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Efficient learning of ground & thermal states within phases of matter - Daniel S. França | TQC 2023
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Daniel Stilck França
Efficient learning of ground & thermal states within phases of matter
We consider two related tasks: (a) estimating a parameterisation of a given Gibbs state and expectation values of Lipschitz observables on this state; and (b) learning the expectation values of local observables within a thermal or quantum phase of matter. In both cases, we wish to minimise the number of samples we use to learn these properties to a given precision. For the first task, we develop new techniques to learn parameterisations of classes of systems, including quantum Gibbs states of non-commuting Hamiltonians with exponential decay of correlations and the approximate Markov property. We show it is possible to infer the expectation values of all extensive properties of the state from a number of copies that not only scales polylogarithmically with the system size, but polynomially in the observable's locality -- an exponential improvement. This set of properties includes expected values of quasi-local observables and entropies. For the second task, we develop efficient algorithms for learning observables in a phase of matter of a quantum system. By exploiting the locality of the Hamiltonian, we show that M local observables can be learned with probability 1−δ to precision ϵ with using only N=O(log(Mδ)epolylog(ϵ−1)) samples -- an exponential improvement on the precision over previous bounds. Our results apply to both families of ground states of Hamiltonians displaying local topological quantum order, and thermal phases of matter with exponential decay of correlations. In addition, our sample complexity applies to the worse case setting whereas previous results only applied on average. Furthermore, we develop tools of independent interest, such as robust shadow tomography algorithms, Gibbs approximations to ground states, and generalisations of transportation cost inequalities for Gibbs states.
July 25, 2023
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TQC 2023 | 24-28 July 2023, University of Aveiro, Portugal
18th Conference on the Theory of Quantum Computation, Communication and Cryptography.
TQC is a leading annual international conference for students and researchers working in the theoretical aspects of quantum information science. The scientific objective is to bring together the theoretical quantum information science community to present and discuss the latest advances in the field.
Organisation:
Squids - Schools for Quantum Information Development
Universidade de Aveiro: Departamento de Matemática, CIDMA & Fábrica
Sponsors:
Phasecraft, UK
Google Quantum AI, USA
QuSoft, The Netherlands
Quantum for Life Centre, Denmark
Technology Innovation Institute, UAE
ML4Q, Germany
Dulwich Quantum
Efficient learning of ground & thermal states within phases of matter
We consider two related tasks: (a) estimating a parameterisation of a given Gibbs state and expectation values of Lipschitz observables on this state; and (b) learning the expectation values of local observables within a thermal or quantum phase of matter. In both cases, we wish to minimise the number of samples we use to learn these properties to a given precision. For the first task, we develop new techniques to learn parameterisations of classes of systems, including quantum Gibbs states of non-commuting Hamiltonians with exponential decay of correlations and the approximate Markov property. We show it is possible to infer the expectation values of all extensive properties of the state from a number of copies that not only scales polylogarithmically with the system size, but polynomially in the observable's locality -- an exponential improvement. This set of properties includes expected values of quasi-local observables and entropies. For the second task, we develop efficient algorithms for learning observables in a phase of matter of a quantum system. By exploiting the locality of the Hamiltonian, we show that M local observables can be learned with probability 1−δ to precision ϵ with using only N=O(log(Mδ)epolylog(ϵ−1)) samples -- an exponential improvement on the precision over previous bounds. Our results apply to both families of ground states of Hamiltonians displaying local topological quantum order, and thermal phases of matter with exponential decay of correlations. In addition, our sample complexity applies to the worse case setting whereas previous results only applied on average. Furthermore, we develop tools of independent interest, such as robust shadow tomography algorithms, Gibbs approximations to ground states, and generalisations of transportation cost inequalities for Gibbs states.
July 25, 2023
---------
TQC 2023 | 24-28 July 2023, University of Aveiro, Portugal
18th Conference on the Theory of Quantum Computation, Communication and Cryptography.
TQC is a leading annual international conference for students and researchers working in the theoretical aspects of quantum information science. The scientific objective is to bring together the theoretical quantum information science community to present and discuss the latest advances in the field.
Organisation:
Squids - Schools for Quantum Information Development
Universidade de Aveiro: Departamento de Matemática, CIDMA & Fábrica
Sponsors:
Phasecraft, UK
Google Quantum AI, USA
QuSoft, The Netherlands
Quantum for Life Centre, Denmark
Technology Innovation Institute, UAE
ML4Q, Germany
Dulwich Quantum