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Contextuality supplies the magic for quantum computation (1401.4174v2)

Published 16 Jan 2014 in quant-ph

Abstract: Quantum computers promise dramatic advantages over their classical counterparts, but the answer to the most basic question "What is the source of the power in quantum computing?" has remained elusive. Here we prove a remarkable equivalence between the onset of contextuality and the possibility of universal quantum computation via magic state distillation. This is a conceptually satisfying link because contextuality provides one of the fundamental characterizations of uniquely quantum phenomena and, moreover, magic state distillation is the leading model for experimentally realizing fault-tolerant quantum computation. Furthermore, this connection suggests a unifying paradigm for the resources of quantum information: the nonlocality of quantum theory is a particular kind of contextuality and nonlocality is already known to be a critical resource for achieving advantages with quantum communication. In addition to clarifying these fundamental issues, this work advances the resource framework for quantum computation, which has a number of practical applications, such as characterizing the efficiency and trade-offs between distinct theoretical and experimental schemes for achieving robust quantum computation and bounding the overhead cost for the classical simulation of quantum algorithms.

Citations (545)

Summary

  • The paper demonstrates that quantum contextuality is equivalent to enabling universal quantum computation through magic state distillation.
  • It employs a graph-theoretical framework to identify noncontextuality inequalities whose violation indicates computational advantages in qudit systems.
  • The findings offer actionable insights for designing fault-tolerant algorithms and refining error-correction protocols in quantum computing.

Contextuality as a Resource in Quantum Computation

The paper "Contextuality supplies the magic for quantum computation" by Howard et al. establishes a pivotal connection between quantum contextuality and the capabilities of quantum computation. The authors present a compelling argument that contextuality, a fundamental quantum mechanical feature, is essential for achieving universal quantum computation through magic state distillation (MSD).

Key Insights

The paper reveals an equivalence between the presence of contextuality in a quantum system and the feasibility of universal computation via MSD, which is critical for fault-tolerant quantum computation. This relationship underscores the role of contextuality as a crucial resource, akin to the role of nonlocality in quantum communication. Contextuality, a broader phenomenon that encompasses nonlocality, serves as a distinguishing feature of quantum mechanics, challenging classical intuitions such as locality and realism.

Theoretical Framework

The paper introduces a framework based on the work of Cabello, Severini, and Winter that leverages graph theory to analyze contextuality. The authors utilize this framework to identify noncontextuality inequalities, where the violation of these inequalities aligns with the onset of quantum computational advantages. Specifically, the authors demonstrate that for qudits (d-dimensional quantum systems, where d is an odd prime), a state exhibits contextuality—and hence can be utilized in MSD—if and only if it violates a noncontextuality inequality and exists outside of the classical simulation boundary, designated as the PSIM polytope.

Implications and Future Directions

The findings of the paper have significant implications for both the theoretical understanding and practical realization of quantum computation. By identifying contextuality as a fundamental resource, the results provide a unified perspective on quantum and classical computational resources. Practically, these insights could inform the design of quantum algorithms and error-correction protocols, specifically by delineating the requirements for achieving quantum computational speedup.

In terms of future directions, the paper opens the door to further exploration of contextuality in various quantum computational frameworks. For qubits, the results suggest that more refined measures of contextuality may be necessary to fully capture the nuances of quantum speedup. Additionally, continued investigation into the role of contextuality in other quantum information tasks, such as communication and cryptography, may uncover further connections and applications.

Conclusion

This work establishes contextuality as a quintessential resource for quantum computation, analogous to other quantum phenomena like entanglement and nonlocality, yet distinct in its operational scope. The theoretical and practical ramifications of this discovery promise to shape the future landscape of quantum computing, offering a consolidated framework for understanding and harnessing the advantages offered by quantum mechanics. The intersection of contextuality, stability through error corrections, and computational power represents a rich area for ongoing and future research endeavors in quantum information science.