Emulating Circuit-Based and Measurement-Based Quantum Computation

This thesis presents classical emulations of circuit-based and measurement-based quantum com-putation. In order to create emulators faithful to the theory of quantum computation we developthe required mathematics, quantum mechanics and quantum computing background. We justifyour implementations by documenting the theory of each of these subjects in detail and showing ourimplementation directly follows from it.In relation to the circuit model we discuss quantum bits (qubits), quantum gates, and quantumcircuits with a view to implementing quantum algorithms. In addition, we develop a quantum circuitcalculus in order to formally specify quantum circuit diagrams.We base our discussion of the measurement-based approach upon the the one-way computer archi-tecture and the measurement calculus. This calculus provides a formal description of measurement-based quantum computation and awards us a quantum assembly language; we structure our im-plementation around this language. Much of the measurement-based model can be equated to thecircuit model, and we do so to help build intuition. We describe von Neumann and generalisedprojective measurement of qubits in non-standard bases; the fundamental operation of measurement-based computation.The evaluation of our implementations covers correctness, accuracy, bounds and efficiency, andlimitations. For both emulators we successfully implement the Deutsch and Deutsch-Jozsa algorithms.For the circuit model we also run Shor’s algorithm for numbers up to 16, requiring 12 combined qubits.With some modification to memory management we are able to represent 13 entangled qubits. Wedescribe the workings of each of these algorithms in detail, and from this we justify the correctness ofour emulators. We give the running times of each algorithm, and we judge the accuracy of probabilisticactions and quantum state representation for each implementation. From these individual evaluationswe provide some comparison between the presented models.Although our emulators are both correct and accurate, we achieve this at the cost of run timeand the number of qubits we are able to represent. In comparison to other circuit model simulationswe achieve fewer combined qubits. For the measurement-based emulator we determine that a muchlower precision representation may be used for qubits. Our investigation reveals that measurementsdestroy any rounding errors introduced into the system. This reduced precision would allow us torepresent a greater number of combined qubits. With this change, our emulations would competewith the majority of existing simulations in therms of qubits represented and run time.