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:

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.

Abstract:

We study the non-unitary dynamics of a class of quantum circuits based on stochastically measuring check operators of subsystem quantum error-correcting codes, such as the Bacon-Shor code and its various generalizations. Our focus is on how properties of the underlying code are imprinted onto the measurement-only dynamics. We find that in a large class of codes with nonlocal stabilizer generators, at late times there is generically a nonlocal contribution to the subsystem entanglement entropy which scales with the subsystem size. The nonlocal stabilizer generators can also induce slow dynamics, since depending on the rate of competing measurements the associated degrees of freedom can take exponentially long (in system size) to purify (disentangle from the environment when starting from a mixed state) and to scramble (become entangled with the rest of the system when starting from a product state). Concretely, we consider circuits for which the nonlocal stabilizer generators of the underlying subsystem code take the form of subsystem symmetries. We present a systematic study of the phase diagrams and relevant time scales in two and three spatial dimensions for both Calderbank-Shor-Steane (CSS) and non-CSS codes, focusing in particular on the link between slow measurement-only dynamics and the geometry of the subsystem symmetry. A key finding of our work is that slowly purifying or scrambling degrees of freedom appear to emerge only in codes whose subsystem symmetries are nonlocally generated, a strict subset of those whose symmetries are simply nonlocal. We comment on the link between our results on subsystem codes and the phenomenon of Hilbert-space fragmentation in light of their shared algebraic structure.

Abstract:

The insulating magnetic pyrochlore ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ has gained attention as a quantum spin-ice candidate with dipole-octupole character that arises from the crystal-electric-field ground-state doublet for the ${\mathrm{Ce}}^{3+}$ Kramers ion. This dipole-octupole character permits both spin-ice phases based on magnetic dipoles and those based on more-exotic octupoles. This work reports low-temperature neutron diffraction measurements on single-crystal ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ with $Q$ coverage both at low $Q$ , where the magnetic form factor for dipoles is near maximal, and at high $Q$ , covering the region where the magnetic form factor for ${\mathrm{Ce}}^{3+}$ octupoles is near maximal. This study was motivated by recent powder neutron diffraction studies of other Ce-based dipole-octupole pyrochlores, ${\mathrm{Ce}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ and ${\mathrm{Ce}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ , which each showed temperature-dependent diffuse diffraction at high $Q$ , interpreted as arising from octupolar correlations. Our measurements use an optimized single-crystal diffuse scattering instrument that allows us to screen against strong Bragg scattering from ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ . The temperature-difference neutron diffraction reveals a low- $Q$ peak consistent with dipolar spin-ice correlations reported in previous work, and an alternation between positive and negative net intensity at higher $Q$ . These features are consistent with our numerical-linked-cluster calculations using pseudospin interaction parameters previously reported for ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ , ${\mathrm{Ce}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ , and ${\mathrm{Ce}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ . Importantly, neither the measured data nor any of the NLC calculations show evidence for increased scattering at high $Q$ resulting from octupolar correlations. We conclude that at the lowest attainable temperature for our measurements ( $T=0.05\mathrm{K}$ ), scattering from octupolar correlations in ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ is not present in the neutron diffraction signal on the level of our observation threshold of around 0.1% of the low- $Q$ dipole scattering. We compare these results to those obtained earlier on powder ${\mathrm{Ce}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ and ${\mathrm{Ce}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ , and to low-energy inelastic neutron scattering from single-crystal ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ .