Model checking quantum protocols

This thesis describes model checking techniques for protocols arising in quantum information theory and quantum cryptography. We discuss the theory and implementation of a practical model checker, QMC, for quantum protocols. In our framework, we assume that the quantum operations performed in a protocol are restricted to those within the stabilizer formalism; while this particular set of operations is not universal for quantum computation, it allows us to develop models of several useful protocols as well as of systems involving both classical and quantum information processing. We detail the syntax, semantics and type system of QMC’s modelling language, the logic QCTL which is used for verification, and the verification algorithms that have been implemented in the tool. We demonstrate our techniques with applications to a number of case studies

Minimally complex ion traps as modules for quantum communication and computing

Optically linked ion traps are promising as components of network-based quantum technologies, including communication systems and modular computers. Experimental results achieved to date indicate that the fidelity of operations within each ion trap module will be far higher than the fidelity of operations involving the links; fortunately internal storage and processing can effectively upgrade the links through the process of purification. Here we perform the most detailed analysis to date on this purification task, using a protocol which is balanced to maximise fidelity while minimising the device complexity and the time cost of the process. Moreover we 'compile down' the quantum circuit to device-level operations including cooling and shutting events. We find that a linear trap with only five ions (two of one species, three of another) can support our protocol while incorporating desirable features such as 'global control', i.e. laser control pulses need only target an entire zone rather than differentiating one ion from its neighbour. To evaluate the capabilities of such a module we consider its use both as a universal communications node for quantum key distribution, and as the basic repeating unit of a quantum computer. For the latter case we evaluate the threshold for fault tolerant quantum computing using the surface code, finding acceptable fidelities for the 'raw' entangling link as low as 83% (or under 75% if an additional ion is available).Comment: 15 pages, 8 figure

Advances in Bosonic Quantum Error Correction with Gottesman-Kitaev-Preskill Codes: Theory, Engineering and Applications

Encoding quantum information into a set of harmonic oscillators is considered a hardware efficient approach to mitigate noise for reliable quantum information processing. Various codes have been proposed to encode a qubit into an oscillator -- including cat codes, binomial codes and Gottesman-Kitaev-Preskill (GKP) codes. These bosonic codes are among the first to reach a break-even point for quantum error correction. Furthermore, GKP states not only enable close-to-optimal quantum communication rates in bosonic channels, but also allow for error correction of an oscillator into many oscillators. This review focuses on the basic working mechanism, performance characterization, and the many applications of GKP codes, with emphasis on recent experimental progress in superconducting circuit architectures and theoretical progress in multimode GKP qubit codes and oscillators-to-oscillators (O2O) codes. We begin with a preliminary continuous-variable formalism needed for bosonic codes. We then proceed to the quantum engineering involved to physically realize GKP states. We take a deep dive into GKP stabilization and preparation in superconducting architectures and examine proposals for realizing GKP states in the optical domain (along with a concise review of GKP realization in trapped-ion platforms). Finally, we present multimode GKP qubits and GKP-O2O codes, examine code performance and discuss applications of GKP codes in quantum information processing tasks such as computing, communication, and sensing.Comment: 77+5 pages, 31 figures. Minor bugs fixed in v2. comments are welcome

Topological Code Architectures for Quantum Computation

This dissertation is concerned with quantum computation using many-body quantum systems encoded in topological codes. The interest in these topological systems has increased in recent years as devices in the lab begin to reach the fidelities required for performing arbitrarily long quantum algorithms. The most well-studied system, Kitaev\u27s toric code, provides both a physical substrate for performing universal fault-tolerant quantum computations and a useful pedagogical tool for explaining the way other topological codes work. In this dissertation, I first review the necessary formalism for quantum information and quantum stabilizer codes, and then I introduce two families of topological codes: Kitaev\u27s toric code and Bombin\u27s color codes. I then present three chapters of original work. First, I explore the distinctness of encoding schemes in the color codes. Second, I introduce a model of quantum computation based on the toric code that uses adiabatic interpolations between static Hamiltonians with gaps constant in the system size. Lastly, I describe novel state distillation protocols that are naturally suited for topological architectures and show that they provide resource savings in terms of the number of required ancilla states when compared to more traditional approaches to quantum gate approximation

Mapping quantum circuits to shallow-depth measurement patterns based on graph states

The paradigm of measurement-based quantum computing (MBQC) starts from a highly entangled resource state on which unitary operations are executed through adaptive measurements and corrections ensuring determinism. This is set in contrast to the more common quantum circuit model, in which unitary operations are directly implemented through quantum gates prior to final measurements. In this work, we incorporate concepts from MBQC into the circuit model to create a hybrid simulation technique, permitting us to split any quantum circuit into a classically efficiently simulatable Clifford-part and a second part consisting of a stabilizer state and local (adaptive) measurement instructions, a so-called standard form, which is executed on a quantum computer. We further process the stabilizer state with the graph state formalism, thus enabling a significant decrease in circuit depth for certain applications. We show that groups of fully commuting operators can be implemented using fully-parallel, i.e., non-adaptive, measurements within our protocol. In addition, we discuss how such circuits can be implemented in constant quantum depths by employing quantum teleportation. Finally, we demonstrate the utility of our technique on two examples of high practical relevance: the Quantum Approximate Optimization Algorithm (QAOA) and the Variational Quantum Eigensolver (VQE)