Quantum Information

Quantum error correction -- originally invented for quantum computing -- has proven itself useful in a variety of non-computational physical systems, as the ideas of QEC are broadly applicable. In this talk, I'll mention a few examples of error correction in the wild, including the recent discovery that the AdS/CFT correspondence implements quantum error correction.  We will then study the hypothesis that any local bulk operator in AdS can be reconstructed using only a causally disconnected subregion of the CFT.  This hypothesis has been proven under the assumption that error correction in AdS/CFT is exact, but this assumption is not expected to be true.  Fortunately, recent advances in the theory of approximate quantum error correction have emerged.  We will review these results on recoverability and approximate quantum error correction, as well as AdS/CFT and the so-called entanglement wedge reconstruction hypothesis.  We will then prove the entanglement wedge hypothesis robustly and find an explicit formula for reconstructed bulk operators.  If time permits, we will explore a generalization of the theory of universal recovery channels to the case of finite-dimensional von Neumann algebras.

We study 't Hooft anomalies of discrete groups in the framework of (1+1)-dimensional multiscale entanglement renormalization ansatz states on the lattice. Using matrix product operators, general topological restrictions on conformal data are derived. An ansatz class allowing for optimization of MERA with an anomalous symmetry is introduced. We utilize this class to numerically study a family of Hamiltonians with a symmetric critical line. Conformal data is obtained for all irreducible projective representations of each anomalous symmetry twist, corresponding to definite topological sectors. It is numerically demonstrated that this line is a protected gapless phase. Finally, we implement a duality transformation between a pair of critical lines using our subclass of MERA.

arXiv:1703.07782

Generally speaking, physicists still experience that computing with paper and pencil is in most cases simpler than computing with a Computer Algebra System.  Although that is true in some cases, the working paradigm is changing: developments in CAS, and particularly recent ones in the Maple system, have resulted in the implementation of most of the mathematical objects and mathematics used in theoretical physics computations, and have dramatically approximated the notation used in the computer to the one used with paper and pencil, diminishing the learning gap and computer-syntax distraction to a strict minimum.  In this talk, the Physics project at Maplesoft will be presented and the resulting Physics package will be illustrated through simple problems in classical field theory, quantum mechanics and general relativity, and through tackling the computations of some recent Physical Review papers in those areas.  In addition to the 10:00 am lecture (taking place in Alice), there will be an afternoon hands-on workshop taking place in the Time Room.

We present an in-depth study of the domain walls available in the color code. We begin by presenting new boundaries which gives rise to a new family of color codes. Interestingly, the smallest example of such a code consists of just 4 qubits and weight three parity check measurements, making it an accessible playground for today's experimentalists interested in small scale experiments on topological codes. Secondly, we catalogue the twist defects that are accessible with the color code model. We give lattice representations of these twists and investigate how they interact with one another, and how they interact with the anyons of the system. Our categorisation allows us to explore new approaches for the fault-tolerant storage and manipulation of quantum information in color codes. This research combines and extends recent work with the surface code [1,2] to the color code models, whose continuous domain walls have been studied in generality in [3]. [1] Delfosse, Nicolas, Pavithran Iyer, and David Poulin. "Generalized surface codes and packing of logical qubits." arXiv preprint arXiv:1606.07116 (2016). [2] Brown, Benjamin J., et al. "Poking holes and cutting corners to achieve Clifford gates with the surface code." Physical Review X 7.2 (2017): 021029. [3] Yoshida, Beni. "Topological color code and symmetry-protected topological phases." Physical Review B 91.24 (2015): 245131.

We first summarize background on the quantum capacity of a quantum channel, and explain why we know very little about this fundamental quantity, even for the qubit depolarizing channel (the quantum analogue of the binary symmetric channel) despite 20 years of effort by the community.

Then, we focus on low-noise quantum channels, and present recent results on the quantum capacity to leading order in the noise parameter.  This in particular solves the quantum capacity problem (to leading order) for the qubit depolarizing channel, and provides a structure theorem for the capacity achieving codes.  For low-noise channels, degenerate codes provide negligible superadditivity effect.

Analoguous results on the private capacity will be presented.  Our results imply that for low-noise channels, there is negligible difference between coherence and privacy, and a key rate approaching the capacity can already be obtained using classical error correction and privacy amplification.

Joint work with Felix Leditzky and Graeme Smith

While originally motivated by quantum computation, quantum error correction (QEC) is currently providing valuable insights into many-body quantum physics such as topological phases of matter. Furthermore, mounting evidence originating from holography research (AdS/CFT), indicates that QEC should also be pertinent for conformal field theories. With this motivation in mind, we introduce quantum source-channel codes, which combine features of lossy-compression and approximate quantum error correction, both of which are predicted in holography. Through a recent construction for approximate recovery maps, we derive guarantees on its erasure decoding performance from calculations of an entropic quantity called conditional mutual information. As an example, we consider Gibbs states of the transverse field Ising model at criticality and provide evidence that they exhibit non-trivial protection from local erasure. This gives rise to the first concrete interpretation of a bona fide conformal field theory as a quantum error correcting code. We argue that quantum source-channel codes are of independent interest beyond holography.