How does interference help to bias the measurement of a qubit towards a desired state?

Question

I just read that :

A fundamental idea in quantum computing is to control the probability a system of qubits collapses into particular measurement states. Quantum interference, a byproduct of superposition, is what allows us to bias the measurement of a qubit toward a desired state or set of states. Please help me to understand how interference helps?

since the amplitudes is added during the constructive interference and resulting amplitude cannot be greater than 1 then what happens if the individual amplitude of waves is 0.6 each or something whose sum is greater than 1?

Answer 1

Maybe a simple example (modified from https://quantumcomputing.stackexchange.com/a/16701/14597) can help answer your question. Let us take a single qubit and see how interference can lead to a bias of the measurement outcome.

We start with a $0$ state for the qubit which can be represented as $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$. We then put it into a $50$-$50$ state by acting on it with the $H = \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix}$ gate which will give us $\begin{pmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{pmatrix}$. This means that there is no bias i.e. that you have exactly the same chance of measuring '$0$' and '$1$' if you make a measurement on this state. Now we act on this new state again with the $H$ operator which gives us

$\begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix}\begin{pmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{pmatrix} = \begin{pmatrix} \frac{1}{2} + \frac{1}{2}\\ \frac{1}{2} - \frac{1}{2} \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$

Here you can see the effect of constructive interference ($\frac{1}{2} + \frac{1}{2}$) and destructive interference ($\frac{1}{2} - \frac{1}{2}$), leading to this new state which will have a measurement outcome of '$0$' with a $100\%$ probability.

To answer your question about the sum of amplitudes being greater than 1, as mentioned by @Ohad, the probability is the square of the absolute value of the amplitude, so as long as the sum of the square of the absolute value of the amplitudes are equal to 1, then you are fine. It turns out that in quantum mechanics, since we always use unitary operators to represent the operator acting on our qubits, this will always be the case.

Answer 2

Generally speaking:

Each entry of a quantum statevector is a probability amplitude of a basis state. The probability to measure each basis state equals to the squared magnitude of the probability amplitude. All squared magnitudes must sum to 1 (It is called the normalization condition, and it makes a lot of sense because probabilities sums to 1 = 100%). The probability amplitudes, unlike regular probability (which varies from 0 to 1), can be positive, negative, or complex - in fact they are complex numbers.

By applying quantum gates we are manipulating the amplitudes of the statevector - Mathematically it is a vector (the statevector) multiplied by a matrix (the gate matrix operator). Each entry now is being manipulated by the corresponding row in the matrix. For example, If we let 2 amplitudes of 1/2 and -1/2 interfere with each other then destructive interference occurs and there will be 0 chance to measure the basis state that corresponds with that specific amplitude (probability amplitude of 0 = 0 probability). If we let 2 amplitudes of 1/2 and 1/2 interfere with each other then constructive interference occurs and there will be 100% chance to measure the basis state that corresponds with that specific amplitude.

Question:

Quantum interference, a byproduct of superposition, is what allows us to bias the measurement of a qubit toward a desired state or set of states. Please help me to understand how interference helps?

Answer:

Using superposition we can create very large quantum statevectors using very few qubits and gates. For example by applying Hadamard gates on just 20 qubits we create a quantum statevector with more than 1 million amplitudes and corresponding basis states. But, when we measure this 20-qubits register we get 1 unique result. For this result to be valuable we need to set things up such that the desired interferences will occur and a measurement will yield the right result.

Question:

since the amplitudes is added during the constructive interference and resulting amplitude cannot be greater than 1 then what happens if the individual amplitude of waves is 0.6 each or something whose sum is greater than 1?

Answer:

All quantum gates (put aside measure and reset gates which are something different) are unitary gates. One of the properties of unitary operators is that they preserves the norm, i.e the magnitude of the vectors being applied upon. So if the initial state has a norm of 1, applying quantum gates will always yield valid quantum statevectors with magnitude of 1.