$TimeEvolver$: A Program for Time Evolution With Improved Error Bound

• URL: http://arxiv.org/abs/2205.15346v1
• Date: Mon, 30 May 2022 18:00:16 GMT
• Title: $TimeEvolver$: A Program for Time Evolution With Improved Error Bound
• Authors: Marco Michel, Sebastian Zell
• Abstract summary: We present $TimeEvolver$, a program for computing time evolution in a generic quantum system. It relies on Krylov subspace techniques to tackle the problem of multiplying the exponential of a large sparse matrix $i H$. The fact that $H$ is Hermitian makes it possible to provide an easily computable bound on the accuracy of the Krylov approximation.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We present $TimeEvolver$, a program for computing time evolution in a generic quantum system. It relies on well-known Krylov subspace techniques to tackle the problem of multiplying the exponential of a large sparse matrix $i H$, where $H$ is the Hamiltonian, with an initial vector $v$. The fact that $H$ is Hermitian makes it possible to provide an easily computable bound on the accuracy of the Krylov approximation. Apart from effects of numerical roundoff, the resulting a posteriori error bound is rigorous, which represents a crucial novelty as compared to existing software packages such as $Expokit$ (R. Sidje, ACM Trans. Math. Softw. 24 (1) 1998). On a standard notebook, $TimeEvolver$ allows to compute time evolution with adjustable precision in Hilbert spaces of dimension greater than $10^6$. Additionally, we provide routines for deriving the matrix $H$ from a more abstract representation of the Hamiltonian operator.

Related papers

• Graph-Theoretic Analysis of $n$-Replica Time Evolution in the Brownian Gaussian Unitary Ensemble [3.9404852133765083]
  We investigate the $n$-replica time evolution operator $mathcalU_n(t)equiv emathcalL_nt $ for the Brownian Gaussian Unitary Ensemble (BGUE) using a graph-theoretic approach. Explicit representations for the cases of $n = 2$ and $n = 3$ are derived, emphasizing the role of graph categorization in simplifying calculations.
  arXiv  Detail & Related papers  (2025-02-13T12:24:50Z)
• Learning the structure of any Hamiltonian from minimal assumptions [2.810160553339817]
  We study the problem of learning an unknown quantum many-body Hamiltonian $H$ from black-box queries to its time evolution. We present efficient algorithms to learn any $n$-qubit Hamiltonian, assuming only a bound on the number of Hamiltonian terms.
  arXiv  Detail & Related papers  (2024-10-29T00:43:33Z)
• Improved Time-independent Hamiltonian Simulation [0.0]
  We describe a simple method for simulating time-independent Hamiltonian $H$ that could be decomposed as $H = sum_i=1m H_i$. We employ the recently introduced quantum singular value transformation framework to utilize the ability to simulate $H_i$ in an alternative way.
  arXiv  Detail & Related papers  (2024-10-20T02:49:14Z)
• Optimal Sketching for Residual Error Estimation for Matrix and Vector Norms [50.15964512954274]
  We study the problem of residual error estimation for matrix and vector norms using a linear sketch. We demonstrate that this gives a substantial advantage empirically, for roughly the same sketch size and accuracy as in previous work. We also show an $Omega(k2/pn1-2/p)$ lower bound for the sparse recovery problem, which is tight up to a $mathrmpoly(log n)$ factor.
  arXiv  Detail & Related papers  (2024-08-16T02:33:07Z)
• Enhancing the Harrow-Hassidim-Lloyd (HHL) algorithm in systems with large condition numbers [0.0]
  We demonstrate the ability of Psi-HHL to handle situations involving large $mathcalkappa$ matrices. We consider matrices up to size $256 times 256$ that reach $mathcalkappa$ of about 466.
  arXiv  Detail & Related papers  (2024-07-31T14:41:30Z)
• Projection by Convolution: Optimal Sample Complexity for Reinforcement Learning in Continuous-Space MDPs [56.237917407785545]
  We consider the problem of learning an $varepsilon$-optimal policy in a general class of continuous-space Markov decision processes (MDPs) having smooth Bellman operators. Key to our solution is a novel projection technique based on ideas from harmonic analysis. Our result bridges the gap between two popular but conflicting perspectives on continuous-space MDPs.
  arXiv  Detail & Related papers  (2024-05-10T09:58:47Z)
• Ground State Preparation via Qubitization [0.0]
  We describe a protocol for preparing the ground state of a Hamiltonian $H$ on a quantum computer. The method relies on the so-called qubitization'' procedure of Low and Chuang. We illustrate our method on two models: the transverse field Ising model and a single qubit toy model.
  arXiv  Detail & Related papers  (2023-06-26T18:20:48Z)
• Cryptographic Hardness of Learning Halfspaces with Massart Noise [59.8587499110224]
  We study the complexity of PAC learning halfspaces in the presence of Massart noise. We show that no-time Massart halfspace learners can achieve error better than $Omega(eta)$, even if the optimal 0-1 error is small.
  arXiv  Detail & Related papers  (2022-07-28T17:50:53Z)
• Learning a Latent Simplex in Input-Sparsity Time [58.30321592603066]
  We consider the problem of learning a latent $k$-vertex simplex $KsubsetmathbbRdtimes n$, given access to $AinmathbbRdtimes n$. We show that the dependence on $k$ in the running time is unnecessary given a natural assumption about the mass of the top $k$ singular values of $A$.
  arXiv  Detail & Related papers  (2021-05-17T16:40:48Z)
• Hardness of Learning Halfspaces with Massart Noise [56.98280399449707]
  We study the complexity of PAC learning halfspaces in the presence of Massart (bounded) noise. We show that there an exponential gap between the information-theoretically optimal error and the best error that can be achieved by a SQ algorithm.
  arXiv  Detail & Related papers  (2020-12-17T16:43:11Z)
• Optimal Robust Linear Regression in Nearly Linear Time [97.11565882347772]
  We study the problem of high-dimensional robust linear regression where a learner is given access to $n$ samples from the generative model $Y = langle X,w* rangle + epsilon$ We propose estimators for this problem under two settings: (i) $X$ is L4-L2 hypercontractive, $mathbbE [XXtop]$ has bounded condition number and $epsilon$ has bounded variance and (ii) $X$ is sub-Gaussian with identity second moment and $epsilon$ is
  arXiv  Detail & Related papers  (2020-07-16T06:44:44Z)

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.