Learning the eigenstructure of quantum dynamics using classical shadows

• URL: http://arxiv.org/abs/2309.12631v1
• Date: Fri, 22 Sep 2023 05:56:58 GMT
• Title: Learning the eigenstructure of quantum dynamics using classical shadows
• Authors: Atithi Acharya, Siddhartha Saha, Shagesh Sridharan, Yanis Bahroun and Anirvan M. Sengupta
• Abstract summary: We show that for moderate size Hilbert spaces, low Kraus rank of the channel, and short time steps, the eigenvalues of the Choi matrix corresponding to the channel have a special structure. We use tools from random matrix theory to understand the effect of estimation noise in the eigenspectrum of the estimated Choi matrix.
• Score: 6.12834448484556
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Learning dynamics from repeated observation of the time evolution of an open quantum system, namely, the problem of quantum process tomography is an important task. This task is difficult in general, but, with some additional constraints could be tractable. This motivates us to look at the problem of Lindblad operator discovery from observations. We point out that for moderate size Hilbert spaces, low Kraus rank of the channel, and short time steps, the eigenvalues of the Choi matrix corresponding to the channel have a special structure. We use the least-square method for the estimation of a channel where, for fixed inputs, we estimate the outputs by classical shadows. The resultant noisy estimate of the channel can then be denoised by diagonalizing the nominal Choi matrix, truncating some eigenvalues, and altering it to a genuine Choi matrix. This processed Choi matrix is then compared to the original one. We see that as the number of samples increases, our reconstruction becomes more accurate. We also use tools from random matrix theory to understand the effect of estimation noise in the eigenspectrum of the estimated Choi matrix.

Related papers

• Lindbladian reverse engineering for general non-equilibrium steady states: A scalable null-space approach [49.1574468325115]
  We introduce a method for reconstructing the corresponding Lindbaldian master equation given any target NESS. The kernel (null-space) of the correlation matrix corresponds to Lindbladian solutions. We illustrate the method in different systems, ranging from bosonic Gaussian to dissipative-driven collective spins.
  arXiv  Detail & Related papers  (2024-08-09T19:00:18Z)
• Quantization of Large Language Models with an Overdetermined Basis [73.79368761182998]
  We introduce an algorithm for data quantization based on the principles of Kashin representation. Our findings demonstrate that Kashin Quantization achieves competitive or superior quality in model performance.
  arXiv  Detail & Related papers  (2024-04-15T12:38:46Z)
• Vectorization of the density matrix and quantum simulation of the von Neumann equation of time-dependent Hamiltonians [65.268245109828]
  We develop a general framework to linearize the von-Neumann equation rendering it in a suitable form for quantum simulations. We show that one of these linearizations of the von-Neumann equation corresponds to the standard case in which the state vector becomes the column stacked elements of the density matrix. A quantum algorithm to simulate the dynamics of the density matrix is proposed.
  arXiv  Detail & Related papers  (2023-06-14T23:08:51Z)
• Robust Dequantization of the Quantum Singular value Transformation and Quantum Machine Learning Algorithms [0.0]
  We show how many techniques from randomized linear algebra can be adapted to work under this weaker assumption. We also apply these results to obtain a robust dequantization of many quantum machine learning algorithms.
  arXiv  Detail & Related papers  (2023-04-11T02:09:13Z)
• Third quantization of open quantum systems: new dissipative symmetries and connections to phase-space and Keldysh field theory formulations [77.34726150561087]
  We reformulate the technique of third quantization in a way that explicitly connects all three methods. We first show that our formulation reveals a fundamental dissipative symmetry present in all quadratic bosonic or fermionic Lindbladians. For bosons, we then show that the Wigner function and the characteristic function can be thought of as ''wavefunctions'' of the density matrix.
  arXiv  Detail & Related papers  (2023-02-27T18:56:40Z)
• Trimmed Sampling Algorithm for the Noisy Generalized Eigenvalue Problem [0.0]
  Solving the generalized eigenvalue problem is a useful method for finding energy eigenstates of large quantum systems. The problem is especially bad when matrix elements are evaluated using methods and have significant error bars. We introduce the trimmed sampling algorithm in order to solve this problem.
  arXiv  Detail & Related papers  (2022-09-05T18:10:12Z)
• On the Benefits of Large Learning Rates for Kernel Methods [110.03020563291788]
  We show that a phenomenon can be precisely characterized in the context of kernel methods. We consider the minimization of a quadratic objective in a separable Hilbert space, and show that with early stopping, the choice of learning rate influences the spectral decomposition of the obtained solution.
  arXiv  Detail & Related papers  (2022-02-28T13:01:04Z)
• Machine Learning approach to the Floquet--Lindbladian problem [0.0]
  We try to find a Lindbladian which generates an evolution identical -- when monitored at discrete instances of time -- to the one induced by a quantum map. We use different Machine Learning methods as a tool to check the hypothesis that the answer to the question is encoded in spectral properties of the so-called Choi matrix. The outcome of our experiment is that, for a given map, the property of being generated by a time-independent Lindbladian is encoded both in the eigenvalues and the eigenstates of the corresponding Choi matrix.
  arXiv  Detail & Related papers  (2022-01-01T22:36:15Z)
• A theory of quantum subspace diagonalization [3.248953303528541]
  We show that a quantum subspace diagonalization algorithm can accurately compute the smallest eigenvalue of a large Hermitian matrix. Our results can be of independent interest to solving eigenvalue problems outside the context of quantum computation.
  arXiv  Detail & Related papers  (2021-10-14T16:09:07Z)
• Hamiltonian-Driven Shadow Tomography of Quantum States [0.0]
  We study the scenario in which the unitary channel can be shallow and is generated by a quantum chaotic Hamiltonian via time evolution. We find that it can be more efficient than the unitary-2-design-based shadow tomography in a sequence of intermediate time windows. In particular, the efficiency of predicting diagonal observables is improved by a factor of $D$ without sacrificing the efficiency of predicting off-diagonal observables.
  arXiv  Detail & Related papers  (2021-02-19T19:30:18Z)
• Provably End-to-end Label-Noise Learning without Anchor Points [118.97592870124937]
  We propose an end-to-end framework for solving label-noise learning without anchor points. Our proposed framework can identify the transition matrix if the clean class-posterior probabilities are sufficiently scattered.
  arXiv  Detail & Related papers  (2021-02-04T03:59:37Z)

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.