Quantum computers have the theoretical potential to solve problems intractable for classical computers. However, realizing this potential requires dealing with the noise inherent in near and far-term devices. One way of doing this is to redundantly encode the quantum information in a quantum error-correcting code and manipulate the encoded states to do computation. Protecting the quantum information in this way incurs additional space overhead in the form of extra qubits; this is problematic since qubits are a scarce resource, especially for near-term quantum computers. Reducing these overheads could significantly accelerate the arrival of large-scale, fault-tolerant quantum computation.

In this thesis, we address this topic of research and present techniques which aim to practically reduce the space and time overheads of implementing quantum error correction. The overarching motivation for the works presented in this thesis is the conception that it is advantageous, perhaps even essential, to measure every stabilizer generator when performing quantum error correction. To address this claim, we introduce partial quantum error correction, which we broadly define to be using incomplete syndrome information from the code or neglecting to correct errors on some part of the system. We show that it is not necessary to measure every stabilizer generator in order to obtain a threshold, and we will describe several situations where we obtain better logical performance and/or reduced overheads by not doing so.

In particular, we present an error correction protocol built on a bilayer architecture that aims to reduce operational overheads when restricted to 2D local gates by measuring some generators less frequently than others. We show through numerical simulations that high-rate quantum error correcting codes implemented with this protocol achieve logical error rates comparable to the surface code while using fewer physical qubits. We then introduce adaptive syndrome extraction as a scheme to improve code performance and reduce the quantum error correction cycle time by measuring only the stabilizer generators that are likely to provide useful syndrome information. We describe and numerically evaluate a concrete example of the scheme instantiated using a concatenated code and a syndrome extraction cycle that uses quantum error detection to modify the syndrome extraction circuits in real time.