21 Jan 2022

Post Quantum Cryptography - What is Quantum Computing?

Author: TCC  /  Categories: Blogs

Introduction
The purpose of this blog is to raise awareness of the impact of quantum computing on information and data security. For that matter, we plan to review, in a series of articles, topics about quantum computing, its possible impact and effectiveness on breaking current cryptographic algorithms, and a viable plan to mitigate the effects of the security collapse when the technology for producing powerful enough quantum computers becomes available.

What is Quantum Computing?
Present day computers are limited in speed by how much technology can shrink a transistor, so the chip makers can squeeze more and more transistors on a computer chip. If we could replace the smallest achievable transistor with another technology in which the classical laws of physics that govern today's computational capabilities are described by quantum physics at a nanoscopic level, we would be able to build another type of computers. These quantum computers that obey the laws of quantum mechanics would calculate things in ways that are unimaginable today. They will perform special quantum computations and will potentially solve certain problems (e.g., integer factorization) much faster than classical computers. Quantum computing will rely on the unique behavior of quantum physics and will take advantage of the properties of quantum states — superposition, interference, and entanglement. A quantum in physics is the smallest possible discrete unit of any physical property. Usually, it refers to properties of atomic or subatomic particles (e.g., electrons, neutrinos, and photons).

In classical computing, information is stored in basic units called bits. A bit holds a binary value, either 0 or 1. One analogy is flipping a coin — we get heads or tails. In quantum computing, the basic units known as qubits can hold a superposition of all possible states. To use the same analogy, we would be able to look at the coin and see both heads and tails at the same time, as well as every state in between; the coin would be in superposition. Thus, even though there are two possible outcomes for the measurement of a qubit (0 or 1), the general state of a qubit according to quantum mechanics is a coherent superposition of both. This property is fundamental to quantum computing.

To visualize a qubit, we need to introduce the concept of Bloch sphere [1].
Bloch sphere

In quantum mechanics, the general quantum state of a pure qubit is a coherent superposition of the basis states. In other words, it can be represented by a linear superposition of its two orthonormal basis states (or basis vectors) and (written in conventional Dirac notation): , where and are the probability amplitudes (complex numbers). When we measure the qubit in the standard basis, the probability of the outcome with value "0" is and the probability of the outcome with value "1" is , where  =1. Thus, the state space of the pure qubit states is represented by any point of the surface of the Bloch sphere. This state space has two local degrees of freedom, represented by the two angles and . Coherence is essential for a qubit to be in a superposition state. With interactions and decoherence (described in Part 2 of our blog), it is possible to put the qubit in a mixed state, a statistical combination or incoherent mixture of different pure states. Mixed states, represented by points inside the Bloch sphere (i.e., the Bloch ball), have three degrees of freedom: the angles and , as well as the length of the vector that represents the mixed state.

The second important property of quantum states is quantum interference, which is an intrinsic behavior of a qubit (due to superposition) that influences the probability of it "collapsing" one way or another (i.e., selecting a 0 or a 1 value). The problem is that measuring a classical bit does not disturb its state whereas measuring a qubit destroys its coherence and disturbs the superposition state. Therefore, quantum computers need to be designed to reduce this interference as much as possible in order to provide accurate results. Interesting enough, the and described above encode more than just the probabilities of the outcomes of a measurement — the relative phase between and is responsible for quantum interference.

Finally, entanglement is a distinguishing feature between qubits and classical bits. It is the ability of quantum particles to correlate their measurement results with each other. When qubits are entangled, they form a single system and influence each other. Measurements from one qubit draw conclusions about other qubits, such that adding and entangling more qubits allows quantum computers to solve more complicated problems. It has been proved that for any quantum algorithm operating with pure states, the presence of entanglement is necessary to offer a computational speedup over classical computing. However, some computational speedup was also possible for algorithms operating with mixed states even in the total absence of entanglement. These findings are important because it is still not very clear what the sufficient condition is to reach computational advantage. Moreover, real quantum systems are rarely in pure states and they are continuously interacting with their environments.

The first part of our blog explained the difference between bits used in classical computers and qubits used in quantum computers. Also, it briefly mentioned the properties of quantum states (e.g., superposition, interference, and entanglement) that make qubits so different from regular bits, but so useful to quantum algorithms because qubits can hold an exponential number of states using superposition and entanglement.

Stay tuned for Part 2 of our blog where we talk about the challenges of building quantum computers.

Bibliography
[1] "Wikipedia," Dec 2021. [Online]. Available: https://en.wikipedia.org/wiki/Bloch_sphere

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