Quantum Information

Based on the immense popularity of causal Bayesian networks and structural causal models, one might expect that these representations are appropriate to describe the causal semantics of any real-world system, at least in principle. In this talk, I will argue that this is not the case, and motivate the study of more general causal modeling frameworks. In particular, I will discuss bipartite graphical causal models. Real-world complex systems are often modelled by systems of equations with endogenous and independent exogenous random variables. Such models have a long tradition in physics and engineering. The structure of such systems of equations can be encoded by a bipartite graph, with variable and equation nodes that are adjacent if a variable appears in an equation. I will show how one can use Simon’s causal ordering algorithm and the Dulmage-Mendelsohn decomposition to derive a Markov property that states the conditional independence for (distributions of) solutions of the equations in terms of the bipartite graph. I will then show how this Markov property gives rise to a do-calculus for bipartite graphical causal models, providing these with a refined causal interpretation.

In the Statistics literature there are three main frameworks for causal modeling: counterfactuals (aka potential outcomes), non-parametric structural equation models (NPSEMs) and graphs (aka path diagrams or causal Bayes nets). These approaches are similar and, in certain specific respects, equivalent. However, there are important conceptual differences and each formulation has its own strengths and weaknesses. These divergences are of relevance both in theory and when the approaches are applied in practice. This talk will introduce the different frameworks, and describe, through examples, both the commonalities and dissimilarities. In particular, we will see that the “default” assumptions within these frameworks lead to different identification results when quantifying mediation and, more generally, path-specific effects.

Can the effectiveness of a medical treatment be determined without the expense of a randomized controlled trial? Can the impact of a new policy be disentangled from other factors that happen to vary at the same time? Questions such as these are the purview of the field of causal inference, a general-purpose science of cause and effect, applicable in domains ranging from epidemiology to economics. Researchers in this field seek in particular to find techniques for extracting causal conclusions from statistical data. Meanwhile, one of the most significant results in the foundations of quantum theory—Bell’s theorem—can also be understood as an attempt to disentangle correlation and causation. Recently, it has been recognized that Bell’s result is an early foray into the field of causal inference and that the insights derived from almost 60 years of research on his theorem can supplement and improve upon state-of-the-art causal inference techniques. In the other direction, the conceptual framework developed by causal inference researchers provides a fruitful new perspective on what could possibly count as a satisfactory causal explanation of the quantum correlations observed in Bell experiments. Efforts to elaborate upon these connections have led to an exciting flow of techniques and insights across the disciplinary divide. This tutorial will highlight some of what is happening at the intersection of these two fields.

In this talk, I’ll discuss the deep connection between emergent k-form symmetries and transversal logical gates in quantum low-density parity-check (LDPC) codes. I’ll then present a parallel fault-tolerant quantum computing scheme for families of homological quantum LDPC codes defined on 3-manifolds with constant or almost-constant encoding rate using the underlying higher symmetries in our recent work. We derive a generic formula for a transversal T gate on color codes defined on general 3-manifolds, which acts as collective non-Clifford logical CCZ gates on any triplet of logical qubits with their logical-X membranes having a Z2 triple intersection at a single point. The triple intersection number is a topological invariant, which also arises in the path integral of the emergent higher symmetry operator in a topological quantum field theory (TQFT): the (Z2) 3 gauge theory. Moreover, the transversal S gate of the color code corresponds to a higher-form symmetry supported on a codimension-1 submanifold, giving rise to exponentially many addressable and parallelizable logical CZ gates. Both symmetries are related to gauged SPT defects in the (Z2) 3 gauge theory. We have then developed a generic formalism to compute the triple intersection invariants for general 3- manifolds. We further develop three types of LDPC codes supporting such logical gates with constant or almost-constant encoding rate and logarithmic distance. Finally, I’ll point out a connection between the gauged SPT defects in the 6D color code and a recently discovered non-Abelian self-correcting quantum memory in 5D. Reference: arXiv:2310.16982, arXiv:2208.07367, arXiv:2405.11719.

In this talk, I will explain how to implement fault-tolerant non-Clifford gates in copies of toric code in two dimensions achieved by transiently switching to a non-Abelian topologically ordered phase by expanding earlier results by Bombin [arXiv.1810.09571] and Brown [SciAdv.aay4929]. This addresses the challenge of performing universal fault-tolerant quantum computation in purely two spatial dimensions and shows a new approach to quantum computation using non-Abelian phases. This talk is based on upcoming work in collaboration with A. Bauer, B.Brown, J. Magdalena de la Fuente, M. Webster and D. Williamson.

Quantum error correction methods for qubit technologies such as ions, photons, or superconducting qubits can appear very different at first glance. Moreover, as more detailed error models are accounted for, the relationship to the abstract models of fault tolerance can appear to become more distant. In this talk I will discuss two unifying frameworks which connect hardware specific models more closely to the underlying code structures, which can help enable QEC development. First I will introduce a unifying framework for fault tolerance based on the ZX calculus (arXiv:2303.08829) and show how it allows us to view circuit-based, measurement-based, fusion-based quantum computation, and Floquet codes as different flavors of the same underlying stabilizer fault-tolerance structure. Secondly I will introduce fusion complexes (arXiv:2308.07844) which allows a topological interpretation of fault tolerance even under circuit level error models. Both of these frameworks are tools that can aid in the design of quantum error correction methods under hardware-focussed models, and I will give some examples of this applied to the design of photonic architectures.