Thresholds and Resource Estimates

10.1 Why a threshold can exist

10.1.1 Suppress faults faster than we add locations

A distance-three fault-tolerant gadget is designed so that one physical fault cannot cause logical failure on valid inputs. Under a suitable local stochastic model, its leading dangerous events therefore involve at least two faults. A simplified bound takes the form p₁≤Ap², where A accounts for dangerous pairs and any higher-order contributions absorbed into the bound’s range of validity.

If the same construction can be concatenated consistently, each encoded location is replaced by another level of protection. The illustrative recursion pₗ₊₁≤Apₗ² gives

Check the first step: at ℓ=0 this is p; at ℓ=1 it is Ap². If Ap<1, repeated squaring drives the bound down rapidly. If Ap>1, this bound provides no suppression guarantee. For A=100 and p=0.001, the successive bounds are 10⁻⁴, 10⁻⁶, and 10⁻¹⁰. These numbers illustrate the mechanism, not a certified threshold for a specific architecture.

A threshold theorem makes such an argument rigorous for specified gadgets and noise assumptions, controlling dependencies and composing corrections with logical gates. For local stochastic noise, a useful assumption bounds the probability that every location in any chosen set S is faulty by p^|S|; this allows some correlations but excludes arbitrary common-cause failures of large sets at probability p. Other threshold theorems use other models. See Aliferis, Gottesman, and Preskill for a distance-three construction and its analysis.

Neither the simple recurrence nor a theorem supplies a universal experimental threshold. Leakage, biased noise, long-range correlations, scheduling, decoding, and available operations can change the applicable model. The first task in using a threshold is to identify which assumptions your system actually satisfies.

10.2 Budgeting an entire computation

10.2.1 Invert a scaling model with units attached

Suppose a computation has N logical failure opportunities, each with probability at most pL. The union bound gives total failure probability at most NpL. Independence is not required for that inequality. To allocate an error budget ε, a sufficient condition is pL≤ε/N. Other errors, such as gate-synthesis approximation and state-preparation inaccuracies, need their own allocations; the budget cannot be spent twice.

For a worked example, assume—not measure—the per-round scaling law

for odd-distance rotated patches. Set A=0.1, p=10⁻³, pth=10⁻², N=10⁸ block-round opportunities, and ε=0.01. Here pL is failure probability per block per correction round; N already sums rounds over all logical blocks. The target is pL≤10⁻¹⁰. Substitution gives 10^(−1−(d+1)/2)≤10⁻¹⁰, hence d≥17. At d=17 the model exactly exhausts this budget; choosing d=19 would leave a factor-ten margin within the model.

A distance-17 rotated patch has 17²=289 data qubits and 288 dedicated check ancillas in the minimal count used earlier, totaling 577. One hundred such patches would use 57,700 qubits before routing, spare capacity, logical-operation space, or magic-state factories. This is an illustrative arithmetic result, not a hardware requirement forecast. Its value lies in making every assumption inspectable.

If pL instead referred to a logical gate lasting several rounds, N would have to count gates. Mixing those units can change an estimate by orders of magnitude. Write “per what?” next to every error probability and every time estimate before comparing architectures.

10.3 A reproducible resource study

10.3.1 Report what the model establishes

Build a small resource-estimation study around the preceding example. Choose a range of physical error probabilities below the hypothetical threshold, solve for the smallest odd distance meeting a fixed budget, and tabulate distance, data qubits, check ancillas, and block-round volume. Then vary N or ε one at a time. The discrete jumps in distance are real features of the chosen model, while its parameters remain assumptions.

Add a separate resource-state budget. If a computation uses M injected states and each fails with probability at most q, then Mq is a union-bound contribution. Include the acceptance yield from Chapter 9, factory throughput, storage while waiting, and space occupied by factories. If logical gates require patches to move, merge, or interact, account for their extra area and duration rather than treating every gate as a free single-patch operation.

The course verification script reproduces small code syndromes, CSS ranks, logical distances, parity extraction, injection branches, and the numerical estimates. Use it as a baseline. A complete project submission should specify its noise and circuit models, equations, decoder assumptions, executable calculation, endpoint checks, and uncertainty or sensitivity analysis. It should clearly separate exact finite-code certificates, theorem statements, numerical observations, and hypothetical planning models.

We began by asking whether an error could be identified without exposing an unknown state. That led to syndrome subspaces, algebraic correction conditions, local code geometry, controlled fault propagation, and a resource budget. Further study can now branch toward quantum LDPC codes, bosonic codes, device-specific noise, or decoder design. In each direction, keep the same discipline: identify the operational task, prove or verify the relevant claim, and state the assumptions that make it true.