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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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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 ...
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)$...
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 ...
1 vote
1 answer
94 views

Matrix representation of the symmetric subspace for two copies

Consider two copies of an $n$ qubit Haar random state, given by: \begin{equation} \rho = \mathbb{E}_{U \sim \mathsf{Haar}}\left[U |0^n\rangle \langle 0^n| U^{*}\otimes U |0^n\rangle \langle 0^n| U^{*}\...
0 votes
1 answer
52 views

Density Matrix for a Quantum Circuit with Clifford Gates and a $T$ Gate in Qiskit

I am trying to analyze the impact of a single $T$ gate within a quantum circuit that primarily consists of Clifford gates. My goal is to understand the $T$ gate's role in $T$-design and Anti-...
1 vote
1 answer
60 views

Simulating Large Quantum Systems with Single T-Gate in Qiskit: Memory Error Beyond Certain Qubit Threshold

I'm currently conducting experiments on unitary t-designs, utilizing random Clifford and T gates within the Qiskit framework. My goal is to simulate quantum circuits that involve the application of a ...
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} (\...
1 vote
1 answer
96 views

Two qubit Pauli expectation value of $\underset{U}{\mathbb{E}}[U^{\otimes 2} (P_1 \otimes P_2)^{\otimes 2} U^{*\otimes 2}]$

I want to find a value for the expression: $$\underset{U}{\mathbb{E}}[U^{\otimes 2} (P_1 \otimes P_2)^{\otimes 2} U^{*\otimes 2}],$$ where $U$ is a two-qubit unitary operator chosen Haar randomly, $...
1 vote
1 answer
62 views

Expected trace distance between two types of random ensembles

Consider a Haar random state on $n$ qubits, and denote it by $|\psi\rangle$. Now consider the following state $$|\phi\rangle = \frac{1}{\sqrt{k}} \sum_{i=1}^{k} |\phi_{1, i} \rangle \otimes |\phi_{2, ...
1 vote
0 answers
17 views

How to calculate the volume of a point set with parameters go over the Haar distribution?

Speically, how to calculate the volume of the set $\{(|\langle\psi|M_1|\psi\rangle|^2,...,|\langle\psi|M_s|\psi\rangle|^2)|\rho \in \mathbb{H}^n\}$ in the space $\mathbb{R}^{s}$, in which $\mathbb{H}^...
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}{...
5 votes
1 answer
162 views

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 ...
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) = \...
0 votes
0 answers
80 views

What is the expectation value of the overlap of two uniformly random pure states? [duplicate]

Let $\psi$ and $\phi$ be two uniformly random pure state $\psi, \phi \sim\mathbb{C}^d$. The the following equality holds \begin{align} \mathbb{E}_{\psi, \phi \sim \mathbb{C}^d} {\rm Tr}[\vert \phi \...
5 votes
2 answers
372 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 ...
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 ...
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 ...
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 ...
1 vote
1 answer
43 views

Does integrating w.r.t. the Haar measure commute with taking partial trace?

Consider a density matrix $\rho(U)$ which depends on $U \in SU(2^n)$, corresponding to a state of a composite, finite-dimensional Hilbert space $\mathcal{H} \cong \bigotimes_{i=1}^{2^n} \mathbb{C}^2$ ...
1 vote
0 answers
44 views

Measure on the unitary space and complexity

I'm currently studying various quantum supremacy protocols and i'm struggling to have a clear and well defined view on the rôle of approximating the Haar-measure (through k-designs ...) and the ...
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}\...
0 votes
1 answer
169 views

Twirling of quantum states: Maximally entangled states

I have been reading the paper "Resource theory of unextendibility and non-asymptotic quantum capacity" (https://arxiv.org/pdf/1803.10710.pdf) by Kaur et.al, I have two questions ...
1 vote
1 answer
255 views

Expressibility and Entanglement Capability of the Parameterized Quantum Circuits

I am trying to calculate the expressibility and entangling capability of a quantum state resulting from a circuit as defined in reference I. One of my attempts was to follow reference II which gives ...
1 vote
1 answer
57 views

Werner Twirling Channel - How to Retrieve Prefactors?

In Watrous' Theory of Quantum Information, Example 7.25 discusses the Werner Twirling Channel: $$\Xi(X) = \int (U \otimes U) X (U \otimes U)^* \mathrm{d}\eta(U)$$ where $\eta$ denotes the Haar measure ...
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 ...
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 ...
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 ...
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(\...
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 $$\sigma = V \rho V^{*}, $$ ...
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 (...
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 ...
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 $\...
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 ...
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,...
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 ...
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}(...
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 ...
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} ...
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 ...
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?
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\...
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 |...
1 vote
0 answers
176 views

How to compute Haar average over the unitary group of a ratio of homogeneous polynomials?

I am interested in the following Haar average over the unitary group: $\mathbb{E}_U\Big[\frac{tr(U^{\otimes p}|j\rangle\langle j|(U^\dagger)^{\otimes p}\rho \otimes \sigma ...)}{tr(U^{\otimes q}|j\...
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 \...
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\}^{...
14 votes
2 answers
7k views

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.
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 ...
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 ...
6 votes
1 answer
188 views

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 ...
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 (...