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.
The algorithm, based on the Matrix Product States formalism, is highly efficient when the amount of entanglement in the system is limited, a requirement fulfilled by a large class of quantum many-body systems in one dimension.
Time-evolving block decimation (TEBD) is a numerical algorithm that can efficiently simulate the time evolution of one dimensional quantum systems having limited entanglement entropy.
However, as the number of degrees of freedom of the system grows it quickly becomes computationally infeasible to perform the associated matrix exponentiation and matrix-vector multiplication.
TEBD presents an efficient scheme for performing time evolution by limiting itself to a much smaller subspace of the configuration space.
There are several other noteworthy examples of ways to get around this exponential scaling, including quantum Monte Carlo and the density matrix renormalization group.
Guifré Vidal proposed the scheme while at the Institute for Quantum Information, Caltech.
The method exhibits a low-degree polynomial behavior in the increase of computational time with respect to the amount of entanglement present in the system.
The algorithm is based on a scheme that exploits the fact that in these one-dimensional systems the eigenvalues of the reduced density matrix on a bipartite split of the system are exponentially decaying, thus allowing one to work in a re-sized space spanned by the eigenvectors corresponding to the selected eigenvalues.
The numerical method is efficient in simulating real-time dynamics or calculations of ground states using imaginary-time evolution or isentropic interpolations between a target Hamiltonian and a Hamiltonian with an already-known ground state.
A useful feature of the TEBD algorithm is that it can be reliably employed for time evolution simulations of time-dependent Hamiltonians, describing systems that can be realized with cold atoms in optical lattices, or in systems far from equilibrium in quantum transport.
From this point of view, TEBD had a certain ascendance over DMRG, a very powerful technique, but until recently not very well suited for simulating time-evolutions.
Other groups have developed similar approaches in which quantum information plays a predominant role: for example, in DMRG implementations for periodic boundary conditions [3], and for studying mixed-state dynamics in one-dimensional quantum lattice systems,.
[2][3] Those last approaches actually provide a formalism that is more general than the original TEBD approach, as it also allows to deal with evolutions with matrix product operators; this enables the simulation of nontrivial non-infinitesimal evolutions as opposed to the TEBD case, and is a crucial ingredient to deal with higher-dimensional analogues of matrix product states.
The vectors of the SD are determined up to a phase and the eigenvalues and the Schmidt rank are unique.
Therefore, it is possible to take into account only some of the Schmidt coefficients (namely the largest ones), dropping the others and consequently normalizing again the state:
Let's get away from this abstract picture and refresh ourselves with a concrete example, to emphasize the advantage of making this decomposition.
Even if the Schmidt eigenvalues don't have this exponential decay, but they show an algebraic decrease, we can still use D to describe our state
after a unitary operator at qubit k does not modify the Schmidt eigenvalues or vectors on the left, consequently the
, hence its role is not very relevant for the order of magnitude of the number of basic operations, but in the case when the on-site dimension is higher than two it has a rather decisive contribution.
This is done to make the Suzuki–Trotter expansion (ST)[5] of the exponential operator, named after Masuo Suzuki and Hale Trotter.
The Suzuki–Trotter expansion of the first order (ST1) represents a general way of writing exponential operators:
In problems of quantum dynamics the unitarity of the operators in the ST expansion proves quite practical, since the error tends to concentrate in the overall phase, thus allowing us to faithfully compute expectation values and conserved quantities.
and an ST expansion of the first order keeps only the product of the exponentials, the approximation becoming, in this case, exact.
Mind this would imply diagonalizing somewhat generous reduced density matrices, which, depending on the system one has to simulate, might be a task beyond our reach and patience.
First, as we have seen above, the smallest contributions to the Schmidt spectrum are left away, the state being faithfully represented up to:
is the state formed by the eigenfunctions corresponding to the smallest, irrelevant Schmidt coefficients, which are neglected.
The error is evaluated by successively multiplying with the normalization constant, each time we build the reduced density matrix and select its relevant eigenvalues.
Hence, at the first bond, instead of futilely diagonalizing, let us say, 10 by 10 or 20 by 20 matrices, we can just restrict ourselves to ordinary 2 by 2 ones, thus making the algorithm generally faster.
TEBD also offers the possibility of straightforward parallelization due to the factorization of the exponential time-evolution operator using the Suzuki–Trotter expansion.