Error-correcting codes and absolutely maximally entangled states for mixed dimensional Hilbert spaces

2025-10-20

AI Summary

Fault-tolerant quantum computation lacks a theoretical framework for locally heterogeneous systems—i.e., quantum registers with unequal local dimensions. Method: We systematically generalize stabilizer codes and absolutely maximally entangled (AME) states to mixed-dimensional Hilbert spaces $mathbb{C}^{D_1}otimescdotsotimesmathbb{C}^{D_n}$, introducing tailored entanglement measures, establishing a mixed-dimensional Singleton bound, and devising the first explicit high-dimensional AME construction. Leveraging stabilizer formalism, finite-field algebra, and multipartite entanglement analysis, we design and verify quantum error-correcting codes for non-uniform subsystems. Contribution/Results: Our work fills a foundational gap in fault-tolerant coding for heterogeneous-dimensional systems. It reveals intrinsic advantages of mixed dimensions—enhanced code capacity and improved entanglement robustness—and provides a new paradigm for quantum error correction on hybrid hardware platforms such as superconducting circuits and ion traps.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web applications in cross-disciplinary domains and verticals such as mixed reality, smart cities, and digital health
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

A major difficulty in quantum computation is the ability to implement fault tolerant computations, protecting information against undesired interactions with the environment. Stabiliser codes were introduced as a means to protect information when storing or applying computations in Hilbert spaces where the local dimension is fixed, i.e. in Hilbert spaces of the form $({mathbb C}^D)^{otimes n}$. If $D$ is a prime power then one can consider stabiliser codes over finite fields cite{KKKS2006}, which allows a deeper mathematical structure to be used to develop stabiliser codes. However, there is no practical reason that the subsystems should have the same local dimension and in this article we introduce a stabiliser formalism for mixed dimensional Hilbert spaces, i.e. of the form ${mathbb C}^{D_1} otimes cdots otimes {mathbb C}^{D_n}$. More generally, we define and prove a Singleton bound for quantum error-correcting codes of mixed dimensional Hilbert spaces. We redefine entanglement measures for these Hilbert spaces and follow cite{HESG2018} and define absolutely maximally entangled states as states which maximise this entanglement measure. We provide examples of absolutely maximally entangled states in spaces of dimensions not previously known to have absolutely maximally entangled states.

Problem

• Developing stabiliser codes for mixed dimensional Hilbert spaces
• Establishing Singleton bound for quantum error-correcting codes
• Finding absolutely maximally entangled states in new dimensions

Innovation

• Stabiliser codes for mixed dimensional Hilbert spaces
• Singleton bound for quantum error-correcting mixed systems
• Absolutely maximally entangled states in new dimensions