Quantum pseudorandom state generators (PRSGs) have stimulated exciting
developments in recent years. A PRSG, on a fixed initial (e.g., all-zero)
state, produces an output state that is computationally indistinguishable from
a Haar random state. However, pseudorandomness of the output state is not
guaranteed on other initial states. In fact, known PRSG constructions provably
fail on some initial state.
In this work, we propose and construct quantum Pseudorandom State Scramblers
(PRSSs), which can produce a pseudorandom state on an arbitrary initial state.
In the information-theoretical setting, we obtain a scrambler which maps an
arbitrary initial state to a distribution of quantum states that is close to
Haar random in total variation distance. As a result, our PRSS exhibits a
dispersing property. Loosely, it can span an ϵ-net of the state space.
This significantly strengthens what standard PRSGs can induce, as they may only
concentrate on a small region of the state space as long as the average output
state approximates a Haar random state in total variation distance.
Our PRSS construction develops a parallel extension of the famous Kac's walk,
and we show that it mixes exponentially faster than the standard Kac's walk.
This constitutes the core of our proof. We also describe a few applications of
PRSSs. While our PRSS construction assumes a post-quantum one-way function,
PRSSs are potentially a weaker primitive and can be separated from one-way
functions in a relativized world similar to standard PRSGs