By Phillip Kaye, Raymond Laflamme, Michele Mosca
This concise, available textual content presents an intensive advent to quantum computing - an exhilarating 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 special and is obviously illustrated all through with diagrams and routines. a few past wisdom of linear algebra is believed, together with vector areas and internal items. in spite of the fact that, past familiarity with subject matters resembling tensor items and spectral decomposition isn't required, because the important fabric is reviewed within the textual content.
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Extra info for An Introduction to Quantum Computing
Aij B1q ⎢Aij B21 Aij B22 . . Aij B2q ⎥ ⎢ ⎥ Aij [B] = ⎢ . 9) .. .. ⎥ . ⎣ .. . ⎦ Aij Bp1 Aij Bp2 . . Aij Bpq The matrix representation for the tensor product of two vectors, or two operators, is the left Kronecker product of the matrix representation of the two vectors or TEAM LinG THE SCHMIDT DECOMPOSITION THEOREM 35 operators being ‘tensored’ together. 10) ⎝α1 β0 ⎠ . α1 β1 α1 β1 A ﬁnal important word about notation. We often leave the ⊗ symbol out of expressions, and thus |ψ ⊗ |ϕ is often written as just |ψ |ϕ , or sometimes even |ψϕ .
For these, the spin state is indeed described by a vector in a 2-dimensional Hilbert space H. A convenient basis for this space consists of a unit vector for the ‘spin-up’ state of the particle, and an orthogonal unit vector for the ‘spin-down’ state of the particle. We can label these basis vectors by |0 and |1 , respectively. The general spin state of a spin- 12 particle is a superposition of spin-up and spin-down. As another example of a physical system whose state can be described by a vector in a 2-dimensional Hilbert space, consider an electron orbiting a nucleus.
Since we should ﬁnd the photon in exactly one path, we must have |α0 |2 + |α1 |2 = 1. When the photon passes through the beam splitter, we multiply its ‘state vector’ by the matrix 1 1i √ . 5) 0 . 6) This result corresponds with the observed behaviour that, after going through the ﬁrst beam splitter, we would measure the photon in path ‘0’ with probability | √12 |2 = 12 , and in path ‘1’ with probability | √i2 |2 = 12 . If we do not measure which path the photon is in, immediately after it passes through the ﬁrst beam splitter, then its state remains 1 1 √ .
An Introduction to Quantum Computing by Phillip Kaye, Raymond Laflamme, Michele Mosca