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1500 questions
14
votes
3 answers
What is the "Stinespring Dilation"?
I've consulted Nielsen and Chuang to understand the Stinespring Dilation, but wasn't able to find anything useful. How does this operation relate to partial trace, Kraus operators, and purification?
Jimit Bavishi
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14
votes
2 answers
Why is an entangled qubit shown at the origin of a Bloch sphere?
I'm unclear why the Bloch sphere representation of a maximally entangled qubit shows the state of the bit as being at the origin of the sphere.
For example, this illustration
shows the effect of the simple circuit
over time, with $q_0$ on the…
orome
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14
votes
1 answer
How many logical qubits are needed to run Shor's algorithm efficiently on large integers ($n > 2^{1024}$)?
First, I know there are differences in logical qubits and physical qubits. It takes more physical qubits for each logical qubit due to quantum error.
Wikipedia states that it takes quantum gates of order $\mathcal{O}((\log N)^2(\log \log N)(\log…
LeWoody
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14
votes
3 answers
Density matrix after measurement on density matrix
Let's say Alice wants to send Bob a $|0\rangle$ with probability .5 and $|1\rangle$ also with probability .5. So after a qubit Alice prepares leaves her lab, the system could be represented by the following density matrix: $$\rho = .5 |0\rangle…
QuestionEverything
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14
votes
2 answers
Automatic compilation of quantum circuits
A recent question here asked how to compile the 4-qubit gate CCCZ (controlled-controlled-controlled-Z) into simple 1-qubit and 2-qubit gates, and the only answer given so far requires 63 gates!
The first step was to use the C$^n$U construction given…
user1271772 No more free time
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14
votes
0 answers
What is the Generalized Quantum Stein's Lemma and why is it important?
I'm sensing a lot of buzz about potential re-proofs of the Generalized Quantum Stein's Lemma - a generalization of the quantum counterpart to the classical Stein's Lemma, which is of some importance in statistical inference and hypothesis…
Mark Spinelli
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14
votes
2 answers
What are the possible non-entangling two-qubit gates?
The non-entangling gates in $ SU_4 $ contains the entire group of gates of the form
$
SU_2 \otimes SU_2.
$
It also contains
$$
\zeta_8 SWAP= \zeta_8 \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1…
Ian Gershon Teixeira
- 5,358
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14
votes
2 answers
How to prove/disprove universality for a set of gates?
A universal set of gates are able to mimic the operation of any other gate type, given enough gates. For example, a universal set of quantum gates are the Hadamard ( $H$ ), the $\pi/8$ phase shift ( $T$ ), and the $\mathrm{CNOT}$ gate. How would one…
chuster
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14
votes
2 answers
What kind of real-world problems (excluding cryptography) can be solved efficiently by a quantum algorithm?
This question is very similar as Is there any general statement about what kinds of problems can be solved more efficiently using a quantum computer?
But the answers provided to that questions mainly looked at it from a theoretical/mathematical…
JanVdA
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14
votes
2 answers
Quantum algorithm for linear systems of equations (HHL09): Step 1 - Confusion regarding the usage of phase estimation algorithm
I have been trying to get my head around the famous(?) paper Quantum algorithm for linear systems of equations (Harrow, Hassidim & Lloyd, 2009) (more popularly known as the HHL09 algorithm paper) for some time, now.
On the very first page, they say:…
Sanchayan Dutta
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- 112
14
votes
3 answers
How to think about the Z gate in a Bloch sphere?
I am confused about how to understand the $Z$ gate in a Bloch sphere.
Considering the matrix $Z = \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}$ it is understandable that $Z|0\rangle = |0\rangle$ and $Z|1\rangle = -|1\rangle$.
It is explained here…
Bick
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14
votes
5 answers
What is the difference between a qudit system with d=4 and a two-qubit system?
I understand that a qudit is a quantum $d$-state system. If $d=4$, is this exactly the same as a two-qubit system, which also presents $4$ quantum states? The Hilbert space is the same, right? Are there any theoretical or practical differences?
Daniel Tordera
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14
votes
1 answer
Why exactly are variational algorithms considered promising?
There is obviously a great deal of work happening at the moment on variational quantum algorithms. However, I'm struggling to understand why exactly are they considered promising? Looking through some papers and review articles (such as this one…
Nikita Nemkov
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14
votes
2 answers
Why is the efficiency of Ekert 91 Protocol 25%?
In Cabello's paper Quantum key distribution without alternative measurements, the author said "the number of useful random bits shared by Alice and Bob by transmitted qubit, before checking for eavesdropping, is 0.5 bits by transmitted qubit, both…
Lynn
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14
votes
2 answers
Why is the Pauli group used for stabilizers?
When it comes to error correction, we take our stabilizers to be members of the Pauli group. Why is the Pauli group used for this and not, say, the group of all unitary matrices?
Quantum spaghettification
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