Adam Wills

The development of quantum codes with good error correction parameters and useful sets of transversal gates is an area of major interest in quantum error correction. Abundant prior works have studied transversal gates which are restricted to acting on all logical qubits simultaneously. In this work, we study codes that support transversal gates which induce addressable logical gates, i.e., the logical gates act only on logical qubits of our choice. As we consider scaling from low-rate to high-rate codes, the study and design of low-overhead, addressable logical operations presents an important problem for both theoretical and practical purposes. In this work, we construct the first quantum codes to support transversally addressable non-Clifford gates. Concretely, given any three logical qubits across one or multiple codeblocks, one can execute the logical CCZ on those qubits via a depth-one physical circuit of CCZ gates. We present a simple, explicit construction based on Reed-Solomon codes that is nearly asymptotically good, and a more involved, asymptotically good construction based on transitive, iso-orthogonal algebraic geometry codes. We go on to develop a powerful theory of quantum codes supporting a rich class of transversally addressable gates in the Clifford hierarchy, going far beyond just the CCZ gate. We call this framework addressable orthogonality, and show that it can be used to construct asymptotically good quantum codes supporting an arbitrary product of multiply-controlled Z gates transversally and addressably, enabling major adaptivity to particular algorithms. Our constructions mark the first quantum codes to support any multi-qubit gate transversally and addressably. Accordingly, our results have major implications for the general addressabilitiy problem in error correction. This is a merged submission based on arXiv:2502.01864 and arXiv:2507.05392.

In this work, we continue the search for quantum locally testable codes (qLTCs) of new parameters by presenting three constructions that can make new qLTCs from old. The first analyses the soundness of a quantum code under Hastings' weight reduction construction for qLDPC codes to give a weight reduction procedure for qLTCs. Secondly, we describe a novel `soundness amplification' procedure for qLTCs which can increase the soundness of any qLTC to a constant while preserving its distance and dimension, with an impact only felt on its locality. Finally, we apply the AEL distance amplification construction to the case of qLTCs for the first time which can turn a high-distance qLTC into one with linear distance, at the expense of other parameters. These constructions can be used on as-yet undiscovered qLTCs to obtain new parameters, but we also find a number of present applications. Applying these constructions in various combinations to recent advancements yields near-optimal quantum locally testable codes.

Port-based teleportation (PBT) is a form of quantum teleportation in which no corrective unitary is required on the part of the receiver. Two primary regimes exist - deterministic PBT in which teleportation is always successful, but is imperfect, and probabilistic PBT, in which teleportation succeeds with probability less than one, but teleportation is perfect upon a success. Two further regimes exist within each of these in which the resource state used for the teleportation is fixed to a maximally entangled state, or free to be optimised. Recently, works resolved the long-standing problem of efficiently implementing port-based teleportation, tackling the two deterministic cases for qudits. Here, we provide algorithms in all four regimes for qubits. Emphasis is placed on the practicality of these algorithms, where we give polynomial improvements in the known gate complexity for PBT, as well as an exponential improvement in the required number of ancillas (albeit in separate protocols). Our approach to the implementation of the square-root measurement in PBT can be directly generalised to other highly symmetric state ensembles. For certain families of states, such a framework yields efficient algorithms in the case that the Petz recovery algorithm for the square-root measurement runs in exponential time.