Abstract Random tensor models have applications in a variety of fields, such as quantum gravity, quantum information theory, mathematics of modern technologies, etc., and studying their statistical properties, e.g. tensor eigenvalue/vector distributions, is interesting and useful. Recently some tensor eigenvalue/vector distributions have been computed by expressing them as partition functions of 0D quantum field theories. In this paper, using this method, we compute three cases of complex eigenvalue/vector distributions of symmetric order-three random tensors, where the three cases can be characterized by the Lie-group invariances, $O(N,\mathbb {R})$ , $O(N,\mathbb {C})$ , and $U(N,\mathbb {C})$ , respectively. Exact closed-form expressions of the distributions are obtained by computing partition functions of four-fermi theories, where the last case is of the “signed” distribution, which counts the distribution with a sign factor coming from a Hessian matrix. As an application, we compute the injective norm of the complex symmetric order-three random tensor in the large-N limit by computing the edge of the last signed distribution, obtaining agreement with an earlier numerical result in the literature.Abstract Random tensor models have applications in a variety of fields, such as quantum gravity, quantum information theory, mathematics of modern technologies, etc., and studying their statistical properties, e.g. tensor eigenvalue/vector distributions, is interesting and useful. Recently some tensor eigenvalue/vector distributions have been computed by expressing them as partition functions of 0D quantum field theories. In this paper, using this method, we compute three cases of complex eigenvalue/vector distributions of symmetric order-three random tensors, where the three cases can be characterized by the Lie-group invariances, $O(N,\mathbb {R})$, $O(N,\mathbb {C})$, and $U(N,\mathbb {C})$, respectively. Exact closed-form expressions of the distributions are obtained by computing partition functions of four-fermi theories, where the last case is of the “signed” distribution, which counts the distribution with a sign factor coming from a Hessian matrix. As an application, we compute the injective norm of the complex symmetric order-three random tensor in the large-N limit by computing the edge of the last signed distribution, obtaining agreement with an earlier numerical result in the literature.

arXiv:2408.01030Progress of Theoretical and Experimental PhysicsAug 31, 20240 citations

A Parallel Tensor Network Contraction Algorithm and Its Applications in Quantum Computation

Fang Zhang

Tensors are a natural generalization of matrices, and tensor networks are a natural generalization of matrix products. Despite the simple definition of tensor networks, they are versatile enough to represent many different kinds of "products" that arise in various theoretical and practical problems. In particular, the powerful computational model of quantum computation can be defined almost entirely in terms of matrix products and tensor products, both of which are special cases of tensor networks. As such, (classical) algorithms for evaluating tensor networks have profound importance in the study of quantum computation. In this thesis, we design and implement a parallel algorithm for tensor network contraction. In addition to finding efficient contraction orders for a tensor network, we also dynamically slice it into multiple sub-tasks with lower space and time costs, in order to evaluate the tensor network in parallel. We refer to such an evaluation strategy as a contraction scheme for the tensor network. In addition, we introduce a local optimization procedure that improves the efficiency of the contraction schemes we find. We also investigate the applications of our parallel tensor network contraction algorithm in quantum computation. The most ready application is the simulation of random quantum supremacy circuits, where we benchmark our algorithm to demonstrate its advantage over other similar tensor network based simulators. Other applications we found include evaluating the energy function of a Quantum Approximate Optimization Algorithm (QAOA), and simulating surface codes under a realistic error model with crosstalk.Tensors are a natural generalization of matrices, and tensor networks are a natural generalization of matrix products. Despite the simple definition of tensor networks, they are versatile enough to represent many different kinds of "products" that arise in various theoretical and practical problems. In particular, the powerful computational model of quantum computation can be defined almost entirely in terms of matrix products and tensor products, both of which are special cases of tensor networks. As such, (classical) algorithms for evaluating tensor networks have profound importance in the study of quantum computation. In this thesis, we design and implement a parallel algorithm for tensor network contraction. In addition to finding efficient contraction orders for a tensor network, we also dynamically slice it into multiple sub-tasks with lower space and time costs, in order to evaluate the tensor network in parallel. We refer to such an evaluation strategy as a contraction scheme for the tensor network. In addition, we introduce a local optimization procedure that improves the efficiency of the contraction schemes we find. We also investigate the applications of our parallel tensor network contraction algorithm in quantum computation. The most ready application is the simulation of random quantum supremacy circuits, where we benchmark our algorithm to demonstrate its advantage over other similar tensor network based simulators. Other applications we found include evaluating the energy function of a Quantum Approximate Optimization Algorithm (QAOA), and simulating surface codes under a realistic error model with crosstalk.

Jan 1, 20200 citations

Quantum algorithms with tensor approach

Lejia Gu

Jan 1, 20200 citations

Tensorial tools for quantum computing

Yannick Meurice

Bulletin of the American Physical SocietyMar 6, 20190 citations

Quantum Computing Applications and Use-cases

Francesco Di Marcantonio, Ema Puljak

Abstract This talk with cover a foundational understanding of tensor networks, starting with the basics of tensors and their contractions. Students will learn simple forms of tensor networks and their connection to quantum circuits. Additionally, the course gives an example of the application of tensor networks in machine learning. By the end of the course, participants will grasp the core concepts and practical uses of tensor networks. Bio Ema is a computer scientist pursuing PhD in physics at CERN and Universitat Autonoma de Barcelona, specializing in quantum and quantum-inspired algorithms for anomaly detection with applications in high-energy physics and medical imaging. Francesco is a PhD student at CERN and University of the Basque Country. He studied at KTH and Politecnico di Torino and focuses on the simulation of Quantum Matter with Tensor Network Methods and other techniques related to Quantum Computation. Abstract This talk with cover a foundational understanding of tensor networks, starting with the basics of tensors and their contractions. Students will learn simple forms of tensor networks and their connection to quantum circuits. Additionally, the course gives an example of the application of tensor networks in machine learning. By the end of the course, participants will grasp the core concepts and practical uses of tensor networks. Bio Ema is a computer scientist pursuing PhD in physics at CERN and Universitat Autonoma de Barcelona, specializing in quantum and quantum-inspired algorithms for anomaly detection with applications in high-energy physics and medical imaging. Francesco is a PhD student at CERN and University of the Basque Country. He studied at KTH and Politecnico di Torino and focuses on the simulation of Quantum Matter with Tensor Network Methods and other techniques related to Quantum Computation.

Jan 1, 20240 citations

O(N,\mathbb {R})

O(N,\mathbb {C})

, and

U(N,\mathbb {C})

, respectively. Exact closed-form expressions of the distributions are obtained by computing partition functions of four-fermi theories, where the last case is of the “signed” distribution, which counts the distribution with a sign factor coming from a Hessian matrix. As an application, we compute the injective norm of the complex symmetric order-three random tensor in the large-N limit by computing the edge of the last signed distribution, obtaining agreement with an earlier numerical result in the literature.