{
 "slug": "qft",
 "title": "Quantum Fourier Transform",
 "symbol": "\\mathrm{QFT}_n",
 "aliases": [
  "qft"
 ],
 "notations": [],
 "description": "Applies the discrete Fourier transform to the amplitudes of an $n$-qubit state.",
 "arity": "n",
 "dimension": 2,
 "matrix": [
  [
   "1/sqrt(8)",
   "1/sqrt(8)",
   "1/sqrt(8)",
   "1/sqrt(8)",
   "1/sqrt(8)",
   "1/sqrt(8)",
   "1/sqrt(8)",
   "1/sqrt(8)"
  ],
  [
   "1/sqrt(8)",
   "exp(2 pi i*1/8)/sqrt(8)",
   "exp(2 pi i*2/8)/sqrt(8)",
   "exp(2 pi i*3/8)/sqrt(8)",
   "exp(2 pi i*4/8)/sqrt(8)",
   "exp(2 pi i*5/8)/sqrt(8)",
   "exp(2 pi i*6/8)/sqrt(8)",
   "exp(2 pi i*7/8)/sqrt(8)"
  ],
  [
   "1/sqrt(8)",
   "exp(2 pi i*2/8)/sqrt(8)",
   "exp(2 pi i*4/8)/sqrt(8)",
   "exp(2 pi i*6/8)/sqrt(8)",
   "1/sqrt(8)",
   "exp(2 pi i*2/8)/sqrt(8)",
   "exp(2 pi i*4/8)/sqrt(8)",
   "exp(2 pi i*6/8)/sqrt(8)"
  ],
  [
   "1/sqrt(8)",
   "exp(2 pi i*3/8)/sqrt(8)",
   "exp(2 pi i*6/8)/sqrt(8)",
   "exp(2 pi i*1/8)/sqrt(8)",
   "exp(2 pi i*4/8)/sqrt(8)",
   "exp(2 pi i*7/8)/sqrt(8)",
   "exp(2 pi i*2/8)/sqrt(8)",
   "exp(2 pi i*5/8)/sqrt(8)"
  ],
  [
   "1/sqrt(8)",
   "exp(2 pi i*4/8)/sqrt(8)",
   "1/sqrt(8)",
   "exp(2 pi i*4/8)/sqrt(8)",
   "1/sqrt(8)",
   "exp(2 pi i*4/8)/sqrt(8)",
   "1/sqrt(8)",
   "exp(2 pi i*4/8)/sqrt(8)"
  ],
  [
   "1/sqrt(8)",
   "exp(2 pi i*5/8)/sqrt(8)",
   "exp(2 pi i*2/8)/sqrt(8)",
   "exp(2 pi i*7/8)/sqrt(8)",
   "exp(2 pi i*4/8)/sqrt(8)",
   "exp(2 pi i*1/8)/sqrt(8)",
   "exp(2 pi i*6/8)/sqrt(8)",
   "exp(2 pi i*3/8)/sqrt(8)"
  ],
  [
   "1/sqrt(8)",
   "exp(2 pi i*6/8)/sqrt(8)",
   "exp(2 pi i*4/8)/sqrt(8)",
   "exp(2 pi i*2/8)/sqrt(8)",
   "1/sqrt(8)",
   "exp(2 pi i*6/8)/sqrt(8)",
   "exp(2 pi i*4/8)/sqrt(8)",
   "exp(2 pi i*2/8)/sqrt(8)"
  ],
  [
   "1/sqrt(8)",
   "exp(2 pi i*7/8)/sqrt(8)",
   "exp(2 pi i*6/8)/sqrt(8)",
   "exp(2 pi i*5/8)/sqrt(8)",
   "exp(2 pi i*4/8)/sqrt(8)",
   "exp(2 pi i*3/8)/sqrt(8)",
   "exp(2 pi i*2/8)/sqrt(8)",
   "exp(2 pi i*1/8)/sqrt(8)"
  ]
 ],
 "matrix_note": "n = 3",
 "groups": [],
 "properties": [],
 "url": "/gates/qft/"
}