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Results tagged with haar-distribution
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user 1351
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.
6
votes
1
answer
255
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 …
- 1,515
6
votes
1
answer
1k
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 \ …
- 1,515
6
votes
1
answer
594
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 s …
- 1,515
5
votes
2
answers
222
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 classica …
- 1,515
5
votes
0
answers
326
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. No …
- 1,515
5
votes
1
answer
408
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)
\begin{equation}
\mathbb{E}\left[\left|\langle 0^{n} |U^{2} |0^{n}\rangle\right|^{2}\right].
\end{equation}
The …
- 1,515
5
votes
2
answers
1k
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 th …
- 1,515
4
votes
2
answers
780
views
Computing expectation value of $|\langle z|C|0^n\rangle|^2$ over Haar random circuit
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 …
- 1,515
4
votes
1
answer
181
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 tak …
- 1,515
4
votes
1
answer
284
views
Compute the large $n$ distribution of $|\langle z_i|\psi\rangle|^2$ over Haar random quantum...
Let $|\psi\rangle$ be a $n$ qubit Haar-random quantum state. I am trying to show that in the limit of large $n$, for each $z_{i} \in \{0, 1\}^{n}$,
$$ |\langle 0|\psi\rangle|^{2}, |\langle 1|\psi\rang …
- 1,515
3
votes
1
answer
117
views
Independence in state prepared by independently drawn Haar random gates
Consider independently drawn $2 \times 2$ Haar random unitaries $U_1, U_2, \ldots, U_n$ and
$$V = U_1 \otimes U_2 \otimes \cdots U_n.$$ Consider the state $\sigma$ given by
$$\sigma = V \rho V^{*}, $$ …
- 1,515
3
votes
0
answers
175
views
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) = \sum …
- 1,515
2
votes
0
answers
262
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
a …
- 1,515
2
votes
1
answer
445
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\rang …
- 1,515
2
votes
2
answers
213
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 m …
- 1,515