Integrating quantum sensors with bespoke quantum error correction

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Physical quantities such as time, phase, and entanglement cannot be measured directly, but instead must be inferred through indirect measurements. An important category of such indirect measurements is parameter estimation. Ideal quantum sensors would estimate physical quantities with unprecedented precision, but practical quantum sensors lose their quantum advantage because of noise. Incorporating quantum error correction codes into quantum sensors is an attractive theoretical approach to reduce noise, but is beset with practical difficulties. Namely, most quantum error correction codes (1) cannot be readily prepared in actual physical systems, (2) would introduce more errors than they correct during imperfect quantum error correction, and (3) can destroy the signal meant to be measured during quantum error correction.Most quantum error correction schemes are studied by abstracting away the physics of sensors, while quantum sensors are typically studied in the absence of quantum error correction. Mainstream approaches treat both quantum sensors and quantum error correction components as black boxes to be optimised separately. This project aims to break down the boundary between the quantum error correction black box and the quantum sensor black box, and integrate them to make an overall quantum error correction-integrated quantum sensor, by optimising over bespoke quantum error correction codes.A critical problem that using bespoke quantum error correction codes in optimising quantum sensors can overcome is the intractability of current numerical approaches in optimising quantum error correction codes for quantum sensors. These numerical methods impose no apriori structure on quantum error correction codes, and suffer from a runtime that increases exponentially in the number of particles. By choosing bespoke quantum codes that can be described with a tractable number of parameters, quantum sensors can be numerically optimised with respect to these codes in a scalable way.A prominent family of bespoke quantum error correction codes that this project will consider are symmetric codes. These codes are invariant under any permutation of the underlying particles, and have other practical advantages apart from the scalability in their numerical optimisations. First symmetric codes are very promising candidates for near-term implementation in physical devices, because their controllability by global fields could allow for their scalable physical implementations in near-term devices where addressability without cross-talk is difficult. Second, such symmetric codes can correct untracked particle losses, which are impossible to correct using conventional quantum error correction codes.This project will find optimal bespoke quantum error correction codes that maximise the quantum advantage attainable in the quantum estimation of classical fields, while also being easy to prepare in actual physical systems. The performance of symmetric codes will be compared with the performance of other families of bespoke quantum error correction codes. In the mathematical optimisation of the quantum sensor's precision, the project will take an integrated approach. Namely, the physical constraints of the quantum sensor such as the number of allowed qubits, operating temperature, and energy budget will be fixed, and the best quantum error correction codes for quantum sensors will be found. In doing so, this project will provide theoretical blueprints on how sensitivities of quantum sensors may be improved using existing quantum hardware.