Exact solution of the many-body problem with a $\mathcal{O}\left(n^6\right)$ complexity

• URL: http://arxiv.org/abs/2111.15281v3
• Date: Thu, 9 Jun 2022 13:06:12 GMT
• Title: Exact solution of the many-body problem with a $\mathcal{O}\left(n^6\right)$ complexity
• Authors: Thierry Deutsch
• Abstract summary: We define a new mathematical object, called a pair $=left(ACright)$ of anti-commutation matrices (ACMP) We show that we can have a compact and exact parametrization with a $mathcalOleft(n6right)$ complexity.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this article, we define a new mathematical object, called a pair $D=\left(A,C\right)$ of anti-commutation matrices (ACMP) based on the anti-commutation relation $a^\dag_{i}a_{j} + a_{j}a^\dag_{i} = \delta_{ij}$ applied to the scalar product between the many-body wavefunctions. This ACMP explicitly separates the different levels of correlation. The one-body correlations are defined by a ACMP $D^0=\left(A^0,C^0\right)$ and the two-body ones by a set of $n$ ACMPs $D^i=\left(A^i,C^i\right)$ where $n$ is the number of states. We show that we can have a compact and exact parametrization with $n^4$ parameters of the two-body reduced density matrix (\TRDM) of any pure or mixed $N$-body state to determine the ground state energy with a $\mathcal{O}\left(n^6\right)$ complexity.

Related papers

• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• LevAttention: Time, Space, and Streaming Efficient Algorithm for Heavy Attentions [54.54897832889028]
  We show that for any $K$, there is a universal set" $U subset [n]$ of size independent of $n$, such that for any $Q$ and any row $i$, the large attention scores $A_i,j$ in row $i$ of $A$ all have $jin U$. We empirically show the benefits of our scheme for vision transformers, showing how to train new models that use our universal set while training as well.
  arXiv  Detail & Related papers  (2024-10-07T19:47:13Z)
• Partially Unitary Learning [0.0]
  An optimal mapping between Hilbert spaces $IN$ of $left|psirightrangle$ and $OUT$ of $left|phirightrangle$ is presented. An iterative algorithm for finding the global maximum of this optimization problem is developed.
  arXiv  Detail & Related papers  (2024-05-16T17:13:55Z)
• Solving Dense Linear Systems Faster Than via Preconditioning [1.8854491183340518]
  We show that our algorithm has an $tilde O(n2)$ when $k=O(n0.729)$. In particular, our algorithm has an $tilde O(n2)$ when $k=O(n0.729)$. Our main algorithm can be viewed as a randomized block coordinate descent method.
  arXiv  Detail & Related papers  (2023-12-14T12:53:34Z)
• $\ell_p$-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)
• Exactly solvable piecewise analytic double well potential $V_{D}(x)=min[(x+d)^2,(x-d)^2]$ and its dual single well potential $V_{S}(x)=max[(x+d)^2,(x-d)^2]$ [0.0]
  Two piecewise analytic quantum systems with a free parameter $d>0$ are obtained. Their eigenvalues $E$ for the even and odd parity sectors are determined. vivid pictures unfold showing the tunneling effects between the two wells.
  arXiv  Detail & Related papers  (2022-09-20T03:46:03Z)
• 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)
• 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)
• An Algorithm for Reversible Logic Circuit Synthesis Based on Tensor Decomposition [0.0]
  An algorithm for reversible logic synthesis is proposed. Map can be written as a tensor product of a rank-($2n-2$) tensor and the $2times 2$ identity matrix.
  arXiv  Detail & Related papers  (2021-07-09T08:18:53Z)
• The Average-Case Time Complexity of Certifying the Restricted Isometry Property [66.65353643599899]
  In compressed sensing, the restricted isometry property (RIP) on $M times N$ sensing matrices guarantees efficient reconstruction of sparse vectors. We investigate the exact average-case time complexity of certifying the RIP property for $Mtimes N$ matrices with i.i.d. $mathcalN(0,1/M)$ entries.
  arXiv  Detail & Related papers  (2020-05-22T16:55:01Z)

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.