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arXiv:1708.09349

One-dimensional quantum systems at finite temperatures can be simulated efficiently on classical computers

Thomas Barthel

Published Aug 30, 2017• 21 citations
arXivHTMLPDF

Abstract

It is by now well-known that ground states of gapped one-dimensional (1d) quantum-many body systems with short-range interactions can be studied efficiently using classical computers and matrix product state techniques. A corresponding result for finite temperatures was missing. Using the replica trick in 1+1d quantum field theory, it is shown here that the cost for the classical simulation of 1d systems at finite temperatures grows in fact only polynomially with the inverse temperature and is system-size independent -- even for gapless systems. In particular, we show that the thermofield double state (TDS), a purification of the equilibrium density operator, can be obtained efficiently in matrix-product form. The argument is based on the scaling behavior of Rényi entanglement entropies in the TDS. At finite temperatures, they obey the area law. For gapless systems with central charge $c$, the entanglement is found to grow only logarithmically with inverse temperature, $S_α\sim \frac{c}{6}(1+1/α)\log β$. The field-theoretical results are tested and confirmed by quasi-exact numerical computations for integrable and non-integrable spin systems, and interacting bosons.

DOI
10.48550/arxiv.1708.09349

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It is by now well-known that ground states of gapped one-dimensional (1d) quantum-many body systems with short-range interactions can be studied efficiently using classical computers and matrix product state techniques. A corresponding result for finite temperatures was missing. Using the replica trick in 1+1d quantum field theory, it is shown here that the cost for the classical simulation of 1d systems at finite temperatures grows in fact only polynomially with the inverse temperature and is system-size independent -- even for gapless systems. In particular, we show that the thermofield double state (TDS), a purification of the equilibrium density operator, can be obtained efficiently in matrix-product form. The argument is based on the scaling behavior of Rényi entanglement entropies in the TDS. At finite temperatures, they obey the area law. For gapless systems with central charge ccc, the entanglement is found to grow only logarithmically with inverse temperature, Sα∼c6(1+1/α)log⁡βS_α\sim \frac{c}{6}(1+1/α)\log βSα​∼6c​(1+1/α)logβ. The field-theoretical results are tested and confirmed by quasi-exact numerical computations for integrable and non-integrable spin systems, and interacting bosons.
We discuss the computational efficiency of the finite-temperature simulation with minimally entangled typical thermal states (METTS). To argue that METTS can be efficiently represented as matrix product states, we present an analytic upper bound for the average entanglement R\'enyi entropy of METTS for a R\'enyi index $0<q\ensuremath{\le}1$. In particular, for one-dimensional (1D) gapless systems described by conformal field theories, the upper bound scales as $O(c{N}^{0}log\ensuremath{\beta})$ where ccc is the central charge and NNN is the system size. Furthermore, we numerically find that the average R\'enyi entropy exhibits a universal behavior characterized by the central charge and is roughly given by half of the analytic upper bound. Based on these results, we show that METTS can provide a speedup compared to employing the purification method to analyze thermal equilibrium states at low temperatures in 1D gapless systems.
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We study the problem of computing energy density in one-dimensional quantum systems. We show that the ground-state energy per site or per bond can be computed in time (i) independent of the system size and subexponential in the desired precision if the ground state satisfies area laws for the Renyi entanglement entropy (this is the first rigorous formulation of the folklore that area laws imply efficient matrix-product-state algorithms); (ii) independent of the system size and polynomial in the desired precision if the system is gapped. As a by-product, we prove that in the presence of area laws (or even an energy gap) the ground state can be approximated by a positive semidefinite matrix product operator of bond dimension independent of the system size and subpolynomial in the desired precision of local properties.
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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 or matrix product state formalism exploited by numerical renormalization group algorithms like the density matrix renormalization group. There are two immediate consequences of this result. The first is that Vidal's time-dependent density matrix renormalization 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 discretization, such circuits are of polynomial depth.