Quantum error-correcting codes are essential in the realization of a scalable fault-tolerant quantum computation. Traditionally, these codes encodes logical information in a fixed subspace of a many-body quantum system which allow correction of errors by performing commuting measurements to determine appropriate corrections. By allowing non-commuting measurements, one obtain the so called "subsystem’’ code which allow for simpler measurements and the ability to perform universal fault-tolerant computation by switching across logical subspaces. Moreover, one could also obtain a "dynamical’’ code that traverses multiple logical subspaces over time as a sequence of these non-commuting measurements is performed. It has been shown that dynamical codes allow for more information to be encoded over space and time with improved error-correction performance in multiple cases. This perspective opens a plethora of possibilities in error-correction and fault-tolerance, while revealing interesting temporal properties of quantum error-correcting codes.
In this talk I’m going to present four results on quantum error-correcting codes in the spatio-temporal perspective. First, I will present how non-classicality manifests in quantum error-correcting codes involving non-commuting measurements, particularly in the form of quantum contextuality. Second, I will present a construction of a family of dynamical codes consists of higher-dimensional quantum systems (qudits) with a high encoding-rate dynamically, but do not encode any information when viewed as a subsystem code. Third, I will discuss a proposal of a universal framework for quantum error-correcting codes, called the "strategic code’’ framework, unifying all codes with fixed or dynamic logical subspace, as well as more general codes which logical subspaces may be determined adaptively in real-time. Particularly, I will demonstrate the application of this framework in obtaining necessary and sufficient conditions to guarantee correction of errors from any physically plausible noise model, including those with spatial and temporal (non-Markovian) correlations. Lastly, I’m going to show an optimization-based construction of dynamical codes under non-Markovian noise model and show their performances. I will conclude by discussing some open questions regarding constructions of better codes, fault-tolerant computation, and their relationships to quantum foundations.
This talk is based on: arXiv:2405.17567, arXiv:2410.02022, arXiv:2502.02553, and arXiv:2502.09177.
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