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The geometric measure, the logarithmic robustness and the relative entropy of entanglement are proved to be equal for a stabilizer quantum codeword. The entanglement upper and lower bounds are determined with the generators of code. The…
We establish the connection between a recent new construction technique for quantum error correcting codes, based on graphs, and the so-called stabilizer codes: Each stabilizer code can be realized as a graph code and vice versa.
Surface and color codes are two forms of topological quantum error correction in two spatial dimensions with complementary properties. Surface codes have lower-depth error detection circuits and well-developed decoders to interpret and…
We describe a method to use measurements and correction operations in order to implement the Clifford group in a stabilizer code, generalising a result from [Bombin,2011] for topological subsystem colour codes. In subsystem stabilizer codes…
We introduce a methodology for generating random multi-qubit stabilizer codes based on solving a constraint satisfaction problem (CSP) on random bipartite graphs. This framework allows us to enforce stabilizer commutation, $X/Z$ balancing,…
Topological codes have many desirable properties that allow fault-tolerant quantum computation with relatively low overhead. A core challenge for these codes, however, is to achieve a low-overhead universal gate set with limited…
We introduce a high-level graphical framework for designing and analysing quantum error correcting codes, centred on what we term the coherent parity check (CPC). The graphical formulation is based on the diagrammatic tools of the…
Quantum error-correcting codes are essential to the implementation of fault-tolerant quantum computation. Homological products of classical codes offer a versatile framework for constructing quantum error-correcting codes with desirable…
Magic state distillation and the Shor factoring algorithm make essential use of logical diagonal gates. We introduce a method of synthesizing CSS codes that realize a target logical diagonal gate at some level $l$ in the Clifford hierarchy.…
Braiding defects in topological stabiliser codes has been widely studied as a promising approach to fault-tolerant quantum computing. Here, we explore the potential and limitations of such schemes in codes of all spatial dimensions. We…
Recent discoveries in asymptotically good quantum codes have intensified research on their application in quantum computation and fault-tolerant operations. This study focuses on the addressability problem within CSS codes: what circuits…
By defining projective error models we study the mathematical structure of Clifford codes and stabilizer codes using tools from projective representation theory. Furthermore, we introduce a new class of codes which we have called weak…
A quantum error correcting code protects encoded logical information against errors. Transversal gates are a naturally fault-tolerant way to manipulate logical qubits but cannot be universal themselves. Protocols such as magic state…
Storing quantum information in a quantum error correction code can protect it from errors, but the ability to transform the stored quantum information in a fault tolerant way is equally important. Logical Pauli group operators can be…
Braiding defects in topological stabiliser codes has been widely studied as a promising approach to fault-tolerant quantum computing. We present a no-go theorem that places very strong limitations on the potential of such schemes for…
The Clifford hierarchy is a set of gates that appears in the theory of fault-tolerant quantum computation, but its precise structure remains elusive. We give a complete characterization of the diagonal gates in the Clifford hierarchy for…
We formulate a bounded distance decoding strategy applicable to all stabilizer codes including both CSS and non-CSS code-families. The framework emerges out of the local Clifford equivalence between arbitrary stabilizer states and graph…
A long-standing open problem in fault-tolerant quantum computation has been to find a universal set of transversal gates. As three of us proved in arXiv: 0706.1382, such a set does not exist for binary stabilizer codes. Here we generalize…
Hypergraph product codes are a class of quantum low density parity check (LDPC) codes discovered by Tillich and Z\'emor. These codes have a constant encoding rate and were recently shown to have a constant fault-tolerant error threshold.…
We propose a scheme for detecting and correcting faults in any Clifford circuit. The scheme is based on the observation that the set of all possible outcome bit-strings of a Clifford circuit is a linear code, which we call the outcome code.…