Markov chains for probability distributions related to matrix product states and 1D Hamiltonians are introduced. With appropriate inverse temperature schedules, these chains can be combined into a random approximation scheme for ground states of such Hamiltonians. Numerical experiments suggest that a linear, i.e. fast, schedule is possible in non-trivial cases. A natural extension of these chains to 2D settings is next presented and tested. The obtained results compare well with Euclidean evolution. The proposed Markov chains are easy to implement and are inherently sign problem free (even for fermionic degrees of freedom).
A scheme for measuring complex temperature partition functions of Ising models is introduced. In the context of ordered qubit registers this scheme finds a natural translation in terms of global operations, and single particle measurements on the edge of the array. Two applications of this scheme are presented. First, through appropriate Wick rotations, those amplitudes can be analytically continued to yield estimates for partition functions of Ising models. Bounds on the estimation error, valid with high confidence, are provided through a central-limit theorem, which validity extends beyond the present context. It holds for example for estimations of the Jones polynomial. Interestingly, the kind of state preparations and measurements involved in this application can in principle be made instantaneous, i.e. independent of the system size or the parameters being simulated. Second, the scheme allows to accurately estimate some non-trivial invariants of links. A third result concerns the computational power of estimations of partition functions for real temperature classical ferromagnetic Ising models on a square lattice. We provide conditions under which estimating such partition functions allows one to reconstruct scattering amplitudes of quantum circuits making the problem BQP-hard. Using this mapping, we show that fidelity overlaps for ground states of quantum Hamiltonians, which serve as a witness to quantum phase transitions, can be estimated from classical Ising model partition functions. Finally, we show that the ability to accurately measure corner magnetizations on thermal states of two-dimensional Ising models with magnetic field leads to fully polynomial random approximation schemes (FPRAS) for the partition function. Each of these results corresponds to a section of the text that can be essentially read independently.
A study of the thermal properties of two-dimensional topological lattice models is presented. This work is relevant to assess the usefulness of these systems as a quantum memory. For our purposes, we use the topological mutual information $I_{mathrm{topo}}$ as a topological order parameter. For Abelian models, we show how $I_{mathrm{topo}}$ depends on the thermal topological charge probability distribution. More generally, we present a conjecture that $I_{mathrm{topo}}$ can (asymptotically) be written as a Kullback-Leitner distance between this probability distribution and that induced by the quantum dimensions of the model at hand. We also explain why $I_{mathrm{topo}}$ is more suitable for our purposes than the more familiar entanglement entropy $S_{mathrm{topo}}$. A scaling law, encoding the interplay of volume and temperature effects, as well as different limit procedures, are derived in detail. A non-Abelian model is next analysed and similar results are found. Finally, we also consider, in the case of a one-plaquette toric code, an environment model giving rise to a simulation of thermal effects in time.
Topological systems, such as fractional quantum Hall liquids, promise to successfully combat environmental decoherence while performing quantum computation. These highly correlated systems can support non-Abelian anyonic quasiparticles that can encode exotic entangled states. To reveal the non-local character of these encoded states we demonstrate the violation of suitable Bell inequalities. We provide an explicit recipe for the preparation, manipulation and measurement of the desired correlations for a large class of topological models. This proposal gives an operational measure of non-locality for anyonic states and it opens up the possibility to violate the Bell inequalities in quantum Hall liquids or spin lattices.
The power of matrix product states to describe infinite-size translational-invariant critical spin chains is investigated. At criticality, the accuracy with which they describe ground state properties of a system is limited by the size $chi$ of the matrices that form the approximation. This limitation is quantified in terms of the scaling of the half-chain entanglement entropy. In the case of the quantum Ising model, we find $S sim {1/6}log chi$ with high precision. This result can be understood as the emergence of an effective finite correlation length $xi_chi$ ruling of all the scaling properties in the system. We produce five extra pieces of evidence for this finite-$chi$ scaling, namely, the scaling of the correlation length, the scaling of magnetization, the shift of the critical point, and the scaling of the entanglement entropy for a finite block of spins. All our computations are consistent with a scaling relation of the form $xi_chisim chi^{kappa}$, with $kappa=2$ for the Ising model. In the case of the Heisenberg model, we find similar results with the value $kappasim 1.37$. We also show how finite-$chi$ scaling allow to extract critical exponents. These results are obtained using the infinite time evolved block decimation algorithm which works in the thermodynamical limit and are verified to agree with density matrix renormalization group results.
Understanding the behaviour of topologically ordered lattice systems at finite temperature is a way of assessing their potential as fault-tolerant quantum memories. We compute the natural extension of the topological entanglement entropy for T > 0, namely the subleading correction $I_{textrm{topo}}$ to the area law for mutual information. Its dependence on T can be written, for Abelian Kitaev models, in terms of information-theoretic functions and readily identifiable scaling behaviour, from which the interplay between volume, temperature, and topological order, can be read. These arguments are extended to non-Abelian quantum double models, and numerical results are given for the $D(S_3)$ model, showing qualitative agreement with the Abelian case.
