By Phillip Kaye, Raymond Laflamme, Michele Mosca
This concise, obtainable textual content offers a radical creation 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 specific 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. besides the fact that, earlier familiarity with themes comparable to tensor items and spectral decomposition isn't really required, because the helpful fabric is reviewed within the textual content.
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Additional info for An Introduction to Quantum Computing
7) From this information, we can construct the matrix for the not gate (in the computational basis): 01 . 8) 10 The gate acts on the state of a qubit by matrix multiplication from the left: not|0 ≡ 01 10 1 0 = 0 1 ≡ |1 . 9) The not gate is often identiﬁed with the symbol X, and is one of the four Pauli gates: σ0 ≡ I ≡ σ2 ≡ σ y ≡ Y ≡ 10 01 σ1 ≡ σx ≡ X ≡ 01 10 0 −i i 0 σ 3 ≡ σz ≡ Z ≡ 1 0 . 10) COMPOSITE SYSTEMS 45 As we will see in Chapter 4, the Pauli gates X, Y , and Z correspond to rotations about the x-, y- and z-axes of the Bloch sphere, respectively.
That is, we would like to know how to describe the state of a closed system of n qubits, how such a state evolves TEAM LinG 46 QUBITS AND THE FRAMEWORK OF QUANTUM MECHANICS in time, and what happens when we measure it. Treating a larger system as a composition of subsystems (of bounded size) allows for an exponentially more eﬃcient description of operations acting on a small number of subsystems. 4. This brings us to our third postulate. Composition of Systems Postulate When two physical systems are treated as one combined system, the state space of the combined physical system is the tensor product space H1 ⊗ H2 of the state spaces H1 , H2 of the component subsystems.
Amn B1q ⎥ ⎢ ⎥ ⎢ ⎥ .. .. .. .. ⎣ ⎦ . . . . Am1 Bp1 . . Am1 Bpq . . . Amn Bp1 . . Amn Bpq This matrix is sometimes written more compactly in ‘block form’ as ⎤ ⎡ A11 [B] A12 [B] . . A1n [B] ⎢ A21 [B] A22 [B] . . A2n [B] ⎥ ⎥ ⎢ A⊗B =⎢ . ⎥. .. ⎦ ⎣ .. . 8) Am1 [B] Am2 [B] . . Amn [B] Here, [B] represents the p × q submatrix B. Then each block entry Aij [B] above is the matrix [B] multiplied by the single entry in row i, column j, of matrix A. ⎡ ⎤ Aij B11 Aij B12 . . Aij B1q ⎢Aij B21 Aij B22 .
An Introduction to Quantum Computing by Phillip Kaye, Raymond Laflamme, Michele Mosca