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What happens when we permute the vertices of a (small) graph by unitarily walking along a (larger) graph of the symmetric group?
Let $G$ be an unweighted, undirected graph on $n$ vertices with vertex labels $\{1,2,\cdots,n\}$; let $A$ be its $n\times n$ adjacency matrix. We can prepare a state $|\psi\rangle$ representing $A$ with $n^2$ qubits; for example, with $n=10$ we…
Mark Spinelli
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Are the coefficients of Clifford gates in the Pauli basis always of equal magnitude?
I'm currently interested in writing Clifford gates in the Pauli basis i.e. $C = \sum_{P}\alpha_P P$, where we are summing over all pauli strings of fixed length. For example:
$H =\frac{1}{\sqrt{2}}(X+Z)$
$CX = \frac{1}{2}(II+XI+IZ-XZ)$
$HS =…
Ethan Davies
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6
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Simple example for the sub-optimality of MWPM vs Maximum likelihood decoder
I understand that the Minimum Weight Perfect Matching (MWPM) decoder tries to identify the single most probable error configuration, while a maximum likelihood decoder aims to find the most probable coset of errors (those equivalent up to a…
Shoham Jacoby
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6
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3 answers
Does the matrix representation of quantum gates depend on the basis?
I have read that I can chose any orthonormal basis $\{|0⟩, |1⟩\}$ of $\mathbb{C}^2$ as my basis states for a single qubit. Usually $|0⟩ = \begin{pmatrix}1\\0\end{pmatrix}$ and $|1⟩ = \begin{pmatrix}0\\1\end{pmatrix}$ is chosen, but say I chose…
Hebol
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6
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Can we learn anything interesting about a claw by taking the square-root-of-NOT of each qubit?
Consider being given a circuit for a two-to-one Boolean function $f$ from $n$ (qu)bits to $m\ge n-1$ (qu)bits, and prepare the following state:
$$\frac{1}{\sqrt {2^n}}\sum_0^{2^n-1}|x\rangle|f(x)\rangle.$$
Upon measuring the second register in the…
Mark Spinelli
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6
votes
3 answers
Are Quantum Algorithms better than classical algorithms for all problems?
My question basically boils down whether Quantum Computers are better computers or are they only different computers?
Is there a quantum algorithm for every problem which is at least as fast as the classical algorithm? If yes, then the fact that it…
user93353
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6
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1 answer
Total number of (unique) moments of the Haar distribution
This is probably a standard fact but I cannot find it in my usual references. Let $G$ be one of the classic matrix Lie groups $\mathrm{U}(N), \mathrm{SU}(N), \mathrm{O}(N), \mathrm{SO}(N)$, equipped with the Haar measure $\mu$. One way to study the…
Banach space fan
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2 answers
Is it possible to have a trace fidelity of 1 even if two unitary operations are different?
The gate fidelity of two quantum unitary operations is often described using $\frac{1}{2^n}|\text{tr}(U^\dagger V)|$. Is it ever possible that $U^\dagger V \ne I$ however $\frac{1}{2^n}|\text{tr}(U^\dagger V)| = 1$? Specifically, is it possible to…
smi
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6
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Is there a rank-1 POVM on qubits with $4$ outcomes that is not extremal but such that its resulting shape on the Bloch sphere has a non-zero volume?
Let us consider a rank-1 POVM acting on qubits with $4$ outcomes (that is, all its elements are rank-1). Furthermore, let us assume that this POVM is unbiased, meaning that $\mathrm{Tr}\left[M_i\right]=\frac12$ for all $i\in[4]$. In particular, it…
Tristan Nemoz
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What exactly are "quantum rotors"?
The Wikipedia entry on the subject is rather short. I am also curious about generalizations of quantum rotors in n-dimensions. An introductory explanation with at least one resource for further reading would be greatly appreciated.
user820789
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6
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Do the Towers of Hanoi admit any perfect state transfer?
The Towers of Hanoi is an old puzzle that involves moving $n$ discs of different radii among $k$ pegs, requiring each stack of discs to be monotonically increasing along the stack. Conventionally $n=6$ or so and $k=3$; it's also often studied in…
Mark Spinelli
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6
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2 answers
Why do completely positive maps satisfy ${\rm Tr}[\Psi(\rho)_++\Psi(-\rho)_+]\leq{\rm Tr}[\Psi(\rho_+)]+{\rm Tr}[\Psi((-\rho)_+)]$?
I am studying a paper of M. Plenio, "Logarithmic Negativity: A Full Entanglement Monotone That is not Convex", PRL 2005 [arXiv:quant-ph/0505071].
In the paper, I do not fully understand the Eq.$(7)$. He said that
Employing linearity of operation…
Acpil
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Why does the partial transpose of an entangled state have at most one negative eigenvalue?
I came over this unclear claim which i wondered someone could clarify:
"The partial transpose of an entangled state has at most one negative eigenvalue."
I wondered if this holds for all states or just some, searched around but couldn’t find some…
Pink Elephants
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Are quantum simulators like Microsoft Q# actually using quantum mechanics in their chips?
Unlike Google's Bristlecone or IBM's Qbit computer, do simulators like Q# or Alibaba really use quantum mechanics anywhere in their physical chips? Are they just defining properties using a classical computer and trying to achieve quantum…
Yashank
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6
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Does the need of many quantum algorithms to be repeated several times impair the efficiency gains?
As I understand so far, in some algorithms such as Simon's algorithm, swap-test algorithm or quantum k-means algorithm, we repetitively perform a measurement yielding a classical result. Consequently, this pushes us to run the whole algorithm again…
Aman
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