Having to do with continuous distributions, which are 'uniform' in the sense of being invariant under some relevant symmetry. In the case of quantum information theory, this usually relates to unitarily invariant distributions on single-qubit states, many-qubit states, single-qubit unitary transformations, or many-qubit unitary transformations.
Questions tagged [haar-distribution]
75 questions
27
votes
1 answer
How to understand the Haar measure from a quantum information perspective?
I found it a little difficult to understand it using Wikipedia and some mathematical documents. How to understand the Haar measure from a quantum information theory perspective? Are there any materials that explain it?
raycosine
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15
votes
1 answer
What is a Haar random quantum state?
Can somebody please explain me what is a Haar random state? I am not able to find any friendly resource to read about it.
Shweta Aggrawal
- 349
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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
- 193
- 4
11
votes
2 answers
On the distribution of the fidelity of a random product state with an arbitrary many-qubit state
Consider an arbitrary $n$-qubit state $\lvert \psi \rangle$. How much do we understand about the probability distribution of the fidelity of $\lvert \psi \rangle$ with a tensor product $\lvert \alpha \rangle = \lvert \alpha_1 \rangle \lvert \alpha_2…
Niel de Beaudrap
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8
votes
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
- 433
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7
votes
0 answers
Complexity of a superposition of two random circuits
$\def\ket#1{|{#1}\rangle}$
Question
What is the minimum number of gates required to create the $N$-qubit state $\ket{\psi} = \frac{\ket{a} + \ket{b}}{\sqrt2}$ from the all-zeroes state, where one term in the superposition, $\ket{a}$, can be created…
Jordan Taylor
- 171
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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
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7
votes
3 answers
Show that, averaging over uniformly random unitaries, $\mathbb{E}[UXU^\dagger]=\operatorname{Tr}(X)\frac{I}{d}$
As mentioned e.g. in this answer, if we compute the average
$$\Phi(X)\equiv \mathbb{E}_U[UXU^\dagger] \equiv \int_{\mathbf U(d)} d\mu(U) UXU^\dagger,$$
where $d\mu(U)$ is the Haar measure over the unitary group $\mathbf U(d)$, we get…
glS
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7
votes
1 answer
Is the Haar measure invariant under conjugation?
Denote the Haar measure on the unitary group $U(\mathcal X)$ by $\eta$. Does this equation hold (assuming the integral exists):
$$\int d\eta(U) f(U) = \int d\eta(U) f(U^\dagger)$$
Intuitively this makes sense because choosing a random $U$ seems to…
dmitryk
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- 2
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
Is there a concentration inequality for the quantum gate fidelity $F(C,U)$ for a channel $C$ such that $\int dU F(C,U)=X$?
For a fixed quantum channel $N$ and a unitary channel $U$, we define $N$'s gate fidelity as
$$ F(N,U) = \int \langle \psi| U \, N(| \psi \rangle \langle \psi |) \, U^\dagger| \psi \rangle d\mu_H(\psi)$$
where $\mu_H$ is the Haar measure over…
Davide Li Calsi
- 63
- 3
6
votes
1 answer
How to show that the integral over all Haar states vanishes: $\int|\psi\rangle\,{\rm d}\psi = 0 $?
How can one show that the integral over all Haar states $|\psi \rangle $ is
$$
\int |\psi \rangle \, \mathrm{d}\psi = 0\ ?
$$
qc6518
- 173
- 2
6
votes
1 answer
Random quantum states and Schur-Weyl duality
Consider the following density matrix over $n$ qubits, with $C$ being a single qubit operator:
$$
\rho_{n} = \int_{C \sim \text{Haar}} \big(C|0\rangle\langle0|C^\dagger\big)^{\otimes n} dC.
$$
Let's say I measure the qubit with respect to orthogonal…
BlackHat18
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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
2 answers
Is the column vector of a uniformly sampled random unitary matrix a uniformly sampled random state vector?
I am wondering if a random unitary matrix taken from a Haar measure (as in, it is uniformly sampled at random) can yield a uniformly sampled random state vector.
In section 3 of this paper it says "It is worthwhile mentioning that, although not…
Quantum Guy 123
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