Error-Correction Transitions in Finite-Depth Quantum Channels

Abstract

We study error correction type protocols in which a quantum channel encodes logical information into an enlarged Hilbert space. Specifically, we consider channels realized by one dimensional random noisy quantum circuits with spatially local interaction gates. We analyze both noise acting after the encoding and noise affecting the encoding circuit itself. Using the coherent information as a metric, we show that in both cases the infinite depth limit is governed by random matrix theory, which predicts a universal phase transition at a critical noise rate. This critical point separates an error correcting phase, in which encoded information is preserved, from a phase in which it is irretrievably lost. Going beyond the infinite depth limit, we characterize the systematic finite depth deviations from random matrix universality. In particular, we show that these deviations behave parametrically differently depending on whether the noise acts after the encoding or also affects the encoding itself. For noiseless encoders, the approach is exponential in circuit depth, although boundary effects can delay perfect encoding relative to the circuit design time. For noisy encoders, we find that the circuit fidelity effectively replaces the Hashing bound, and perfect encoding is approached polynomially with depth.

Figures (6)

• Figure 1: We study the coherent information of a random encoding circuit at finite depth $t$ and investigate its ability to protect quantum information against noise, whether the noise acts after the encoder or within the encoding circuit itself.
• Figure 2: Coherent information in setups I (a,b) and II (c,d). In both cases, we set the encoding rate to $r=1/4$ and use depolarizing noise channels (see Fig. \ref{['fig:info_ampdamp']} for similar results with amplitude damping noise). Panel (a) shows the coherent information in setup I for system size $N=512$ as a function of the Hashing bound $H_2$. Panel (c) displays the coherent information for setup II with system size $N=32$ as a function of the parameter $f_2$ (see Eq. \ref{['eq:smallf']}) and for different depths. In both panels (a),(c) a range of values of the depolarizing strength $\gamma$ have been explored. Panels (b) and (d) display finite-time corrections to the RM result for setup I and II respectively. Three distinct noise strengths, two within the recovery regime and one at the transition point are considered (see arrows in panels (a),(c)). In setup I, panel (b), in the recovery regime, corrections initially decay as $N e^{-2t/\tau}$, the same scaling that governs the approach to a quantum state design, before crossing over to a slower decay $e^{-t/\tau}$; the early-time scaling is evidenced by the collapse of the corresponding curves. At the transition point, the corrections exhibit a scaling precisely of the form $N e^{-t/\tau}$, as demonstrated by the overlap of all curves. In setup II, panel (d), the corrections always follow a scaling proportional to $N/t$. For higher-replica numerical data see Fig. \ref{['fig:info_3_replicas']}.
• Figure 3: Setup I (a) and II (b) with depolarizing noise and encoding rate $r=1/4$. We show the Holevo information at different depths and noise strength and show the finite-time corrections to the RM result for three distinct level of noise per qubits, two within the recovery regime and one at the transition point. While the general shape of the Holevo information is different than the coherent information, the critical noise strength at which the transition away from the recovery phase occurs and the scaling of the finite-time corrections are identical.
• Figure 4: Setups I (a) and II (b) with an encoding rate of $r=1/4$ and depolarizing noise where the coherent information now involves $3-$Rényi entropies. In both cases, we show that the corrections to the RM result are scaling in the exact same way as for $2-$Rényi entropies. The Hashing bound $H_3$ is defined in Eq. \ref{['eq:depo_hashing_bound']} while $f_3$ is defined in Eq. \ref{['eq:f2higherreplica']}.
• Figure 5: Setups I (a) and II (b) with an encoding rate of $r=1/4$ and amplitude damping noise. In both cases, we show that the corrections to the RM result are scaling in the exact same way as the depolarizing noise (see Fig. \ref{['fig:info_depolarizing']}).