We introduce a semistochastic implementation of the power method to compute, for very large matrices, the dominant eigenvalue and expectation values involving the corresponding eigenvector. The method is semistochastic in that the matrix multiplication is partially implemented numerically exactly and partially stochastically with respect to expectation values only. Compared to a fully stochastic method, the semistochastic approach significantly reduces the computational time required to obtain the eigenvalue to a specified statistical uncertainty. This is demonstrated by the application of the semistochastic quantum Monte Carlo method to systems with a sign problem: the fermion Hubbard model and the carbon dimer. Introduction.-Consider the computation of the dominant eigenvalue of an N Â N matrix, with N so large that the matrix cannot be stored. Transformation methods cannot be used in this case, but one can still proceed with the power method, also known as the projection method, as long as one can compute and store the result of multiplication of an arbitrary vector by the matrix. When, for sufficiently large N, this is no longer feasible, Monte Carlo methods can be used to represent stochastically both the vector and multiplication by the matrix. This suffices to implement the power method to compute the dominant eigenvalue and averages involving its corresponding eigenvector.In this Letter, we propose a hybrid method consisting of numerically exact representation and multiplication in a small deterministic subspace, complemented by stochastic treatment of the rest of the space. This semistochastic projection method combines the advantages of both approaches: it greatly reduces the statistical uncertainty of averages relative to purely stochastic projection while allowing N to be large. These advantages are realized if one succeeds in choosing a deterministic subspace that carries a substantial fraction of the total spectral weight of the dominant eigenstate.Semistochastic projection has numerous potential applications: transfer matrix [1] and quantum Monte Carlo (QMC) [2][3][4] calculations, respectively for classical statistical mechanical and quantum mechanical systems, and the calculation of subdominant eigenvalues [5].In this Letter we apply the semistochastic method to compute the ground state energy of quantum mechanical Hamiltonians represented in a discrete basis. In this context, deterministic projection is known as full configuration interaction (FCI) to chemists and as exact diagonalization to physicists, whereas stochastic projection is the essence of various projector QMC methods [2,3]. Hence, semistochastic projection shall be referred to as the SQMC method. The benefit of the SQMC method over the

We introduce an algorithm for sampling many-body quantum states in Fock space. The algorithm efficiently samples states with probability approximately proportional to an arbitrary function of the second-quantized Hamiltonian matrix element connecting the sampled state to the current state. We apply the new sampling algorithm to the recently developed semistochastic full configuration interaction quantum Monte Carlo (S-FCIQMC) method, a semistochastic implementation of the power method for projecting out the ground state energy in a basis of Slater determinants. Our new sampling method requires modest additional computational time and memory compared to uniform sampling but results in newly spawned weights that are approximately of the same magnitude, thereby greatly improving the efficiency of projection. A comparison in efficiency between our sampling algorithm and uniform sampling is performed on the all-electron nitrogen dimer at equilibrium in Dunning's cc-pVXZ basis sets with X ∈ {D, T, Q, 5}, demonstrating a large gain in efficiency that increases with basis set size. In addition, a comparison in efficiency is performed on three all-electron first-row dimers, B2, N2, and F2, in a cc-pVQZ basis, demonstrating that the gain in efficiency compared to uniform sampling also increases dramatically with the number of electrons.

We describe correlator product states, a class of numerically efficient many-body wave functions to describe strongly correlated wave functions in any dimension. Correlator product states introduce direct correlations between physical degrees of freedom in a simple way, yet provide the flexibility to describe a wide variety of systems. We show that many interesting wave functions can be mapped exactly onto correlator product states, including Laughlin's quantum Hall wave function, Kitaev's toric code states, and Huse and Elser's frustrated spin states. We also outline the relationship between correlator product states and other common families of variational wave functions such as matrix product states, tensor product states, and resonating valence-bond states. Variational calculations for the Heisenberg and spinless Hubbard models demonstrate the promise of correlator product states for describing both two-dimensional and fermion correlations. Even in onedimensional systems, correlator product states are competitive with matrix product states for a fixed number of variational parameters.

Parafermions are exotic quasiparticles with non-Abelian fractional statistics that can be realized and stabilized in 1-dimensional models that are generalizations of the Kitaev p-wave wire. We study the simplest generalization, i.e. the Z3 parafermionic chain. Using a Jordan-Wigner transform we focus on the equivalent three-state chiral clock model, and study its rich phase diagram using the density matrix renormalization group technique. We perform our analyses using quantum entanglement diagnostics which allow us to determine phase boundaries, and the nature of the phase transitions. In particular, we study the transition between the topological and trivial phases, as well as to an intervening incommensurate phase which appears in a wide region of the phase diagram. The phase diagram is predicted to contain a Lifshitz type transition which we confirm using entanglement measures. We also attempt to locate and characterize a putative tricritical point in the phase diagram where the three above mentioned phases meet at a single point.Introduction-There has been concerted effort to engineer systems with stable Majorana bound states, and other anyonic quasiparticles, for use in the topological quantum computation architecture [1][2][3][4][5][6][7]. For example, there has been recent progress in attempts to isolate Majorana bound states in quantum nanowires [5,[8][9][10] and in superconductor surfaces implanted with a line of magnetic impurities [11]. These quasi-1D systems effectively realize a version of the Kitaev p-wave wire model [12], and are predicted to have a gapped topological phase which supports characteristic Majorana bound states at the ends of the wire.While the boundary modes in these heterostructure systems are non-Abelian anyons, they are unfortunately known to be insufficient for universal quantum computation. A possible remedy for this problem has been to look for more exotic nonAbelian excitations. For example, Fendley has recently suggested exploring one-dimensional Z N para-fermionic models which support topological phases with more computationally efficient non-Abelian anyon bound states [13]. Still, the Z N non-Abelian anyons are not able to perform universal quantum computation, however they can be leveraged to create a 2D phase with Fibonaccci anyons, which are universal [14]. These promising features have spurred wide spread interest in these models, and has led to many analytical and numerical studies, including several experimental proposals for realizing these topological phases .In this work, we continue along these lines of research by exploring the rich phase diagram of the Z 3 para-fermionic chain; though for ease of calculation we actually study the Jordan-Wigner transformed para-fermionic chain [40], including chiral interactions. The resulting model is the three state chiral clock model. This model re-surfaced in this context in Ref. 13 as a candidate for exhibiting non-Abelian bound states beyond Majorana fermions. It was shown analytically that para-fermionic boundary zero mod...

Frustrated quantum magnets are a central theme in condensed matter physics due to the richness of their phase diagrams. They support a panoply of phases including various ordered states and topological phases. Yet, this problem has defied a solution for a long time due to the lack of controlled approximations which make it difficult to distinguish between competing phases. Here we report the discovery of a special quantum macroscopically degenerate point in the XXZ model on the spin-1/2 kagome quantum antiferromagnet for the ratio of Ising to antiferromagnetic transverse coupling J_{z}/J=-1/2. This point is proximate to many competing phases explaining the source of the complexity of the phase diagram. We identify five phases near this point including both spin-liquid and broken-symmetry phases and give evidence that the kagome Heisenberg antiferromagnet is close to a transition between two phases.