Quantum Computing Basics (Part II)

To understand quantum computing, we must first understand the tricky subject of quantum mechanics. Don't worry though, it isn't as hard as it seems.

In order for us to appreciate the complexities of quantum computing and how it might benefit us, we must first understand a few interesting mechanics of very small objects. Rather than providing a lengthy introduction to quantum mechanics, the elements necessary for the understanding of this series will be included in this article and the next.

The Qubit and its Superposition State


In a classical bit, high and low voltages are used to denote the binary data of 1’s and 0’s. However, a hypothetical quantum bit can represent such data by choosing any quantum system that have two states. The two frequently discussed possibilities are the two linear polarization states of a photon (horizontal and vertical):

Or the upward and downward spin state of an electron:

Choosing any one of these quantum systems, it can be noted that due to the prominent nature of the wave-particle duality in very small objects, the interference experienced by the wave-nature of a particle causes the particle to be in a superposition state where all possible states of that particle is simultaneously in existence. The time-varying characteristics of a quantum system can be roughly approximated by using Schrödinger’s equation as demonstrated by Hey [1, p.6], this would allow us to write the general state of a two-state quantum system as a superposition of its eigenstates:

Where different coefficients of α and ß allows us to represent any state of the qubit [2].

Wave Function Collapse and Qubit Measurements


In quantum mechanics, the fragile nature of a quantum system means that any interaction of the superposition with the external world would reduce all existing eigenstates to only one eigenstate. The “observation” of a wave function in order to associate it with classical properties such as position and momentum would therefore result in a thermodynamically irreversible collapse [3].


Based on this phenomenon, measuring a qubit will always produce an unpredictable eigenvalue that is from its original superposition. If, however, we were to prepare and repeat several identical systems, quantum mechanics assures us that the probability of observing a result of ‘0’ and ‘1’ on a qubit each time would be and respectively as the normalization of the wave function guarantees:

As such, it would be essential to know that the observation of a computation made out of any number of qubits and quantum algorithms would only return the right result with a certain probability. It is therefore appropriate for calculations to be repeated in order to assure accuracy. Although the notion that quantum computation doesn’t always provide the right result all the time may seem counter-intuitive in improving information technology, the right exploitation of superposition and quantum entanglement (which will be covered in section 3) would allow the computational process of quantum computers to be exponentially more efficient than normal computers [4].

Quantum Entanglement of Qubits


Perhaps one of the most peculiar mechanics of the quantum realm is the entanglement of particles where the quantum state of the two particles cannot be independently described. An example of this arises from the decay of an elementary particle at rest with a charge and spin of zero - such as a neutral pion. As this particle disintegrates, a spin ½ electron and spin ½ positron would be created (Fig. 1).

Figure 1: Entangled qubit pair created by neutral pion decay [1, p.9]

By conserving the angular momentum of the system, the two particles having half spins must together combine to form a spin state of zero. So, if we were to measure the positron spin to be down, the electron spin must be up - and vice versa. In this manner, by carefully setting up a system of entangled qubits, one can make use of this property to perform more complex computational algorithms.


End of Part II

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References

[1] T. Hey, “Quantum Computing: An Introduction”, Computing & Control Engineering Journal, vol. 10, no. 3, pp. 105-112, Jun. 1999.

[2] K. Pham, Z. Wang, G. Chen, D. Shen, E. Blasch and B. Jia, “Quantum Technology for Aerospace Applications”, Proceedings of SPIE, vol. 9085, Jun. 2014.

[3] M. Schlosshauer, “Decoherence, the measurement problem, and interpretations of quantum mechanics”, Reviews of Modern Physics, vol. 76, no. 4, pp. 1267-1305, Feb. 23, 2005.

[4] C. O’Connell, Quantum computing for the qubit curious, Cosmos Magazine, Jul. 6, 2019.

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