Quantum Inspire is a quantum computing platform provided by QuTech. It consists of two real quantum processors - Starmon-5 and Spin-2. Whereas it is possible to use rotation gates $Rx$, $Ry$ and $Rz$ on Spin-2 processor, Starmon-5 gate set consist only of:
- Pauli gates $\text{X}$, $\text{Y}$, $\text{Z}$ and $\text{I}$
- Hadamard gate $\text{H}$
- Phase gate ($\text{S}$) and $\pi/8$ ($\text{T}$) gate and their conjugate transpose gates
- rotation around axes $x$ and $y$ for angle $\pm\frac{\pi}{2}$ gates
- $\text{CNOT}$, $\text{CZ}$ and $\text{SWAP}$ gates
My question is how can I construct gates $Rx$, $Ry$ and $Rz$ with rotation angle $\theta$ from gate listed above.
EDIT:
Based on advice from JSdJ and the Nielsen and Chuang book, I found out that:
- $HTH = Rx(\pi/4)$ which together with the fact that $T=Rz(\pi/4)$ allows to build gate $$R_{\hat{n}}=Rz(\pi/4)Rx(\pi/4)= \text{exp}(-i\frac{\pi}{8}Z)\text{exp}(-i\frac{\pi}{8}X) = \\ \cos^2\frac{\pi}{8}I-i(\cos\frac{\pi}{8}(X+Z)+\sin\frac{\pi}{8}Y)\sin\frac{\pi}{8}$$ (note that $Y=-iZX$)
- this is a rotation around axis defined by vector $\hat{n}=(\cos(\pi/8);\sin(\pi/8);\cos(\pi/8))$. Rotation angle is given by equation $\cos(\theta/2) =\cos^2(\pi/8)$
- angle $\theta$ is $2\arccos[\cos^2(\pi/8)] \approx 1.096$ which is irrational multiple of $2\pi$
- since $\theta$ is irrational, repeated application of the gate $R_{\hat{n}}$ never leads to rotation by an angle which sum up to $2\pi$
- as a result we are able to reach any rotation by angle $\alpha$ around $\hat{n}$ axis by repeated application of $R_{\hat{n}}$, i.e. we can construct $R_{\hat{n}}(\alpha)=R_{\hat{n}}^{n_1}$, where $n_1$ is an integer
- also it is true that $HR_{\hat{n}}(\alpha)H = R_{\hat{m}}(\alpha)$ where $\hat{m}$ is axis defined by vector $(\cos(\pi/8);-\sin(\pi/8);\cos(\pi/8))$, so the same procedure allows to find rotation for $\alpha$ around axis $\hat{m}$
- it can also be proven that any arbitrary single qubit unitary matrix $U$ can be written (up to global phase) as $U = R_{\hat{n}}(\beta_1)R_{\hat{m}}(\gamma_1)R_{\hat{n}}(\beta_2)R_{\hat{m}}(\gamma_2)\dots$
- together this leads to conclusion that any $U$ (including $Rx$, $Ry$ and $Rz$ rotations) can be approximated by repeated application of $R_{\hat{n}}$ and Hadamards, i.e. $U \approx R_{\hat{n}}^{n_1}HR_{\hat{n}}^{n_2}HR_{\hat{n}}^{n_3}HR_{\hat{n}}^{n_4}H \dots$, where $n_i$ are integers
To sum up, only with $T$ and $H$ gates we can construct any rotation.
Only question is how to find integers $n_i$ for construction of arbitrary $Rx$, $Ry$ and $Rz$ rotation. Can anybody give me an example of such circuit, for exaple gate $Ry(\pi/8)$?
