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What is the expectation value of $|\langle \psi|U|\psi \rangle|$ over Haar random states $|\psi\rangle$?

This partial answer calculates the integral for $d=2$. In this case, every traceless unitary $U$ is equivalent to the Pauli $Z$ up to similarity and global phase, so, by rotational invariance of the ...
Adam Zalcman's user avatar
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3 votes
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Schur's lemma for quantum states

Most generally, Schur's Lemma is used as a tool in representation theory. In this answer, I'll try to explain it without talking about said theory. Preliminaries: Schur's Lemma Let us consider a ...
Tristan Nemoz's user avatar
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Two qubit Pauli expectation value of $\underset{U}{\mathbb{E}}[U^{\otimes 2} (P_1 \otimes P_2)^{\otimes 2} U^{*\otimes 2}]$

We can prove a slightly more general statement, which works for any Hermitian "sandwiched" matrix $P$ in any dimension $d$... but only for $t=2$ in the number of copies :'( Let us call $V$ ...
Tristan Nemoz's user avatar
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Moments of Pauli coefficients of Haar-random states

Let us compute the value for $\alpha=4$, averaged over Haar-random states. We have the following identity: $$ \sum_{P\in\mathcal{P}_n} \mathrm{tr}(\rho P)^4= \sum_{P\in\mathcal{P}_n} \mathrm{tr}(\rho^...
Markus Heinrich's user avatar
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How to calculate the Haar measure for the middle SU(2), in an SU(3) factorization?

Let us consider the $SU(3)$ decomposition as such: Assuming the matrix we create is Haar-random, it means that when it is applied to $|0\rangle$ it should yield a Haar-random state. So let's see how ...
Tristan Nemoz's user avatar
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1 vote

Density Matrix for a Quantum Circuit with Clifford Gates and a $T$ Gate in Qiskit

Maybe you can use approximate or Simplified Models Stabilizer Rank Methods: For circuits that predominantly contain Clifford gates and a small number of T gates, you can use techniques based on the ...
Bram's user avatar
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1 vote
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Simulating Large Quantum Systems with Single T-Gate in Qiskit: Memory Error Beyond Certain Qubit Threshold

On IBM Quantum platform, a universal simulator is limited by 32 qubits. There is also 64-qubit simulator Clifford+T which could be suitable for your problem. You can also run your circuit on 127-qubit ...
Martin Vesely's user avatar
1 vote

Expected trace distance between two types of random ensembles

Assume each of these states to be independently distributed. The question is too challenging as stated, so let us look at the expected fidelity between the two states, where trace distance is the ...
Quantum Mechanic's user avatar
1 vote

Does integrating w.r.t. the Haar measure commute with taking partial trace?

Yes. We can check this explicitly as follows. $$\int dU \ \text{Tr}_\mathcal{E}(\rho(U)) = \int dU \ \sum_i (\mathbb{I} \otimes \langle i \lvert) \rho(U) (\lvert i \rangle \otimes \mathbb{I})\\ = \...
Silly Goose's user avatar
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What is the expectation value of $|\langle \psi|U|\psi \rangle|$ over Haar random states $|\psi\rangle$?

As a follow-up discussion with Adam, I evaluated the integral till $d=4$ for high-dimensional unitary $Z$, with ${Z}_{kl}{=}\exp(\frac{i2\pi k}{d})\delta_{kl}$. For $d{=}4$, I haven't considered the ...
Abir's user avatar
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Symmetric subspaces and Haar averaging over the Unitary group

You have a syntactic mistake in your argument. The dimensions in the equation $$(V^\dagger)^{\otimes 2} \otimes V^{\otimes 2}D(x)(V^\dagger)^{\otimes 2} \otimes V^{\otimes 2} = D(x) \,,$$ don't match. ...
Markus Heinrich's user avatar
1 vote

Show that, averaging over uniformly random unitaries, $\mathbb{E}[UXU^\dagger]=\operatorname{Tr}(X)\frac{I}{d}$

Another intuitive way of looking at it is to note that the Haar average of all pure states is a maximally mixed state, $\int |\psi\rangle\langle\psi| d\psi=\frac{\mathbb{I}}{d}$. Considering $|\psi\...
Abir's user avatar
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