arXiv subjects

M. B. Hastings

Publications and source records attributed to M. B. Hastings.

At least 55 records · Page 3Linked to original sources

arXiv2013

Quantum Systems on Non-$k$-Hyperfinite Complexes: A Generalization of Classical Statistical Mechanics on Expander Graphs

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[1] of Tillich and Zémor are generalized using NH complexes.

quant-ph↗

arXiv2013

Obstructions To Classically Simulating The Quantum Adiabatic Algorithm

We consider the adiabatic quantum algorithm for systems with "no sign problem", such as the transverse field Ising mode, and analyze the equilibration time for quantum Monte Carlo (QMC) on these systems. We ask: if the spectral gap is only inverse polynomially small, will equilibration methods based on slowly changing the Hamiltonian parameters in the QMC simulation succeed in a polynomial time? We show that this is not true, by constructing counter-examples. Some examples are Hamiltonians where the space of configurations where the wavefunction has non-negligible amplitude has a nontrivial fundamental group, causing the space of trajectories in imaginary time to break into disconnected components, with only negligible probability outside these components. For the simplest example we give with an abelian fundamental group, QMC does not equilibrate but still solves the optimization problem. More severe effects leading to failure to solve the optimization can occur when the fundamental group is a free group on two generators. Other examples where QMC fails have a trivial fundamental group, but still use ideas from topology relating group presentations to simplicial complexes. We define gadgets to realize these Hamiltonians as the effective low-energy dynamics of a transverse field Ising model. We present some analytic results on equilibration times which may be of some independent interest in the theory of equilibration of Markov chains. Conversely, we show that a small spectral gap implies slow equilibration at low temperature for some initial conditions and for a natural choice of local QMC updates.

quant-ph↗

arXiv2012

Trivial Low Energy States for Commuting Hamiltonians, and the Quantum PCP Conjecture

We consider whether or not Hamiltonians which are sums of commuting projectors have "trivial" ground states which can be constructed by a local quantum circuit of bounded depth and range acting on a product state. While the toric code only has nontrivial ground states, commuting projector Hamiltonians which are sums of two-body interactions have trivial ground states. We define an "interaction complex" for a Hamiltonian, generalizing the interaction graph, and we show that if this 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 Hamiltonian (this condition holds for all stabilizer Hamiltonians, and we also prove the result for all Hamiltonians under an assumption on the 1-complex). While this includes cases considered by Ref., it also includes other Hamiltonians whose interaction complexes cannot be coarse-grained into the case of Ref. One motivation for this is the quantum PCP conjecture. 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, a trivial ground state for the Hamiltonian with those sites removed is a low energy trivial state for the original Hamiltonian. Such states can act as a classical witness to the existence of a low energy state. While this result applies only to commuting Hamiltonians, it suggests that to prove a quantum PCP conjecture one should consider interaction complexes which cannot be mapped to 1-complexes after removing a small fraction of cells. We define this more precisely below; in a sense this generalizes the idea of an expander graph. Surprisingly, such complexes do exist as will be shown elsewhere, and have useful properties in quantum coding theory.

quant-ph↗

arXiv2012

Solving Gapped Hamiltonians Locally

We show that any short-range Hamiltonian with a gap between the ground and excited states can be written as a sum of local operators, such that the ground state is an approximate eigenvector of each operator separately. We then show that the ground state of any such Hamiltonian is close to a generalized matrix product state. The range of the given operators needed to obtain a good approximation to the ground state is proportional to the square of the logarithm of the system size times a characteristic "factorization length". Applications to many-body quantum simulation are discussed. We also consider density matrices of systems at non-zero temperature.

cond-mat.str-el↗

arXiv2011

Topological Order at Non-zero Temperature

We propose a definition for topological order at nonzero temperature in analogy to the usual zero temperature definition that a state is topologically ordered, or "nontrivial", if it cannot be transformed into a product state (or a state close to a product state) using a local (or approximately local) quantum circuit. We prove that any two dimensional Hamiltonian which is a sum of commuting local terms is not topologically ordered at $T>0$. We show that such trivial states cannot be used to store quantum information using certain stringlike operators. This definition is not too restrictive, however, as the four dimensional toric code does have a nontrivial phase at nonzero temperature.

quant-ph↗

arXiv2011

Making Almost Commuting Matrices Commute

Suppose two Hermitian matrices $A,B$ almost commute ($\Vert [A,B] \Vert \leq δ$). Are they close to a commuting pair of Hermitian matrices, $A',B'$, with $\Vert A-A' \Vert,\Vert B-B'\Vert \leq ε$? A theorem of H. Lin shows that this is uniformly true, in that for every $ε>0$ there exists a $δ>0$, independent of the size $N$ of the matrices, for which almost commuting implies being close to a commuting pair. However, this theorem does not specify how $δ$ depends on $ε$. We give uniform bounds relating $δ$ and $ε$. We provide tighter bounds in the case of block tridiagonal and tridiagonal matrices and a fully constructive method in that case. Within the context of quantum measurement, this implies an algorithm to construct a basis in which we can make a {\it projective} measurement that approximately measures two approximately commuting operators simultaneously. Finally, we comment briefly on the case of approximately measuring three or more approximately commuting operators using POVMs (positive operator-valued measures) instead of projective measurements.

quant-ph↗

arXiv2010

Disordered Topological Insulators via $C^*$-Algebras

The theory of almost commuting matrices can be used to quantify topological obstructions to the existence of localized Wannier functions with time-reversal symmetry in systems with time-reversal symmetry and strong spin-orbit coupling. We present a numerical procedure that calculates a Z_2 invariant using these techniques, and apply it to a model of HgTe. This numerical procedure allows us to access sizes significantly larger than procedures based on studying twisted boundary conditions. Our numerical results indicate the existence of a metallic phase in the presence of scattering between up and down spin components, while there is a sharp transition when the system decouples into two copies of the quantum Hall effect. In addition to the Z_2 invariant calculation in the case when up and down components are coupled, we also present a simple method of evaluating the integer invariant in the quantum Hall case where they are decoupled.

cond-mat.mes-hall↗

arXiv2010

Topological Insulators and C^*-Algebras: Theory and Numerical Practice

We apply ideas from $C^*$-algebra to the study of disordered topological insulators. We extract certain almost commuting matrices from the free Fermi Hamiltonian, describing band projected coordinate matrices. By considering topological obstructions to approximating these matrices by exactly commuting matrices, we are able to compute invariants quantifying different topological phases. We generalize previous two dimensional results to higher dimensions; we give a general expression for the topological invariants for arbitrary dimension and several symmetry classes, including chiral symmetry classes, and we present a detailed $K$-theory treatment of this expression for time reversal invariant three dimensional systems. We can use these results to show non-existence of localized Wannier functions for these systems. We use this approach to calculate the index for time-reversal invariant systems with spin-orbit scattering in three dimensions, on sizes up to $12^3$, averaging over a large number of samples. The results show an interesting separation between the localization transition and the point at which the average index (which can be viewed as an "order parameter" for the topological insulator) begins to fluctuate from sample too sample, implying the existence of an unsuspected quantum phase transition separating two different delocalized phases in this system. One of the particular advantages of the $C^*$-algebraic technique that we present is that it is significantly faster in practice than other methods of computing the index, allowing the study of larger systems. In this paper, we present a detailed discussion of numerical implementation of our method.

cond-mat.mes-hall↗

arXiv2010

Locality in Quantum Systems

These lecture notes focus on the application of ideas of locality, in particular Lieb-Robinson bounds, to quantum many-body systems. We consider applications including correlation decay, topological order, a higher dimensional Lieb-Schultz-Mattis theorem, and a nonrelativistic Goldstone theorem. The emphasis is on trying to show the ideas behind the calculations. As a result, the proofs are only sketched with an emphasis on the intuitive ideas behind them, and in some cases we use techniques that give very slightly weaker bounds for simplicity. This is a preliminary version of the lecture notes, with the goal of getting the notes out close to the end of the school. Comments welcome.

math-ph↗

arXiv2010

Entanglement vs. gap for one-dimensional spin systems

We study the relationship between entanglement and spectral gap for local Hamiltonians in one dimension. The area law for a one-dimensional system states that for the ground state, the entanglement of any interval is upper-bounded by a constant independent of the size of the interval. However, the possible dependence of the upper bound on the spectral gap Delta is not known, as the best known general upper bound is asymptotically much larger than the largest possible entropy of any model system previously constructed for small Delta. To help resolve this asymptotic behavior, we construct a family of one-dimensional local systems for which some intervals have entanglement entropy which is polynomial in 1/Delta, whereas previously studied systems, such as free fermion systems or systems described by conformal field theory, had the entropy of all intervals bounded by a constant times log(1/Delta).

quant-ph↗

arXiv2010

Quasi-adiabatic Continuation for Disordered Systems: Applications to Correlations, Lieb-Schultz-Mattis, and Hall Conductance

We present a possible definition of a mobility gap for a many-body quantum system, in analogy to definitions of dynamical localization for single particle systems. Using this definition, we construct "corrected" quasi-adiabatic continuation operators. Under an appropriate definition of a unique ground state, we show how to introduce virtual fluxes. Armed with these results, we can directly carry over previous results in the case of a spectral gap. We present a proof of decay of correlation functions and we present a proof of Hall conductance quantization under very mild density-of-states assumptions defined later. We also generalize these definitions to the case of a "bulk mobility gap", in the case of a system with boundaries, and present a proof of Hall conductance quantization on an annulus under appropriate assumptions. Further, we present a new "optimized" quasi-adiabatic continuation operator which simplifies previous estimates and tightens bounds in certain cases. This is presented in an appendix which can be read independently of the rest of the paper as it also improves estimates in the case of systems with a spectral gap. This filter function used decays in time at least as fast as ${\cal O}(\exp(-t^α))$ for all $α<1$, a class of decay called subexponential (a tighter description of what is possible is below). Using this function it is possible to tighten recent estimates of the Hall conductance quantization for gapped systems\cite{hall} to a decay which is subexponential in system size.

math-ph↗

arXiv2010

A short proof of stability of topological order under local perturbations

Recently, the stability of certain topological phases of matter under weak perturbations was proven. Here, we present a short, alternate proof of the same result. We consider models of topological quantum order for which the unperturbed Hamiltonian $H_0$ can be written as a sum of local pairwise commuting projectors on a $D$-dimensional lattice. We consider a perturbed Hamiltonian $H=H_0+V$ involving a generic perturbation $V$ that can be written as a sum of short-range bounded-norm interactions. We prove that if the strength of $V$ is below a constant threshold value then $H$ has well-defined spectral bands originating from the low-lying eigenvalues of $H_0$. These bands are separated from the rest of the spectrum and from each other by a constant gap. The width of the band originating from the smallest eigenvalue of $H_0$ decays faster than any power of the lattice size.

math-ph↗

arXiv2009

Superadditivity of communication capacity using entangled inputs

The design of error-correcting codes used in modern communications relies on information theory to quantify the capacity of a noisy channel to send information [1]. This capacity can be expressed using the mutual information between input and output for a single use of the channel: although correlations between subsequent input bits are used to correct errors, they cannot increase the capacity. For quantum channels, it has been an open question whether entangled input states can increase the capacity to send classical information [2]. The additivity conjecture [3,4] states that entanglement does not help, making practical computations of the capacity possible. While additivity is widely believed to be true, there is no proof. Here we show that additivity is false, by constructing a random counter-example. Our results show that the most basic question of classical capacity of a quantum channel remains open, with further work needed to determine in which other situations entanglement can boost capacity.

quant-ph↗

arXiv2009

Almost Commuting Matrices, Localized Wannier Functions, and the Quantum Hall Effect

For models of non-interacting fermions moving within sites arranged on a surface in three dimensional space, there can be obstructions to finding localized Wannier functions. We show that such obstructions are $K$-theoretic obstructions to approximating almost commuting, complex-valued matrices by commuting matrices, and we demonstrate numerically the presence of this obstruction for a lattice model of the quantum Hall effect in a spherical geometry. The numerical calculation of the obstruction is straightforward, and does not require translational invariance or introducing a flux torus. We further show that there is a $Z_2$ index obstruction to approximating almost commuting self-dual matrices by exactly commuting self-dual matrices, and present additional conjectures regarding the approximation of almost commuting real and self-dual matrices by exactly commuting real and self-dual matrices. The motivation for considering this problem is the case of physical systems with additional antiunitary symmetries such as time reversal or particle-hole conjugation. Finally, in the case of the sphere--mathematically speaking three almost commuting Hermitians whose sum of square is near the identity--we give the first quantitative result showing this index is the only obstruction to finding commuting approximations. We review the known non-quantitative results for the torus.

math-ph↗

arXiv2009

Matrix Product States for dynamical simulation of infinite chains

We propose a new method for computing the ground state properties and the time evolution of infinite chains based on a transverse contraction of the tensor network. The method does not require finite size extrapolation and avoids explicit truncation of the bond dimension along the evolution. By folding the network in the time direction prior to contraction, time dependent expectation values and dynamic correlation functions can be computed after much longer evolution time than with any previous method. Moreover, the algorithm we propose can be used for the study of some non-invariant infinite chains, including impurity models.

quant-ph↗

arXiv2009

Light Cone Matrix Product

We show how to combine the light-cone and matrix product algorithms to simulate quantum systems far from equilibrium for long times. For the case of the XXZ spin chain at $Δ=0.5$, we simulate to a time of $\approx 22.5$. While part of the long simulation time is due to the use of the light-cone method, we also describe a modification of the iTEBD algorithm with improved numerical stability, and we describe how to incorporate symmetry into this algorithm. While statistical sampling error means that we are not yet able to make a definite statement, the behavior of the simulation at long times indicates the appearance of either "revivals" in the order parameter as predicted previously or of a distinct shoulder in the decay of the order parameter.

quant-ph↗

arXiv2009

Quantum Adiabatic Computation With a Constant Gap is Not Useful in One Dimension

We show that it is possible to use a classical computer to efficiently simulate the adiabatic evolution of a quantum system in one dimension with a constant spectral gap, starting the adiabatic evolution from a known initial product state. The proof relies on a recently proven area law for such systems, implying the existence of a good matrix product representation of the ground state, combined with an appropriate algorithm to update the matrix product state as the Hamiltonian is changed. This implies that adiabatic evolution with such Hamiltonians is not useful for universal quantum computation. Therefore, adiabatic algorithms which are useful for universal quantum computation either require a spectral gap tending to zero or need to be implemented in more than one dimension (we leave open the question of the computational power of adiabatic simulation with a constant gap in more than one dimension).

quant-ph↗

arXiv2008

Classical and Quantum Tensor Product Expanders

We introduce the concept of quantum tensor product expanders. These are expanders that act on several copies of a given system, where the Kraus operators are tensor products of the Kraus operator on a single system. We begin with the classical case, and show that a classical two-copy expander can be used to produce a quantum expander. We then discuss the quantum case and give applications to the Solovay-Kitaev problem. We give probabilistic constructions in both classical and quantum cases, giving tight bounds on the expectation value of the largest nontrivial eigenvalue in the quantum case.

quant-ph↗