Classical Algorithms for Constant Approximation of the Ground State Energy of Local Hamiltonians

• URL: http://arxiv.org/abs/2410.21833v1
• Date: Tue, 29 Oct 2024 07:56:38 GMT
• Title: Classical Algorithms for Constant Approximation of the Ground State Energy of Local Hamiltonians
• Authors: François Le Gall,
• Abstract summary: We construct classical algorithms computing an approximation of the ground state energy of an arbitrary $k$-local Hamiltonian acting on $n$ qubits. We show that a constant approximation of the ground state energy can be computed classically in $mathrmpolyleft (1/chi,nright)$ time and $mathrmpoly(n)$ space.
• Score: 0.39886149789339326
• License:
• Abstract: We construct classical algorithms computing an approximation of the ground state energy of an arbitrary $k$-local Hamiltonian acting on $n$ qubits. We first consider the setting where a good ``guiding state'' is available, which is the main setting where quantum algorithms are expected to achieve an exponential speedup over classical methods. We show that a constant approximation of the ground state energy can be computed classically in $\mathrm{poly}\left(1/\chi,n\right)$ time and $\mathrm{poly}(n)$ space, where $\chi$ denotes the overlap between the guiding state and the ground state (as in prior works in dequantization, we assume sample-and-query access to the guiding state). This gives a significant improvement over the recent classical algorithm by Gharibian and Le Gall (SICOMP 2023), and matches (up a to polynomial overhead) both the time and space complexities of quantum algorithms for constant approximation of the ground state energy. We also obtain classical algorithms for higher-precision approximation. For the setting where no guided state is given (i.e., the standard version of the local Hamiltonian problem), we obtain a classical algorithm computing a constant approximation of the ground state energy in $2^{O(n)}$ time and $\mathrm{poly}(n)$ space. To our knowledge, before this work it was unknown how to classically achieve these bounds simultaneously, even for constant approximation. We also discuss complexity-theoretic aspects of our results and their implications for the quantum PCP conjecture.

Related papers

• Sum-of-Squares inspired Quantum Metaheuristic for Polynomial Optimization with the Hadamard Test and Approximate Amplitude Constraints [76.53316706600717]
  Recently proposed quantum algorithm arXiv:2206.14999 is based on semidefinite programming (SDP) We generalize the SDP-inspired quantum algorithm to sum-of-squares. Our results show that our algorithm is suitable for large problems and approximate the best known classicals.
  arXiv  Detail & Related papers  (2024-08-14T19:04:13Z)
• Beating Grover search for low-energy estimation and state preparation [0.23034630097498876]
  Estimating ground state energies of many-body Hamiltonians is a central task in many areas of quantum physics. In this work, we give quantum algorithms which, given any $k$-body Hamiltonian $H$, compute an estimate for the ground state energy.
  arXiv  Detail & Related papers  (2024-07-03T12:47:06Z)
• Predicting Ground State Properties: Constant Sample Complexity and Deep Learning Algorithms [48.869199703062606]
  A fundamental problem in quantum many-body physics is that of finding ground states of local Hamiltonians. We introduce two approaches that achieve a constant sample complexity, independent of system size $n$, for learning ground state properties.
  arXiv  Detail & Related papers  (2024-05-28T18:00:32Z)
• A polynomial-time dissipation-based quantum algorithm for solving the ground states of a class of classically hard Hamiltonians [4.500918096201963]
  We give a quantum algorithm for solving the ground states of a class of Hamiltonians. The mechanism of the exponential speedup that appeared in our algorithm comes from dissipation in open quantum systems.
  arXiv  Detail & Related papers  (2024-01-25T05:01:02Z)
• When can you trust feature selection? -- I: A condition-based analysis of LASSO and generalised hardness of approximation [49.1574468325115]
  We show how no (randomised) algorithm can determine the correct support sets (with probability $> 1/2$) of minimisers of LASSO when reading approximate input. For ill-posed inputs, the algorithm runs forever, hence, it will never produce a wrong answer. For any algorithm defined on an open set containing a point with infinite condition number, there is an input for which the algorithm will either run forever or produce a wrong answer.
  arXiv  Detail & Related papers  (2023-12-18T18:29:01Z)
• On the approximability of random-hypergraph MAX-3-XORSAT problems with quantum algorithms [0.0]
  A feature of constraint satisfaction problems in NP is approximation hardness, where in the worst case, finding sufficient-quality approximate solutions is exponentially hard. For algorithms based on Hamiltonian time evolution, we explore this question through the prototypically hard MAX-3-XORSAT problem class. We show that, for random hypergraphs in the approximation-hard regime, if we define the energy to be $E = N_mathrmunsat-N_mathrmsat$, spectrally filtered quantum optimization will return states with $E leq q_m.
  arXiv  Detail & Related papers  (2023-12-11T04:15:55Z)
• Sparse random Hamiltonians are quantumly easy [105.6788971265845]
  A candidate application for quantum computers is to simulate the low-temperature properties of quantum systems. This paper shows that, for most random Hamiltonians, the maximally mixed state is a sufficiently good trial state. Phase estimation efficiently prepares states with energy arbitrarily close to the ground energy.
  arXiv  Detail & Related papers  (2023-02-07T10:57:36Z)
• Entanglement and coherence in Bernstein-Vazirani algorithm [58.720142291102135]
  Bernstein-Vazirani algorithm allows one to determine a bit string encoded into an oracle. We analyze in detail the quantum resources in the Bernstein-Vazirani algorithm. We show that in the absence of entanglement, the performance of the algorithm is directly related to the amount of quantum coherence in the initial state.
  arXiv  Detail & Related papers  (2022-05-26T20:32:36Z)
• Beating Random Assignment for Approximating Quantum 2-Local Hamiltonian Problems [3.553493344868414]
  k-Local Hamiltonian problem is a generalization of classical constraint satisfaction problems (k-CSP) For strictly quadratic instances, which are maximally entangled instances, we provide a classical 0.764-time 0.764-approximation. We conjecture these are the hardest instances to approximate. Our work employs recently developed techniques for analyzing classical approximations of CSPs and is intended to be accessible to both quantum information scientists and classical computer scientists.
  arXiv  Detail & Related papers  (2020-12-22T20:41:34Z)
• Quantum Assisted Eigensolver [0.0]
  We propose a hybrid quantum-classical algorithm for approxing the ground state and ground state energy of a Hamiltonian. The output from the quantum part of the algorithm is utilized as input for the classical computer.
  arXiv  Detail & Related papers  (2020-09-23T08:33:18Z)
• Quantum Gram-Schmidt Processes and Their Application to Efficient State Read-out for Quantum Algorithms [87.04438831673063]
  We present an efficient read-out protocol that yields the classical vector form of the generated state. Our protocol suits the case that the output state lies in the row space of the input matrix. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure.
  arXiv  Detail & Related papers  (2020-04-14T11:05:26Z)

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.