Quantum Talagrand, KKL and Friedgut's theorems and the learnability of quantum Boolean functions

• URL: http://arxiv.org/abs/2209.07279v3
• Date: Wed, 3 Apr 2024 19:45:46 GMT
• Title: Quantum Talagrand, KKL and Friedgut's theorems and the learnability of quantum Boolean functions
• Authors: Cambyse Rouzé, Melchior Wirth, Haonan Zhang,
• Abstract summary: We extend three related results from the analysis of influences of Boolean functions to the quantum inequality setting. Our results are derived by a joint use of recently studied hypercontractivity and gradient estimates. We comment on the implications of our results as regards to noncommutative extensions of isoperimetric type inequalities, quantum circuit complexity lower bounds and the learnability of quantum observables.
• Score: 9.428671399402553
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We extend three related results from the analysis of influences of Boolean functions to the quantum setting, namely the KKL Theorem, Friedgut's Junta Theorem and Talagrand's variance inequality for geometric influences. Our results are derived by a joint use of recently studied hypercontractivity and gradient estimates. These generic tools also allow us to derive generalizations of these results in a general von Neumann algebraic setting beyond the case of the quantum hypercube, including examples in infinite dimensions relevant to quantum information theory such as continuous variables quantum systems. Finally, we comment on the implications of our results as regards to noncommutative extensions of isoperimetric type inequalities, quantum circuit complexity lower bounds and the learnability of quantum observables.

Related papers

• Unexpected consequences of Post-Quantum theories in the graph-theoretical approach to correlations [0.0]
  This work explores the implications of the Exclusivity Principle (EP) in the context of quantum and post-quantum correlations. We first establish a key technical result demonstrating that given the set of correlations for a complementary experiment, the EP restricts the maximum set of correlations for the original experiment to the anti-blocking set. Based on it, we can prove our central result: if all quantum behaviors are accessible in Nature, the EP guarantees that no post-quantum behaviors can be realized.
  arXiv  Detail & Related papers  (2024-11-13T21:29:13Z)
• Generalized geometric speed limits for quantum observables [1.451121761055173]
  We derive generalized quantum speed limits on the rate of change of the expectation values of observables. These bounds subsume and, for Hilbert space dimension $geq 3$, tighten existing bounds. The generalized bounds can be used to design "fast" Hamiltonians that enable the rapid driving of the expectation values of observables.
  arXiv  Detail & Related papers  (2024-09-06T18:15:58Z)
• Quantum channels, complex Stiefel manifolds, and optimization [45.9982965995401]
  We establish a continuity relation between the topological space of quantum channels and the quotient of the complex Stiefel manifold. The established relation can be applied to various quantum optimization problems.
  arXiv  Detail & Related papers  (2024-08-19T09:15:54Z)
• Entropic Quantum Central Limit Theorem and Quantum Inverse Sumset Theorem [0.0]
  We establish an entropic, quantum central limit theorem and quantum inverse sumset theorem in discrete-variable quantum systems. A byproduct of this work is a magic measure to quantify the nonstabilizer nature of a state, based on the quantum Ruzsa divergence.
  arXiv  Detail & Related papers  (2024-01-25T18:43:24Z)
• Theory of Quantum Generative Learning Models with Maximum Mean Discrepancy [67.02951777522547]
  We study learnability of quantum circuit Born machines (QCBMs) and quantum generative adversarial networks (QGANs) We first analyze the generalization ability of QCBMs and identify their superiorities when the quantum devices can directly access the target distribution. Next, we prove how the generalization error bound of QGANs depends on the employed Ansatz, the number of qudits, and input states.
  arXiv  Detail & Related papers  (2022-05-10T08:05:59Z)
• Quantum Fluctuation-Response Inequality and Its Application in Quantum Hypothesis Testing [6.245537312562826]
  We find a bound for the mean difference of an observable at two different quantum states. When the spectrum of the observable is bounded, the sub-Gaussian property is used to link the bound with the sub-Gaussian norm of the observable. We show the versatility of our results by their applications in problems like thermodynamic inference and speed limit.
  arXiv  Detail & Related papers  (2022-03-20T09:10:54Z)
• Experimental violations of Leggett-Garg's inequalities on a quantum computer [77.34726150561087]
  We experimentally observe the violations of Leggett-Garg-Bell's inequalities on single and multi-qubit systems. Our analysis highlights the limits of nowadays quantum platforms, showing that the above-mentioned correlation functions deviate from theoretical prediction as the number of qubits and the depth of the circuit grow.
  arXiv  Detail & Related papers  (2021-09-06T14:35:15Z)
• Experimental Validation of Fully Quantum Fluctuation Theorems Using Dynamic Bayesian Networks [48.7576911714538]
  Fluctuation theorems are fundamental extensions of the second law of thermodynamics for small systems. We experimentally verify detailed and integral fully quantum fluctuation theorems for heat exchange using two quantum-correlated thermal spins-1/2 in a nuclear magnetic resonance setup.
  arXiv  Detail & Related papers  (2020-12-11T12:55:17Z)
• Using Quantum Metrological Bounds in Quantum Error Correction: A Simple Proof of the Approximate Eastin-Knill Theorem [77.34726150561087]
  We present a proof of the approximate Eastin-Knill theorem, which connects the quality of a quantum error-correcting code with its ability to achieve a universal set of logical gates. Our derivation employs powerful bounds on the quantum Fisher information in generic quantum metrological protocols.
  arXiv  Detail & Related papers  (2020-04-24T17:58:10Z)
• Error bounds for constrained dynamics in gapped quantum systems: Rigorous results and generalizations [0.0]
  We introduce a universal error bound for constrained unitary dynamics within a well-gapped energy band of an isolated quantum system. We generalize the result to isolated quantum many-body systems by employing the local Schrieffer-Wolff transformation.
  arXiv  Detail & Related papers  (2020-01-03T06:32:47Z)
• A refinement of Reznick's Positivstellensatz with applications to quantum information theory [72.8349503901712]
  In Hilbert's 17th problem Artin showed that any positive definite in several variables can be written as the quotient of two sums of squares. Reznick showed that the denominator in Artin's result can always be chosen as an $N$-th power of the squared norm of the variables.
  arXiv  Detail & Related papers  (2019-09-04T11:46:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.