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As an extension of a famous description of non-local Hamiltonian simulation in section 4.7.3 of Nielsen and Chuang, the welded-trees paper of Childs, et al. provides the following circuit for use in the quantum walk along their graph:

Circuit in Childs et al.

Here, $a$ and $b$ are two different $2n$-bit labels for the nodes of the graph, while $W$ appears somehow related to a SWAP gate used in the traversal of the graph.

In particular, the eigenvalues of SWAP are $\pm 1$ because SWAP$\ne$SWAP$^2=\mathbb 1$, while the unique eigenvector of the SWAP gate having eigenvalue $-1$ is $\frac{1}{\sqrt 2}(|01\rangle-|10\rangle)$.

Childs et al. define $W$ as:

$$\begin{eqnarray} W|00\rangle=|00\rangle\\ W\frac{1}{\sqrt 2}(|01\rangle+|10\rangle)=|01\rangle\\ W\frac{1}{\sqrt 2}(|01\rangle-|10\rangle)=|10\rangle\\ W|11\rangle=|11\rangle \end{eqnarray}$$

But what, then, are the entries of this gate $W$?

Are they something like: $$\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & e^{i\pi q/2}\cos\frac{\pi q}{2} & -ie^{i\pi q/2}\sin\frac{\pi q}{2} & \: 0 \ \\ 0 & -ie^{i\pi q/2}\sin\frac{\pi q}{2} & e^{i\pi q/2}\cos \frac{\pi q}{2} & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) $$

for some $q$?

I'm really just trying to understand that paper in general, and this circuit in particular, in more detail. Childs et al. trotterizes this circuit over all the different edge-colorings of the graph to do the Hamiltonian simulation thereof; after about a linear amount of time there's a decent odds that the particle, which was initially at node $|\psi\rangle=$ ENTRY, has found the EXIT.

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    $\begingroup$ You can get $W|01\rangle$ by summing the second and third equation and dividing by $\sqrt{2}$ and $W|10\rangle$ by taking the difference between the second equation and the third one and then once again dividing by $\sqrt{2}$ $\endgroup$ May 15, 2023 at 7:40

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According to the definition you gave, $$ W=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1/\sqrt{2} & 1/\sqrt{2} & 0 \\ 0 & 1/\sqrt{2} & -1/\sqrt{2} & 0 \\ 0 & 0 & 0 & 1 \end{array}\right). $$ (Just think of the central $2\times 2$ block like a Hadamard, which would convert $|+\rangle$ to $|0\rangle$ and $|-\rangle$ to $|1\rangle$ except that, in this case, the basis elements are $|01\rangle$ and $|10\rangle$ instead of $|0\rangle$ and $|1\rangle$.)

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  • $\begingroup$ Awesome! Do you know why the circuit shows that the $|10\rangle$ output controls the CCNOT parity count of the bottom ancilla? $\endgroup$ May 15, 2023 at 12:27
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    $\begingroup$ $T$ is basically the SWAP operator in parallel. Note that $S=I-2|\psi\rangle\langle\psi|$ where $|\psi\rangle=(|01\rangle-|10\rangle)/\sqrt{2}=W^\dagger|10\rangle$. This means that $e^{iSt}=e^{iSt}$ effectively adds a relative phase to the component in $|\psi\rangle$ compared to everything else. So, the idea is that $W$ converts the singlet state into $|10\rangle$. Then controlled off those qubits being $|10\rangle$, it adds a phase. $\endgroup$
    – DaftWullie
    May 15, 2023 at 12:36
  • $\begingroup$ Oh I see! That's awesome! The H'nian in § 4.7.3 of Mike and Ike is a tensor product of $Z$ gates, where the unique eigenvector having an eigenvalue of $-1$ is $|1\rangle$ - this is why their circuit uses CNOT gates with a filled-in bubble to count the parity. But, the H'nian in Childs et al. is a tensor product of a bunch of SWAP gates, and the singlet state is the one having an eigenvalue of $-1$ - this maps to $|10\rangle$. Hence the circuit in Childs et al. has a CCNOT with a filled-in bubble for the first qubit and an open bubble for the second. How neat is that! $\endgroup$ May 17, 2023 at 20:21

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