What Is the Fault-Tolerance Threshold?
The fault-tolerance threshold is the physical error rate below which quantum error correction can make the error rate of a whole computation as small as needed, by adding more redundancy. Above it, that particular error-correction scheme no longer gives asymptotic error suppression; increasing the code size may fail to help and often makes the logical error rate worse.
The threshold theorem turns this into a statement about fault-tolerant computing: if the noise in each operation is below a constant value, and the noise meets certain assumptions, arbitrarily long quantum computations can run reliably with only polylogarithmic overhead. Versions were proved by Aharonov and Ben-Or (1997), Kitaev (1997) and Knill, Laflamme and Zurek (1998), building on the fault-tolerant constructions of Shor (1996).
Why a Threshold Exists
Encoding a logical qubit in many physical qubits creates two competing effects. More qubits and more operations mean more places for errors to occur. A code with larger distance \(d\) can correct more of them. Which effect wins depends on how noisy the hardware is.
For the surface code, a common rule of thumb for the logical error rate is
\[ p_L \approx A\left(\frac{p}{p_{\mathrm{th}}}\right)^{\lfloor (d+1)/2 \rfloor} \]
where \(p\) is the physical error rate, \(p_{\mathrm{th}}\) is the threshold and \(A\) is a fitted constant. If \(p/p_{\mathrm{th}} = 0.1\), each step up in distance (3, 5, 7) cuts \(p_L\) by a factor of 10. If the ratio is 0.5, each step gains only a factor of 2. Once the ratio exceeds 1, larger codes fail more often.
What Sets the Number: Code, Noise Model and Decoder
The error threshold is a property of a code, a noise model and a decoder together, not of a machine. The same surface code gives very different numbers depending on what is assumed (approximate values):
| Noise model | What is assumed | Approximate threshold |
|---|---|---|
| Code capacity | Only data qubits suffer independent bit-flip and phase-flip errors; syndrome readout is perfect | About 10% (optimal decoding) |
| Phenomenological | Data and measurement errors | About 3% |
| Circuit level | Every gate, measurement and idle step is noisy | Around 1% or somewhat below |
These figures trace to Dennis, Kitaev, Landahl and Preskill (2002), Wang, Harrington and Preskill (2003) and Fowler and coauthors (2012). The quality of decoding also shifts the result: optimal decoders reach higher thresholds than fast, simple ones. Correlated errors lower the effective threshold, and noise biased toward one error type can raise it.
Common Misreadings
- Gate fidelity is not the whole story. A high gate fidelity for one operation does not put a system below threshold. Measurement, idling, atom movement and loss all count.
- Below threshold is necessary, not sufficient. Near the threshold, the redundancy required is enormous. Practical machines need to sit well below it.
- The theorem has assumptions. Typical threshold statements specify assumptions about the noise model and allowed operations—such as local or weakly correlated noise, sufficient parallelism and ancilla preparation; schemes with measurement-based correction also require classical processing fast enough to keep up with the correction cycle.
- Demonstrations are specific. Google Quantum AI (2024, published in Nature in 2025) reported a surface-code memory operating below threshold on a superconducting processor, with logical error falling by roughly a factor of two as distance grew from 3 to 5 to 7. That shows the scaling for one code on one device, not a universal result.
Thresholds on Neutral-Atom Hardware
QuEra builds computers from neutral atoms held in optical tweezers, and three platform features matter for threshold analysis:
- Reconfigurable connectivity. Atoms can be moved during a computation, so codes need not be limited to nearest-neighbor layouts. This makes qLDPC codes practical to consider. Bravyi and coauthors (2024) presented a fault-tolerant memory protocol based on bivariate-bicycle qLDPC codes, with a reported circuit-level threshold of about 0.7% and substantially lower qubit overhead than a surface-code memory in their examples.
- Atom loss. Loss is a distinct error type. Where it can be detected, it can be treated as erasure, a known-location error that codes tolerate better than errors at unknown locations. Threshold estimates must model it explicitly.
- Gate quality. Evered and coauthors (2023), a Harvard, MIT and QuEra team, reported 99.5% two-qubit gate fidelity on a neutral-atom processor. As above, that figure alone does not decide threshold status.
Crossing the threshold is a precondition for the megaquop regime, where a machine runs about a million operations before an uncorrected error is likely. Whether a given system is below threshold has to be measured for its own code, noise and decoder.
FAQ
Is the threshold the same as gate fidelity?
No. Gate fidelity describes one kind of operation. The threshold compares the combined error rate of all operations in a code cycle, including measurement, idling and movement, with a value set by the code, noise model and decoder.
Is there one threshold number for quantum computers?
No. The figure of around 1% is often quoted for the surface code under circuit-level noise, but code-capacity models give about 10%, and other codes and decoders give different values. Any quoted threshold should come with its noise model.
Does being below threshold mean a useful machine exists?
Not by itself. Below threshold, adding redundancy suppresses logical errors, but the overhead can be large unless the physical error rate is well under the threshold. Reaching useful logical error rates also requires many more qubits and operations.
What does the threshold theorem assume?
Typical threshold statements assume a specified noise model and allowed operations, often including local or weakly correlated noise, enough parallelism and ancilla preparation; measurement-based schemes also require classical processing fast enough to keep up with the correction cycle. If these fail, the guarantee does not hold.
Key Takeaways
- The fault-tolerance threshold is the error rate below which more redundancy lowers logical errors and above which it raises them.
- The threshold theorem guarantees reliable arbitrarily long computation when noise is below a constant value and meets stated assumptions.
- The error threshold depends on the code, noise model and decoder, so quoted numbers such as 1% or 10% are not interchangeable.
- Being below threshold is necessary but not sufficient: overhead, loss, correlated noise and every operation in the cycle still matter.
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