Rapid mixing of path integral Monte Carlo for 1D stoquastic Hamiltonians
Path integral quantum Monte Carlo (PIMC) is a method for estimating thermal equilibrium properties of stoquastic quantum spin systems by sampling from a classical Gibbs distribution using Markov chain Monte Carlo. The PIMC method has been widely used to study the physics of materials and for simulat...
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| Format: | Article |
| Language: | English |
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Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften
2022
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| Online Access: | https://hdl.handle.net/1721.1/141897 |
| _version_ | 1826215087211282432 |
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| author | Crosson, Elizabeth Harrow, Aram W |
| author2 | Massachusetts Institute of Technology. Center for Theoretical Physics |
| author_facet | Massachusetts Institute of Technology. Center for Theoretical Physics Crosson, Elizabeth Harrow, Aram W |
| author_sort | Crosson, Elizabeth |
| collection | MIT |
| description | Path integral quantum Monte Carlo (PIMC) is a method for estimating thermal equilibrium properties of stoquastic quantum spin systems by sampling from a classical Gibbs distribution using Markov chain Monte Carlo. The PIMC method has been widely used to study the physics of materials and for simulated quantum annealing, but these successful applications are rarely accompanied by formal proofs that the Markov chains underlying PIMC rapidly converge to the desired equilibrium distribution. In this work we analyze the mixing time of PIMC for 1D stoquastic Hamiltonians, including disordered transverse Ising models (TIM) with long-range algebraically decaying interactions as well as disordered XY spin chains with nearest-neighbor interactions. By bounding the convergence time to the equilibrium distribution we rigorously justify the use of PIMC to approximate partition functions and expectations of observables for these models at inverse temperatures that scale at most logarithmically with the number of qubits. The mixing time analysis is based on the canonical paths method applied to the single-site Metropolis Markov chain for the Gibbs distribution of 2D classical spin models with couplings related to the interactions in the quantum Hamiltonian. Since the system has strongly nonisotropic couplings that grow with system size, it does not fall into the known cases where 2D classical spin models are known to mix rapidly. |
| first_indexed | 2024-09-23T16:16:20Z |
| format | Article |
| id | mit-1721.1/141897 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T16:16:20Z |
| publishDate | 2022 |
| publisher | Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften |
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| spelling | mit-1721.1/1418972023-01-10T15:17:13Z Rapid mixing of path integral Monte Carlo for 1D stoquastic Hamiltonians Crosson, Elizabeth Harrow, Aram W Massachusetts Institute of Technology. Center for Theoretical Physics Path integral quantum Monte Carlo (PIMC) is a method for estimating thermal equilibrium properties of stoquastic quantum spin systems by sampling from a classical Gibbs distribution using Markov chain Monte Carlo. The PIMC method has been widely used to study the physics of materials and for simulated quantum annealing, but these successful applications are rarely accompanied by formal proofs that the Markov chains underlying PIMC rapidly converge to the desired equilibrium distribution. In this work we analyze the mixing time of PIMC for 1D stoquastic Hamiltonians, including disordered transverse Ising models (TIM) with long-range algebraically decaying interactions as well as disordered XY spin chains with nearest-neighbor interactions. By bounding the convergence time to the equilibrium distribution we rigorously justify the use of PIMC to approximate partition functions and expectations of observables for these models at inverse temperatures that scale at most logarithmically with the number of qubits. The mixing time analysis is based on the canonical paths method applied to the single-site Metropolis Markov chain for the Gibbs distribution of 2D classical spin models with couplings related to the interactions in the quantum Hamiltonian. Since the system has strongly nonisotropic couplings that grow with system size, it does not fall into the known cases where 2D classical spin models are known to mix rapidly. 2022-04-13T18:29:30Z 2022-04-13T18:29:30Z 2021 2022-04-13T18:16:36Z Article http://purl.org/eprint/type/JournalArticle 2521-327X https://hdl.handle.net/1721.1/141897 Crosson, Elizabeth and Harrow, Aram W. 2021. "Rapid mixing of path integral Monte Carlo for 1D stoquastic Hamiltonians." Quantum, 5. en 10.22331/Q-2021-02-11-395 Quantum Creative Commons Attribution 4.0 International license https://creativecommons.org/licenses/by/4.0/ application/pdf Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften Quantum |
| spellingShingle | Crosson, Elizabeth Harrow, Aram W Rapid mixing of path integral Monte Carlo for 1D stoquastic Hamiltonians |
| title | Rapid mixing of path integral Monte Carlo for 1D stoquastic Hamiltonians |
| title_full | Rapid mixing of path integral Monte Carlo for 1D stoquastic Hamiltonians |
| title_fullStr | Rapid mixing of path integral Monte Carlo for 1D stoquastic Hamiltonians |
| title_full_unstemmed | Rapid mixing of path integral Monte Carlo for 1D stoquastic Hamiltonians |
| title_short | Rapid mixing of path integral Monte Carlo for 1D stoquastic Hamiltonians |
| title_sort | rapid mixing of path integral monte carlo for 1d stoquastic hamiltonians |
| url | https://hdl.handle.net/1721.1/141897 |
| work_keys_str_mv | AT crossonelizabeth rapidmixingofpathintegralmontecarlofor1dstoquastichamiltonians AT harrowaramw rapidmixingofpathintegralmontecarlofor1dstoquastichamiltonians |