Quantum Algorithms for Ground-State Preparation and Green's Function Calculation

• URL: http://arxiv.org/abs/2112.05731v1
• Date: Fri, 10 Dec 2021 18:39:55 GMT
• Title: Quantum Algorithms for Ground-State Preparation and Green's Function Calculation
• Authors: Trevor Keen, Eugene Dumitrescu, Yan Wang
• Abstract summary: We present projective quantum algorithms for ground-state preparation and calculations of the many-body Green's functions in frequency domain. The algorithms are based on the linear combination of unitary (LCU) operations and essentially only use quantum resources.
• Score: 5.28670135448572
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We propose quantum algorithms for projective ground-state preparation and calculations of the many-body Green's functions directly in frequency domain. The algorithms are based on the linear combination of unitary (LCU) operations and essentially only use quantum resources. To prepare the ground state, we construct the operator ${\exp}(-\tau \hat{H}^2)$ using Hubbard-Stratonovich transformation by LCU and apply it on an easy-to-prepare initial state. Our projective state preparation procedure saturates the near-optimal scaling, $\mathcal{O}(\frac{1}{\gamma\Delta} \log \frac{1}{\gamma\eta})$, of other currently known algorithms, in terms of the spectral-gap lower bound $\Delta$, the additive error $\eta$ in the state vector, and the overlap lower bound $\gamma$ between the initial state and the exact ground state. It is straightforward to combine our algorithm with spectral-gap amplification technique to achieve quadratically improved scaling $\mathcal{O}(1/\sqrt{\Delta})$ for ground-state preparation of frustration-free Hamiltonians, which we demonstrate with numerical results of the $q$-deformed XXZ chain. To compute the Green's functions, including the single-particle and other response functions, we act on the prepared ground state with the retarded resolvent operator $R(\omega + i\Gamma; \hat{H})$ in the LCU form derived from the Fourier-Laplace integral transform (FIT). Our resolvent algorithm has $\mathcal{O}(\frac{1}{\Gamma^2} \log\frac{1}{\Gamma\epsilon})$ complexity scaling for the frequency resolution $\Gamma$ of the response functions and the targeted error $\epsilon$, while classical algorithms for FIT usually have polynomial scaling over the error $\epsilon$. To illustrate the complexity scaling of our algorithms, we provide numerical results for their application to the paradigmatic Fermi-Hubbard model on a one-dimensional lattice with different numbers of sites.

Related papers

• Multi-level quantum signal processing with applications to ground state preparation using fast-forwarded Hamiltonian evolution [0.8359711817610189]
  Preparation of the ground state of a Hamiltonian $H$ with a large spectral radius has applications in many areas. We develop a multi-level Quantum Signal Processing (QSP)-based algorithm that exploits the fast-forwarding feature.
  arXiv  Detail & Related papers  (2024-06-04T08:09:51Z)
• Low-depth Gaussian State Energy Estimation [0.0]
  Ground state energy estimation (GSEE) is an important subroutine in quantum chemistry and materials. We detail a new GSEE algorithm which uses a number of operations scaling as $O(logDelta)$. We adapt this algorithm to interpolate between the low-depth and full-depth regime by replacing $Delta$ with anything between $Delta$ and $epsilon$.
  arXiv  Detail & Related papers  (2023-09-28T18:29:14Z)
• Efficiently Learning One-Hidden-Layer ReLU Networks via Schur Polynomials [50.90125395570797]
  We study the problem of PAC learning a linear combination of $k$ ReLU activations under the standard Gaussian distribution on $mathbbRd$ with respect to the square loss. Our main result is an efficient algorithm for this learning task with sample and computational complexity $(dk/epsilon)O(k)$, whereepsilon>0$ is the target accuracy.
  arXiv  Detail & Related papers  (2023-07-24T14:37:22Z)
• Near-Optimal Bounds for Learning Gaussian Halfspaces with Random Classification Noise [50.64137465792738]
  We show that any efficient SQ algorithm for the problem requires sample complexity at least $Omega(d1/2/(maxp, epsilon)2)$. Our lower bound suggests that this quadratic dependence on $1/epsilon$ is inherent for efficient algorithms.
  arXiv  Detail & Related papers  (2023-07-13T18:59:28Z)
• An Oblivious Stochastic Composite Optimization Algorithm for Eigenvalue Optimization Problems [76.2042837251496]
  We introduce two oblivious mirror descent algorithms based on a complementary composite setting. Remarkably, both algorithms work without prior knowledge of the Lipschitz constant or smoothness of the objective function. We show how to extend our framework to scale and demonstrate the efficiency and robustness of our methods on large scale semidefinite programs.
  arXiv  Detail & Related papers  (2023-06-30T08:34:29Z)
• Quantum Simulation of the First-Quantized Pauli-Fierz Hamiltonian [0.5097809301149342]
  We show that a na"ive partitioning and low-order splitting formula can yield, through our divide and conquer formalism, superior scaling to qubitization for large $Lambda$. We also give new algorithmic and circuit level techniques for gate optimization including a new way of implementing a group of multi-controlled-X gates.
  arXiv  Detail & Related papers  (2023-06-19T23:20:30Z)
• ReSQueing Parallel and Private Stochastic Convex Optimization [59.53297063174519]
  We introduce a new tool for BFG convex optimization (SCO): a Reweighted Query (ReSQue) estimator for the gradient of a function convolved with a (Gaussian) probability density. We develop algorithms achieving state-of-the-art complexities for SCO in parallel and private settings.
  arXiv  Detail & Related papers  (2023-01-01T18:51:29Z)
• Learning many-body Hamiltonians with Heisenberg-limited scaling [3.460138063155115]
  We propose the first algorithm to achieve the Heisenberg limit for learning interacting $N$-qubit local Hamiltonian. After a total evolution time of $mathcalO(epsilon-1)$, the proposed algorithm can efficiently estimate any parameter in the $N$-qubit Hamiltonian to $epsilon$-error with high probability.
  arXiv  Detail & Related papers  (2022-10-06T16:30:51Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• How to simulate quantum measurement without computing marginals [3.222802562733787]
  We describe and analyze algorithms for classically computation measurement of an $n$-qubit quantum state $psi$ in the standard basis. Our algorithms reduce the sampling task to computing poly(n)$ amplitudes of $n$-qubit states.
  arXiv  Detail & Related papers  (2021-12-15T21:44:05Z)
• Spectral density estimation with the Gaussian Integral Transform [91.3755431537592]
  spectral density operator $hatrho(omega)=delta(omega-hatH)$ plays a central role in linear response theory. We describe a near optimal quantum algorithm providing an approximation to the spectral density.
  arXiv  Detail & Related papers  (2020-04-10T03:14:38Z)

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.