Abstract
Quantum Computing is a fascinating new field at the intersection of computer science, mathematics and physics. This field studies how to harness some of the strange aspects of quantum physics for use in computer science. Many of the texts to this field require knowledge of a large corpus of advanced mathematics or physics. We try to remedy this situation by presenting the basic ideas of quantum computing understandable to anyone who has had a course in pre-calculus or discrete structures. (A good course in linear algebra would help, but, the reader is reminded of many definitions in the footnotes.)
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Notes
- 1.
Although most texts might have \(M[i,j]=1\) if and only if there is an arrow from vertex i to vertex j, we shall need it to be the other way for reasons which will become apparent later. The difference is trivial.
- 2.
Although the theory works with any \(r\in [0,1]\), we shall deal only with fractions.
- 3.
The matrix B is not a doubly stochastic matrix. The sum of the weights entering vertex 0 is not 1. The sum of weights leaving vertices 3, 4, 5, 6, and 7 are more than 1. This fact should not bother you. We are interested in demonstrating the way probabilities behave with respect to these matrices.
- 4.
We remind the reader that if \(c=a+bi\) is a complex number, then its modulus is \(|c|=\sqrt{a^2+b^2}\) and \(|c|^2=a^2+b^2\).
- 5.
The important point here is that the modulus squared is positive. For simplicity of calculations, we have chosen easy complex numbers.
- 6.
Let us just remember: a matrix U is unitary if \(U\star U^\dag = I = U^\dag \star U\). The adjoint of U, denoted as U †, is defined as \(U^\dag=(\overline{U})^T= \overline{(U^T)}\) or \(U^\dag [j,k] = \overline{U[k,j]}\).
- 7.
The actual complex number weights are not our interest here. If we wanted to calculate the actual numbers, we would have to measure the width of the slits, the distance between the slits, the distance from the slits to the measuring devices etc. However, our goal here is to clearly demonstrate the interference phenomenon. And so we chose the above complex numbers simply because the modulus squared are exactly the same as the bullets case.
- 8.
This matrix is not a unitary matrix. Looking carefully at row 0, one can immediately see that P is not unitary. In our graph, there is nothing entering vertex 0. The reason why this matrix fails to be unitary is because we have not put in all the arrows in our graph. There are many more possible ways the photon can travel in a real-life physical situation. In particular, the photon might go from the right to the left. The diagram and matrix would become too complicated if we put in all the transitions. We are simply trying to demonstrate the interference phenomenon and we can accomplish that even with a matrix that is not quite unitary.
- 9.
Formally, the tensor product of matrices is a function
$$\otimes: \mathbb{C}^{m \times m} \times \mathbb{C}^{n \times n} \longrightarrow \mathbb{C}^{m \times m} \otimes \mathbb{C}^{n \times n}=\mathbb{C}^{mn \times mn}$$and it is defined as: \((A \otimes B)[j,k]=A[j/n,k/m]\times B[j \mbox{ MOD }n,k \mbox{ MOD }m].\)
- 10.
“Ket” is the second half of “bracket”. However we shall not use the “bra” part in our exposition.
References
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Acknowledgements
I am grateful to Dr. Mirco Mannucci for many helpful discussions and cheery editing sessions.
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Yanofsky, N.S. (2011). An Introduction to Quantum Computing. In: van Benthem, J., Gupta, A., Parikh, R. (eds) Proof, Computation and Agency. Synthese Library, vol 352. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0080-2_10
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