Perimeter Institute Quantum Discussions

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 high energy and many-body physics. This work investigates the interplay between the degrees of entanglement and randomness in pure states and unitary channels, by employing tools from random matrix theory, representation theory, combinatorics, Weingarten calculus etc. We reveal strong connections between designs (distributions of states or unitaries that match certain moments of the uniform Haar measure) and generalized entropies (entropic functions that depend on certain powers of the density operator), by showing that Renyi entanglement entropies averaged over designs of the same order are almost maximal. This strengthens the celebrated Page's theorem. Moreover, we find that designs of an order that is logarithmic in the dimension maximize all Renyi entanglement entropies, and so are completely random in terms of the entanglement spectrum. Our results relate the behaviors of Renyi entanglement entropies to the complexity beyond scrambling/thermalization/chaos in terms of the degree of randomness, and suggest a generalization of the fast scrambling conjecture.  (Refs: 1703.08104, 1709.04313)

(1) Entanglement-enhanced quantum sensing: parameter estimation and hypothesis testing.
(2) Security from entanglement: quantum key distribution.
(3) Entanglement enhanced communication: channel capacity and additivity issues.
(4) Some open problems.

We discuss some applications of a result on the convex combination of the quantum states (that we refer to as convex-split technique) and its variants. In the framework of Quantum Resource theory, we provide an operational way of characterizing the amount of resource in a given quantum state, for a large class of resource theories. Our results use the convex-split technique in the achievability proof, with a matching converse in the one-shot setting.  Building upon the ideas involved in this achievability proof, we show how the technique (and its close counterpart of position-based decoding) can lead to a large family of entanglement-assisted protocols for quantum communication. We sketch close connections between these protocols and the port-based teleportation scheme of Ishizaka and Hiroshima [Phys. Rev. Lett. 101, 240501]. We exploit these connections to obtain a new protocol for entanglement assisted point to point quantum channel coding in the asymptotic and i.i.d. setting, with the rate of entanglement cost and achievable communication matching that of Bennett et. al. [IEEE Trans. Inf. Theory, 48, 2002].  Based on the works arXiv:1410.3031,  arXiv:1708.00381,   arXiv:1702.01940.

From a quantum information perspective, we will study universal features of chaotic quantum systems. Recent progress has made evident that quantifying chaos is a useful way to gain insight into strongly-coupled field theories, quantum many-body systems, as well as the quantum nature of black holes. We will derive relations between different diagnostics of chaos and scrambling (OTOCs, spectral functions, and frame potentials) and define a quantity to capture the onset of a random matrix description. We will review and use tools from quantum information and random matrix theory, but our goal will be to understand strongly-interacting systems.

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.

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.

The surface code is currently the leading proposal to achieve fault-tolerant quantum computation. Among its strengths are the plethora of known ways in which fault-tolerant Clifford operations can be performed, namely, by deforming the topology of the surface, by the fusion and splitting of codes, and even by braiding engineered Majorana modes using twist defects. Here, we present a unified framework to describe these methods, which can be used to better compare different schemes and to facilitate the design of hybrid schemes. Our unification includes the identification of twist defects with the corners of the planar code. This identification enables us to perform single-qubit Clifford gates by exchanging the corners of the planar code via code deformation. We analyze ways in which different schemes can be combined and propose a new logical encoding. We also show how all of the Clifford gates can be implemented with the planar code, without loss of distance, using code deformations, thus offering an attractive alternative to ancilla-mediated schemes to complete the Clifford group with lattice surgery.

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.

The theory of alphabits is a natural generalisation of approximate quantum error correction that proves fundamental to the study of asymptotic quantum resources. In particular, it leads to an asymptotically reversible variation on quantum teleportation, called zerobit teleportation, which decomposes qubits of communication into correlation and transmission components. They also naturally arise in the study of black holes with significant consequences for the nature of quantum error correction in AdS/CFT.

Transversality is one of the most desirable features of fault-tolerant circuits because it automatically limits the propagation of errors. However, it was shown by Eastin & Knill that no universal set of quantum gates on any quantum code is transversal. In this talk, we strengthen this result for stabilizer codes to say that transversal gates must in fact be contained in the Clifford hierarchy. Moreover, we present new circuits on Bacon-Shor codes that saturate our bounds. In particular, we show how a k-qubit controlled-Z gate can be implemented by a transversal circuit on m-by-m^k Bacon-Shor codes and provide some estimates of its performance in terms of pseudo-thresholds and resource overhead.