A fault-tolerant resource estimate connects an algorithm's logical requirements to a proposed physical implementation. It should state the code, operation set, target failure probability, scheduling assumptions, and supporting hardware model. A single number of “required qubits” hides too many choices to serve as an admission decision.
The project includes a synthetic resource calculation to make those choices visible. Its values are arithmetic teaching inputs, not forecasts for a provider, validated hardware fits, or evidence that the required logical operations are available. Use the calculation to learn how constraints interact, then replace its assumptions with justified estimates for a real system.
Separate space, time, and failure risk
Logical data patches occupy physical resources, but so can routing, ancillary operations, syndrome extraction, and state factories. A count that includes only stored logical data can miss the resources needed to make progress. Time also matters: longer execution exposes stored information and supporting operations to more opportunities for failure.
Admission therefore has several dimensions. A workload must fit spatially, meet its scheduling and throughput requirements, and satisfy its allocated failure budget under the chosen model. Passing one test does not waive the others. A decoder or factory bottleneck can make a qubit-feasible plan unusable.
Logical error estimates require a definition of the event being bounded. A per-patch-per-cycle model cannot be combined indiscriminately with a measured failure rate for a whole memory experiment. The QEC fixture elsewhere in this book reports the latter and does not calibrate this synthetic resource model.
The declared teaching model
The calculation assigns 2d² physical resource units to a logical patch at distance d. Request A needs 100 patches plus a fixed allowance of 40,000 units. Request B needs 60 patches plus 30,000 units. Available capacity is 230,000 units. The fixed allowances deliberately compress other resources; they are not detailed factory layouts or guarantees of factory throughput.
For request A, the example assumes 100 patches exposed for 100 million cycles. It assigns each modeled patch-cycle a failure-probability bound using the illustrative expression:
p_L = 0.1 × (0.001 / 0.01)^((d + 1)/2).
The numbers resemble a common form of below-threshold scaling expression, but no hardware evidence has been fitted here. Assuming this bound applies to every counted event, the union bound gives P(any modeled failure) ≤ 100 × 100,000,000 × p_L. Independence is not required for a union bound; valid bounds on the counted events are required.
Worked example: capacity and assurance disagree
| Distance | Request A | Request B | Combined | Modeled union bound for A |
|---|---|---|---|---|
| 21 | 128,200 | 82,920 | 211,120 | 0.01 |
| 25 | 165,000 | 105,000 | 270,000 | 0.0001 |
| 29 | 208,200 | 130,920 | 339,120 | 0.000001 |
At distance 21, both requests fit in the available 230,000 units. But if A requires the modeled failure bound to be at most 0.0001, this distance fails that requirement. Spatial feasibility has not produced an admissible A workload.
At distance 25, A meets that illustrative failure requirement, but both requests no longer fit together. A alone fits. Serializing work, changing resource allocation, or revisiting justified assumptions may produce a feasible plan. Quietly reducing the distance to satisfy capacity would trade away the requirement that motivated fault tolerance.
At distance 29, the modeled bound is smaller still, but resource use rises. “Choose the largest distance” is not an optimization policy: the operator must consider the actual assurance objective, execution time, opportunity cost, and any omitted failure mechanisms.
Turn an estimate into an operating decision
An admission record should identify the estimate's inputs and version, the constraints checked, the selected schedule, and the reason for acceptance or refusal. It should also identify evidence gaps. A synthetic scaling expression cannot establish the effects of leakage, correlated faults, calibration drift, or failures in omitted components.
For a real application, allocate failure risk among the relevant operations and resources rather than spending the whole budget on one convenient term. Check sensitivity to uncertain parameters. A small change near a feasibility boundary can alter the selected distance, factory count, or schedule.
Exercise: explain the refusal
At distance 25, an operator has already admitted A and is asked to admit B concurrently. How much additional capacity would the synthetic spatial model require, and would supplying it settle the complete admission decision?
Answer. Combined use is 270,000 against 230,000 available, a shortage of 40,000 units. Adding that capacity resolves only the stated spatial constraint. Scheduling, factory throughput, decoder/control requirements, and B's own failure analysis still need checks. The model has not established those properties by counting patches.
Return to physical and logical qubits when interpreting the experimental evidence that would be needed to replace these teaching assumptions.