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Download Lecture 2: Quantum Math Basics 1 Complex Numbers
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Transcript
Im(z · z 0 ) 6 |z| b0 r b r BMB Im(z 0 ) Im(z) 3 θ1 - a Re(z) r θ2 6 |z 0 | a0 |z| · |z 0 | BB B - B Re(z 0 ) B (b) z 0 = a0 + b0 i (a) z = a + bi 6 θ1 + θ2 - Re(z · z 0 ) (c) z · z 0 Figure 2: Geometric representation of z · z 0 As a convention for complex numbers, we call the product of a complex number with its complex conjugate the square of that complex number, which is essentially the square of its magnitude: z · z â = (a + bi)(a â bi) = a2 + b2 = |z|2 Notice that the result is always a real number, which becomes obvious when we realize that z â is basically a reflection of z about the real axis. Since the sum of their angles in the complex plane is 0, z · z â always lands on the axis. We can therefore naturally generalize the inner product for complex vectors. Definition 1.4. The inner product (or dot product) of two d-dimensional vectors is defined as (z1 , . . . , zd ) · (w1 , . . . , wd ) = z1â w1 + · · · + zdâ wd . The dot product of a vector with itself now becomes: (z1 , . . . , zd ) · (z1 , . . . , zd ) = |z1 |2 + · · · + |zd |2 . 2 Quantum Bits Just as a classical bit can have a state of either 0 or 1, the two most common states for a qubit (quantum bit) are the states |0i and |1i. For now, letâs just see the notation â| iâ as a way of distinguishing qubits from classical bits. The actual difference though is that a qubit can be in linear combinations of states, also know as superpositions. In other words, we can write a quantum state in a more general form: |Ïi = α |0i + β |1i , C where α, β â , and |α|2 + |β|2 = 1. Two other famous states that we will see very often in this class are: 1 1 1 1 |+i = â |0i + â |1i , |âi = â |0i â â |1i . 2 2 2 2 We can also think of |Ïi as a vector in the two-dimensional complex plane spanned by the two basis states |0i and |1i. As mentioned last time, often we can view α and β as real numbers without losing much. The reason we can sometimes ignore the fact that they are complex numbers is that can be easily simulated by 2 . Thatâs in fact exactly what we C R 2
