Phased discrete sinc state

As explained in Building Quantum Software with Python

A phased discrete sinc state is a quantum state whose computational-basis amplitudes follow a discretely sampled sinc profile with associated phase progression.

A phased discrete sinc state is produced when the inverse quantum Fourier transform (IQFT) acts on a geometric sequence state. Its computational-basis amplitude magnitudes follow a discretely sampled sinc profile; non-integer encoded frequencies add a phase progression. section 8.2.2 section 8.5

Read the measurement distribution through the encoded frequency v:

• For an integer v, the matching basis outcome has magnitude 1 and every other outcome has magnitude 0. section 8.5
• For a non-integer v, the two nearest integer outcomes have the largest magnitudes, while their amplitudes retain phase information. section 8.2.2 section 8.5
• With n qubits, evaluate outcomes k from 0 through 2^n − 1 and compare amplitudes with products of cosine terms. section 8.2.2

Compute and verify the state

For n = 3 and v = 4.3, construct the amplitudes directly and compare them with an IQFT-generated state: "def phased_sincd(n, v):\n N = 2 ** n\n return [prod(\n cos((v - k) * pi / 2 ** (j + 1)) * cis((v - k) * pi / 2 ** (j + 1))\n for j in range(n)) for k in range(N)]" The equivalent combined form separates the cosine product from a single complex phase and produces the same state. section 8.2.2

Figure 8.18 The general form of a phased discrete sinc state with n qubits, where 0 ≤ k < 2ⁿ, and an encoded frequency v, where 0 ≤ v < 2ⁿ

Use the general amplitude form to inspect how n, k, and v determine each basis amplitude. section 8.2.2

Reversing qubit order changes the amplitude indexing and lets the IQFT omit final swaps, while preserving the resulting phased discrete sinc state. section 8.2.3

Quantum phase estimation uses a single phased discrete sinc state to estimate an encoded frequency; without eigenstate uncomputation, it produces a superposition of such states. section 9.3

How Building Quantum Software with Python covers it

How deep it goes

Derives product and complex-phase forms, implements them in Python, verifies equivalence, and defers broader applications to later algorithms.

Examples

Demonstrates three-qubit states with v = 4.3, v = 4.7, product-of-cosines checks, and reversed-qubit circuits without swaps.

What it assumes

Assumes familiarity with qubits, amplitudes, geometric sequence states, and the IQFT while relating the pattern to the sinc function.

What you can do after

Build it from scratch with phased_sincd, verify amplitudes against an IQFT circuit, and interpret measurements as frequency estimates.

How it connects

Connects the discrete sinc function, geometric sequence states, the quantum Fourier transform, frequency estimation, and quantum phase estimation.

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Related concepts

• quantum phase estimationproduced by · Quantum phase estimation is a quantum algorithm that estimates the eigenphase of a unitary operator given a corresponding eigenstate.
• quantum Fourier transformproduced by · The quantum Fourier transform is a quantum circuit implementing the discrete Fourier transform on the amplitudes of a quantum state.
• frequencyuses · Frequency is the count or rate at which an item, event, outcome, or periodic cycle occurs within a specified collection or interval.
• inverse quantum Fourier transformproduced by · The inverse quantum Fourier transform is the inverse unitary transformation of the quantum Fourier transform, mapping encoded phase information toward computational-basis information.

This explainer was generated by Manning's knowledge graph from the text of Building Quantum Software with Python. The section links point to the parts of the book it draws on.