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Monte Carlo studies of many quantum systems face exponentially severe signal-to-noise problems. We show that noise arising from complex phase fluctuations of observables can be reduced without introducing bias using path integral contour deformation techniques. A numerical study of contour deformations for correlation functions in Abelian gauge theory and complex scalar field theory demonstrates that variance can be reduced by orders of magnitude without modifying Monte Carlo sampling.
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Path integral contour deformations have been shown to mitigate sign and signal-to-noise problems associated with phase fluctuations in lattice field theories. We define a family of contour deformations applicable to $SU(N)$ lattice gauge theory that can reduce sign and signal-to-noise problems associated with complex actions and complex observables. For observables, these contours can be used to define deformed observables with identical expectation value but different variance. As a proof-of-principle, we apply machine learning techniques to optimize the deformed observables associated with Wilson loops in two dimensional $SU(2)$ and $SU(3)$ gauge theory. We study loops consisting of up to 64 plaquettes and achieve variance reduction of up to 4 orders of magnitude.
The Wilson action for Euclidean lattice gauge theory defines a positive-definite transfer matrix that corresponds to a unitary lattice gauge theory time-evolution operator if analytically continued to real time. Hoshina, Fujii, and Kikukawa (HFK) recently pointed out that applying the Wilson action discretization to continuum real-time gauge theory does not lead to this, or any other, unitary theory and proposed an alternate real-time lattice gauge theory action that does result in a unitary real-time transfer matrix. The character expansion defining the HFK action is divergent, and in this work we apply a path integral contour deformation to obtain a convergent representation for U(1) HFK path integrals suitable for numerical Monte Carlo calculations. We also introduce a class of real-time lattice gauge theory actions based on analytic continuation of the Euclidean heat-kernel action. Similar divergent sums are involved in defining these actions, but for one action in this class this divergence takes a particularly simple form, allowing construction of a path integral contour deformation that provides absolutely convergent representations for U(1) and SU(N) real-time lattice gauge theory path integrals. We perform proof-of-principle Monte Carlo calculations of real-time U(1) and SU(3) lattice gauge theory and verify that exact results for unitary time evolution of static quark-antiquark pairs in (1 + 1)D are reproduced.
In this work, we formulate a path-integral optimization for two dimensional conformal field theories perturbed by relevant operators. We present several evidences how this optimization mechanism works, based on calculations in free field theories as well as general arguments of RG flows in field theories. Our optimization is performed by minimizing the path-integral complexity functional that depends on the metric and also on the relevant couplings. Then, we compute the optimal metric perturbatively and find that it agrees with the time slice of the hyperbolic metric perturbed by a scalar field in the AdS/CFT correspondence. Last but not the least, we estimate contributions to complexity from relevant perturbations.
We propose an optimization procedure for Euclidean path-integrals that evaluate CFT wave functionals in arbitrary dimensions. The optimization is performed by minimizing certain functional, which can be interpreted as a measure of computational complexity, with respect to background metrics for the path-integrals. In two dimensional CFTs, this functional is given by the Liouville action. We also formulate the optimization for higher dimensional CFTs and, in various examples, find that the optimized hyperbolic metrics coincide with the time slices of expected gravity duals. Moreover, if we optimize a reduced density matrix, the geometry becomes two copies of the entanglement wedge and reproduces the holographic entanglement entropy. Our approach resembles a continuous tensor network renormalization and provides a concrete realization of the proposed interpretation of AdS/CFT as tensor networks. The present paper is an extended version of our earlier report arXiv:1703.00456 and includes many new results such as evaluations of complexity functionals, energy stress tensor, higher dimensional extensions and time evolutions of thermofield double states.
We present a tensor network representation of the path integral for the one-component real scalar field theory in 1+1 dimensional Minkowski space-time. It is numerically verified by comparing with the exact result in the non-interacting case.
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