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1500 questions
7
votes
1 answer
How exactly does modular exponentiation in Shor's algorithm work?
Consider the modular exponentiation part of Shor's algorithm which in many works is just referred to as
$$U_{f}\sum^{N-1}_{x = 0}\vert x\rangle\vert 0\rangle = \vert x\rangle\vert a^{x}\text{ mod }N\rangle$$
where $a$ is random number between $1 < a…
Poramet Pathumsoot
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7
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3 answers
Forming states of the form $\sqrt{p}\vert 0\rangle+\sqrt{1-p}\vert 1\rangle$
I'm curious about how to form arbitrary-sized uniform superpositions, i.e.,
$$\frac{1}{\sqrt{N}}\sum_{x=0}^{N-1}\vert x\rangle$$
for $N$ that is not a power of 2.
If this is possible, then one can use the inverse of such a circuit to produce…
Sam Jaques
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7
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What resources are available for learning QCL?
I'm struggling to find much about the language QCL, rather than about quantum computing itself.
Is there anything out there like that? It doesn't have to be free.
Katie
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7
votes
1 answer
What happens with first phase factor in QFT?
I'm using Mermin's Quantum Computer Science book to understand Shor's algorithm, but I can't figure out why one of the phase factors drops out of the probability for measuring a certain y.
This is the application of the QFT on the superposition of…
jvdh
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7
votes
3 answers
Hadamard gate as a product of $R_x$, $R_z$ and a phase
I am having problems with this task.
Since the Hadamard gate rotates a state $180°$ about the $\hat{n} = \frac{\hat{x} + \hat{z}}{\sqrt{2}}$ axis, I imagine the solution can be found the following way:
First rotate $\hat{n}$ so it lies in the…
QCQCQC
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7
votes
2 answers
Quantum proof for the group non-membership problem
Group non-membership problem:
Input: Group elements $g_1,..., g_k$ and $h$ of $G$.
Yes: $h \not\in \langle g_1, ..., g_k\rangle$
No: $h\in \langle g_1, ..., g_k\rangle$
Notation: $\langle g_1, ..., g_k\rangle$ is the subgroup generated by…
Sanchayan Dutta
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7
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2 answers
How to translate matrix back into Dirac notation?
In Circuit composition and entangled states section of Wikipedia's article on Quantum logic gates the final result of a combined Hadamard gate and identity gate on $|\Phi^{+}\rangle$ state is:
$$ M \frac{|00\rangle + |11\rangle}{\sqrt{2}} =…
Michał Zając
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7
votes
1 answer
Status of software packages for quantum compiling
By "quantum compiling", what I mean is classical algorithms to solve the following problem: given a $SU(D)$ matrix $U$ (the goal) and a set of $SU(D)$ unitary matrices $V_1 \cdots V_N$ (the gates), find a string $i_1\cdots i_K$ such that
$$
U…
Scott Lawrence
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7
votes
3 answers
Partial trace over a product of matrices - prove that ${\rm Tr}(\rho^{AB}(\sigma^A\otimes I))={\rm Tr}(\rho^A\sigma^A)$
$$Tr(\rho^{AB} (\sigma^A \otimes I/d)) = Tr(\rho^A \sigma^A)$$
I came across the above, but I'm not sure how it's true. I figured they first partial traced out the B subsystem, and then trace A, but I don't see how you are allowed to partial trace…
Mahathi Vempati
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7
votes
2 answers
Is the set of all states with negative conditional Von Neumann entropy convex?
I have read somewhere / heard that the set of all states that have non-negative conditional Von Neumann entropy forms a convex set. Is this true? Is there a proof for it?
Can anything be said about the reverse - set of all states that have negative…
Mahathi Vempati
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7
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2 answers
How does $\mathcal E(\rho)=\mathrm{Tr}_{env}[U(\rho\otimes\rho_{env})U^\dagger]$ turn into $P_0\rho P_0+P_1\rho P_1$?
In the Quantum Operations section in Nielsen and Chuang, (page 358 in the 2002 edition), they have the following equation:
$$\mathcal E(\rho) = \mathrm{Tr}_{env} [U(\rho \otimes \rho_{env})U^\dagger]$$
They show an example with
$\rho_{env} =…
Mahathi Vempati
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7
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1 answer
Problem with the mathematical formulation of "qubitization"
In this research paper (arXiv), the authors introduce a new algorithm to perform Hamiltonian simulation.
The beginning of their abstract is
Given a Hermitian operator $\hat{H} = \langle G\vert \hat{U} \vert G\rangle$ that is the projection of an…
Adrien Suau
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7
votes
1 answer
How does the CNOT gate operate when the control qubit is a superposition?
If a control qubit is in superposition, how it will affect target qubit if it is collapsed or in superposition? Is it true that CNOT works only if the control bit collapsed to 1? Also, is it possible to collapse or Hadamard control qubit “on the…
Olexander Korenyuk
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7
votes
1 answer
Stabilizer for quantum error correction code
I have some very basic questions about stabilizers.
What I understood:
To describe a state $|\psi \rangle$ that lives in an $n$-qubit Hilbert space, we can either give the wavefunction (so the expression of $|\psi\rangle$), either give a set of…
Marco Fellous-Asiani
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7
votes
1 answer
What is the quantum bandwidth of a planar array of noisy qubits, assuming free classical communication?
A common task to perform during quantum computation on the surface code is moving qubits from one place to another. There are standard ways to do this within the surface code, but I was wondering what the actual fundamental limits are. If we forget…
Craig Gidney
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