Circle Notation
Will it go round in circles?
Just finished reading a really impressive book on the fundamentals of quantum computing. In my experience many books that cover this territory get somewhat lost in the formality of quantum notations and linear algebra formulations without imparting any real intuition on the mechanics behind quantum algorithms. This book, Programming Quantum Computers by Eric Johnston, Nic Hurrigan, and Mercedes Gimeno-Segovia, turned out to be the single most helpful book I’ve come across for clearly articulating what is taking place in fundamental algorithms like Grover’s search and Shor’s factoring algorithms. Even better than Mike & Ike, which if you are familiar with the literature will understand that this is high praise indeed.
I think what made this book so helpful was the broad disregard for articulating linear algebra operations, where as an alternative nearly all discussions were illustrated in three distinct and corresponding forms: quantum circuit diagrams, coding implementations, and circle notation diagrams depicting associated qubit states. The coding implementations were nice, but their primary benefit was just to reinforce the simplicity of gate applications and such, especially in the high level QCEngine framework serving as basis. The real spark for me though was in getting exposure to the circle notation diagrams in the context of corresponding circuit diagrams serving as their source of derivation.
This circle notation really helped to clear up some ambiguities in conceptualizing Bloch sphere properties, an alternative convention for representing qubit states that this blog has addressed previously. While the Bloch sphere is useful for visualizing the superposition of a single qubit in isolation, the representation can become somewhat unwieldy as circuit width is expanded, leaving it hard to visualize multi-qubit interactions resulting from gate applications, at least to this observer. The circle notation representation on the other hand overcomes such ambiguities by allowing a visual representation of each distinct segment of a multi-qubit superposition, where “segment” as used here is referring to each potential set of measurement outcomes. (For a single qubit this would be two segments as |0> and |1>, for a pair of qubits this would instead be four segments as |00>, |01>, |10>, and |11>, three qubits would require eight segments, and similarly n qubits would require 2^n segments for the representation.)
The convention used in the book, and I’m not sure I like this particular aspect, is that the measurement set labeling (such as |00>, |01>, |10>, and |11>) are discarded for use of ranged integer labels (such as |0>, |1>, |2>, and |3>). I suspect the justification here is to accommodate higher width states with shorter labels. So yeah if you need to convert from the integers back to the 0/1 sets, the convention is rightward bits are incremented before leftward, for example a three qubit state with segment labels 0–7 inclusive would follow 000, 001, 010, 011, 100, 101, 110, 111.
The segment labels aside, the visualizations of circle notation are realized by a kind of, well obviously, circles — demonstrating two distinct properties associated with a superposition state segment, that of a magnitude and a phase. Here the magnitude can be visualized by the ratio of the circle with shading, and the phase by a kind of dial that may rotate from 0–2π radians, with 0 radians corresponding to an upward facing dial and increasing rotations conducted counter-clockwise.
The magnitude and phase of each potential measurement set segment are a way of characterizing the probability amplitude of that measurement set, which is a complex number quantum dynamics construct which can be translated to a classical probability as Pr(x + iy) = x² + y², subject to the further constraint that the sum of classical probabilities across corresponding measurement sets sum to unity. It turns out that this same complex number amplitude can be represented in several equivalent fashions, either as a pair of real / imaginary components or other forms as a function of magnitude and phase (also known by their more formal titles as modulus and argument).
The |z|e^iθ form may not be quite as intuitive for those less mathematically inclined. There’s a related video [Proving the Most Beautiful Equation Bob Ross Style] shared by the charming YouTuber Tibees that I found helpful.
The translation of amplitudes to their magnitude and phase turns out to be really helpful when you consider that there are some quantum gates that leave a qubit’s magnitude intact and only transform the phase, or more particularly the relative phase between different measurement set segments. Here’s a simple example similar to the Bell state demonstrated above, with the addition of an extra Z gate resulting in a change of phase to register |3>.
The only point here, if I have one, is that found this circle notation quite helpful and decided to share in case may be of any benefit to others trying to reason their way through quantum computation. For much more check out Programming Quantum Computers, seriously it’s that good. Until next time.
Books that were referenced here or otherwise inspired this post:
Programming Quantum Computers — Eric Johnston, Nic Hurrigan, and Mercedes Gimeno-Segovia
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For further readings please check out the Table of Contents, Book Recommendations, and Music Recommendations. For more on Automunge: automunge.com
