I’ve finished the third chapter of the Introduction to Classical and Quantum Computing which was all about Linear Algebra. In the previous section, we learned about qubits from a trigonometric perspective, while in this chapter, we learned about some of the linear algebra prerequisites to understand qubits and how to represent and calculate different states using linear algebra.

Lessons learned

We started off by learning how kets can be represented as column vectors, and how bras are obtained by calculating the conjugate transpose of a ket. We then learned about the inner product of two kets, performed by multiplying a bra and a ket, and how it is always a scalar value. We then learned about the idea of orthonormal states, i.e. states that are both normalized as well as orthogonal to each other. We say how inner products can be used to find the amplitudes and orthonormality of a state in a certain basis.

We also learned how quantum gates are just matrices that act on a certain state (vector) to transform it. We saw matrix representations of common one-qubit quantum gates, and how we can apply them sequentially to perform some computation. We then looked at two of the most important properties of quantum gates, i.e. Unitarity (quantum gates are unitary matrices) and reversibility (a quantum gate is always reversible). In the end, we learned about outer products of two states (which always returns a matrix) and the completeness relation i.e. $\ket a \bra a + \ket b \bra b = I$.

Moving forward

I was hoping to finish this chapter in a single day, but slight fever and headache came in my way and I wasn’t able to ☹️. Next chapter in the book is quite a hefty one, and I think it’ll take me at least 2 days to finish (note taking takes up most of the time). I’ll update you guys once I’m done with that, be it 2 days or 3. Talk to you then.

PDF Notes

My notes on the second Chapter of the Thomas Wong’s Introduction to Classical and Quantum Computing book.