Efficient witnessing and testing of magic in mixed quantum states (2504.18098v1)

Abstract: Nonstabilizerness or `magic' is a crucial resource for quantum computers which can be distilled from noisy quantum states. However, determining the magic of mixed quantum has been a notoriously difficult task. Here, we provide efficient witnesses of magic based on the stabilizer R\'enyi entropy which robustly indicate the presence of magic and quantitatively estimate magic monotones. We also design efficient property testing algorithms to reliably distinguish states with high and low magic, assuming the entropy is bounded. We apply our methods to certify the number of noisy T-gates under a wide class of noise models. Additionally, using the IonQ quantum computer, we experimentally verify the magic of noisy random quantum circuits. Surprisingly, we find that magic is highly robust, persisting even under exponentially strong noise. Our witnesses can also be efficiently computed for matrix product states, revealing that subsystems of many-body quantum states can contain extensive magic despite entanglement. Finally, our work also has direct implications for cryptography and pseudomagic: To mimic high magic states with as little magic as possible, one requires an extensive amount of entropy. This implies that entropy is a necessary resource to hide magic from eavesdroppers. Our work uncovers powerful tools to verify and study the complexity of noisy quantum systems.

Overview of Magic in Mixed Quantum States

The research paper under review presents a comprehensive paper of nonstabilizerness, colloquially referred to as 'magic', and its pivotal role in the field of quantum computing. Magic is an essential quantum resource for achieving universal quantum computation, particularly in the domain of fault-tolerant quantum computers. These computers rely on the integration of non-universal Clifford operations with non-Clifford gates, with the latter being indispensable for universal operations. The ability to distill magic from noisy quantum states is crucial, yet poses significant challenges due to the detrimental effect of noise on magic.

Key Contributions

The paper provides several major contributions to the field:

1. Efficient Magic Witnesses: The authors introduce efficient witnesses of magic based on the stabilizer Rényi entropy (SRE). These witnesses offer robust indicators of magic presence and enable quantitative estimations of magic monotones. Thoroughly designed property testing algorithms are delineated to distinguish states based on their level of magic efficiently, given bounded entropy conditions.
2. Experimental Demonstrations: Implementations on the IonQ quantum computer reveal that magic persists even under exponentially strong noise, showcasing surprising levels of robustness. This experimental verification underscores the practical viability of the proposed method.
3. Evaluation of T-Gates: The paper outlines a method to certify the number of noisy T-gates across a broad array of noise models. This has significant implications for fault-tolerant quantum computing, where T-gates serve as vital resource states.
4. Potential in Cryptography: The paper identifies implications for cryptography and pseudomagic, highlighting entropy as a necessary resource for concealing magic from potential eavesdroppers. This introduces new perspectives on secure quantum communication.

Numerical Insights and Implications

The authors demonstrate several strong numerical results, indicating the robustness and efficiency of the proposed witnesses. Notable findings include:

• Matrix Product State Analysis: The witnesses have been efficiently applicable to matrix product states, highlighting that subsystems of many-body quantum states can harbor extensive magic despite existing entanglement.
• Magic Robustness Under Noise: Experimental findings show that even random quantum circuits containing noise exhibit significant amounts of magic, reaching critical depths irrespective of qubit numbers. This suggests potential scalability and practical implementations in noisy quantum technologies.

Theoretical and Practical Implications

The paper's insights have profound theoretical and practical implications:

• Enhanced Quantum System Complexity Analysis: The tools provided enable deeper understanding and verification of complexity within noisy quantum systems, aiding both experimental endeavours and theoretical developments.
• Advancements in Quantum Cryptography: The paper opens new avenues for enhancing cryptographic protocols by leveraging the properties of magic and entropy.
• Future Quantum Technologies: The findings pave the way for further exploration into quantum magic's role in cryptography, security, and advanced computational frameworks, potentially influencing future quantum technology designs and applications.

Conclusion

The paper significantly advances our understanding of magic in quantum systems, providing robust methodologies for witnessing and testing magic in mixed states. These contributions not only foster experimental validation but also strategically connect quantum computing and cryptographical innovations. The pioneering witness methods proposed might inform future research on the interplay between quantum resource theories and noise, promoting the development of scalable quantum technologies.