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The quantum walk on the line:
Scaling and the role of initial conditions
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Authors
T. De Coster
M. Fannes
In this thesis, I studied a concept from quantum physics known as the “Quantum Walk.” It’s a quantum version of the classic idea of a random walk—a simple process where, say, a person flips a coin and moves left or right depending on the result. But unlike the classical version, a quantum walk includes the weirdness of quantum mechanics, like superposition (being in two places at once) and interference.
My focus was on quantum walks along a straight line, where each position is represented by a number on the number line. In the quantum case, we can’t just move left or right by flipping a coin. We also need to include a kind of internal property—like spin—called a “qubit,” which adds a layer of complexity to how the walker moves.
I examined how the walker behaves over time when starting from different initial setups. The most common one is where the walker starts from a single position. But I went beyond this and looked at what happens when the walker starts in a superposition—meaning it’s in two or more positions at once.
Using mathematical tools like Fourier analysis and the stationary phase method, I derived new results for how the probability distribution of the walk develops over time. I proved a limit theorem that shows the overall behavior in the long run, including what it looks like when the starting point is a superposition of two positions. I also showed how this result connects to earlier, well-known findings (the Hadamard case).
Figure 1: The main principles underlying a classical (left) and quantum (right) random walk.
Why is this important?
Quantum walks aren’t just theoretical fun—they’re useful. They form the backbone of some quantum algorithms, which could run much faster than traditional algorithms. For example, quantum walks are key components in search algorithms for quantum computers. The fact that a quantum walker spreads out faster than a classical one means quantum algorithms can, in theory, find answers more quickly.
But up to now, most quantum algorithms have been designed assuming the walker starts from a single position. That’s a bit unrealistic since real quantum systems often start in more complex states, like superpositions. By figuring out what happens in these more general cases, my work helps us get closer to using quantum walks in practical, real-world quantum computing scenarios.
Figure 2: An example of how a quantum walk can create interference in the probability distribution, thereby creating regions with high probability alternating with regions with very low probability to end up in.
Making Sense of the Quantum World
One of the fascinating things I observed is how superposition leads to oscillations in the probability of finding the walker at certain positions. In the classical world, probabilities just add up, but in the quantum world, they can interfere and cancel each other out. This interference creates places where the walker is never found, even though you might expect it to be.
My results show how this interference plays out when the walker starts in two positions at once. The patterns it creates are unique to quantum mechanics and show just how differently quantum systems behave.
In summary, this thesis explores how the initial setup of a quantum system affects its long-term behavior, and it expands our ability to model and use quantum walks in future quantum technologies.
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