# 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.

66 questions
Filter by
Sorted by
Tagged with
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.
283 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 ...
• 183
1k views

• 1,363
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 ...
• 163
244 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 ...
• 1,363
515 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 ...
• 61
1k views

### Expected value of a Haar random quantum state multiplied by a unitary

Consider a quantity $$\mathbb{E}\big[\langle z|\rho|z\rangle\big],$$ where $\rho = |\psi \rangle \langle \psi|$ is a Haar-random state $n$-qubit quantum state and $z$ is ...
• 1,363
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) $$\mathbb{E}\left[\left|\langle 0^{n} |U^{2} |0^{n}\rangle\right|^{2}\right].$$ The ...
• 1,363
120 views

• 1,363
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?
179 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 ...
• 1,363
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 ...
79 views

• 6,712
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 $$\sigma = V \rho V^{*},$$ ...
• 1,363
330 views

• 163