently, the quantum Ising chain in a transverse field); (iii) a subset of these MPS states can be particularly efficiently processed on a noisy intermediate-scale quantum computer.

cond-mat.str-el↗

arXiv2021

Orthogonal Quantum Many-body Scars

Quantum many-body scars have been put forward as counterexamples to the Eigenstate Thermalization Hypothesis. These atypical states are observed in a range of correlated models as long-lived oscillations of local observables in quench experiments starting from selected initial states. The long-time memory is a manifestation of quantum non-ergodicity generally linked to a sub-extensive generation of entanglement entropy, the latter of which is widely used as a diagnostic for identifying quantum many-body scars numerically as low entanglement outliers. Here we show that, by adding kinetic constraints to a fractionalized orthogonal metal, we can construct a minimal model with orthogonal quantum many-body scars leading to persistent oscillations with infinite lifetime coexisting with rapid volume-law entanglement generation. Our example provides new insights into the link between quantum ergodicity and many-body entanglement while opening new avenues for exotic non-equilibrium dynamics in strongly correlated multi-component quantum systems.

cond-mat.stat-mech↗

arXiv2021

Identifying Correlation Clusters in Many-Body Localized Systems

We introduce techniques for analysing the structure of quantum states of many-body localized (MBL) spin chains by identifying correlation clusters from pairwise correlations. These techniques proceed by interpreting pairwise correlations in the state as a weighted graph, which we analyse using an established graph theoretic clustering algorithm. We validate our approach by studying the eigenstates of a disordered XXZ spin chain across the MBL to ergodic transition, as well as the non-equilibrium dyanmics in the MBL phase following a global quantum quench. We successfully reproduce theoretical predictions about the MBL transition obtained from renormalization group schemes. Furthermore, we identify a clear signature of many-body dynamics analogous to the logarithmic growth of entanglement. The techniques that we introduce are computationally inexpensive and in combination with matrix product state methods allow for the study of large scale localized systems. Moreover, the correlation functions we use are directly accessible in a range of experimental settings including cold atoms.

cond-mat.dis-nn↗

arXiv2021

When is Memorization of Irrelevant Training Data Necessary for High-Accuracy Learning?

Modern machine learning models are complex and frequently encode surprising amounts of information about individual inputs. In extreme cases, complex models appear to memorize entire input examples, including seemingly irrelevant information (social security numbers from text, for example). In this paper, we aim to understand whether this sort of memorization is necessary for accurate learning. We describe natural prediction problems in which every sufficiently accurate training algorithm must encode, in the prediction model, essentially all the information about a large subset of its training examples. This remains true even when the examples are high-dimensional and have entropy much higher than the sample size, and even when most of that information is ultimately irrelevant to the task at hand. Further, our results do not depend on the training algorithm or the class of models used for learning. Our problems are simple and fairly natural variants of the next-symbol prediction and the cluster labeling tasks. These tasks can be seen as abstractions of text- and image-related prediction problems. To establish our results, we reduce from a family of one-way communication problems for which we prove new information complexity lower bounds. Additionally, we present synthetic-data experiments demonstrating successful attacks on logistic regression and neural network classifiers.

cs.LG↗

arXiv2021

Non-parametric Differentially Private Confidence Intervals for the Median

Differential privacy is a restriction on data processing algorithms that provides strong confidentiality guarantees for individual records in the data. However, research on proper statistical inference, that is, research on properly quantifying the uncertainty of the (noisy) sample estimate regarding the true value in the population, is currently still limited. This paper proposes and evaluates several strategies to compute valid differentially private confidence intervals for the median. Instead of computing a differentially private point estimate and deriving its uncertainty, we directly estimate the interval bounds and discuss why this approach is superior if ensuring privacy is important. We also illustrate that addressing both sources of uncertainty--the error from sampling and the error from protecting the output--simultaneously should be preferred over simpler approaches that incorporate the uncertainty in a sequential fashion. We evaluate the performance of the different algorithms under various parameter settings in extensive simulation studies and demonstrate how the findings could be applied in practical settings using data from the 1940 Decennial Census.

cs.CR↗

arXiv2021

Unlocking capacities of viral genomics for the COVID-19 pandemic response

More than any other infectious disease epidemic, the COVID-19 pandemic has been characterized by the generation of large volumes of viral genomic data at an incredible pace due to recent advances in high-throughput sequencing technologies, the rapid global spread of SARS-CoV-2, and its persistent threat to public health. However, distinguishing the most epidemiologically relevant information encoded in these vast amounts of data requires substantial effort across the research and public health communities. Studies of SARS-CoV-2 genomes have been critical in tracking the spread of variants and understanding its epidemic dynamics, and may prove crucial for controlling future epidemics and alleviating significant public health burdens. Together, genomic data and bioinformatics methods enable broad-scale investigations of the spread of SARS-CoV-2 at the local, national, and global scales and allow researchers the ability to efficiently track the emergence of novel variants, reconstruct epidemic dynamics, and provide important insights into drug and vaccine development and disease control. Here, we discuss the tremendous opportunities that genomics offers to unlock the effective use of SARS-CoV-2 genomic data for efficient public health surveillance and guiding timely responses to COVID-19.

q-bio.GN↗

arXiv2021

Topological two-dimensional Floquet lattice on a single superconducting qubit

Previous theoretical and experimental research has shown that current NISQ devices constitute powerful platforms for analogue quantum simulation. With the exquisite level of control offered by state-of-the-art quantum computers, we show that one can go further and implement a wide class of Floquet Hamiltonians, or timedependent Hamiltonians in general. We then implement a single-qubit version of these models in the IBM Quantum Experience and experimentally realize a temporal version of the Bernevig-Hughes-Zhang Chern insulator. From our data we can infer the presence of a topological transition, thus realizing an earlier proposal of topological frequency conversion by Martin, Refael, and Halperin. Our study highlights promises and limitations when studying many-body systems through multi-frequency driving of quantum computers.

quant-ph↗

arXiv2020

Empirical Risk Minimization in the Non-interactive Local Model of Differential Privacy

In this paper, we study the Empirical Risk Minimization (ERM) problem in the non-interactive Local Differential Privacy (LDP) model. Previous research on this problem \citep{smith2017interaction} indicates that the sample complexity, to achieve error $α$, needs to be exponentially depending on the dimensionality $p$ for general loss functions. In this paper, we make two attempts to resolve this issue by investigating conditions on the loss functions that allow us to remove such a limit. In our first attempt, we show that if the loss function is $(\infty, T)$-smooth, by using the Bernstein polynomial approximation we can avoid the exponential dependency in the term of $α$. We then propose player-efficient algorithms with $1$-bit communication complexity and $O(1)$ computation cost for each player. The error bound of these algorithms is asymptotically the same as the original one. With some additional assumptions, we also give an algorithm which is more efficient for the server. In our second attempt, we show that for any $1$-Lipschitz generalized linear convex loss function, there is an $(ε, δ)$-LDP algorithm whose sample complexity for achieving error $α$ is only linear in the dimensionality $p$. Our results use a polynomial of inner product approximation technique. Finally, motivated by the idea of using polynomial approximation and based on different types of polynomial approximations, we propose (efficient) non-interactive locally differentially private algorithms for learning the set of k-way marginal queries and the set of smooth queries.

cs.LG↗

arXiv2020

Intrinsic sign problems in topological quantum field theories

The sign problem is a widespread numerical hurdle preventing us from simulating the equilibrium behavior of various problems at the forefront of physics. Focusing on an important sub-class of such problems, bosonic $(2+1)$-dimensional topological quantum field theories, here we provide a simple criterion to diagnose intrinsic sign problems---that is, sign problems that are inherent to that phase of matter and cannot be removed by any local unitary transformation. Explicitly, \textit{if the exchange statistics of the anyonic excitations do not form complete sets of roots of unity, then the model has an intrinsic sign problem}. This establishes a concrete connection between the statistics of anyons, contained in the modular $S$ and $T$ matrices, and the presence of a sign problem in a microscopic Hamiltonian. Furthermore, it places constraints on the phases that can be realised by stoquastic Hamiltonians. We prove this and a more restrictive criterion for the large set of gapped bosonic models described by an abelian topological quantum field theory at low-energy, and offer evidence that it applies more generally with analogous results for non-abelian and chiral theories.

cond-mat.str-el↗

arXiv2020

Real- and imaginary-time evolution with compressed quantum circuits

The current generation of noisy intermediate scale quantum computers introduces new opportunities to study quantum many-body systems. In this paper, we show that quantum circuits can provide a dramatically more efficient representation than current classical numerics of the quantum states generated under non-equilibrium quantum dynamics. For quantum circuits, we perform both real- and imaginary-time evolution using an optimization algorithm that is feasible on near-term quantum computers. We benchmark the algorithms by finding the ground state and simulating a global quench of the transverse field Ising model with a longitudinal field on a classical computer. Furthermore, we implement (classically optimized) gates on a quantum processing unit and demonstrate that our algorithm effectively captures real time evolution.

quant-ph↗

arXiv2020

Noninteractive Locally Private Learning of Linear Models via Polynomial Approximations

Minimizing a convex risk function is the main step in many basic learning algorithms. We study protocols for convex optimization which provably leak very little about the individual data points that constitute the loss function. Specifically, we consider differentially private algorithms that operate in the local model, where each data record is stored on a separate user device and randomization is performed locally by those devices. We give new protocols for \emph{noninteractive} LDP convex optimization---i.e., protocols that require only a single randomized report from each user to an untrusted aggregator. We study our algorithms' performance with respect to expected loss---either over the data set at hand (empirical risk) or a larger population from which our data set is assumed to be drawn. Our error bounds depend on the form of individuals' contribution to the expected loss. For the case of \emph{generalized linear losses} (such as hinge and logistic losses), we give an LDP algorithm whose sample complexity is only linear in the dimensionality $p$ and quasipolynomial in other terms (the privacy parameters $ε$ and $δ$, and the desired excess risk $α$). This is the first algorithm for nonsmooth losses with sub-exponential dependence on $p$. For the Euclidean median problem, where the loss is given by the Euclidean distance to a given data point, we give a protocol whose sample complexity grows quasipolynomially in $p$. This is the first protocol with sub-exponential dependence on $p$ for a loss that is not a generalized linear loss . Our result for the hinge loss is based on a technique, dubbed polynomial of inner product approximation, which may be applicable to other problems. Our results for generalized linear losses and the Euclidean median are based on new reductions to the case of hinge loss.

cs.LG↗

arXiv2020

Differentially Private Simple Linear Regression

Economics and social science research often require analyzing datasets of sensitive personal information at fine granularity, with models fit to small subsets of the data. Unfortunately, such fine-grained analysis can easily reveal sensitive individual information. We study algorithms for simple linear regression that satisfy differential privacy, a constraint which guarantees that an algorithm's output reveals little about any individual input data record, even to an attacker with arbitrary side information about the dataset. We consider the design of differentially private algorithms for simple linear regression for small datasets, with tens to hundreds of datapoints, which is a particularly challenging regime for differential privacy. Focusing on a particular application to small-area analysis in economics research, we study the performance of a spectrum of algorithms we adapt to the setting. We identify key factors that affect their performance, showing through a range of experiments that algorithms based on robust estimators (in particular, the Theil-Sen estimator) perform well on the smallest datasets, but that other more standard algorithms do better as the dataset size increases.

cs.LG↗

arXiv2020

Disorder-free localization in a simple $U(1)$ lattice gauge theory

Localization due to the presence of disorder has proven crucial for our current understanding of relaxation in isolated quantum systems. The many-body localized phase constitutes a robust alternative to the thermalization of complex interacting systems, but recently the importance of disorder has been brought into question. Starting from translationally invariant $(1 + 1)$-dimensional quantum electrodynamics, we modify the dynamics of the gauge field and reveal a mechanism of disorder-free localization. We consider two different discretizations of the continuum model resulting in a free-fermion soluble model in one case and an interacting model in the other. We diagnose the localization in the far-from-equilibrium dynamics following a global quantum quench.

cond-mat.str-el↗

arXiv2020

Intrinsic sign problem in fermionic and bosonic chiral topological matter

The infamous sign problem leads to an exponential complexity in Monte Carlo simulations of generic many-body quantum systems. Nevertheless, many phases of matter are known to admit a sign-problem-free representative, allowing efficient simulations on classical computers. Motivated by long standing open problems in many-body physics, as well as fundamental questions in quantum complexity, the possibility of intrinsic sign problems, where a phase of matter admits no sign-problem-free representative, was recently raised but remains largely unexplored. Here, we establish the existence of an intrinsic sign problem in a broad class of gapped, chiral, topological phases of matter. Within this class, we exclude the possibility of stoquastic Hamiltonians for bosons (or 'qudits'), and of sign-problem-free determinantal Monte Carlo algorithms for fermions. The intrinsically sign-problematic class of phases we identify is defined in terms of topological invariants with clear observable signatures: the chiral central charge, and the topological spins of anyons. We obtain analogous results for phases that are spontaneously chiral, and present evidence for an extension of our results that applies to both chiral and non-chiral topological matter.

cond-mat.str-el↗