Homological analysis of multi-qubit entanglement

We propose the usage of persistent homologies to characterize multipartite entanglement. On a multi-qubit data set we introduce metric-like measures defined only in terms of bipartite entanglement and then we derive barcodes. We show that they are able to provide a good classification of entangled states, at least for a small number of qubit

Quantum Approaches to Data Science and Data Analytics

In this thesis are explored different research directions related to both the use of classical data analysis techniques for the study of quantum systems and the employment of quantum computing to speed up hard Machine Learning task

Probing multipartite entanglement through persistent homology

We propose a study of multipartite entanglement through persistent homology, a tool used in topological data analysis. In persistent homology, a 1-parameter filtration of simplicial complexes called persistence complex is used to reveal persistent topological features of the underlying data set. This is achieved via the computation of homological invariants that can be visualized as a persistence barcode encoding all relevant topological information. In this work, we apply this technique to study multipartite quantum systems by interpreting the individual systems as vertices of a simplicial complex. To construct a persistence complex from a given multipartite quantum state, we use a generalization of the bipartite mutual information called the deformed total correlation. Computing the persistence barcodes of this complex yields a visualization or `topological fingerprint' of the multipartite entanglement in the quantum state. The barcodes can also be used to compute a topological summary called the integrated Euler characteristic of a persistence complex. We show that in our case this integrated Euler characteristic is equal to the deformed interaction information, another multipartite version of mutual information. When choosing the linear entropy as the underlying entropy, this deformed interaction information coincides with the $n$-tangle, a well-known entanglement measure. The persistence barcodes thus provide more fine-grained information about the entanglement structure than its topological summary, the $n$-tangle, alone, which we illustrate with examples of pairs of states with identical $n$-tangle but different barcodes. Furthermore, a variant of persistent homology computed relative to a fixed subset yields an interesting connection to strong subadditivity and entropy inequalities. We also comment on a possible generalization of our approach to arbitrary resource theories.Comment: 22 pages, 3 figures. Code available at https://github.com/felixled/entanglement_persistent_homolog

The Small Stellated Dodecahedron Code and Friends

We explore a distance-3 homological CSS quantum code, namely the small stellated dodecahedron code, for dense storage of quantum information and we compare its performance with the distance-3 surface code. The data and ancilla qubits of the small stellated dodecahedron code can be located on the edges resp. vertices of a small stellated dodecahedron, making this code suitable for 3D connectivity. This code encodes 8 logical qubits into 30 physical qubits (plus 22 ancilla qubits for parity check measurements) as compared to 1 logical qubit into 9 physical qubits (plus 8 ancilla qubits) for the surface code. We develop fault-tolerant parity check circuits and a decoder for this code, allowing us to numerically assess the circuit-based pseudo-threshold.Comment: 19 pages, 14 figures, comments welcome! v2 includes updates which conforms with the journal versio

Fault-tolerance in two-dimensional topological systems

This thesis is a collection of ideas with the general goal of building, at least in the abstract, a local fault-tolerant quantum computer. The connection between quantum information and topology has proven to be an active area of research in several fields. The introduction of the toric code by Alexei Kitaev demonstrated the usefulness of topology for quantum memory and quantum computation. Many quantum codes used for quantum memory are modeled by spin systems on a lattice, with operators that extract syndrome information placed on vertices or faces of the lattice. It is natural to wonder whether the useful codes in such systems can be classified. This thesis presents work that leverages ideas from topology and graph theory to explore the space of such codes. Homological stabilizer codes are introduced and it is shown that, under a set of reasonable assumptions, any qubit homological stabilizer code is equivalent to either a toric code or a color code. Additionally, the toric code and the color code correspond to distinct classes of graphs. Many systems have been proposed as candidate quantum computers. It is very desirable to design quantum computing architectures with two-dimensional layouts and low complexity in parity-checking circuitry. Kitaev\u27s surface codes provided the first example of codes satisfying this property. They provided a new route to fault tolerance with more modest overheads and thresholds approaching 1%. The recently discovered color codes share many properties with the surface codes, such as the ability to perform syndrome extraction locally in two dimensions. Some families of color codes admit a transversal implementation of the entire Clifford group. This work investigates color codes on the 4.8.8 lattice known as triangular codes. I develop a fault-tolerant error-correction strategy for these codes in which repeated syndrome measurements on this lattice generate a three-dimensional space-time combinatorial structure. I then develop an integer program that analyzes this structure and determines the most likely set of errors consistent with the observed syndrome values. I implement this integer program to find the threshold for depolarizing noise on small versions of these triangular codes. Because the threshold for magic-state distillation is likely to be higher than this value and because logical CNOT gates can be performed by code deformation in a single block instead of between pairs of blocks, the threshold for fault-tolerant quantum memory for these codes is also the threshold for fault-tolerant quantum computation with them. Since the advent of a threshold theorem for quantum computers much has been improved upon. Thresholds have increased, architectures have become more local, and gate sets have been simplified. The overhead for magic-state distillation has been studied, but not nearly to the extent of the aforementioned topics. A method for greatly reducing this overhead, known as reusable magic states, is studied here. While examples of reusable magic states exist for Clifford gates, I give strong reasons to believe they do not exist for non-Clifford gates

Multipartite entanglement via the Mayer-Vietoris theorem

The connection between entanglement and topology manifests itself in the form of the ER-EPR duality. This statement however refers to the maximally entangled states only. In this article I study the multipartite entanglement and the way in which it relates to the topological interpretation of the ER-EPR duality. The $2$ dimensional genus $1$ torus will be generalised to a $n$-dimensional general torus, where the information about the multipartite entanglement will be encoded in the higher inclusion maps of the Mayer-Vietorist sequence.Comment: 2 figure

Fault-tolerant complexes

Fault-tolerant complexes describe surface-code fault-tolerant protocols from a single geometric object. We first introduce fusion complexes that define a general family of fusion-based quantum computing (FBQC) fault-tolerant quantum protocols based on surface codes. We show that any 3-dimensional cell complex where each edge has four incident faces gives a valid fusion complex. This construction enables an automated search for fault tolerance schemes, allowing us to identify 627 examples within a moderate search time. We implement this using the open-source software tool Gavrog and present threshold results for a variety of schemes, finding fusion networks with higher erasure and Pauli thresholds than those existing in the literature. We then define more general structures we call fault-tolerant complexes that provide a homological description of fault tolerance from a large family of low-level error models, which include circuit-based computation, floquet-based computation, and FBQC with multi-qubit measurements. This extends the applicability of homological descriptions of fault tolerance, and enables the generation of many new schemes which have not been previously identified. We also define families of fault-tolerant complexes for color codes and 3d single-shot subsystem codes, which enables similar constructive methods, and we present several new examples of each

The Quantum PCP Conjecture

The classical PCP theorem is arguably the most important achievement of classical complexity theory in the past quarter century. In recent years, researchers in quantum computational complexity have tried to identify approaches and develop tools that address the question: does a quantum version of the PCP theorem hold? The story of this study starts with classical complexity and takes unexpected turns providing fascinating vistas on the foundations of quantum mechanics, the global nature of entanglement and its topological properties, quantum error correction, information theory, and much more; it raises questions that touch upon some of the most fundamental issues at the heart of our understanding of quantum mechanics. At this point, the jury is still out as to whether or not such a theorem holds. This survey aims to provide a snapshot of the status in this ongoing story, tailored to a general theory-of-CS audience.Comment: 45 pages, 4 figures, an enhanced version of the SIGACT guest column from Volume 44 Issue 2, June 201