[
 {
  "slug": "pauli-x",
  "title": "Pauli-X",
  "symbol": "X",
  "aliases": [
   "x"
  ],
  "notations": [
   "X",
   "\\sigma_x",
   "\\sigma^x",
   "\\sigma_1",
   "\\mathrm{NOT}"
  ],
  "description": "Bit-flip gate that swaps $|0\\rangle$ and $|1\\rangle$.",
  "arity": 1,
  "dimension": 2,
  "matrix": [
   [
    0,
    1
   ],
   [
    1,
    0
   ]
  ],
  "groups": [
   "pauli",
   "clifford",
   "orthogonal",
   "permutation"
  ],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/pauli-x/"
 },
 {
  "slug": "pauli-y",
  "title": "Pauli-Y",
  "symbol": "Y",
  "aliases": [
   "y"
  ],
  "notations": [
   "Y",
   "\\sigma_y",
   "\\sigma^y",
   "\\sigma_2"
  ],
  "description": "Bit-and-phase flip combining $X$ and $Z$ with a phase.",
  "arity": 1,
  "dimension": 2,
  "matrix": [
   [
    0,
    "-i"
   ],
   [
    "i",
    0
   ]
  ],
  "groups": [
   "pauli",
   "clifford"
  ],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/pauli-y/"
 },
 {
  "slug": "pauli-z",
  "title": "Pauli-Z",
  "symbol": "Z",
  "aliases": [
   "z"
  ],
  "notations": [
   "Z",
   "\\sigma_z",
   "\\sigma^z",
   "\\sigma_3"
  ],
  "description": "Phase-flip gate that leaves $|0\\rangle$ unchanged and flips the phase of $|1\\rangle$.",
  "arity": 1,
  "dimension": 2,
  "matrix": [
   [
    1,
    0
   ],
   [
    0,
    -1
   ]
  ],
  "groups": [
   "pauli",
   "clifford",
   "diagonal",
   "orthogonal",
   "number-preserving"
  ],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/pauli-z/"
 },
 {
  "slug": "identity",
  "title": "Identity",
  "symbol": "I",
  "aliases": [
   "i"
  ],
  "notations": [
   "I",
   "I_n",
   "1_n",
   "\\mathbb{1}",
   "\\mathrm{Id}"
  ],
  "description": "Leaves the quantum state unchanged.",
  "arity": "n",
  "dimension": 2,
  "matrix": [
   [
    1,
    0
   ],
   [
    0,
    1
   ]
  ],
  "groups": [
   "pauli",
   "clifford",
   "diagonal",
   "orthogonal",
   "permutation",
   "number-preserving"
  ],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/identity/"
 },
 {
  "slug": "hadamard",
  "title": "Hadamard",
  "symbol": "H",
  "aliases": [
   "h"
  ],
  "notations": [],
  "description": "Maps computational basis states to equal superpositions and swaps the $X$ and $Z$ bases.",
  "arity": 1,
  "dimension": 2,
  "matrix": [
   [
    "1/sqrt(2)",
    "1/sqrt(2)"
   ],
   [
    "1/sqrt(2)",
    "-1/sqrt(2)"
   ]
  ],
  "groups": [
   "clifford",
   "orthogonal"
  ],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/hadamard/"
 },
 {
  "slug": "sx",
  "title": "Square Root of X",
  "symbol": "\\sqrt{X}",
  "aliases": [
   "sx",
   "sqrt-x",
   "sqrtx"
  ],
  "notations": [
   "\\sqrt{X}",
   "\\mathrm{S}X",
   "V"
  ],
  "description": "A gate whose square is the Pauli-$X$ (NOT) gate.",
  "arity": 1,
  "dimension": 2,
  "matrix": [
   [
    "(1+i)/2",
    "(1-i)/2"
   ],
   [
    "(1-i)/2",
    "(1+i)/2"
   ]
  ],
  "groups": [
   "clifford"
  ],
  "properties": [],
  "url": "/gates/sx/"
 },
 {
  "slug": "s",
  "title": "Phase",
  "symbol": "S",
  "aliases": [],
  "notations": [
   "S",
   "\\sqrt{Z}",
   "Z^{1/2}",
   "P(\\pi/2)",
   "K"
  ],
  "description": "$\\pi/2$ phase gate, equal to $\\sqrt{Z}$.",
  "arity": 1,
  "dimension": 2,
  "matrix": [
   [
    1,
    0
   ],
   [
    0,
    "i"
   ]
  ],
  "groups": [
   "clifford",
   "diagonal",
   "number-preserving"
  ],
  "properties": [],
  "url": "/gates/s/"
 },
 {
  "slug": "t",
  "title": "T",
  "symbol": "T",
  "aliases": [],
  "notations": [
   "\\pi/8"
  ],
  "description": "A $\\pi/8$ phase gate which is a non-Clifford rotation about $Z$.",
  "arity": 1,
  "dimension": 2,
  "matrix": [
   [
    1,
    0
   ],
   [
    0,
    "exp(i pi/4)"
   ]
  ],
  "groups": [
   "diagonal",
   "number-preserving"
  ],
  "properties": [],
  "url": "/gates/t/"
 },
 {
  "slug": "toff-decomp",
  "title": "unnamed",
  "symbol": "G",
  "aliases": [],
  "notations": [],
  "description": "A single-qubit rotation used in optimal Toffoli decompositions.",
  "arity": 1,
  "dimension": 2,
  "matrix": [
   [
    "cos(pi/8)",
    "-sin(pi/8)"
   ],
   [
    "sin(pi/8)",
    "cos(pi/8)"
   ]
  ],
  "groups": [
   "orthogonal"
  ],
  "properties": [],
  "url": "/gates/toff-decomp/"
 },
 {
  "slug": "u2",
  "title": "U2",
  "symbol": "U_2",
  "aliases": [
   "u2"
  ],
  "notations": [
   "U_2(\\phi, \\lambda)",
   "\\mathrm{U2}(\\phi, \\lambda)"
  ],
  "description": "Legacy IBM gate performing a fixed $\\pi/2$ rotation dressed by two $z$ rotations, $U_2(\\phi, \\lambda) = U_3(\\pi/2, \\phi, \\lambda)$.",
  "arity": 1,
  "dimension": 2,
  "params": {
   "phi": {
    "label": "\\phi",
    "default": "1/2",
    "range": [
     0,
     2
    ]
   },
   "lambda": {
    "label": "\\lambda",
    "default": "1/4",
    "range": [
     0,
     4
    ]
   }
  },
  "matrix": [
   [
    "1/sqrt(2)",
    "-exp(i lambda)/sqrt(2)"
   ],
   [
    "exp(i phi)/sqrt(2)",
    "exp(i (phi+lambda))/sqrt(2)"
   ]
  ],
  "groups": [],
  "properties": [],
  "url": "/gates/u2/"
 },
 {
  "slug": "rx",
  "title": "x Rotation",
  "symbol": "R_x",
  "aliases": [],
  "notations": [
   "R_x(\\theta)",
   "\\mathrm{Rx}(\\theta)",
   "X(\\theta)",
   "X_\\theta"
  ],
  "description": "Rotation about the $x$ axis of the Bloch sphere.",
  "arity": 1,
  "dimension": 2,
  "params": {
   "theta": {
    "label": "\\theta",
    "default": "1/2",
    "range": [
     0,
     2
    ]
   }
  },
  "matrix": [
   [
    "cos(theta/2)",
    "-i sin(theta/2)"
   ],
   [
    "-i sin(theta/2)",
    "cos(theta/2)"
   ]
  ],
  "groups": [],
  "properties": [],
  "url": "/gates/rx/"
 },
 {
  "slug": "ry",
  "title": "y Rotation",
  "symbol": "R_y",
  "aliases": [],
  "notations": [
   "R_y(\\theta)",
   "\\mathrm{Ry}(\\theta)",
   "Y(\\theta)",
   "Y_\\theta"
  ],
  "description": "Rotation about the $y$ axis of the Bloch sphere.",
  "arity": 1,
  "dimension": 2,
  "params": {
   "theta": {
    "label": "\\theta",
    "default": "1/2",
    "range": [
     0,
     2
    ]
   }
  },
  "matrix": [
   [
    "cos(theta/2)",
    "-sin(theta/2)"
   ],
   [
    "sin(theta/2)",
    "cos(theta/2)"
   ]
  ],
  "groups": [
   "orthogonal"
  ],
  "properties": [],
  "url": "/gates/ry/"
 },
 {
  "slug": "rz",
  "title": "z Rotation",
  "symbol": "R_z",
  "aliases": [],
  "notations": [
   "R_z(\\theta)",
   "\\mathrm{Rz}(\\theta)",
   "Z(\\theta)",
   "Z_\\theta"
  ],
  "description": "Rotation about the $z$ axis of the Bloch sphere.",
  "arity": 1,
  "dimension": 2,
  "params": {
   "theta": {
    "label": "\\theta",
    "default": "1/2",
    "range": [
     0,
     2
    ]
   }
  },
  "matrix": [
   [
    "exp(-i theta/2)",
    0
   ],
   [
    0,
    "exp(i theta/2)"
   ]
  ],
  "groups": [
   "diagonal",
   "number-preserving"
  ],
  "properties": [],
  "url": "/gates/rz/"
 },
 {
  "slug": "phase",
  "title": "Phase Shift",
  "symbol": "P",
  "aliases": [
   "phase",
   "p",
   "u1"
  ],
  "notations": [
   "U_1(\\phi)"
  ],
  "description": "Applies a relative phase $e^{i\\phi}$ to the $|1\\rangle$ component.",
  "arity": 1,
  "dimension": 2,
  "params": {
   "phi": {
    "label": "\\phi",
    "default": "1/4",
    "range": [
     0,
     2
    ]
   }
  },
  "matrix": [
   [
    1,
    0
   ],
   [
    0,
    "exp(i phi)"
   ]
  ],
  "groups": [
   "diagonal",
   "number-preserving"
  ],
  "properties": [],
  "url": "/gates/phase/"
 },
 {
  "slug": "r",
  "title": "Phased Rotation",
  "symbol": "R(\\theta, \\phi)",
  "aliases": [
   "r",
   "phased-x"
  ],
  "notations": [
   "R(\\theta, \\phi)",
   "R_\\phi(\\theta)",
   "\\mathrm{PhasedX}"
  ],
  "description": "Rotation by $\\theta$ about the equatorial Bloch-sphere axis at azimuthal angle $\\phi$.",
  "arity": 1,
  "dimension": 2,
  "params": {
   "theta": {
    "label": "\\theta",
    "default": "1/2",
    "range": [
     0,
     2
    ]
   },
   "phi": {
    "label": "\\phi",
    "default": "1/4",
    "range": [
     0,
     4
    ]
   }
  },
  "matrix": [
   [
    "cos(theta/2)",
    "-i exp(-i phi) sin(theta/2)"
   ],
   [
    "-i exp(i phi) sin(theta/2)",
    "cos(theta/2)"
   ]
  ],
  "groups": [],
  "properties": [],
  "url": "/gates/r/"
 },
 {
  "slug": "rot",
  "title": "Arbitrary Rotation",
  "symbol": "R_{\\mathbf{n}}",
  "aliases": [
   "rot"
  ],
  "notations": [
   "\\mathrm{Rot}(\\vec{n}, \\theta)",
   "R_{\\vec{n}}(\\theta)"
  ],
  "description": "Rotation by angle $\\theta$ about the Bloch-sphere axis $\\vec{n}$.",
  "arity": 1,
  "dimension": 2,
  "params": {
   "theta": {
    "label": "\\theta",
    "default": "1/2",
    "range": [
     0,
     2
    ]
   },
   "alpha": {
    "label": "\\alpha",
    "default": "1/4",
    "range": [
     0,
     2
    ]
   },
   "beta": {
    "label": "\\beta",
    "default": "acos(1/sqrt(3))/pi",
    "range": [
     0,
     1
    ]
   }
  },
  "matrix": [
   [
    "cos(theta/2) - i sin(theta/2) cos(beta)",
    "-i sin(theta/2) sin(beta) exp(-i alpha)"
   ],
   [
    "-i sin(theta/2) sin(beta) exp(i alpha)",
    "cos(theta/2) + i sin(theta/2) cos(beta)"
   ]
  ],
  "groups": [],
  "properties": [],
  "url": "/gates/rot/"
 },
 {
  "slug": "u3",
  "title": "U3",
  "symbol": "U_3",
  "aliases": [
   "u3"
  ],
  "notations": [
   "U_3(\\theta, \\phi, \\lambda)",
   "\\mathrm{U3}(\\theta, \\phi, \\lambda)",
   "U(\\theta, \\phi, \\lambda)"
  ],
  "description": "General single-qubit gate parameterized by three Euler angles.",
  "arity": 1,
  "dimension": 2,
  "params": {
   "theta": {
    "label": "\\theta",
    "default": "1/3",
    "range": [
     0,
     2
    ]
   },
   "phi": {
    "label": "\\phi",
    "default": "1/4",
    "range": [
     0,
     4
    ]
   },
   "lambda": {
    "label": "\\lambda",
    "default": "1/6",
    "range": [
     0,
     6
    ]
   }
  },
  "matrix": [
   [
    "cos(theta/2)",
    "-exp(i lambda) sin(theta/2)"
   ],
   [
    "exp(i phi) sin(theta/2)",
    "exp(i (phi+lambda)) cos(theta/2)"
   ]
  ],
  "groups": [],
  "properties": [],
  "url": "/gates/u3/"
 },
 {
  "slug": "rxx",
  "title": "xx Rotation",
  "symbol": "R_{xx}",
  "aliases": [
   "XX"
  ],
  "notations": [
   "R_{xx}(\\theta)",
   "\\mathrm{RXX}(\\theta)",
   "XX(\\theta)"
  ],
  "description": "Two-qubit rotation generated by $X \\otimes X$.",
  "arity": 2,
  "dimension": 2,
  "params": {
   "theta": {
    "label": "\\theta",
    "default": "1/2",
    "range": [
     0,
     2
    ]
   }
  },
  "matrix": [
   [
    "cos(theta/2)",
    0,
    0,
    "-i sin(theta/2)"
   ],
   [
    0,
    "cos(theta/2)",
    "-i sin(theta/2)",
    0
   ],
   [
    0,
    "-i sin(theta/2)",
    "cos(theta/2)",
    0
   ],
   [
    "-i sin(theta/2)",
    0,
    0,
    "cos(theta/2)"
   ]
  ],
  "groups": [
   "matchgate"
  ],
  "properties": [],
  "url": "/gates/rxx/"
 },
 {
  "slug": "ryy",
  "title": "yy Rotation",
  "symbol": "R_{yy}",
  "aliases": [
   "YY"
  ],
  "notations": [
   "R_{yy}(\\theta)",
   "\\mathrm{RYY}(\\theta)",
   "YY(\\theta)"
  ],
  "description": "Two-qubit rotation generated by $Y \\otimes Y$.",
  "arity": 2,
  "dimension": 2,
  "params": {
   "theta": {
    "label": "\\theta",
    "default": "1/2",
    "range": [
     0,
     2
    ]
   }
  },
  "matrix": [
   [
    "cos(theta/2)",
    0,
    0,
    "i sin(theta/2)"
   ],
   [
    0,
    "cos(theta/2)",
    "-i sin(theta/2)",
    0
   ],
   [
    0,
    "-i sin(theta/2)",
    "cos(theta/2)",
    0
   ],
   [
    "i sin(theta/2)",
    0,
    0,
    "cos(theta/2)"
   ]
  ],
  "groups": [
   "matchgate"
  ],
  "properties": [],
  "url": "/gates/ryy/"
 },
 {
  "slug": "rzz",
  "title": "zz Rotation",
  "symbol": "R_{zz}",
  "aliases": [
   "ZZ"
  ],
  "notations": [
   "R_{zz}(\\theta)",
   "\\mathrm{RZZ}(\\theta)",
   "ZZ(\\theta)"
  ],
  "description": "Two-qubit rotation generated by $Z \\otimes Z$.",
  "arity": 2,
  "dimension": 2,
  "params": {
   "theta": {
    "label": "\\theta",
    "default": "1/2",
    "range": [
     0,
     2
    ]
   }
  },
  "matrix": [
   [
    "exp(-i theta/2)",
    0,
    0,
    0
   ],
   [
    0,
    "exp(i theta/2)",
    0,
    0
   ],
   [
    0,
    0,
    "exp(i theta/2)",
    0
   ],
   [
    0,
    0,
    0,
    "exp(-i theta/2)"
   ]
  ],
  "groups": [
   "diagonal",
   "number-preserving"
  ],
  "properties": [],
  "url": "/gates/rzz/"
 },
 {
  "slug": "rzx",
  "title": "zx Rotation",
  "symbol": "R_{zx}",
  "aliases": [
   "rzx",
   "cross-resonance"
  ],
  "notations": [
   "R_{zx}(\\theta)",
   "\\mathrm{ZX}(\\theta)",
   "\\mathrm{CR}(\\theta)"
  ],
  "description": "Rotation generated by the $Z \\otimes X$ interaction, the native cross-resonance interaction of fixed-frequency superconducting qubits.",
  "arity": 2,
  "dimension": 2,
  "params": {
   "theta": {
    "label": "\\theta",
    "default": "1/2",
    "range": [
     0,
     2
    ]
   }
  },
  "matrix": [
   [
    "cos(theta/2)",
    "-i sin(theta/2)",
    0,
    0
   ],
   [
    "-i sin(theta/2)",
    "cos(theta/2)",
    0,
    0
   ],
   [
    0,
    0,
    "cos(theta/2)",
    "i sin(theta/2)"
   ],
   [
    0,
    0,
    "i sin(theta/2)",
    "cos(theta/2)"
   ]
  ],
  "groups": [],
  "properties": [],
  "url": "/gates/rzx/"
 },
 {
  "slug": "crx",
  "title": "Controlled x Rotation",
  "symbol": "\\mathrm{C}R_x",
  "aliases": [
   "crx"
  ],
  "notations": [
   "\\mathrm{CR}_x(\\theta)",
   "\\Lambda(R_x(\\theta))"
  ],
  "description": "Rotates the target about the $x$ axis by $\\theta$ when the control qubit is in the $|1\\rangle$ state.",
  "arity": 2,
  "dimension": 2,
  "params": {
   "theta": {
    "label": "\\theta",
    "default": "1",
    "range": [
     0,
     4
    ]
   }
  },
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    1,
    0,
    0
   ],
   [
    0,
    0,
    "cos(theta/2)",
    "-i sin(theta/2)"
   ],
   [
    0,
    0,
    "-i sin(theta/2)",
    "cos(theta/2)"
   ]
  ],
  "groups": [],
  "properties": [],
  "url": "/gates/crx/"
 },
 {
  "slug": "cry",
  "title": "Controlled y Rotation",
  "symbol": "\\mathrm{C}R_y",
  "aliases": [
   "cry"
  ],
  "notations": [
   "\\mathrm{CR}_y(\\theta)",
   "\\Lambda(R_y(\\theta))"
  ],
  "description": "Rotates the target about the $y$ axis by $\\theta$ when the control qubit is in the $|1\\rangle$ state.",
  "arity": 2,
  "dimension": 2,
  "params": {
   "theta": {
    "label": "\\theta",
    "default": "1",
    "range": [
     0,
     4
    ]
   }
  },
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    1,
    0,
    0
   ],
   [
    0,
    0,
    "cos(theta/2)",
    "-sin(theta/2)"
   ],
   [
    0,
    0,
    "sin(theta/2)",
    "cos(theta/2)"
   ]
  ],
  "groups": [
   "orthogonal"
  ],
  "properties": [],
  "url": "/gates/cry/"
 },
 {
  "slug": "crz",
  "title": "Controlled z Rotation",
  "symbol": "\\mathrm{C}R_z",
  "aliases": [
   "crz"
  ],
  "notations": [
   "\\mathrm{CR}_z(\\theta)",
   "\\Lambda(R_z(\\theta))"
  ],
  "description": "Rotates the target about the $z$ axis by $\\theta$ when the control qubit is in the $|1\\rangle$ state.",
  "arity": 2,
  "dimension": 2,
  "params": {
   "theta": {
    "label": "\\theta",
    "default": "1",
    "range": [
     0,
     4
    ]
   }
  },
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    1,
    0,
    0
   ],
   [
    0,
    0,
    "exp(-i theta/2)",
    0
   ],
   [
    0,
    0,
    0,
    "exp(i theta/2)"
   ]
  ],
  "groups": [
   "diagonal",
   "number-preserving"
  ],
  "properties": [],
  "url": "/gates/crz/"
 },
 {
  "slug": "cnot",
  "title": "Controlled Not",
  "symbol": "\\mathrm{C}X",
  "aliases": [
   "cx"
  ],
  "notations": [
   "\\mathrm{C}X",
   "\\mathrm{CNOT}",
   "\\mathrm{CNOT}_{ij}",
   "\\text{controlled-}X",
   "\\Lambda(X)",
   "\\mathrm{XOR}[i, j]"
  ],
  "description": "Flips the target qubit when the control qubit is in the $|1\\rangle$ state.",
  "arity": 2,
  "dimension": 2,
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    1,
    0,
    0
   ],
   [
    0,
    0,
    0,
    1
   ],
   [
    0,
    0,
    1,
    0
   ]
  ],
  "groups": [
   "clifford",
   "orthogonal",
   "permutation"
  ],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/cnot/"
 },
 {
  "slug": "cy",
  "title": "Controlled-Y",
  "symbol": "\\mathrm{C}Y",
  "aliases": [
   "cy"
  ],
  "notations": [
   "\\Lambda(Y)",
   "\\text{controlled-}Y"
  ],
  "description": "Applies $Y$ to the target qubit when the control qubit is in the $|1\\rangle$ state.",
  "arity": 2,
  "dimension": 2,
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    1,
    0,
    0
   ],
   [
    0,
    0,
    0,
    "-i"
   ],
   [
    0,
    0,
    "i",
    0
   ]
  ],
  "groups": [
   "clifford"
  ],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/cy/"
 },
 {
  "slug": "cz",
  "title": "Controlled-Z",
  "symbol": "\\mathrm{C}Z",
  "aliases": [
   "cz"
  ],
  "notations": [
   "\\Lambda(Z)",
   "\\text{controlled-}Z"
  ],
  "description": "Applies a $-1$ phase to $|11\\rangle$.",
  "arity": 2,
  "dimension": 2,
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    1,
    0,
    0
   ],
   [
    0,
    0,
    1,
    0
   ],
   [
    0,
    0,
    0,
    -1
   ]
  ],
  "groups": [
   "clifford",
   "diagonal",
   "orthogonal",
   "number-preserving"
  ],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/cz/"
 },
 {
  "slug": "controlled-phase",
  "title": "Controlled Phase",
  "symbol": "\\mathrm{C}P",
  "aliases": [
   "cphase",
   "cp"
  ],
  "notations": [
   "\\mathrm{CPhase}(\\phi)",
   "\\Lambda(\\mathrm{e}^{i\\phi})"
  ],
  "description": "Applies a phase $\\mathrm{e}^{i\\phi}$ to $|11\\rangle$ and leaves other basis states unchanged.",
  "arity": 2,
  "dimension": 2,
  "params": {
   "phi": {
    "label": "\\phi",
    "default": "1",
    "range": [
     0,
     2
    ]
   }
  },
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    1,
    0,
    0
   ],
   [
    0,
    0,
    1,
    0
   ],
   [
    0,
    0,
    0,
    "exp(i phi)"
   ]
  ],
  "groups": [
   "diagonal",
   "number-preserving"
  ],
  "properties": [],
  "url": "/gates/controlled-phase/"
 },
 {
  "slug": "ch",
  "title": "Controlled-Hadamard",
  "symbol": "\\mathrm{C}H",
  "aliases": [
   "ch"
  ],
  "notations": [
   "\\Lambda(H)",
   "\\text{controlled-}H"
  ],
  "description": "Applies a Hadamard to the target qubit when the control qubit is in the $|1\\rangle$ state.",
  "arity": 2,
  "dimension": 2,
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    1,
    0,
    0
   ],
   [
    0,
    0,
    "1/sqrt(2)",
    "1/sqrt(2)"
   ],
   [
    0,
    0,
    "1/sqrt(2)",
    "-1/sqrt(2)"
   ]
  ],
  "groups": [
   "orthogonal"
  ],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/ch/"
 },
 {
  "slug": "cs",
  "title": "Controlled-S",
  "symbol": "\\mathrm{C}S",
  "aliases": [
   "cs"
  ],
  "notations": [
   "\\Lambda(S)",
   "\\sqrt{\\mathrm{C}Z}",
   "\\mathrm{CP}(\\pi/2)"
  ],
  "description": "Applies a phase of $i$ to the $|11\\rangle$ state, the square root of controlled-$Z$.",
  "arity": 2,
  "dimension": 2,
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    1,
    0,
    0
   ],
   [
    0,
    0,
    1,
    0
   ],
   [
    0,
    0,
    0,
    "i"
   ]
  ],
  "groups": [
   "diagonal",
   "number-preserving"
  ],
  "properties": [],
  "url": "/gates/cs/"
 },
 {
  "slug": "csx",
  "title": "Controlled sqrt(X)",
  "symbol": "\\mathrm{C}\\sqrt{X}",
  "aliases": [
   "csx",
   "controlled-sqrt-x"
  ],
  "notations": [
   "\\mathrm{CS}X",
   "\\mathrm{C}V",
   "\\text{controlled-}\\sqrt{X}"
  ],
  "description": "Applies $\\sqrt{X}$ to the target when the control qubit is $|1\\rangle$.",
  "arity": 2,
  "dimension": 2,
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    1,
    0,
    0
   ],
   [
    0,
    0,
    "(1+i)/2",
    "(1-i)/2"
   ],
   [
    0,
    0,
    "(1-i)/2",
    "(1+i)/2"
   ]
  ],
  "groups": [],
  "properties": [],
  "url": "/gates/csx/"
 },
 {
  "slug": "toffoli",
  "title": "Toffoli",
  "symbol": "\\mathrm{CC}X",
  "aliases": [
   "toffoli",
   "ccx"
  ],
  "notations": [
   "\\mathrm{CCX}",
   "\\mathrm{CCNOT}",
   "\\mathrm{TOFF}",
   "C^2X",
   "\\Lambda^2(X)",
   "\\text{Toffoli}"
  ],
  "description": "Controlled-controlled-NOT gate that flips a target qubit when both controls are $|1\\rangle$.",
  "arity": 3,
  "dimension": 2,
  "matrix": [
   [
    1,
    0,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    1,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    1,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    1,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    1,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    1,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    1
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    1,
    0
   ]
  ],
  "groups": [
   "orthogonal",
   "permutation"
  ],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/toffoli/"
 },
 {
  "slug": "ccz",
  "title": "Controlled-CZ",
  "symbol": "\\mathrm{CC}Z",
  "aliases": [
   "ccz"
  ],
  "notations": [
   "C^2Z",
   "\\Lambda^2(Z)"
  ],
  "description": "Applies a $-1$ phase to $|111\\rangle$ and leaves all other basis states unchanged.",
  "arity": 3,
  "dimension": 2,
  "matrix": [
   [
    1,
    0,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    1,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    1,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    1,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    1,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    1,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    1,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    -1
   ]
  ],
  "groups": [
   "diagonal",
   "orthogonal",
   "number-preserving"
  ],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/ccz/"
 },
 {
  "slug": "cswap",
  "title": "Fredkin",
  "symbol": "\\mathrm{CSWAP}",
  "aliases": [
   "fredkin"
  ],
  "notations": [
   "\\mathrm{CSWAP}",
   "\\text{Fredkin}",
   "\\text{controlled-}SWAP"
  ],
  "description": "Controlled swap of two target qubits conditioned on a single control qubit.",
  "arity": 3,
  "dimension": 2,
  "matrix": [
   [
    1,
    0,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    1,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    1,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    1,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    1,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    1,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    1,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    1
   ]
  ],
  "groups": [
   "orthogonal",
   "permutation",
   "number-preserving"
  ],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/cswap/"
 },
 {
  "slug": "mcx",
  "title": "Multi-Controlled X",
  "symbol": "\\mathrm{C}^nX",
  "aliases": [
   "mcx",
   "c3x",
   "c4x"
  ],
  "notations": [
   "\\mathrm{C}^nX",
   "\\Lambda^n(X)",
   "\\mathrm{MCX}",
   "\\mathrm{C}^3X"
  ],
  "description": "Flips the target qubit when all $n$ control qubits are in the $|1\\rangle$ state.",
  "arity": "n+1",
  "dimension": 2,
  "matrix": [
   [
    1,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    1,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    1,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    1,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    1,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    1,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    1,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    1,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    1,
    0,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    1,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    1,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    1,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    1,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    1,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    1
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    1,
    0
   ]
  ],
  "matrix_note": "n = 3",
  "groups": [
   "orthogonal",
   "permutation"
  ],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/mcx/"
 },
 {
  "slug": "deutsch",
  "title": "Deutsch",
  "symbol": "D(\\theta)",
  "aliases": [
   "deutsch"
  ],
  "notations": [
   "D(\\theta)",
   "\\Lambda^2(iR_x(2\\theta))"
  ],
  "description": "The original universal three-qubit gate, applying $iR_x(2\\theta)$ to the target when both controls are $|1\\rangle$.",
  "arity": 3,
  "dimension": 2,
  "params": {
   "theta": {
    "label": "\\theta",
    "default": "1/4",
    "range": [
     0,
     2
    ]
   }
  },
  "matrix": [
   [
    1,
    0,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    1,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    1,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    1,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    1,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    1,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    "i cos(theta)",
    "sin(theta)"
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    "sin(theta)",
    "i cos(theta)"
   ]
  ],
  "groups": [],
  "properties": [],
  "url": "/gates/deutsch/"
 },
 {
  "slug": "rccx",
  "title": "Margolus",
  "symbol": "\\mathrm{RCC}X",
  "aliases": [
   "rccx",
   "margolus"
  ],
  "notations": [
   "\\mathrm{RCCX}",
   "\\widetilde{\\mathrm{CC}X}"
  ],
  "description": "Simplified Toffoli gate, equal to the Toffoli up to a $-1$ phase on $|101\\rangle$, costing only three CNOTs.",
  "arity": 3,
  "dimension": 2,
  "matrix": [
   [
    1,
    0,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    1,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    1,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    1,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    1,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    -1,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    1
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    1,
    0
   ]
  ],
  "groups": [
   "orthogonal"
  ],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/rccx/"
 },
 {
  "slug": "swap",
  "title": "SWAP",
  "symbol": "\\mathrm{SWAP}",
  "aliases": [],
  "notations": [
   "\\mathrm{SWAP}",
   "\\mathrm{SWAP}_{ij}"
  ],
  "description": "Exchanges the quantum states of two qubits.",
  "arity": 2,
  "dimension": 2,
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    0,
    1,
    0
   ],
   [
    0,
    1,
    0,
    0
   ],
   [
    0,
    0,
    0,
    1
   ]
  ],
  "groups": [
   "clifford",
   "orthogonal",
   "permutation",
   "number-preserving"
  ],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/swap/"
 },
 {
  "slug": "iswap",
  "title": "iSWAP",
  "symbol": "i\\mathrm{SWAP}",
  "aliases": [],
  "notations": [],
  "description": "Swaps $|01\\rangle$ and $|10\\rangle$ and adds a phase of $i$.",
  "arity": 2,
  "dimension": 2,
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    0,
    "i",
    0
   ],
   [
    0,
    "i",
    0,
    0
   ],
   [
    0,
    0,
    0,
    1
   ]
  ],
  "groups": [
   "clifford",
   "matchgate",
   "number-preserving"
  ],
  "properties": [],
  "url": "/gates/iswap/"
 },
 {
  "slug": "sqrt-swap",
  "title": "Square Root SWAP",
  "symbol": "\\sqrt{\\mathrm{SWAP}}",
  "aliases": [
   "sqrtswap"
  ],
  "notations": [],
  "description": "Applies half of a SWAP interaction, creating maximal entanglement from a product state.",
  "arity": 2,
  "dimension": 2,
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    "(1+i)/2",
    "(1-i)/2",
    0
   ],
   [
    0,
    "(1-i)/2",
    "(1+i)/2",
    0
   ],
   [
    0,
    0,
    0,
    1
   ]
  ],
  "groups": [
   "number-preserving"
  ],
  "properties": [],
  "url": "/gates/sqrt-swap/"
 },
 {
  "slug": "sqrt-iswap",
  "title": "Square Root iSWAP",
  "symbol": "\\sqrt{i\\mathrm{SWAP}}",
  "aliases": [
   "sqrt-iswap",
   "siswap",
   "sqisw"
  ],
  "notations": [
   "\\mathrm{SISWAP}",
   "\\mathrm{SQISW}"
  ],
  "description": "Applies half of an iSWAP interaction, native to superconducting hardware with $XY$ coupling.",
  "arity": 2,
  "dimension": 2,
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    "1/sqrt(2)",
    "i/sqrt(2)",
    0
   ],
   [
    0,
    "i/sqrt(2)",
    "1/sqrt(2)",
    0
   ],
   [
    0,
    0,
    0,
    1
   ]
  ],
  "groups": [
   "matchgate",
   "number-preserving"
  ],
  "properties": [],
  "url": "/gates/sqrt-iswap/"
 },
 {
  "slug": "fswap",
  "title": "fSWAP",
  "symbol": "\\mathrm{fSWAP}",
  "aliases": [
   "fswap",
   "fermionic-swap"
  ],
  "notations": [],
  "description": "Swaps $|01\\rangle$ and $|10\\rangle$ while applying a $-1$ phase to $|11\\rangle$.",
  "arity": 2,
  "dimension": 2,
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    0,
    1,
    0
   ],
   [
    0,
    1,
    0,
    0
   ],
   [
    0,
    0,
    0,
    -1
   ]
  ],
  "groups": [
   "clifford",
   "orthogonal",
   "matchgate",
   "number-preserving"
  ],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/fswap/"
 },
 {
  "slug": "dcx",
  "title": "Double CX",
  "symbol": "\\mathrm{DC}X",
  "aliases": [
   "dcnot"
  ],
  "notations": [
   "\\mathrm{DCNOT}"
  ],
  "description": "Two back-to-back CNOTs with alternating control and target qubits.",
  "arity": 2,
  "dimension": 2,
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    0,
    1,
    0
   ],
   [
    0,
    0,
    0,
    1
   ],
   [
    0,
    1,
    0,
    0
   ]
  ],
  "groups": [
   "clifford",
   "orthogonal",
   "permutation"
  ],
  "properties": [],
  "url": "/gates/dcx/"
 },
 {
  "slug": "gpi",
  "title": "GPi",
  "symbol": "\\mathrm{GPi}",
  "aliases": [
   "gpi"
  ],
  "notations": [
   "\\mathrm{GPi}(\\phi)",
   "\\mathrm{GPI}(\\phi)"
  ],
  "description": "IonQ's native $\\pi$ pulse, a bit flip about the equatorial Bloch-sphere axis at angle $\\phi$.",
  "arity": 1,
  "dimension": 2,
  "params": {
   "phi": {
    "label": "\\phi",
    "default": "1/4",
    "range": [
     0,
     2
    ]
   }
  },
  "matrix": [
   [
    0,
    "exp(-i phi)"
   ],
   [
    "exp(i phi)",
    0
   ]
  ],
  "groups": [],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/gpi/"
 },
 {
  "slug": "gpi2",
  "title": "GPi2",
  "symbol": "\\mathrm{GPi2}",
  "aliases": [
   "gpi2"
  ],
  "notations": [
   "\\mathrm{GPi2}(\\phi)",
   "\\mathrm{GPI2}(\\phi)"
  ],
  "description": "IonQ's native $\\pi/2$ pulse, a half rotation about the equatorial Bloch-sphere axis at angle $\\phi$.",
  "arity": 1,
  "dimension": 2,
  "params": {
   "phi": {
    "label": "\\phi",
    "default": "1/4",
    "range": [
     0,
     2
    ]
   }
  },
  "matrix": [
   [
    "1/sqrt(2)",
    "-i exp(-i phi)/sqrt(2)"
   ],
   [
    "-i exp(i phi)/sqrt(2)",
    "1/sqrt(2)"
   ]
  ],
  "groups": [],
  "properties": [],
  "url": "/gates/gpi2/"
 },
 {
  "slug": "molmer-sorenson",
  "title": "Mølmer-Sørensen",
  "symbol": "MS",
  "aliases": [],
  "notations": [
   "\\mathrm{MS}",
   "\\mathrm{MS}(\\chi_{ij})",
   "\\mathrm{GMS}"
  ],
  "description": "A native entangling interaction in ion-trap systems based on collective spin-motion coupling.",
  "arity": "n",
  "dimension": 2,
  "params": {
   "chi": {
    "label": "\\chi",
    "default": "1/2",
    "range": [
     0,
     2
    ]
   }
  },
  "matrix": [
   [
    "cos(chi/2)",
    0,
    0,
    "-i sin(chi/2)"
   ],
   [
    0,
    "cos(chi/2)",
    "-i sin(chi/2)",
    0
   ],
   [
    0,
    "-i sin(chi/2)",
    "cos(chi/2)",
    0
   ],
   [
    "-i sin(chi/2)",
    0,
    0,
    "cos(chi/2)"
   ]
  ],
  "matrix_note": "n = 2",
  "groups": [],
  "properties": [],
  "url": "/gates/molmer-sorenson/"
 },
 {
  "slug": "ecr",
  "title": "Echoed Cross-Resonance",
  "symbol": "ECR",
  "aliases": [],
  "notations": [],
  "description": "A native two-qubit Clifford gate for cross-resonance hardware, locally equivalent to CNOT.",
  "arity": 2,
  "dimension": 2,
  "matrix": [
   [
    0,
    "1/sqrt(2)",
    0,
    "i/sqrt(2)"
   ],
   [
    "1/sqrt(2)",
    0,
    "-i/sqrt(2)",
    0
   ],
   [
    0,
    "i/sqrt(2)",
    0,
    "1/sqrt(2)"
   ],
   [
    "-i/sqrt(2)",
    0,
    "1/sqrt(2)",
    0
   ]
  ],
  "groups": [
   "clifford"
  ],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/ecr/"
 },
 {
  "slug": "sycamore",
  "title": "Sycamore",
  "symbol": "\\mathrm{SYC}",
  "aliases": [
   "sycamore",
   "syc"
  ],
  "notations": [
   "\\mathrm{SYC}",
   "\\mathrm{fSim}(\\pi/2, \\pi/6)"
  ],
  "description": "Google's native two-qubit gate, the fSim gate at $\\theta = \\pi/2$, $\\phi = \\pi/6$.",
  "arity": 2,
  "dimension": 2,
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    0,
    "-i",
    0
   ],
   [
    0,
    "-i",
    0,
    0
   ],
   [
    0,
    0,
    0,
    "exp(-i pi/6)"
   ]
  ],
  "groups": [
   "number-preserving"
  ],
  "properties": [],
  "url": "/gates/sycamore/"
 },
 {
  "slug": "xy",
  "title": "XY Interaction",
  "symbol": "\\mathrm{XY}",
  "aliases": [
   "xy",
   "xx-plus-yy"
  ],
  "notations": [
   "\\mathrm{XY}(\\theta)",
   "R_{xx+yy}(\\theta)",
   "\\mathrm{PISWAP}(\\theta)"
  ],
  "description": "Partial iSWAP generated by the $XX + YY$ exchange interaction, rotating within the single-excitation subspace.",
  "arity": 2,
  "dimension": 2,
  "params": {
   "theta": {
    "label": "\\theta",
    "default": "1",
    "range": [
     0,
     4
    ]
   }
  },
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    "cos(theta/2)",
    "i sin(theta/2)",
    0
   ],
   [
    0,
    "i sin(theta/2)",
    "cos(theta/2)",
    0
   ],
   [
    0,
    0,
    0,
    1
   ]
  ],
  "groups": [
   "matchgate",
   "number-preserving"
  ],
  "properties": [],
  "url": "/gates/xy/"
 },
 {
  "slug": "fsim",
  "title": "Fermionic Simulator",
  "symbol": "\\mathrm{FSim}",
  "aliases": [
   "fsim"
  ],
  "notations": [],
  "description": "Parameterized two-qubit gate combining an $XY$ interaction with a controlled-phase, native to Google superconducting hardware.",
  "arity": 2,
  "dimension": 2,
  "params": {
   "theta": {
    "label": "\\theta",
    "default": "1/2",
    "range": [
     0,
     2
    ]
   },
   "phi": {
    "label": "\\phi",
    "default": "1/6",
    "range": [
     0,
     4
    ]
   }
  },
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    "cos(theta)",
    "-i sin(theta)",
    0
   ],
   [
    0,
    "-i sin(theta)",
    "cos(theta)",
    0
   ],
   [
    0,
    0,
    0,
    "exp(-i phi)"
   ]
  ],
  "groups": [
   "number-preserving"
  ],
  "properties": [],
  "url": "/gates/fsim/"
 },
 {
  "slug": "can",
  "title": "Canonical entangling",
  "symbol": "\\mathrm{CAN}",
  "aliases": [
   "can",
   "canonical"
  ],
  "notations": [],
  "description": "Parameterizes every two-qubit interaction up to local unitaries.",
  "arity": 2,
  "dimension": 2,
  "params": {
   "a": {
    "label": "a",
    "default": "3/16",
    "range": [
     0,
     1
    ]
   },
   "b": {
    "label": "b",
    "default": "1/8",
    "range": [
     0,
     2
    ]
   },
   "c": {
    "label": "c",
    "default": "1/16",
    "range": [
     0,
     3
    ]
   }
  },
  "matrix": [
   [
    "exp(i c) cos(a-b)",
    0,
    0,
    "i exp(i c) sin(a-b)"
   ],
   [
    0,
    "exp(-i c) cos(a+b)",
    "i exp(-i c) sin(a+b)",
    0
   ],
   [
    0,
    "i exp(-i c) sin(a+b)",
    "exp(-i c) cos(a+b)",
    0
   ],
   [
    "i exp(i c) sin(a-b)",
    0,
    0,
    "exp(i c) cos(a-b)"
   ]
  ],
  "groups": [],
  "properties": [],
  "url": "/gates/can/"
 },
 {
  "slug": "b",
  "title": "B Gate",
  "symbol": "B",
  "aliases": [],
  "notations": [],
  "description": "Two-qubit entangling gate that can synthesize arbitrary elements of $U(4)$ with only two applications.",
  "arity": 2,
  "dimension": 2,
  "matrix": [
   [
    "(cos(pi/8)+sin(pi/8))/sqrt(2)",
    0,
    0,
    "i (cos(pi/8)-sin(pi/8))/sqrt(2)"
   ],
   [
    0,
    "(cos(pi/8)-sin(pi/8))/sqrt(2)",
    "i (cos(pi/8)+sin(pi/8))/sqrt(2)",
    0
   ],
   [
    0,
    "i (cos(pi/8)+sin(pi/8))/sqrt(2)",
    "(cos(pi/8)-sin(pi/8))/sqrt(2)",
    0
   ],
   [
    "i (cos(pi/8)-sin(pi/8))/sqrt(2)",
    0,
    0,
    "(cos(pi/8)+sin(pi/8))/sqrt(2)"
   ]
  ],
  "groups": [
   "matchgate"
  ],
  "properties": [],
  "url": "/gates/b/"
 },
 {
  "slug": "magic",
  "title": "Magic",
  "symbol": "M",
  "aliases": [
   "m"
  ],
  "notations": [
   "\\mathcal{M}",
   "Q",
   "E"
  ],
  "description": "Changes from the computational basis to the magic (Bell) basis used in two-qubit local invariants.",
  "arity": 2,
  "dimension": 2,
  "matrix": [
   [
    "1/sqrt(2)",
    "i/sqrt(2)",
    0,
    0
   ],
   [
    0,
    0,
    "i/sqrt(2)",
    "1/sqrt(2)"
   ],
   [
    0,
    0,
    "i/sqrt(2)",
    "-1/sqrt(2)"
   ],
   [
    "1/sqrt(2)",
    "-i/sqrt(2)",
    0,
    0
   ]
  ],
  "groups": [
   "clifford"
  ],
  "properties": [],
  "url": "/gates/magic/"
 },
 {
  "slug": "givens",
  "title": "Givens Rotation",
  "symbol": "G_{ij}",
  "aliases": [
   "givens"
  ],
  "notations": [
   "G(\\theta)",
   "G_{ij}(\\theta)"
  ],
  "description": "Two-level rotation that mixes the $|01\\rangle$ and $|10\\rangle$ subspace.",
  "arity": 2,
  "dimension": 2,
  "params": {
   "theta": {
    "label": "\\theta",
    "default": "1/4",
    "range": [
     0,
     2
    ]
   }
  },
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    "cos(theta)",
    "-sin(theta)",
    0
   ],
   [
    0,
    "sin(theta)",
    "cos(theta)",
    0
   ],
   [
    0,
    0,
    0,
    1
   ]
  ],
  "groups": [
   "orthogonal",
   "matchgate",
   "number-preserving"
  ],
  "properties": [],
  "url": "/gates/givens/"
 },
 {
  "slug": "xx-minus-yy",
  "title": "XX−YY Interaction",
  "symbol": "R_{xx-yy}",
  "aliases": [
   "xx-minus-yy"
  ],
  "notations": [
   "R_{xx-yy}(\\theta)",
   "\\mathrm{XX{-}YY}(\\theta)"
  ],
  "description": "Rotation generated by the $XX - YY$ interaction, acting only on the even-parity subspace.",
  "arity": 2,
  "dimension": 2,
  "params": {
   "theta": {
    "label": "\\theta",
    "default": "1",
    "range": [
     0,
     4
    ]
   }
  },
  "matrix": [
   [
    "cos(theta/2)",
    0,
    0,
    "-i sin(theta/2)"
   ],
   [
    0,
    1,
    0,
    0
   ],
   [
    0,
    0,
    1,
    0
   ],
   [
    "-i sin(theta/2)",
    0,
    0,
    "cos(theta/2)"
   ]
  ],
  "groups": [
   "matchgate"
  ],
  "properties": [],
  "url": "/gates/xx-minus-yy/"
 },
 {
  "slug": "shift",
  "title": "Shift",
  "symbol": "X_d",
  "aliases": [
   "shift"
  ],
  "notations": [
   "X_d",
   "\\Sigma_x",
   "\\tau"
  ],
  "description": "Qudit generalization of Pauli-$X$ that cyclically increments the basis state, $|j\\rangle \\mapsto |j+1 \\bmod d\\rangle$.",
  "arity": 1,
  "dimension": "d",
  "matrix": [
   [
    0,
    0,
    1
   ],
   [
    1,
    0,
    0
   ],
   [
    0,
    1,
    0
   ]
  ],
  "matrix_note": "d = 3",
  "groups": [
   "orthogonal",
   "permutation"
  ],
  "properties": [],
  "url": "/gates/shift/"
 },
 {
  "slug": "clock",
  "title": "Clock",
  "symbol": "Z_d",
  "aliases": [
   "clock"
  ],
  "notations": [
   "Z_d",
   "\\Sigma_z",
   "\\sigma"
  ],
  "description": "Qudit generalization of Pauli-$Z$ that tags each basis state with a root-of-unity phase, $|j\\rangle \\mapsto \\omega^j |j\\rangle$.",
  "arity": 1,
  "dimension": "d",
  "matrix": [
   [
    "1",
    0,
    0
   ],
   [
    0,
    "exp(2 pi i/3)",
    0
   ],
   [
    0,
    0,
    "exp(4 pi i/3)"
   ]
  ],
  "matrix_note": "d = 3",
  "groups": [
   "diagonal",
   "number-preserving"
  ],
  "properties": [],
  "url": "/gates/clock/"
 },
 {
  "slug": "chrestenson",
  "title": "Chrestenson",
  "symbol": "C_d",
  "aliases": [
   "chrestenson",
   "chrestenson-transform",
   "generalized-hadamard"
  ],
  "notations": [
   "C_d",
   "\\mathcal{C}_d"
  ],
  "description": "Qudit generalization of the Hadamard gate that maps each basis state to an equal superposition with root-of-unity phases.",
  "arity": 1,
  "dimension": "d",
  "matrix": [
   [
    "1/sqrt(3)",
    "1/sqrt(3)",
    "1/sqrt(3)"
   ],
   [
    "1/sqrt(3)",
    "exp(2 pi i/3)/sqrt(3)",
    "exp(4 pi i/3)/sqrt(3)"
   ],
   [
    "1/sqrt(3)",
    "exp(4 pi i/3)/sqrt(3)",
    "exp(2 pi i/3)/sqrt(3)"
   ]
  ],
  "matrix_note": "d = 3",
  "groups": [],
  "properties": [],
  "url": "/gates/chrestenson/"
 },
 {
  "slug": "controlled-shift",
  "title": "Controlled Shift (SUM)",
  "symbol": "\\mathrm{SUM}_d",
  "aliases": [
   "sum",
   "cadd",
   "qudit-cx",
   "controlled-shift"
  ],
  "notations": [
   "\\mathrm{C}X_d",
   "\\mathrm{ADD}_d",
   "\\mathrm{SUM}_d"
  ],
  "description": "Qudit-controlled shift that adds the control value to the target modulo $d$.",
  "arity": 2,
  "dimension": "d",
  "matrix": [
   [
    1,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    1,
    0,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    1,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    1,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    1,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    1,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    1,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    1
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    1,
    0,
    0
   ]
  ],
  "matrix_note": "d = 3",
  "groups": [
   "orthogonal",
   "permutation"
  ],
  "properties": [],
  "url": "/gates/controlled-shift/"
 },
 {
  "slug": "controlled-clock",
  "title": "Controlled Clock",
  "symbol": "\\mathrm{C}Z_d",
  "aliases": [
   "controlled-clock",
   "qudit-cz",
   "czd",
   "controlled-zd"
  ],
  "notations": [
   "\\mathrm{C}Z_d",
   "\\mathrm{CP}_d",
   "\\Lambda(Z_d)"
  ],
  "description": "Qudit generalization of controlled-$Z$ that applies the phase $\\omega^{ct}$ to $|c,t\\rangle$.",
  "arity": 2,
  "dimension": "d",
  "matrix": [
   [
    "1",
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    "1",
    0,
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    "1",
    0,
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    "1",
    0,
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    "exp(2 pi i/3)",
    0,
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    "exp(4 pi i/3)",
    0,
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    "1",
    0,
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    "exp(4 pi i/3)",
    0
   ],
   [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    "exp(2 pi i/3)"
   ]
  ],
  "matrix_note": "d = 3",
  "groups": [
   "diagonal",
   "number-preserving"
  ],
  "properties": [],
  "url": "/gates/controlled-clock/"
 },
 {
  "slug": "global-phase",
  "title": "Global Phase",
  "symbol": "\\mathrm{Ph}",
  "aliases": [
   "gphase",
   "ph"
  ],
  "notations": [
   "\\mathrm{e}^{i\\phi}",
   "\\mathrm{e}^{i\\phi} I"
  ],
  "description": "Multiplies the entire quantum state by an overall phase factor.",
  "arity": "n",
  "dimension": 2,
  "params": {
   "phi": {
    "label": "\\phi",
    "default": "1/2",
    "range": [
     0,
     2
    ]
   }
  },
  "matrix": [
   [
    "exp(i phi)",
    0
   ],
   [
    0,
    "exp(i phi)"
   ]
  ],
  "matrix_note": "n = 1",
  "groups": [
   "diagonal",
   "number-preserving"
  ],
  "properties": [],
  "url": "/gates/global-phase/"
 },
 {
  "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/"
 },
 {
  "slug": "oracle",
  "title": "Oracle",
  "symbol": "O_f",
  "aliases": [],
  "notations": [
   "U_f",
   "\\mathcal{O}_f"
  ],
  "description": "Implements a classical function as a reversible black-box unitary that can be queried coherently.",
  "arity": "n",
  "dimension": "p",
  "groups": [
   "orthogonal",
   "permutation"
  ],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/oracle/"
 },
 {
  "slug": "phase-oracle",
  "title": "Phase Oracle",
  "symbol": "O_f^\\pm",
  "aliases": [],
  "notations": [
   "U_f^\\pm",
   "\\mathcal{O}_f^\\pm"
  ],
  "description": "Encodes a Boolean function as phases on computational basis states.",
  "arity": "n",
  "dimension": "p",
  "groups": [
   "diagonal",
   "orthogonal",
   "number-preserving"
  ],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/phase-oracle/"
 },
 {
  "slug": "grover-diffuser",
  "title": "Grover Diffuser",
  "symbol": "U_s",
  "aliases": [
   "grover-diffuser",
   "diffuser"
  ],
  "notations": [
   "U_s",
   "2|s\\rangle\\langle s| - I",
   "\\mathrm{D}"
  ],
  "description": "Reflection about the uniform superposition, the \"inversion about the mean\" step of Grover's algorithm.",
  "arity": "n",
  "dimension": 2,
  "matrix": [
   [
    "-1/2",
    "1/2",
    "1/2",
    "1/2"
   ],
   [
    "1/2",
    "-1/2",
    "1/2",
    "1/2"
   ],
   [
    "1/2",
    "1/2",
    "-1/2",
    "1/2"
   ],
   [
    "1/2",
    "1/2",
    "1/2",
    "-1/2"
   ]
  ],
  "matrix_note": "n = 2",
  "groups": [
   "orthogonal"
  ],
  "properties": [
   "hermitian"
  ],
  "url": "/gates/grover-diffuser/"
 },
 {
  "slug": "virtual-distillation",
  "title": "Virtual Distillation",
  "symbol": "B^{(2)}_i",
  "aliases": [],
  "notations": [],
  "description": "A two-qubit entangling gate used in virtual distillation error-mitigation protocols.",
  "arity": 2,
  "dimension": 2,
  "matrix": [
   [
    1,
    0,
    0,
    0
   ],
   [
    0,
    "1/sqrt(2)",
    "-1/sqrt(2)",
    0
   ],
   [
    0,
    "1/sqrt(2)",
    "1/sqrt(2)",
    0
   ],
   [
    0,
    0,
    0,
    1
   ]
  ],
  "groups": [
   "orthogonal"
  ],
  "properties": [],
  "url": "/gates/virtual-distillation/"
 }
]