By Phillip Kaye, Raymond Laflamme, Michele Mosca
This concise, available textual content presents a radical advent to quantum computing - a thrilling emergent box on the interface of the pc, engineering, mathematical and actual sciences. geared toward complex undergraduate and starting graduate scholars in those disciplines, the textual content is technically targeted and is obviously illustrated all through with diagrams and routines. a few previous wisdom of linear algebra is believed, together with vector areas and internal items. in spite of the fact that, earlier familiarity with issues comparable to tensor items and spectral decomposition isn't really required, because the valuable fabric is reviewed within the textual content.
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Additional resources for An Introduction to Quantum Computing
1. In this ﬁgure, the state can be indicated by a point in one of two positions, indicated by the two points labelled 0 and 1. Next consider the slightly more complicated situation of a classical bit whose value is not known exactly, but is known to be either 0 or 1 with corresponding probabilities p0 and p1 . We might call this a probabilistic classical bit. The state of such a probabilistic bit is described by the probabilities p0 and p1 , which satisfy p0 + p1 = 1 (reﬂecting the fact that we know the bit has to be either 0 or 1).
4) Suppose A and B are linear operators on H1 and H2 respectively. Then A ⊗ B is the linear operator on H1 ⊗ H2 deﬁned by (A ⊗ B) |ψ1 ⊗ |ψ2 ≡ A|ψ1 ⊗ B|ψ2 . 5) This deﬁnition extends linearly over the elements of H1 ⊗ H2 : ⎞ ⎛ λij |bi ⊗ |cj ⎠ ≡ (A ⊗ B) ⎝ ij λij A|bi ⊗ B|cj . 6) ij We have presented the tensor product using the Dirac notation. In the matrix representation, this translates as follows. Suppose A is an m × n matrix and B a p × q matrix, then the left Kronecker product of A with B is the mp × nq matrix ⎡ ⎤ A11 B11 .
In light of the above observation, it would seem that we would be unable to simulate such a computer with a probabilistic Turing machine. If we believe that such a quantum computer is ‘realistic’ then it seems to violate the strong Church–Turing Thesis! The ﬁrst formal model of a quantum computer was given by David Deutsch, who proposed a model for a quantum Turing machine as well as the quantum circuit model. That it is possible to design a model of computation based explicitly on the principles of quantum mechanics is very interesting in itself.
An Introduction to Quantum Computing by Phillip Kaye, Raymond Laflamme, Michele Mosca