Computing solution space properties of combinatorial optimization problems via generic tensor networks

We introduce a unified framework to compute the solution space properties of a broad class of combinatorial optimization problems. These properties include finding one of the optimum solutions, counting the number of solutions of a given size, and enumeration and sampling of solutions of a given size. Using the independent set problem as an example, we show how all these solution space properties can be computed in the unified approach of generic tensor networks. We demonstrate the versatility of this computational tool by applying it to several examples, including computing the entropy constant for hardcore lattice gases, studying the overlap gap properties, and analyzing the performance of quantum and classical algorithms for finding maximum independent sets.Comment: Github repo: https://github.com/QuEraComputing/GenericTensorNetworks.j

Long-Time Dynamics in Quantum Spin Lattices:Ergodicity and Hydrodynamic Projections at All Frequencies and Wavelengths

Obtaining rigorous and general results about the non-equilibrium dynamics of extended many-body systems is a difficult task. In quantum lattice models with short-range interactions, the Lieb-Robinson bound tells us that the spatial extent of operators grows at most linearly in time. But what happens within this light-cone? We discuss rigorous results on ergodicity and the emergence of the hydrodynamic scale, which establish fundamental principles at the root of non-equilibrium physics. One key idea of the present work is that general structures of hydrodynamics at the Euler scale follow independently from the details of the microscopic dynamics, and in particular do not necessitate chaos; they are consequences of "extensivity". Another crucial observation is that these apply at arbitrary frequencies and wavelengths. That is, long-time oscillatory behaviours can be reproduced from a natural extension of standard hydrodynamic notions, thus enlarging the hydrodynamic paradigm beyond the zero-frequency / infinite-wavelength point that it traditionally addresses.Comment: v1: 6 pages, 10 pages supplemental material v2: 54 pages, largely extended, including the full proof of hydrodynamic projections in arbitrary dimensions, submitted to Ann. H. Poincar\'e, special issue dedicated to the memory of Krzysztof Gawedsk

Impact of conditional modelling for a universal autoregressive quantum state

We present a generalized framework toadapt universal quantum state approxima-tors, enabling them to satisfy rigorous nor-malization and autoregressive properties.We also introduce filters as analogues toconvolutional layers in neural networks toincorporate translationally symmetrizedcorrelations in arbitrary quantum states.By applying this framework to the Gaus-sian process state, we enforce autoregres-sive and/or filter properties, analyzingthe impact of the resulting inductive bi-ases on variational flexibility, symmetries,and conserved quantities. In doing sowe bring together different autoregressivestates under a unified framework for ma-chine learning-inspired ans ̈atze. Our re-sults provide insights into how the autore-gressive construction influences the abilityof a variational model to describe corre-lations in spin and fermionic lattice mod-els, as well as ab initio electronic structureproblems where the choice of representa-tion affects accuracy. We conclude that,while enabling efficient and direct sam-pling, thus avoiding autocorrelation andloss of ergodicity issues in Metropolis sam-pling, the autoregressive construction ma-terially constrains the expressivity of themodel in many systems

Explaining Low Entropy: An Examination of the Efficacy of Cosmological Explanations for the Second Law of Thermodynamics

Given a low-entropy past state, statistical mechanics purportedly explains the Second Law of Thermodynamics. Modern cosmology furnishes this account with a low-entropy early universe. However, what explains this past state? One approach simply postulates this initial condition as a contingent fact or law. Alternatively, one can seek a dynamical solution. This paper assesses the explanatory efficacy some popular proposals. In particular, Carroll's multiverse hypothesis is critiqued and shown to be subject to similar problems faced by Boltzmann