Quantum computing:  

Introduction: Certain computation tasks can be executed exponentially faster on quantum computers than on classical computers. Noted Physicist Richard Feynman suggested that it was exponentially costly to simulate quantum systems with classical computers. He envisioned a few benchmarks. First, the computational space must be large, and the error rate must be small such that a quantum speedup can be observed. Second, a problem that is hard for classical computers must be easy for quantum computers.  Such questions that frame a benchmark can be executed on a superconducting qubit processor. Algorithms that take full advantage of quantum computing are still a nascent field. 

In quantum computing, the basic unit of quantum information is a qubit. This is a two-state quantum mechanical system. In classical systems a bit must be 0 or 1. Quantum mechanics allows the qubit to be in a coherent superposition of both states simultaneously which allows a qubit to hold a lot more information than just two discrete states. This property is fundamental to quantum mechanics and quantum computing. For example, the polarization of a single photon can have both vertical polarization as well as horizontal polarization. The spin of an electron can be taken as spin up and spin down. The first step in the quantum data storage was demonstrated with the coherent transfer of a superposition state in an electron spin “processing” qubit to a nuclear spin “memory” qubit.  The transmission protocol for two classical bits of information such as 00, 01, 10, or 11 can be transmitted by sending only one qubit from a sender say Alice to a receiver, say Bob. This is called superdense coding. It requires Alice to perform one of four quantum gate operations to prearrange the measurement Bob makes. Next, Bob can decode the information being sent by receiving Alice’s qubit, decoding it, operating on the pair, and measuring both.  Since both qubits are required to decode the information, it eliminates the risk of spoofing, tampering, repudiation, information disclosure, denial of service, and elevation of privilege which is also referred to as the STRIDE model of threat analysis. An eavesdropper, Eve, cannot figure out Alices’ qubit or Bob’s qubit and any such attempt would alert both Alice and Bob.  

A system of n-components requires only n bits in classical physics whereas, in quantum physics, it requires Math.pow(2, n) complex numbers and is referred to as the Hilbert space. The quantum state of a qubit can be represented as a linear superposition of its basis vectors usually called ket0 and ket1 vectors. This transforms the original information on n components into the high-dimensional linear vector Hilbert space.  

Representing vectors and matrices and using factorization for matrices is well-established in classical systems and these can be repurposed in the Hilbert space. Newer algorithms for factoring a number like Shor’s algorithm are faster than their predecessors but realizing the full potential of quantum computing for Shor’s algorithm still requires certain obstacles to be overcome with fault-tolerant logical qubits.