Infinite quantum signal processing

Quantum signal processing (QSP) represents a real scalar polynomial of degree <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math> using a product of unitary matrices of size <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn><mml:mo>&amp;#x00D7;</mml:mo><mml:mn>2</mml:mn></mml:math>, parameterized by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> real numbers called the phase factors. This innovative representation of polynomials has a wide range of applications in quantum computation. When the polynomial of interest is obtained by truncating an infinite polynomial series, a natural question is whether the phase factors have a well defined limit as the degree <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi><mml:mo stretchy="false">&amp;#x2192;</mml:mo><mml:mi mathvariant="normal">&amp;#x221E;</mml:mi></mml:math>. While the phase factors are generally not unique, we find that there exists a consistent choice of parameterization so that the limit is well defined in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>&amp;#x2113;</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:math> space. This generalization of QSP, called the infinite quantum signal processing, can be used to represent a large class of non-polynomial functions. Our analysis reveals a surprising connection between the regularity of the target function and the decay properties of the phase factors. Our analysis also inspires a very simple and efficient algorithm to approximately compute the phase factors in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>&amp;#x2113;</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:math> space. The algorithm uses only double precision arithmetic operations, and provably converges when the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>&amp;#x2113;</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:math> norm of the Chebyshev coefficients of the target function is upper bounded by a constant that is independent of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>. This is also the first numerically stable algorithm for finding phase factors with provable performance guarantees in the limit <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi><mml:mo stretchy="false">&amp;#x2192;</mml:mo><mml:mi mathvariant="normal">&amp;#x221E;</mml:mi></mml:math>.