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Questions tagged [haar-distribution]

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.

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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's user avatar
13 votes
1 answer
283 views

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 ...
Calvin Liu's user avatar
11 votes
2 answers
1k views

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 \...
Niel de Beaudrap's user avatar
8 votes
0 answers
196 views

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 ...
Loic Stoic's user avatar
7 votes
1 answer
178 views

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 (...
nervxxx's user avatar
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6 votes
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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 ...
BlackHat18's user avatar
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6 votes
1 answer
914 views

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 \...
BlackHat18's user avatar
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6 votes
1 answer
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How to show that the integral over all Haar states vanishes: $\int|\psi\rangle\,{\rm d}\psi = 0 $?

Can we show that the integral over all Haar states $|\psi \rangle $ is $$ \int |\psi \rangle \, \mathrm{d}\psi = 0~. $$ This is an integral over Haar vectors Reference to a post about what is Haar ...
qc6518's user avatar
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6 votes
1 answer
244 views

How close or far apart are the distributions generated by two Haar random states?

Consider two $n$ qubit Haar-random quantum states $|\psi\rangle$ and $|\phi\rangle$. Let $D_{|\psi\rangle}$ and $D_{|\phi\rangle}$ be the two probability distributions (over $n$-bit strings) obtained ...
BlackHat18's user avatar
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6 votes
1 answer
515 views

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 ...
dmitryk's user avatar
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Expected value of a Haar random quantum state multiplied by a unitary

Consider a quantity \begin{equation} \mathbb{E}\big[\langle z|\rho|z\rangle\big], \end{equation} where $\rho = |\psi \rangle \langle \psi|$ is a Haar-random state $n$-qubit quantum state and $z$ is ...
BlackHat18's user avatar
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5 votes
1 answer
372 views

Multiplication by a Haar random unitary two times

Consider a Haar random unitary $U$. I am trying to compute the value (or put a bound on) \begin{equation} \mathbb{E}\left[\left|\langle 0^{n} |U^{2} |0^{n}\rangle\right|^{2}\right]. \end{equation} The ...
BlackHat18's user avatar
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5 votes
1 answer
120 views

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)$...
Davide Li Calsi's user avatar
5 votes
2 answers
374 views

What is the expectation value of $|\langle \psi|U|\psi \rangle|$ over Haar random states $|\psi\rangle$?

We know the average unitary fidelity, $\int |\langle \psi|U|\psi \rangle|^2 d\psi$, has a nice closed-form solution: $\frac{1+\frac{1}{d}|Tr (U)|^2}{1+d}$, thanks to Horodecki and Nielsen. However, I ...
Abir's user avatar
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1 answer
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Moments of Pauli coefficients of Haar-random states

I want to evaluate the quantity $\sum_{P\in \rm{P}^n}\text{Tr}^{\alpha}(\rho P)$, where $P$ is an element of n-qubit Pauli group $\rm{P}^n$ and $\rho$ is a density matrix of a Haar random state. It is ...
Feng Pan's user avatar
5 votes
1 answer
89 views

Anticoncentration for two independent random quantum circuits in parallel

Consider two Haar random $n$ qubit unitaries, $U_1$ and $U_2$. Consider the quantum state $$|\psi\rangle = (U_1 \otimes U_2) |0^{2n}\rangle. $$ Let $p_x = |\langle x| \psi \rangle|^{2}$, for $x \in \{...
Tom Clancy's user avatar
5 votes
3 answers
853 views

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 ...
glS's user avatar
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4 votes
2 answers
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Is there a lower bound on the average diamond norm of two uniformly random unitaries U1 and U1 of dimension D that are sampled from haar measure?

I could not find any lower bound on the diamond norm for two uniformly random unitaries of dimension D sampled from the haar measure.
Niraj Kumar's user avatar
4 votes
2 answers
384 views

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 &...
Quantum Guy 123's user avatar
4 votes
2 answers
739 views

Computing expectation value of $|\langle z|C|0^n\rangle|^2$ over Haar random circuit

I am trying to understand the integration on page 4 of this paper. Consider a Haar random circuit $C$ and a fixed basis $z$. Each output probability of a Haar random circuit (given by $|\langle z | C |...
BlackHat18's user avatar
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4 votes
1 answer
335 views

What is the expectation value ${\Bbb E}[\langle\psi,O\psi\rangle]$ over the Haar distribution?

What is the average $\mathbb{E}_{\text{Haar}}|\langle\psi|O\psi\rangle|$ of expectation of an arbitrary observable $O$ over the Haar distribution? Let $d$ be the dimension, i.e, the size of $O$. Do we ...
doug doug's user avatar
4 votes
2 answers
211 views

Quantum hardness of XQUATH conjecture

Consider the XQUATH conjectures, as defined here (https://arxiv.org/abs/1910.12085, Definition 1). (XQUATH, or Linear Cross-Entropy Quantum Threshold Assumption). There is no polynomial-time ...
BlackHat18's user avatar
  • 1,363
4 votes
1 answer
279 views

Compute the large $n$ distribution of $|\langle z_i|\psi\rangle|^2$ over Haar random quantum states

Let $|\psi\rangle$ be a $n$ qubit Haar-random quantum state. I am trying to show that in the limit of large $n$, for each $z_{i} \in \{0, 1\}^{n}$, $$ |\langle 0|\psi\rangle|^{2}, |\langle 1|\psi\...
BlackHat18's user avatar
  • 1,363
4 votes
1 answer
320 views

Prove that uniformly random states have moments ${\bf E}_\psi|\langle x|\psi\rangle|^{2t}\sim1/\binom d t$

Im looking for the moments of Haar random states. Is it true that $\textbf{E}_{\psi\sim \text{Haar}}|\langle x| \psi\rangle|^{2t}\sim \frac{1}{\binom{d}{t}}?$ How does one prove this?
postasguest's user avatar
4 votes
1 answer
179 views

Spoofing XQUATH with the Feynman method

Consider the XQUATH conjecture for random quantum circuits, as mentioned here. (XQUATH, or Linear Cross-Entropy Quantum Threshold Assumption). There is no polynomial-time classical algorithm that ...
BlackHat18's user avatar
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4 votes
1 answer
69 views

How do I calculate the expectation of the rational function, in the sense of the Haar measure?

I want to know the analytical solution of $\mathbb{E}_{\psi}\frac{\langle \psi |A|\psi\rangle}{\langle \psi |A^2|\psi\rangle}$. I see similar questions before approximate average, but it does not ...
Dan David's user avatar
4 votes
0 answers
79 views

Expectation value over random $k$-local Pauli operators for two random quantum states

Suppose we have a uniform distribution $D$ over $k$-local Pauli operators $P_{q_1}\otimes \dotsc \otimes P_{q_k} $, $P_{q_i} \in \{ X, Y, Z, I \}$. Is it possible to calculate $\mathbb{E}_{P_i \sim D} ...
userflux9674's user avatar
4 votes
0 answers
293 views

Reduced density matrix of a Haar random state and its Schmidt decomposition

Consider a Haar random quantum state $|\psi\rangle$. Note that $$\rho =\mathbb{E}[|\psi\rangle\langle \psi|] = \frac{\mathbb{I}_{n}}{2^{n}}.$$ $\mathbb{I}_n$ is the identity operator on $n$ qubits. ...
BlackHat18's user avatar
  • 1,363
3 votes
1 answer
178 views

Schur's lemma for quantum states

I am trying to understand Lemma 2 in this paper. Consider a state $\tau_{H^n}=\int \sigma^{\otimes n}_{H} \mu(\sigma)$ where $\mu(\sigma)$ is the measure on the space of density operators on a single ...
user1936752's user avatar
  • 3,075
3 votes
1 answer
408 views

Integral over Haar measure of squared density matrix of Haar random state is proportional to the identity plus swap operator

I am having some trouble understanding why $\int d\psi (| \psi \rangle \langle \psi | )^{\otimes ^2}\propto \ I+$ SWAP , where $|\psi \rangle =U|\psi _0\rangle$ are Haar random states and $d\psi $ is ...
Andrew Dynneson's user avatar
3 votes
1 answer
439 views

Averaging over a single Haar-random unitary applied $t$ times

I'm trying to compute the following integral: $$\int U^{\otimes t}\left|x_1,\cdots,x_t\middle\rangle\middle\langle x'_1,\cdots,x_n'\right|\left(U^\dagger\right)^{\otimes t}\,\mathrm{d}\mu(U)$$ Where $\...
Tristan Nemoz's user avatar
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3 votes
1 answer
110 views

Independence in state prepared by independently drawn Haar random gates

Consider independently drawn $2 \times 2$ Haar random unitaries $U_1, U_2, \ldots, U_n$ and $$V = U_1 \otimes U_2 \otimes \cdots U_n.$$ Consider the state $\sigma$ given by $$\sigma = V \rho V^{*}, $$ ...
BlackHat18's user avatar
  • 1,363
3 votes
1 answer
330 views

Haar measure : trace of an operator squared and square of the trace of an operator

From doing numerical simulations, I seem to have the following results : $$ \int d \rho \,\, \text{Tr}(\rho M^\dagger M) = \frac{1}{d} \text{Tr}(M^\dagger M) $$ and $$ \int d \rho \,\, \left|\text{Tr}(...
Denis _J's user avatar
3 votes
0 answers
166 views

A question on a subset of projectors onto symmetric subspace

Use $\text{perm}_t$ to denote the set of all permutations among $t$ items. For any particular subset $S\subseteq\{0,1\}^n$ and any $\sigma\in \text{perm}_t$, we define \begin{align} P_S(\sigma) = \...
BlackHat18's user avatar
  • 1,363
2 votes
2 answers
136 views

Computing a ratio involving Haar random unitaries

Consider an $n$-qubit Haar random unitary $U$. I am trying to compute the expression \begin{equation} \mathbb{E}\left[ \frac{\text{Tr}\left(|0^n\rangle \langle 0^n | ~U\rho U^*\right)}{\text{Tr}\left(\...
BlackHat18's user avatar
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2 votes
1 answer
336 views

At what depth and for what architecture are random quantum circuits $1$-designs?

I was confused about something related to quantum $1$ designs. Let us recap two facts we know about random circuit ensembles that form a $1$ design. $1$ design, for a quantum circuit over $n$ qubits, ...
BlackHat18's user avatar
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2 votes
1 answer
83 views

Conditional expectation for Haar random states

Let $U$ be an $n$ qubit Haar random circuit applied to $|0^n \rangle$. Thereafter, the state is measured in the standard basis. Let $p_0$ be the probability of getting $0$ in the first qubit. We know ...
BlackHat18's user avatar
  • 1,363
2 votes
1 answer
397 views

Realizing Haar random unitary matrix on IBM Q

I am thinking about if it is possible to achieve Haar random single qubit unitary matrix on some real quantum computers like IBM Q. I am reading a paper https://arxiv.org/abs/2203.04338. In this paper,...
jisutich's user avatar
2 votes
1 answer
457 views

Density matrices of multiples copies of a single Haar-Random state

In Pseudorandom States, Non-Cloning Theorems and Quantum Money, the authors state that: Let $\rho_\mu^m$ be the density matrix of a random $|\psi\rangle^{\otimes m}$ for $|\psi\rangle$ chosen from ...
Tristan Nemoz's user avatar
  • 6,712
2 votes
1 answer
277 views

How to calculate the Haar measure for the middle SU(2), in an SU(3) factorization?

I read this blog https://pennylane.ai/qml/demos/tutorial_haar_measure#deguise2018 regarding a basic introduction to haar measure. In the "show me more math" section, they said $SU(3)$ can be ...
Việt Nguyễn's user avatar
2 votes
1 answer
159 views

Does the invariance of the Haar measure still hold if you use Clifford gates to approximate the Haar random unitaries?

I am not familiar with the Clifford group - I do know that Clifford unitaries can form a unitary 3-design (from this paper) and can be used to approximate Haar random unitaries, but I don't know how ...
Scott's user avatar
  • 23
2 votes
2 answers
192 views

Random quantum circuits and general efficient POVM measurement

Let's consider a random quantum circuit $C$, applied to the $n$ qubit initial state $|0^{n}\rangle$, producing the state $|\psi\rangle$. Consider a general efficiently implementable $m$-outcome POVM ...
BlackHat18's user avatar
  • 1,363
2 votes
1 answer
403 views

Average output state of random quantum circuits

Let $|\psi\rangle = C_1 |0^{n}\rangle$ be a quantum state, such that $C_1$ is a Haar random unitary circuit. Consider a density matrix $\rho$ as follows \begin{equation} \rho_1 = \mathbb{E}[|\psi\...
BlackHat18's user avatar
  • 1,363
2 votes
1 answer
221 views

Symmetric subspaces and Haar averaging over the Unitary group

I am interested in the following Haar average over the unitary group $D(x) = \int d\mathscr{U} ~(\mathscr{U})^{\otimes 2}(|\tilde{x}_{\mathscr{U}}\rangle\langle \tilde{x}_{\mathscr{U}}|)^{\otimes 2} (\...
Ghost-of-PPPF's user avatar
2 votes
1 answer
116 views

How to compute k-moment of Haar averaging with n qubits

Let us consider the following Haar averaging over $k$ copies of Pauli strings of $n$ qubits: $\mathbb{E}_U \left[ U^{\otimes k}\sigma_{q_1} \otimes … \otimes \sigma_{q_k} (U^{\dagger})^{\otimes k}\...
Emma's user avatar
  • 31
2 votes
2 answers
137 views

Approximating the average of a rational function with respect to the Haar measure

Suppose I have a state $|\psi\rangle = U|0\rangle$ where $U$ is a $d$-dimensional unitary sampled uniformly with respect the Haar measure. I'm interested in computing or approximating analytically an ...
forky40's user avatar
  • 7,183
2 votes
1 answer
52 views

Sampling Haar over two systems

Say $M$ is a matrix acting on $C^r \otimes C^s$. $X$ is the system of dimension $r$, and $Y$ is the system of dimension $s$. With $|\psi\rangle$ sampled from Haar, how can we show that $$ \int (...
qc6518's user avatar
  • 163
2 votes
0 answers
104 views

How to integrate a function with the Haar measure over multiple qubits

I am starting with a product state over multiple qubits. That looks like the expression below. $$ |\psi\rangle = \left(\cos\left(\frac{\theta_1}{2}\right)|0\rangle+e^{i\phi_1}\sin\left(\frac{\theta_1}{...
Endeavour 's user avatar
2 votes
0 answers
112 views

Distribution of partial trace of Haar unitary

I am sure this must have been covered in the mathematical literature, but hoping someone can direct me to the right place. Let us be given random unitaries $U$ on $n$ qubits (so dimension of the space ...
nervxxx's user avatar
  • 540
2 votes
0 answers
107 views

A question on random quantum states and the uniform distribution

Consider an $n$ qubit Haar random quantum state $|\psi\rangle$. Consider a distribution $\mathcal{D}_1$ over $n$ bit strings defined as $$ p_x = |\langle x| \psi \rangle|^{2}, $$ for $x \in \{0, 1\}^{...
Tom Clancy's user avatar