Skip to content

U gate

Identifier Operator Example statement
U \(U(\theta, \phi, \lambda)\) U(pi/2,0,pi) q[0]

Description

The U gate can describe any element of \(U(2)\) through the specification of the input parameters \(\theta\), \(\phi\), and \(\lambda\). Notable examples of single-qubit operations in terms of \(U(\theta, \phi, \lambda)\) are:

Representation

\[\begin{align} U(\theta, \phi, \lambda) = \left(\begin{matrix} \cos\left(\theta / 2\right) & -e^{i\lambda}\sin\left(\theta / 2\right) \\ e^{i\phi}\sin\left(\theta / 2\right) & e^{i\left(\phi + \lambda\right)}\cos\left(\theta / 2\right) \end{matrix}\right) \end{align}\]

Any single-qubit operation in \(U(2)\) (including global phase) can be expressed by 5 parameters in the canonical representation \(R_\hat{\mathbf{n}}\)

\[R_\hat{\mathbf{n}}\left([n_x, n_y, n_z]^T, \theta, \phi\right) = e^{i\phi} \cdot e^{-i\frac{\theta}{2}\left(n_x\cdot\sigma_x + n_y\cdot\sigma_y + n_z\cdot\sigma_z\right)},\]

where \(\hat{\mathbf{n}}=[n_x, n_y, n_z]^T\) denotes the axis of rotation, \(\theta\in(-\pi, \pi]\) the angle of rotation [rad], and \(\phi\in[0,2\pi)\) the global phase angle [rad].

The U gate is given by:

\[\begin{equation} U(\theta, \phi, \lambda) = R_\hat{\mathbf{n}}\left([0, 0, 1]^T, \phi, \frac{\phi + \lambda}{2}\right)\cdot R_\hat{\mathbf{n}}\left([0, 1, 0]^T, \theta, 0\right)\cdot R_\hat{\mathbf{n}}\left([0, 0, 1]^T, \lambda, 0\right), \\ \end{equation}\]

In the Hadamard basis \(\{|+\rangle, |-\rangle\}\), the U gate \(U_H\) is given by:

\[U_H = HUH = \frac{1}{2}\left(\begin{matrix} \,[e^{i \phi} - e^{i \lambda}] \sin(\theta/2) + [1 + e^{i \lambda + i \phi}] \cos(\theta/2) & [e^{i \lambda} + e^{i \phi}] \sin(\theta/2) + [1 - e^{i \lambda + i \phi}] \cos(\theta/2) \\ -[e^{i \lambda} + e^{i \phi}] \sin(\theta/2) + [1 - e^{i \lambda + i \phi}] \cos(\theta/2) & [e^{i \lambda} - e^{i \phi}] \sin(\theta/2) + [1 + e^{i \lambda + i \phi}] \cos(\theta/2) \end{matrix}\right)\]

Operation examples

Standard basis

\[\begin{align} U(\theta, \phi, \lambda)\,|0\rangle &= \cos\left(\theta / 2\right)|0\rangle + e^{i\phi}\sin\left(\theta / 2\right)|1\rangle \\ \\ U(\theta, \phi, \lambda)\,|1\rangle &= -e^{i\lambda}\sin\left(\theta / 2\right)|0\rangle + e^{i\left(\phi + \lambda\right)}\cos\left(\theta / 2\right)|1\rangle \\ \end{align}\]

Hadamard basis

\[\begin{align} U(\theta, \phi, \lambda)\,|+\rangle &= \frac{[e^{i \phi} - e^{i \lambda}] \sin(\theta/2) + [1 + e^{i \lambda + i \phi}] \cos(\theta/2)}{2}|+\rangle - \frac{[e^{i \lambda} + e^{i \phi}] \sin(\theta/2) - [1 - e^{i \lambda + i \phi}] \cos(\theta/2)}{2}|-\rangle \\ \\ U(\theta, \phi, \lambda)\,|-\rangle &= \frac{[e^{i \lambda} + e^{i \phi}] \sin(\theta/2) + [1 - e^{i \lambda + i \phi}] \cos(\theta/2)}{2}|+\rangle + \frac{[e^{i \lambda} - e^{i \phi}] \sin(\theta/2) + [1 + e^{i \lambda + i \phi}] \cos(\theta/2)}{2}|-\rangle \end{align}\]