Typical 1d quantum systems at finite temperatures can be simulated efficiently on classical computers

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.

For 1d systems that can be described by 1+1d field theory, I show that the cost for the classical simulation at finite temperatures grows in fact only polynomially with the inverse temperature and is system-size independent — even for gapless systems. In particular, 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 conformally invariant systems, the Rényi entropies are found to grow only logarithmically with inverse temperature. The field-theoretical results are tested and confirmed by quasi-exact numerical computations for integrable and non-integrable spin systems, and interacting bosons.

Ref: T. Barthel, arXiv:1708.09349 (2017)