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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71 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, ...
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1answer
42 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 ...
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1answer
90 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\...
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0answers
62 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
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1answer
154 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 ...
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1answer
102 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 ...
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1answer
110 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 ...
2
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1answer
203 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
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2answers
80 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 &...
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2answers
278 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 |...
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1answer
162 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 ...
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195 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 ...
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1answer
129 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 ...
12
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1answer
226 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 ...
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2answers
255 views

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.
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338 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 \...