Time-evolving block decimation

The Time-evolving Block Decimation (TEBD) algorithm is a numerical scheme used to simulate one-dimensional quantum many-body systems, characterized by at most nearest-neighbour interactions. It is dubbed Time-evolving Block Decimation because it dynamically identifies the relevant low-dimensional Hilbert subspaces of an exponentially larger original Hilbert space. The algorithm, based on the Matrix Product States formalism, is highly efficient when the amount of entanglement in the system is limited, a requirement which is fulfilled by a large class of quantum many-body systems in one dimension.

Introduction

There is nowadays a considerable interest in the field of quantum theory for computational methods well-suited to the physics of many-body systems. Considering the inherent difficulties of simulating general quantum many-body systems, the exponential increase in parameters with the size of the system, and correspondingly, the high computational costs, one solution would be to look for numerical methods which deal with special cases, where one can profit from the physics of the system. The raw approach, by directly dealing with all the parameters used to fully characterize a quantum many-body system is seriously impeded by the exponential increase in the number of parameters with the size of the system, which leads to unreasonably long computational times and extended use of memory. To get around this problem a number of various methods have been developed and put into practice in the course of time, one of the most successful ones being the Quantum Monte Carlo Method (QMC). Also the density matrix renormalization group (DMRG) method, next to QMC, is a very reliable method, with an expanding community of users and an increasing number of applications to physical systems.

When the first quantum computer will be plugged in and functioning, the perspectives for the field of computational physics will look rather promising, but until that day one has to restrict oneself to the mundane tools offered by classical computers. While experimental physicists are putting a lot of effort in trying to build the first quantum computer, theoretical physicists are searching, in the field of Quantum Information Theory (QIT), for genuine quantum algorithms, appropriate for problems which would perform badly when trying to be solved on a classical computer but pretty fast and successful on a quantum one. The search for such algorithms is still going, the best-known (and almost the only ones found) being the Shor's algorithm, for factoring large numbers, and Grover's search algorithm.

In the field of QIT one has to identify the primary resources necessary for genuine quantum computation. Such a resource may be responsible for the speedup gain in quantum versus classical, identifying them means also identifying systems which can be simulated in a reasonably efficient manner on a classical computer. Such a resource is quantum entanglement; hence, it is possible to establish a distinct lower bound for the entanglement needed for quantum computational speedups.

Guifr Vidal, from Institute for Quantum Information, CalTech, has recently proposed a scheme useful for simulating a certain category of quantum systems.