Improved upper bounds on the stabilizer rank of magic states [article]

Hammam Qassim, Hakop Pashayan, David Gosset
2021 arXiv   pre-print
In this work we improve the runtime of recent classical algorithms for strong simulation of quantum circuits composed of Clifford and T gates. The improvement is obtained by establishing a new upper bound on the stabilizer rank of m copies of the magic state |T⟩=√(2)^-1(|0⟩+e^iπ/4|1⟩) in the limit of large m. In particular, we show that |T⟩^⊗ m can be exactly expressed as a superposition of at most O(2^α m) stabilizer states, where α≤ 0.3963, improving on the best previously known bound α≤
more » ... . This furnishes, via known techniques, a classical algorithm which approximates output probabilities of an n-qubit Clifford + T circuit U with m uses of the T gate to within a given inverse polynomial relative error using a runtime poly(n,m)2^α m. We also provide improved upper bounds on the stabilizer rank of symmetric product states |ψ⟩^⊗ m more generally; as a consequence we obtain a strong simulation algorithm for circuits consisting of Clifford gates and m instances of any (fixed) single-qubit Z-rotation gate with runtime poly(n,m) 2^m/2. We suggest a method to further improve the upper bounds by constructing linear codes with certain properties.
arXiv:2106.07740v2 fatcat:o5rfzp32rbgsdmic5v3556tjze