Quantum Computation and Information Seminar 

16/05/2006, 14:00 — 15:00 — Room P6, Mathematics Building
Ignacio Cirac, Max-Planck-Institut für Quantenoptik

Quantum Simulations in Many-Body Systems

The description of many-body quantum states is, typically, very hard. The reason is that the number of parameters needed to characterize the quantum state of $N$ $d$-level systems scales as $d^N$, so that even for qubits ( $d=2$) already for $N>40$ it is impossible to store all the corresponding coefficients. Furthermore, if one wants to determine the expectation value of any observable one needs to perform a number of basic operations which also scale exponentially with the number of particles. However, in Nature, only some particular states appear, and thus it may happen that different ways of parametrizing quantum states are much more efficient and do not require an exponential scaling. In this talk I presented a new characterization of quantum states, what we call Projected Entangled-Pair States (PEPS). This characterization is based on constructing pairs of maximally entangled states in a Hilbert space of dimension $D^2$, and then projecting those states in subspaces of dimension $d$. In one dimension, one recovers the familiar matrix product states, whereas in higher dimensions this procedure gives rise to other interesting states. We have used this new parametrization to construct numerical algorithms to simulate the ground state properties and dynamics of certain quantum-many body systems in two dimensions. The results are very encouraging, since we have been able to simulate $20\times 20$ spin $1/2$ lattices interacting with the Heisenberg nearest neighbor Hamiltonian, as well as with other frustrated Hamiltonians.