3
$\begingroup$

$\newcommand{\ket}[1]{\vert#1\rangle}\newcommand{\bra}[1]{\langle#1\vert}$

Given a quantum circuit with 2 qubits that executes a controlled gate $CU$ where the control qubit is in the $\ket{+}$ state, and the target qubit is in an arbitrary state $\ket{\psi}$.

Intuitively, I would assume that this circuit can be decomposed into two circuits, one with initial state $\ket{0}\ket{\psi}$ that doesn't execute $U$ on the target qubit, and one with initial state $\ket{1}\otimes\ket{\psi}$ that executes $U$ on the target qubit. The idea is that the control qubit, when measured, will be 0 in 50% of the shots ($U$ is not executed) and 1 in the other 50% of the shots ($U$ is executed).

However, from doing the math, I came to the conclusion that this doesn't work:

Leaving the initial state as $\ket{+}\otimes\ket{\psi}$ does not work either:

I assume the phase kickback is the problem here.

Is there a way to to decompose the $CU$ gate into 1-qubit operations when the control qubit is in state $\ket{+}$?


My calculations:

Let $U=\left[\begin{matrix}u_{0} & u_{1}\\u_{2} & u_{3}\end{matrix}\right]$, $CU=\left[\begin{matrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & u_{0} & u_{1}\\0 & 0 & u_{2} & u_{3}\end{matrix}\right]$, and $\ket{\psi} = \begin{bmatrix} a \\ b \end{bmatrix} \\\\$.

\begin{align*} CU(\ket{+}\otimes\ket{\psi})(\bra{+}\otimes\bra{\psi})CU^\dagger &= \frac{1}{2}\left[\begin{matrix}a \overline{a} & a \overline{b} & a \left(\overline{a} \overline{u_{0}} + \overline{b} \overline{u_{1}}\right) & a \left(\overline{a} \overline{u_{2}} + \overline{b} \overline{u_{3}}\right)\\b \overline{a} & b \overline{b} & b \left(\overline{a} \overline{u_{0}} + \overline{b} \overline{u_{1}}\right) & b \left(\overline{a} \overline{u_{2}} + \overline{b} \overline{u_{3}}\right)\\\left(a u_{0} + b u_{1}\right) \overline{a} & \left(a u_{0} + b u_{1}\right) \overline{b} & \left(a u_{0} + b u_{1}\right) \left(\overline{a} \overline{u_{0}} + \overline{b} \overline{u_{1}}\right) & \left(a u_{0} + b u_{1}\right) \left(\overline{a} \overline{u_{2}} + \overline{b} \overline{u_{3}}\right)\\\left(a u_{2} + b u_{3}\right) \overline{a} & \left(a u_{2} + b u_{3}\right) \overline{b} & \left(a u_{2} + b u_{3}\right) \left(\overline{a} \overline{u_{0}} + \overline{b} \overline{u_{1}}\right) & \left(a u_{2} + b u_{3}\right) \left(\overline{a} \overline{u_{2}} + \overline{b} \overline{u_{3}}\right)\end{matrix}\right] \\ & \neq \frac{1}{2}\left[\begin{matrix}a \overline{a} & a \overline{b} & 0 & 0\\b \overline{a} & b \overline{b} & 0 & 0\\0 & 0 & \left(a u_{0} + b u_{1}\right) \left(\overline{a} \overline{u_{0}} + \overline{b} \overline{u_{1}}\right) & \left(a u_{0} + b u_{1}\right) \left(\overline{a} \overline{u_{2}} + \overline{b} \overline{u_{3}}\right)\\0 & 0 & \left(a u_{2} + b u_{3}\right) \left(\overline{a} \overline{u_{0}} + \overline{b} \overline{u_{1}}\right) & \left(a u_{2} + b u_{3}\right) \left(\overline{a} \overline{u_{2}} + \overline{b} \overline{u_{3}}\right)\end{matrix}\right] \\ & = \frac{1}{2} \left( (\ket{0}\otimes\ket{\psi})(\bra{0}\otimes\bra{\psi}) + (\ket{1}\otimes U\ket{\psi})(\bra{1}\otimes U\bra{\psi}) \right) \end{align*}

\begin{align*} CU(\ket{+}\otimes\ket{\psi})(\bra{+}\otimes\bra{\psi})CU^\dagger &= \frac{1}{2}\left[\begin{matrix}a \overline{a} & a \overline{b} & a \left(\overline{a} \overline{u_{0}} + \overline{b} \overline{u_{1}}\right) & a \left(\overline{a} \overline{u_{2}} + \overline{b} \overline{u_{3}}\right)\\b \overline{a} & b \overline{b} & b \left(\overline{a} \overline{u_{0}} + \overline{b} \overline{u_{1}}\right) & b \left(\overline{a} \overline{u_{2}} + \overline{b} \overline{u_{3}}\right)\\\left(a u_{0} + b u_{1}\right) \overline{a} & \left(a u_{0} + b u_{1}\right) \overline{b} & \left(a u_{0} + b u_{1}\right) \left(\overline{a} \overline{u_{0}} + \overline{b} \overline{u_{1}}\right) & \left(a u_{0} + b u_{1}\right) \left(\overline{a} \overline{u_{2}} + \overline{b} \overline{u_{3}}\right)\\\left(a u_{2} + b u_{3}\right) \overline{a} & \left(a u_{2} + b u_{3}\right) \overline{b} & \left(a u_{2} + b u_{3}\right) \left(\overline{a} \overline{u_{0}} + \overline{b} \overline{u_{1}}\right) & \left(a u_{2} + b u_{3}\right) \left(\overline{a} \overline{u_{2}} + \overline{b} \overline{u_{3}}\right)\end{matrix}\right] \\ & \neq \frac{1}{4}\left[\begin{matrix}a \overline{a} + \left(a u_{0} + b u_{1}\right) \overline{a} \overline{u_{0}} + \left(a u_{0} + b u_{1}\right) \overline{b} \overline{u_{1}} & a \overline{b} + \left(a u_{0} + b u_{1}\right) \overline{a} \overline{u_{2}} + \left(a u_{0} + b u_{1}\right) \overline{b} \overline{u_{3}} & a \overline{a} + \left(a u_{0} + b u_{1}\right) \overline{a} \overline{u_{0}} + \left(a u_{0} + b u_{1}\right) \overline{b} \overline{u_{1}} & a \overline{b} + \left(a u_{0} + b u_{1}\right) \overline{a} \overline{u_{2}} + \left(a u_{0} + b u_{1}\right) \overline{b} \overline{u_{3}}\\b \overline{a} + \left(a u_{2} + b u_{3}\right) \overline{a} \overline{u_{0}} + \left(a u_{2} + b u_{3}\right) \overline{b} \overline{u_{1}} & b \overline{b} + \left(a u_{2} + b u_{3}\right) \overline{a} \overline{u_{2}} + \left(a u_{2} + b u_{3}\right) \overline{b} \overline{u_{3}} & b \overline{a} + \left(a u_{2} + b u_{3}\right) \overline{a} \overline{u_{0}} + \left(a u_{2} + b u_{3}\right) \overline{b} \overline{u_{1}} & b \overline{b} + \left(a u_{2} + b u_{3}\right) \overline{a} \overline{u_{2}} + \left(a u_{2} + b u_{3}\right) \overline{b} \overline{u_{3}}\\a \overline{a} + \left(a u_{0} + b u_{1}\right) \overline{a} \overline{u_{0}} + \left(a u_{0} + b u_{1}\right) \overline{b} \overline{u_{1}} & a \overline{b} + \left(a u_{0} + b u_{1}\right) \overline{a} \overline{u_{2}} + \left(a u_{0} + b u_{1}\right) \overline{b} \overline{u_{3}} & a \overline{a} + \left(a u_{0} + b u_{1}\right) \overline{a} \overline{u_{0}} + \left(a u_{0} + b u_{1}\right) \overline{b} \overline{u_{1}} & a \overline{b} + \left(a u_{0} + b u_{1}\right) \overline{a} \overline{u_{2}} + \left(a u_{0} + b u_{1}\right) \overline{b} \overline{u_{3}}\\b \overline{a} + \left(a u_{2} + b u_{3}\right) \overline{a} \overline{u_{0}} + \left(a u_{2} + b u_{3}\right) \overline{b} \overline{u_{1}} & b \overline{b} + \left(a u_{2} + b u_{3}\right) \overline{a} \overline{u_{2}} + \left(a u_{2} + b u_{3}\right) \overline{b} \overline{u_{3}} & b \overline{a} + \left(a u_{2} + b u_{3}\right) \overline{a} \overline{u_{0}} + \left(a u_{2} + b u_{3}\right) \overline{b} \overline{u_{1}} & b \overline{b} + \left(a u_{2} + b u_{3}\right) \overline{a} \overline{u_{2}} + \left(a u_{2} + b u_{3}\right) \overline{b} \overline{u_{3}}\end{matrix}\right] \\ & = \frac{1}{2} \left( (\ket{+}\otimes\ket{\psi})(\bra{+}\otimes\bra{\psi}) + (\ket{+}\otimes U\ket{\psi})(\bra{+}\otimes U\bra{\psi}) \right) \end{align*}

$\endgroup$

1 Answer 1

1
$\begingroup$

We have, $$|+\rangle = \frac{1}{\sqrt{2}} |0\rangle + \frac{1}{\sqrt{2}} |1\rangle$$ So, you should replace $\frac{1}{2}$ by $\frac{1}{\sqrt{2}}$ here: enter image description here

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.