Symmetry, Phases of Matter, and Resources in Quantum Computing
Consider the task of estimating the expectation value of an n-qubit tensor product observable in the output state of a shallow quantum circuit. This task is a cornerstone of variational quantum algorithms for optimization, machine learning, and the simulation of quantum many-body systems. In this talk I will describe three special cases of this problem which are "easy" for classical computers. This is joint work with Sergey Bravyi and Ramis Movassagh.
I will review the stabiliser rank and associated pure state magic monotone, the extent, [Bravyi et. al 2019]. Then I will discuss several new magic monotones that can be regarded as a generalisation of the extent monotone to mixed states [Campbell et. al., in preparation]. My talk will outline several nice theorems we can prove about these monotones relate to each other and how they are related to the runtime of new classical simulation algorithms.
The variational quantum eigensolver (VQE) is the leading candidate for practical applications of Noisy Intermediate Scale Quantum (NISQ) devices. The method has been widely implemented on small NISQ machines in both superconducting and ion trap implementations. I will review progress to date and discuss two questions . Firstly, how quantum mechanical are small VQE demonstrations? We will analyze this question using strong measurement contextuality. Secondly, can VQE be implemented at the scale of devices capable of exhibiting quantum supremacy, around 50 qubits?
We take a resource-theoretic approach to the problem of quantifying nonclassicality in Bell scenarios. The resources are conceptualized as probabilistic processes from the setting variables to the outcome variables which have a particular causal structure, namely, one wherein the wings are only connected by a common cause. The distinction between classical and nonclassical is then defined in terms of whether or not a classical causal model can explain the correlations.
Measurement-based quantum computation (MBQC) is a computational scheme to simulate spacetime dynamics on the network of entanglement using local measurements and classical communication. The pursuit of a broad class of useful entanglement encountered a concept of symmetry-protected topologically ordered (SPTO) phases in condensed matter physics.
The interplay of symmetry and topology in quantum many-body systems can lead to novel phases of matter, with applications in quantum memories and resources for quantum computing. While we understand the range of phenomena quite well in 2-d systems, there are many open questions for the 3-d case, in particular what kind of symmetries and topology can allow for thermal stability in 3-d models. I’ll present some of the results and open questions in this direction, using the 3-d toric code and the RBH models as examples.