Improvement of quantum walks search algorithm in single-marked vertex graph

Abstract Quantum walks are powerful tools for building quantum search algorithms. However, these success probabilities are far below 1. Amplitude amplification is usually used to amplify success probability, but the soufflé problem follows. Only stop at the right step can we achieve a maximum success probability. Otherwise, as the number of steps increases, the success probability may decrease, which will cause troubles in the practical application of the algorithm when the optimal number of steps is unknown. In this work, we define generalized interpolated quantum walks instead of amplitude amplification, which can not only improve the success probability but also avoid the soufflé problem. We choose a special case of generalized interpolated quantum walks and construct a series of new search algorithms based on phase estimation and quantum fast-forwarding, respectively. Especially, by combining our interpolated quantum walks with quantum fast-forwarding, we both reduce the time complexity of the search algorithm from <?CDATA $\Theta((\varepsilon^{-1})\sqrt{\mathrm{HT}})$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="normal">Θ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>ε</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:msqrt> <mml:mrow> <mml:mi mathvariant="normal">H</mml:mi> <mml:mi mathvariant="normal">T</mml:mi> </mml:mrow> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> </mml:math> to <?CDATA $\Theta(\log(\varepsilon^{-1})\sqrt{\mathrm{HT}})$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>Θ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>log</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>ε</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:msqrt> <mml:mrow> <mml:mtext>HT</mml:mtext> </mml:mrow> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> and reduce the number of ancilla qubits required from <?CDATA $\Theta(\log(\varepsilon^{-1})+\log\sqrt{\mathrm{HT}})$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="normal">Θ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo form="prefix">log</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>ε</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mo form="prefix">log</mml:mo> <mml:msqrt> <mml:mrow> <mml:mi mathvariant="normal">H</mml:mi> <mml:mi mathvariant="normal">T</mml:mi> </mml:mrow> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> </mml:math> to <?CDATA $\Theta(\log\log(\varepsilon^{-1})+\log\sqrt{\mathrm{HT}})$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="normal">Θ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo form="prefix">log</mml:mo> <mml:mo form="prefix">log</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>ε</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mo form="prefix">log</mml:mo> <mml:msqrt> <mml:mrow> <mml:mi mathvariant="normal">H</mml:mi> <mml:mi mathvariant="normal">T</mml:mi> </mml:mrow> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> </mml:math> , where ɛ denotes the precision and <?CDATA $\mathrm{HT}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">H</mml:mi> <mml:mi mathvariant="normal">T</mml:mi> </mml:mrow> </mml:math> denotes the classical hitting time. In addition, we show that our generalized interpolated quantum walks can improve the construction of quantum stationary states corresponding to reversible Markov chains. Finally, we give an application to construct a slowly evolving Markov chain sequence by applying generalized interpolated quantum walks, which is the necessary premise in adiabatic stationary state preparation.