Quantum Computing Simplified Part I: Introduction

Quantum Computing is an area of computing focused on developing computer technologies governed by the laws of Quantum Mechanics (It is a branch of physics dealing with the behavior of matter and light at a subatomic level that are subject to Schrodinger’s Uncertainty Principle) to store data and perform computations.

• A Quantum Computer employs something known as ‘Qubits’.
• A Quantum Computation consists of a sequence of operations performed on ‘Quantum States’.
• The operations and states can be represented by matrices and vectors, respectively, and are required by the underlying physics (Quantum Mechanics) to follow the rules of Linear Algebra.

Linear Algebra is the Language of Quantum Computing

It is therefore crucial to develop a good understanding of the basic mathematical concepts that linear algebra is built upon, in order to arrive at many of the amazing and interesting constructions seen in quantum computation.

Analogy

Quantum Computing/Mechanics -> Linear Algebra

• Wave Function (Quantum State) -> Vector
• Linear Operator -> Matrix
• Eigenstates -> Eigenvectors
• Physical System -> Hilbert Space
• Physical Observable -> Hermitian Matrix

Uses of Quantum Computing

Quantum Computing has a plethora of potential use-cases a few of which are as illustrated below:

Healthcare

• Research & Drug Development
• Diagnostics
• Treatment

Financial Modelling

• Automated, High Frequency Trading

Don’t forget to check my blog on Artificial Intelligence for Trading!

Cybersecurity & Cryptography

Computational Chemistry

Machine Learning & Deep Learning

More on: A Beginner’s Guide for Getting Started with Machine Learning

Marketing

• Big Data Analytics

Meteorology & Weather Forecasting

Logistic Optimization

The history of the universe is, in effect, a huge and ongoing quantum computation. The universe is a quantum computer.

• Quantum computers are great for solving optimization problems from figuring out the best way to schedule flights at an airport to determining the best delivery routes for the FedEx truck.
• Quantum systems are exponentially powerful. A system of 500 particles has 2⁵⁰⁰ “computing power”.
• Quantum Computers provide a neat shortcut for solving a range of mathematical tasks known as NP-complete problems.
• For example, factorization is an exponential time task for classical computers.
• But Shor’s quantum algorithm for factorization is a polynomial time algorithm. It has successfully broken RSA cryptosystem.

Motivations for Quantum Computation

• Faster than light (?) communication.
• Highly parallel and efficient Quantum Algorithms.
• Quantum Cryptography and many more..

How do Quantum Computers process a lot of information at a faster rate?

Quantum Superposition

Bit vs Qubit Illustration

• In classical computation, bits are either represented with 0 or 1.
• In Quantum Computation, these bits are replaced by a superposition of both 0 and 1.
• If a quantum system can be in one of k states, it can also be in any linear superposition of those k states.
• Two level systems are called Qubits (k=2).

These are called the basis states, the superposition is denoted as a linear combination of these basis.

In Quantum Mechanics the “Intensity” (Amplitude Squared) gives the probability of finding the particle in a particular position

Qubits

• In Quantum Computing, a Quantum Bit or a ‘Qubit’ is a building block or a basic unit of Quantum Information.
• We have various physical interpretations of a Qubit in Quantum Mechanics.
• Qubits are represented as |0⟩ and |1⟩ or their linear combination.
• Qubits have been created in the laboratory using photons, ions and certain sorts of atomic nuclei.

Illustration of Hydrogen Atom

• A Hydrogen atom could be interpreted as a Qubit. An electron in its ground state, could be represented as |0⟩ and in its first energy state as |1⟩.
• The electron dwells in some linear superposition of these two energy levels. But during measurement, it can only be found in one of these energy states.

Physical Interpretation of Qubits

Photon Polarization: The orientation of electrical field oscillation is either horizontal or vertical.

Photo Polarization

Electron Spin: The electron spin is either up or down.

Electron Spin

Representation of Qubits

Matrix Representation of |0⟩ and |1⟩:

Qubits: Matrix Representation

A Qubit is Mathematically represented as a Quantum State of the form: ∣ψ⟩ = α∣0⟩ + β∣1⟩

Mathematical Illustration of Quantum State

Non-Determinism in Quantum Mechanics

An Attempt at 3rd Postulate of Quantum Mechanics

• Unlike classical physics, measurement in quantum mechanics is not deterministic.
• Even if we have the complete knowledge of a system, we can at most predict the probability of a certain outcome from a set of possible outcomes.

• If we have a quantum state ∣ψ⟩ = α∣0⟩ + β∣1⟩, then the probability of getting outcome ∣0⟩ is ∣α∣² and that of ∣1⟩ is ∣β∣².
• After measurement, the state of the system collapses to either ∣0⟩ or ∣1⟩ with the said probability.
• However, after measurement if the new state of the system is ∣0⟩ (say) then further measurements in the same basis gives outcome ∣0⟩ with the probability of 1.

The postulate states that a quantum system stays in a superposition when it is not observed. When a measurement is done, it immediately collapses to one of its eigenstates (eigenstate [physics] is a dynamic quantum mechanical state whose wave function is an eigenvector that corresponds to a physical quantity).

Hence, we can never observe what the original superposition of the system was. We merely observe the state after it collapses.

This inherent ambiguity provides an excellent security in Quantum Cryptography.

Imagine to have 500 qubits, then 2⁵⁰⁰ complex coefficients describe their state.

How to store this state?

• 2⁵⁰⁰ is larger than the number of atoms in the universe.
• It is impossible in classical bits.
• This is also why it is hard to simulate quantum systems on classical computers.

A quantum computer would be much more efficient than a classical computer at simulating quantum systems.

Superposition and Entanglement are what give quantum computers the ability to process so much more information so much faster.

Stay tuned for the upcoming parts on Quantum Computing Simplified!

References

• https://en.wikipedia.org/wiki/Quantum_computing
• https://cse.iitkgp.ac.in/~goutam/quantumComputing/lect4part.pdf
• David J. Griffiths Introduction to Quantum Mechanics
• MIT OpenCourseWare
• https://qudev.phys.ethz.ch/static/content/courses/QSIT08/QSIT08_V03_2page.pdf
• http://community.middlebury.edu/~ngraham/math.pdf
• https://www.academia.edu/4425386/A_Brief_Introduction_to_Linear_Algebra_and_Quantum_Mechanics_for_Quantum_Computation
• N. David Mermin, Cornell University Lecture Notes on Quantum Computation