How to check universality of quantum gates?
- URL: http://arxiv.org/abs/2111.03862v6
- Date: Fri, 17 Jun 2022 13:48:06 GMT
- Title: How to check universality of quantum gates?
- Authors: Adam Sawicki, Lorenzo Mattioli and Zolt\'an Zimbor\'as
- Abstract summary: Our first criterion states that $mathcalSsubset G_d:=U(d)$ is universal if and only if $mathcalS$ forms a $delta$-approximate $t(d)$-design.
Our second universality criterion says that $mathcalSsubset G_d$ is universal if and only if the centralizer of $mathcalSt(d),t(d)=Uotimes t(d)otimes t(d)|U
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: We provide two simple universality criteria. Our first criterion states that
$\mathcal{S}\subset G_d:=U(d)$ is universal if and only if $\mathcal{S}$ forms
a $\delta$-approximate $t(d)$-design, where $t(2)=6$ and $t(d)=4$ for $d\geq3$.
Our second universality criterion says that $\mathcal{S}\subset G_d$ is
universal if and only if the centralizer of
$\mathcal{S}^{t(d),t(d)}=\{U^{\otimes t(d)}\otimes \bar{U}^{\otimes t(d)}|U\in
\mathcal{S}\}$ is equal to the centralizer of $G_d^{t(d),t(d)}=\{U^{\otimes
t(d)}\otimes \bar{U}^{\otimes t(d)}|U\in G_d\}$, where $t(2)=3$, and $t(d)=2$
for $d\geq 3$. The equality of the centralizers can be verified by comparing
their dimensions.
Related papers
- On the $O(\frac{\sqrt{d}}{T^{1/4}})$ Convergence Rate of RMSProp and Its Momentum Extension Measured by $\ell_1$ Norm [59.65871549878937]
This paper considers the RMSProp and its momentum extension and establishes the convergence rate of $frac1Tsum_k=1T.
Our convergence rate matches the lower bound with respect to all the coefficients except the dimension $d$.
Our convergence rate can be considered to be analogous to the $frac1Tsum_k=1T.
arXiv Detail & Related papers (2024-02-01T07:21:32Z) - Synthesis and Arithmetic of Single Qutrit Circuits [0.9208007322096532]
We study single qutrit quantum circuits consisting of words over the Clifford+ $mathcalD$ gate set.
We characterize classes of qutrit unit vectors $z$ with entries in $mathbbZ[xi, frac1chi]$.
arXiv Detail & Related papers (2023-11-15T04:50:41Z) - Dimension-free Remez Inequalities and norm designs [48.5897526636987]
A class of domains $X$ and test sets $Y$ -- termed emphnorm -- enjoy dimension-free Remez-type estimates.
We show that the supremum of $f$ does not increase by more than $mathcalO(log K)2d$ when $f$ is extended to the polytorus.
arXiv Detail & Related papers (2023-10-11T22:46:09Z) - Subspace Controllability and Clebsch-Gordan Decomposition of Symmetric
Quantum Networks [0.0]
We describe a framework for the controllability analysis of networks of $n$ quantum systems of an arbitrary dimension $d$, it qudits
Because of the symmetry, the underlying Hilbert space, $cal H=(mathbbCd)otimes n$, splits into invariant subspaces for the Lie algebra of $S_n$-invariant elements in $u(dn)$, denoted here by $uS_n(dn)$.
arXiv Detail & Related papers (2023-07-24T16:06:01Z) - $\ell_p$-Regression in the Arbitrary Partition Model of Communication [59.89387020011663]
We consider the randomized communication complexity of the distributed $ell_p$-regression problem in the coordinator model.
For $p = 2$, i.e., least squares regression, we give the first optimal bound of $tildeTheta(sd2 + sd/epsilon)$ bits.
For $p in (1,2)$,we obtain an $tildeO(sd2/epsilon + sd/mathrmpoly(epsilon)$ upper bound.
arXiv Detail & Related papers (2023-07-11T08:51:53Z) - Realization of an arbitrary structure of perfect distinguishability of
states in general probability theory [0.0]
All subsets with a single element are of course in $mathcal A$, and since smaller collections are easier to distinguish, if $Hin mathcal A$ and $L subset H$ then $Lin mathcal A$; in other words, $mathcal A$ is a so-called $textitindependence system$ on the set of indices $[n]$.
arXiv Detail & Related papers (2023-01-16T18:33:39Z) - Enlarging the notion of additivity of resource quantifiers [62.997667081978825]
Given a quantum state $varrho$ and a quantifier $cal E(varrho), it is a hard task to determine $cal E(varrhootimes N)$.
We show that the one shot distillable entanglement of certain spherically symmetric states can be quantitatively approximated by such an augmented additivity.
arXiv Detail & Related papers (2022-07-31T00:23:10Z) - Matrix concentration inequalities and efficiency of random universal
sets of quantum gates [0.0]
For a random set $mathcalS subset U(d)$ of quantum gates we provide bounds on the probability that $mathcalS$ forms a $delta$-approximate $t$-design.
We show that for $mathcalS$ drawn from an exact $t$-design the probability that it forms a $delta$-approximate $t$-design satisfies the inequality $mathbbPleft(delta geq x right)leq 2D_t
arXiv Detail & Related papers (2022-02-10T23:44:09Z) - Low-Rank Approximation with $1/\epsilon^{1/3}$ Matrix-Vector Products [58.05771390012827]
We study iterative methods based on Krylov subspaces for low-rank approximation under any Schatten-$p$ norm.
Our main result is an algorithm that uses only $tildeO(k/sqrtepsilon)$ matrix-vector products.
arXiv Detail & Related papers (2022-02-10T16:10:41Z) - On the continuous Zauner conjecture [0.0]
In this paper we prove that for any $t in [-frac1d2-1, frac1d+1] setminus0$ the equality $textebr(Phi_t)=d2$ is equivalent to the existence of a pair of informationally complete unit norm tight frames.
arXiv Detail & Related papers (2021-12-11T00:14:35Z) - Spectral properties of sample covariance matrices arising from random
matrices with independent non identically distributed columns [50.053491972003656]
It was previously shown that the functionals $texttr(AR(z))$, for $R(z) = (frac1nXXT- zI_p)-1$ and $Ain mathcal M_p$ deterministic, have a standard deviation of order $O(|A|_* / sqrt n)$.
Here, we show that $|mathbb E[R(z)] - tilde R(z)|_F
arXiv Detail & Related papers (2021-09-06T14:21:43Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.