How to create an Ising coupling gate with Qiskit?

Question

I'm trying to apply a time evolution algorithm for a physical system I'm trying to simulate on QISkit, however, in order to do that, I need to use the so-called Ising coupling gate:

$I=\begin{pmatrix} e^{ia} & 0 & 0 &0 \\ 0 & e^{-ia} & 0 & 0 \\ 0 & 0 & e^{-ia} & 0 \\ 0 & 0 & 0 & e^{ia} \end{pmatrix}$

I've tried performing rotations in the z-axis in both quits with the rz gate, also I've tried combining crz gates, as well as rzz and cu1 gates, but nothing seems to work.

The closest I could get was by implementing a zzz gate followed by a cu1 gate with oposite angle, however $[I]_{1,1}$ still remains at 1, no phase detected by the Aer unitary simulator.

How can I implement this gate?

Thank you very much in advance.

Answer 1

AHusain's answer is absolutely correct, but perhaps lacks some detail. The circuit that you want to implement is Basically, the key is to realise that you want to apply phase $e^{i\alpha}$ to the basis elements $|00\rangle$ and $|11\rangle$, and $e^{-i\alpha}$ otherwise. In other words, you care about the parity of the two bits. If you can compute that parity of the two bits somewhere, you can perform a phase gate on that output, then undo the computation. Controlled-not computes the parity.

Here, I'm assuming that $$ R_z(\alpha)=\left(\begin{array}{cc} e^{i\alpha} & 0 \\ 0 & e^{-i\alpha} \end{array}\right). $$ This might be inconsistent with whatever definition you wish to use by a global phase or by a factor of 2 on the angle.

Answer 2

Conjugate by a CNOT, you'll see a controlled unitary of multiplying by $e^{\pm 2i a}$ depending on which CNOT you do and an overall phase of $e^{\mp i a}$.

Answer 3

I guess what you may need is the Pauli operators:

https://qiskit.org/documentation/_modules/qiskit/quantum_info/operators/pauli.html

https://www.sciencedirect.com/topics/engineering/pauli-operator

Answer 4

rzz gate works, if you want to apply rzz gate between qubit q0 and q1 for a circuit named as circ, you write

$$ \mathrm{circ.rzz}(\theta, q0,q1)$$

This performs $\theta$ rotation for the two-qubit state,by applying the following matrix \begin{equation} R_{Z Z}(\theta)=\exp \left(-i \frac{\theta}{2} Z \otimes Z\right)=\left(\begin{array}{cccc} e^{-i \frac{\theta}{2}} & 0 & 0 & 0 \\ 0 & e^{i \frac{\theta}{2}} & 0 & 0 \\ 0 & 0 & e^{i \frac{\theta}{2}} & 0 \\ 0 & 0 & 0 & e^{-i \frac{\theta}{2}} \end{array}\right) \end{equation}

Of course, not all quantum computers has 'rzz' as a native gate, in that case you need to decompose it in terms of the native gates. But for simulation purpose this works.