New answers tagged resource-request
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can we trade the number of samples required to learn a function for time
In general, the answer to this question will be no, because there are two different scaling behaviors being conflated:
a. Sample complexity: The number of samples sufficient to accurately learn a dataset, from any randomly sampled training set.$^1$
b. Algorithmic complexity: Some ...
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Time irreducible & Inaccessibility (philosophical + analytically):
Interesting question! Let me start antichronological with your lovely example. Neal has 20/20 and Peter 20/80. You are saying that Peter needs to get closer to see the mountains and it's infrastructure clearly. So you are changing the equal base condition both had at the beginning. That's ...
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The article of Devoret and Schoelkopf [1] and an update provided in Section 7.1 of Reagor [2] makes a comparison between Moore's law and an observed trend of exponentially improving $T_1$ and $T_2$ times for superconducting qubits. The trend they present shows a roughly exponential improvement from $10^0$ to $10^6$ nanoseconds for $T_2$ between various ...
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A couple of additional remarks:
The "Bloch sphere" is a representation in $S^2$ (the unit sphere) of qubit (pure) states. Such a representation is possible because there is a bijection $\mathbb{CP}^1\simeq S^2$, via the standard mapping between states and Bloch vectors, given e.g. in this other answer.
"Single-qubit unitaries" are ...
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TL;DR: Yes, ignoring the unobservable global phase, every single-qubit unitary corresponds to a unique rotation of $\mathbb{R}^3$ and vice versa.
Single-qubit unitaries and rotations
Let us first pin down the two objects in question. The first one - the set of single-qubit unitaries - is sometimes imprecisely described as the group $U(2)$ of $2 \times 2$ ...
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Consider a single-qubit unitary. This has two eigenvalues $e^{i\theta_1}$ and $e^{i\theta_2}$. So $V=e^{-i(\theta_1+\theta_2)/2}U$ has eigenvalues $e^{\pm i\phi}$ where $\phi=(\theta_1-\theta_2)/2$. Note that the two eigenvectors of $V$ are (i) orthogonal, meaning that they are anti-parallel on the Bloch sphere, defining a single axis, and (ii) these are the ...
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