By Phillip Kaye, Raymond Laflamme, Michele Mosca
This concise, obtainable textual content presents an intensive 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 unique and is obviously illustrated all through with diagrams and workouts. a few past wisdom of linear algebra is thought, together with vector areas and internal items. despite the fact that, previous familiarity with subject matters corresponding to tensor items and spectral decomposition isn't required, because the useful fabric is reviewed within the textual content.
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Extra info for An Introduction to Quantum Computing
6 Tensor Products The tensor product is a way of combining spaces, vectors, or operators together. Suppose H1 and H2 are Hilbert spaces of dimension n and m respectively. Then the tensor product space H1 ⊗ H2 is a new, larger Hilbert space of dimension n×m. ,m} is an orthonormal basis for H2 . 1) is an orthonormal basis for the space H1 ⊗ H2 . The tensor product of two vectors |ψ1 and |ψ2 from spaces H1 and H2 , respectively, is a vector in H1 ⊗ H2 , and is written |ψ1 ⊗ |ψ2 . The tensor product is characterized by the following axioms: 1.
3. Two real parameters θ and ϕ are suﬃcient to describe a state vector, since state vectors are constrained to have norm 1 and are equivalent up to global phase. Points on the surface of the Bloch sphere can also be expressed in Cartesian coordinates as (x, y, z) = (sin θ cos ϕ, sin θ sin ϕ, cos θ). 5. 4 summarizes the graphical representations of the states of a classical bit, a probabilistic classical bit, and a quantum bit. TEAM LinG TIME-EVOLUTION OF A CLOSED SYSTEM 43 Fig. 3 State of a qubit on the Bloch sphere.
For the moment we will not use the Dirac notation, and write vectors in boldface. For vectors over the complex numbers, an inner product is a function which takes two vectors from the same space and evaluates to a single complex number. We write the inner product of vector v with w as v, w . An inner product is such a function having the following properties. 1. 1) i 2. 2) 3. 3) with equality if and only if v = 0. 2), we use the notation c∗ to denote the complex conjugate1 of a complex number c, as will be our convention throughout this book.
An Introduction to Quantum Computing by Phillip Kaye, Raymond Laflamme, Michele Mosca