Spectral properties and coding transitions of Haar-random quantum codes

TL;DR

The paper analyzes spectral properties and coding transitions of Haar-random quantum codes under depolarizing noise, revealing a banded structure in the density matrix conditioned on error weight. Using fixed-weight ensembles, Weingarten-calculus-based calculations, and perturbative treatments of nonorthogonality, it shows the coding threshold saturates the hashing bound and coincides with random stabilizer codes, with high-weight bands merging beyond the threshold. Beyond threshold, low-weight bands permit postselection up to a higher detection threshold $p_d$, with Rényi-entropy diagnostics and quantum MacWilliams identities providing operational thresholds $p_c^{(\alpha)}$ and insights into the nature of the transition. The study connects information-theoretic limits to spectral features of random codes, suggests possible universality across code families, and raises questions about extensions to stabilizer/topological codes and their error-decomposition structures.

Abstract

A quantum error-correcting code with a nonzero error threshold undergoes a mixed-state phase transition when the error rate reaches that threshold. We explore this phase transition for Haar-random quantum codes, in which the logical information is encoded in a random subspace of the physical Hilbert space. We focus on the spectrum of the encoded system density matrix as a function of the rate of uncorrelated, single-qudit errors. For low error rates, this spectrum consists of well-separated bands, representing errors of different weights. As the error rate increases, the bands for high-weight errors merge. The evolution of these bands with increasing error rate is well described by a simple analytic ansatz. Using this ansatz, as well as an explicit calculation, we show that the threshold for Haar-random quantum codes saturates the hashing bound, and thus coincides with that for random \emph{stabilizer} codes. For error rates that exceed the hashing bound, typical errors are uncorrectable, but postselected error correction remains possible until a much higher \emph{detection} threshold. Postselection can in principle be implemented by projecting onto subspaces corresponding to low-weight errors, which remain correctable past the hashing bound.

Figures (4)

• Figure 1: Schematic circuit diagram of the Haar-random codes studied in this work. $k$ logical qudits are isometrically encoded in an $N$-qudit system via a Haar-random unitary $U$. Each system qudit $i$ is then subjected to a depolarizing channel $\mathcal{N}_{i,p}$ of strength $p$ [see Eq. \ref{['eqn:dep_channel3-p']}].
• Figure 2: Fixed-weight error ensemble, $k=1$ qubit Haar-code. (a) Density of states of the density matrix $\mathcal{N}^{(w)}(\rho_Q)$, for $N=13, w=1,2$ (below hashing) and $w=3,4$ (above hashing), rescaled to the mean nonzero eigenvalue, $\overline{\lambda} = 1/\min(2^N,2\Omega(w))$ in each band. Solid curves are data aggregated over 467 independent states, while dashed curves are the Marchenko-Pastur distribution with $c=2^{N-1}/\Omega(w)$suppmat. (b) Transition in the coherent information of the fixed-weight error ensemble, with $w_c(N) = p_c N$. Each data point is an average over at least 200 independent states, and error bars are smaller than the markers. The dashed curve, shown to guide the eye, is a fit to a sigmoid function using data points with $N\geq 8$.
• Figure 3: Depolarizing error channel, $k=1$ qubit Haar-code. (a) Eigenvalues of the density matrix $\mathcal{N}_p(\rho_Q)$, as a function of depolarization rate $p$. Filled curves are data aggregated from $\approx 100$ samples at $N=13,k=1$. Dashed curves are the ansatz. (b) Coherent information for $k=1$, odd $N$ ranging from 5 to 13. Inset: scaling collapse with $p_c = p_{hash} \approx 0.1893$ for all $N$ in the interval $[7,13]$. Gray curve shows the ansatz for $N=13$.
• Figure 4: (a) Theoretical phase diagram for postselection in a zero-rate qubit Haar-code. Information is recoverable at error rate $p$ by postselecting on band $w$ within the blue-shaded region. Dashed curve shows the threshold under the soft postselection defined in \ref{['eq:M-alpha']}, corresponding to the Rényi threshold plotted as a function of $\alpha$ in (b). The star in (b) marks $\alpha=2, p_c^{(2)} = (3-\sqrt{3})/4$. This threshold manifests as both (c) a nonanalyticity in $S_2(\tilde{\rho}_Q)/N$ (x's) and $S_{\rm vN}(\tilde{\sigma}_{Q,2})/N$ (circles) and (d) a transition in the coherent information of $\tilde{\sigma}_{Q,2}$. The main panels show odd system sizes ranging from $N=5$ to $N=13$. In (c), black lines indicate the leading-$N$ behavior for the two quantities (\ref{['eq:renyi']} and \ref{['eq:reweight']}), while the barely visible red curves are the theoretical calculations for $N=13$. The inset of (d) shows a scaling collapse of the postselected coherent information, with system sizes ranging from $N=7$ to $N=13$.