Translating one-way quantum computation to the circuit model: methods and applications

In this thesis I study the one-way quantum computation (1WQC) model and some applications of the different ways of translating 1WQC algorithms into the circuit model. In a series of recent results, different sets of conditions for implementing a computation deterministically in the one-way model have been proposed, each of them with their own properties. Some of those sets of conditions generically known as flow conditions try to explore the distinct parallel power of the 1WQC model, by increasing the number of operations that can be performed simultaneously. Here I contribute to this line of research by defining a new type of flow, which I call the signal-shifted flow (SSF), which has an interesting parallel structure that equals that of a depth-optimal flow. I also introduce a new framework for translating 1WQC algorithms into the circuit model. This translation preserves not only the computation performed but also some features of the 1WQC algorithm design. Within this framework I give two algorithms, each implementing a different translation procedure: the first gives compact (in space use) circuits for Regular Flow one-way computations, and the second does the same for SSF one-way computations. As an application of the SSF translation procedure, I combine it with other translation and optimization techniques to give an automated quantum circuit optimization procedure. This procedure is based on back-and-forth translation between the 1WQC and the circuit model, using 1WQC techniques to time-optimize computations in the circuit model. In the second part of this thesis, I use 1WQC tools to analyze quantum circuits interacting with closed timelike curves (CTCs). I do so by translating to the 1WQC model CTC-assisted circuits, and then showing that in some cases they can be shown to be equivalent to time-respecting circuits. The predictions obtained in those cases are exactly those of the quantum CTC model based on post-selected teleportation, proposed by Bennett, Schumacher and Svetlichny (BSS). This enabled us to show that the BSS model for quantum CTCs makes predictions which disagree with those of the highly influential CTC model proposed by David Deutsch. Resumo Nesta tese eu estudo o modelo de computação quântica baseada em medições (CQBM) e algumas aplicações das diferentes maneiras de traduzir algoritmos de CQBM para o modelo de circuitos. Em uma série de resultados recentes, vários conjuntos de condições para implementar uma computação deterministicamente no modelo de CQBM têm sido propostas, cada um deles com diferentes propriedades. Alguns desses conjuntos de condições genericamente conhecidos como condições de fluxo (flow) tentam explorar o poder de paralelização do modelo de CQBM, aumentando o número de operações que podem ser realizadas simultaneamente. Aqui eu contribuo para essa linha de pesquisa definindo um novo tipo de fluxo, chamado fluxo de sinal deslocado (FSD), que tem uma estrutura paralela interessante que se iguala ao de um fluxo ótimo, do ponto de vista temporal. Eu também introduzo um novo sistema para traduzir algoritmos de CQBM para o modelo de circuitos. Esta tradução preserva não só a computação, mas também outras características de algoritmos em CQBM. Usando esse sistema eu desenvolvo dois algoritmos, cada um capaz de executar um procedimento de tradução diferente: o primeiro obtém circuitos compactos a partir de computações com fluxo regular, e o segundo faz o mesmo para computações com FSD. Como uma aplicação do procedimento de tradução de computações com FSD, eu combino esse procedimento com outras técnicas de tradução e otimização para desenvolver um procedimento automático de otimização de circuitos quânticos. Esse procedimento é baseado em traduções nos dois sentidos entre os modelos de CQBM e de circuitos, usando técnicas de CQBM para otimizar circuitos quânticos Na segunda parte desta tese, eu uso ferramentas do modelo de CQBM para analisar circuitos quânticos interagindo com curvas temporais fechadas (CTFs). Essa análise é feita traduzindo circuitos interagindo com CTFs para o modelo de CQBM e em seguida mostrando que, em alguns casos, esses circuitos podem ser transcritos como circuitos sem CTFs que realizam a mesma computação. As predições obtidas nesses casos são exatamente as mesmas daquelas obtidas usando o modelo para estudar CTFs proposto por Bennett, Schumacher e Svetlichny (BSS). Isso nos permitiu mostrar que o modelo BSS para CTFs faz predições que não concordam com aquelas dadas pelo influente modelo de CFTs proposto por David Deutsch. List of Figures 1 The application of the unitary evolutions Bell" (a) or Bell# (b) followed by the measurement of both qubits in the computational basis is equivalent to the measurement of those qubits in the Bell basis. The circuits in (a) and (b) will be used later as part of the teleportation protocol. . p. 12 2 The teleportation protocol (Section 2.1.2). . . . . . . . . . . . . . . . . p. 13 3 The double teleporter: two teleportation protocol circuits (Fig. 2) combined to teleport two spatially separated states |ai and |bi to the same place. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 16 4 A double teleporter with a CNOT gate being applied to the teleported states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 17 5 Adapted version of the double teleporter: using the four-qubit state | iabcd, this circuit applies the CNOT gate to the unknown states |ai and |bi using only measurements and classically controlled operations. . p. 18 6 Universal graph states: (a) Square Lattice, (b) Triangular Lattice, (c) Hexagonal Lattice, (d) Kagome Lattice. . . . . . . . . . . . . . . . . . . p. 22 7 Graphs implementing (a) Arbitrary single-qubit unitary, (b) CNOT gate, (c) CZ gate, (d) two arbitrary single-qubit rotations followed by a CZ gate and then other two single-qubit rotations. . . . . . . . . . . . . . . p. 30 8 (a) composed J gate teleportation protocol and (b) 1WQC protocol using adaptive measurements to implement the same unitary as in (a). . . . . p. 32 9 The simulation of a quantum circuit in a cluster state. The black dots represent qubits measured in the Z basis (and hence deleted from the cluster) and the circles with arrows represent qubits measured in the B(✓) basis. Different arrow directions indicate different angles of measurement. Each set of qubits boxed with a dashed line represents the simulation of a given circuit gate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 35 10 Examples of graphs satisfying different flow conditions. In Fig. (a) a graph with regflow is depicted, in (b) a graph with gflow but no regflow and in (c) a graph with three different gflows: a regflow, a generalized flow and an optimal gflow (see Section 3.3.6 for more information about this example). Each vertex represents a qubit, where the black vertices represent measured qubits and the white ones are unmeasured qubits. The input qubits are represented by boxed vertices and the arrows denote the dependency structure, pointing from vertex i to its correcting set. The dashed line with an arrow represent the case where the vertex from the correcting set is a non-neighbouring vertex (vertices not linked by an edge). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 57 11 Generic structure of extended circuits obtained from graphs with flow or gflow. See main text for information about the division into time slices. p. 64 12 Extended circuit for a simple one-way quantum computation protocol. This J-gate identity will be used repeatedly to simplify generic extended circuits in Chapters 5 and 6. Note that the J gate angles in (a) and (b) differ by a minus sign. . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 65 13 Two isomorphic graphs: (a) generically arranged graph and (b) same graph re-arranged to obey, from left to right, the partial ordering induced by regular flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 67 14 Quantum circuit obtained from the graph in Fig. 13-b using the Star Pattern translation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 67 15 Fig. (a) shows a subgraph that works as a building block for the Star Pattern translation. Decomposing a general graph into these subgraphs, we can translate each one to the corresponding circuit (Fig. b) and compose them to obtain the complete translated quantum circuit. . . . p. 68 16 Using the method in Def. 16 to generate an open graph state able to simulate the circuit on the top of the figure. . . . . . . . . . . . . . . . p. 70 17 Graphical representation of the star pattern translation being applied to the gflow pattern associated to the graph in (a). Figures (b) and (c) show the star pattern decomposition for the graph in (a). These subgraphs can be directly translated to the circuit model using the correspondence shown in Fig. 15. As a result we have the circuits in (d) and (e). Finally, composing these subcircuits we obtain the quantum circuit in (f). We can rewrite this circuit, preserving its meaning, to the circuit in (g) by adding a SWAP gate acting on the (newly added) time-traveling qubit [the round wire at the bottom of Fig. (g)] and the left-most wire segment where the anachornical CZ in Fig. (f) were acting upon. By doing so, the quantum state just before the SWAP gate is able to go to the future, be acted upon by the CZ gate and then go back to the past and re-enter the time respecting part of the circuit using the same SWAP gate. The circuit in (g) has no anachronical CZ like the circuit in (f) but instead there is a quantum wire interacting with a closed wire (which represents a qubit traveling in time). . . . . . . . . . . . . . . . . . . . . . . . . . p. 72 18 The J-gate identity. This is the same identity as the one shown in Sec. 4.1, and it is reproduced here for convenience of the reader. . . . . . . . p. 77 19 Extended ci