Echoed Cross-Resonance gate
| Identifier | Operator | Example statement |
|---|---|---|
| ECR | \(ECR\) | ECR q[0], q[1] |
Description
The echoed cross-resonance, or ECR, gate is a two-qubit gate. It is an entangling two-qubit gate commonly used in superconducting QPUs, such as IBM quantum systems.
Properties
- Perfect Entangler (maximally entangles specific product states);
- Maximum Entangling Power over all uniformly random product states;
- Ising gate.
Representation
\[\begin{align}
ECR &= \frac{1}{\sqrt{2}} \left(\begin{matrix}
0 & 0 & 1 & i \\
0 & 0 & i & 1 \\
1 & -i & 0 & 0 \\
-i & 1 & 0 & 0
\end{matrix}\right)
\end{align}\]
Operation examples
Standard basis
\[\begin{align}
ECR\,|00\rangle &= \tfrac{1}{\sqrt{2}} |01\rangle - \tfrac{i}{\sqrt{2}} \,|11\rangle \\
\\
ECR\,|01\rangle &= \tfrac{1}{\sqrt{2}} |00\rangle + \tfrac{i}{\sqrt{2}} \,|10\rangle \\
\\
ECR\,|10\rangle &= -\tfrac{i}{\sqrt{2}} \,|01\rangle + \tfrac{1}{\sqrt{2}} |11\rangle \\
\\
ECR\,|11\rangle &= \tfrac{i}{\sqrt{2}} \,|00\rangle + \tfrac{1}{\sqrt{2}} |10\rangle \\
\end{align}\]
Qubit state ordering convention and matrix representation
Note that qubits in a ket are ordered with qubit indices decreasing from left to right, i.e.,
\[|\psi\rangle = \sum c_i~|q_nq_{n-1}~...q_1q_0\rangle_i\]
Note that for matrices a reversed basis ordering convention is adopted, as is done in most textbooks. For instance, in the case of a two-qubit control gate, the matrix is represented such that \(q_0\) is the control qubit and \(q_1\) is the target qubit; the state on which the matrix is applied should then effectively be written as \(|q_0\rangle \otimes |q_1\rangle\).