A complete theory of the Clifford commutant (2504.12263v1)

Abstract: The Clifford group plays a central role in quantum information science. It is the building block for many error-correcting schemes and matches the first three moments of the Haar measure over the unitary group -a property that is essential for a broad range of quantum algorithms, with applications in pseudorandomness, learning theory, benchmarking, and entanglement distillation. At the heart of understanding many properties of the Clifford group lies the Clifford commutant: the set of operators that commute with $k$-fold tensor powers of Clifford unitaries. Previous understanding of this commutant has been limited to relatively small values of $k$, constrained by the number of qubits $n$. In this work, we develop a complete theory of the Clifford commutant. Our first result provides an explicit orthogonal basis for the commutant and computes its dimension for arbitrary $n$ and $k$. We also introduce an alternative and easy-to-manipulate basis formed by isotropic sums of Pauli operators. We show that this basis is generated by products of permutations -which generate the unitary group commutant- and at most three other operators. Additionally, we develop a graphical calculus allowing a diagrammatic manipulation of elements of this basis. These results enable a wealth of applications: among others, we characterize all measurable magic measures and identify optimal strategies for stabilizer property testing, whose success probability also offers an operational interpretation to stabilizer entropies. Finally, we show that these results also generalize to multi-qudit systems with prime local dimension.

Complete Theory of the Clifford Commutant

This paper presents a comprehensive theory of the Clifford commutant, addressing its dimensions, basis, and applications in quantum information science. The Clifford group is significant due to its role in quantum error correction, quantum simulation, and resource theories. Its characterization as a unitary 3-design positions it as a cornerstone in various quantum algorithms.

Key Contributions

1. Basis for the Clifford Commutant: The authors establish an explicit orthogonal basis for the Clifford commutant for arbitrary numbers of qubits $n$ and tensor powers $k$. They mesh this basis with isotropic sums of Pauli operators, presenting a framework for manipulating these operators diagrammatically.
2. Minimal Generating Set: Beyond permutations used for generating unitary commutants, three additional elements—specific isotropic sums of Pauli operators—are identified as generators for the Clifford commutant across all orders. This finding gracefully extends earlier results that the Clifford group fails to be a unitary 4-design, now shown to hold for any $k$.
3. Dimensional Analysis: The dimension of the Clifford commutant is determined, revealing complexity depending on the number of qubits and moments considered. High permutations and efficient algebraic properties characterize it.
4. Graphical Calculus: A graphical technique for handling Pauli monomials improves the interpretability and manipulation of these bases. This calculus simplifies deriving expressions for Clifford averages, critical for applications in quantum benchmarking and state testing.
5. Applications to Magic-State Resource Theory: The paper aligns measurable magic monotones with expectations of the Clifford commutant, suggesting a conceptual bridge from quantum state properties to structural group properties. Specifically, stabilizer entropies emerge as characterizations of the commutant's basis elements.
6. Property Testing of Stabilizer States: Insights into property testing protocols and stabilizer state testing are provided, emphasizing strategies that analyze quantum state deviations in Clifford settings.
7. Multidimensional Extensions: The framework presented extends naturally to multi-qudit systems with prime dimensions, underscoring the generality and potential reach of the Clifford commutant theory.

Implications for Quantum Computing

1. Quantum Algorithm Design: Understanding the Clifford commutant enhances our ability to design algorithms for quantum state verification, error correction, and magic state resource evaluation, integral to the effective deployment of quantum technology.
2. Theoretical Developments: By filling knowledge gaps about the exact structure of the Clifford commutant, this work aids in further investigations related to quantum speedups and efficiency of quantum circuits, specifically those leveraging Clifford operations.
3. Benchmarking and Verification: The characterization of Clifford averages contributes to robust, randomized benchmarking protocols, essential for quantum device validation and performance assessment.

Conclusion and Future Directions

The paper establishes a deeper understanding of the Clifford group's commutant, vital for advancing quantum computing's theoretical and practical aspects. Future research could focus on exploiting this framework for new quantum algorithms, understanding Clifford orbits, and exploring the physical realization of optimal measurement strategies enabled by these mathematical insights. As quantum technology progresses towards more complex qudit systems, the relevance of this work in foundational quantum mechanics and computational applications is poised to grow.