Session A28: DQI Prize Session

Dannie Heineman Prize for Mathematical Physics Talk: Twelve Tales in Mathematical Physics

I’ll briefly discuss 12 topics on which I worked that fit into the prize citation and then focus on a few including Continuous Symmetry Breaking in Classical and Quantum Spin system and my work on the geometric understanding of Berry’s phase.

  

Wavelet-Based Representations of Quantum Field Theory

Here we present current results from our investigations into wavelet-based representations of quantum field theory. Specifically, we develop representations of one-dimensional free field theories for fermions and scalar bosons using the Daubechies wavelets, which are desirable due to their compact support and vanishing moments. We reproduce entanglement area laws with a resolution-dependent cutoff and generalize to fractal sets.

The ground states of these one-dimensional free field theories have a holographic dual representation in terms of multiscale wavelet degrees of freedom. We show how an emergent geometry can be inferred from the scaling of mutual information between wavelet degrees of freedom in the bulk. At the critical point, the bulk has an anti-de-Sitter geometry with radius of curvature that depends on the Daubechies wavelet index.

Our work has implications for resource-theory-based approaches to quantum field theory as well as applications to the development of quantum algorithms for simulating quantum field theory.   

Quantum Speed Limits for Quantum Information Processing Tasks

Quantum speed limits provide important bounds on quantum information processing tasks. Systems where these limitations can be probed are already being realized experimentally. We show that the Lieb-Robinson bound can be used to derive fundamental speed limits on quantum information processing tasks in local quantum spin systems. Comparison with numerical optimal control results for spin chains suggests that unexplored regions of the dynamical landscape may support enhanced performance of key quantum information processing tasks.   

Exact Holographic Tensor Networks

We present exact holographic tensor network representations for a class of ground states of a continuous family of frustration-free Hamiltonians.
These states, known as area deformed Motzkin states, exhibit a novel quantum phase transition where the entanglement entropy transitions continuously from being bounded to scaling with the volume.
The states provide a test case for recent ideas about holography interpreted as an optimized path integral representation of a state.
In particular, by tuning the parameter we can go from the highly entangled ``Rainbow'' phase where no tree representation (such as MERA) is possible for the state to a Lifshitz point where we present a tree-like structure for the ground state.   

Rolf Landauer and Charles H. Bennett Award talk on the mathematics of quantum information, and the development of new algorithmic primitives for quantum computers

The field of quantum algorithms has developed around a set of basic building blocks such as the quantum Fourier transform, amplitude amplification and so on, which are then combined with classical algorithms in order to achieve quantum speedups. Likewise quantum information combines classical tools such as Shannon's theorem with new elements such as teleportation and super-dense coding. Ultimately we hope that these primitives both tell us what we can do with a quantum computing or communication network, but also something about the nature of quantum information and how it differs from classical information.

In this talk, I will describe some of my own contributions to quantum algorithms and information theory, and will discuss their significance both operationally and conceptually.   

Quantum Circuit Designs for Gate Model Quantum Computers

We define a method to design quantum circuits for gate model quantum computers. Our Quantum Triple Annealing Minimization (QTAM) algorithm provides the quantum circuit minimization on the physical layout (circuit depths and area), the superconducting wire length minimization of the quantum circuit, and the minimization of the input quantum states and the measurements. The QTAM method minimizes the Hamiltonian operator via the maximization of an objective function. We define a multilayer structure for quantum circuit computations using the hardware restrictions of superconducting quantum computers. The results can be straightforwardly applied in experimental superconducting quantum computers, and gate model quantum computations.   

Quantum walks as a path for complexity reduction of multiqubit gates

Quantum walks are known to provide an alternative equivalent approach to quantum algorithms. We show that continuous time quantum walks have potential to reduce complexity class of quantum operations in system where few auxiliary states are available. Unlike classical driving used to execute quantum gates, the walks explore multiple quantum trajectories simultaneously, probing interactions and accumulating the needed phase more effectively.   

Reliable information transmission through Gaussian loss channels using Gottesman-Kitaev-Preskill codes

Generalizing the concept of (classical) channel capacity in Shannon information theory, quantum channel capacity quantifies the amount of quantum bits per channel use that can be transmitted reliably, upon optimal quantum error correction, with vanishing decoding error in the limit of infinitely many channel uses. Among various types of quantum channels, Gaussian thermal loss channels are of particular interest because they model realistic optical communication channels, and their quantum capacities are determined up to (at most) a constant gap between a lower and upper bound. Here, we consider an explicit quantum communication protocol using the Gottesman-Kitaev-Preskill (GKP) code, and report that the GKP code, although not designed to correct loss errors, nevertheless can achieve the quantum capacity of the Gaussian thermal loss channel up to a bounded constant offset.   

Generalized entanglement entropies of quantum designs

The entanglement properties of random quantum states or dynamics are important to the study of a broad spectrum of disciplines of physics, ranging from quantum information to condensed matter to high energy. Ensembles of quantum states or unitaries that reproduce the first $\alpha$ moments of completely random states or unitary channels (drawn from the Haar measure) are called $\alpha$-designs. Entropic functions of the $\alpha$-th power of a density operator are called $\alpha$-entropies (e.g.~R\'enyi and Tsallis). We reveal strong connections between the orders of designs and generalized (in particular R\'enyi) entropies, by showing that the R\'enyi-$\alpha$ entanglement entropies averaged over (approximate) $\alpha$-designs are generically almost maximal. Moreover, we find that the min entanglement entropies become maximal for designs of an order logarithmic in the dimension of the system, which implies that they are indistinguishable from uniformly random by the entanglement spectrum. Our results relate the complexity of scrambling to the degree of randomness by R\'enyi entanglement entropy.   

Computing on quantum shared secrets

A (k,n)-threshold secret-sharing scheme allows for a string to be split into n shares in such a way that any subset of at least k shares suffices to recover the secret string, but such that any subset of at most k−1 shares contains no information about the secret. Quantum secret-sharing schemes extend this idea to the sharing of quantum states. Here we propose a method of performing computation on quantum shared secrets. We introduce a (n,n)-quantum secret sharing scheme together with a set of protocols that allow quantum circuits to be evaluated on the shared secret without the need to decode the secret. We consider a multipartite setting, with each participant holding a share of the secret. We show that if there exists at least one honest participant, no group of dishonest participants can recover any information about the shared secret, independent of their deviations from the protocol.   

A versatile quantum data hiding protocol with enhanced security

One of the fashionable applications of quantum technology recent years is to use quantum states and measurements to communicate which offers more reliable security promises. Quantum data hiding, which gives the source party the ability to share data among multiple receivers and reveal it at a later time depending on his will, is one of the promising information sharing schemes which may address a lot of security related issues nowadays, for instance, the bit commitment problem. In this work, we propose a novel quantum data hiding protocol which covers a variate of hiding scenarios. We improve the security level compared to the existing protocols by embedding the security into different layers of the protocol. The hiding of the data is secure against not only local operations and classical communications (LOCC), but also any nonlocal measurement. In addition, our protocol equips the source party the ability to abort the hiding protocol while it is still in the hiding stage, which is not possible with previous works. We also show that the unwanted information leakage at any stage in our protocol can be made asymptotically small with the increase of the dimension of the Hilbert space.