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To measure noise and speed in a quantum Fourier transform (QFT) circuit, define the exact circuit and ideal result, run it on a named quantum backend, and report an explicit output-agreement or process-fidelity estimate alongside its shot count. For speed, report compiled circuit resources and state exactly what the timing includes. Calibration, connectivity, compilation, measurement, and mitigation all affect results, so a single fidelity or runtime number is not a general measure of “the QFT.”
If you mean a classical fast Fourier transform (FFT) in software, measure numerical error against a high-precision reference and time repeated transforms separately from setup; that is a different kind of error from quantum gate or readout noise.
Define what the QFT circuit is supposed to do
Before measuring anything, specify whether the workload is a QFT unitary by itself or a QFT followed immediately by measurement. Record the number of qubits, the tested input states, and the ideal output expected for each input. A result for one measured output distribution is not automatically a process-fidelity measurement; name the statistic you actually estimate.
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For one documented approach, IBM’s Orbit tutorial prepares selected inverse-QFT input states, runs the noisy QFT-plus-measurement implementation, and estimates the probability of obtaining the corresponding ideal output. This is a sampled estimate under a particular test protocol, not a universal score for every QFT circuit.
Measure noise against an explicit reference
Report an explicitly defined ideal-process fidelity or output-agreement statistic, the inputs used, and the number of shots. Also state whether measurement-error mitigation, dynamical decoupling, or another correction or suppression method was enabled. Without those details, two reported values may not measure the same thing.
Include available backend calibration context, such as gate-error and readout or measurement-fidelity metrics. IBM’s QPU information guide describes layered two-qubit gate error and a measurement-fidelity metric commonly calculated from preparation and readout error probabilities. These hardware metrics help explain a circuit result; they do not replace testing the QFT workload itself.
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Compilation and qubit connectivity also matter: a logical circuit may map to different physical qubits and acquire different gate counts or depth. IBM notes that representative Orbit results depend on the device, calibration state, circuit, and execution settings. An isolated fidelity number is therefore not a timeless property of a QFT design or a hardware provider.
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For a hardware run, record the backend, qubit count, compiled depth and gate counts, shot count, and the interval your timing covers. These are distinct measures and should not be presented as interchangeable:
- Circuit execution time: the time for the compiled workload to run on the device.
- Execution including measurement and control: specify whether measurement, reset, or classical control is included.
- Total job elapsed time: state whether submission and queueing are part of the clock.
- Throughput: a platform-level rate, not necessarily the duration of your particular QFT.
IBM’s QPU guide defines maximum circuits per second (MCPS) around a circuit that includes measurement, reset, and reinitialization. MCPS is a platform metric; do not treat it as the measured execution time of a specific QFT.
Disclose QFT choices that change resources or results
The standard QFT construction uses Hadamard gates and controlled phase rotations, with an optional final swap layer. Qiskit’s QFT documentation notes that final swaps may be omitted when the QFT is at the end of a circuit and output bit reordering is handled classically. If swaps are omitted, explain how you interpret or reorder measured bits.
An approximate QFT can omit small controlled-phase rotations to reduce resources such as circuit depth, but it is no longer the same implementation as an exact QFT. For a fair comparison, hold the swap and approximation settings constant or disclose the differences, and report the resulting compiled resources.
Use a like-for-like comparison checklist
When comparing QFT implementations or devices, align the task and protocol first, then record the following for each result:
Best Value
- Circuit variant: unitary or dynamic; exact or approximate.
- Qubit count, logical task, and tested input states.
- Backend, calibration timestamp, topology, connectivity, and qubit mapping.
- Compiled one- and two-qubit gate counts and circuit depth.
- Fidelity or error estimator, ideal reference, inputs, and shot count.
- Readout and gate-error context, plus any mitigation or suppression settings.
- Timing boundary and whether the reported speed is execution time, job elapsed time, or throughput.
IBM’s Orbit tutorial compares unitary and dynamic QFT-plus-measurement variants using sampled process-fidelity estimates, illustrating why the circuit definition and measurement protocol need to accompany the result. A 2024 paper, Quantum Fourier Transform using Dynamic Circuits, reports certified process fidelities above 50% up to 16 qubits and above 1% up to 37 qubits on IBM superconducting hardware. Those are results for the authors’ protocol and hardware, not expected values for arbitrary circuits or current backends. The paper also reports that, for QFT followed immediately by measurement, its standard unitary formulation uses O(n²) two-qubit gates under all-to-all connectivity, while its dynamic counterpart uses O(n) mid-circuit measurements without connectivity constraints. Those scaling claims apply to the studied formulations and task, not to every QFT use.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.If you mean a classical FFT in software
Classical FFT benchmarking measures numerical discrepancy and runtime, not quantum hardware noise or process fidelity. For accuracy, compare outputs against an appropriately precise reference and report normalized error measures. benchFFT’s accuracy methodology compares FFT results with an arbitrary-precision FFT and reports normalized L1, L2, and maximum-norm errors. These quantify numerical output error.
For timing, separate initialization or planning from repeated transforms. FFTW’s benchmark methodology batches repeated transforms until timing is accurate, repeats the averaging process eight times, and reports the minimum average; it treats initialization separately. Input/output formats must be comparable for a fair benchmark. Its performance-scaling measure is useful for comparison, but is not a literal operation count.
Quick Recap
- Compare the same transform size and type, precision, data layout, and compiler/build settings.
- State whether plans or initialization are included, and whether any warm-up is part of the method.
- Report repeated execution time separately from setup and pair it with normalized error against the reference.
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