aiqbench analysis
This paper is a mathematical study of the third tensor power of the Clifford representation on qutrit systems, the structure underlying stabilizer-based testing. It identifies the invariants that govern Clifford twirling of three-copy states and gives explicit bounds for stabilizer testing and unknown-observable testing. The results are mainly useful to researchers working on quantum property testing and the representation theory of finite Clifford groups, though they remain theoretical and do not yet translate into hardware or algorithmic benchmarks.
Clifford Invariants and Three-Copy Quantum Tests
arXiv:2610.14758v1 Announce Type: new Abstract: We give an explicit treatment of the third tensor power of the Clifford representation on $n$ qutrits, for $n\geq2$. This representation has five pairwise inequivalent irreducible constituents, with multiplicities $1,1,1,1,2$. The decomposition reduces the Clifford twirl of a tensor cube to four real parameters. Two of them are spectral moments of the input. The other two are cubic invariants expressed as sums of cubes of the Weyl characteristic function and the discrete Wigner function. On trace-one Hermitian matrices, the invariant polynomials of degree at most three are spanned by the constant function and four algebraically independent functions. These functions do not separate Clifford orbits. For pure inputs a single parameter suffices. We recover its range and the characterization of its maximum by stabilizer states. We bound the stabilizer fidelity, determine full separability across the three copies in the symmetric family, and recover the criterion for a complex projective $3$-design. We classify the jointly invariant quantum Boolean functions. For stabilizer testing and for testing unknown observables we give explicit bounds on copies and queries. We also give character formulas for the finite projective Clifford groups and record the complete character table for two qutrits.
Abstract:We give an explicit treatment of the third tensor power of the Clifford representation on $n$ qutrits, for $n\geq2$. This representation has five pairwise inequivalent irreducible constituents, with multiplicities $1,1,1,1,2$. The decomposition reduces the Clifford twirl of a tensor cube to four real parameters. Two of them are spectral moments of the input. The other two are cubic invariants expressed as sums of cubes of the Weyl characteristic function and the discrete Wigner function. On trace-one Hermitian matrices, the invariant polynomials of degree at most three are spanned by the constant function and four algebraically independent functions. These functions do not separate Clifford orbits. For pure inputs a single parameter suffices. We recover its range and the characterization of its maximum by stabilizer states. We bound the stabilizer fidelity, determine full separability across the three copies in the symmetric family, and recover the criterion for a complex projective $3$-design. We classify the jointly invariant quantum Boolean functions. For stabilizer testing and for testing unknown observables we give explicit bounds on copies and queries. We also give character formulas for the finite projective Clifford groups and record the complete character table for two qutrits.
Quantum Physics (quant-ph); Group Theory (math.GR); Representation Theory (math.RT)
arXiv:2610.14758 [quant-ph]
(or arXiv:2610.14758v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2610.14758 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 293 words
