Norbert Schuch

Classifying quantum phases using matrix product states and projected entangled pair states

We give a classification of gapped quantum phases of one-dimensional systems in the framework of Matrix Product States (MPS) and their associated parent Hamiltonians, for systems with unique as well as degenerate ground states, and both in the absence and presence of symmetries. We find that without symmetries, all systems are in the same phase, up to accidental ground state degeneracies. If symmetries are imposed, phases without symmetry breaking (i.e., with unique ground states) are classified by the cohomology classes of the symmetry group, this is, the equivalence classes of its projective representations, a result first derived in [X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 83, 035107 (2011); arXiv:1008.3745]. For phases with symmetry breaking (i.e., degenerate ground states), we find that the symmetry consists of two parts, one of which acts by permuting the ground states, while the other acts on individual ground states, and phases are labelled by both the permutation action of the former and the cohomology class of the latter. Using Projected Entangled Pair States (PEPS), we subsequently extend our framework to the classification of twodimensional phases in the neighborhood of a number of important cases, in particular systems with unique ground states, degenerate ground states with a local order parameter, and topological order. We also show that in two dimensions, imposing symmetries does not constrain the phase diagram in the same way it does in one dimension. As a central tool, we introduce the isometric form, a normal form for MPS and PEPS which is a renormalization fixed point. Transforming a state to its isometric form does not change the phase, and thus, we can focus on to the classification of isometric forms.

Entropy scaling and simulability by matrix product states.

We investigate the relation between the scaling of block entropies and the efficient simulability by matrix product states (MPSs) and clarify the connection both for von Neumann and Rényi entropies. Most notably, even states obeying a strict area law for the von Neumann entropy are not necessarily approximable by MPSs. We apply these results to illustrate that quantum computers might outperform classical computers in simulating the time evolution of quantum systems, even for completely translational invariant systems subject to a time-independent Hamiltonian.

PEPS as ground states: Degeneracy and topology

We introduce a framework for characterizing Matrix Product States (MPS) and Projected Entangled Pair States (PEPS) in terms of symmetries. This allows us to understand how PEPS appear as ground states of local Hamiltonians with finitely degenerate ground states and to characterize the ground state subspace. Subsequently, we apply our framework to show how the topological properties of these ground states can be explained solely from the symmetry: We prove that ground states are locally indistinguishable and can be transformed into each other by acting on a restricted region, we explain the origin of the topological entropy, and we discuss how to renormalize these states based on their symmetries. Finally, we show how the anyonic character of excitations can be understood as a consequence of the underlying symmetries.

Entanglement spectrum and boundary theories with projected entangled-pair states

In many physical scenarios, close relations between the bulk properties of quantum systems and theories associated with their boundaries have been observed. In this work, we provide an exact duality mapping between the bulk of a quantum spin system and its boundary using projected entangled-pair states. This duality associates to every region a Hamiltonian on its boundary, in such a way that the entanglement spectrum of the bulk corresponds to the excitation spectrum of the boundary Hamiltonian. We study various specific models: a deformed AKLT model [I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys. Rev. Lett. 59, 799 (1987)], an Ising-type model [F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac, Phys. Rev. Lett. 96, 220601 (2006)], and Kitaev’s toric code [A. Kitaev, Ann. Phys. 303, 2 (2003)], both in finite ladders and in infinite square lattices. In the second case, some of those models display quantum phase transitions. We find that a gapped bulk phase with local order corresponds to a boundary Hamiltonian with local interactions, whereas critical behavior in the bulk is reflected on a diverging interaction length of the boundary Hamiltonian. Furthermore, topologically ordered states yield nonlocal Hamiltonians. Because our duality also associates a boundary operator to any operator in the bulk, it in fact provides a full holographic framework for the study of quantum many-body systems via their boundary.

Computational Complexity of Projected Entangled Pair States

● What is the power of creating ground states? The Projected Entangled Pair States (PEPS) formalism which underlies DMRG has proven very successful for the description of complex many-body systems. While in one dimension, PEPS can be created as well as simulated efficiently, the situation in two and more dimensions is less clear. In this work, we determine the power of creating PEPS as well as the complexity of classically simulating PEPS. We also consider ground states of gapped Hamiltonians and show how they can be approximated by PEPS. Finally, we prove that the hardness of creating ground states is lower than for arbitrary PEPS. The central tool for our proofs is a new duality between PEPS and postselection. Abstract

Natural two-qubit gate for quantum computation using the XY interaction

The two-qubit interaction Hamiltonian of a given physical implementation determines whether or not a two-qubit gate such as the controlled-NOT (cNOT) gate can be realized easily. It can be shown that, e.g., with the XY interaction more than one two-qubit operation is required in order to realize the CNOT operation. Here we propose a two-qubit gate for the X Y interaction which combines the CNOT and SWAP operations. By using this gate quantum circuits can be implemented efficiently, even if only nearest-neighbor coupling between the qubits is available.

Fermionic projected entangled pair states

We introduce a family of states, the fermionic projected entangled pair states (fPEPS), which describe fermionic systems on lattices in arbitrary spatial dimensions. It constitutes the natural extension of another family of states, the PEPS, which efficiently approximate ground and thermal states of spin systems with short-range interactions. We give an explicit mapping between those families, which allows us to extend previous simulation methods to fermionic systems. We also show that fPEPS naturally arise as exact ground states of certain fermionic Hamiltonians. We give an example of such a Hamiltonian, exhibiting criticality while obeying an area law.

Simulation of Quantum Many-Body Systems with Strings of Operators and Monte Carlo Tensor Contractions

We introduce string-bond states, a class of states obtained by placing strings of operators on a lattice, which encompasses the relevant states in Quantum Information. For string-bond states, expectation values of local observables can be computed efficiently using Monte Carlo sampling, making them suitable for a variational abgorithm which extends DMRG to higher dimensional and irregular systems. Numerical results demonstrate the applicability of these states to the simulation of many-body sytems.

Quantum States on Harmonic Lattices

We investigate bosonic Gaussian quantum states on an infinite cubic lattice in arbitrary spatial dimensions. We derive general properties of such states as ground states of quadratic Hamiltonians for both critical and non-critical cases. Tight analytic relations between the decay of the interaction and the correlation functions are proven and the dependence of the correlation length on band gap and effective mass is derived. We show that properties of critical ground states depend on the gap of the point-symmetrized rather than on that of the original Hamiltonian. For critical systems with polynomially decaying interactions logarithmic deviations from polynomially decaying correlation functions are found.

Topological Order in the Projected Entangled-Pair States Formalism: Transfer Operator and Boundary Hamiltonians

Norbert Schuch, 2 Didier Poilblanc, J. Ignacio Cirac, and David Pérez-Garćıa Institut für Quanteninformation, RWTH Aachen, 52056 Aachen, Germany Institute for Quantum Information, California Institute of Technology, MC 305-16, Pasadena CA 91125, U.S.A. Laboratoire de Physique Théorique, C.N.R.S. and Université de Toulouse, 31062 Toulouse, France Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany Dpto. Analisis Matematico and IMI, Universidad Complutense de Madrid, E-28040 Madrid, Spain