A polynomial-time classical decision algorithm exactly characterizes which multivariable Laurent polynomial pairs are realizable by M-QSP and supplies a constructive implementation when the answer is yes.
Proof of Lemma 1 (⊂) First, assume that ( P, Q) can be constructed by m-variable M-QSP in ( n − 2) steps
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Polynomial time constructive decision algorithm for multivariable quantum signal processing
A polynomial-time classical decision algorithm exactly characterizes which multivariable Laurent polynomial pairs are realizable by M-QSP and supplies a constructive implementation when the answer is yes.