Abstract
Randomness is an essential tool in many disciplines of modern sciences, such as cryptography, black hole physics, random matrix theory and Monte Carlo sampling. In quantum systems, random operations can be obtained via random circuits thanks to so-called q-designs, and play a central role in the fast scrambling conjecture for black holes. Here we consider a more physically motivated way of generating random evolutions by exploiting the many-body dynamics of a quantum system driven with stochastic external pulses. We combine techniques from quantum control, open quantum systems and exactly solvable models (via the Bethe-Ansatz) to generate Haar-uniform random operations in driven many-body systems. We show that any fully controllable system converges to a unitary q-design in the long-time limit. Moreover, we study the convergence time of a driven spin chain by mapping its random evolution into a semigroup with an integrable Liouvillean and finding its gap. Remarkably, we find via Bethe-Ansatz techniques that the gap is independent of q. We use mean-field techniques to argue that this property may be typical for other controllable systems, although we explicitly construct counter-examples via symmetry breaking arguments to show that this is not always the case. Our findings open up new physical methods to transform classical randomness into quantum randomness, via a combination of quantum many-body dynamics and random driving.
Highlights
Randomness generating quantum operations play a central role in our understanding of many various physical phenomena [1]
Random operations can be obtained via random circuits thanks to so-called q-designs and play a central role in condensedmatter physics and in the fast scrambling conjecture for black holes
In this paper we study the quantum dynamics resulting from a stochastic driving of quantum many-body systems, and we answer the following questions: when, and how rapidly, the dynamics of a driven quantum system is equivalent to a fully uniform random evolution, namely under unitaries sampled from the Haar measure
Summary
Randomness generating quantum operations play a central role in our understanding of many various physical phenomena [1]. One of the central result of this paper is that driving a controllable quantum system with stochastic control pulses offers a natural approach to generate random unitary operations with physical processes. Within this picture, the estimation of the mixing time is the crucial theoretical aspect. This implies that, apart from measure zero sets, at this time any evolution is achievable with a suitable choice of the control field Another motivation for the present work is for the problem of fast scrambling of quantum information.
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