dorsal/arxiv
View SchemaQuantum Wavelet Transforms: Fast Algorithms and Complete Circuits
| Authors | Amir Fijany, Colin P. Williams |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9809004 |
| URL | https://arxiv.org/abs/quant-ph/9809004 |
Abstract
The quantum Fourier transform (QFT), a quantum analog of the classical Fourier transform, has been shown to be a powerful tool in developing quantum algorithms. However, in classical computing there is another class of unitary transforms, the wavelet transforms, which are every bit as useful as the Fourier transform. Wavelet transforms are used to expose the multi-scale structure of a signal and are likely to be useful for quantum image processing and quantum data compression. In this paper, we derive efficient, complete, quantum circuits for two representative quantum wavelet transforms, the quantum Haar and quantum Daubechies $D^{(4)}$ transforms. Our approach is to factor the operators for these transforms into direct sums, direct products and dot products of unitary matrices. In so doing, we find that permutation matrices, a particular class of unitary matrices, play a pivotal role. Surprisingly, we find that operations that are easy and inexpensive to implement classically are not always easy and inexpensive to implement quantum mechanically, and vice versa. In particular, the computational cost of performing certain permutation matrices is ignored classically because they can be avoided explicitly. However, quantum mechanically, these permutation operations must be performed explicitly and hence their cost enters into the full complexity measure of the quantum transform. We consider the particular set of permutation matrices arising in quantum wavelet transforms and develop efficient quantum circuits that implement them. This allows us to design efficient, complete quantum circuits for the quantum wavelet transform.
{
"annotation_id": "594855b3-e99b-4e13-840e-fb927334414d",
"date_created": "2026-03-02T18:02:45.254000Z",
"date_modified": "2026-03-02T18:02:45.254000Z",
"file_hash": "5a3705d712450f2a900e8b9c8f550074c9260477085a2e852fe3fd0b976c2211",
"private": false,
"record": {
"abstract": "The quantum Fourier transform (QFT), a quantum analog of the classical\nFourier transform, has been shown to be a powerful tool in developing quantum\nalgorithms. However, in classical computing there is another class of unitary\ntransforms, the wavelet transforms, which are every bit as useful as the\nFourier transform. Wavelet transforms are used to expose the multi-scale\nstructure of a signal and are likely to be useful for quantum image processing\nand quantum data compression. In this paper, we derive efficient, complete,\nquantum circuits for two representative quantum wavelet transforms, the quantum\nHaar and quantum Daubechies $D^{(4)}$ transforms. Our approach is to factor the\noperators for these transforms into direct sums, direct products and dot\nproducts of unitary matrices. In so doing, we find that permutation matrices, a\nparticular class of unitary matrices, play a pivotal role. Surprisingly, we\nfind that operations that are easy and inexpensive to implement classically are\nnot always easy and inexpensive to implement quantum mechanically, and vice\nversa. In particular, the computational cost of performing certain permutation\nmatrices is ignored classically because they can be avoided explicitly.\nHowever, quantum mechanically, these permutation operations must be performed\nexplicitly and hence their cost enters into the full complexity measure of the\nquantum transform. We consider the particular set of permutation matrices\narising in quantum wavelet transforms and develop efficient quantum circuits\nthat implement them. This allows us to design efficient, complete quantum\ncircuits for the quantum wavelet transform.",
"arxiv_id": "quant-ph/9809004",
"authors": [
"Amir Fijany",
"Colin P. Williams"
],
"categories": [
"quant-ph"
],
"title": "Quantum Wavelet Transforms: Fast Algorithms and Complete Circuits",
"url": "https://arxiv.org/abs/quant-ph/9809004"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "214929e2-64cd-401a-a735-8abc24170424",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}