What Is an Error Rate?
An error rate is the figure of merit for how often, or by how much, a specified quantum operation deviates from the ideal. Depending on context it may be a fault probability, an average gate infidelity, a readout assignment error, an idle-error probability per unit time or cycle, or a logical failure probability. It is usually written as a number like \(10^{-3}\); for a stochastic per-operation failure probability, that means about one fault per thousand operations on average.
The term covers several different quantities, so a bare number says little. A useful quote names the operation, the qubits involved and how it was measured. The quantum error rate of a processor is a set of numbers, not one.
Error rate is closely tied to fidelity. For the average gate fidelity \(F_{\mathrm{avg}}\), the average gate infidelity is \(r = 1 - F_{\mathrm{avg}}\). Thus an average gate fidelity of 99.5% corresponds to an average gate infidelity of 0.5%, or \(5\times10^{-3}\).
Physical and Logical Error Rates
A quantum computing error rate depends on what is being counted. These are the ones most often reported:
| Kind | What it counts | Usual unit |
|---|---|---|
| Single-qubit gate error | Deviation from the ideal one-qubit operation, often reported as average gate infidelity or Pauli error probability | Per gate |
| Two-qubit gate error | Deviation from the ideal entangling operation, often reported as average gate infidelity or Pauli error probability | Per gate |
| Readout error | Misreading a 0 or 1 at measurement | Per measurement |
| Idle error | Decoherence while waiting | Per unit time or per cycle |
| Logical error rate | Failure of an encoded qubit | Per correction round or per logical operation |
The first four are physical-level error rates, determined by the hardware, controls, calibration and circuit context. The logical error rate is the probability that an encoded qubit or logical operation fails after syndrome extraction, decoding and recovery by quantum error correction. Always check its unit: per round, per logical operation and per whole circuit give very different numbers.
How Error Rates Are Measured
Randomized benchmarking is a standard way to estimate average gate error. The device runs random sequences of Clifford gates of increasing length, and the probability of returning to the expected state decays roughly exponentially with length. The fitted decay parameter is converted to an average error per Clifford gate; for Hilbert-space dimension \(D\), \(r=(D-1)(1-\alpha)/D\), where \(\alpha\) is the decay parameter. Magesan, Gambetta and Emerson gave the widely used formulation in 2011. The method is largely insensitive to state preparation and measurement (SPAM) errors, which mainly enter the fit as nuisance parameters; detailed SPAM characterization requires separate measurements.
A common logical-memory benchmark prepares an encoded state, runs rounds of error correction, decodes, reads out, and counts how often the decoded answer is wrong. Logical gates and full circuits require corresponding logical-operation or circuit-level benchmarks. Repeating this under comparable noise conditions at several code sizes can show whether the logical error rate decreases with code distance.
Three cautions apply to any published number:
- An average can hide the worst case. Coherent errors, such as a small systematic over-rotation, can accumulate faster than random errors of the same average size.
- A rate measured on isolated gates may change when the whole array runs at once, because of crosstalk.
- A benchmark circuit may not predict the error in a real application.
Why Error Rates Limit a Computation
Errors compound. If each of \(N\) operations fails independently with probability \(p\), the chance that none fails is
\[ (1-p)^N \approx e^{-pN} \]
With \(p = 10^{-3}\) and \(N = 1000\), about 37% of runs are error-free. At \(N = 10{,}000\), almost none are. An unprotected circuit reaches an order-unity chance of at least one fault after about \(1/p\) operations, so a high target success probability allows fewer operations than that. A gigaquop, a billion operations, needs a logical failure probability of order \(10^{-9}\) per operation merely to keep the expected number of failures near one, and about \(10^{-11}\) per operation for a 1% total failure budget, far below typical physical gate error rates.
Fault-tolerant error correction is designed to close that gap, but only below a threshold. For the surface code, simulations by Fowler and colleagues in 2012 support the approximate scaling
\[ p_L \approx A\left(\frac{p}{p_{\mathrm{th}}}\right)^{\lfloor (d+1)/2 \rfloor} \]
where \(d\) is the code distance, \(p_{\mathrm{th}}\) is the threshold (on the order of 1%, depending on the noise model) and \(A\) is a fitted constant. Below threshold, each increase in \(d\) lowers the logical error rate exponentially. Above it, larger codes make things worse.
Error Rates on Neutral-Atom Hardware
On neutral atoms, Evered and colleagues reported in 2023 a two-qubit gate fidelity of 99.5%, with CZ gates benchmarked on up to 60 atoms in parallel.
Two features matter when reading error rates on this platform. First, atoms can be lost from their optical tweezers, a failure that is neither a bit flip nor a phase flip and has to be tracked as its own error type. Second, atoms can be rearranged to bring selected qubits into Rydberg-interaction range, so a circuit may need fewer gates, which means fewer operations exposed to error.
Comparing platforms therefore calls for error rates read alongside gate speed, connectivity and compilation, not a single headline figure.
FAQ
What is a good error rate for a quantum computer?
There is no single target. It depends on the error-correcting code, the noise model and the application. For the surface code the threshold is on the order of 1%, but overhead falls steeply as physical error rates drop further below it, and each application sets its own logical error target.
What is the quantum bit error rate?
Quantum bit error rate (QBER) is a term from quantum key distribution: the fraction of sifted key bits that differ between sender and receiver. It signals channel noise or eavesdropping. Shor and Preskill proved in 2000 that ideal BB84 is secure, and the usual asymptotic analysis with one-way classical post-processing gives a positive secret-key rate up to a QBER of about 11%. It is a different quantity from a gate error rate.
How does a logical error rate differ from a physical one?
A physical error rate describes one operation on the hardware. A logical error rate describes an encoded qubit protected by a code. It can be far lower than the physical rate if the physical rate is below threshold and the code is large enough, and it can be worse if the physical rate is above threshold.
Is a lower error rate worth more than more qubits?
Neither is enough alone. Very low error rates on too few qubits cannot run a large algorithm, and many noisy qubits cannot run a deep circuit reliably. Error-corrected computing needs both, along with fast decoding.
Key Takeaways
- An error rate is a specified failure probability or error metric, and it means little unless the operation, unit and measurement method are named.
- For average gate fidelity, the average gate infidelity is \(r = 1 - F_{\mathrm{avg}}\); other error rates are not always equal to \(1-F\).
- Errors compound roughly as \(e^{-pN}\), so an unprotected circuit is limited to about \(1/p\) operations.
- Logical error rate falls with code size only when physical error rates are below the threshold.
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