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Recently, quantum analogs of classical Gibbs samplers have been introduced—quantum Markov chains that generalize Glauber or Metropolis dynamics, and serve as models of nature’s thermalization process. In this work, we show that every one-dimensional quantum Hamiltonian with short-range interactions admits a quantum Gibbs sampler with a system-size–independent, optimal spectral gap at all finite temperatures.
Goldreich-Levin reductions are ubiquitous in cryptography: they convert an algorithm capable of guessing mod 2, for a hidden string m and a random challenge r, into one that is capable of extracting the entirety of m. Here, I will describe a...
In this talk, I will talk about field dynamics, explaining its definition and why I like it. Field dynamics is a continuous analogue of the down-up walk on a simplicial complex. It has a lot of applications in discrete sampling, including sampling from the hardcore model (i.e., independent sets) on graphs of unbounded degree, up to and at the critical point.
In this talk, we will review entropic independence, the analog of spectral independence when replacing variance with entropy, and show how to use entropic independence to derive tight bounds for the mixing time of Markov chains.
I will review the Knabe method for spectral gap lower bounds and show its application to random unitary circuits. This leads to the optimal (up to a constant) estimate of the spectral gap of the 1d brickwork unitary random circuit over qubits.
Based on the joint work with Tim Baer.
Aldous’ spectral-gap conjecture, proved by Caputo, Liggett and Richthammer, states that the interchange process on the symmetric group and the underlying random walk have the same spectral gap. We formulate a unitary analogue: a weighted hypergraph generates a random walk on the unitary group U(n) by Haar-randomizing the coordinate subspaces associated with its hyperedges. The resulting spectrum is surprisingly rich: it contains the spectra of all the associated discrete KMP processes, with arbitrary numbers of particles, as well as the entire spectrum of the corresponding symmetric-group walk, and much more. We show that the spectral gap can nevertheless be sought entirely in the torus-invariant, or zero-weight, parts of mixed tensor powers having equally many standard and dual factors. We furthermore conjecture that the spectral gap of the unitary walk always equals that of the two-particle KMP process. We prove the conjecture in the mean-field and codimension-one cases. Joint work with Doron Puder.