Download E-books An Introduction to Quantum Computing PDF

By Phillip Kaye

This concise, available textual content offers an intensive creation to quantum computing - an exhilarating emergent box on the interface of the pc, engineering, mathematical and actual sciences. geared 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 workouts. a few past wisdom of linear algebra is thought, together with vector areas and internal items. although, earlier familiarity with subject matters reminiscent of tensor items and spectral decomposition isn't required, because the useful fabric is reviewed within the text.

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2 Quantum Gates four. 2. 1 1-Qubit Gates In part three. 2, we acknowledged that any unitary operator performing on a 2-dimensional quan- tum process (a qubit) is named a ‘1-qubit quantum gate’. We gave the quantum now not gate (sometimes known as the Pauli X gate) for instance (and pointed out the different Pauli gates). remember the Bloch sphere from part three. 1. each 1-qubit natural country is represented as some extent at the floor of the Bloch sphere, or equivalently as a unit vector whose beginning is fastened on the centre of the Bloch sphere. A 1-qubit quantum gate U transforms a quantum nation |ψ into one other quantum country U |ψ . when it comes to the Bloch sphere, the motion of U on |ψ should be regarded as a rotation of the Bloch vector for |ψ to the Bloch vector for U |ψ . for instance, the now not gate takes the nation | zero to the kingdom | 1 (and takes | 1 to | zero ). In phrases of the Bloch sphere, this motion could be visualized as a rotation via an perspective π concerning the x axis, as illustrated in determine four. four. keep in mind in part 2. five we observed easy methods to compute the exponential (and different func- tions) of operators. If we exponentiate the Pauli gates, we get unitary operators akin to extremely important periods of 1-qubit gates. those are the rotation group LinG sixty four A QUANTUM version OF COMPUTATION gates, which correspond to rotations concerning the x-, y-, and z- axes of the Bloch sphere. they're outlined by way of the Pauli gates, and so for comfort, we remind you currently of the definitions of the Pauli gates: I ≡ 1 zero X ≡ zero 1 zero 1 1 zero Y ≡ zero −i Z ≡ 1 zero . (4. 2. 1) i zero zero − 1 The rotation gates are outlined as follows: −iθX Rx( θ) ≡ e 2 −iθY Ry( θ) ≡ e 2 −iθZ Rz( θ) ≡ e 2 . (4. 2. 2) workout four. 2. 1 allow x be a true quantity and A a matrix such 2 = I. convey that eiAx = cos( x) I + i sin( x) A. you can cost that the Pauli operators X, Y , and Z fulfill the stipulations X 2 = I, Y 2 = I, and Z 2 = I, and so utilizing the results of workout four. 2. 7 we will be able to write the rotation gates as: −iθX Rx( θ) ≡ e 2 = cos θ I − i sin θ X 2 2 −iθY Ry( θ) ≡ e 2 = cos θ I − i sin θ Y 2 2 −iθZ Rz( θ) ≡ e 2 = cos θ I − i sin θ Z. (4. 2. three) 2 2 figuring out the matrices for I, X, Y , and Z within the computational foundation, we will now write the rotation gates as matrices within the computational foundation: cos θ −i sin θ R 2 2 x( θ) = −i sin θ cos θ 2 2 cos θ − sin θ R 2 2 y ( θ) = sin θ cos θ 2 2 e−i θ 2 zero Rz( θ) = . (4. 2. four) zero ei θ 2 group LinG QUANTUM GATES sixty five ponder an arbitrary 1-qubit country, written by way of its Bloch vector angles σ and τ : σ σ cos | zero + eiτ sin | 1 . (4. 2. five) 2 2 within the computational foundation, this is written because the column vector cos σ 2 . (4. 2. 6) eiτ sin σ 2 The influence of making use of Rz( θ) in this country may be visible through acting a matrix multiplication: e−i θ 2 zero cos σ e−i θ 2 cos σ 2 = 2 zero ei θ 2 eiτ sin σ 2 ei θ 2 eiτ sin σ 2 cos σ = e−i θ 2 2 eiθeiτ sin σ 2 σ σ = e−i θ 2 cos | zero + ei( τ+ θ) sin | 1 . (4. 2. 7) 2 2 when you consider that an international part is insignificant, we now have the country σ σ cos | zero + ei( τ+ θ) sin | 1 .

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