Stabilizer Circuits, Quadratic Forms, and Computing Matrix Rank
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We show that a form of strong simulation for n-qubit quantum stabilizer circuits C is computable in O(s + n^ω) time, where ω is the exponent of matrix multiplication. Solution counting for quadratic forms over F_2 is also placed into O(n^ω) time. This improves previous O(n^3) bounds. Our methods in fact show an O(n^2)-time reduction from matrix rank over F_2 to computing p = |〈 0^n | C | 0^n 〉|^2 (hence also to solution counting) and a converse reduction that is O(s + n^2) except for matrix multiplications used to decide whether p > 0. The current best-known worst-case time for matrix rank is O(n^ω) over F_2, indeed over any field, while ω is currently upper-bounded by 2.3728... Our methods draw on properties of classical quadratic forms over Z_4. We study possible distributions of Feynman paths in the circuits and prove that the differences in +1 vs. -1 counts and +i vs. -i counts are always 0 or a power of 2. Further properties of quantum graph states and connections to graph theory are discussed.
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