The Dynamics of 1D Quantum Spin Systems Can Be Approximated Efficiently

Details The Dynamics of 1D Quantum Spin Systems Can Be Approximated Efficiently The Dynamics of 1D Quantum Spin Systems Can Be Approximated Efficiently

2005-08-03

arxiv.org/abs/quant-ph/0508031v2

Phys. Rev. Lett. 97, 157202 (2006)

Physics

Quantum Physics

4 pages, 2 figures. Simplified argument

Scientific paper

10.1103/PhysRevLett.97.157202

In this Letter we show that an arbitrarily good approximation to the propagator e^{itH} for a 1D lattice of n quantum spins with hamiltonian H may be obtained with polynomial computational resources in n and the error \epsilon, and exponential resources in |t|. Our proof makes use of the finitely correlated state/matrix product state formalism exploited by numerical renormalisation group algorithms like the density matrix renormalisation group. There are two immediate consequences of this result. The first is that the Vidal's time-dependent density matrix renormalisation group will require only polynomial resources to simulate 1D quantum spin systems for logarithmic |t|. The second consequence is that continuous-time 1D quantum circuits with logarithmic |t| can be simulated efficiently on a classical computer, despite the fact that, after discretisation, such circuits are of polynomial depth.

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