Quantum gates
This is a list of gates that qujax supports natively. You can also define custom operations by directly passing an array or function instead of a string, as documented in get_params_to_statetensor_func and get_params_to_densitytensor_func.
Name(s) |
String |
Matrix representation |
|---|---|---|
Pauli X gate NOT gate Bit flip gate |
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\[\begin{split}X = NOT = \begin{bmatrix}0 & 1\\ 1 & 0 \end{bmatrix}\end{split}\]
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Pauli Y gate |
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\[\begin{split}Y = \begin{bmatrix}0 & -i\\ i & 0 \end{bmatrix}\end{split}\]
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Pauli Z gate Phase flip gate |
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\[\begin{split}Z = \begin{bmatrix}1 & 0\\ 0 & -1 \end{bmatrix}\end{split}\]
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Hadamard gate |
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\[\begin{split}H = \frac{1}{\sqrt{2}}\begin{bmatrix}1 & 1\\ 1 & -1 \end{bmatrix}\end{split}\]
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S gate P (phase) gate |
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\[\begin{split}S = P = \sqrt{Z} = \begin{bmatrix}1 & 0\\ 0 & i \end{bmatrix}\end{split}\]
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Conjugated S gate |
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\[\begin{split}S^\dagger = \begin{bmatrix}1 & 0\\ 0 & -i \end{bmatrix}\end{split}\]
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T gate |
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\[\begin{split}T = \sqrt[4]{Z} = \begin{bmatrix}1 & 0\\ 0 & \exp(\frac{\pi i}{4}) \end{bmatrix}\end{split}\]
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Conjugated T gate |
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\[\begin{split}T^\dagger = \begin{bmatrix}1 & 0\\ 0 & -\exp(\frac{\pi i}{4}) \end{bmatrix}\end{split}\]
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V gate |
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\[\begin{split}V = \sqrt{X} = \frac{1}{\sqrt{2}}\begin{bmatrix}1 & -i\\ -i & 1 \end{bmatrix}\end{split}\]
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Conjugated V gate |
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\[\begin{split}V^\dagger = \frac{1}{\sqrt{2}}\begin{bmatrix}1 & i\\ i & 1 \end{bmatrix}\end{split}\]
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SX gate |
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\[\begin{split}SX = \sqrt{X} = \frac{1}{2}\begin{bmatrix}1 + i & 1 - i\\ 1 - i & 1 + i \end{bmatrix}\end{split}\]
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Conjugated SX gate |
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\[\begin{split}SX^\dagger = \frac{1}{2}\begin{bmatrix}1 - i & 1 + i\\ 1 + i & 1 - i \end{bmatrix}\end{split}\]
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CX (Controlled X) gate CNOT gate |
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\[\begin{split}CX = CNOT = \begin{bmatrix}I & 0\\ 0 & X \end{bmatrix} = \begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix}\end{split}\]
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CY (Controlled Y) gate |
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\[\begin{split}CY = \begin{bmatrix}I & 0\\ 0 & Y \end{bmatrix} = \begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \end{bmatrix}\end{split}\]
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Controlled Z gate |
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\[\begin{split}CZ = \begin{bmatrix}I & 0\\ 0 & Z \end{bmatrix} = \begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix}\end{split}\]
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Controlled Hadamard gate |
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\[\begin{split}CH = \begin{bmatrix}I & 0\\ 0 & H \end{bmatrix} = \frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & -1 \end{bmatrix}\end{split}\]
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Controlled V gate |
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\[\begin{split}CV = \begin{bmatrix}I & 0\\ 0 & V \end{bmatrix} = \frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -i \\ 0 & 0 & -i & 1 \end{bmatrix}\end{split}\]
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Conjugated controlled V gate |
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\[\begin{split}CVdg = \begin{bmatrix}I & 0\\ 0 & V^\dagger \end{bmatrix} = \frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & i \\ 0 & 0 & i & 1 \end{bmatrix}\end{split}\]
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Controlled SX gate |
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\[\begin{split}CSX = \begin{bmatrix}I & 0\\ 0 & SX \end{bmatrix} = \frac{1}{2}\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1+i & 1-i \\ 0 & 0 & 1-i & 1+i \end{bmatrix}\end{split}\]
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Conjugated controlled SX gate |
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\[\begin{split}CSX^\dagger = \begin{bmatrix}I & 0\\ 0 & SX^\dagger \end{bmatrix} = \frac{1}{2}\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1-i & 1+i \\ 0 & 0 & 1+i & 1-i \end{bmatrix}\end{split}\]
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Toffoli gate CCX CCNOT |
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\[\begin{split}CCX = \begin{bmatrix}I & 0 & 0 & 0\\ 0 & I & 0 & 0 \\ 0 & 0 & I & 0 \\ 0 & 0 & 0 & X \end{bmatrix} = \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ \end{bmatrix}\end{split}\]
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Echoed cross-resonance gate |
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\[\begin{split}ECR = \begin{bmatrix}0 & V^\dagger \\ V & 0 \end{bmatrix} = \frac{1}{2}\begin{bmatrix}0 & 0 & 1 & i \\ 0 & 0 & i & 1 \\ 1 & -i & 0 & 0 \\ i & 1 & 0 & 0 \end{bmatrix}\end{split}\]
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Swap gate |
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\[\begin{split}SWAP = \begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\end{split}\]
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Controlled swap gate |
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\[\begin{split}CSWAP = \begin{bmatrix}I & 0 \\ 0 & SWAP \end{bmatrix} = \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}\end{split}\]
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Rotation around X axis |
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\[\begin{split}R_X(\theta) = \exp\left(-i \frac{\pi}{2} \theta X\right) = \begin{bmatrix} \cos( \frac{\pi}{2} \theta) & - \sin( \frac{\pi}{2} \theta) \\ - \sin( \frac{\pi}{2} \theta) & \cos( \frac{\pi}{2} \theta) \end{bmatrix}\end{split}\]
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Rotation around X axis |
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\[\begin{split}R_Y(\theta) = \exp\left(-i \frac{\pi}{2} \theta Y\right) = \begin{bmatrix} \cos( \frac{\pi}{2} \theta) & i \sin( \frac{\pi}{2} \theta) \\ - i \sin( \frac{\pi}{2} \theta) & \cos( \frac{\pi}{2} \theta) \end{bmatrix}\end{split}\]
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Rotation around Z axis |
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\[\begin{split}R_Z(\theta) = \exp\left(-i \frac{\pi}{2} \theta Z\right) = \begin{bmatrix} \cos( \frac{\pi}{2} \theta) + \sin( \frac{\pi}{2} \theta) & 0 \\ 0 & \cos( \frac{\pi}{2} \theta) - \sin( \frac{\pi}{2} \theta) \end{bmatrix}\end{split}\]
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Controlled rotation around X axis |
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\[\begin{split}CR_X(\theta) = \begin{bmatrix}I & 0\\ 0 & RX(\theta) \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos( \frac{\pi}{2} \theta) & - \sin( \frac{\pi}{2} \theta) \\ 0 & 0 & - \sin( \frac{\pi}{2} \theta) & \cos( \frac{\pi}{2} \theta) \end{bmatrix}\end{split}\]
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Controlled rotation around Y axis |
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\[\begin{split}CR_Y(\theta) = \begin{bmatrix}I & 0\\ 0 & RY(\theta) \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos( \frac{\pi}{2} \theta) & i \sin( \frac{\pi}{2} \theta) \\ 0 & 0 & - i \sin( \frac{\pi}{2} \theta) & \cos( \frac{\pi}{2} \theta) \end{bmatrix}\end{split}\]
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Controlled rotation around Z axis |
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\[\begin{split}CR_Z(\theta) = \begin{bmatrix}I & 0\\ 0 & RZ(\theta)\end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos( \frac{\pi}{2} \theta) + \sin( \frac{\pi}{2} \theta) & 0 \\ 0 & 0 & 0 & \cos( \frac{\pi}{2} \theta) - \sin( \frac{\pi}{2} \theta) \end{bmatrix}\end{split}\]
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U3 |
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\[U3(\alpha,\beta,\gamma) = \exp((\alpha + \beta) i \frac{\pi}{2}) R_Z(\beta) R_Y(\alpha) R_Z(\gamma)\]
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U1 |
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\[U1(\gamma) = U3(0, 0, \gamma)\]
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U2 |
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\[U2(\beta, \gamma) = U3(0.5, \beta, \gamma)\]
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Controlled U3 |
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\[\begin{split}CU3(\alpha,\beta,\gamma) = \begin{bmatrix}I & 0\\ 0 & U3(\alpha,\beta,\gamma)\end{bmatrix}\end{split}\]
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Controlled U1 |
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\[\begin{split}CU1(\gamma) = \begin{bmatrix}I & 0\\ 0 & U1(\gamma)\end{bmatrix}\end{split}\]
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Controlled U2 |
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\[\begin{split}CU2(\beta, \gamma) = \begin{bmatrix}I & 0\\ 0 & U2(\beta, \gamma)\end{bmatrix}\end{split}\]
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Imaginary swap |
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\[\begin{split}iSWAP(\theta) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos( \frac{\pi}{2} \theta) & i \sin( \frac{\pi}{2} \theta) & 0 \\ 0 & i \sin( \frac{\pi}{2} \theta) & \cos( \frac{\pi}{2} \theta) & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\end{split}\]
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Phased imaginary swap |
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\[\begin{split}PhasedISWAP(\phi, \theta) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos( \frac{\pi}{2} \theta) & \exp(2i\pi \phi) i \sin( \frac{\pi}{2} \theta) & 0 \\ 0 & \exp(- 2i\pi \phi) i \sin( \frac{\pi}{2} \theta) & \cos( \frac{\pi}{2} \theta) & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\end{split}\]
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XXPhase XX interaction |
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\[\begin{split}R_{XX}(\theta) = \exp\left(\frac{\pi}{2} \theta X\otimes X\right) = \begin{bmatrix} \cos( \frac{\pi}{2} \theta) & 0 & 0 & -i \sin( \frac{\pi}{2} \theta) \\ 0 & \cos( \frac{\pi}{2} \theta) & -i \sin( \frac{\pi}{2} \theta) & 0 \\ 0 & -i \sin( \frac{\pi}{2} \theta) & \cos( \frac{\pi}{2} \theta) & 0 \\ -i \sin( \frac{\pi}{2} \theta) & 0 & 0 & \cos( \frac{\pi}{2} \theta) \end{bmatrix}\end{split}\]
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YYPhase YY interaction |
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\[\begin{split}R_{YY}(\theta) = \exp\left(\frac{\pi}{2} \theta Y\otimes Y\right) = \begin{bmatrix} \cos( \frac{\pi}{2} \theta) & 0 & 0 & i \sin( \frac{\pi}{2} \theta) \\ 0 & \cos( \frac{\pi}{2} \theta) & -i \sin( \frac{\pi}{2} \theta) & 0 \\ 0 & -i \sin( \frac{\pi}{2} \theta) & \cos( \frac{\pi}{2} \theta) & 0 \\ i \sin( \frac{\pi}{2} \theta) & 0 & 0 & \cos( \frac{\pi}{2} \theta) \end{bmatrix}\end{split}\]
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ZZPhase ZZ interaction |
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\[\begin{split}R_{ZZ}(\theta) = \exp\left(\frac{\pi}{2} \theta Z\otimes Z\right) = \begin{bmatrix} \exp( -i \frac{\pi}{2} \theta) & 0 & 0 & 0 \\ 0 & \exp( i \frac{\pi}{2} \theta) & 0 & 0 \\ 0 & 0 & \exp( i \frac{\pi}{2} \theta) & 0 \\ 0 & 0 & 0 & \exp( -i \frac{\pi}{2} \theta) \end{bmatrix}\end{split}\]
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ZZMax |
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\[ZZMax = R_{ZZ}(0.5)\]
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PhasedX |
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\[PhasedX(\theta, \phi) = R_Z(\phi)R_X(\theta)R_Z(-\phi)\]
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