Quantum Fourier Transform: QFT and Inverse QFT

The Quantum Fourier Transform (QFT) is the quantum analogue of the discrete Fourier transform. It maps computational basis states to the Fourier basis: QFT|j⟩ = (1/√2^n) Σ_k exp(2πijk/2^n)|k⟩. In practice you use the QFT — almost always its inverse — to read out a phase from a quantum register. It is the final step in Quantum Phase Estimation, and it appears in Shor’s order-finding circuit and in any algorithm that encodes information in the eigenvalue of a unitary.

Imports

from b01t.zoo.qft import qft, inverse_qft

Both functions are @parametric because they use continuous rotation gates (CRZ, Rz). This means you can call them from inside other @parametric or @adaptive functions, but not from @coherent bodies (which require exact/permutation gates only).

`qft`

qft(reg: QReg)

Applies the Quantum Fourier Transform to the register reg in-place. Processes qubits from LSB to MSB without a final bit-reversal swap. This ordering matches the QPE convention, where counting[k] controls U^{2^k}, so the inverse QFT can decode the phase directly without additional permutations. Gate decomposition: For each qubit j, applies H(reg[j]) followed by controlled-phase gates CP(π/2^(k−j)) from qubit k to qubit j, for all k > j. Each CP(θ) is decomposed as CRZ(θ, ctrl, tgt) followed by Rz(θ/2, ctrl).

`inverse_qft`

inverse_qft(reg: QReg)

Applies the inverse QFT — the adjoint of qft — to the register reg in-place. Use this at the end of a QPE circuit to convert phase information in the counting register into a readable binary value.

Building a circuit with QFT

Import and define a register

from b01t import QReg, parametric, h
from b01t.zoo.qft import qft, inverse_qft

Apply the QFT inside a @parametric function

@parametric
def prepare_fourier_basis(reg: QReg):
    """Put each qubit in superposition then apply QFT."""
    for q in reg:
        h(q)
    qft(reg)

Build and compile to Qiskit

from b01t import QiskitBackend

prog = prepare_fourier_basis.build(("reg", 4))
qc = QiskitBackend().emit(prog)
print(qc.draw())

Using inverse_qft in a QPE circuit

The most common use of inverse_qft is as the final step of Quantum Phase Estimation. b01t’s zoo.qpe module wraps this pattern for you, but you can also build it directly:

from b01t import QReg, parametric, h
from b01t.zoo.qft import inverse_qft


def my_controlled_unitary(counting: QReg, work: QReg):
    """Apply controlled-U^(2^k) for each counting bit k."""
    ...


@parametric
def qpe(counting: QReg, work: QReg):
    # Step 1: uniform superposition on counting register
    for i in range(len(counting)):
        h(counting[i])

    # Step 2: controlled unitaries (your implementation)
    my_controlled_unitary(counting, work)

    # Step 3: inverse QFT decodes the phase into the counting register
    inverse_qft(counting)

Or use make_qpe from zoo.qpe directly:

from b01t.zoo.qpe import make_qpe

qpe = make_qpe(my_controlled_unitary)
# qpe(counting, work) does all three steps automatically

Round-trip example

You can verify that inverse_qft(qft(reg)) is the identity by building the round-trip circuit and checking the qubit count:

from b01t import QReg, parametric
from b01t.zoo.qft import qft, inverse_qft
from b01t import QiskitBackend

@parametric
def roundtrip(reg: QReg):
    qft(reg)
    inverse_qft(reg)

prog = roundtrip.build(("reg", 3))
qc = QiskitBackend().emit(prog)
print(f"Qubits: {qc.num_qubits}, Gates: {qc.size()}")

The b01t QFT does not include bit-reversal swaps. This is deliberate: it matches the QPE convention where counting[k] controls U^{2^k}. If you need the standard DFT bit ordering for a different application, you will need to add the swap layer manually.

Gate count for the QFT on n qubits is n + n*(n-1)/2 two-qubit gates, which grows as O(n²). For large registers this can be significant — consider approximate QFT (truncating small rotation angles) if circuit depth is a constraint.

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Quantum Fourier Transform: QFT and Inverse QFT

Imports

`qft`

`inverse_qft`

Building a circuit with QFT

Using inverse_qft in a QPE circuit

Round-trip example

​Imports

​qft

​inverse_qft

​Building a circuit with QFT

​Using inverse_qft in a QPE circuit

​Round-trip example

Imports

`qft`

`inverse_qft`

Building a circuit with QFT

Using inverse_qft in a QPE circuit

Round-trip example