Requires the physical error rate $p$ to sit strictly below a threshold that depends on the code variant, the syndrome-extraction circuit, the noise model AND the decoder — a threshold quoted without all four means nothing. The encoding rate is poor: one logical qubit per patch. The code distance is not a property of a machine; it is solved for from $p$ and the target logical error rate the algorithm's total operation count demands. Real-time syndrome decoding is a separate engineering problem with a hard latency budget, and the decoder is part of what sets the observed threshold. One caution on reading the Google result: the once-an-hour correlated-error floor sometimes quoted beside it was measured with repetition codes run to probe the limits, not observed as the limit of the surface-code memories.

A physical error rate $p$ and noise model; a target logical error rate $P_L$; a connectivity constraint; a measurement and feedback cycle time.

1. Physical qubits → Logical qubitschain X and Z operators across the boundaries

   On the 2D lattice each measure-Z (measure-X) qubit forces its four neighbouring data qubits into an eigenstate of the weight-4 stabilizer $\hat Z_a\hat Z_b\hat Z_c\hat Z_d$ ($\hat X_a\hat X_b\hat X_c\hat X_d$). On a patch with 41 data qubits and 40 measure qubits the data qubits' $2\times41$ real degrees of freedom exceed the $2\times40$ independent stabilizer constraints by two, defining one logical qubit. The logical operators $\hat X_L$, $\hat Z_L$ are chains of single-qubit $\hat X$, $\hat Z$ operators connecting opposite lattice boundaries that commute with every stabilizer and anticommute with each other; the code distance $d$ is the minimum such chain weight. assumption: the physical error rate $p$ per step lies strictly below a threshold $p_{th}$ fixed jointly by the noise model, syndrome-extraction circuit and matching decoder, below which $P_L \sim 0.03(p/p_{th})^{d_e}$, with $d_e=(d+1)/2$ for odd $d$ and $d_e=d/2$ rounded down for even $d$, an approximation: empirical fit to matching-decoder simulation data, not a derived bound.

   approximationassumption

Logical qubits, together with the code and code distance $d$ that were chosen for them, a physical-qubits-per-logical-qubit figure, and a decoding latency requirement.

given  a physical error rate p and noise model, a target logical error rate
       P_L, a connectivity constraint, and a measurement and feedback cycle
       time

require  p strictly below a threshold that depends on the code variant, the
         syndrome-extraction circuit, the noise model AND the decoder
# a threshold quoted without all four means nothing

encode a logical qubit in the homology of a two-dimensional lattice of
    physical qubits
measure the weight-4 stabilizers with nearest-neighbour circuits
# only a 2D nearest-neighbour grid is needed, and the code tolerates a
# comparatively high physical error rate -- which is why it is the dominant
# fault-tolerant code

solve for the code distance d from p and the target logical error rate P_L
    that the algorithm's total operation count demands
# the code distance is not a property of a machine
# Fowler, Mariantoni, Martinis and Cleland give the empirical scaling
#     P_L approx 0.03 (p/p_th)^d_e,  error dimension d_e = (d+1)/2 for odd d
#     (rounded down to d/2 for even d), and measure p_th = 0.57% for their
#     circuit and noise model

decode the syndromes in real time
# a separate engineering problem with a hard latency budget, and the decoder
# is part of what sets the observed threshold

return the logical qubits, the distance d solved for, the physical qubits
    they cost per logical qubit, and the decoding latency to be met
# the encoding rate is poor: one logical qubit per patch
# in Fowler et al.'s defect-based construction a logical qubit costs
#     2.5 x 1.25 x (2d)^2 approx 12.5 d^2 physical qubits -- about 3600 at
#     d = 17 and about 14500 at d = 34

# measured: Google reports logical error suppressed by Lambda = 2.14 +/- 0.02
# per two units of distance, reaching 0.143% +/- 0.003% per cycle on a
# 101-qubit distance-7 code, beyond break-even by a factor 2.4 +/- 0.3
# one caution on reading that result: the once-an-hour correlated-error floor
# sometimes quoted beside it was measured with repetition codes run to probe
# the limits, not observed as the limit of the surface-code memories

Fowler, Mariantoni, Martinis and Cleland give the empirical scaling $P_L ≈ 0.03 (p/p_th)^d_e$, with error dimension $d_e = (d+1)/2$ for odd $d$ (rounded down to $d/2$ for even $d$), and measure $p_th = 0.57$% for their circuit and noise model. In their defect-based construction a logical qubit costs $2.5 x 1.25 x (2d)^2 ≈ 12.5 d^2$ physical qubits — about 3600 at $d = 17$ and about 14500 at $d = 34$. Google reports logical error suppressed by $Λ = 2.14 ± 0.02$ per two units of distance, reaching $0.143$% $± 0.003$% per cycle on a 101-qubit distance-7 code, beyond break-even by a factor $2.4 ± 0.3$.

• The two below-threshold surface-code memories

  • Quantum error correction below the surface code threshold

    Rajeev Acharya, Laleh Aghababaie-Beni, Igor Aleiner, Trond I. Andersen, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Nikita Astrakhantsev, Juan Atalaya, Ryan Babbush, Dave Bacon, Brian Ballard, Joseph C. Bardin, Johannes Bausch, Andreas Bengtsson, Alexander Bilmes, Sam Blackwell, Sergio Boixo, Gina Bortoli, Alexandre Bourassa, Jenna Bovaird, Leon Brill, Michael Broughton, David A. Browne, Brett Buchea, Bob B. Buckley, David A. Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Anthony Cabrera, Juan Campero, Hung-Shen Chang, Yu Chen, Zijun Chen, Ben Chiaro, Desmond Chik, Charina Chou, Jahan Claes, Agnetta Y. Cleland, Josh Cogan, Roberto Collins, Paul Conner, William Courtney, Alexander L. Crook, Ben Curtin, Sayan Das, Alex Davies, Laura De Lorenzo, Dripto M. Debroy, Sean Demura, Michel Devoret, Agustin Di Paolo, Paul Donohoe, Ilya Drozdov, Andrew Dunsworth, Clint Earle, Thomas Edlich, Alec Eickbusch, Aviv Moshe Elbag, Mahmoud Elzouka, Catherine Erickson, Lara Faoro, Edward Farhi, Vinicius S. Ferreira, Leslie Flores Burgos, Ebrahim Forati, Austin G. Fowler, Brooks Foxen, Suhas Ganjam, Gonzalo Garcia, Robert Gasca, Élie Genois, William Giang, Craig Gidney, Dar Gilboa, Raja Gosula, Alejandro Grajales Dau, Dietrich Graumann, Alex Greene, Jonathan A. Gross, Steve Habegger, John Hall, Michael C. Hamilton, Monica Hansen, Matthew P. Harrigan, Sean D. Harrington, Francisco J. H. Heras, Stephen Heslin, Paula Heu, Oscar Higgott, Gordon Hill, Jeremy Hilton, George Holland, Sabrina Hong, Hsin-Yuan Huang, Ashley Huff, William J. Huggins, Lev B. Ioffe, Sergei V. Isakov, Justin Iveland, Evan Jeffrey, Zhang Jiang, Cody Jones, Stephen Jordan, Chaitali Joshi, Pavol Juhas, Dvir Kafri, Hui Kang, Amir H. Karamlou, Kostyantyn Kechedzhi, Julian Kelly, Trupti Khaire, Tanuj Khattar, Mostafa Khezri, Seon Kim, Paul V. Klimov, Andrey R. Klots, Bryce Kobrin, Pushmeet Kohli, Alexander N. Korotkov, Fedor Kostritsa, Robin Kothari, Borislav Kozlovskii, John Mark Kreikebaum, Vladislav D. Kurilovich, Nathan Lacroix, David Landhuis, Tiano Lange-Dei, Brandon W. Langley, Pavel Laptev, Kim-Ming Lau, Loïck Le Guevel, Justin Ledford, Kenny Lee, Yuri D. Lensky, Shannon Leon, Brian J. Lester, Wing Yan Li, Yin Li, Alexander T. Lill, Wayne Liu, William P. Livingston, Aditya Locharla, Erik Lucero, Daniel Lundahl, Aaron Lunt, Sid Madhuk, Fionn D. Malone, Ashley Maloney, Salvatore Mandrá, Leigh S. Martin, Steven Martin, Orion Martin, Cameron Maxfield, Jarrod R. McClean, Matt McEwen, Seneca Meeks, Anthony Megrant, Xiao Mi, Kevin C. Miao, Amanda Mieszala, Reza Molavi, Sebastian Molina, Shirin Montazeri, Alexis Morvan, Ramis Movassagh, Wojciech Mruczkiewicz, Ofer Naaman, Matthew Neeley, Charles Neill, Ani Nersisyan, Hartmut Neven, Michael Newman, Jiun How Ng, Anthony Nguyen, Murray Nguyen, Chia-Hung Ni, Thomas E. O'Brien, William D. Oliver, Alex Opremcak, Kristoffer Ottosson, Andre Petukhov, Alex Pizzuto, John Platt, Rebecca Potter, Orion Pritchard, Leonid P. Pryadko, Chris Quintana, Ganesh Ramachandran, Matthew J. Reagor, David M. Rhodes, Gabrielle Roberts, Eliott Rosenberg, Emma Rosenfeld, Pedram Roushan, Nicholas C. Rubin, Negar Saei, Daniel Sank, Kannan Sankaragomathi, Kevin J. Satzinger, Henry F. Schurkus, Christopher Schuster, Andrew W. Senior, Michael J. Shearn, Aaron Shorter, Noah Shutty, Vladimir Shvarts, Shraddha Singh, Volodymyr Sivak, Jindra Skruzny, Spencer Small, Vadim Smelyanskiy, W. Clarke Smith, Rolando D. Somma, Sofia Springer, George Sterling, Doug Strain, Jordan Suchard, Aaron Szasz, Alex Sztein, Douglas Thor, Alfredo Torres, M. Mert Torunbalci, Abeer Vaishnav, Justin Vargas, Sergey Vdovichev, Guifre Vidal, Benjamin Villalonga, Catherine Vollgraff Heidweiller, Steven Waltman, Shannon X. Wang, Brayden Ware, Kate Weber, Theodore White, Kristi Wong, Bryan W. K. Woo, Cheng Xing, Z. Jamie Yao, Ping Yeh, Bicheng Ying, Juhwan Yoo, Noureldin Yosri, Grayson Young, Adam Zalcman, Yaxing Zhang, Ningfeng Zhu, Nicholas Zobrist · 2024

  About

  The first reported surface-code memories running *below* threshold — the condition the code's whole promise rests on, and the one this record's `conditions` field states as a requirement rather than an achievement. Published as Nature 638 (2025) 920–926.

  Methods

  Two memories: a distance-7 code, and a distance-5 code integrated with a real-time decoder. Repetition codes were run out to distance-29 separately, to probe where the error-correction performance stops improving.

  Data

  None found yet.

  Code

  None found yet.

  Results

  The logical error rate is suppressed by $\Lambda = 2.14 \pm 0.02$ per two units of code distance, reaching $0.143\% \pm 0.003\%$ error per cycle on a 101-qubit distance-7 code — beyond break-even, exceeding the best physical qubit's lifetime by $2.4 \pm 0.3$. Below-threshold performance held while decoding in real time, at an average decoder latency of 63 μs at distance 5 over a million cycles, with a 1.1 μs cycle time. The distance-29 repetition codes were limited by rare correlated error events about once an hour, or every $3 \times 10^9$ cycles — a figure measured while probing the limits, not a floor observed on the surface-code memories themselves.