Most Popular
1500 questions
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
- 844
- 1
- 8
- 14
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: $$\rho = .5 |0\rangle…
QuestionEverything
- 1,785
- 11
- 22
14
votes
0 answers
Relation between quantum entanglement and quantum state complexity
Both quantum entanglement and quantum state complexity are important in quantum information processing. They are usually highly correlated, i.e., roughly a state with a higher entanglement corresponds to a higher quantum state complexity or a…
XXDD
- 333
- 1
- 7
14
votes
2 answers
How to prove/disprove universality for a set of gates?
A universal set of gates are able to mimic the operation of any other gate type, given enough gates. For example, a universal set of quantum gates are the Hadamard ( $H$ ), the $\pi/8$ phase shift ( $T$ ), and the $\mathrm{CNOT}$ gate. How would one…
chuster
- 255
- 3
- 8
14
votes
2 answers
What kind of real-world problems (excluding cryptography) can be solved efficiently by a quantum algorithm?
This question is very similar as Is there any general statement about what kinds of problems can be solved more efficiently using a quantum computer?
But the answers provided to that questions mainly looked at it from a theoretical/mathematical…
JanVdA
- 1,148
- 8
- 17
14
votes
3 answers
How to think about the Z gate in a Bloch sphere?
I am confused about how to understand the $Z$ gate in a Bloch sphere.
Considering the matrix $Z = \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}$ it is understandable that $Z|0\rangle = |0\rangle$ and $Z|1\rangle = -|1\rangle$.
It is explained here…
Bick
- 842
- 4
- 13
14
votes
4 answers
Grover's Algorithm and its relation to complexity classes?
I am getting confused about Grover's algorithm and it's connection to complexity classes.
The Grover's algorithm finds and element $k$ in a database of $N=2^n$ (such that $f(k)=1$) of elements with $$\sim \sqrt{N}=2^{n/2}$$
calls to the oracle.
So…
Quantum spaghettification
- 1,452
- 10
- 27
14
votes
3 answers
Is acting with a positive map on a state not part of a larger system allowed?
In the comments to a question I asked recently, there is a discussion between user1271772 and myself on positive operators.
I know that for a positive trace-preserving operator $\Lambda$ (e.g. the partial transpose) if acting on a mixed state…
Quantum spaghettification
- 1,452
- 10
- 27
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
- 885
- 5
- 13
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
- 1,605
- 5
- 18
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
- 311
- 2
- 7
14
votes
1 answer
What is a stabilizer state?
I am reading through the paper "Direct Fidelity Estimation from Few Pauli Measurements" (arXiv:1104.4695) and it mentions 'stabilizer state'.
"The number of repetitions depends on the desired
state $\rho$. In the worst case, it is $O(d)$, but in…
Quantum Guy 123
- 1,371
- 5
- 18
14
votes
3 answers
What is the difference between "code space", "code word" and "stabilizer code"?
I keep reading (e.g. Nielsen and Chuang, 2010; pg. 456 and 465) the following three phases; "code space", "code word" and "stabilizer code" - but am having a difficult time finding definitions of them and more importantly how they differ from one…
Quantum spaghettification
- 1,452
- 10
- 27
14
votes
2 answers
What does quantum error correction code notation stand for?
I understand the notation for classical error correcting codes. E.g., "Hamming(7,4)" stands for a Hamming code that uses 7 bits to encode blocks of 4 bits.
What does the notation for quantum error correcting codes mean? E.g., there is a paper that…
Alexander Pozdneev
- 411
- 3
- 10
14
votes
3 answers
Are spin-glass problems NP (-complete)?
It is well known that finding ground states for spin-glass systems (Ising, XY...) is NP-hard (at least as hard as the hardest NP-problems) so that they can be efficiently used to solve other NP problems like the Traveling Salesman Problem.
My…
Wouter
- 291
- 1
- 9