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arXiv:2402.00145 · PRX Quantum

Randomly Monitored Quantum Codes

Dongjin Lee, Beni Yoshida

Published Apr 16, 2026• 0 citations
arXivHTMLPDF

Abstract

Recent studies of monitored quantum dynamics have revealed that projective measurements, traditionally viewed as decohering operations, can instead generate and sustain long-range entanglement. Motivated by these, we ask how many physical qubits must be measured in random basis to irreversibly destroy quantum information encoded in a quantum error-correcting code. We study this problem for a broad class of stabilizer and subsystem codes, derive necessary and sufficient conditions for measurement-induced information destruction, and show that many codes, including concatenated and topological codes, achieve the maximal measurement threshold p m th = 1 , meaning that the encoded information survives as long as arbitrarily small but finite fraction of physical qubits remain unmeasured. Beyond this surprising robustness, we find a structural relation underlying maximal thresholds. Namely, we prove that if the Pauli basis of the logical operator measured at full measurement does not concentrate on a single Pauli, then the measurement threshold always satisfies p m th = 1 . This result uncovers a structural relation between logical measurement statistics and stability under partial measurement, revealing a general mechanism by which access to measurement outcomes enhances decodability under monitored dynamics.

DOI
10.1103/6s65-ylvw

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Recent studies of monitored quantum dynamics have revealed that projective measurements, traditionally viewed as decohering operations, can instead generate and sustain long-range entanglement. Motivated by these, we ask how many physical qubits must be measured in random basis to irreversibly destroy quantum information encoded in a quantum error-correcting code. We study this problem for a broad class of stabilizer and subsystem codes, derive necessary and sufficient conditions for measurement-induced information destruction, and show that many codes, including concatenated and topological codes, achieve the maximal measurement threshold p m th = 1 , meaning that the encoded information survives as long as arbitrarily small but finite fraction of physical qubits remain unmeasured. Beyond this surprising robustness, we find a structural relation underlying maximal thresholds. Namely, we prove that if the Pauli basis of the logical operator measured at full measurement does not concentrate on a single Pauli, then the measurement threshold always satisfies p m th = 1 . This result uncovers a structural relation between logical measurement statistics and stability under partial measurement, revealing a general mechanism by which access to measurement outcomes enhances decodability under monitored dynamics.
Random projective measurements are an effective tool to model the interlinked phenomenology of monitoring, inferring, and decoding highly entangled states such as quantum error-correcting codes. At the same time, the study of such monitored dynamics has opened a new arena for critical phenomena, renewing longstanding questions in disordered statistical mechanics, including percolation, random-bond models, and Nishimori physics. Here we investigate the robustness of stabilizer quantum error-correcting codes under a single round of independently sampled single-qubit Pauli measurements and demonstrate that their threshold behavior falls into two broad categories - geometric and homological percolation transitions, with the toric code and color code being the principal examples for each class. We establish the distinct character of these two threshold theories by analyzing the logical-support criteria of these codes and by providing numerical estimates of their respective critical exponents obtained from optimized stabilizer simulations of large-scale systems with over one million qubits. Our framework extends to non-CSS, subsystem, higher-dimensional, and Floquet stabilizer codes and allows us to determine their respective threshold phase diagrams. We exemplify how the latter provide quantitative and qualitative guidance for the computationally more demanding problems of tunable learning and maximum-likelihood decoding. As a practical application of our framework, we discuss the robustness of the gross code.
We investigate the stability of logical information in quantum stabilizer codes subject to coherent unitary errors. Beginning with a logical state, we apply a random unitary error channel and subsequently measure stabilizer checks, resulting in a syndrome-dependent post-measurement state. By examining both this \emph{syndrome state} and the associated syndrome distribution, we identify a phase transition in the behavior of the logical information. In the Clifford/stabilizer setting, the change of logical stabilizer structure in a syndrome branch determines the state-level MAP Pauli-frame return probability for a chosen logical-basis input. We separately use quantum coherent information to diagnose recoverability of arbitrary encoded inputs at the channel level. Below a critical error threshold pcp_cpc​, the syndrome branches remain compatible with the original logical sector, enabling a high state-return probability. Above pcp_cpc​, the data are consistent with effective scrambling of the encoded subspace relative to the stabilizer checks, where the syndrome-resolved logical maps approach a global-Clifford-induced stabilizer instrument. This common post-threshold picture organizes both the toric-code and finite-rate random-stabilizer-code results within the same scrambling mechanism. The syndrome distribution supplies complementary local and global structural diagnostics whose scaling depends on the code family. We refer to this phenomenon as a \emph{coherent error induced phase transition}. To illustrate this transition, we study two classes of quantum error-correcting codes, the toric code and finite-rate random stabilizer codes, thereby shedding light on the design and performance limits of quantum error correction under coherent errors.
Measurements underpin active quantum error correction (QEC) and have been recognized as a source of novel measurement-induced many-body phenomena. Here, we study the statistical properties of post-measurement logical states arising in QEC on topological codes subject to deterministic transversal unitary gates. Upon syndrome extraction followed by maximum-likelihood decoding, a Born-weighted ensemble arises which we dub the "projected logical ensemble" (PLE). Focusing on surface codes subject to uniform single-qubit Pauli-XXX rotations, we characterize the measurement-induced randomness of the PLE. To this end, we show that for a code with a single logical qubit, the PLE is isomorphic to an ensemble of scattering matrices describing mesoscopic quantum dots obtained from a 2D Majorana network model with suitable boundary conditions. We uncover regimes where these quantum dots are chaotic such that their scattering matrices are well-described by random matrix theory. In these regimes, the PLE approaches a universal ensemble that is maximally random up to symmetry and decoder-induced constraints. The symmetry constraints, set by stabilizer and logical operator weights, realize Altland-Zirnbauer classes D or DIII, which we both illustrate. Our results establish a fundamental connection between emergent universality concepts in mesoscopic physics, quantum many-body systems, and QEC.
Abstract We examine the robustness of a logical qubit in the planar surface code subject to ‘measurement-errors’, i.e. to local Pauli measurements at known positions. This yields a measurement-only dynamics, which is driven by the competition between local Pauli measurements and stabilizer measurements. The setup differs from the conventional surface code for which errors are caused by decoherence and their positions are unknown. Yet, our setting allows us to examine the dynamics of the encoded logical qubit by using a combination of analytical arguments based on percolation theory and numerical simulations. Firstly, we show that for a single round of Pauli measurements only, the threshold corresponding to the irreversible loss of the logical qubit depends only on the rate of X ^ - and Z ^ -measurements, and that this loss of logical quantum information is equivalent to the bond percolation transition in a 2D square lattice. Local Y ^ -measurements, which affect both X and Z stabilizers, erase the logical qubit only if all physical qubits along one of the diagonals are measured, and are thus negligible at large code distance. Secondly, we examine the dynamics in the code by considering the interplay between rounds of Pauli measurements and rounds of stabilizer measurements. Focusing on the lifetime of the logical qubit, we obtain a rich phase diagram featuring different dynamical regimes. We argue that the loss of the logical qubit in this setting can still be understood by percolation theory and underpin our arguments with numerical simulations.
We study the protection of information in nearly critical topological quantum codes, constructed by perturbing topological stabilizer codes towards continuous quantum phase transitions. Our focus is on the transverse-field toric code subjected to local Pauli decoherence. Despite the strong quantum fluctuations of anyons when the transverse field is tuned infinitesimally close to the critical point, we show that a finite strength of Pauli decoherence remains necessary to irreversibly destroy information encoded in the ground-state manifold. Using a replica statistical physics mapping for the coherent information, we show that decoherence can be understood as introducing a two-dimensional inter-replica defect within a three-dimensional replica statistical physics model. A field theoretical analysis shows that this defect is perturbatively irrelevant to the bulk critical point, and cannot renormalize the transverse field strength, leading to a finite error threshold. We argue that a qualitatively similar conclusion can be drawn for a broad class of nearly critical topological codes, under a variety of decoherence channels.
We examine the robustness of a logical qubit in the planar surface code subject to 'measurement-errors', i.e., to local Pauli measurements at known positions. This yields a measurement-only dynamics, which is driven by the competition between local Pauli measurements and stabilizer measurements. The setup differs from the conventional surface code for which errors are caused by decoherence and their positions are unknown. Yet, our setting allows us to examine the dynamics of the encoded logical qubit by using a combination of analytical arguments based on percolation theory and numerical simulations. Firstly, we show that for a single round of Pauli measurements only, the threshold corresponding to the irreversible loss of the logical qubit depends only on the rate of $ \hat X$- and Z^\hat ZZ^-measurements, and that this loss of logical quantum information is equivalent to the bond percolation transition in a 2D square lattice. Local Y^\hat YY^-measurements, which affect both XXX and ZZZ stabilizers, erase the logical qubit only if all physical qubits along one of the diagonals are measured, and are thus negligible at large code distance. Secondly, we examine the dynamics in the code by considering the interplay between rounds of Pauli measurements and rounds of stabilizer measurements. Focusing on the lifetime of the logical qubit, we obtain a rich phase diagram featuring different dynamical regimes. We argue that the loss of the logical qubit in this setting can still be understood by percolation theory and underpin our arguments with numerical simulations.