116 research outputs found
Functional completeness of planar Rydberg blockade structures
The construction of Hilbert spaces that are characterized by local
constraints as the low-energy sectors of microscopic models is an important
step towards the realization of a wide range of quantum phases with long-range
entanglement and emergent gauge fields. Here we show that planar structures of
trapped atoms in the Rydberg blockade regime are functionally complete: Their
ground state manifold can realize any Hilbert space that can be characterized
by local constraints in the product basis. We introduce a versatile framework,
together with a set of provably minimal logic primitives as building blocks, to
implement these constraints. As examples, we present lattice realizations of
the string-net Hilbert spaces that underlie the surface code and the Fibonacci
anyon model. We discuss possible optimizations of planar Rydberg structures to
increase their geometrical robustness.Comment: 33 pages, 14 figures, v2: fixed typos, added additional references
and comment
Non-ergodic phenomena in many-body quantum systems
The assumption of ergodicity is the cornerstone of conventional thermodynamics, connecting the equilibrium properties of macroscopic systems to the chaotic nature of the underlying microscopic dynamics, which eventuates in thermalization and the scrambling of information contained in any generic initial condition. The modern understanding of ergodicity in a quantum mechanical framework is encapsulated in the so-called eigenstate thermalization hypothesis, which asserts that thermalization of an isolated quantum system is a manifestation of the random-like character of individual eigenstates in the bulk of the spectrum of the system's Hamiltonian.
In this work, we consider two major exceptions to the rule of generic thermalization in interacting many-body quantum systems: many-body localization, and quantum spin glasses. In the first part, we debate the possibility of localization in a system endowed with a non-Abelian symmetry. We show that, in line with proposed theoretical arguments, such a system is probably delocalized in the thermodynamic limit, but the ergodization length scale is anomalously large, explaining the non-ergodic behavior observed in previous experimental and numerical works. A crucial feature of this system is the quasi-tensor-network nature of its eigenstates, which is dictated by the presence of nontrivial symmetry multiplets. As a consequence, ergodicity may only be restored by extensively large cascades of resonating spins, explaining the system's resistance to delocalization. In the second part, we study the effects of non-ergodic behavior in glassy systems in relation to the possibility of speeding up classical algorithms via quantum resources, namely tunneling across tall free energy barriers. First, we define a pseudo-tunneling event in classical diffusion Monte Carlo (DMC) and characterize the corresponding tunneling rate. Our findings suggest that DMC is very efficient at tunneling in stoquastic problems even in the presence of frustrated couplings, asymptotically outperforming incoherent quantum tunneling. We also analyze in detail the impact of importance sampling, finding that it does not alter the scaling. Next, we study the so-called population transfer (PT) algorithm applied to the problem of energy matching in combinatorial problems. After summarizing some known results on a simpler model, we take the quantum random energy model as a testbed for a thorough, model-agnostic numerical characterization of the algorithm, including parameter setting and quality assessment. From the accessible system sizes, we observe no meaningful asymptotic speedup, but argue in favor of a better performance in more realistic energy landscapes
Differential KO-theory: constructions, computations, and applications
We provide a systematic and detailed treatment of differential refinements of
KO-theory. We explain how various flavors capture geometric aspects in
different but related ways, highlighting the utility of each. While general
axiomatics exist, no explicit constructions seem to have appeared before. This
fills a gap in the literature in which K-theory is usually worked out leaving
KO-theory essentially untouched, with only scattered partial information in
print. We compare to the complex case, highlighting which constructions follow
analogously and which are much more subtle. We construct a pushforward and
differential refinements of genera, leading to a Riemann-Roch theorem for
-theory. We also construct the corresponding
Atiyah-Hirzebruch spectral sequence (AHSS) and explicitly identify the
differentials, including ones which mix geometric and topological data. This
allows us to completely characterize the image of the Pontrjagin character.
Then we illustrate with examples and applications, including higher tangential
structures, Adams operations, and a differential Wu formula.Comment: 105 pages, very minor changes, comments welcom
Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end
We present a quantum algorithm that has rigorous runtime guarantees for
several families of binary optimization problems, including Quadratic
Unconstrained Binary Optimization (QUBO), Ising spin glasses (-spin model),
and -local constraint satisfaction problems (-CSP). We show that either
(a) the algorithm finds the optimal solution in time for an
-independent constant , a advantage over Grover's algorithm; or
(b) there are sufficiently many low-cost solutions such that classical random
guessing produces a approximation to the optimal cost value in
sub-exponential time for arbitrarily small choice of . Additionally, we
show that for a large fraction of random instances from the -spin model and
for any fully satisfiable or slightly frustrated -CSP formula, statement (a)
is the case. The algorithm and its analysis is largely inspired by Hastings'
short-path algorithm [ (2018) 78].Comment: 49 pages, 3 figure
- …