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1500 questions
8
votes
2 answers
Quantum error correction references
I would like to understand more about QEC, such as surface codes, error codes, and qLDPC codes.
For quantum algorithms beyond the Nielsen and Chuang book the lecture notes by Andrew Childs are a very good resource, and I'm looking for something…
NYG
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8
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Happy 30th Anniversary to Shor's Algorithm! How challenging was the review process when it was first announced?
Shor's algorithm dropped 30 years ago sometime in April, 1994. Peter Shor has given many wonderful accounts of the early history of the field and of what sparked his particular interest.
Shor has been very consistent in stating that he was inspired…
Mark Spinelli
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8
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2 answers
How can quantum error correction correct small rotations/continuous errors?
I'm having trouble understanding the so-called "digitization of errors" argument in QEC.
Suppose I have to encode my logical qubit into $n$ physical qubits to do error correction. I will use some encoder - let's call it $E$ - and note that the…
Joan
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8
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2 answers
What is the definition of physical gate error rate?
The fidelity of two quantum states $\rho$ and $\sigma$ is a well-defined (up to discussions about a square):
$$
F(\rho, \sigma) = \left(\text{Tr} \sqrt{ \sqrt{\rho} \sigma \sqrt{\rho}}\right)^2.
$$
Infidelity based error rate
For a noisy quantum…
Frederik Ravn Klausen
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8
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3 answers
In Schur-Weyl's duality, why is the commutant of $\pi_k(S_k)$ spanned by $U(d)^{\otimes k}$ matrices?
I'm reading this tutorial paper about quantum state certification. However, I'm confused about the concept of Schur-Weyl duality, explicitly Theorem 35 of the paper. Let $S_k$ denotes the symmetric group and $\pi_k$ its unitary representation, i.e.,…
Sherlock
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8
votes
2 answers
Are classical shadows useful?
according to the paper https://doi.org/10.1038/s42254-022-00535-2 , the advantage of classical shadows is doing measurements first and asking questions later.
But in real experiments, who would do measurement first and ask questions later, being…
yuanyi_thu
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8
votes
1 answer
Is there a fast sparse Hadamard transform?
Suppose I give you an $n$-qubit state vector as a classical list of numbers (or as an oracle that can query the amplitudes). I tell you this state vector will contain exactly $k$ non-zero amplitudes, after you apply a Hadamard transform to it. You…
Craig Gidney
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8
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1 answer
Which Clifford groups are 2-designs?
Let $ X $ be the $ q \times q $ shift matrix sending $ |y \rangle \mapsto |y+1 \rangle $ where the ket index $ y=0,\dots, q-1 $ is taken mod $ q $. Let $ Z $ be the diagonal $ q \times q $ clock matrix sending $ |y \rangle \mapsto (e^{2 \pi i…
Ian Gershon Teixeira
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8
votes
0 answers
Can we distill magic states with arbitrary angle $\theta$?
There seems to be numerous work about the distillation protocol of the $T$-magic state
$$
\frac{1}{\sqrt{2}}(|0\rangle+e^{i\pi/4}|1\rangle).
$$
Similarly, I am wondering if it is possible to distill a $\theta$-magic…
Yunzhe
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8
votes
1 answer
Smallest qudit error correcting/detecting codes
Consider encoding a qubit into $n$ qubits. It is well known that the smallest error detecting code is in $n=4$ qubits and the smallest error correcting code is in $n=5$ qubits.
Is there a similar result for qudits? If we encode a qudit into $n$…
Eric Kubischta
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8
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1 answer
Is it possible to perform search with $O(\sqrt{N})$ copies of a "resource" state rather than an oracle?
Suppose that we wish to find $x$ s.t. $f(x) = 1$. Instead of having access to an oracle like $U_f: |i\rangle \mapsto (-1)^{f(i)}|i\rangle$ or $U_f: |i\rangle|z\rangle \mapsto |i\rangle|z\oplus f(i)\rangle$, suppose we have access to copies of a…
shashvat
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8
votes
1 answer
Were quantum computers conjectured to factor large numbers before Shor developed his algorithm?
Peter Shor has given wonderful accounts of the development of his algorithm, with a lot of detail on the activity in the field at around the early-mid 90's. He's been very free about emphasizing that his algorithm was inspired by Umesh Vazirani's…
Mark Spinelli
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8
votes
1 answer
Polynomial time reductions vs. Quantum Polynomial time reductions
In computer science, a language $A$ reduces to a language $B$ if there exists a computable function (one that can be computed by a Turing machine) $f_{AB} \colon \Sigma^* \mapsto \Sigma^*$ such that $x \in A \iff f_{AB}(x) \in B$.
We call the…
Andrew Baker
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8
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0 answers
Optimal estimation of quantum state overlap - Circuit implementation?
I've been reading this paper, but don't understand what their optimal method really is, and how it can be realized as a quantum circuit.
The paper mentions the "Schur transform" which has a circuit provided in this paper. But is this Schur transform…
Loic Stoic
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8
votes
1 answer
Parametrization of a two-qubit state
A single-qubit state can be parametrized with real $\theta$ and $\phi$ as follows:
$$|\psi(\theta, \phi)\rangle = \cos\frac{\theta}{2}|0\rangle + e^{i \phi} \sin\frac{\theta}{2}|1\rangle.$$
I would like to know how to parameterize an arbitrary…
MonteNero
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