Lecture 2: Quantum Math Basics 1 Complex Numbers

Quantum computation (CMU 18-859BB, Fall 2015) Lecture 2: Quantum math BasicsSeptember 14, 2015 Lecturer: John WrightScribe: Yongshan Ding1 Complex NumbersFrom last Lecture , we have seen some of the essentials of the Quantum circuit model of compu- tation , as well as their strong connections with classical randomized model of , we will characterize the Quantum model in a more formal way. Let s get started withthe very Basics , Complex numberz Cis a number of the forma+bi, wherea,b R,andiis the imaginary unit, satisfyingi2= s always convenient to picture a Complex numberz=a+bias a point (a,b) in thetwo-dimensionalcomplex plane, where the horizontal axis is the real part and the verticalaxis is the imaginary part:-Re(z)6Im(z)r 3|z|= a2+b2 abFigure 1: Geometric representation of Complex numberz=a+biAnother common way of parametrizing a point in the Complex plane, instead of usingCartesian coordinatesaandb, is to use theradial coordinater(the Euclidean distance of thepoint from the origin|z|= a2+b2), together with theangular coordinate (the angle fromthe real axis).

Quantum Computation (CMU 18-859BB, Fall 2015) Lecture 2: Quantum Math Basics September 14, 2015 Lecturer: John Wright Scribe: Yongshan Ding 1 Complex Numbers From last lecture, we have seen some of the essentials of the quantum circuit model of compu-tation, as well as their strong connections with classical randomized model of computation.

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