Geometric Approach Towards Complete Logarithmic Sobolev Inequalities

Gao, Li; Junge, Marius; Li, Haojian

doi:10.48550/arxiv.2102.04434

Cited by 4 publications

(6 citation statements)

References 22 publications

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“…The following result was recently proved in [38] (see also [37] in the tracial setting): Theorem 2.4 ( [38]). For any GNS-symmetric QMS (e tL ) t≥0 over the algebra B(H) of linear operators on a finite dimensional Hilbert space H, α c (L) > 0.…”

Section: Modified Logarithmic Sobolev Inequalitymentioning

confidence: 96%

Entropy decay for Davies semigroups of a one dimensional quantum lattice

Bardet¹,

Capel²,

Li³

et al. 2021

Preprint

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Given a finite-range, translation-invariant commuting system Hamiltonians on a spin chain, we show that the Davies semigroup describing the reduced dynamics resulting from the joint Hamiltonian evolution of a spin chain weakly coupled to a large heat bath thermalizes rapidly at any temperature. More precisely, we prove that the relative entropy between any evolved state and the equilibrium Gibbs state contracts exponentially fast with an exponent that scales logarithmically with the length of the chain. Our theorem extends a seminal result of Holley and Stroock [40] to the quantum setting, up to a logarithmic overhead, as well as provides an exponential improvement over the non-closure of the gap proved by Brandao and Kastoryano [43]. This has wide-ranging applications to the study of many-body in and out-of-equilibrium quantum systems. Our proof relies upon a recently derived strong decay of correlations for Gibbs states of one dimensional, translation-invariant local Hamiltonians, and tools from the theory of operator spaces.

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Section: Modified Logarithmic Sobolev Inequalitymentioning

confidence: 96%

Entropy decay for Davies semigroups of a one dimensional quantum lattice

Bardet¹,

Capel²,

Li³

et al. 2021

Preprint

Self Cite

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“…Furthermore, our particular form has a unique and usually full rank fixed point state as long as depolarizing strength is non-zero, a useful property in theoretical analysis and in estimating rates of decay to equilibrium [91,92]. While the D 4 form contains some unitary rotation, because this rotation commutes with the restriction to the fixed point algebra, one may still apply recent results showing exponential decay of relative entropy to the fixed point [93][94][95].…”

Section: Modeling Qubit Movements Via Physically Inferred Channelsmentioning

confidence: 99%

Modeling Short-Range Microwave Networks to Scale Superconducting Quantum Computation

LaRacuente¹,

Smith²,

Imany³

et al. 2022

Preprint

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A core challenge for superconducting quantum computers is to scale up the number of qubits in each processor without increasing noise or cross-talk. Distributing a quantum computer across nearby small qubit arrays, known as chiplets, could solve many problems associated with size. We propose a chiplet architecture over microwave links with potential to exceed monolithic performance on near-term hardware. We model and evaluate the chiplet architecture in a way that bridges the physical and network layers. We find concrete evidence that distributed quantum computing may accelerate the path toward useful and ultimately scalable quantum computers. In the long-term, short-range networks may underlie quantum computers just as local area networks underlie classical datacenters and supercomputers today.

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“…This result was derived in a self-contained way in [GR21]. It was simultaneously derived in [GJL21] for semigroups that are self-adjoint with respect to the trace, which via [JLR19] also extends to all finite-dimensional semigroups with detailed balance. An important tool is the multiplicative relative entropy comparison from earlier work.…”

Section: 1mentioning

confidence: 99%

“…Iterating completes the Remark as we take the limit τ → ∞, using the fact that CMLSI holds for tracially self-adjoint Lindbladians as shown in [GR21] and [GJL21].…”

Section: Appendix C Proofs Of Entrpy Decay Boundsmentioning

confidence: 99%

“…In contrast to Hamiltonians are Lindbladians that have detailed balance, a property related to time reversal symmetry [CM17]. Recently, a series of results [JLR19, GJL21,GR21] confirmed that all Lindbladians having detailed balance with respect to a GNS inner product obey a complete, modified logarithmic-Sobolev inequality (CMLSI) based on the modified logarithmic Sobolev inequality defined in [AMTU98, BT03,KT13]. For a semigroup (Φ t : t ∈ R + ) in terms of the quantum relative entropy D(• •), λ-CMLSI states that…”

Section: Introductionmentioning

confidence: 99%

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Self-restricting Noise and Quantum Relative Entropy Decay

LaRacuente¹

2022

Preprint

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Open quantum systems as modeled by quantum channels and quantum Markov semigroups usually decay to subspaces that are invariant under environmental interactions. It is known that finite-dimensional semigroups with detailed balance decay exponentially under modified logarithmic-Sobolev inequalities (MLSIs). Here we analyze discrete and continuous processes that include unitary components, breaking the detailed balance assumption. We find counter-examples to analogs of MLSIs for these systems. The generalized quantum Zeno effect appears for many Lindbladians that combine a decay process with unitary drift. As incompatible long-time and Zeno limits compete, strong noise often protects subsystems and subspaces from its own spread. We observe this interplay between decay and Zeno-like effects experimentally on superconducting qubits using IBM Q devices. Nonetheless, by combining MLSIs for effective self-adjoint decay processes across different times, we obtain eventual exponential decay. We similarly obtain decay rate lower bounds for discrete compositions of quantum channels.

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Geometric Approach Towards Complete Logarithmic Sobolev Inequalities

Cited by 4 publications

References 22 publications

Entropy decay for Davies semigroups of a one dimensional quantum lattice

Entropy decay for Davies semigroups of a one dimensional quantum lattice

Modeling Short-Range Microwave Networks to Scale Superconducting Quantum Computation

Self-restricting Noise and Quantum Relative Entropy Decay

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