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Initial State Memory in Finite Random Brickwork Circuits

Jakob Bannister, Katja Klobas, Colin Rylands, Bruno Bertini

Abstract

We ask under what conditions a finite brickwork circuit of random gates retains local information about the initial state. To answer this question we measure the averaged Frobenius distance between the reduced states obtained by evolving two arbitrary initial states and tracing out a portion of the system. By characterising this distance exactly at all times we find that the information is retained if the environment -- the subsystem traced out -- is smaller than half of the system and washed away otherwise. We also find that, while the dynamics of the Frobenius distance depends on the specific initial states chosen, this dependence becomes increasingly weak for large scales and eventually the Frobenius distance attains a universal form as a function of time. Finally, we show that by introducing weak enough boundary dissipation, one can observe a phase transition between a memory preserving phase and one where the information is completely lost.

Initial State Memory in Finite Random Brickwork Circuits

Abstract

We ask under what conditions a finite brickwork circuit of random gates retains local information about the initial state. To answer this question we measure the averaged Frobenius distance between the reduced states obtained by evolving two arbitrary initial states and tracing out a portion of the system. By characterising this distance exactly at all times we find that the information is retained if the environment -- the subsystem traced out -- is smaller than half of the system and washed away otherwise. We also find that, while the dynamics of the Frobenius distance depends on the specific initial states chosen, this dependence becomes increasingly weak for large scales and eventually the Frobenius distance attains a universal form as a function of time. Finally, we show that by introducing weak enough boundary dissipation, one can observe a phase transition between a memory preserving phase and one where the information is completely lost.
Paper Structure (16 sections, 96 equations, 4 figures)

This paper contains 16 sections, 96 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. Annealed average under unitary time-evolution
  4. Exact expression for the annealed average
  5. Infinite time regime
  6. Mixed states
  7. Finite-time dynamics
  8. Exponential scaling
  9. Constant overlap
  10. Open systems
  11. Markov Matrix Formalism
  12. Conclusions
  13. Simplification of $A_{\infty}(x)$
  14. Downward evolving state for an all-to-all random unitary matrix
  15. Short-time behaviour of $\ket{w}$ states.
  16. ...and 1 more sections

Figures (4)

  • Figure 1: Infinite time limit of $\left\langle \Delta_2(t)^2\right\rangle_a$ given by Eq. \ref{['eq:long_time_full']} for the W-states in Eq. \ref{['eq:W_state_2']} as a function of $x$ with $2L=200$ and for different values of $\omega$ (cf. Eq. \ref{['eq:omega']}). We see a sharp transition from a regime where memory of the initial state is lost $x/L<1$ to where it is retained for $x/L>1$.
  • Figure 2: Time dependence of $\left\langle \Delta_2(t)^2\right\rangle_a$ from the initial pair product state in Eq. \ref{['eq:pairproductstate']} and different subsystem sizes $x$. We take $2L=200,q=2,\beta=0.7$. For $x<L$ the curves drop to zero while for $x>L$ they stay finite for all times. The difference between the curves for the two largest values of $x$ is not distinguishable on this scale.
  • Figure 3: Time dependence of $\left\langle \Delta_2(t)^2\right\rangle_a$ from the pair of states in \ref{['eq:W_state_2']} for different subsystem sizes $x$. We take $2L=200,q=2,\omega=0.7$.
  • Figure 4: Values of $\left\langle \Delta_2(T)\right\rangle^2$ in the presence of boundary dissipation for the W states from \ref{['eq:open_long_time']} as a function of dissipation strength $a$ and for different values of subsystem size. We take $q=2,2L=100,T=20L$ and $\omega=0.7$. Dashed lines indicate value of $\Bar{a}_c$ predicted with first order perturbation theory.