Complex Paths Around The Sign Problem

Abstract

A. Single Thimble Methods Contraction Algorithm HMC on thimbles Langevin on thimbles Case study: bosonic gases B. Generalized thimble method Case study: 0+1D Thirring model Case study: 1+1D Thirring model C. Trapping and tempered algorithms D. Algorithms for the Jacobian Case study: real time field theory E. Gauge theories Case study: Heavy-Dense QCD Case study: 2D QED IV. Other manifolds and the algorithms that can find them A. Well beyond thimbles B. Learnifolds Case study: 1+1D Thirring model revisited C. Path optimization Case study: 2+1D Thirring model V. Conclusion and prospects

Complex, Lorentzian, and Euclidean simplicial quantum gravity: numerical methods and physical prospects

“…In this work we propose to generalize simplicial quantum gravity to the complex domain. This allows us to apply the techniques of complex contour deformation developed in recent years to alleviate the sign problem [19,20]. By a higher dimensional version of Cauchy's integration theorem, a path integral with a real integration contour can equally be evaluated along a complex contour if the two contours are related across a region where the integrand is holomorphic.…”

“…The sign problem could be milder on the deformed contour. As reviewed in [20], this idea has been successfully applied to various lattice field theories of matter. It has also been applied to analyze gravitational propagators for spin-foam models in the large spin limit [21].…”

“…Here we show that the complex contour deformation method also works for Lorentzian simplicial quantum gravity. Monte Carlo simulations are performed to compute the expectation value of spacetime lengths in 1 + 1D using the holomorphic gradient flow algorithm (also called the generalized thimble algorithm) [22,23,20]. It is found that the sign fluctuations are largely suppressed on suitable complex contours.…”

Normalizing flows for the real-time sign problem

“…A similar issue occurs for many fermionic models, including QCD at non-zero baryon density. While there are several proposal for alleviating sign problems proposed, one long-standing method which successfully tamed sign problems for some models of our interest is the so-called manifold deformation method [1]. The idea of the method is simple: we deform the contour of integration is the path integral, R 𝑁 to the complex plane C 𝑁 of the field variables 𝜙, aiming for a larger average sign, and thus milder sign problem.…”

Fermion sign bounds theory in quantum Monte Carlo simulation

“…In fact, there are cases the sign problem can be cured by choosing new bases where certain symmetries or mathematical operations, such as time reversal or anti-unitary symmetry [7][8][9][10], meron cluster [11], Fermi bags [12], split-orthogonal group [13], Majorana time-reversal symmetry [14][15][16], Majorana positivity [17], semigroup approach [18] and pesudo-unitary group [19], can be used to prove the simulation is sign problem free. In some other cases, sign problem can be alleviated by an optimal choice of basis [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34], or through adiabatic method [35].…”