A Quantum Approximate Optimization Algorithm for Local Hamiltonian Problems

• URL: http://arxiv.org/abs/2412.09221v1
• Date: Thu, 12 Dec 2024 12:22:08 GMT
• Title: A Quantum Approximate Optimization Algorithm for Local Hamiltonian Problems
• Authors: Ishaan Kannan, Robbie King, Leo Zhou,
• Abstract summary: Local Hamiltonian Problems (LHPs) are important problems that are computationally-complete and physically relevant for many-body quantum systems. We propose and analyze a quantum approximation which we call the Hamiltonian Quantum Approximate Optimization Algorithm (HamQAOA) Our results indicate that the linear-depth HamQAOA can deterministically prepare exact ground states of 1-dimensional antiferromagnetic Heisenberg spin chains.
• Score: 0.0
• License:
• Abstract: Local Hamiltonian Problems (LHPs) are important problems that are computationally QMA-complete and physically relevant for many-body quantum systems. Quantum MaxCut (QMC), which equates to finding ground states of the quantum Heisenberg model, is the canonical LHP for which various algorithms have been proposed, including semidefinite programs and variational quantum algorithms. We propose and analyze a quantum approximation algorithm which we call the Hamiltonian Quantum Approximate Optimization Algorithm (HamQAOA), which builds on the well-known scheme for combinatorial optimization and is suitable for implementations on near-term hardware. We establish rigorous performance guarantees of the HamQAOA for QMC on high-girth regular graphs, and our result provides bounds on the ground energy density for quantum Heisenberg spin glasses in the infinite size limit that improve with depth. Furthermore, we develop heuristic strategies with which to efficiently obtain good HamQAOA parameters. Through numerical simulations, we show that the HamQAOA empirically outperforms prior algorithms on a wide variety of QMC instances. In particular, our results indicate that the linear-depth HamQAOA can deterministically prepare exact ground states of 1-dimensional antiferromagnetic Heisenberg spin chains described by the Bethe ansatz, in contrast to the exponential depths required in previous protocols for preparing Bethe states.

Related papers

• Bias-field digitized counterdiabatic quantum optimization [39.58317527488534]
  We call this protocol bias-field digitizeddiabatic quantum optimization (BF-DCQO) Our purely quantum approach eliminates the dependency on classical variational quantum algorithms. It achieves scaling improvements in ground state success probabilities, increasing by up to two orders of magnitude.
  arXiv  Detail & Related papers  (2024-05-22T18:11:42Z)
• Quantum Subroutine for Variance Estimation: Algorithmic Design and Applications [80.04533958880862]
  Quantum computing sets the foundation for new ways of designing algorithms. New challenges arise concerning which field quantum speedup can be achieved. Looking for the design of quantum subroutines that are more efficient than their classical counterpart poses solid pillars to new powerful quantum algorithms.
  arXiv  Detail & Related papers  (2024-02-26T09:32:07Z)
• A Review on Quantum Approximate Optimization Algorithm and its Variants [47.89542334125886]
  The Quantum Approximate Optimization Algorithm (QAOA) is a highly promising variational quantum algorithm that aims to solve intractable optimization problems. This comprehensive review offers an overview of the current state of QAOA, encompassing its performance analysis in diverse scenarios. We conduct a comparative study of selected QAOA extensions and variants, while exploring future prospects and directions for the algorithm.
  arXiv  Detail & Related papers  (2023-06-15T15:28:12Z)
• A Universal Quantum Algorithm for Weighted Maximum Cut and Ising Problems [0.0]
  We propose a hybrid quantum-classical algorithm to compute approximate solutions of binary problems. We employ a shallow-depth quantum circuit to implement a unitary and Hermitian operator that block-encodes the weighted maximum cut or the Ising Hamiltonian. Measuring the expectation of this operator on a variational quantum state yields the variational energy of the quantum system.
  arXiv  Detail & Related papers  (2023-06-10T23:28:13Z)
• End-to-end resource analysis for quantum interior point methods and portfolio optimization [63.4863637315163]
  We provide a complete quantum circuit-level description of the algorithm from problem input to problem output. We report the number of logical qubits and the quantity/depth of non-Clifford T-gates needed to run the algorithm.
  arXiv  Detail & Related papers  (2022-11-22T18:54:48Z)
• Improved iterative quantum algorithm for ground-state preparation [4.921552273745794]
  We propose an improved iterative quantum algorithm to prepare the ground state of a Hamiltonian system. Our approach has advantages including the higher success probability at each iteration, the measurement precision-independent sampling complexity, the lower gate complexity, and only quantum resources are required when the ancillary state is well prepared.
  arXiv  Detail & Related papers  (2022-10-16T05:57:43Z)
• Efficient Classical Computation of Quantum Mean Values for Shallow QAOA Circuits [15.279642278652654]
  We present a novel graph decomposition based classical algorithm that scales linearly with the number of qubits for the shallow QAOA circuits. Our results are not only important for the exploration of quantum advantages with QAOA, but also useful for the benchmarking of NISQ processors.
  arXiv  Detail & Related papers  (2021-12-21T12:41:31Z)
• Quantum Approximate Optimization Algorithm Based Maximum Likelihood Detection [80.28858481461418]
  Recent advances in quantum technologies pave the way for noisy intermediate-scale quantum (NISQ) devices. Recent advances in quantum technologies pave the way for noisy intermediate-scale quantum (NISQ) devices.
  arXiv  Detail & Related papers  (2021-07-11T10:56:24Z)
• Fixed Depth Hamiltonian Simulation via Cartan Decomposition [59.20417091220753]
  We present a constructive algorithm for generating quantum circuits with time-independent depth. We highlight our algorithm for special classes of models, including Anderson localization in one dimensional transverse field XY model. In addition to providing exact circuits for a broad set of spin and fermionic models, our algorithm provides broad analytic and numerical insight into optimal Hamiltonian simulations.
  arXiv  Detail & Related papers  (2021-04-01T19:06:00Z)
• Iterative Quantum Assisted Eigensolver [0.0]
  We provide a hybrid quantum-classical algorithm for approximating the ground state of a Hamiltonian. Our algorithm builds on the powerful Krylov subspace method in a way that is suitable for current quantum computers.
  arXiv  Detail & Related papers  (2020-10-12T12:25:16Z)
• K-spin Hamiltonian for quantum-resolvable Markov decision processes [0.0]
  We derive a pseudo-Boolean cost function that is equivalent to a K-spin Hamiltonian representation of the discrete, finite, discounted Markov decision process. This K-spin Hamiltonian furnishes a starting point from which to solve for an optimal policy using quantum algorithms.
  arXiv  Detail & Related papers  (2020-04-13T16:15:25Z)

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.