Qubit

 There’s a famous scene in The Big Bang Theory where Penny asks Sheldon if she should pursue relationship with Leonard. Sheldon explains Penny about the Schrodinger’s cat. It was a thought experiment proposed by Schrodinger to explain superposition in Quantum mechanics. So, what’s Schrodinger’s cat? Imagine a sealed box containing a cat and vile of poison. As long as the box is sealed, we don’t know if the cat is dead because the vile broke or alive. But, if we open the box, the vile of poison could break due to our action and kill the cat. It might not break and the cat could be still alive. So, as long as the box remains closed, the cat can be thought as being both dead and alive at the same time. This peculiar concept is called superposition. It is a key concept we need to know to understand about Qubits.

 

Wait, what’re qubits? Just as how classical computer works using bits, a quantum computer works using Qubits. Qubits are units of information which when measured can give one of the two states. For example, in the Schrodinger’s cat thought experiment, cat being alive is one state and it being dead is another. Let’s denote these states as |0> and |1>(This is Dirac’s notation to represent quantum states). Just as in the experiment, a Qubit can stay in superposition of both the states. What do I mean when I say it can stay as a superposition of two states? Imagine someone created a million copies of the same Qubit. If you measure the value of all the Qubits, some of them can become |0>, others |1>. The state of this Qubit can be represented by

 

S = a |0> + b |1>

 

Quantum mechanics states that the probability that measurement yield |0> is given by square of a and the probability of |1> is b squared. Since the measurement yields either |0> or |1> and probabilities should add up to 1, 

 

a^2 + b^2 = 1. 

 

Also, it is impossible to predict before hand if the measurement yields a |0> or |1> if neither of a nor b is 0. In the above equation, a and b are both complex numbers. So, a Qubit can technically be in infinite possible states even if the measurement always yields a |0> or |1>

 

Why should a and b be complex? Why not real numbers? The state of any entity(electron/photon/muon etc) is given by a wave function. We can imagine wave function as a function of position and time. What it means is that, the probability of finding an entity at a position depends upon the time when it is observed and position where we’re observing. In the young’s double slit experiment conducted with electrons(photons), these entities behaved as if they’re waves. Any wave is characterised by two properties, amplitude and phase. That’s the reason why a complex number is better suited to represent the coefficients of wave function rather than a real number as it represents two properties(real component and imaginary component). Alternatively, you can use two real numbers for each state(one cosine component and one sine component). We are choosing the former.

 

These Qubits are visualised using a Bloch sphere. What’s Bloch sphere and how to understand it will be discussed in the next post