New PDF release: An Introduction to Quantum Computing

By Phillip Kaye, Raymond Laflamme, Michele Mosca

ISBN-10: 0198570007

ISBN-13: 9780198570004

This concise, available textual content presents a radical creation to quantum computing - an exhilarating emergent box on the interface of the pc, engineering, mathematical and actual sciences. aimed toward complicated undergraduate and starting graduate scholars in those disciplines, the textual content is technically certain and is obviously illustrated all through with diagrams and routines. a few past wisdom of linear algebra is thought, together with vector areas and internal items. although, past familiarity with themes reminiscent of tensor items and spectral decomposition isn't really required, because the helpful fabric is reviewed within the textual content.

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Note that U † = U −1 implies U † U = I, where I is the identity operator. The unitary operators preserve inner products between vectors, and in particular, preserve the norm of vectors. 4). e. it is equal to its own Hermitean conjugate). 7 A projector on a vector space H is a linear operator P that satisfies P 2 = P . An orthogonal projector is a projector that also satisfies P† = P. Recall the following definition from basic linear algebra. 13) for some constant c. The constant c is called the eigenvalue of T corresponding to the eigenvector |ψ .

E. the ‘Hermitean conjugate’ of the column matrix for |χ ). Then the inner product of |ψ with |ϕ is ψ|ϕ , which in the matrix representation is computed as the single element of the matrix product of the row matrix representing ψ| with the column matrix representing |ϕ . This is equivalent to taking the dot product of the column vector associated with |ψ with the column vector associated with |ϕ . 6) and |ϕ = 1 |10 2 1 |11 2 + . 8) 1 2 respectively. 9) TEAM LinG DUAL VECTORS 25 and so the inner product of |ψ with |ϕ is −i ψ|ϕ = √ .

G. a|b ). We now carefully review the definitions of the main algebraic objects of interest, using the Dirac notation. The vector spaces we consider will be over the complex numbers, and are finitedimensional, which significantly simplifies the mathematics we need. Such vector spaces are members of a class of vector spaces called Hilbert spaces. Nothing substantial is gained at this point by defining rigorously what a Hilbert space is, but virtually all the quantum computing literature refers to a finite-dimensional complex vector space by the name ‘Hilbert space’, and so we will follow this convention.

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An Introduction to Quantum Computing by Phillip Kaye, Raymond Laflamme, Michele Mosca


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