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A low-noise quantum Fourier transform (QFT) circuit has not been shown by the evidence here to beat a classical digital fast Fourier transform (FFT) at computing the same classical input’s full spectrum. They do different jobs: an FFT calculates explicit frequency-domain values, while a QFT transforms the amplitudes of a quantum state. Quantum circuit improvements can matter inside a larger quantum algorithm, but gate counts or circuit depth alone do not establish an end-to-end speed advantage.
What does a QFT do that an FFT does not?
A classical FFT is an efficient way to calculate the discrete Fourier transform of a classical sequence. For N samples, its familiar computational cost is on the order of N log N, rather than the direct DFT’s quadratic work. The result is a list of classical Fourier coefficients that can be inspected, stored, or used in later calculations.
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A QFT applies the corresponding transform to the amplitudes of a quantum state. For example, a state can encode amplitudes across computational basis states, and a QFT changes those amplitudes. Its output remains a quantum state: measuring it does not reveal every transformed amplitude. QFTs are useful as subroutines in algorithms such as phase estimation and Shor’s algorithm, where the transformed state is used as part of a larger computation.
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One free scan finds every outdated or missing driver and matches the right update for your exact hardware.Free scan · exact hardware matchThere is also a distinct construction: a reversible quantum circuit that processes classical data encoded in basis states. That is not the same operation model as a QFT over amplitudes. A 2020 circuit paper discusses this distinction and flags data encoding and readout as important costs and challenges for quantum processing of classical data.
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So an apparent comparison between an O(n2)-gate QFT on n qubits and an O(N log N) FFT on N explicit samples is not, by itself, a comparison of the same workload. A quantum state on n qubits has a space of 2n basis states, but preparing the relevant state, using the output, and extracting information all affect the value of the circuit.
What do low-noise and resource-efficient QFT results show?
These results concern the cost or fidelity of quantum circuits under specified designs and assumptions. They are meaningful for quantum algorithms, but they are not direct runtime comparisons against classical FFT software.
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Digital-analog circuits and noise
A 2020 Physical Review Research paper proposed a digital-analog QFT algorithm. Under the paper’s stated noise-model assumptions, its authors reported that fidelity improved considerably as the number of qubits grew. That is a conditional result for the proposed architecture and noise models, not a general ranking of quantum hardware.
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A 2024 Communications Physics study compared digital and digital-analog approaches to QFT and phase estimation under single- and two-qubit noise sources on superconducting-processor models. In that study’s setup, the authors reported that digital-analog circuits consistently achieved higher fidelity than the digital approaches. They also reported fidelities above 0.95 for 8 qubits with zero-noise extrapolation, which reduced computation errors to the order of 10−3. Those figures describe the study’s conditions; they should not be read as general performance figures for current processors.
Dynamic circuits when measurement follows the QFT
A standard unitary QFT circuit has O(n2) two-qubit-gate scaling. When the QFT is followed immediately by measurement, a dynamic formulation can use mid-circuit measurements and classical feed-forward instead. A 2024 Physical Review Letters paper by Bäumer and colleagues reported superconducting-hardware demonstrations of this approach, with certified process-fidelity results up to 16 qubits and demonstrations up to 37 qubits. The two qubit counts describe different claims: the certified-fidelity result reached 16 qubits, while the demonstrations reached 37. Neither is a classical FFT timing benchmark.
Approximate fault-tolerant circuits
An approximate QFT can omit sufficiently small rotations to reduce circuit resources, at the cost of approximation error. A 2020 npj Quantum Information paper reported an approximate construction with T-count O(n log n) for fixed approximation error, improving on the standard O(n log2 n) approach discussed in that paper. The stated asymptotic count fixes the error and omits its dependence; practical cost also depends on the error target, gate synthesis, connectivity, and implementation choices. T-count is a resource measure, not a wall-clock runtime.
Depth bounds are not runtime results
A 2026 review reports an approximate-QFT depth upper bound of O(log n + log log(1/ε)) and an Ω(log n) lower bound for constant error. These are circuit-complexity bounds under their stated mathematical framing. They do not show how quickly a particular device completes an end-to-end task, and they do not mean the QFT produces the full classical spectrum.
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| Reader’s task | Relevant approach | What the result provides |
|---|---|---|
| Calculate the Fourier coefficients of a classical signal and inspect the full spectrum | Classical FFT | Explicit classical frequency-domain values |
| Transform amplitudes within a quantum algorithm | QFT circuit | A transformed quantum state used by later algorithm steps; not a printed list of all amplitudes |
| Reversibly process classically encoded data in a quantum circuit | Quantum FFT-style construction | A separate circuit model whose encoding and readout costs must be included |
If the desired output is every coefficient of an ordinary classical signal, use a classical FFT as the baseline. A quantum comparison is meaningful only if it solves the same problem and accounts for preparing or accessing the data and obtaining the requested output.
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How to judge a claimed speed advantage
For a fair comparison, first state the task and the output precisely. Then compare total cost rather than selecting a favorable circuit metric in isolation.
- Match the input and output: Say whether the input is a classical sequence, a prepared quantum state, or classical data encoded into a quantum circuit. Specify whether the output must include every Fourier coefficient or only information used by a quantum algorithm.
- Name the resource being measured: Wall-clock runtime, circuit depth, two-qubit-gate count, T-count, qubit count, and measurement count answer different questions. Do not treat one as a substitute for another.
- State the accuracy target: Give the error definition and target ε. For approximate circuits, say which rotations are omitted and include gate-synthesis error where relevant.
- Specify the hardware assumptions: Connectivity, native gates, noise channels, calibration, mid-circuit measurement and feed-forward quality, and any assumed error correction can change the outcome.
- Include data movement and repetition: Count state preparation or data encoding, circuit executions, error mitigation or correction, measurements, and classical post-processing needed to obtain the result.
- Identify the evidence type: A complexity bound, a noise-model study, a hardware demonstration, and a same-workload wall-clock benchmark are different kinds of evidence. Only the last directly times a matched end-to-end workload.
What is the practical verdict?
Low-noise, dynamic, and approximate QFT designs can improve quantum-circuit performance for particular algorithms, error targets, and hardware conditions. The cited studies establish circuit-level advances and conditional fidelity results, not that a quantum computer running a QFT is faster end to end than a digital system running an FFT on the same classical data and returning the same explicit spectrum.
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