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Why can quantum walks not approach a stationary distribution

Suppose for contradiction that there is some limiting state $|f\rangle$ that an initial state $|s_1\rangle \neq |f\rangle$ approaches as a unitary operation $U$ is repeatedly applied. So there is ...
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Accepted

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Why is the triangle finding problem important?

According to Le Gall and Nakajima's paper Quantum Algorithm for Triangle Finding in Sparse Graphs, Williams and Williams have shown a surprising reduction from Boolean matrix multiplication to ...
Accepted

Quantum circuit for Szegedy quantum walk on a cyclic graph

It's been a long while since I've looked at my thesis, so a lot of my knowledge is pretty rusty, but here goes. If you're looking for the circuit representation of $C_8$, here's the Quirk simulation ...
Accepted

Implementing Quantum Walks at IBM

Let's explain what this circuit do : A coin quantum walk is a process in which a walker will evolve through the differents vertices of a graph. Its steps will be guided by a coin, a quantum register ...
Accepted

How does a Hadamard discrete-time quantum walk result in a skewed distribution?

You get a skewed distribution because you start with a "skewed" coin state (I'm assuming the system you are considering starts with the walker state in a single fixed state). In fact, you ...
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How does a Hadamard discrete-time quantum walk result in a skewed distribution?

I did the math for the first three steps when the coin qubit is initialized to $|\uparrow\rangle$ using the link @Mark S commented. We use as the coin flip operator the Hadamard gate, and the ...
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What is meant by "perfect state transfer"?

Perfect state transfer is generally discussed in the context of continuous time evolution. For example, you might be evolving under the influence of a Hamiltonian $H$. Particularly when one is ...
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Relationship of Adiabatic Quantum Computing speedup to Quantum Random Walk hit time

In response to my own question, I'm thinking Aaronson stated somewhere (I can't find the reference) that the quadratic speedup of e.g. Grover search arises from the fact that probability is amplitude ...
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Accepted

What does the notation $|\psi(0)\rangle = |0\rangle|n=0\rangle$ mean?

This is common shorthand for the tensor product. That is, you should read it as $|0 \rangle \otimes | n=0\rangle$.
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Quantum Walk Study Resource for Non-regular Graph

Some important references could be the following: Quantum Walks On Graphs, Quantum walks: a comprehensive review, Quantum random walks - an introductory overview.
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Quantum circuit implementation of shift operator in quantum walk

Once you have some operator form, one method is just to start multiplying it by unitary operators (pre- or post- or both, doesn't matter) until you can make it into identity. For example, maybe you ...
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How do we compute quantum walks for a graph?

The graph is the Hamiltonian I think. I'll start off by saying I only understand portions of Childs' paper and I am far from familiar with graph theory, much less spectral graph theory - much of what'...
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Why is Grover's Algorithm considered to be a Quantum Walk?

A quantum walk can actually be defined in any graph. For each defined graph, it is only true that the operator must be local (considering the graph edges) and unitary. So we must consider the Grover's ...
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Accepted

What are the entries in the 2x2 $W$ gate used for walking along the welded-trees graph of Childs et al.?

According to the definition you gave,  W=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1/\sqrt{2} & 1/\sqrt{2} & 0 \\ 0 & 1/\sqrt{2} & -1/\sqrt{2} & 0 \\ 0 & ...
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1 vote

finding subsets which meets conditions

I've put an implementation here. The idea is to construct a Hamiltonian, which would be a diagonal matrix, and then use a variational algorithm to see whether there exists a state with eigenvalue 0, ...
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1 vote

continuous time quantum walk on a cycle - transition matrix

(Not much of an answer yet, just some tentative ideas about how to simulate a continuous-time random walk on a cycle for now. I describe how to take a small quantum walk for a short amount of time, ...
• 12.5k
1 vote

What is this equation for coin operator is trying to do in this quantum walk for Non-regular graph? This coin operator is called Fourier coin

The operator Fourier Coin is $k$-point Discrete Fourier Transform (DFT) of node $i$. The matrix representation of a general $N$-point DFT can be found here. The implementation of DFT on the quantum ...
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1 vote

Use of Position Hilbert Space in Quantum Walk

Now, my question is what is the use of above representation, if all the operation are defined on initial state. This means very little. Like in any kind of dynamics, you start with an initial state ...
• 25.4k
1 vote

Why, in a discrete-time quantum walk,we first apply the tensor product of the coin operation with the identity?

The idea is that by applying Hadamard just to the coin and doing nothing to the position state, this is the equivalent of tossing the coin. To see this in a very crude way, imagine applying Hadamard ...
• 58.7k
1 vote

Why, in a discrete-time quantum walk,we first apply the tensor product of the coin operation with the identity?

A (discrete-time) quantum walk can be thought of as the direct "quantization" of a classical random walk. In a classical random walk, at each iteration, you flip a coin and move the walker ...
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1 vote

Quantum walk with binary tree

Quantum Walk may not perform in optimal ways over a conventional and determinisitc binary tree data structure. In a quantum walk experimental setup, the coin will not follow a binomial distribution. ...
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1 vote

Quantum Walk: Why the need of adding "tail" nodes to the root?

Following up on and inspired by the comments from Rob, I sense that there's a bit of a similarity between, on the one hand, the boolean tree evaluation of Farhi and Gutmann (and of Ambainis et al.), ...
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