Computing the partition function of the Sherrington-Kirkpatrick model is hard on average

We establish the average-case hardness of the algorithmic problem of exact computation of the partition function associated with the Sherrington-Kirkpatrick model of spin glasses with Gaussian couplings and random external field. In particular, we establish that unless $P= \#P$, there does not exist a polynomial-time algorithm to exactly compute the partition function on average. This is done by showing that if there exists a polynomial time algorithm, which exactly computes the partition function for inverse polynomial fraction ($1/n^{O(1)}$) of all inputs, then there is a polynomial time algorithm, which exactly computes the partition function for all inputs, with high probability, yielding $P=\#P$. The computational model that we adopt is {\em finite-precision arithmetic}, where the algorithmic inputs are truncated first to a certain level $N$ of digital precision. The ingredients of our proof include the random and downward self-reducibility of the partition function with random external field; an argument of Cai et al. \cite{cai1999hardness} for establishing the average-case hardness of computing the permanent of a matrix; a list-decoding algorithm of Sudan \cite{sudan1996maximum}, for reconstructing polynomials intersecting a given list of numbers at sufficiently many points; and near-uniformity of the log-normal distribution, modulo a large prime $p$. To the best of our knowledge, our result is the first one establishing a provable hardness of a model arising in the field of spin glasses. Furthermore, we extend our result to the same problem under a different {\em real-valued} computational model, e.g. using a Blum-Shub-Smale machine \cite{blum1988theory} operating over real-valued inputs.Comment: 31 page

Random Max-CSPs Inherit Algorithmic Hardness from Spin Glasses

We study random constraint satisfaction problems (CSPs) in the unsatisfiable regime. We relate the structure of near-optimal solutions for any Max-CSP to that for an associated spin glass on the hypercube, using the Guerra-Toninelli interpolation from statistical physics. The noise stability polynomial of the CSP's predicate is, up to a constant, the mixture polynomial of the associated spin glass. We prove two main consequences: 1) We relate the maximum fraction of constraints that can be satisfied in a random Max-CSP to the ground state energy density of the corresponding spin glass. Since the latter value can be computed with the Parisi formula, we provide numerical values for some popular CSPs. 2) We prove that a Max-CSP possesses generalized versions of the overlap gap property if and only if the same holds for the corresponding spin glass. We transfer results from Huang et al. [arXiv:2110.07847, 2021] to obstruct algorithms with overlap concentration on a large class of Max-CSPs. This immediately includes local classical and local quantum algorithms.Comment: 41 pages, 1 tabl

Random Max-CSPs Inherit Algorithmic Hardness from Spin Glasses

We study random constraint satisfaction problems (CSPs) in the unsatisfiable regime. We relate the structure of near-optimal solutions for any Max-CSP to that for an associated spin glass on the hypercube, using the Guerra-Toninelli interpolation from statistical physics. The noise stability polynomial of the CSP's predicate is, up to a constant, the mixture polynomial of the associated spin glass. We prove two main consequences: 1) We relate the maximum fraction of constraints that can be satisfied in a random Max-CSP to the ground state energy density of the corresponding spin glass. Since the latter value can be computed with the Parisi formula, we provide numerical values for some popular CSPs. 2) We prove that a Max-CSP possesses generalized versions of the overlap gap property if and only if the same holds for the corresponding spin glass. We transfer results from Huang et al. [arXiv:2110.07847, 2021] to obstruct algorithms with overlap concentration on a large class of Max-CSPs. This immediately includes local classical and local quantum algorithms.Comment: 41 pages, 1 tabl

Non-ergodic phenomena in many-body quantum systems

The assumption of ergodicity is the cornerstone of conventional thermodynamics, connecting the equilibrium properties of macroscopic systems to the chaotic nature of the underlying microscopic dynamics, which eventuates in thermalization and the scrambling of information contained in any generic initial condition. The modern understanding of ergodicity in a quantum mechanical framework is encapsulated in the so-called eigenstate thermalization hypothesis, which asserts that thermalization of an isolated quantum system is a manifestation of the random-like character of individual eigenstates in the bulk of the spectrum of the system's Hamiltonian. In this work, we consider two major exceptions to the rule of generic thermalization in interacting many-body quantum systems: many-body localization, and quantum spin glasses. In the first part, we debate the possibility of localization in a system endowed with a non-Abelian symmetry. We show that, in line with proposed theoretical arguments, such a system is probably delocalized in the thermodynamic limit, but the ergodization length scale is anomalously large, explaining the non-ergodic behavior observed in previous experimental and numerical works. A crucial feature of this system is the quasi-tensor-network nature of its eigenstates, which is dictated by the presence of nontrivial symmetry multiplets. As a consequence, ergodicity may only be restored by extensively large cascades of resonating spins, explaining the system's resistance to delocalization. In the second part, we study the effects of non-ergodic behavior in glassy systems in relation to the possibility of speeding up classical algorithms via quantum resources, namely tunneling across tall free energy barriers. First, we define a pseudo-tunneling event in classical diffusion Monte Carlo (DMC) and characterize the corresponding tunneling rate. Our findings suggest that DMC is very efficient at tunneling in stoquastic problems even in the presence of frustrated couplings, asymptotically outperforming incoherent quantum tunneling. We also analyze in detail the impact of importance sampling, finding that it does not alter the scaling. Next, we study the so-called population transfer (PT) algorithm applied to the problem of energy matching in combinatorial problems. After summarizing some known results on a simpler model, we take the quantum random energy model as a testbed for a thorough, model-agnostic numerical characterization of the algorithm, including parameter setting and quality assessment. From the accessible system sizes, we observe no meaningful asymptotic speedup, but argue in favor of a better performance in more realistic energy landscapes

Computational complexity of the landscape I

We study the computational complexity of the physical problem of finding vacua of string theory which agree with data, such as the cosmological constant, and show that such problems are typically NP hard. In particular, we prove that in the Bousso-Polchinski model, the problem is NP complete. We discuss the issues this raises and the possibility that, even if we were to find compelling evidence that some vacuum of string theory describes our universe, we might never be able to find that vacuum explicitly. In a companion paper, we apply this point of view to the question of how early cosmology might select a vacuum.Comment: JHEP3 Latex, 53 pp, 2 .eps figure

Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems

This article reviews recent developments in the theoretical understanding and the numerical implementation of variational renormalization group methods using matrix product states and projected entangled pair states.Comment: Review from 200

Unconventional computing platforms and nature-inspired methods for solving hard optimisation problems

The search for novel hardware beyond the traditional von Neumann architecture has given rise to a modern area of unconventional computing requiring the efforts of mathematicians, physicists and engineers. Many analogue physical systems, including networks of nonlinear oscillators, lasers, condensates, and superconducting qubits, are proposed and realised to address challenging computational problems from various areas of social and physical sciences and technology. Understanding the underlying physical process by which the system finds the solutions to such problems often leads to new optimisation algorithms. This thesis focuses on studying gain-dissipative systems and nature-inspired algorithms that form a hybrid architecture that may soon rival classical hardware. Chapter 1 lays the necessary foundation and explains various interdisciplinary terms that are used throughout the dissertation. In particular, connections between the optimisation problems and spin Hamiltonians are established, their computational complexity classes are explained, and the most prominent physical platforms for spin Hamiltonian implementation are reviewed. Chapter 2 demonstrates a large variety of behaviours encapsulated in networks of polariton condensates, which are a vivid example of a gain-dissipative system we use throughout the thesis. We explain how the variations of experimentally tunable parameters allow the networks of polariton condensates to represent different oscillator models. We derive analytic expressions for the interactions between two spatially separated polariton condensates and show various synchronisation regimes for periodic chains of condensates. An odd number of condensates at the vertices of a regular polygon leads to a spontaneous formation of a giant multiply-quantised vortex at the centre of a polygon. Numerical simulations of all studied configurations of polariton condensates are performed with a mean-field approach with some theoretically proposed physical phenomena supported by the relevant experiments. Chapter 3 examines the potential of polariton graphs to find the low-energy minima of the spin Hamiltonians. By associating a spin with a condensate phase, the minima of the XY model are achieved for simple configurations of spatially-interacting polariton condensates. We argue that such implementation of gain-dissipative simulators limits their applicability to the classes of easily solvable problems since the parameters of a particular Hamiltonian depend on the node occupancies that are not known a priori. To overcome this difficulty, we propose to adjust pumping intensities and coupling strengths dynamically. We further theoretically suggest how the discrete Ising and $n$-state planar Potts models with or without external fields can be simulated using gain-dissipative platforms. The underlying operational principle originates from a combination of resonant and non-resonant pumping. Spatial anisotropy of pump and dissipation profiles enables an effective control of the sign and intensity of the coupling strength between any two neighbouring sites, which we demonstrate with a two dimensional square lattice of polariton condensates. For an accurate minimisation of discrete and continuous spin Hamiltonians, we propose a fully controllable polaritonic XY-Ising machine based on a network of geometrically isolated polariton condensates. In Chapter 4, we look at classical computing rivals and study nature-inspired methods for optimising spin Hamiltonians. Based on the operational principles of gain-dissipative machines, we develop a novel class of gain-dissipative algorithms for the optimisation of discrete and continuous problems and show its performance in comparison with traditional optimisation techniques. Besides looking at traditional heuristic methods for Ising minimisation, such as the Hopfield-Tank neural networks and parallel tempering, we consider a recent physics-inspired algorithm, namely chaotic amplitude control, and exact commercial solver, Gurobi. For a proper evaluation of physical simulators, we further discuss the importance of detecting easy instances of hard combinatorial optimisation problems. The Ising model for certain interaction matrices, that are commonly used for evaluating the performance of unconventional computing machines and assumed to be exponentially hard, is shown to be solvable in polynomial time including the Mobius ladder graphs and Mattis spin glasses. In Chapter 5 we discuss possible future applications of unconventional computing platforms including emulation of search algorithms such as PageRank, realisation of a proof-of-work protocol for blockchain technology, and reservoir computing