MethodLayer 1
Surface code
Encode a logical qubit in the homology of a two-dimensional lattice of physical qubits, with weight-4 stabilizers measured by nearest-neighbour circuits. It is the dominant fault-tolerant code because it needs only a 2D nearest-neighbour grid and tolerates a comparatively high physical error rate.
A physical error rate and noise model; a target logical error rate ; a connectivity constraint; a measurement and feedback cycle time.
Logical qubits, together with the code and code distance that were chosen for them, a physical-qubits-per-logical-qubit figure, and a decoding latency requirement.
Same contract as the slot it fills.
This one, drawn
From Physical qubits to Logical qubits
A circle is an object you are holding. This method is drawn heavier, opened into its own steps; the other lines between the same two ends are the alternatives recorded for the same slot. Circles are named on hover, and each one is a link.
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What it fills
- Build logical qubits at a target logical error rate
Encode physical qubits whose error rate sits below a code- and decoder-specific threshold into logical qubits meeting a target logical error rate per round, by spending qubits and time on redundancy and decoding syndromes in real time. Which code sits underneath reaches the layers above only as a physical-qubit count and a demand on connectivity.
When it applies
Requires the physical error rate 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 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.
Requires
Every step this method names moves its route along, so there is nothing it needs alongside them.
Example
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 memoriesCost, as the source states it
Fowler, Mariantoni, Martinis and Cleland give the empirical scaling , with error dimension for odd (rounded down to for even ), and measure % for their circuit and noise model. In their defect-based construction a logical qubit costs physical qubits — about 3600 at and about 14500 at . Google reports logical error suppressed by per two units of distance, reaching % % per cycle on a 101-qubit distance-7 code, beyond break-even by a factor .
Implementations
The two below-threshold surface-code memories
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.
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.
The logical error rate is suppressed by per two units of code distance, reaching error per cycle on a 101-qubit distance-7 code — beyond break-even, exceeding the best physical qubit's lifetime by . 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 cycles — a figure measured while probing the limits, not a floor observed on the surface-code memories themselves.
What it needs
Nobody has taken this apart yet. That is a gap in this graph, not a claim that the method has no parts.
Other ways to fill the same slot
Different approaches
- Quantum LDPC codes (bivariate bicycle family)
Trade the surface code's strictly planar layout for slightly richer connectivity, in exchange for a much better encoding rate. Many logical qubits live in one code block instead of one per patch.
In the Atlas
No record in the Atlas covers this yet. The catalogue is circuits and primitives; this part of the literature is not in it.