By Phillip Kaye, Raymond Laflamme, Michele Mosca

This concise, available textual content offers a radical advent to quantum computing - a thrilling emergent box on the interface of the pc, engineering, mathematical and actual sciences. aimed toward complex undergraduate and starting graduate scholars in those disciplines, the textual content is technically particular and is obviously illustrated all through with diagrams and workouts. a few previous wisdom of linear algebra is believed, together with vector areas and internal items. besides the fact that, earlier familiarity with subject matters resembling tensor items and spectral decomposition isn't required, because the useful fabric is reviewed within the textual content.

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We can label these states with |0 and |1 , respectively. 3) where α0 and α1 are complex coeﬃcients, often called the amplitudes of the basis states |0 and |1 , respectively. Note that a complex amplitude α can be 2 However, it is common to use inﬁnite dimensional state spaces to model the physical systems used to implement quantum (as well as classical) information processing. TEAM LinG 40 QUBITS AND THE FRAMEWORK OF QUANTUM MECHANICS decomposed unique as a product eiθ |α| where |α| is the non-negative real number α has norm 1.

That is, ψ|ψ . 11) |ψ ≡ The quantity |ψ is called the Euclidean norm of |ψ . A vector is called a unit vector if it has norm 1. A set of unit vectors that are mutually orthogonal is called an orthonormal set. The Kronecker delta function, δi,j , is deﬁned to be equal to 1 whenever i = j, and 0 otherwise. We use the Kronecker delta function in our deﬁnition of an orthonormal basis. 3 Consider a Hilbert space H of dimension 2n . 12) and every |ψ ∈ H can be written as ψn |bn , for some ψn ∈ C. 13) bn ∈B The values of ψn satisfy ψn = bn |ψ , and are called the ‘coeﬃcients of |ψ with respect to basis {|bn }’.

A familiar example of an inner product is the dot product for column vectors. The dot product of v with w is written v · w and is deﬁned as follows. 4) ⎜ .. ⎟ · ⎜ .. ⎟ = v1∗ v2∗ . . vn∗ ⎜ .. ⎟ = ⎝ . ⎠ ⎝ . ⎠ i=1 vn 1 The wn wn complex conjugate of c = a + bi (where a and b are real) is c∗ = a − bi. TEAM LinG 24 LINEAR ALGEBRA AND THE DIRAC NOTATION We now return to the Dirac notation, and deﬁne the dual vector space and dual vectors. 1 Let H be a Hilbert space. The Hilbert space H∗ is deﬁned as the set of linear maps H → C.