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Variational quantum Monte Carlo simulations with tensor-network states · Around

2007

Variational quantum Monte Carlo simulations with tensor-network states

Sandvik, A. W., Vidal, G.

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We show that the formalism of tensor-network states, such as the matrix product states (MPS), can be used as a basis for variational quantum Monte Carlo simulations.

Using a stochastic optimization method, we demonstrate the potential of this approach by explicit MPS calculations for the transverse Ising chain with up to N=256 spins at criticality, using periodic boundary conditions and D*D matrices with D up to 48.
The computational cost of our scheme formally scales as ND^3, whereas standard MPS approaches and the related density matrix renromalization group method scale as ND^5 and ND^6, respectively, for periodic systems.

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2007
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F. Verstraete and J. I. Cirac, Arxiv:cond-mat/0407066

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J. Jordan, R. Orús, G. Vidal, F. Verstraete, and J. I. Cirac, ArXiv:cond-mat/0703788

Cited in the paper.

The scaling with MPS is reduced from D 3 D^{3} and D 5 D^{5} [ 25 ] for open and periodic boundaries, respectively, to D 2 D^{2} and D 3 D^{3} . For PEPS with open boundaries the sacling goes from D 8 D ~ 2 ≈ D 12 D^{8}\tilde{D}^{2}\approx D^{12} [ 17 ] to D 4 D ′ 2 + D 2 D ′ 3 ≈ D 6 D^{4}D^{\prime 2}+D^{2}D^{\prime 3}\approx D^{6} , where D ~ ≈ D 2 \tilde{D}\approx D^{2} and D ′ ≈ D D^{\prime}\approx D are the ranks of the boundary MPS employed in the contraction. A reduction of several powers of D D is also achieved in the case of PEPS with cylinder and torus boundary conditions

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Stochastic optimization takes advantage of noise [ 21 ] , but there is some limit beyond which too many errors in the signs of the derivatives are detrimental

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N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, ArXiv:0708.1567

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2007

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