36 research outputs found
Computational complexity of the ground state energy density problem
We study the complexity of finding the ground state energy density of a local Hamiltonian on a lattice in the thermodynamic limit of infinite lattice size. We formulate this rigorously as a function problem, in which we request an estimate of the ground state energy density to some specified precision; and as an equivalent promise problem, GSED, in which we ask whether the ground state energy density is above or below specified thresholds. The ground state energy density problem is unusual, in that it concerns a single, fixed Hamiltonian in the thermodynamic limit, whose ground state energy density is just some fixed, real number. The only input to the computational problem is the precision to which to estimate this fixed real number, corresponding to the ground state energy density. Hardness of this problem for a complexity class therefore implies that the solutions to all problems in the class are encoded in this single number (analogous to Chaitin's constant in computability theory). This captures computationally the type of question most commonly encountered in condensed matter physics, which is typically concerned with the physical properties of a single Hamiltonian in the thermodynamic limit. We show that for classical, translationally invariant, nearest neighbour Hamiltonians on a 2D square lattice, PNEEXPâ EXPGSEDâ EXPNEXP, and for quantum Hamiltonians PNEEXPâ EXPGSEDâ EXPQMAEXP. With some technical caveats on the oracle definitions, the EXP in some of these results can be strengthened to PSPACE. We also give analogous complexity bounds for the function version of GSED
Uncomputability of phase diagrams
The phase diagram of a material is of central importance in describing the properties and behaviour of a condensed matter system. In this work, we prove that the task of determining the phase diagram of a many-body Hamiltonian is in general uncomputable, by explicitly constructing a continuous one-parameter family of Hamiltonians H(Ï), where Ïâ R, for which this is the case. The H(Ï) are translationally-invariant, with nearest-neighbour couplings on a 2D spin lattice. As well as implying uncomputablity of phase diagrams, our result also proves that undecidability can hold for a set of positive measure of a Hamiltonianâs parameter space, whereas previous results only implied undecidability on a zero measure set. This brings the spectral gap undecidability results a step closer to standard condensed matter problems, where one typically studies phase diagrams of many-body models as a function of one or more continuously varying real parameters, such as magnetic field strength or pressure
Uncomputably complex renormalisation group flows
Renormalisation group methods are among the most important techniques for analysing the physics of many-body systems: by iterating a renormalisation group map, which coarse-grains the description of a system and generates a flow in the parameter space, physical properties of interest can be extracted. However, recent work has shown that important physical features, such as the spectral gap and phase diagram, may be impossible to determine, even in principle. Following these insights, we construct a rigorous renormalisation group map for the original undecidable many-body system that appeared in the literature, which reveals a renormalisation group flow so complex that it cannot be predicted. We prove that each step of this map is computable, and that it converges to the correct fixed points, yet the resulting flow is uncomputable. This extreme form of unpredictability for renormalisation group flows had not been shown before and goes beyond the chaotic behaviour seen previously
The Un(solv)able Problem
After a years-long intellectual journey, three mathematicians have discovered that a problem of central importance in physics is impossible to solveâand that means other big questions may be undecidable, too.
In Brief:
Kurt Gödel famously discovered in the 1930s that some statements are impossible to prove true or falseâthey will always be âundecidable.â
Mathematicians recently set out to discover whether a certain fundamental problem in quantum physicsâthe so-called spectral gap questionâfalls into this category. The spectral gap refers to the energy difference between the lowest energy state a material can occupy and the next state up.
After three years of blackboard brainstorming, midnight calculating and much theorizing over coffee, the mathematicians produced a 146-page proof that the spectral gap problem is, in fact, undecidable. The result raises the possibility that other important questions may likewise be unanswerable
Undecidability of the Spectral Gap in One Dimension
The spectral gap problemâdetermining whether the energy spectrum of a system has an energy gap above ground state, or if there is a continuous range of low-energy excitationsâpervades quantum many-body physics. Recently, this important problem was shown to be undecidable for quantum-spin systems in two (or more) spatial dimensions: There exists no algorithm that determines in general whether a system is gapped or gapless, a result which has many unexpected consequences for the physics of such systems. However, there are many indications that one-dimensional spin systems are simpler than their higher-dimensional counterparts: For example, they cannot have thermal phase transitions or topological order, and there exist highly effective numerical algorithms such as the density matrix renormalization groupâand even provably polynomial-time onesâfor gapped 1D systems, exploiting the fact that such systems obey an entropy area law. Furthermore, the spectral gap undecidability construction crucially relied on aperiodic tilings, which are not possible in 1D. So does the spectral gap problem become decidable in 1D? In this paper, we prove this is not the case by constructing a family of 1D spin chains with translationally invariant nearest-neighbor interactions for which no algorithm can determine the presence of a spectral gap. This not only proves that the spectral gap of 1D systems is just as intractable as in higher dimensions, but it also predicts the existence of qualitatively new types of complex physics in 1D spin chains. In particular, it implies there are 1D systems with a constant spectral gap and nondegenerate classical ground state for all systems sizes up to an uncomputably large size, whereupon they switch to a gapless behavior with dense spectrum
Size-driven quantum phase transitions
Can the properties of the thermodynamic limit of a many-body quantum system be extrapolated by analyzing a sequence of finite-size cases? We present models for which such an approach gives completely misleading results: translationally invariant, local Hamiltonians on a square lattice with open boundary conditions and constant spectral gap, which have a classical product ground state for all system sizes smaller than a particular threshold size, but a ground state with topological degeneracy for all system sizes larger than this threshold. Starting from a minimal case with spins of dimension 6 and threshold lattice size 15Ă1515Ă15, we show that the latter grows faster than any computable function with increasing local spin dimension. The resulting effect may be viewed as a unique type of quantum phase transition that is driven by the size of the system rather than by an external field or coupling strength. We prove that the construction is thermally robust, showing that these effects are in principle accessible to experimental observation
Detection of multipartite entanglement with two-body correlations
We show how to detect entanglement with criteria built from simple two-body
correlation terms. Since many natural Hamiltonians are sums of such correlation
terms, our ideas can be used to detect entanglement by energy measurement. Our
criteria can straightforwardly be applied for detecting different forms of
multipartite entanglement in familiar spin models in thermal equilibrium.Comment: 5 pages including 2 figures, LaTeX; for the proceedings of the DPG
spring meeting, Berlin, March 200
Simulation Methodology for Electron Transfer in CMOS Quantum Dots
The construction of quantum computer simulators requires advanced software
which can capture the most significant characteristics of the quantum behavior
and quantum states of qubits in such systems. Additionally, one needs to
provide valid models for the description of the interface between classical
circuitry and quantum core hardware. In this study, we model electron transport
in semiconductor qubits based on an advanced CMOS technology. Starting from 3D
simulations, we demonstrate an order reduction and the steps necessary to
obtain ordinary differential equations on probability amplitudes in a
multi-particle system. We compare numerical and semi-analytical techniques
concluding this paper by examining two case studies: the electron transfer
through multiple quantum dots and the construction of a Hadamard gate simulated
using a numerical method to solve the time-dependent Schrodinger equation and
the tight-binding formalism for a time-dependent Hamiltonian
Entanglement distribution and quantum discord
Establishing entanglement between distant parties is one of the most
important problems of quantum technology, since long-distance entanglement is
an essential part of such fundamental tasks as quantum cryptography or quantum
teleportation. In this lecture we review basic properties of entanglement and
quantum discord, and discuss recent results on entanglement distribution and
the role of quantum discord therein. We also review entanglement distribution
with separable states, and discuss important problems which still remain open.
One such open problem is a possible advantage of indirect entanglement
distribution, when compared to direct distribution protocols.Comment: 7 pages, 2 figures, contribution to "Lectures on general quantum
correlations and their applications", edited by Felipe Fanchini, Diogo
Soares-Pinto, and Gerardo Adess
Dissipative and Non-dissipative Single-Qubit Channels: Dynamics and Geometry
Single-qubit channels are studied under two broad classes: amplitude damping
channels and generalized depolarizing channels. A canonical derivation of the
Kraus representation of the former, via the Choi isomorphism is presented for
the general case of a system's interaction with a squeezed thermal bath. This
isomorphism is also used to characterize the difference in the geometry and
rank of these channel classes. Under the isomorphism, the degree of decoherence
is quantified according to the mixedness or separability of the Choi matrix.
Whereas the latter channels form a 3-simplex, the former channels do not form a
convex set as seen from an ab initio perspective. Further, where the rank of
generalized depolarizing channels can be any positive integer upto 4, that of
amplitude damping ones is either 2 or 4. Various channel performance parameters
are used to bring out the different influences of temperature and squeezing in
dissipative channels. In particular, a noise range is identified where the
distinguishability of states improves inspite of increasing decoherence due to
environmental squeezing.Comment: 12 pages, 4 figure