All pure fermionic non-Gaussian states are magic states for matchgate computations (1905.08584v2)

Abstract: Magic states were introduced in the context of Clifford circuits as a resource that elevates classically simulatable computations to quantum universal capability, while maintaining the same gate set. Here we study magic states in the context of matchgate (MG) circuits, where the notion becomes more subtle, as MGs are subject to locality constraints and also the SWAP gate is not available. Nevertheless a similar picture of gate-gadget constructions applies, and we show that every pure fermionic state which is non-Gaussian, i.e. which cannot be generated by MGs from a computational basis state, is a magic state for MG computations. This result has significance for prospective quantum computing implementation in view of the fact that MG circuit evolutions coincide with the quantum physical evolution of non-interacting fermions.

• The paper establishes that every pure fermionic non-Gaussian state qualifies as a magic state for matchgate computations.
• It generalizes magic state theory by showing that only non-Gaussian fermionic states, not Gaussian ones, provide the computational advantage.
• The study demonstrates a deterministic SWAP via a 4-qubit magic state, offering a practical route to universal quantum computation.

Overview of "All Pure Fermionic Non-Gaussian States are Magic States for Matchgate Computations"

The research paper "All pure fermionic non-Gaussian states are magic states for matchgate computations" offers a significant advancement in quantum computing by exploring the concept of magic states within the context of matchgate (MG) circuits. In quantum information theory, magic states provide the necessary resources to convert classically simulatable computation systems into quantum universal ones, doing so while maintaining a consistent gate set. This work extends the scope of magic states beyond Clifford circuits to MG circuits, which have distinct structural constraints not present in the former, such as locality limits on gate operations.

Core Contributions

1. Subtle Role of Magic States in MG Circuits: This paper establishes that for MG computations, every non-Gaussian pure fermionic state qualifies as a magic state. This finding is supported by the theoretical framework set forth in the computation of non-interacting fermions, which relate directly to MG circuits via the Jordan-Wigner transform.
2. Generalization of Magic State Theory to Non-Gaussian States: The research elegantly broadens the classical notion of magic states (i.e., for Clifford gates) by focusing on the Gaussian and non-Gaussian distinctions among quantum states. The paper identifies that Gaussian states, by virtue of being generable by MG circuits, do not suffice as magic states—a role fulfilled instead by non-Gaussian states.
3. Deterministic Implementation of SWAP Gates: The authors provide practical demonstration through a specific 4-qubit magic state, which is leveraged to deterministically implement a SWAP gate—a resourceful gate—via MGs and adaptive measurements. This construction provides paths to extend log-space bounded computational capabilities of MG circuits to full universal quantum computation.
4. Constructive Procedures and Proofs: The paper presents detailed lemmas that guide the transformation of any non-Gaussian fermionic state into a computationally resourceful state within the MG circuit model. Additionally, these transformations are shown to be efficient, with provable bounded error probabilities that ensure their feasibility in practical quantum computing.

Theoretical and Practical Implications

From a theoretical vantage point, this work delineates the boundary between Gaussian and non-Gaussian operations within MG contexts, elucidating the subtle transformative role of fermionic non-Gaussian states as computational resources. Practically, this research guides the development of quantum algorithms and hardware that can exploit these magic states to transcend classical simulation barriers, paving the way for refined designs of quantum circuits that harness MG-based evolution paths.

Speculative Future Developments

Building on the implications drawn in this paper, future work will undoubtedly focus on efficiently leveraging these magic states in near-term quantum computing devices and developing robust error-correction schemes to manage inherent quantum noise. Additionally, uncovering generalized states for broader classes of resourceful operations and intending strategies for state distillation will enhance the practicality of these findings. Furthermore, the insights into compressed quantum circuits suggest potential areas of innovation in quantum algorithm optimization and scalability.

By expanding the theory of magic states into the domain of MG computations, this paper constitutes a foundational step towards exploiting limited quantum systems more effectively, offering new directions for achieving universality in near-term quantum devices.