For questions associated with the calculation of probability, expected value, variance, standard deviation, or similar statistical quantities.
Questions tagged [probability]
131 questions
14
votes
1 answer
What is the probability $\Pr(\|U-I\|_{\rm op}<\varepsilon)$ of a Haar-random unitary being close to the identity?
If one generates an $n\times n$ Haar random unitary $U$, then clearly $\Pr(U=I)=0$. However, for every $\epsilon>0$, the probability
$$\Pr(\|U-I\|_{\rm op}<\varepsilon)$$
should be positive.
How can this quantity be computed?
Calvin Liu
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10
votes
2 answers
Quantum Amplitude Estimation vs Quantum Phase Estimation
Quick question concerning the probability of success after a phase estimation algorithm vs an amplitude estimation algorithm.
Given the calculation on the wikipedia page, the probability of measuring the desired output in a phase estimation…
sheesymcdeezy
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9
votes
1 answer
Understanding a quantum algorithm to estimate inner products
While reading the paper "Compiling basic linear algebra subroutines for quantum computers", here (arXiv), in the Appendix, the author/s have included a section on quantum inner product estimation.
Consider two vectors $x,y \in \mathbb{C}^n, x= (x_1,…
IntegrateThis
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- 14
7
votes
1 answer
Generating random, but non-uniform state
I would like an algorithm that generates a random state, sampled according to some probability distribution which is not uniform in Hilbert space. Assume though that I have at my disposal a uniform (i.e. Haar) random state generator. How do I do…
nervxxx
- 560
- 2
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6
votes
1 answer
Total number of (unique) moments of the Haar distribution
This is probably a standard fact but I cannot find it in my usual references. Let $G$ be one of the classic matrix Lie groups $\mathrm{U}(N), \mathrm{SU}(N), \mathrm{O}(N), \mathrm{SO}(N)$, equipped with the Haar measure $\mu$. One way to study the…
Banach space fan
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6
votes
1 answer
Do entangled measurements across multiple copies help in state distinguishability?
Consider two density matrices $\rho$ and $\sigma$. The task is to distinguish between these two states, given one of them --- you do not know beforehand which one.
There is an optimal measurement to distinguish between these two states --- the…
BlackHat18
- 1,527
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6
votes
1 answer
Confusion about the output distribution of Haar random quantum states
Consider a Haar random quantum state $|\psi \rangle$. I was confused between two facts about $|\psi \rangle$, which appear related:
Consider the output distribution of a particular $n$-qubit $|\psi \rangle$. For a large enough $n$, the probability…
BlackHat18
- 1,527
- 9
- 22
6
votes
1 answer
What does $M_m |\psi_i\rangle$ mean in the equation $p(m|i)=\langle\psi_i|M_m^\dagger M_m|\psi_i\rangle$?
I have trouble understanding two equations in the Nielsen & Chuang textbook. Suppose we perform a measurement described by the operator $M_m$. If the initial state is $|\psi_i\rangle$, then the probability of getting result m…
ZR-
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- 24
6
votes
0 answers
In QAOA why do we need $m \log(m)$ repetitions to get at least $F_{p}(\beta , \gamma) - 1$ with probability of $1 - 1/m$?
In the original QAOA paper from Farhi
https://arxiv.org/pdf/1411.4028.pdf,
it is stated in chapter 2 last paragraph (page 6) that: when measuring $F_{p}(\beta , \gamma)$ we get an outcome of at least $F_{p}(\beta , \gamma) - 1$ with probability of…
Hannah
- 529
- 2
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5
votes
2 answers
Simultaneous measurements and Bell basis measurements to estimate $\lvert\text{Tr}(\sigma \rho)\rvert^2$ in Huang et al. paper
Theorem 2 of this paper says if one is able to prepare $\rho^{\otimes k}$ then it is possible to predict expectation values of all $n$-qubit Pauli observables using $O(n)$ number of copies of $\rho$. It then gives an explicit procedure in Appendix…
user8183310
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5
votes
1 answer
What is the quantum analogue of $P_{XY} = P_{Y|X}P_X$
A standard trick in probability manipulation is to take some joint distribution $P_{XY}$ and express it as $P_{Y|X}P_X$. This trick is useful because when one looks at things like the ratio of $\frac{P_{XY}}{P_XP_Y}$, one can rewrite it as…
user1936752
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5
votes
1 answer
Why is sampling from probability distributions generated by specific quantum circuits classically intractable?
I was reading a paper by Benedetti et al. titled Parameterized quantum circuits as machine learning models. Its authors state the following:
We also know that sampling from the probability distribution
generated by instantaneous quantum…
karolyzz
- 299
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- 7
5
votes
1 answer
Can all mixed states be written as a convex combination $\rho=\sum_j p_j |\psi_j\rangle\langle \psi_j|$?
States belonging to some space $\mathcal H$ can be described by density operators $\rho\in L(\mathcal H)$ that are positive and have trace one. Pure states are the ones that can be written as $\rho=|\psi\rangle\langle \psi|$ for some…
Balter 90s
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5
votes
1 answer
How to prepare a quantum state of the form $\frac1{2^{n/2}}\sum_{x \in \{0, 1\}^{n}} |x\rangle |y_x\rangle$ with $y_x$ random variables?
Let's say I am given an efficiently samplable probability distribution $D$, over $n$ bit strings.
I want to efficiently prepare the following state
\begin{equation}
|\psi\rangle = \frac{1}{\sqrt{2^{n}}}\sum_{x \in \{0, 1\}^{n}} |x\rangle…
BlackHat18
- 1,527
- 9
- 22
5
votes
1 answer
Why do probablity distribution with orthogonal suppor have maximal Kolmogorov distance?
Can anyone explain why the $l_1$ distance has the property that probability distributions $P,Q$ with orthogonal support (meaning that the product $p_iq_i$ vanishes for each value of $i$) are at a maximal distance from each other?
Consider $$…
Sakh10
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