Lifting quantum Markov dynamics to speed up mixing
Bowen Li | City University of Hong Kong
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 over damped 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 over damped 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 offer quantitative convergence guarantees for hypocoercive quantum Markov processes, and also characterize the potential and limitations of accelerating the convergence through lifting. 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.
Location
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QNC 1201
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Meeting ID: 926 8095 1108
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Passcode: 493142
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