Exploring Qiskit (1)

Purpose

I studied using Quantum Computing with IBM Quantum (written by blueqat members) as the main text, with the Qiskit Textbook as a supplement. Along the way, I also briefly looked through Quantum Computation and Quantum Information and the Quantum Circuit Learning paper, but since I only ended up with a pile of messy notes, I want to reconstruct them with the goal of achieving hybrid quantum-classical machine learning.

The purpose of this article is to explore a starting point while describing the vague feelings and personal theories I've wanted to write about in a "100% poem" (musing) format. I'll just jot down some symbols and relationships that I might use later. I will conclude by sharing my impression that Nielsen & Chuang is indeed a great book.

Background

I was wondering if I could utilize mathematics and physics more, and "Quantum Computing" caught my eye as a suitable topic, so I started reading books on it. Since I was already doing machine learning, I set my goal to Quantum Machine Learning and began reading the aforementioned books. Although quantum mechanics as physics is not my specialty, as I read through, I realized that "for now, it's fine to forget about quantum mechanics; it's enough to read on the premise that computers satisfying the requirements of quantum computation simply exist," which allowed me to finish the books^[1].

The next section is more like a memory of the trial-and-error process leading up to that realization. It's probably unnecessary, but I'll leave it as an entry point since it might be useful for later articles.

The Double Helix of Physics and Mathematics

I quite like the book The Double Helix of Physics and Mathematics (Japanese title: 物理と数学の2重らせん), and I believe physics and mathematics should move closer together. For example, I think symbols should be more integrated and allowed to transition freely between the two.

Bra-ket Notation

When you read slightly advanced quantum mechanics books like Modern Quantum Mechanics (Vol. 1) 2nd Edition, you encounter the bra-ket notation introduced by P. A. M. Dirac. Since books on quantum information and quantum computing use it extensively, it can be a bit overwhelming. It involves writing expressions like $\braket{\beta|A|\alpha}$ using a ket vector $\ket{\alpha}$, a bra vector $\bra{\beta}$, and an observable $A$ to represent values.
Even in quantum information books, stories about Hadamard gates $H$ and qubits $\ket{0}$ and $\ket{1}$ where $H \ket{0} = \frac{1}{\sqrt{2}}(\ket{0} + \ket{1})$ appear and can be a bit discouraging^[2].

This bra-ket notation can be re-examined through the eyes of mathematics.

Inner Product in Functional Analysis

The book Hilbert Spaces and Quantum Mechanics (Japanese title: ヒルベルト空間と量子力学) is categorized as a mathematics book, but it is like a much gentler explanation of the mathematical foundations of quantum mechanics by J. von Neumann. It is a compact yet informative and interesting book.

Let's write out the symbols found in such books closer to physics notation. Suppose there is a space called a Hilbert space $\mathcal{H}$ where certain vectors belong, and there is a mapping $\braket{\cdot,\cdot}$ from $\mathcal{H} \times \mathcal{H}$ to complex numbers $\mathbb{C}$ that satisfies the following properties:

$$\begin{align*} \braket{\alpha,\alpha} &\geq 0 \\ \braket{\alpha,\alpha} = 0 &\Rightarrow \alpha = 0 \\ \braket{\beta,c_\alpha \alpha + c_{\alpha^\prime} \alpha^\prime} &= c_\alpha \braket{\beta,\alpha} + c_{\alpha^\prime} \braket{\beta,\alpha^\prime} \\ \braket{\beta,\alpha} &= \braket{\alpha,\beta}^* \end{align*}$$

Here, $\alpha, \alpha^\prime, \beta \in \mathcal{H}$, $c_\alpha, c_{\alpha^\prime} \in \mathbb{C}$, and $^*$ denotes the complex conjugate.
In this case, $\braket{\cdot,\cdot}$ is called the inner product of $\mathcal{H}$.

What's the Benefit?

It helps clear up some of that vague, fuzzy feeling.

Once you grasp this framework, the observable $A$ in quantum mechanics corresponds to an operator (infinite-dimensional matrix) in the Hilbert space $\mathcal{H}$, and $\braket{\beta|A|\alpha}$ in bra-ket notation corresponds to the inner product $\braket{\beta,A\alpha}$.

Feeling overwhelmed by seeing $\ket{\beta}\!\bra{\alpha}$ after just barely understanding $\braket{\beta|\alpha}$ in a quantum mechanics book also becomes much easier when translated into mathematics. The former corresponds to the inner product $\braket{\beta,\alpha}$ of vectors $\alpha$ and $\beta$. The latter corresponds to the projection operator $\braket{\alpha,\cdot} \beta$, although it's a bit of an unusual notation. In other words, it's a projection onto the one-dimensional subspace spanned by $\beta$. Forcing bra-ket notation into mathematical terms might look like $\braket{\alpha|\cdot} \ket{\beta}$, which could still feel a bit strange.

The frequently seen $\sum_a \ket{a}\!\bra{a} = 1$ can also be viewed through the eyes of mathematics as an expansion using a complete orthonormal system, or in a broader sense, a Fourier series expansion, which feels reassuring. It’s not an entirely new concept but a common one that already exists.

In any case, being able to give a clearer, more concrete identity to symbols whose true nature is unknown provides a sense of security, which I personally find rewarding. For this reason, I think it’s beneficial to go a step further into Hilbert spaces rather than just peeking through the entrance and leaving quickly—even if it's a bit of a detour, it leads to better clarification...

Quantum Computation and Quantum Information I is highly commendable in this regard, as Section 2.1, "Linear Algebra" within Chapter 2, "Introduction to Quantum Mechanics," provides a thorough explanation using a substantial yet appropriate amount of mathematics. When mathematics is utilized as a tool to this extent, the somewhat fuzzy feeling of quantum mechanics suddenly boils down to just linear algebra, allowing one to focus on quantum computation.

Summary

Since I felt hesitant to jump straight into articles about the Deutsch-Jozsa algorithm or Quantum Fourier Transforms, I decided to use this "poem" as an entry point. In future articles, I hope to organize the topics I am interested in.

脚注

1. In other words, up to a certain point, I was exploring the geometric properties of the Bloch sphere or reviewing quantum mechanics by reading J.J. Sakurai, but I realized those weren't necessary for my immediate goals.

2. Thinking back, it was sufficient to interpret it as computational logic in an abstract computer where such mysterious bits and logic gates are implemented. For physical implementation, it seems enough to understand that it can actually be realized using things like nuclear spin.