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1500 questions
7
votes
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Understanding classical vs. quantum channel capacities
The classical channel capacity ($C_{ea}$) and the quantum channel capacity ($Q$) as defined here (eqs. 1 and 2) are given by
\begin{equation}
C_{ea} = \text{sup}_{\rho} \Big[S(\rho) + S(\Phi_t \rho) - S(\rho,t)\Big],
\end{equation}
and…
Tobias Fritzn
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7
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2 answers
Why can't a quantum computer strongly simulate itself?
We can strongly simulate a quantum circuit if we can estimate the probability of measuring $|0^n\rangle$, say, at the end to within some fixed relative error.
Now, by measurement, a quantum computer can certainly weakly simulate itself, i.e.…
wdc
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7
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Proof of the optimality of Grover's algorithm
I am currently working on the proof of Grover's algorithm, which states that the runtime is optimal.
In Nielsen they say, the idea is to check whether $D_k$ is restricted and does not grow faster than $O(k^2)$. Now in Nielsen, an inductive proof is…
P_Gate
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7
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2 answers
What is maximum circuit depth and size IBM Q5 and Q16 could handle?
I'm trying to implement a couple of algorithms on ibmq_16_melbourne, so I need to know if this device is able to handle with depth and size of my current circuit or not.
For example, the circuit size is 300 and depth is 99.
C-Roux
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7
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1 answer
Quantum speedup without entanglement
Is there an instance of a quantum algorithm that is faster than its classical counterpart, but doesn't use entanglement, only superposition?
psitae
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What role do Hecke operators and ideal classes perform in “Quantum Money from Modular Forms?”
Cross-posted on MO
The original ideas from the 70's/80's - that begat the [BB84] quantum key distribution - concerned quantum money that is unforgeable by virtue of the no-cloning theorem. A limitation was that the quantum money required the bank…
Mark Spinelli
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7
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1 answer
Quantum algorithm for linear system of equations (HHL) - Final Step: How can I find my vector of solution $|x\rangle$?
I'm working on solving a linear system with the quantum algorithm HHL. I don't understand how I can recover my vector $|x\rangle$ of real solution of the system starting from the states measured with ancilla qubit in $|1\rangle$.
I found something…
Nicolò Cangini
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7
votes
1 answer
Interaction of an RF pulse with transmon qubit
When an RF pulse is interacting on resonance with a transmon qubit, it leads to rotation of the qubit around an axis in the XY plane (in a reference frame rotating in the transition frequency of the qubit). What is the exact relation between the…
Qexp
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7
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A two qubit state in a special form
How can a pure two-qubit state $|\psi\rangle = a |00\rangle + b|01\rangle + c|10\rangle + d|11\rangle$, be written in the following form
\begin{equation}
|\psi_{\alpha}\rangle = \sqrt{\alpha}|01\rangle + \sqrt{1-\alpha}…
Tobias Fritzn
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7
votes
0 answers
Construction of optimal ensemble to show quantum steerability
In Wiseman et al. (2007), in the process of deriving necessary and sufficient conditions for the steerability of some classes of states, the authors show (lemma 1, page 3) how to construct an optimal ensemble $F^\star=\{\rho_\xi^\star\mathscr…
glS
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7
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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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votes
1 answer
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
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How is Grover's operator represented as a rotation matrix?
I have seen that it is possible to represent the Grover iterator as a rotation matrix $G$. My question is, how can you do that exactly?
So we say that $|\psi\rangle$ is a superposition of the states of searched and not searched elements, that can be…
user4961
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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