Practical viability of a surface code architecture for quantum logic using donors and dots in silicon

Abstract

This project deals with the theory of a practical implementation of quantum information processing in solid state systems. I explore the practical feasibility of a proposal for a solid-state architecture for surface codes that uses a combination of bismuth donors and quantum dots to serve as physical qubits.

I begin with a brief summary of surface codes, which subsequent chapters will be motivated by and link back to. Each section of this chapter illustrates a fundamental component or function of surface codes, each of which motivate and contextualize a proceeding chapter. Where relevant or necessary I also touch upon elements of surface codes which are beyond the scope of the investigation for completeness.

The second chapter describes the physical architecture of the scheme, in addition to the states that are used to define the physical qubits, and the protocol envisioned for performing the crucial operations required for establishing a surface code array upon them. Much of this foundation is already laid out in the original proposal, and so I highlight the few areas where the concept has evolved from its initial concept.

The third chapter examines the coupling between the donor and dot electrons which facilitates adiabatic state transfer between the two, forming the basis of an entangling CNOT operation. I establish a mathematical description for the two states, and explore how the coupling strength may be evaluated as a function of donor-dot separation. I demonstrate also how the variation of coupling can be 'mapped' algebraically in order to be easily and quickly reproduced for use in simulations of the scheme.

The fourth chapter describes the principal investigation carried out in this work, which involved simulating the entangling CNOT operation between the dot and donor qubits. I first describe how constructing a time evolution operator for the system enables an approximation to be found for the fidelity of the adiabatic exchange. I then introduce a theoretical tool for obtaining a superoperator representation of the gate output, which can be applied to a master equation simulation of the gate to obtain a weighted error profile to be expressed as a weighted sum of pure Pauli operations applied to the desired CNOT gate, enabling statistical modeling of the architecture.

In the last of the core chapters, I discuss how the ability to model an array of donor and dot qubits statistically enables the resilience of the scheme to error to be evaluated. I then extend this to practical considerations of implementing the scheme in reality, culminating with the accuracy required for the placement of the donor and dot when fabricating a device.

The final chapter contrasts the threshold accuracy found with current experimental limitations, leading to the conclusion that building such a device may be possible in the near future. I end by discussing what the next logical steps for developing the scheme would be, and what else is needed to achieve universal quantum computation.