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Speeding up quantum Markov processes through lifting

Bowen Li, Jianfeng Lu

TL;DR

This work develops a second-order lifting framework for detailed-balanced quantum Markov semigroups to accelerate convergence to equilibrium. It proves that the $L^2$ relaxation rate of lifted dynamics is bounded by the square root of the overdamped spectral gap, establishing a fundamental diffusive-to-ballistic speed-up limit, and complements this with a space-time hypocoercivity-based lower bound. An abstract Hilbert-space lifting theory is formulated to unify lifting analyses across classical and quantum dynamics, enabling construction of optimal lifts for several detailed-balanced processes such as depolarizing semigroups, Schur multipliers, and group von Neumann algebras. Together, these results provide quantitative convergence guarantees for hypocoercive QMS and delineate the potential and limits of accelerating quantum dynamics via lifting, with implications for quantum Gibbs sampling and open-system simulations.

Abstract

We generalize the concept of non-reversible lifts for reversible diffusion processes initiated by Eberle and Lorler (2024) to quantum Markov dynamics. The lifting operation, which naturally results in hypocoercive processes, can be formally interpreted as, though not restricted to, the reverse of the overdamped limit. We prove that the $L^2$ convergence rate of the lifted process is bounded above by the square root of the spectral gap of its overdamped dynamics, indicating that the lifting approach can at most achieve a transition from diffusive to ballistic mixing speeds. Further, using the variational hypocoercivity framework based on space-time Poincare inequalities, we derive a lower bound for the convergence rate of the lifted dynamics. These findings not only offer quantitative convergence guarantees for hypocoercive quantum Markov processes but also characterize the potential and limitations of accelerating the convergence through lifting. In addition, we develop an abstract lifting framework in the Hilbert space setting applicable to any symmetric contraction $C_0$-semigroup, thereby unifying the treatment of classical and quantum dynamics. As applications, we construct optimal lifts for various detailed balanced classical and quantum processes, including the symmetric random walk on a chain, the depolarizing semigroup, Schur multipliers, and quantum Markov semigroups on group von Neumann algebras.

Speeding up quantum Markov processes through lifting

TL;DR

This work develops a second-order lifting framework for detailed-balanced quantum Markov semigroups to accelerate convergence to equilibrium. It proves that the relaxation rate of lifted dynamics is bounded by the square root of the overdamped spectral gap, establishing a fundamental diffusive-to-ballistic speed-up limit, and complements this with a space-time hypocoercivity-based lower bound. An abstract Hilbert-space lifting theory is formulated to unify lifting analyses across classical and quantum dynamics, enabling construction of optimal lifts for several detailed-balanced processes such as depolarizing semigroups, Schur multipliers, and group von Neumann algebras. Together, these results provide quantitative convergence guarantees for hypocoercive QMS and delineate the potential and limits of accelerating quantum dynamics via lifting, with implications for quantum Gibbs sampling and open-system simulations.

Abstract

We generalize the concept of non-reversible lifts for reversible diffusion processes initiated by Eberle and Lorler (2024) to quantum Markov dynamics. The lifting operation, which naturally results in hypocoercive processes, can be formally interpreted as, though not restricted to, the reverse of the overdamped limit. We prove that the convergence rate of the lifted process is bounded above by the square root of the spectral gap of its overdamped dynamics, indicating that the lifting approach can at most achieve a transition from diffusive to ballistic mixing speeds. Further, using the variational hypocoercivity framework based on space-time Poincare inequalities, we derive a lower bound for the convergence rate of the lifted dynamics. These findings not only offer quantitative convergence guarantees for hypocoercive quantum Markov processes but also characterize the potential and limitations of accelerating the convergence through lifting. In addition, we develop an abstract lifting framework in the Hilbert space setting applicable to any symmetric contraction -semigroup, thereby unifying the treatment of classical and quantum dynamics. As applications, we construct optimal lifts for various detailed balanced classical and quantum processes, including the symmetric random walk on a chain, the depolarizing semigroup, Schur multipliers, and quantum Markov semigroups on group von Neumann algebras.
Paper Structure (9 sections, 16 theorems, 86 equations)

This paper contains 9 sections, 16 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Contribution and Outline
  3. Other related works
  4. Lifting quantum Markov semigroups
  5. Hypocoercive Quantum Markov semigroups
  6. Overdamped limit and lifting of quantum dynamics
  7. Lower and upper bounds of convergence rate
  8. Lifting in Hilbert spaces
  9. General framework of lifting

Key Result

Lemma 2.2

Let $\mathcal{P}_t = \exp(t \mathcal{L})$ be a QMS with invariant state $\sigma \in \mathcal{D}_+(\mathcal{M})$. Then, we have with $E_{\mathcal{N}}$ being the conditional expectation to $\mathcal{N}(\mathcal{L})$, and the limit exists and defines the conditional expectation $E_{\mathcal{F}}$ onto $\mathcal{F}(\mathcal{L})$ with respect to $\sigma$. Moreover, the following convergence holds: if

Theorems & Definitions (37)

  • Definition 2.1
  • Lemma 2.2: frigerio1982long*Theorems 3.3 and 3.4
  • Definition 2.3
  • Lemma 2.4
  • Corollary 2.5
  • Lemma 2.6
  • Definition 2.7
  • Lemma 2.8
  • proof
  • Lemma 2.9
  • ...and 27 more