[
 {
  "slug": "clifford",
  "title": "Clifford Group",
  "description": "Unitaries that map Pauli operators to Pauli operators: efficiently simulable, and the backbone of error correction.",
  "gates": [
   "cnot",
   "cy",
   "cz",
   "dcx",
   "ecr",
   "fswap",
   "hadamard",
   "identity",
   "iswap",
   "magic",
   "pauli-x",
   "pauli-y",
   "pauli-z",
   "s",
   "swap",
   "sx"
  ],
  "url": "/groups/clifford/"
 },
 {
  "slug": "diagonal",
  "title": "Diagonal Group",
  "description": "Gates that only attach a phase to each computational basis state, so they all commute with one another.",
  "gates": [
   "ccz",
   "clock",
   "controlled-clock",
   "controlled-phase",
   "crz",
   "cs",
   "cz",
   "global-phase",
   "identity",
   "pauli-z",
   "phase-oracle",
   "phase",
   "rz",
   "rzz",
   "s",
   "t"
  ],
  "url": "/groups/diagonal/"
 },
 {
  "slug": "matchgate",
  "title": "Matchgate Group",
  "description": "Two-qubit gates that act separately on the even- and odd-parity subspaces with matching determinants: free-fermion evolutions.",
  "gates": [
   "b",
   "fswap",
   "givens",
   "iswap",
   "rxx",
   "ryy",
   "sqrt-iswap",
   "xx-minus-yy",
   "xy"
  ],
  "url": "/groups/matchgate/"
 },
 {
  "slug": "number-preserving",
  "title": "Number-Preserving Group",
  "description": "Gates that conserve the number of excitations, moving them around and adjusting phases without creating or destroying any.",
  "gates": [
   "ccz",
   "clock",
   "controlled-clock",
   "controlled-phase",
   "crz",
   "cs",
   "cswap",
   "cz",
   "fsim",
   "fswap",
   "givens",
   "global-phase",
   "identity",
   "iswap",
   "pauli-z",
   "phase-oracle",
   "phase",
   "rz",
   "rzz",
   "s",
   "sqrt-iswap",
   "sqrt-swap",
   "swap",
   "sycamore",
   "t",
   "xy"
  ],
  "url": "/groups/number-preserving/"
 },
 {
  "slug": "orthogonal",
  "title": "Orthogonal Group",
  "description": "The real unitaries. Circuits built from them keep every amplitude real, and that costs surprisingly little.",
  "gates": [
   "ccz",
   "ch",
   "cnot",
   "controlled-shift",
   "cry",
   "cswap",
   "cz",
   "dcx",
   "fswap",
   "givens",
   "grover-diffuser",
   "hadamard",
   "identity",
   "mcx",
   "oracle",
   "pauli-x",
   "pauli-z",
   "phase-oracle",
   "rccx",
   "ry",
   "shift",
   "swap",
   "toff-decomp",
   "toffoli",
   "virtual-distillation"
  ],
  "url": "/groups/orthogonal/"
 },
 {
  "slug": "pauli",
  "title": "Pauli Group",
  "description": "Bit flip, phase flip and their product, together with the phases that close them under multiplication.",
  "gates": [
   "identity",
   "pauli-x",
   "pauli-y",
   "pauli-z"
  ],
  "url": "/groups/pauli/"
 },
 {
  "slug": "permutation",
  "title": "Permutation Group",
  "description": "Gates that shuffle computational basis states: classical reversible logic inside quantum computing.",
  "gates": [
   "cnot",
   "controlled-shift",
   "cswap",
   "dcx",
   "identity",
   "mcx",
   "oracle",
   "pauli-x",
   "shift",
   "swap",
   "toffoli"
  ],
  "url": "/groups/permutation/"
 },
 {
  "slug": "unitary",
  "title": "Unitary Group",
  "description": "Every gate. The group of all unitary matrices, which is exactly what closed-system quantum evolution allows.",
  "gates": [
   "b",
   "can",
   "ccz",
   "ch",
   "chrestenson",
   "clock",
   "cnot",
   "controlled-clock",
   "controlled-phase",
   "controlled-shift",
   "crx",
   "cry",
   "crz",
   "cs",
   "cswap",
   "csx",
   "cy",
   "cz",
   "dcx",
   "deutsch",
   "ecr",
   "fsim",
   "fswap",
   "givens",
   "global-phase",
   "gpi",
   "gpi2",
   "grover-diffuser",
   "hadamard",
   "identity",
   "iswap",
   "magic",
   "mcx",
   "molmer-sorenson",
   "oracle",
   "pauli-x",
   "pauli-y",
   "pauli-z",
   "phase-oracle",
   "phase",
   "qft",
   "r",
   "rccx",
   "rot",
   "rx",
   "rxx",
   "ry",
   "ryy",
   "rz",
   "rzx",
   "rzz",
   "s",
   "shift",
   "sqrt-iswap",
   "sqrt-swap",
   "swap",
   "sx",
   "sycamore",
   "t",
   "toff-decomp",
   "toffoli",
   "u2",
   "u3",
   "virtual-distillation",
   "xx-minus-yy",
   "xy"
  ],
  "url": "/groups/unitary/"
 }
]