109 research outputs found

### Complexity of commuting Hamiltonians on a square lattice of qubits

We consider the computational complexity of Hamiltonians which are sums of
commuting terms acting on plaquettes in a square lattice of qubits, and we show
that deciding whether the ground state minimizes the energy of each local term
individually is in the complexity class NP. That is, if the ground states has
this property, this can be proven using a classical certificate which can be
efficiently verified on a classical computer. Different to previous results on
commuting Hamiltonians, our certificate proves the existence of such a state
without giving instructions on how to prepare it.Comment: 16 pages, 12 figures. v2: Minor corrections. Accepted version,
Journal-Ref adde

### Simplex Z_2 spin liquids on the Kagome lattice with Entangled Pair States: spinon and vison coherence lengths, topological entropy and gapless edge modes

Gapped Z_2 spin liquids have been proposed as candidates for the ground-state
of the S=1/2 quantum antiferromagnet on the Kagome lattice. We extend the use
of Projected Entangled Pair States to construct (on the cylinder)Resonating
Valence Bond (RVB) states including both nearest-neighbor and next-nearest
neighbor singlet bonds. Our ansatz -- dubbed "simplex spin liquid" -- allows
for an asymmetry between the two types of triangles (of order 2-3% in the
energy density after optimization) leading to the breaking of inversion
symmetry. We show that the topological Z_2 structure is still preserved and, by
considering the presence or the absence of spinon and vison lines along an
infinite cylinder, we explicitly construct four orthogonal RVB Minimally
Entangled States. The spinon and vison coherence lengths are extracted from a
finite size scaling w.r.t the cylinder perimeter of the energy splittings of
the four sectors and are found to be of the order of the lattice spacing. The
entanglement spectrum of a partitioned (infinite) cylinder is found to be
gapless suggesting the occurrence, on a cylinder with {\it real} open
boundaries, of gapless edge modes formally similar to Luttinger liquid
(non-chiral) spin and charge modes. When inversion symmetry is spontaneously
broken, the RVB spin liquid exhibits an extra Ising degeneracy, which might
have been observed in recent exact diagonalisation studies.Comment: 5 pages, 6 figures - v2 with moderate revision of abstract, text and
conclusio

### Gapped $Z_2$ spin liquid in the breathing kagome Heisenberg antiferromagnet

We investigate the spin-1/2 Heisenberg antiferromagnet on the kagome lattice
with breathing anisotropy (i.e. with weak and strong triangular units),
constructing an improved simplex Resonating Valence Bond (RVB) ansatz by
successive applications (up to three times) of local quantum gates which
implement a filtering operation on the bare nearest-neighbor RVB state. The
resulting Projected Entangled Pair State involves a small number of variational
parameters (only one at each level of application) and preserves full lattice
and spin-rotation symmetries. Despite its simple analytic form, the simplex RVB
provides very good variational energies at strong and even intermediate
breathing anisotropy. We show that it carries $Z_2$ topological order which
does not fade away under the first few applications of the quantum gates,
suggesting that the RVB topological spin liquid becomes a competing ground
state candidate for the kagome antiferromagnet at large breathing anisotropy

### Quantum proofs can be verified using only single qubit measurements

QMA (Quantum Merlin Arthur) is the class of problems which, though
potentially hard to solve, have a quantum solution which can be verified
efficiently using a quantum computer. It thus forms a natural quantum version
of the classical complexity class NP (and its probabilistic variant MA,
Merlin-Arthur games), where the verifier has only classical computational
resources. In this paper, we study what happens when we restrict the quantum
resources of the verifier to the bare minimum: individual measurements on
single qubits received as they come, one-by-one. We find that despite this
grave restriction, it is still possible to soundly verify any problem in QMA
for the verifier with the minimum quantum resources possible, without using any
quantum memory or multiqubit operations. We provide two independent proofs of
this fact, based on measurement based quantum computation and the local
Hamiltonian problem, respectively. The former construction also applies to
QMA$_1$, i.e., QMA with one-sided error.Comment: 7 pages, 1 figur

### Semionic resonating valence bond states

The nature of the kagome Heisenberg antiferromagnet (HAFM) is under ongoing
debate. While recent evidence points towards a Z_2 topological spin liquid, the
exact nature of the topological phase is still unclear. In this paper, we
introduce semionic Resonating Valence Bond (RVB) states, this is, Resonating
Valence Bond states which are in the Z_2 ordered double-semion phase, and study
them using Projected Entangled Pair States (PEPS). We investigate their physics
and study their suitability as an ansatz for the HAFM, as compared to a
conventional RVB state which is in the Toric Code Z_2 topological phase. In
particular, we find that a suitably optimized "semionic simplex RVB"
outperforms the equally optimized conventional "simplex RVB" state, and that
the entanglement spectrum (ES) of the semionic RVB behaves very differently
from the ES of the conventional RVB, which suggests to use the ES to
discriminate the two phases. Finally, we also discuss the possible relevance of
space group symmetry breaking in valence bond wavefunctions with double-semion
topological order.Comment: 14 pages, 21 figures. v2: minor correction

### $\mathbb{Z}_N$ symmetry breaking in Projected Entangled Pair State models

We consider Projected Entangled Pair State (PEPS) models with a global
$\mathbb Z_N$ symmetry, which are constructed from $\mathbb Z_N$-symmetric
tensors and are thus $\mathbb Z_N$-invariant wavefunctions, and study the
occurence of long-range order and symmetry breaking in these systems. First, we
show that long-range order in those models is accompanied by a degeneracy in
the so-called transfer operator of the system. We subsequently use this
degeneracy to determine the nature of the symmetry broken states, i.e., those
stable under arbitrary perturbations, and provide a succinct characterization
in terms of the fixed points of the transfer operator (i.e.\ the different
boundary conditions) in the individual symmetry sectors. We verify our findings
numerically through the study of a $\mathbb Z_3$-symmetric model, and show that
the entanglement Hamiltonian derived from the symmetry broken states is
quasi-local (unlike the one derived from the symmetric state), reinforcing the
locality of the entanglement Hamiltonian for gapped phases.Comment: 11 page

### Study of anyon condensation and topological phase transitions from a $\mathbb{Z}_4$ topological phase using Projected Entangled Pair States

We use Projected Entangled Pair States (PEPS) to study topological quantum
phase transitions. The local description of topological order in the PEPS
formalism allows us to set up order parameters which measure condensation and
deconfinement of anyons, and serve as a substitute for conventional order
parameters. We apply these order parameters, together with anyon-anyon
correlation functions and some further probes, to characterize topological
phases and phase transitions within a family of models based on a $\mathbb Z_4$
symmetry, which contains $\mathbb Z_4$ quantum double, toric code, double
semion, and trivial phases. We find a diverse phase diagram which exhibits a
variety of different phase transitions of both first and second order which we
comprehensively characterize, including direct transitions between the toric
code and the double semion phase.Comment: 21+6 page

### The computational complexity of density functional theory

Density functional theory is a successful branch of numerical simulations of
quantum systems. While the foundations are rigorously defined, the universal
functional must be approximated resulting in a `semi'-ab initio approach. The
search for improved functionals has resulted in hundreds of functionals and
remains an active research area. This chapter is concerned with understanding
fundamental limitations of any algorithmic approach to approximating the
universal functional. The results based on Hamiltonian complexity presented
here are largely based on \cite{Schuch09}. In this chapter, we explain the
computational complexity of DFT and any other approach to solving electronic
structure Hamiltonians. The proof relies on perturbative gadgets widely used in
Hamiltonian complexity and we provide an introduction to these techniques using
the Schrieffer-Wolff method. Since the difficulty of this problem has been well
appreciated before this formalization, practitioners have turned to a host
approximate Hamiltonians. By extending the results of \cite{Schuch09}, we show
in DFT, although the introduction of an approximate potential leads to a
non-interacting Hamiltonian, it remains, in the worst case, an NP-complete
problem.Comment: Contributed chapter to "Many-Electron Approaches in Physics,
Chemistry and Mathematics: A Multidisciplinary View

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