scholarly journals Trivial low energy states for commuting Hamiltonians, and the quantum PCP conjecture

2013 ◽  
Vol 13 (5&6) ◽  
pp. 393-429
Author(s):  
Matthew Hastings
Keyword(s):  
Ground State ◽  
Coding Theory ◽  
Energy State ◽  
Ground States ◽  
Coarse Graining ◽  
Quantum Circuit ◽  
Technical Condition ◽  
Two Dimensions ◽  
Low Energy ◽  
Coarse Grained

We consider the entanglement properties of ground states of Hamiltonians which are sums of commuting projectors (we call these commuting projector Hamiltonians), in particular whether or not they have ``trivial" ground states, where a state is trivial if it is constructed by a local quantum circuit of bounded depth and range acting on a product state. It is known that Hamiltonians such as the toric code only have nontrivial ground states in two dimensions. Conversely, commuting projector Hamiltonians which are sums of two-body interactions have trivial ground states\cite{bv}. Using a coarse-graining procedure, this implies that any such Hamiltonian with bounded range interactions in one dimension has a trivial ground state. In this paper, we further explore the question of which Hamiltonians have trivial ground states. We define an ``interaction complex" for a Hamiltonian, which generalizes the notion of interaction graph and we show that if the interaction complex can be continuously mapped to a $1$-complex using a map with bounded diameter of pre-images then the Hamiltonian has a trivial ground state assuming one technical condition on the Hamiltonians holds (this condition holds for all stabilizer Hamiltonians, and we additionally prove the result for all Hamiltonians under one assumption on the $1$-complex). While this includes the cases considered by Ref.~\onlinecite{bv}, we show that it also includes a larger class of Hamiltonians whose interaction complexes cannot be coarse-grained into the case of Ref.~\onlinecite{bv} but still can be mapped continuously to a $1$-complex. One motivation for this study is an approach to the quantum PCP conjecture. We note that many commonly studied interaction complexes can be mapped to a $1$-complex after removing a small fraction of sites. For commuting projector Hamiltonians on such complexes, in order to find low energy trivial states for the original Hamiltonian, it would suffice to find trivial ground states for the Hamiltonian with those sites removed. Such trivial states can act as a classical witness to the existence of a low energy state. While this result applies for commuting Hamiltonians and does not necessarily apply to other Hamiltonians, it suggests that to prove a quantum PCP conjecture for commuting Hamiltonians, it is worth investigating interaction complexes which cannot be mapped to $1$-complexes after removing a small fraction of points. We define this more precisely below; in some sense this generalizes the notion of an expander graph. Surprisingly, such complexes do exist as will be shown elsewhere\cite{fh}, and have useful properties in quantum coding theory.

CRYSTAL STRUCTURE AND LUMINESCENT PROPERTIES OF Sr2B2O5:Eu2+ BLUE PHOSPHORS

2013 ◽  
Vol 06 (03) ◽  
pp. 1350034 ◽  
Author(s):  
QUAN-SHENG LIU ◽  
ZHAO-YANG ZHENG ◽  
LI-QUN CHENG ◽  
XI-YAN ZHANG ◽  
YUE SONG ◽  
...  
Keyword(s):  
Crystal Structure ◽  
Energy Level ◽  
Ground State ◽  
Luminescence Band ◽  
Energy State ◽  
Luminescent Properties ◽  
Low Energy ◽  
Monoclinic System ◽  
Broad Luminescence Band ◽  
Ion Doping

The Sr2B2O5:Eu2+ phosphor was synthesized by high-temperature solid state reaction. The crystal structure of Sr2B2O5:Eu2+ phosphor is monoclinic system with space group of P21/a(14). Due to Eu2+ ion doping, the lattice parameters and unit cell volume decreased. The luminescence spectrum is a broad luminescence band peaking at 468 nm and 478 nm, which corresponds to the transition of 5d-low energy state in Eu2 and Eu1 to the ground state. The excitation spectrum is at UV region less than 264 nm corresponding to the band gap transition of borate host. The low energy level of 5d-band locates 20921 cm-1, the band width is 1 eV. The electronegativity of O2- anions around Eu1 and Eu2 is 3.147 eV and 3.326 eV respectively, which determines the split of 5d energy level.

EFFECTIVE ACTION OF ANTIFERROMAGNETISM IN TWO DIMENSIONS: PARITY VIOLATING GROUND STATE AND HALL EFFECT

1989 ◽  
Vol 03 (18) ◽  
pp. 1383-1392 ◽  
Author(s):  
D. V. KHVESHCHENKO ◽  
P. B. WIEGMANN
Keyword(s):  
Hall Effect ◽  
Ground State ◽  
Effective Action ◽  
Ground States ◽  
Two Dimensions ◽  
Fractional Spin ◽  
Large N ◽  
Chern Simons ◽  
Spin And Statistics

Antiferromagnetic ground states with abnormal parity characterized by "magnetic" translations are discussed. Their excitations are described by a nonlinear σ-model with the Chern-Simons term with integer ϑ and obey fractional spin and statistics with even denominators. The effective action is derived on the basis of a large N-approximation. The large N reduces the problem to the Hall effect and realizes the parity anomaly.

scholarly journals Locally accurate MPS approximations for ground states of one-dimensional gapped local Hamiltonians

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 187 ◽  
Author(s):  
Alexander M. Dalzell ◽  
Fernando G. S. L. Brandão
Keyword(s):  
Ground State ◽  
State Energy ◽  
Nearest Neighbor ◽  
Ground States ◽  
Quantum Circuit ◽  
Local Approximation ◽  
Matrix Product ◽  
One Dimensional ◽  
Local Approximations ◽  
Reduced Density

A key feature of ground states of gapped local 1D Hamiltonians is their relatively low entanglement --- they are well approximated by matrix product states (MPS) with bond dimension scaling polynomially in the length N of the chain, while general states require a bond dimension scaling exponentially. We show that the bond dimension of these MPS approximations can be improved to a constant, independent of the chain length, if we relax our notion of approximation to be more local: for all length-k segments of the chain, the reduced density matrices of our approximations are ϵ-close to those of the exact state. If the state is a ground state of a gapped local Hamiltonian, the bond dimension of the approximation scales like (k/ϵ)1+o(1), and at the expense of worse but still poly(k,1/ϵ) scaling of the bond dimension, we give an alternate construction with the additional features that it can be generated by a constant-depth quantum circuit with nearest-neighbor gates, and that it applies generally for any state with exponentially decaying correlations. For a completely general state, we give an approximation with bond dimension exp⁡(O(k/ϵ)), which is exponentially worse, but still independent of N. Then, we consider the prospect of designing an algorithm to find a local approximation for ground states of gapped local 1D Hamiltonians. When the Hamiltonian is translationally invariant, we show that the ability to find O(1)-accurate local approximations to the ground state in T(N) time implies the ability to estimate the ground state energy to O(1) precision in O(T(N)log⁡(N)) time.

scholarly journals Quantum Systems on Non-$k$-Hyperfinite Complexes: a generalization of classical statistical mechanics on expander graphs

2014 ◽  
Vol 14 (1&2) ◽  
pp. 144-180
Author(s):  
Michael H. Freedman ◽  
Matthew B. Hastings
Keyword(s):  
Coding Theory ◽  
Energy State ◽  
Local System ◽  
Quantum Systems ◽  
Low Energy ◽  
Topological Order ◽  
Expander Graphs ◽  
Cell Complexes ◽  
Classical Statistical Mechanics ◽  
Quantum Coding

We construct families of cell complexes that generalize expander graphs. These families are called non-$k$-hyperfinite, generalizing the idea of a non-hyperfinite (NH) family of graphs. Roughly speaking, such a complex has the property that one cannot remove a small fraction of points and be left with an object that looks $k-1$-dimensional at large scales. We then consider certain quantum systems on these complexes. A future goal is to construct a family of Hamiltonians such that every low energy state has topological order as part of an attempt to prove the quantum PCP conjecture. This goal is approached by constructing a toric code Hamiltonian with the property that every low energy state without vertex defects has topological order, a property that would not hold for any local system in any lattice $Z^d$ or indeed on any $1$-hyperfinite complex. Further, such NH complexes find application in quantum coding theory. The hypergraph product codes\cite{hpc} of Tillich and Z\'{e}mor are generalized using NH complexes.

scholarly journals Scaling of variational quantum circuit depth for condensed matter systems

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 272 ◽  
Author(s):  
Carlos Bravo-Prieto ◽  
Josep Lumbreras-Zarapico ◽  
Luca Tagliacozzo ◽  
José I. Latorre
Keyword(s):  
Ground State ◽  
Condensed Matter ◽  
Finite Size ◽  
Ground States ◽  
Finite Depth ◽  
Quantum Circuit ◽  
Critical Number ◽  
Conformally Invariant ◽  
Number Of Layers ◽  
Circuit Depth

We benchmark the accuracy of a variational quantum eigensolver based on a finite-depth quantum circuit encoding ground state of local Hamiltonians. We show that in gapped phases, the accuracy improves exponentially with the depth of the circuit. When trying to encode the ground state of conformally invariant Hamiltonians, we observe two regimes. A finite-depth regime, where the accuracy improves slowly with the number of layers, and a finite-size regime where it improves again exponentially. The cross-over between the two regimes happens at a critical number of layers whose value increases linearly with the size of the system. We discuss the implication of these observations in the context of comparing different variational ansatz and their effectiveness in describing critical ground states.

"MUTUAL FRACTIONAL STATISTICS" AND "RELATIVE ANYONS"

1991 ◽  
Vol 05 (10) ◽  
pp. 1771-1778 ◽  
Author(s):  
Chia-Ren Hu
Keyword(s):  
Quantum Hall ◽  
Ground States ◽  
Two Dimensions ◽  
Statistical Properties ◽  
Phase Factor ◽  
Fractional Statistics ◽  
Fractional Quantum ◽  
Fractional Quantum Hall ◽  
General Properties ◽  
Topological Argument

A topological argument similar to that of Leinaas and Myrheim implies that a non-trivial statistical phase factor can also arise from exchanging twice a pair of distinguishable particles in two dimensions. Some general properties of this phase factor are deduced. Wilczek's model for anyons and the Laughlin theory for the quasiparticles in the fractional quantum Hall ground states are examined in light of these properties, and the former is generalized for systems containing many species of anyons. The statistical properties of holons and spinons relative to each other are briefly discussed as an example.

GROUND STATE CHARGE OF FERMION SOLITON SYSTEM IN 1+1 AND 3+1 DIMENSIONS

1993 ◽  
Vol 08 (04) ◽  
pp. 705-721
Author(s):  
M. RAVENDRANADHAN ◽  
M. SABIR
Keyword(s):  
Ground State ◽  
Energy State ◽  
Flow Analysis ◽  
Background Field ◽  
Bound State ◽  
Fermion Mass ◽  
Spectral Flow ◽  
Bound State Spectrum ◽  
Spectral Asymmetry ◽  
State Charge

Ground state charge of some fermion soliton system without C-invariance is calculated in 1+1 and 3+1 dimensions by a combination of adiabatic method and spectral flow analysis. Induced charge is calculated by evolving adiabatically the fields from a vacuum having a background field which has a zero energy state and spectral symmetry. The spectral flow is calculated by an analysis of the bound state spectrum. In 1+1 dimension our calculations are in agreement with the results already found in the literature. In 3+1 dimension we study the interaction of fermions with monopoles and dyons. In the case of monopoles, even though there is spectral asymmetry, ground state charge is found to be ±1/2. It is shown that ground state charge gets contribution only from the lowest angular momentum states and is discontinuous at the fermion mass.

scholarly journals On the three-dimensional spatial correlations of curved dislocation systems

2021 ◽  
Vol 5 (1) ◽  
Author(s):  
Joseph Pierre Anderson ◽  
Anter El-Azab
Keyword(s):  
Correlation Function ◽  
Slip System ◽  
Correlation Functions ◽  
Mean Field ◽  
Coarse Graining ◽  
Dislocation Dynamics ◽  
Coarse Grained ◽  
Spatial Correlation Function ◽  
Line System ◽  
Dislocation Densities

AbstractCoarse-grained descriptions of dislocation motion in crystalline metals inherently represent a loss of information regarding dislocation-dislocation interactions. In the present work, we consider a coarse-graining framework capable of re-capturing these interactions by means of the dislocation-dislocation correlation functions. The framework depends on a convolution length to define slip-system-specific dislocation densities. Following a statistical definition of this coarse-graining process, we define a spatial correlation function which will allow the arrangement of the discrete line system at two points—and thus the strength of their interactions at short range—to be recaptured into a mean field description of dislocation dynamics. Through a statistical homogeneity argument, we present a method of evaluating this correlation function from discrete dislocation dynamics simulations. Finally, results of this evaluation are shown in the form of the correlation of dislocation densities on the same slip-system. These correlation functions are seen to depend weakly on plastic strain, and in turn, the dislocation density, but are seen to depend strongly on the convolution length. Implications of these correlation functions in regard to continuum dislocation dynamics as well as future directions of investigation are also discussed.

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