[PhD] Efficient Fault-Tolerant Quantum Computation with Quantum LDPC Codes
The thesis will be hosted in the MOCQUA team under Emmanuel Jeandel and Christophe Vuillot’s supervision. There will be frequent meetings and interactions with other members of the team will be encouraged.
Quantum computers, if built, would be able to solve some problems that are as of today intractable in practice for classical computers, and also speed up significantly some others. These remarkable theoretical results kick-started a global effort, academic as well as industrial, to address the roadblocks still preventing the construction of a large scale universal quantum computer. One of the main roadblocks is the sensitivity of real physical quantum systems to uncontrolled perturbation from their environment as well as imperfect control of the system, ever present in any real physical machine. So-called threshold theorems proved that this issue can be tackled by the implementation of fault-tolerant quantum computation [1, 2, 3]. Different schemes for universal and fault-tolerant quantum computation combine in some ways some families of quantum error correcting codes with techniques to obtain a set of universal fault-tolerant gates acting on them . These schemes detect and remove errors from the computation at the cost of an overhead in physical qubits used and length of the computation. The magnitude of these overheads with state of the art techniques remain today prohibitive and comes at odds with the possibility of building a quantum computer . This subject proposes to address this issue by exploring new schemes with the potential to reduce these overheads both from an asymptotic point of view and a finite-size practical one.
Most schemes for fault-tolerant quantum computation such as concatenated codes as well as the leading candidate, surface code  with magic state distillation  all have a poly-logarithmic overhead in both time and space. The study of single-shot error correction and quantum error correcting codes admitting transversal gates yielded schemes with only a constant time overhead . Yet, all the above techniques suffer from one drawback of the underlying quantum error correcting code families: namely their encoding rate is asymptotically vanishing. Hence they cannot achieve better than poly-logarithmic space overhead.
The field of quantum low-density parity-check (LDPC) codes has attracted a lot of interest in recent years since such codes can have constant encoding rates with growing minimum distance. Moreover their LDPC property makes them suitable for fault-tolerant implementations. The possibility for having to pay only a constant overhead in space using such codes was put forward in  and recently proved to be achievable using quantum expander codes . However by encoding several logical qubits within the same block of code these codes render difficult to act independently on different logical qubits of the same block. Indeed the scheme in [9, 10] can only work by applying one gate at a time and therefore cannot work on parallel circuits which need to be first linearized. More recently it was proposed to use some 2D hyperbolic
codes with constant space overhead and efficient code deformation techniques which could have a poly-logarithmic time overhead (including for parallel circuits) . Yet the question of being able to achieve a constant overhead in both time and space remains, to this day, open.
A first approach is to design code families with built-in capabilities for fault-tolerant gates, for instance generalizing ideas from quantum color codes . A first step in this direction was taken by generalizing quantum color codes to quantum pin codes  which are block codes retaining some structure from quantum color codes responsible for their interesting computing properties. Finding good instances of quantum pin codes as well as a full scheme using them for fault-tolerant quantum computation is a promising direction.
A second approach is to apply general techniques to already known good families of code such as code deformation techniques . First results along this direction were taken for hyperbolic codes and hypergraph-product codes [14, 11, 15]. Extending these techniques is also a promising direction.
Candidates should posses a creative as well as mathematically rigorous mindset, a master degree or equivalent in either theoretical physics, computer science or mathematics. Previous work or interest in the theory of quantum computing and/or error correction would be a strong asset.
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