Phase transitions in sampling and error correction in local Brownian circuits

We study the emergence of anticoncentration and approximate unitary design behavior in local Brownian circuits. The dynamics of circuit-averaged moments of the probability distribution and entropies of the output state can be represented as imaginary-time evolution with an effective local Hamiltonian in the replica space. This facilitates large-scale numerical simulation of the dynamics in $1+1$ dimensions of such circuit-averaged quantities using tensor network tools as well as identifying the various regimes of the Brownian circuit as distinct thermodynamic phases. In particular, we identify the emergence of anticoncentration as a sharp transition in the collision probability at $lnN$ timescale, where $N$ is the number of qubits. We also show evidence for a specific classical approximation algorithm undergoing a computational hardness transition at the same timescale. In the presence of noise, we show there is a noise-induced first-order phase transition in the linear cross entropy benchmark when the noise rate is scaled down as $1/N$. At longer times, the Brownian circuits approximate a unitary 2-design in $O\left(N\right)$ time. We directly probe the feasibility of quantum error correction by such circuits and identify a first-order transition at $O\left(N\right)$ timescales. The scaling behaviors for all these phase transitions are obtained from the large-scale numerics and corroborated by analyzing the spectrum of the effective replica Hamiltonian.

Figure 1

We consider local Brownian circuits acting on qubits in a chain. Averaging the replica dynamics $U\otimes U^{*}\otimes U\otimes U^{*}$ leads to an imaginary-time evolution with a local Hamiltonian in replica space, $H_{\text{eff}}$. We can simulate this imaginary-time evolution in $1+1$ dimensions efficiently with tensor networks. We study three scenarios, i.e., anticoncentration, noisy circuit benchmarking, and approximate unitary design (and consequently error correction), and find distinct quantum informational phases in different time and noise regimes, separated by phase transitions. These results are corroborated using large-scale numerics and analyzing the spectrum of $H_{\text{eff}}$.

Figure 2

Anticoncentration in the Brownian circuit: (a) $Z\sim 2^{-N}$ in $t/lnN={\tau }^{*}\approx 1.5$ time and (b) data collapse, which is consistent with Eq. (18).

Figure 3

Phase transition in the annealed Rényi-2 CMI of $p_U\left(s\right)$. We consider an equal tripartition of the spin chain such that $\left|A\right|=\left|B\right|=\left|C\right|=N/3$, with $B$ separating regions $A$ and $C$, and we plot Rényi-2 CMI $I^{\left(2\right)}(A:C|B)$. As $N$ is increased, for $t<{\tau }^{*}lnN$ (${\tau }^{*}\approx 1.2$), the CMI decays with $N$ proportionally to $e^{-N}$. For larger times $t>{\tau }^{*}lnN$ the CMI does not decay with $N$ and asymptotes to $ln2$.

Figure 4

In the presence of noise, there is a depth-driven crossover at $t^{*}\sim O\left(1\right)$; for $t<t^{*}$ the linear XEB $\chi \sim e^N$ and for $t>t^{*}, \chi \sim e^{-N\epsilon t}$. We consider a constant noise strength of $\lambda =0.01$.

Figure 5

Noise-driven transition for scaled noise $\lambda \sim c/N$. (a) Plot of $\chi$ for $N=40$ and different values of $\lambda N$. We find that for $\lambda N\le {\lambda }^{*}N\approx 0.84, \chi$ approximates the fidelity $F$ after an initial fast decay. For $\lambda N>{\lambda }^{*}N$, however, $\chi$ behaves differently from $F$, which results in a sharp phase transition as shown in (b) of the order parameter $F/\chi$, evaluated at a time $t>t^{*}=O\left(1\right)$ of initial decay. The scaling collapse is shown in the inset.

Figure 6

Dynamics of the annealed Rényi-2 CMI of the classical output distribution of the noisy Brownian circuit for noise levels (a) $\lambda =0.1/N$ and (b) $\lambda =2.0/N$.

Figure 7

Unitary encoding $U_{\text{enc}}$ of reference $R$ into system $A$. Any noise in $A$ after the encoding can be represented by a unitary operation $U_{\text{err}}$ coupling the part of $A$ where the error acted ($A_4$ in the figure), with an environment $E$.

Figure 8

The 2-design transition in the Brownian circuit. (a) We encode one qubit on the left end and have erasures (depolarization channel of strength $\lambda =0.75$) act on a fraction $p$ of qubits on the left of the chain. (b) Mutual purity for the setup in (a) with $p=0.25$, plotted as a function of time. (c) Here $\mathcal{F}$ saturates to ${\mathcal{F}}_{\text{Haar}}\approx O\left(2^{-N}\right)$ after $t\sim O\left(N\right)$ time. We scale the time linearly with $N$ and find that there is a transition at $t={\tau }^{*}N$, with ${\tau }^{*}\approx 0.77$. The scaling collapse of this transition is shown in the inset. (d) Plot of the right-hand side of the error correction bound of Eq. (43) for different values $p$, with the saturated mutual purity after $t\sim N$ time of encoding by the Brownian circuit. There is a transition $p_c\approx 0.17$ below which the error is correctable and can thus be identified as a lower bound to the true error threshold of the code. The scaling collapse of this transition is shown in the inset. (e) Transition in the mutual purity can be related to a first-order pinning transition between two domain-wall configurations separating domains of two ordered states consistent with the boundary conditions set by the definition of mutual purity. As described in the text, the trace structure in the definition of mutual purity forces the boundary state to certain ordered states. In the initial-time boundary, the state is swap ordered in the part where the reference qubit is encoded (red) and has the free-boundary condition elsewhere (no ordered state). The timelike boundaries on the left and right are also free by the open-boundary condition. In the final state, the swap (red) part refers to the erasure errors and the id (blue) part refers to the partial trace in the definition of mutual purity.

Figure 9

Mutual purity with erasures of a fraction $p=0.25$ of qubits on random locations. The data presented are averaged over 80 and 20 random instances of the erasure locations for $N=20,24,28$ and $N=32$ chains.

Figure 10

(a) Single-qubit reference encoded on the right end of chain and erasures of a fraction $p=0.25$ of qubits on the left of chain. Only (c) a specific domain-wall configuration contributes to the mutual purity dynamics as plotted in (b).