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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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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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 ...
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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 $?
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 ...
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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
$$
\int (...
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What is a Haar random quantum state?
Can somebody please explain me what is a Haar random state? I am not able to find any friendly resource to read about it.
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Anticoncentration for two independent random quantum circuits in parallel
Consider two Haar random $n$ qubit unitaries, $U_1$ and $U_2$. Consider the quantum state
$$|\psi\rangle = (U_1 \otimes U_2) |0^{2n}\rangle. $$
Let $p_x = |\langle x| \psi \rangle|^{2}$, for $x \in \{...
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Reduced density matrix of a Haar random state and its Schmidt decomposition
Consider a Haar random quantum state $|\psi\rangle$. Note that
$$\rho =\mathbb{E}[|\psi\rangle\langle \psi|] = \frac{\mathbb{I}_{n}}{2^{n}}.$$
$\mathbb{I}_n$ is the identity operator on $n$ qubits. ...
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Random quantum states and Schur-Weyl duality
Consider the following density matrix over $n$ qubits, with $C$ being a single qubit operator:
$$
\rho_{n} = \int_{C \sim \text{Haar}} \big(C|0\rangle\langle0|C^\dagger\big)^{\otimes n} dC.
$$
Let's ...
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Optimality of the SWAP test versus weak Schur sampling for testing unitarily invariant properties
Consider the following setting.
I am either given the density matrix $|\psi\rangle \langle \psi|^{\otimes k}$ or the density matrix $\frac{\mathbb{I}^{\otimes k}}{2^{nk}}$, where $\mathbb{I}$ is the $...
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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, ...
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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 ...
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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\...
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Spreading of entanglement with depth for Haar random states
Consider a Haar random quantum state of depth $d$. Consider any bipartition of this state. According to this paper (page $2$):
Haar-random states on $n$ qudits are nearly maximally entangled across
...
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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 ...
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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 ...
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Quantum supremacy: shallow depth Haar random circuits and unitary designs
I had a confusion about shallow depth Haar random quantum circuits. In this paper, in Section B (related works), it is mentioned that Haar random quantum circuits form approximate $2$-designs only ...
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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 \...
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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 &...
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