Fault-tolerant quantum computation

Abstract

Quantum computing is a field that shows tremendous possibilities and promise. It can provide an exponential speedup compared to classical computers in many computational problems, including simulations of general quantum mechanical systems, pattern finding and solving linear systems. Quantum computations can be performed by making use of qubits and performing operations, called gates, on them. A physical realisation of a qubit will be faulty, and to combat the errors that will inadvertently but unavoidably occur, the use of a quantum error correction code (QECC) is needed. A QECC is able to correct static errors; errors that occur on the states. However, it can not correct errors that occur on and during the operations. To make the computer resilient to these errors, the implementation of the QECC has to be performed fault-tolerantly. A fault-tolerant implementation of a quantum operation is designed in such a way that errors that happen during the operation, through faulty gates, do not propagate to errors on the states that are not correctable any more by the QECC. Hence, error propagation is a key concept in fault-tolerance. Every element of a general quantum circuit needs to be designed fault-tolerant. One of the elements that will be needed in any realistic quantum computer architecture is the possibility of exchanging the states of two qubits, so that an arbitrary pair of qubits can be brought physically close, allowing the implementation of multi-qubit gates. Exchanging the states of two qubits can be performed by a fault-tolerant SWAP gate, that makes use of one ancilla qubit. In this thesis, a fault-tolerant SWAP operation is implemented on the IBM 5-qubit ‘Tenerife’ device. It is characterized as a quantum channel by performing quantum process tomography (QPT). The CPTP constraints of quantum channels in the framework of QPT are discussed and a method of obtaining a CPTP estimate of an initial QPT reconstruction is given. The error process of the fault-tolerant SWAP is analysed and subsequently compared to a normal implementation of the SWAP gate, on which tomography is performed as well. The main measure that is used in the comparison is the ratio r of multi-qubit errors (or correlated) to single-qubit (or uncorrelated) errors in the two implementations. SPAM errors are an important source of estimated infidelity. To filter these SPAM errors and obtain a better estimate of only the circuits, SPAM errors are approximated to be only measurement errors. The errors are characterized by performing tomography on an arbitrary short identity channel, and subsequently using this as an estimate of the measurement error map. This gives a method to estimate a representation of the circuit itself. We can report on an error ratio r of 0.170±0.0029 for the fault-tolerant SWAP operation and an error ratio r of 1.256 ± 0.0129 for the non-fault-tolerant SWAP operation. However, the total error in the fault-tolerant SWAP is much higher, resulting in a process fidelity Fp = 0.608 for the fault-tolerant implementation and Fp = 0.742 for the normal implementation. This is due to the larger number of gates in the fault-tolerant circuit. This research shows that a SWAP operation can be implemented fault-tolerantly, but that the error rates of the devices need to be reduced before their use becomes viable.