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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Property of Haar random state
Let $|\psi\rangle$ be a Haar random state and let $|\psi^{\perp}\rangle$ be any state that is perpendicular to $|\psi\rangle$. Let us define
$$p_x = |\langle x| \psi \rangle|^2,$$
and $$q_x = |\...
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Finding a density matrix for a distirbution of pure states
Let $\theta$ be a Gaussian variable with mean 0 and variance 1. Then for $t>0$, the variable $\theta \sqrt{t}$ is also Gaussian with mean $0$ and variance $t$. Let $|\psi_0\rangle$ be an arbitrary ...
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Averaging over Haar measure with projectors in the context of 2-qubit states
The paper https://arxiv.org/abs/2407.20184 deals with calculating the so-called filter function capturing the sensitivity of the fidelity of a quantum channel to the power-spectral-density of a noise ...
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Sampling a Haar Random State Conditioned on Having Low Entanglement Entropy
After applying exponentially, or even polynomially many random local gates to a fixed input state, the resulting distribution of the output state $\lvert\psi_{out}\rangle$ will (be very close) to Haar ...
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Is there a concentration inequality for the quantum gate fidelity $F(C,U)$ for a channel $C$ such that $\int dU F(C,U)=X$?
For a fixed quantum channel $N$ and a unitary channel $U$, we define $N$'s gate fidelity as
$$ F(N,U) = \int \langle \psi| U \, N(| \psi \rangle \langle \psi |) \, U^\dagger| \psi \rangle d\mu_H(\psi)$...
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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^{*}\...
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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-...
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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 ...
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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, $...
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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
$$|\phi\rangle = \frac{1}{\sqrt{k}} \sum_{i=1}^{k} |\phi_{1, i} \rangle \otimes |\phi_{2, ...
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How to calculate the volume of a point set with parameters go over the Haar distribution?
Speically, how to calculate the volume of the set $\{(|\langle\psi|M_1|\psi\rangle|^2,...,|\langle\psi|M_s|\psi\rangle|^2)|\rho \in \mathbb{H}^n\}$ in the space $\mathbb{R}^{s}$, in which $\mathbb{H}^...
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How do I calculate the expectation of the rational function, in the sense of the Haar measure?
I want to know the analytical solution of $\mathbb{E}_{\psi}\frac{\langle \psi |A|\psi\rangle}{\langle \psi |A^2|\psi\rangle}$. I see similar questions before approximate average, but it does not ...
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How to integrate a function with the Haar measure over multiple qubits
I am starting with a product state over multiple qubits. That looks like the expression below.
$$
|\psi\rangle = \left(\cos\left(\frac{\theta_1}{2}\right)|0\rangle+e^{i\phi_1}\sin\left(\frac{\theta_1}{...
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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 ...
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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 \...
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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) = \...
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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 ...
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Why does Schur's lemma imply that $\int \sigma^{\otimes n}_{HK} d(\sigma)$ must be a multiple of the identity on the symmetric subspace?
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 ...
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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 $\mathcal{H} \cong \bigotimes_{i=1}^{2^n} \mathbb{C}^2$ ...
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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 ...
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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 ...
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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} (\...
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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 ...
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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 ...
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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}\...
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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 ...
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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 ...
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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 ...
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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 (...
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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
$$\sigma = V \rho V^{*}, $$
...
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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(\...
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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 $\...
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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 ...
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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,...
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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 ...
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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
$$ \int d \rho \,\, \left|\text{Tr}(...
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Density matrices of multiples copies of a single Haar-Random state
In Pseudorandom States, Non-Cloning Theorems and Quantum Money, the authors state that:
Let $\rho_\mu^m$ be the density matrix of a random $|\psi\rangle^{\otimes m}$ for $|\psi\rangle$ chosen from ...
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Expectation value over random $k$-local Pauli operators for two random quantum states
Suppose we have a uniform distribution $D$ over $k$-local Pauli operators $P_{q_1}\otimes \dotsc \otimes P_{q_k} $, $P_{q_i} \in \{ X, Y, Z, I \}$. Is it possible to calculate $\mathbb{E}_{P_i \sim D} ...
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Does the invariance of the Haar measure still hold if you use Clifford gates to approximate the Haar random unitaries?
I am not familiar with the Clifford group - I do know that Clifford unitaries can form a unitary 3-design (from this paper) and can be used to approximate Haar random unitaries, but I don't know how ...
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Prove that uniformly random states have moments ${\bf E}_\psi|\langle x|\psi\rangle|^{2t}\sim1/\binom d t$
Im looking for the moments of Haar random states. Is it true that
$\textbf{E}_{\psi\sim \text{Haar}}|\langle x| \psi\rangle|^{2t}\sim \frac{1}{\binom{d}{t}}?$ How does one prove this?
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Show that, averaging over uniformly random unitaries, $\mathbb{E}[UXU^\dagger]=\operatorname{Tr}(X)\frac{I}{d}$
As mentioned e.g. in this answer, if we compute the average
$$\Phi(X)\equiv \mathbb{E}_U[UXU^\dagger] \equiv \int_{\mathbf U(d)} d\mu(U) UXU^\dagger,$$
where $d\mu(U)$ is the Haar measure over the ...
glS♦
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How to compute Haar average over the unitary group of a ratio of homogeneous polynomials?
I am interested in the following Haar average over the unitary group:
$\mathbb{E}_U\Big[\frac{tr(U^{\otimes p}|j\rangle\langle j|(U^\dagger)^{\otimes p}\rho \otimes \sigma ...)}{tr(U^{\otimes q}|j\...
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A question on random quantum states and the uniform distribution
Consider an $n$ qubit Haar random quantum state $|\psi\rangle$. Consider a distribution $\mathcal{D}_1$ over $n$ bit strings defined as
$$
p_x = |\langle x| \psi \rangle|^{2},
$$
for $x \in \{0, 1\}^{...
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Multiplication by a Haar random unitary two times
Consider a Haar random unitary $U$.
I am trying to compute the value (or put a bound on)
\begin{equation}
\mathbb{E}\left[\left|\langle 0^{n} |U^{2} |0^{n}\rangle\right|^{2}\right].
\end{equation}
The ...
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Approximating the average of a rational function with respect to the Haar measure
Suppose I have a state $|\psi\rangle = U|0\rangle$ where $U$ is a $d$-dimensional unitary sampled uniformly with respect the Haar measure. I'm interested in computing or approximating analytically an ...
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What is the expectation value ${\Bbb E}[\langle\psi,O\psi\rangle]$ over the Haar distribution?
What is the average $\mathbb{E}_{\text{Haar}}|\langle\psi|O\psi\rangle|$ of expectation of an arbitrary observable $O$ over the Haar distribution? Let $d$ be the dimension, i.e, the size of $O$. Do we ...
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How to show that the integral over all Haar states vanishes: $\int|\psi\rangle\,{\rm d}\psi = 0 $?
How can one show that the integral over all Haar states $|\psi \rangle $ is
$$
\int |\psi \rangle \, \mathrm{d}\psi = 0\ ?
$$
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Sampling Haar over two systems
Say $M$ is a matrix acting on $\mathbb C^r \otimes \mathbb C^s$. $X$ is the system of dimension $r$, and $Y$ is the system of dimension $s$.
With $|\psi\rangle$ Haar-randomly sampled, how can one show ...
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What is a Haar random quantum state?
Can somebody please explain me what is a Haar random state? I am not able to find any friendly resource to read about it.
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Anticoncentration for two independent random quantum circuits in parallel
Consider two Haar random $n$ qubit unitaries, $U_1$ and $U_2$. Consider the quantum state
$$|\psi\rangle = (U_1 \otimes U_2) |0^{2n}\rangle. $$
Let $p_x = |\langle x| \psi \rangle|^{2}$, for $x \in \{...
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