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.
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.
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.
Nishimori Threshold Estimation for Bayesian Inference and Zq Surface Code Decoding
In quantum error correction, the error threshold provides essential quantitative guidance for the ability to bring about fault-tolerance through decoding the effects of incoherent noise, weak measurement or inference. However, the numerical value of an error threshold is typically only accessible through large-scale numerical simulations of the underlying noise model. Here we introduce an analytical estimate of error thresholds falling into the Nishimori universality class via a Fourier--Walsh projection scheme that maps the critical point of the underlying disorder-free statistical-mechanics model to the Born-disordered Nishimori critical point. Using a minimal replica theory approach, this closed-form estimate is obtained from a projection of the exact replicated single-bond weight which we find to reproduce (within a percentage point) the known numerical thresholds of random-bond and random-plaquette Ising models / Z2 stabilizer codes in spatial dimensions d=2−5, and extends to Potts variables with q≤4. The main application of our projection scheme is to Zq surface codes, whose decoding problem maps to the disordered q-state clock model. For q≥5 the clean clock model has \textit{two} Berezinskii--Kosterlitz--Thouless transitions, which the projection maps to two Nishimori temperatures that bound an intermediate information-critical phase. The resulting threshold values not only accurately agree with recent decohered-Zq-toric-code numerics, but are found to satisfy the Gilbert--Varshamov self-dual entropy relation lnq≃Hq(T1∗)+Hq(T2∗), although no duality condition is imposed in the construction. Our approach thereby points to a deeper connection between the clean and Born-disordered models, while allowing for instant analytical estimates of error thresholds for a variety of stabilizer codes.
We explore a class of quantum circuit models with local decoherence (noise) and local projective measurements, each respecting a global Z_2 symmetry. The model supports a spin glass phase where the Z_2 symmetry is spontaneously broken (not possible in an equilibrium 1d system), a paramagnetic phase characterized by a divergent susceptibility, and an intermediate trivial phase. All three phases are also stable to Z_2-symmetric local unitary gates, and the dynamical phase transitions between the phases are in the percolation universality class. The open circuit dynamics can be purified by explicitly introducing a bath with its own scrambling dynamics, as in [Bao, Choi, Altman, arXiv:2102.09164], which does not change any of the universal physics. Within the spin glass phase the circuit dynamics can be interpreted as a quantum repetition code, with each stabilizer of the code measured stochastically at a finite rate, and the decoherences as effective bit-flip errors. Motivated by the geometry of the spin glass phase, we devise a novel decoding algorithm for recovering an arbitrary initial qubit state in the code space, assuming knowledge of the history of the measurement outcomes, and the ability of performing local Pauli measurements and gates on the final state. For a circuit with L^d qubits running for time T, the time needed to execute the decoder scales as O(L^d T) (with dimensionality d). With this decoder in hand, we find that the information of the initial encoded qubit state can be retained (and then recovered) for a time logarithmic in L for a 1d circuit, and for a time at least linear in L in 2d below a finite error threshold. For both the repetition and toric codes, we compare and contrast our decoding algorithm with earlier algorithms that map the error model to the random bond Ising model.
Floquet quantum error-correcting codes provide an operationally economical route to fault tolerance by dynamically generating stabilizer structures using only two-body Pauli measurements. But while it is well established that stabilizer codes in higher spatial dimensions gain additional levels of intrinsic robustness, higher-dimensional Floquet codes have hitherto been explored only in limited scope. Here we introduce a 3d generalization of a Floquet code whose instantaneous stabilizer group realizes a 3d fermionic toric code, while crucially preserving all three logical qubits throughout the entire measurement sequence. One central ingredient is the identification of a 3d lattice geometry that generalizes the features of the Kekulé lattice underlying the 2d Hastings-Haah code - specifically, a structure where deleting any one edge color yields a two-color subgraph that decomposes into short, closed loops rather than homologically nontrivial chains. This loop property avoids the collapse of logical information that plagues naive sequential two-color measurement schedules on many 3d lattices. Although, for our lattice geometry, a simple 3-round cycle that sequentially measures the three types of parity checks does not expose the full error syndrome set, we show that one can append a measurement sequence to extract the missing syndromes without disturbing the logical subspace. Beyond code design, 3d tricoordinated lattice geometries define a family of 3d monitored Kitaev models, in which random measurements of the non-commuting parity checks give rise to dynamically created entangled phases with nontrivial topology. In discussing the general structure of their underlying phase diagrams and, in particular, the existence of certain quantum critical points, we again make a connection to the general preservation of logical information in time-ordered Floquet protocols.
Quantum error correcting (QEC) codes protect quantum information from decoherence, as long as error rates fall below critical error thresholds. In general, obtaining thresholds implies simulating the QEC procedure using, in general, sub-optimal decoding strategies. In a few cases and for sufficiently simple noise models, optimal decoding of QEC codes can be framed as a phase transition in disordered classical spin models. In both situations, accurate estimation of thresholds demands intensive computational resources. Here we use the coherent information of the mixed state of noisy QEC codes to accurately estimate the associated optimal QEC thresholds already from small-distance codes at moderate computational cost. We show the effectiveness and versatility of our method by applying it first to the topological surface and color code under bit-flip and depolarizing noise. We then extend the coherent information based methodology to phenomenological and quantum circuit level noise settings. For all examples considered we obtain highly accurate estimates of optimal error thresholds from small, low-distance instances of the codes, in close accordance with threshold values reported in the literature. Our findings establish the coherent information as a reliable competitive practical tool for the calculation of optimal thresholds of state-of-the-art QEC codes under realistic noise models.
A CSS quantum error-correcting code is, on the nose, a length-three chain complex over F2 : the commutation law HZ H⊤ X = 0 that makes the code valid is exactly the condition ∂ 2 = 0 that makes a chain complex. We take this identification literally and show that the entire working theory of a quantum code is one homological object the same engine computes, at scale, and certifies. The number of logical qubits is the first Betti number, k = dim H1 = n − rank HX − rank HZ = b1 ; the code distance is the systole, the shortest non-contractible cycle. We reproduce the published parameters of state-of-the-art codes exactly—the toric and hypergraph-product families and the IBM bivariate-bicycle codes [[72, 12, 6]], [[144, 12, 12]], [[90, 8, 10]]—cross-validated four independent ways and kernelcertified for the small cases in Cubical Agda, on CPU and GPU. The two scale certificates—the Betti count b1 and the rank-derived logical count k—are themselves kernel-proved to be a single first-homology certificate (a DescentCertificate both readings factor through), closed terminally through a fixed-point engine. We then run the code, not just measure it. (i) The Union-Find decoder is the engine's connected-components reduction, and "a logical error occurred" is the engine's coboundary decision (the residual cycle has nonzero winding); decoding the toric code at growing distance recovers the accuracy threshold near 9.5%, and the optimal (maximum-likelihood) threshold near 11% is the engine's two homology sectors competing as the domain-wall free energy of a random-bond Ising model on the Nishimori line. (ii) A code's distance scaling is its Tanner-graph spectral expansion: the second eigenvalue λ2 versus the Ramanujan bound is a computable certificate (Sipser–Spielman), exact on the topological tier where the spectral gap obeys γ d2 → 4π 2 . (iii) The full chain complex carries metachecks—redundancies among the parity checks—and we compute them: the bivariate-bicycle codes have six, the engines of single-shot error correction; the 4-dimensional toric code's homology (1, 4, 6, 4, 1), with k = b2 = 6, is the tensor of four circle complexes. (iv) The logical error-rate prefactor is the multiplicity of the systole, A = 2L for the toric code, which we compute exactly. (v) Finally the engine measures a topological phase transition: as random noise thickens, torus-wrapping "giant" cycles appear in a sharp homological-percolation transition at pc ≈ 0.5 that sharpens with system size—the same homology machinery now detecting when noise spontaneously writes a logical operator. (vi) The engine runs over any coefficients: qudit (Fp ) codes are chain complexes over Fp , and the integral homology, computed by the Smith normal form, exposes torsion—so a code's logical-qudit count can depend on the qudit dimension, an extra logical appearing exactly when p divides a torsion coefficient (reproducing the known homology of the torus, the Moore spaces, the Klein bottle, and RP2 ). We are careful to fence what is not claimed: minimum distance is NP-hard in general and we compute it only on the systolic tier; the giant-cycle transition is the random-cluster one, distinct from the maximum-likelihood decoding threshold of the random-bond Ising model. The thesis is singular and operational: a quantum code's parameters, its decoder, its quality, its higher-dimensional structure, its performance prefactor, and its phase transition are not six theories but one—homology you can run. Published by The Institute for Applied Ontological Mathematics.