Perimeter Institute Quantum Discussions

This talk is a progress report on ongoing research. I will explain what resource theories have to do with real algebraic geometry, and then present a preliminary result in real algebraic geometry which can be interpreted as a theorem on asymptotic and catalytic resource orderings.

It reproves the known criterion for asymptotic and catalytic majorization in terms of Rényi entropies, and generalizes it to any resource theory which satisfies a mild boundedness hypothesis. I will sketch the case of matrix majorization as an example.

We show that a generic many-body Hamiltonian can be uniquely reconstructed from a single pair of initial-final states under the unitary time evolution. Interesting it is, this method is not practical due to its high complexity. We then propose a practical method for Hamiltonian reconstruction from multiple pairs of initial-final states. The stability of this method is mathematically proved and numerically verified.

This work is joint with Liujun Zou and Timothy Hsieh.

Results from the first digital quantum simulation of an SU(2) gauge theory are presented. This was done by analytically constructing gauge-invariant states and implementing a Trotterized time evolution operator for that basis on superconducting hardware. By using error mitigation techniques, electric energy measurements could be reliably extracted following one Trotter-Suzuki time step. This work is a small but important step toward determining what field-theoretic calculations will be possible using near-term devices.

In quantum spin systems, the existence of a spectral gap above the ground state has strong implications for the low-energy physics. We survey recent results establishing spectral gaps in various frustration-free spin systems by verifying finite-size criteria. The talk is based on collaborations with Abdul-Rahman, Lucia, Mozgunov, Nachtergaele, Sandvik, Yang, Young, and Wang.

Connections between 2D gapped quantum phases and the anyon fusion theory have been proven in various ways under different settings. In this work, we introduce a new framework connecting them by only assuming a conjectured form of entanglement area law for 2D gapped systems. We show that one can systematically define topological charges and fusion rules from the area law alone, in a well-defined way. We then derive the fusion rules of charges satisfy all the axioms required in the algebraic theory of anyons. Moreover, even though we make no assumption about the exact value of the constant sub-leading term of the entanglement entropy, this term is shown to be equal to the logarithm of the total quantum dimension of the anyon theory we defined. 

 Holographic quantum error correcting codes (HQECC) have been proposed as toy models for the AdS/CFT correspondence, and exhibit many of the features of the duality. HQECC give a mapping of states and observables. However, they do not map local bulk Hamiltonians to local Hamiltonians on the boundary. In this work, we combine HQECC with Hamiltonian simulation theory to construct a bulk-boundary mapping between local Hamiltonians, whilst retaining all the features of the HQECC duality. This allows us to construct a duality between models, encompassing the relationship between bulk and boundary energy scales and time dynamics.

It also allows us to construct a map in the reverse direction: from local boundary Hamiltonians to the corresponding local Hamiltonian in the bulk. Under this boundary-to- bulk mapping, the bulk geometry emerges as an approximate, low-energy, effective theory living in the code-space of an (approximate) HQECC on the boundary. At higher energy scales, this emergent bulk geometry is modified in a way that matches the toy models of black holes proposed previously for HQECC. Moreover, the duality on the level of dynamics shows how these toy-model black holes can form dynamically.

We give a quantum speedup for solving the canonical semidefinite programming relaxation for binary quadratic optimization. The class of relaxations for combinatorial optimization has so far eluded quantum speedups. Our methods combine ideas from quantum Gibbs sampling and matrix exponent updates. A de-quantization of the algorithm also leads to a faster classical solver. For generic instances, our quantum solver gives a nearly quadratic speedup over state-of-the-art algorithms.
This is joint work with Fernando Brandao (Caltech) and Daniel Stilck Franca (QMATH, Copenhagen).

The mixing time of Markovian dissipative evolutions of open quantum many-body systems can be bounded using optimal constants of certain quantum functional inequalities, such as the logarithmic Sobolev constant. For classical spin systems, the positivity of such constants follows from a mixing condition for the Gibbs measure, via quasi-factorization results for the entropy.

Inspired by the classical case, we present a strategy to derive the positivity of the logarithmic Sobolev constant associated to the dynamics of certain quantum systems from some clustering conditions on the Gibbs state of a local, commuting Hamiltonian. In particular we address this problem for the heat-bath dynamics in 1D and the Davies dynamics, showing that the first one is positive under the assumptions of a mixing condition on the Gibbs state and a strong quasi-factorization of the relative entropy, and the second one under some strong clustering of correlations.

Quantum complexity is a notion characterizing the universality of the entanglement arising from a quantum evolution. A universal evolution will result in a complex entanglement. At the same time, this also corresponds to small fluctuations and to unlearnability from the point of view of machine learning. All these aspects are connected to the different features of k-designs, which are under-samplings of the Hilbert space. 

We study the transition in complexity due to the doping of a quantum circuit by universal gates and show that the transition to complex entanglement can be obtained by just a single gate. 
These results are relevant for the notions of scrambling, quantum chaos, OTOCs and operator spreading.

We conjecture that the transition to 4−design, W-D and unlearnability are one and the same.

I will describe some connections between the Eigenstate Thermalization Hypothesis (ETH), the entanglement structure of generic excited eigenstates of chaotic quantum systems ("EE", arXiv:1906.04295), and the "bound on chaos" limiting the growth rate of the out-of-time-order four-point correlator in such systems ("OTOC", arXiv:1906.10808).

In quantum error correcting codes, there is a distinction
between coherent and incoherent noise. Coherent noise can cause the
average infidelity to accumulate quadratically when a fixed channel is
applied many times in succession, rather than linearly as in the case
of incoherent noise. I will present a proof that unitary single qubit
noise in the 2D toric code with minimum weight decoding is mapped to
less coherent logical noise, and as the code size grows, the coherence
of the logical noise channel is suppressed. In the process, I will
describe how to characterize the coherence of noise using either the
growth of infidelity or the relation between the diamond distance from
identity and the average infidelity. I will explain how coherence in
the noise on physical qubits is transformed by error correction in
stabilizer codes. Then, I will sketch the proof that coherence is
suppressed for the 2D toric code. The result holds even when the
single qubit unitary rotation are allowed to have arbitrary directions
and angles, so long as the angles are below a threshold, and even when
the rotations are correlated. Joint work with John Preskill.

The out-of-time-ordered correlator (OTOC) and entanglement are two physically motivated and widely used probes of the ``scrambling'' of quantum information, which has drawn great interest recently in quantum gravity and many-body physics. By proving upper and lower bounds for OTOC saturation on graphs with bounded degree and a lower bound for entanglement on general graphs, we show that the time scales of scrambling as given by the growth of OTOC and entanglement entropy can be asymptotically separated in a random quantum circuit model defined on graphs with a tight bottleneck. Our result counters the intuition that a random quantum circuit mixes in time proportional to the diameter of the underlying graph of interactions.  It also serves as a more rigorous justification for an argument of [Shor, 1807.04363], that black holes may be very slow scramblers in terms of entanglement generation. Such observations may be of fundamental importance in the understanding of the black hole information problem. The bound we obtained for OTOC is interesting in its own right in that it generalized previous studies of OTOC on lattices to the geometries on graphs and proved it rigorously.   Based on [Harrow-Kong-Liu-Mehraban-Shor, 1906.02219]