Quantum learning algorithms imply circuit lower bounds

Arunachalam, Srinivasan; Grilo, Alex B.; Gur, Tom; Oliveira, Igor Carboni; Sundaram, Aarthi

doi:10.1109/focs52979.2021.00062

Cited by 3 publications

(6 citation statements)

References 43 publications

(66 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For more discussion on quantum Gibbs sampling, see Section VI. As with learning, a similar polynomial efficiency is expected for variational quantum Gibbs sampling combined with vanilla stochastic gradient descent, if we use the ansatz (13) given the strong convexity [121]. However, again, our method allows flexibility in the choice of model.…”

Section: Sample-efficient Modelling Of Gibbs Statesmentioning

confidence: 81%

“…We require Φ to be a contrast functional. We say that Φ is a constrast functional 13 if it is a non-negative smooth function [101] such that…”

Section: Background a Loss Functions For Density Operatorsmentioning

confidence: 99%

“…see from (QPRGF) that when we use the update rule (9) then the same guarantee holds approximately under any smooth, invertible re-parameterization. The ability to re-parameterize, i.e., choose a different model which parameterizes the same subset of M (N ) spanned by (13), unlocks new routes to tackling time-inefficiency. For example, we will be able to consider models with friendlier gradients that can be more tractably estimated classically.…”

Section: Sample-efficient Learning Of Many-body Statesmentioning

confidence: 99%

“…However, we may expect similar behavior given a sufficient approximation. In this way, when we do not have the strong prior of knowing the relevant { Êl }, we may still be able to posit a reasonably approximate reparameterization of the space spanned by (13). We note that for high temperature (above the phase transition point) a separate approach has been analyzed which delivers optimal sample-and time-complexity with sample complexity S = O(log n/(βε) 2 ) and time-complexity O(nS) [25].…”

Section: Sample-efficient Learning Of Many-body Statesmentioning

confidence: 99%

“…In particular, many speedups vanish if one has classical l 2 -sampling access to the same data (which is at least as easy to obtain as quantum state preparation). Several additional theoretical [7,8] and practical [2,9,10] barriers have emerged along this direction, though key milestones in understanding have been reached [11][12][13].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Provably efficient variational generative modeling of quantum many-body systems via quantum-probabilistic information geometry

Sbahi¹,

Martínez²,

Patel³

et al. 2022

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Sample-efficient Modelling Of Gibbs Statesmentioning

confidence: 81%

“…We require Φ to be a contrast functional. We say that Φ is a constrast functional 13 if it is a non-negative smooth function [101] such that…”

Section: Background a Loss Functions For Density Operatorsmentioning

confidence: 99%

Section: Sample-efficient Learning Of Many-body Statesmentioning

confidence: 99%

Section: Sample-efficient Learning Of Many-body Statesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Provably efficient variational generative modeling of quantum many-body systems via quantum-probabilistic information geometry

Sbahi¹,

Martínez²,

Patel³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

A survey on the complexity of learning quantum states

Anshu,

Arunachalam

2023

Nat Rev Phys

View full text Add to dashboard Cite

Expressivity of Variational Quantum Machine Learning on the Boolean Cube

Herman

Raymond

et al. 2023

IEEE Trans. Quantum Eng.

View full text Add to dashboard Cite

Categorical data plays an important part in machine learning research and appears in a variety of applications. Models that can express large classes of real-valued functions on the Boolean cube are useful for problems involving discrete-valued data types, including those which are not Boolean. To this date, the commonly used schemes for embedding classical data into variational quantum machine learning models encode continuous values. Here we investigate quantum embeddings for encoding Boolean-valued data into parameterized quantum circuits used for machine learning tasks. We narrow down representability conditions for functions on the n-dimensional Boolean cube with respect to previously known results, using two quantum embeddings: a phase embedding and an embedding based on quantum random access codes. We show that for any real-valued function on the n-dimensional Boolean cube, there exists a variational linear quantum model based on a phase embedding using n qubits that can represent it and an ensemble of such models using d < n qubits that can express any function with degree at most d. Additionally, we prove that variational linear quantum models that use the quantum random access code embedding can express functions on the Boolean cube with degree d ≤ ⌈ n 3 ⌉ using ⌈ n 3 ⌉ qubits, and that an ensemble of such models can represent any function on the Boolean cube with degree d ≤ ⌈ n 3 ⌉. Furthermore, we discuss the potential benefits of each embedding and the impact of serial repetitions. Lastly, we demonstrate the use of the embeddings presented by performing numerical simulations and experiments on IBM quantum processors using the Qiskit machine learning framework.

show abstract

Quantum learning algorithms imply circuit lower bounds

Abstract: How to cite:Please refer to published version for the most recent bibliographic citation information.

Cited by 3 publications

References 43 publications

Provably efficient variational generative modeling of quantum many-body systems via quantum-probabilistic information geometry

Provably efficient variational generative modeling of quantum many-body systems via quantum-probabilistic information geometry

A survey on the complexity of learning quantum states

Expressivity of Variational Quantum Machine Learning on the Boolean Cube

Contact Info

Product

Resources

About