Quantum computing -- UCC ansatz

Unitary Coupled Clusters (UCC) ansatz

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The UCC ansatz writes:

\[\begin{equation} \ket{\Psi}_{\text{UCC}} = {e}^{\hat{T} - \hat{T}^{\dagger}} \ket{\Psi}_{\text{HF}} \end{equation}\]

where \({ \hat{T} = \hat{T}_{1} + \hat{T}_{2} + ... }\) with \({ \hat{T}_{i} }\) is an operator exciting \({ i }\) particles out of the Hartree-Fock (HF) reference state. In the “single” case (\({ i = 1 }\)) and for only two shells labeled by \({ n = 0,1 }\), one has:

\[\begin{equation} {e}^{\hat{T} - \hat{T}^{\dagger}} = {e}^{ {a}_{0}^{\dagger} {a}_{1} - {a}_{1}^{\dagger} {a}_{0} } \end{equation}\]

As such, this ansatz cannot be used together with the VQE method (see link) and must be parametrized. To this end, one simply adds a parameter \({ \theta }\) in the exponential.

\[\begin{equation} U(\theta) = {e}^{ \theta ( a_0^{\dagger} a_1 - a_1^{\dagger} a_0 ) } \end{equation}\]

Moreover, the creation and annihilation operators can transformed into the \({ X }\), \({ Y }\), and \({ Z }\) gates acting on qubits using the Jordan-Wigner transformation:

\begin{equation} \begin{aligned} & \hat{a}_n^{\dagger} \to \frac{1}{2} \left[ \prod_{j=0}^{n-1} (-Z_j) \right] ( X_n - i Y_n ) \\\
& \hat{a}_n \to \frac{1}{2} \left[ \prod_{j=0}^{n-1} (-Z_j) \right] ( X_n + i Y_n ) \end{aligned} \end{equation}

It follows that:

\begin{equation} \begin{aligned} & \hat{a}_0^{\dagger} \to \frac{1}{2} ( X_0 - i Y_0 ) \\\
& \hat{a}_0 \to \frac{1}{2} ( X_0 + i Y_0 ) \\\
& \hat{a}_1^{\dagger} \to \frac{1}{2} (-Z_0) ( X_1 - i Y_1 ) \\\
& \hat{a}_1 \to \frac{1}{2} (-Z_0) ( X_1 + i Y_1 ) \end{aligned} \end{equation}

and hence:

\begin{equation} \begin{aligned} {a}_0^{\dagger} a_1 - a_1^{\dagger} a_0 &\to \frac{1}{2} ( X_0 - i Y_0 ) \frac{1}{2} (-Z_0) ( X_1 + i Y_1 ) - \frac{1}{2} (-Z_0) ( X_1 - i Y_1 ) \frac{1}{2} ( X_0 + i Y_0 ) \\\
&= \frac{1}{4} \left[ ( X_0 X_1 + i X_0 Y_1 - i Y_0 X_1 + Y_0 Y_1 ) (-Z_0) + Z_0 ( X_1 X_0 + i X_1 Y_0 - i Y_1 X_0 + Y_1 Y_0 ) \right] \\\
&= \frac{1}{4} \left[ X_1 ( Z_0 X_0 - X_0 Z_0 ) + Y_1 ( Z_0 Y_0 - Y_0 Z_0 ) + i Y_1 ( Z_0 Y_0 + Y_0 Z_0 ) - i Y_1 ( Z_0 Y_0 + Y_0 Z_0 ) \right] \\\
&= \frac{i}{2} ( X_1 Y_0 - Y_1 X_0 ) \end{aligned} \end{equation}

It appears that the operator in the exponential writes:

\begin{equation} \begin{aligned} X_1 Y_0 - Y_1 X_0 = -2 \begin{pmatrix} 0 & 0 & 0 & 0 \\\
0 & 0 & -i & 0 \\\
0 & i & 0 & 0 \\\
0 & 0 & 0 & 0 \end{pmatrix} \end{aligned} \end{equation}

This is equivalent to a \({ Y }\) gate in the subspace defined by \({ \ket{01} }\) and \({ \ket{10} }\), and so the exponential operator is equivalent to a rotation operator around the \({ y }\) axis in this subspace. One can then emulate the effect of this operator by starting from the HF reference state \({ \ket{00} }\), flip one qubit to \({ \ket{1} }\) using a \({ X }\) gate, apply a rotation \({ {R}_{Y}(\theta) = {e}^{-i \frac{\theta}{2} Y } }\) on the second qubit (also noted \({ Y(\theta) }\)), and finally flip the first qubit to \({ \ket{0} }\) only if the second qubit is \({ \ket{1} }\) using a CNOT gate from 2 to 1 (also noted \({ \text{CNOT}_{01} }\) instead of \({ \text{CNOT}_{10} }\) for usual one). This is how the UCC ansatz was simplified in Phys. Rev. Lett. 120, 210501 (2018) (for details see link).

This results in the following wave function after the first operations:

\begin{equation} \begin{aligned} \ket{\Psi(t_1)} &= X \ket{0} \otimes {R}_{y}(\theta) \ket{0} \\\
&= \begin{pmatrix} 0 & 1 \\\
1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\\
0 \end{pmatrix} \otimes \begin{pmatrix} \cos(\frac{\theta}{2}) & -\sin(\frac{\theta}{2}) \\\
\sin(\frac{\theta}{2}) & \sin(\frac{\theta}{2}) \end{pmatrix} \begin{pmatrix} 1 \\\
0 \end{pmatrix}\\\
&= \ket{1} \otimes \left( \cos(\frac{\theta}{2}) \ket{0} + \sin(\frac{\theta}{2}) \ket{1} \right) \\\
&= \cos(\frac{\theta}{2}) \ket{10} + \sin(\frac{\theta}{2}) \ket{11} \end{aligned} \end{equation}

and then after the two-quibit gate one has:

\begin{equation} \begin{aligned} \ket{\Psi(t_2)} &= \text{CNOT}_{01} \ket{\Psi(t_1)} \\\
&= \begin{pmatrix} 1 & 0 & 0 & 0 \\\
0 & 0 & 0 & 1 \\\
0 & 0 & 1 & 0 \\\
0 & 1 & 0 & 0 \end{pmatrix} \left[ \cos(\frac{\theta}{2}) \begin{pmatrix} 0 \\\
0 \\\
1 \\\
0 \end{pmatrix} + \sin(\frac{\theta}{2}) \begin{pmatrix} 0 \\\
0 \\\
0 \\\
1 \end{pmatrix} \right] \\\
&= \cos(\frac{\theta}{2}) \begin{pmatrix} 0 \\\
0 \\\
1 \\\
0 \end{pmatrix} + \sin(\frac{\theta}{2}) \begin{pmatrix} 0 \\\
1 \\\
0 \\\
0 \end{pmatrix}\\\
&= \cos(\frac{\theta}{2}) \ket{10} + \sin(\frac{\theta}{2}) \ket{01} \end{aligned} \end{equation}

The UCC ansatz wave function looks like a parametrized Bell state:

\begin{equation} \ket{\Psi}_{\text{UCC}} = \cos(\frac{\theta}{2}) \ket{10} + \sin(\frac{\theta}{2}) \ket{01} \end{equation}