Academic Awards 2025 booklet

17 Algebraic Characterisation of Bivariate Bicycle Codes Quantum computers promise great advancements in the future but are currently infeasible to build. One of the main issues is the probabilistic nature of quantum bits (figure 1), which necessitates error correction. No practically implementable scalable schemes had been found, until bivariate bicycle codes (figure 2) were introduced last year. Although originally these codes were characterised using mathematical parameters, I showed that a geometric characterisation can reveal additional information, such as restrictions and symmetries, greatly reducing the time needed to search for good parameters. Alongside this, I started algebraic research on which mathematical parameters produce good codes, finding a lower bound on the number of logical qubits, further reducing the search time. Next, to physically implement bivariate bicycle codes, data and check qubits need to interact according to the non-local edges. Whereas a non-local approach already exists, it is non-scalable. Therefore, I researched the possibility of moving qubits as to only require local interactions. The resulting transpilation scheme exploits the bicyclic property of the code, allowing the non-local edges to be placed as spokes in a wheel of local edges (figure 3). Though slower, its scalability makes moving qubits a relevant part of the implementation of large-scale bivariate bicycle codes. Figure 1: A visualisation of the values a quantum bit (qubit) can take, called the Bloch sphere. Due to external influences and the continuous spectrum of possible values, minor errors can occur in two degrees of freedom, conventionally taken to be the X and Z direction. Figure 2: The Tanner graph for a bivariate bicycle code for a 12-qubit quantum computer, which contains 72 physical qubits and can correct any 2 errors. Grey qubits store data, whereas blue and green qubits check for mistakes in the two degrees of freedom, Z and X, respectively. Lines indicate a connection between a data and a check qubit, meaning that that check qubit incorporates the data from that data qubit in its value. Note that this Tanner graph has periodic boundary conditions.