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arXiv:2311.14338

Robustness and measurement-induced percolation of the surface code

Thomas Botzung, Michael Buchhold, Sebastian Diehl, Markus Müller

Published Nov 24, 2023• 0 citations
arXivHTMLPDF

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 $ \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.

DOI
10.48550/arxiv.2311.14338

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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^ \hat XX^- 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.
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.
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 examine the dynamics of a (1+1)(1+1)(1+1)-dimensional measurement-only circuit defined by the stabilizers of the [[5,1,3]] quantum error correcting code interrupted by single-qubit Pauli measurements. The code corrects arbitrary single-qubit errors and it stabilizes an area law entangled state with a ${D}_{2}={\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry protected topological (SPT) order, as well as a symmetry breaking (SB) order from a twofold bulk degeneracy. The Pauli measurements break the topological order and induce a phase transition into a trivial area law phase. Allowing more than one type of Pauli measurement increases the measurement-induced frustration and the SPT and SB order can be broken either simultaneously or separately at nonzero measurement rate. This yields a rich phase diagram and unanticipated critical behavior at the phase transitions. Although the correlation length exponent $\ensuremath{\nu}=\frac{4}{3}$ and the dynamical critical exponent z=1z=1z=1 are consistent with bond percolation, the prefactor of the logarithmic entanglement growth may take noninteger multiples of the percolation value. Remarkably, we identify a robust transient scaling regime for the purification dynamics of LLL qubits. It reveals a modified dynamical critical exponent ${z}^{*}\ensuremath{\ne}z$, which is observable up to times $t\ensuremath{\sim}{L}^{{z}^{*}}$ and is reminiscent of the relaxation of critical systems into a prethermal state.
In fault-tolerant quantum computation, the preparation of logical states is a ubiquitous subroutine, yet significant challenges persist even for the simplest states required. In the present work, we present a unitary, scalable, distance-preserving encoding scheme for preparing Pauli eigenstates in surface codes. Unlike previous unitary approaches whose fault-distance remains constant with increasing code distance, our scheme ensures that the protection offered by the code is preserved during state preparation. Building on strategies discovered by reinforcement learning for the surface-17 code, we generalize the construction to arbitrary code distances and both rotated and unrotated surface codes. The proposed encoding relies only on geometrically local gates, and is therefore fully compatible with planar 2D qubit connectivity, and it achieves circuit depth scaling as O(d)\mathcal{O}(d)O(d), consistent with fundamental entanglement-generation bounds. We design explicit stabilizer-expanding circuits with and without ancilla-mediated connectivity and analyze their error-propagation behavior. Numerical simulations under depolarizing noise show that our unitary encoding without ancillas outperforms standard stabilizer-measurement-based schemes, reducing logical error rates by up to an order of magnitude. These results make the scheme particularly relevant for platforms such as trapped ions and neutral atoms, where measurements are costly relative to gates and idling noise is considerably weaker than gate noise. Our work bridges the gap between measurement-based and unitary encodings of surface-code states and opens new directions for distance-preserving state preparation in fault-tolerant quantum computation.
Quantum information theory has shown strong connections with classical statistical physics. For example, quantum error correcting codes like the surface and the color code present a tolerance to qubit loss that is related to the classical percolation threshold of the lattices where the codes are defined. Here we explore such connection to study analytically the tolerance of the color code when the protocol introduced in Vodola et al. [Phys. Rev. Lett. 121, 060501 (2018)] to correct qubit losses is applied. This protocol is based on the removal of the lost qubit from the code, a neighboring qubit, and the lattice edges where these two qubits reside. We first obtain analytically the average fraction of edges r(p)r(p)r(p) that the protocol erases from the lattice to correct a fraction ppp of qubit losses. Then, the threshold pc{p}_{c}pc​ below which the logical information is protected corresponds to the value of ppp at which r(p)r(p)r(p) equals the bond-percolation threshold of the lattice. Moreover, we prove that the logical information is protected if and only if the set of lost qubits does not include the entire support of any logical operator. The results presented here open a route to an analytical understanding of the effects of qubit losses in topological quantum error codes.
We consider the surface code under errors featuring both coherent and incoherent components and study the coherence of the corresponding logical noise channel and how this impacts information-theoretic measures of code performance, namely coherent information and quantum relative entropy. Using numerical simulations and developing a phenomenological field theory, focusing on the most general single-qubit X -error channel, we show that, for any nonzero incoherent noise component, the coherence of the logical noise is exponentially suppressed with the code distance. We also find that the information-theoretic measures require this suppression to detect optimal thresholds for Pauli recovery; for this they thus require increasingly large distances for increasing error coherence and ultimately break down for fully coherent errors. To obtain our results, we develop a statistical mechanics mapping and a corresponding matrix-product-state algorithm for approximate syndrome sampling. These methods enable the large-scale simulation of these non-Pauli errors, including their maximum-likelihood thresholds, away from the limits captured by previous approaches.