Visualization of fault-tolerant operations

Abstract

Current quantum computers are prone to errors and one method to reduce these errors is based on a technique called quantum error correction. In quantum error correction it is possible to construct a logical qubit from multiple physical qubits to protect the quantum information within it on the effects of physical errors. In this thesis we introduce the three qubit code, the five qubit code and the seven qubit Steane code. The Steane code, which is often marked as a [[7,1,3]] code, encodes a single logical qubit from seven physical qubits and can correct a single error that has happened during a time evolution. In order to perform useful quantum computation using current and near future quantum computers, which are still noisy, logical qubits and logical gates with very low error rates are needed. To achieve this, different methods for quantum error correction codes and fault-tolerant quantum computation are very topical right now and they are being researched actively. In this thesis, we visualize fault-tolerant quantum computation using the Steane code. In fault-tolerant quantum computation, we try to find quantum circuits that detect possible error propagations from a physical qubit to another physical qubit. Error propagation can be a serious problem, because if the errors propagate too widely in the physical qubit registers, the quantum error correction codes will not be able to correct the errors. In quantum computation there are three important operations that are needed and that are studied in a fault-tolerant manner in this thesis, which are state preparation, quantum gates and measurements. We also examine how to obtain a universal gate set for the Steane code using fault-tolerant operations. Using programming languages Python and Qiskit, we visualize how to do these operations fault-tolerantly in the Steane code. We benchmark the fault-tolerant state preparation by using a quantum computer simulator with a bit-flip noise model. The bit-flip noise model is used to model a noisy quantum computer, where a bit-flip error can happen in any of the operations of the quantum computer. We compare the results of the fault-tolerant state preparation scheme to a non-fault-tolerant state preparation scheme by calculating relative Hamming distances and logical fidelities to obtain numerical results. We show that the relative amount of errors with distances two or three are reduced when the fault-tolerant Steane logical state preparation circuit is used, compared to the non-fault-tolerant counterpart. In addition, the fault-tolerant scheme has also a lower logical infidelity compared to the non-fault-tolerant scheme, which shows that there is a benefit in doing the state preparation in a fault-tolerant fashion.