Seminar

Lecture 1:  Introduction to quantum error correction

I provide a brief introduction to the tenets of quantum error correction using the four-qubit code, making contact with concatenated, CSS, stabilizer, and rotated surface codes. 

Noncommuting conserved quantities have recently launched a subfield of quantum thermodynamics. In conventional thermodynamics, a system of interest and an environment exchange quantities—energy, particles, electric charge, etc.—that are globally conserved and are represented by Hermitian operators. These operators were implicitly assumed to commute with each other, until a few years ago. Freeing the operators to fail to commute has enabled many theoretical discoveries—about reference frames, entropy production, resource-theory models, etc.

Recently, we presented a systematic recipe for generating duality transformations in one dimensional lattice models. Our construction is based on a detailed understanding of the most general kind of symmetry a one-dimensional lattice model can exhibit: categorical symmetries. These symmetries are conveniently described in the language of tensor networks, where they are represented as matrix product operators.

Huang, Kueng, Preskill introduced the learning task now known as “classical shadows”: given few copies of an unknown state ρ, construct a classical description of the state from independent measurements that can be used to predict certain properties of the state. Specifically, they show Θ(B/epsilon^2) samples of ρ suffice to approximate the expectation value Tr(Oρ) of any Hermitian observable O to within additive error epsilon provided Tr(O^2) ≤ B and the eigenvalues of O are contained in [-1,1].

The quantum marginals problem (QMP) aims to understand how the various marginals of a joint quantum state are related to one another by deciding whether or not a given collection of marginals is compatible with some joint quantum state. Although existing techniques for the QMP are well developed for the special case of disjoint marginals, the same is not true for the generic case of overlapping marginals. The leading technique for the generic QMP, published by Yu et. al. (2021), resorts to evaluating a hierarchy of semidefinite programs.

In this talk, we consider two fundamental tasks in quantum state estimation, namely, quantum tomography and quantum state certification. In the former, we are given n copies of an unknown mixed state rho, and the goal is to learn it to good accuracy in trace norm. In the latter, the goal is to distinguish if rho is equal to some specified state, or far from it. When we are allowed to perform arbitrary (possibly entangled) measurements on our copies, then the exact sample complexity of these problems is well-understood.

The ability to perform fast and robust operations on multi-qubit quantum systems is a necessity for realizing reliable quantum computation. Unfortunately, the inevitable interaction between a quantum system and its environment presents an obstacle for achieving such operations. Despite this challenge, when used in tandem, quantum noise characterization and quantum control provide a means for engineering targeted control protocols that achieve noise-robust quantum logic operations informed by knowledge of the underlying noise properties.

Quantum devices hold promise to outperform classical computers in performing some physical simulations in the nearest future, making them a valuable tool for physics research. In this talk, Oles will focus on quantum simulation of the topological states of matter hosting Majorana modes -- the exotic "half-electron" states. He will show the results obtained from noisy quantum hardware provide us with accurate prediction of Majorana mode wavefunctions. This experiment also allows us to verify the topological nature of observed modes.

An interesting problem in the field of quantum error correction involves finding a physical system that hosts a "self-correcting quantum memory," defined  as an encoded qubit  coupled to an environment that naturally wants to correct errors.  To date, a quantum memory stable against finite-temperature effects is only known in four spatial dimensions or higher. Here, we take a different approach to realize a  stable  quantum memory by relying on a driven-dissipative environment.

In this talk, I'll motivate studying the complexity of quantum states and transformations. I'll discuss how this general study is related to a seminal theoretical computer science concept: search vs. decision. I'll show how to construct a form of search-to-decision reductions for QMA problems and show why it is unlikely that we can do (much) better. I'll conclude by discussing a parametrized notion of QMA and the notion of QMA solutions in this context.