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arXiv:2510.07396 · PRX Quantum

Spectral Properties and Coding Transitions of Haar-Random Quantum Codes

Grace M. Sommers, J. Alexander Jacoby, Zack Weinstein, David A. Huse, Sarang Gopalakrishnan

Published Apr 3, 2026• 1 citation
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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 codes. For error rates that exceed the hashing bound, typical errors are uncorrectable, but postselected error correction remains possible until a much higher threshold. Postselection can in principle be implemented by projecting onto subspaces corresponding to low-weight errors, which remain correctable past the hashing bound.

DOI
10.1103/d4wh-hqcp

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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 codes. For error rates that exceed the hashing bound, typical errors are uncorrectable, but postselected error correction remains possible until a much higher threshold. Postselection can in principle be implemented by projecting onto subspaces corresponding to low-weight errors, which remain correctable past the hashing bound.
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.
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In this work, we study the task of encoding logical information via a noisy quantum circuit. It is known that at superlogarithmic depth, the output of any noisy circuit without reset gates or intermediate measurements becomes indistinguishable from the maximally mixed state, implying that all input information is destroyed. This raises the question of whether there is a low-depth regime where information is preserved as it is encoded into an error-correcting codespace by the circuit. When considering noisy random encoding circuits, our numerical simulations show that there is a sharp phase transition at a critical depth of order p−1p^{-1}p−1, where ppp is the noise rate, such that below this depth threshold quantum information is preserved, whereas after this threshold it is lost. Furthermore, we rigorously prove that this is the best achievable trade-off between depth and noise rate for any noisy circuit encoding a constant rate of information. Thus, random circuits are optimal noisy encoders in this sense.
We study the error correcting properties of Haar random codes, in which a KKK-dimensional code space C⊆CN\boldsymbol{C} \subseteq \mathbb{C}^NC⊆CN is chosen at random from the Haar distribution. Our main result is that Haar random codes can approximately correct errors up to the quantum Hamming bound, meaning that a set of mmm Pauli errors can be approximately corrected so long as mK≪NmK \ll NmK≪N. This is the strongest bound known for any family of quantum error correcting codes (QECs), and continues a line of work showing that approximate QECs can significantly outperform exact QECs [LNCY97, CGS05, BGG24]. Our proof relies on a recent matrix concentration result of Bandeira, Boedihardjo, and van Handel.
Quantum thermalization explains how an isolated many-body system naturally evolves towards a thermal state, rendering information about the initial conditions inaccessible to local measurements. This is precisely the mechanism utilized in quantum error correction, where information is protected by design through a nonlocal encoding. In this work, we leverage this connection to port the rigorous framework of (approximate) quantum error correction to the study of quantum thermalization. Treating typical late-time states as codewords, we characterize the error-correcting properties of generic thermalizing dynamics. We numerically uncover a universal relationship between the encoding rate, distance, and thermal entropy density of the emergent code. At infinite temperature, this universal curve saturates the quantum Singleton bound, achieving the same optimal limit as Haar-random codes. At finite temperature, we introduce a code family based on the Scrooge ensemble, the natural thermal analogue of the Haar ensemble, and prove it saturates the entropic quantum Singleton bound, establishing this family as optimal within entropic constraints. Our extracted universal curve independently saturates this same bound, revealing that finite-temperature thermalization is itself optimal. Finally, we show how conserved quantities limit the error-correcting behavior of thermalization: codewords with differing energies, or other conserved charges, leak only classical information, and correctability persists until the difference reaches the scale of thermal fluctuations. Our results reveal a universal optimal coding structure in thermalizing dynamics, while introducing new optimal codes that achieve fundamental limits of approximate quantum error correction.
Quantum error correction (QEC) is the cornerstone of fault‑tolerant quantum computation, enabling logical qubits to be protected against physical noise by encoding them into highly entangled states. In this paper we propose that the very ability of a QEC code to protect against errors is itself subject to a phase transition. We formulate a theory of the Quantum Error Correction Phase Transition (QECP), in which the system undergoes a sharp change from a regime of successful error suppression to a regime where logical errors proliferate, as a control parameter – such as the physical error rate, the code distance, or the connectivity of the underlying hardware – is varied. Drawing analogies to conventional symmetry‑breaking and topological phase transitions, we construct a statistical mechanical model that captures the competition between error‑correction resources and noise. We analyze the critical behavior, derive scaling relations, and discuss implications for the design of scalable quantum architectures. The existence of a QECP provides a new physical lens through which to view quantum information processing and suggests that fault‑tolerance thresholds may be understood as critical points in an extended parameter space.