Quantum Mechanics and quantum computation

First of all, Why quantum computing? Haven’t classical computers achieved great speed and efficiency already?

Well the answer is we need more speed and efficiency. Quantum computing provides more speed and efficiency, allows reversible computation and even solves algorithms faster. In fact there exists a field of study called quantum algorithms which focus on the computational power of quantum systems and in some cases these algorithms can solve problems in polynomial time while classical algorithms require super polynomial time to solve problems.

In the given article, I will try to give a brief introduction to quantum computing and the basis of creating a quantum computer. I’ll try being as less technical as possible, without going into much derivations and detailed analysis , rather an overview of some of the basic principles like the concept of qubits and move on to quantum gates with a brief outline of universal gates.

Given below are some odd features of quantum particles:

• Complete knowledge of a system’s state is forbidden - A measurement reveals only a small amount of information about the quantum state of the system.

• The act of measuring a particle fundamentally disturbs its state.

• Quantum entities do not have trajectories. All that we can say for an elementary particle is that it started at A and was measured later at B. We cannot say anything about the trajectory or path it took from A to B.

• Quantum Mechanics is inherently probabilistic. If we prepare two elementary particles in identical states and measure them, the results may be different for each particle.

• Quantum entities behave in some ways like particles and in others like waves. But they really behave in their own unique way, neither particles nor waves.

These are the reasons that led Richard Feynman to say “I can safely say no one understands quantum mechanics”

To begin with lets start with some basic notations. I’ll just give some basis to show the concepts of quantum gates and an example of quantum algorithm to show it’s computational efficiency.

Qubit

Let us consider a thought experiment of considering an electron as a bit such that the ground state is considered as 0 and excited state as 1. The state of the system can be written in Dirac notation as
|ψ> = α |0> + β |1> with α, β ∈ C and |α|² + |β^|2 = 1

Here α and β are the complex amplitudes of the system and the square of the magnitude represents the probability of finding the electron at a particular state.
(Simple I hope… just the basic thing that the sum of the probabilities of finding an object is always =1).
The system can be represented on the complex vector space as shown:

Let |ψ>  be the vector which is being measured. In order to measure we need some reference, which is called the basis vector. The most common basis vectors used for measurement are the |0>, |1> basis and also the |+>, |-> basis which are at 45 degrees to the |0>, |1> basis.
It can be carried out in any arbitrary basis. I’m not going It is important to note that any state of the system is obtained by a rotation of the vectors in the circle, i.e. if we can obtain the future state or rather the evolution of the qubit in time by rotating it in the circle shown above.

Quantum Gates

I’ll try to keep the discussion simple by focusing on single qubit gates, just to give an understanding of how such systems work. Some examples of quantum gates are as follows:

Bit Flip Gate (X)

This flips a bit from 0 to 1 and vice versa, something like the NOT gate in the classical case.

Phase Flip Gate (Z)

 

It is like a NOT gate but unlike bit flip it acts in the +, - basis I ‘d shown earlier.

Hadamard Transform (H)

In addition, there are 2 qubit gates like CNOT and CSWAP gates which act as quantum gates.

Universal Gates

Just like the universal NAND gate, the quantum analogue lies in the gates like CNOT,X, Z, H, π/8 rotations through which we can construct any given circuit.[1]

Quantum Algorithms and efficiency

To round up this article, let’s look at an example of the improvement in efficiency of by using quantum algorithms.

Parity Problem: Given a function f: {0, 1}ⁿ -> {0,1} as a black box such that f(x)=u ^. x for u € {0,1}ⁿ, the problem is to figure out u with as few queries to f as possible.

Complexity of Solution:
For a recursive version of the parity problem,

• ·    Classical algorithms satisfy the recursion in T(n) > nT(n/2) + ni.e. T(n)=Ω (n^logn)
• ·         Quantum algorithm satisfies recursion in  T(n) = 2T(n/2) + O(n)i.e. T(n)= O (n log n)