# 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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### 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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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 $$\int (... 0 votes 0 answers 29 views ### Convergence of measure of products of random unitaries I'm trying to read this paper by Emerson, Livine, Llyod (https://journals.aps.org/pra/abstract/10.1103/PhysRevA.72.060302), arXiv version: (https://arxiv.org/pdf/quant-ph/0503210.pdf). Essentially I ... 7 votes 1 answer 943 views ### What is a Haar random quantum state? Can somebody please explain me what is a Haar random state? I am not able to find any friendly resource to read about it. 5 votes 1 answer 47 views ### Anticoncentration for two independent random quantum circuits in parallel Consider two Haar random n qubit unitaries, U_1 and U_2. Consider the quantum state$$|\psi\rangle = (U_1 \otimes U_2) |0^{2n}\rangle. $$Let p_x = |\langle x| \psi \rangle|^{2}, for x \in \{... 3 votes 0 answers 125 views ### Reduced density matrix of a Haar random state and its Schmidt decomposition Consider a Haar random quantum state |\psi\rangle. Note that$$\rho =\mathbb{E}[|\psi\rangle\langle \psi|] = \frac{\mathbb{I}_{n}}{2^{n}}.$$\mathbb{I}_n is the identity operator on n qubits. ... 4 votes 1 answer 118 views ### Random quantum states and Schur-Weyl duality Consider the following density matrix over n qubits, with C being a single qubit operator:$$ \rho_{n} = \int_{C \sim \text{Haar}} \big(C|0\rangle\langle0|C^\dagger\big)^{\otimes n} dC. $$Let's ... 1 vote 0 answers 32 views ### 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 ... 2 votes 1 answer 127 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, ... 2 votes 1 answer 79 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 ... 2 votes 1 answer 138 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\... 2 votes 0 answers 87 views ### 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 ... 4 votes 1 answer 160 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 ... 3 votes 1 answer 116 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 ... 0 votes 1 answer 196 views ### 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 ... 3 votes 1 answer 314 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 \... 4 votes 2 answers 125 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 &... 3 votes 2 answers 495 views ### Question regarding integration of Haar random state 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 |... 5 votes 1 answer 182 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 ... 4 votes 2 answers 347 views ### 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 ... 6 votes 1 answer 181 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 ... 13 votes 1 answer 232 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 ...
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 \...