Tensor network non-zero testing

We show how fermionic statistics can be naturally incorporated in tensor networks on arbitrary graphs through the use of graded Hilbert spaces. This formalism allows the use of tensor network methods for fermionic lattice systems in a local way, avoiding the need of a Jordan-Wigner transformation or the explicit tracking of leg crossings by swap gates in 2D tensor networks. The graded Hilbert spaces can be readily integrated with other internal and lattice symmetries, and only require minor extensions to an existing tensor network software package. We review and benchmark the fermionic versions of common algorithms for matrix product states and projected entangled-pair states.

The realization of topological quantum phases of matter remains a key challenge to condensed matter physics and quantum information science. In this work, we demonstrate that progress in this direction can be made by combining concepts of tensor network theory with Majorana device technology. Considering the topological double semion string-net phase as an example, we exploit the fact that the representation of topological phases by tensor networks can be significantly simpler than their description by lattice Hamiltonians. The building blocks defining the tensor network are tailored to realization via simple units of capacitively coupled Majorana bound states. In the case under consideration, this yields a remarkably simple blueprint of a synthetic double semion string-net, and one may be optimistic that the required device technology will be available soon. Our results indicate that the implementation of tensor network structures via mesoscopic quantum devices opens up a powerful novel avenue toward the realization and quantum simulation of synthetic topological quantum matter.

Tensor network states and methods have erupted in recent years. Originally developed in the context of condensed matter physics and based on renormalization group ideas, tensor networks lived a revival thanks to quantum information theory and the understanding of entanglement in quantum many-body systems. Moreover, it has been not-so-long realized that tensor network states play a key role in other scientific disciplines, such as quantum gravity and artificial intelligence. In this context, here we provide an overview of basic concepts and key developments in the field. In particular, we briefly discuss the most important tensor network structures and algorithms, together with a sketch on advances related to global and gauge symmetries, fermions, topological order, classification of phases, entanglement Hamiltonians, AdS/CFT, artificial intelligence, the 2d Hubbard model, 2d quantum antiferromagnets, conformal field theory, quantum chemistry, disordered systems, and many-body localization.

We investigate how to count the number of solutions in the Boolean satisfiability (SAT) problem, a fundamental problem in theoretical computer science that has applications in various domains. We convert the problem to a spin-glass problem in statistical physics and approximately compute the entropy of the problem, which corresponds to the logarithm of the number of solutions. We propose a new method for the entropy computing problem based on a combination of the tensor network method and the message-passing algorithm. The significance of the proposed method is its ability to consider the effects of both long loops and short loops present in the factor graph of the SAT problem. We validate the efficacy of our approach using 3-SAT problems defined on the random graphs and structured graphs, and show that the proposed method gives more accurate results on the number of solutions than the standard belief propagation algorithms. We also discuss the applications of our method across a wide range of combinatorial optimization problems.

The intensive pursuit for quantum advantage in terms of computational complexity has further led to a modernized crucial question of when and how will quantum computers outperform classical computers. The next milestone is undoubtedly the realization of quantum acceleration in practical problems. Here we provide a clear evidence and arguments that the primary target is likely to be condensed matter physics. Our primary contributions are summarized as follows: 1) Proposal of systematic error/runtime analysis on state-of-the-art classical algorithm based on tensor networks; 2) Dedicated and high-resolution analysis on quantum resource performed at the level of executable logical instructions; 3) Clarification of quantum-classical crosspoint for ground-state simulation to be within runtime of hours using only a few hundreds of thousand physical qubits for 2d Heisenberg and 2d Fermi-Hubbard models, assuming that logical qubits are encoded via the surface code with the physical error rate of p = 10−3. To our knowledge, we argue that condensed matter problems offer the earliest platform for demonstration of practical quantum advantage that is order-of-magnitude more feasible than ever known candidates, in terms of both qubit counts and total runtime.

Anyons and matrix product operator algebras

The Affleck-Kennedy-Lieb-Tasaki (AKLT) states have significant applications in condensed matter physics and quantum computation. An adiabatic path that enables the deterministic preparation of a broad range of many-body states, including AKLT states on arbitrary lattice geometries, is proposed. The numerical results on 1D and 2D AKLT states demonstrate the high efficiency of this approach.

Quantum simulation promises to have wide applications in many fields where problems are hard to model with classical computers. Various quantum devices of different platforms have been built to tackle the problems in, say, quantum chemistry, condensed matter physics, and high-energy physics. Here, we report an experiment towards the simulation of quantum gravity by simulating the holographic entanglement entropy. On a six-qubit nuclear magnetic resonance quantum simulator, we demonstrate a key result of Anti-de Sitter/conformal field theory (AdS/CFT) correspondence—the Ryu-Takayanagi formula is demonstrated by measuring the relevant entanglement entropies on the perfect tensor state. The fidelity of our experimentally prepared the six-qubit state is 85.0% via full state tomography and reaches 93.7% if the signal-decay due to decoherence is taken into account. Our experiment serves as the basic module of simulating more complex tensor network states that exploring AdS/CFT correspondence. As the initial experimental attempt to study AdS/CFT via quantum information processing, our work opens up new avenues exploring quantum gravity phenomena on quantum simulators.

We present a unifying framework for understanding and developing SAT-based decision procedures for Satisfiability Modulo Theories (SMT). The framework is based on a reduction of the decision problem to propositional logic by means of a deductive system. The two commonly used techniques, eager encodings (a direct reduction to propositional logic) and lazy encodings (a family of techniques based on an interplay between a SAT solver and a decision procedure) are identified as special cases. This framework offers the first generic approach for eager encodings, and a simple generalization of various lazy techniques that are found in the literature.

We describe an incremental algorithm for computing interpolants for a pair ϕ A , ϕ B of formulas in propositional logic. In contrast with the common approaches, our method does not require a proof of unsatisfiability of ϕ A ∧ ϕ B , and can be realized using any SAT solver as a black box. We achieve this by combining model enumeration with the ability to easily generate interpolants in the special case that one of the formulas is a cube.

We state a general conjecture that T-duality trivialises a model for the bulk-boundary correspondence in the parametrised context. We give evidence that it is valid by proving it in a special interesting case, which is relevant both to String Theory and to the study of topological insulators with defects in Condensed Matter Physics.

We study several problems related to properties of nonnegative matrices that arise at the boundary between quantum and classical probabilistic computation. Our results are twofold. First, we identify a large class of quantum Hamiltonians describing systems of qubits for which the adiabatic evolution can be efficiently simulated on a classical probabilistic computer. These are stoquastic local Hamiltonians with a “frustration-free” ground-state. A Hamiltonian belongs to this class iff it can be represented as $H=\sum_{a}H_{a}$ where (1) every term $H_{a}$ acts nontrivially on a constant number of qubits, (2) every term $H_{a}$ has real nonpositive off-diagonal matrix elements in the standard basis, and (3) the ground-state of $H$ is a ground-state of every term $H_{a}$. Second, we generalize the Cook-Levin theorem proving NP-completeness of the satisfiability problem to the complexity class MA (Merlin-Arthur games)—a probabilistic analogue of NP. Specifically, we construct a quantum version of the $k$-SAT problem which we call “stoquastic $k$-SAT” such that stoquastic $k$-SAT is contained in MA for any constant $k$, and any promise problem in MA is Karp-reducible to stoquastic 6-SAT. This result provides the first nontrivial example of a MA-complete promise problem.

The ground state degeneracy of an SU(N)k topological phase with n quasiparticle excitations is a relevant quantity for quantum computation, condensed matter physics, and knot theory. It is an open question to find a closed formula for this degeneracy for any N > 2. Here we present the problem in an explicit combinatorial way and analyze the case N = 3. While not finding a complete closed-form solution, we obtain generating functions and solve some special cases.

Understanding the microscopic mechanism of the transition of glass remains one of the most challenging topics in Condensed Matter Physics. What controls the sharp slowing down of molecular motion upon approaching the glass transition temperature Tg, whether there is an underlying thermodynamic transition at some finite temperature below Tg, what the role of cooperativity and heterogeneity are, and many other questions continue to be topics of active discussions. This review focuses on the mechanisms that control the steepness of the temperature dependence of structural relaxation (fragility) in glass-forming liquids. We present a brief overview of the basic theoretical models and their experimental tests, analyzing their predictions for fragility and emphasizing the successes and failures of the models. Special attention is focused on the connection of fast dynamics on picosecond time scales to the behavior of structural relaxation on much longer time scales. A separate section discusses the specific case of polymeric glass-forming liquids, which usually have extremely high fragility. We emphasize the apparent difference between the glass transitions in polymers and small molecules. We also discuss the possible role of quantum effects in the glass transition of light molecules and highlight the recent discovery of the unusually low fragility of water. At the end, we formulate the major challenges and questions remaining in this field.

A problem of cooperative path-planning is addressed from the perspective of propositional satisfiability in this paper. Two new encodings of the problem as SAT are proposed and evaluated. Together with the existent solution optimization method which locally improves a sub-optimal solution of the problem through SAT solving, one of the new encodings constitute a state-of-the-art method for cooperative path-planning in highly occupied environments.