Phase Kickback

Part of Womanium Quantum + AI 2024 program

Definition

Phase kickback is a unique quantum effect, where the controlled bit in controlled gates gets affected by the change to the target bit, this change happens to the phase of the control bit.

Phase Kickback with a CNOT Gate

When applying a CNOT gate to a system in the state $| + - ⟩$ such that the first bit, $| + ⟩$ , is the control bit and the second bit, $| - ⟩$ , is the target bit, a phase kickback occurs flipping the phase of the control bit.

This can be shown by manually applying the gate to the system as follows:

(\begin{matrix} \frac{1}{\sqrt{2}} | 0 ⟩ + \frac{1}{\sqrt{2}} | 1 ⟩ \end{matrix}) \otimes (\begin{matrix} \frac{1}{\sqrt{2}} | 0 ⟩ - \frac{1}{\sqrt{2}} | 1 ⟩ \end{matrix})

The state of the second bit will only change when the control bit is in the $| 1 ⟩$ state, so we rewrite the system state to show that effect:

= \frac{1}{\sqrt{2}} | 0 ⟩ \otimes (\begin{matrix} \frac{1}{\sqrt{2}} | 0 ⟩ - \frac{1}{\sqrt{2}} | 1 ⟩ \end{matrix}) + \frac{1}{\sqrt{2}} | 1 ⟩ \otimes (\begin{matrix} \frac{1}{\sqrt{2}} | 0 ⟩ - \frac{1}{\sqrt{2}} | 1 ⟩ \end{matrix})

Now we apply the CNOT gate

\begin{aligned} \overset{I \otimes CNOT}{\to} & \frac{1}{\sqrt{2}} | 0 ⟩ \otimes (\begin{array}{c} \frac{1}{\sqrt{2}} | 0 ⟩ - \frac{1}{\sqrt{2}} | 1 ⟩ \end{array}) + \frac{1}{\sqrt{2}} | 1 ⟩ \otimes (\begin{array}{c} \frac{1}{\sqrt{2}} | 1 ⟩ - \frac{1}{\sqrt{2}} | 0 ⟩ \end{array}) \\ = & \frac{1}{\sqrt{2}} | 0 ⟩ \otimes (\begin{array}{c} \frac{1}{\sqrt{2}} | 0 ⟩ - \frac{1}{\sqrt{2}} | 1 ⟩ \end{array}) + \frac{1}{\sqrt{2}} | 1 ⟩ \otimes (\begin{array}{c} \frac{1}{\sqrt{2}} | 1 ⟩ - \frac{1}{\sqrt{2}} | 0 ⟩ \end{array}) \\ = & \frac{1}{\sqrt{2}} | 0 ⟩ \otimes (\begin{array}{c} \frac{1}{\sqrt{2}} | 0 ⟩ - \frac{1}{\sqrt{2}} | 1 ⟩ \end{array}) - \frac{1}{\sqrt{2}} | 1 ⟩ \otimes (\begin{array}{c} \frac{1}{\sqrt{2}} | 1 ⟩ - \frac{0}{\sqrt{2}} | 1 ⟩ \end{array}) \\ = & (\begin{array}{c} \frac{1}{\sqrt{2}} | 0 ⟩ - \frac{1}{\sqrt{2}} | 1 ⟩ \end{array}) \otimes (\begin{array}{c} \frac{1}{\sqrt{2}} | 0 ⟩ - \frac{1}{\sqrt{2}} | 1 ⟩ \end{array}) \\ = & | - ⟩ \otimes | - ⟩ \\ = & | - - ⟩ \end{aligned}

We can see that the phase of the target was kicked back to the control bit, since we changed the global phase of the target when the control was in the $| 1 ⟩$ state.

Finally we apply the Hadamard gate, and observe the change in the state:

\overset{HH}{\to} | 11 ⟩

The following shows a circuit for phase kickback with a CNOT gate.

Phase Kickback with a Unitary Operator

Phase kickback also shows in other places, like when using a function implemented as a unitary operator.

Assume we have an operator, $B_{f}$ , that applies it's function on $n$ bits and outputs 1 bit.

We usually feed the operator one extra bits in the $| 0 ⟩$ state to store the output in. However what if this extra bit was in $| - ⟩$ state?

Let's apply that and observe the result.

\begin{aligned} | a ⟩ | - ⟩ \overset{B_{f}}{\to} & B_{f} | a ⟩ | - ⟩ = B_{f} | a ⟩ \frac{1}{\sqrt{2}} (| 0 ⟩ - | 1 ⟩) = \frac{1}{\sqrt{2}} (B_{f} | a ⟩ | 0 ⟩ - B_{f} | a ⟩ | 1 ⟩) \\ = & \frac{1}{\sqrt{2}} (| a ⟩ | 0 \oplus f (a) ⟩ - | a ⟩ | 1 \oplus f (a) ⟩) \\ = & \frac{1}{\sqrt{2}} | a ⟩ (| 0 \oplus f (a) ⟩ - | 1 \oplus f (a) ⟩) \end{aligned}

Now let's look at the possible values of $f (a)$ and their effect

f (a) = {\begin{cases} 0 & \to | 0 ⟩ - | 1 ⟩ = | - ⟩ \\ 1 & \to | 1 ⟩ - | 0 ⟩ = - | - ⟩ \end{cases}

We notice that the function won't have an effect on the state of the $| - ⟩$ state, but it will change it's global phase if $f (a) = 1$ . Let's represent this change in the phase as follows:

= (- 1)^{f (a)} | a ⟩ | - ⟩

Info

The following is used as direct formula to apply phase kickback

B_{f} | a ⟩ | - ⟩ = (- 1)^{f (a)} | a ⟩ | - ⟩

Again, we see the phase is being kicked back to the "control" without affecting the target. We can benefit from that by neglecting the extra bit and keeping the result of the function stored in the phase of $| a ⟩$ .

The following shows a generalization for applying $B_{f}$ to a system.

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