Benedikt Placke

Abstract:

We prove an analog of the “bottleneck theorem,” well-known for classical Markov chains, for Markovian quantum channels. In particular, we show that if two regions (subspaces) of Hilbert space are separated by a region that has very low weight in the channel’s steady state, then states initialized on one side of this barrier will take a long time to relax, putting a lower bound on the mixing time in terms of an appropriately defined “quantum bottleneck ratio.” Importantly, this bottleneck ratio involves not only the probabilities of the relevant subspaces, but also the size of off-diagonal matrix elements between them. For low temperature quantum many-body systems, we use the bottleneck theorem to bound the performance of any quasilocal Gibbs sampler. This leads to a new perspective on thermally stable quantum phases in terms of a decomposition of the Gibbs state into multiple components separated by bottlenecks. As a concrete application, we show rigorously that weakly perturbed commuting projector models with extensive energy barriers (which include certain classical and quantum expander codes) have exponentially large mixing times.

Abstract:

<jats:p> Preparing the thermal density matrix <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:msub> <a:mi>ρ</a:mi> <a:mi>β</a:mi> </a:msub> <a:mo>∝</a:mo> <a:msup> <a:mi>e</a:mi> <a:mrow> <a:mo>−</a:mo> <a:mi>β</a:mi> <a:mi>H</a:mi> </a:mrow> </a:msup> </a:mrow> </a:math> corresponding to a given Hamiltonian <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>H</b:mi> </b:math> is a task of central interest across quantum many-body physics, and is particularly salient when attempting to study it with quantum computers. Although solved in principle by recent constructions of efficiently simulable Lindblad master equations—that provably have <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:msub> <c:mi>ρ</c:mi> <c:mi>β</c:mi> </c:msub> </c:math> as a steady state [C.-F. Chen , ]—the implementation of these “exact Gibbs samplers” requires large-scale quantum computing resources and is hence challenging in practice on current or even near-term quantum devices. Here, we propose a scheme for approximately simulating an exact Gibbs sampler up to a rigorously bounded error that only requires the (repeated) implementation of three readily available ingredients: (a) analog simulation of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mi>H</d:mi> </d:math> ; (b) strictly local but time-dependent couplings to ancilla qubits; and (c) reset of the ancillas. We give rigorous guarantees on the difference between the fixed point reached by our protocol and the exact thermal state, which only depend on parameters of the protocol and its . The procedure is efficiently implementable on near-term devices if <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mi>H</e:mi> </e:math> is local and the mixing time scales mildly with both system size and protocol parameters. While guaranteeing the latter for Hamiltonians of interest remains an important problem for future work, here we lay the groundwork for developing fully efficient thermal state preparation protocols on quantum simulators. </jats:p>

Abstract:

We devise tractable models of unitary quantum many-body dynamics on tree graphs, as a first step toward a deeper understanding of dynamics in non-Euclidean spaces. To this end, we first demonstrate how to construct strictly local quantum circuits that preserve the symmetries of trees, such that their dynamical light cones grow isotropically. For trees with coordination number 𝑧, such circuits can be built from 𝑧-site gates. We then introduce a family of gates for which the dynamics are exactly solvable; these satisfy a set of constraints that we term “tree-unitarity.” Notably, tree-unitarity reduces to the previously established notion of dual-unitarity for 𝑧 =2, when the tree reduces to a line. Among the unexpected features of tree-unitarity is a trade-off between “maximum butterfly velocity” dynamics of out-of-time-order correlators and the existence of nonvanishing correlation functions in multiple directions, a tension absent in one-dimensional dual-unitary models and their Euclidean generalizations. We connect the existence of (a wide class of) solvable dynamics with nonmaximal butterfly velocity directly to a property of the underlying circuit geometry called 𝛿-hyperbolicity, and argue that such dynamics can only arise in non-Euclidean geometries. We give various examples of tree-unitary gates, discuss dynamical correlations, out-of-time-order correlators, and entanglement growth, and show that the kicked Ising model on a tree is a physically motivated example of maximum-velocity tree-unitary dynamics.

Abstract:

Magnetic heat capacity measurements of a high-quality single crystal of the dipole-octupole pyrochlore Ce2Hf2O7 down to a temperature of T=0.02 K are reported. These show a two-peaked structure, with a Schottky-like peak at T1∼0.065 K, similar to what is observed in its sister Ce pyrochlores Ce2Zr2O7 and Ce2Sn2O7. However, a second sharper peak is observed at T2∼0.025 K, signifying the entrance to the ground state. The ground state appears to have gapped excitations, as even the most abrupt extrapolation to CP=0 at T=0 K fully accounts for the Rln(2) entropy associated with the pseudospin-1/2 doublet for Ce3+ in this environment. The ground state could be conventionally ordered, although theory predicts a much larger anomaly in CP at much higher temperatures than the measured T2 for expectations from an all-in, all-out ground state of the XYZ Hamiltonian for Ce2Hf2O7. The sharp low-temperature peak could also signify a crossover from a classical spin liquid to a quantum spin liquid (QSL). For both scenarios, comparison of the measured CP with NLC calculations suggests that weak interactions beyond the nearest-neighbor XYZ Hamiltonian become relevant below T∼0.25 K. The diffuse magnetic neutron scattering observed from Ce2Hf2O7 at low temperatures between T2 and T1 resembles that observed from Ce2Zr2O7, which is well established as a π-flux quantum spin ice (QSI). Together with the peak in the heat capacity at T2, this diffuse scattering from Ce2Hf2O7 is suggestive of a classical spin liquid regime above T2 that is distinct from the zero-entropy quantum ground state below T2.