Mixing properties of stochastic quantum Hamiltonians

TL;DR

<3-5 sentence high-level summary> The paper addresses how time-fluctuating local Hamiltonians—formulated as Brownian motion on the unitary group—mix quantum information and approximate Haar randomness. It develops a rigorous framework connecting continuous-time diffusion to discrete random circuits, using representation theory and the Casimir element to bound spectral gaps and prove that local stochastic evolutions implement approximate unitary k-designs and tensor product expanders. A key result is a decoupling bound with almost linear scaling in system size, along with concrete analyses (e.g., Pauli-noise) and applications to dissipative dynamics and fast scrambling. This unifies disparate random quantum processes under a single mathematical approach and offers practical routes to pseudo-randomness for quantum information tasks and quantum many-body physics.

Abstract

Random quantum processes play a central role both in the study of fundamental mixing processes in quantum mechanics related to equilibration, thermalisation and fast scrambling by black holes, as well as in quantum process design and quantum information theory. In this work, we present a framework describing the mixing properties of continuous-time unitary evolutions originating from local Hamiltonians having time-fluctuating terms, reflecting a Brownian motion on the unitary group. The induced stochastic time evolution is shown to converge to a unitary design. As a first main result, we present bounds to the mixing time. By developing tools in representation theory, we analytically derive an expression for a local k-th moment operator that is entirely independent of k, giving rise to approximate unitary k-designs and quantum tensor product expanders. As a second main result, we introduce tools for proving bounds on the rate of decoupling from an environment with random quantum processes. By tying the mathematical description closely with the more established one of random quantum circuits, we present a unified picture for analysing local random quantum and classes of Markovian dissipative processes, for which we also discuss applications.

Figures (2)

• Figure 1: In the decoupling theorem, an initial bipartite state $\rho_{AE}$ is affected by a unitary evolution $U_A$ chosen at according to a certain distribution $\mu$. Then, subsystem $A$ is mapped to another subsystem $B$ through a completely positive map $\mathcal{T}_{A \rightarrow B}$. Finally, the distance between the final state and the product state $\tau_B \otimes \rho_E$ is characterised by entropy measures.
• Figure 2: A quantum memory system $M$ is initially entangled with a reference system denoted by $R$ and is subsequently thrown into a black hole $A$ (left picture). As the black holes leaks out Hawking radiation, it shrinks into a smaller system $B$. When a controlled subystem of the irradiated environment $E'$, has become maximally entangled with the reference system $R$, the initial information $M$ has been mirrored (right picture).

Theorems & Definitions (42)

• definition 1: Unitary designs
• definition 2: Tensor product expanders
• lemma 1: Criterion for being an approximate unitary design PhDLow
• definition 3: Universal distribution
• lemma 2: Lemma 3.7 in ref. RandomCircuitsLow
• definition 4: Brownian motion on the unitary group
• remark 1: Orthonormal basis
• remark 2: Overcomplete sets of operators
• theorem 1: Local Brownian motions on $\mathds{U} (d^n)$ are quantum $(\lambda,k)$-tensor product expanders
• corollary 1: Approximate unitary $k$-designs