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.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.

Journal of Physics A Mathematical and TheoreticalMay 1, 20254 citations

The surface code: processing experimental data

Austin G. Fowler

Bulletin of the American Physical SocietyMar 2, 20150 citations

Robustness and measurement-induced percolation of the surface code

Thomas Botzung, Michael Buchhold, Sebastian Diehl +1 more

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 $\hat Z$ -measurements, and that this loss of logical quantum information is equivalent to the bond percolation transition in a 2D square lattice. Local $\hat Y$ -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 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 $\hat Z$-measurements, and that this loss of logical quantum information is equivalent to the bond percolation transition in a 2D square lattice. Local $\hat Y$-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.

arXiv:2311.14338Nov 24, 20230 citations

Percolation Thresholds Unify Quantum Error Correction and Phase Transitions — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Surface-code error correction thresholds align with percolation-theoretic critical probabilities in 2D lattices, establishing a deep equivalence between quantum error correction and classical statistical-mechanics phase transitions. | MATH: The surface code threshold is governed by the bond-percolation threshold $p_c$ on the square lattice, $p_c = 1/2$ . For the Fibonacci Turaev–Viro code (non-Abelian anyons), thresholds map to golden-ratio-related critical exponents. The Kitaev toric code threshold satisfies $p_{\text{th}} = p_c(\text{square}) = 0.5$ for perfect syndrome measurement; with faulty measurements, the threshold drops to $p_{\text{th}} \approx 0.109$ (matching the 2D random-bond Ising model multicritical point). LDPC codes (Delfosse) exhibit thresholds tied to the percolation transition on expander graphs, where the critical density satisfies $p_c \sim 1/d$ for degree- $d$ graphs. | CONNECTION: The square-lattice percolation threshold $ p_c = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Surface-code error correction thresholds align with percolation-theoretic critical probabilities in 2D lattices, establishing a deep equivalence between quantum error correction and classical statistical-mechanics phase transitions. | MATH: The surface code threshold is governed by the bond-percolation threshold \( p_c $ on the square lattice, $ p_c = 1/2 $. For the Fibonacci Turaev–Viro code (non-Abelian anyons), thresholds map to golden-ratio-related critical exponents. The Kitaev toric code threshold satisfies $ p_{\text{th}} = p_c(\text{square}) = 0.5 $ for perfect syndrome measurement; with faulty measurements, the threshold drops to $ p_{\text{th}} \approx 0.109 $ (matching the 2D random-bond Ising model multicritical point). LDPC codes (Delfosse) exhibit thresholds tied to the percolation transition on expander graphs, where the critical density satisfies $ p_c \sim 1/d $ for degree-$ d $ graphs. | CONNECTION: The square-lattice percolation threshold \( p_c = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 23, 20260 citations

The intrinsic error thresholds of the surface code with correlated errors

Pejman Jouzdani, Eduardo R. Mucciolo, E. Novais

Bulletin of the American Physical SocietyMar 5, 20140 citations

\hat Z

-measurements, and that this loss of logical quantum information is equivalent to the bond percolation transition in a 2D square lattice. Local

\hat 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.

p_c

on the square lattice,

p_c = 1/2

. For the Fibonacci Turaev–Viro code (non-Abelian anyons), thresholds map to golden-ratio-related critical exponents. The Kitaev toric code threshold satisfies

p_{\text{th}} = p_c(\text{square}) = 0.5

for perfect syndrome measurement; with faulty measurements, the threshold drops to

p_{\text{th}} \approx 0.109

(matching the 2D random-bond Ising model multicritical point). LDPC codes (Delfosse) exhibit thresholds tied to the percolation transition on expander graphs, where the critical density satisfies

p_c \sim 1/d

for degree-

d

graphs. | CONNECTION: The square-lattice percolation threshold \( p_c = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com