Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems
Lin Lin, Yu Tong
Abstract
We present a quantum eigenstate filtering algorithm based on quantum signal processing (QSP) and minimax polynomials. The algorithm allows us to efficiently prepare a target eigenstate of a given Hamiltonian, if we have access to an initial state with non-trivial overlap with the target eigenstate and have a reasonable lower bound for the spectral gap. We apply this algorithm to the quantum linear system problem (QLSP), and present two algorithms based on quantum adiabatic computing (AQC) and quantum Zeno effect respectively. Both algorithms prepare the final solution as a pure state, and achieves the near optimal<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi><mml:mo class="MJX-tex-caligraphic" mathvariant="script">~</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mi>κ</mml:mi><mml:mi>log</mml:mi><mml:mo></mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi>ϵ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math>query complexity for a<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>-sparse matrix, where<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>κ</mml:mi></mml:math>is the condition number, and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ϵ</mml:mi></mml:math>is the desired precision. Neither algorithm uses phase estimation or amplitude amplification.