Quantum algorithm to calculate path integrals in quantum multi-body problems
Path integral Monte Carlo (PIMC) is aquantum Monte Carlo method used to solvequantum statistical mechanics problems numerically within thepath integral formulation. The application of Monte Carlo methods to path integral simulations of condensed matter systems was first pursued in a key paper by John A. Barker.[1][2]
The method is typically (but not necessarily) applied under the assumption that symmetry or antisymmetry under exchange can be neglected, i.e., identical particles are assumed to be quantum Boltzmann particles, as opposed tofermion andboson particles. The method is often applied to calculate thermodynamic properties[3] such as theinternal energy,[4] heat capacity,[5] orfree energy.[6][7] As with allMonte Carlo method based approaches, a large number of points must be calculated.
In principle, as more path descriptors are used (these can be "replicas", "beads," or "Fourier coefficients," depending on what strategy is used to represent the paths),[8] the more quantum (and the less classical) the result is. However, for some properties the correction may cause model predictions to initially become less accurate than neglecting them if a small number of path descriptors are included. At some point the number of descriptors is sufficiently large and the corrected model begins to converge smoothly to the correct quantum answer.[5] Because it is a statistical sampling method, PIMC can takeanharmonicity fully into account, and because it is quantum, it takes into account important quantum effects such astunneling andzero-point energy (while neglecting theexchange interaction in some cases).[6]
The basic framework was originally formulated within the canonical ensemble,[9] but has since been extended to include thegrand canonical ensemble[10] and themicrocanonical ensemble.[11] Its use has been extended to fermion systems[12] as well as systems of bosons.[13]
An early application was to the study of liquid helium.[14] Numerous applications have been made to other systems, including liquid water[15] and the hydrated electron.[16] The algorithms and formalism have also been mapped onto non-quantum mechanical problems in the field offinancial modeling, includingoption pricing.[17]
^Barker, J. A. (1979). "A quantum-statistical Monte Carlo method; path integrals with boundary conditions".The Journal of Chemical Physics.70 (6):2914–2918.Bibcode:1979JChPh..70.2914B.doi:10.1063/1.437829.
^abGlaesemann, Kurt R.; Fried, Laurence E. (2002). "Improved heat capacity estimator for path integral simulations".The Journal of Chemical Physics.117 (7):3020–3026.Bibcode:2002JChPh.117.3020G.doi:10.1063/1.1493184.
^Feynman, Richard P.; Hibbs, Albert R. (1965).Quantum Mechanics and Path Integrals. New York: McGraw-Hill.
^Wang, Q.; Johnson, J. K.; Broughton, J. Q. (1997). "Path integral grand canonical Monte Carlo".The Journal of Chemical Physics.107 (13):5108–5117.Bibcode:1997JChPh.107.5108W.doi:10.1063/1.474874.
