The Interesting Phenomenon Of Phase Kickback

A lesson on a secret quantum trick that powers everything from Shor's Algorithm to Quantum Machine Learning.

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We learned about the CNOT gate a few weeks ago.

The CNOT gate, also known as the Controlled-NOT gate, is a two-qubit gate.

It receives two inputs and gives out two outputs.

The first input is called the Control qubit, and the second one is called the Target qubit.

The CNOT gate flips the state of the target qubit only if the control qubit is in the state |1>. Otherwise, it does nothing to the target.

In other words, it acts as a NOT gate or Pauli X-gate when the control qubit is |1>.

Operation of the CNOT gate on a two-qubit system, where the first qubit acts as a control qubit while the second is the target qubit.

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A Mind-Boggling Finding With The CNOT Gate

There’s something interesting that occurs when we apply the CNOT gate to qubits in a superposition state.

Consider the following equal superposition states:

These are alternatively written as:

A two-qubit system composed of these states |+-> is represented by the tensor product as follows:

Let’s see what happens when we apply the CNOT gate to |+->.

(The first qubit |+> is the control qubit, and the second qubit |-> is the target qubit in this case.)

The following changes take place due to the gate.

This results in:

This result can be further simplified to:

We know that:

So:

Substituting this in the tensor product:

Finally:

Did you notice what happened there?

Applying the CNOT gate to |+-> resulted in |-->.

This is weird because the control qubit just got flipped instead of the target qubit.

This phenomenon is called Phase Kickback. It is when the phase of the target qubit “kicks back” to the control qubit and changes it.

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When Does Phase Kickback Occur?

Phase kickback isn’t limited to the CNOT gate, but it requires three specific conditions to occur.

These are as follows:

1. The Control qubit must be in a superposition

In the case of applying the CNOT gate to |+->, the control qubit is |+>.

If it were |0> or |1> instead, the gate operation would end up changing its global phase, which would not be evident in the measurement probabilities.

It is only when the control qubit is in superposition that the relative phase between its components changes, which can be observed as Phase kickback.