Recent work shows that quantum signal processing (QSP) and its multi-qubit lifted version, quantum singular value transformation (QSVT), unify and improve the presentation of most quantum algorithms. QSP/QSVT characterize the ability, by alternating ansätze, to obliviously transform the singular values of subsystems of unitary matrices by polynomial functions; these algorithms are numerically stable and analytically well-understood. That said, QSP/QSVT require consistent access to aoracle, saying nothing about computingof two or more oracles; these can be far cheaper to determine given an ability to pit oracles against one another coherently. This work introduces a corresponding theory of QSP over multiple variables: M-QSP. Surprisingly, despite the non-existence of the fundamental theorem of algebra for multivariable polynomials, there exist necessary and sufficient conditions under which a desiredmultivariable polynomial transformation is possible. Moreover, the classical subroutines used by QSP protocols survive in the multivariable setting for non-obvious reasons, and remain numerically stable and efficient. Up to a well-defined conjecture, we give proof that the family of achievable multivariable transforms is as loosely constrained as could be expected. The unique ability of M-QSP toapproximateof multiple variables coherently leads to novel speedups incommensurate with those of other quantum algorithms, and provides a bridge from quantum algorithms to algebraic geometry.
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Martyn, John M. ; Rossi, Zane M. ; Tan, Andrew K. ; Chuang, Isaac L. ( , PRX Quantum)
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Rossi, Zane M. ; Chuang, Isaac L. ( , Physical Review A)null (Ed.)