7 research outputs found
Exponential separations between classical and quantum learners
Despite significant effort, the quantum machine learning community has only
demonstrated quantum learning advantages for artificial cryptography-inspired
datasets when dealing with classical data. In this paper we address the
challenge of finding learning problems where quantum learning algorithms can
achieve a provable exponential speedup over classical learning algorithms. We
reflect on computational learning theory concepts related to this question and
discuss how subtle differences in definitions can result in significantly
different requirements and tasks for the learner to meet and solve. We examine
existing learning problems with provable quantum speedups and find that they
largely rely on the classical hardness of evaluating the function that
generates the data, rather than identifying it. To address this, we present two
new learning separations where the classical difficulty primarily lies in
identifying the function generating the data. Furthermore, we explore
computational hardness assumptions that can be leveraged to prove quantum
speedups in scenarios where data is quantum-generated, which implies likely
quantum advantages in a plethora of more natural settings (e.g., in condensed
matter and high energy physics). We also discuss the limitations of the
classical shadow paradigm in the context of learning separations, and how
physically-motivated settings such as characterizing phases of matter and
Hamiltonian learning fit in the computational learning framework.Comment: this article supersedes arXiv:2208.0633
Towards quantum advantage via topological data analysis
Even after decades of quantum computing development, examples of generally
useful quantum algorithms with exponential speedups over classical counterparts
are scarce. Recent progress in quantum algorithms for linear-algebra positioned
quantum machine learning (QML) as a potential source of such useful exponential
improvements. Yet, in an unexpected development, a recent series of
"dequantization" results has equally rapidly removed the promise of exponential
speedups for several QML algorithms. This raises the critical question whether
exponential speedups of other linear-algebraic QML algorithms persist. In this
paper, we study the quantum-algorithmic methods behind the algorithm for
topological data analysis of Lloyd, Garnerone and Zanardi through this lens. We
provide evidence that the problem solved by this algorithm is classically
intractable by showing that its natural generalization is as hard as simulating
the one clean qubit model -- which is widely believed to require
superpolynomial time on a classical computer -- and is thus very likely immune
to dequantizations. Based on this result, we provide a number of new quantum
algorithms for problems such as rank estimation and complex network analysis,
along with complexity-theoretic evidence for their classical intractability.
Furthermore, we analyze the suitability of the proposed quantum algorithms for
near-term implementations. Our results provide a number of useful applications
for full-blown, and restricted quantum computers with a guaranteed exponential
speedup over classical methods, recovering some of the potential for
linear-algebraic QML to become one of quantum computing's killer applications.Comment: 29 pages, 3 figures. New results added and improved expositio
Shadows of quantum machine learning
Quantum machine learning is often highlighted as one of the most promising
uses for a quantum computer to solve practical problems. However, a major
obstacle to the widespread use of quantum machine learning models in practice
is that these models, even once trained, still require access to a quantum
computer in order to be evaluated on new data. To solve this issue, we suggest
that following the training phase of a quantum model, a quantum computer could
be used to generate what we call a classical shadow of this model, i.e., a
classically computable approximation of the learned function. While recent
works already explore this idea and suggest approaches to construct such shadow
models, they also raise the possibility that a completely classical model could
be trained instead, thus circumventing the need for a quantum computer in the
first place. In this work, we take a novel approach to define shadow models
based on the frameworks of quantum linear models and classical shadow
tomography. This approach allows us to show that there exist shadow models
which can solve certain learning tasks that are intractable for fully classical
models, based on widely-believed cryptography assumptions. We also discuss the
(un)likeliness that all quantum models could be shadowfiable, based on common
assumptions in complexity theory.Comment: 7 + 16 pages, 5 figure
Shadows of quantum machine learning
Quantum machine learning is often highlighted as one of the most promising practical applications for which quantum computers could provide a computational advantage. However, a major obstacle to the widespread use of quantum machine learning models in practice is that these models, even once trained, still require access to a quantum computer in order to be evaluated on new data. To solve this issue, we introduce a class of quantum models where quantum resources are only required during training, while the deployment of the trained model is classical. Specifically, the training phase of our models ends with the generation of a âshadow modelâ from which the classical deployment becomes possible. We prove that: (i) this class of models is universal for classically-deployed quantum machine learning; (ii) it does have restricted learning capacities compared to âfully quantumâ models, but nonetheless (iii) it achieves a provable learning advantage over fully classical learners, contingent on widely believed assumptions in complexity theory. These results provide compelling evidence that quantum machine learning can confer learning advantages across a substantially broader range of scenarios, where quantum computers are exclusively employed during the training phase. By enabling classical deployment, our approach facilitates the implementation of quantum machine learning models in various practical contexts
Analyzing Prospects for Quantum Advantage in Topological Data Analysis
Lloyd et al. were first to demonstrate the promise of quantum algorithms for
computing Betti numbers, a way to characterize topological features of data
sets. Here, we propose, analyze, and optimize an improved quantum algorithm for
topological data analysis (TDA) with reduced scaling, including a method for
preparing Dicke states based on inequality testing, a more efficient amplitude
estimation algorithm using Kaiser windows, and an optimal implementation of
eigenvalue projectors based on Chebyshev polynomials. We compile our approach
to a fault-tolerant gate set and estimate constant factors in the Toffoli
complexity. Our analysis reveals that super-quadratic quantum speedups are only
possible for this problem when targeting a multiplicative error approximation
and the Betti number grows asymptotically. Further, we propose a dequantization
of the quantum TDA algorithm that shows that having exponentially large
dimension and Betti number are necessary, but insufficient conditions, for
super-polynomial advantage. We then introduce and analyze specific problem
examples which have parameters in the regime where super-polynomial advantages
may be achieved, and argue that quantum circuits with tens of billions of
Toffoli gates can solve seemingly classically intractable instances.Comment: 54 pages, 7 figures. Added a number of theorems and lemmas to clarify
findings and also a discussion in the main text and new appendix about
variants of our problems with high Betti numbers that are challenging for
recent classical algorithm
High Dimensional Quantum Machine Learning With Small Quantum Computers
Quantum computers hold great promise to enhance machine learning, but their
current qubit counts restrict the realisation of this promise. In an attempt to
placate this limitation techniques can be applied for evaluating a quantum
circuit using a machine with fewer qubits than the circuit naively requires.
These techniques work by evaluating many smaller circuits on the smaller
machine, that are then combined in a polynomial to replicate the output of the
larger machine. This scheme requires more circuit evaluations than are
practical for general circuits. However, we investigate the possibility that
for certain applications many of these subcircuits are superfluous, and that a
much smaller sum is sufficient to estimate the full circuit. We construct a
machine learning model that may be capable of approximating the outputs of the
larger circuit with much fewer circuit evaluations. We successfully apply our
model to the task of digit recognition, using simulated quantum computers much
smaller than the data dimension. The model is also applied to the task of
approximating a random 10 qubit PQC with simulated access to a 5 qubit
computer, even with only relatively modest number of circuits our model
provides an accurate approximation of the 10 qubit PQCs output, superior to a
neural network attempt. The developed method might be useful for implementing
quantum models on larger data throughout the NISQ era.Comment: 13 pages, 7 figure
Structural risk minimization for quantum linear classifiers
Quantum machine learning (QML) models based on parameterized quantum circuits are often highlighted as candidates for quantum computingâs near-term âkiller applicationâ. However, the understanding of the empirical and generalization performance of these models is still in its infancy. In this paper we study how to balance between training accuracy and generalization performance (also called structural risk minimization) for two prominent QML models introduced by HavlĂÄek et al. [1], and Schuld and Killoran [2]. Firstly, using relationships to well understood classical models, we prove that two model parameters â i.e., the dimension of the sum of the images and the Frobenius norm of the observables used by the model â closely control the modelsâ complexity and therefore its generalization performance. Secondly, using ideas inspired by process tomography, we prove that these model parameters also closely control the modelsâ ability to capture correlations in sets of training examples. In summary, our results give rise to new options for structural risk minimization for QML models