Revisiting fixed-point quantum search: proof of the quasi-Chebyshev lemma

TL;DR

This work provides a rigorous, detailed proof of the key quasi-Chebyshev lemma underlying fixed-point quantum search, showing that the recursive polynomials$a_L^{(\gamma)}(x)$ collapse to the Chebyshev form $T_L(x/\gamma)/T_L(1/\gamma)$. By relating the algorithm’s failure amplitude to these polynomials, the authors derive the closed-form success bound $\|\Pi_M|\psi_l\rangle\| \ge \sqrt{1-\delta^2}$ when the iteration count $l$ and lower bound $w$ meet $l \ge \ln(2/\delta)/(2w)$. The core technical contribution uses a rich set of tools, including combinatorial tiling interpretations on the $L$-star, Euler’s formula, and Vieta’s relations, to prove $N_L^{\gamma}(x)=\gamma^L T_L(x/\gamma)$ and hence $a_L^{(\gamma)}(x)=T_L(x/\gamma)/T_L(1/\gamma)$. This solidifies the fixed-point method as a sound, sometimes essential component in quantum search on graphs and related subroutines, with potential wider applicability to quantum walks and quantum phase discrimination.

Abstract

The original Grover's algorithm suffers from the souffle problem, which means that the success probability of quantum search decreases dramatically if the iteration time is too small or too large from the right time. To overcome the souffle problem, the fixed-point quantum search with an optimal number of queries was proposed [Phys. Rev. Lett. 113, 210501 (2014)], which always finds a marked state with a high probability when a lower bound of the proportion of marked states is given. The fixed-point quantum search relies on a key lemma regarding the explicit formula of recursive quasi-Chebyshev polynomials, but its proof is not given explicitly. In this work, we give a detailed proof of this lemma, thus providing a sound foundation for the correctness of the fixed-point quantum search. This lemma may be of independent interest as well, since it expands the mathematical form of the recursive relation of Chebyshev polynomials of the first kind, and it also constitutes a key component in overcoming the souffle problem of quantum walk-based search algorithms, for example, robust quantum walk search on complete bipartite graphs [Phys. Rev. A 106, 052207 (2022)]. The lemma is also central to a recently proposed quantum algorithm named quantum phase discrimination, which has become a fundamental subroutine in quantum search on graphs [arxiv: 2504.15194]. Hopefully, more applications of the lemma will be found in the future.

Figures (4)

• Figure 1: Geometric interpretation of Grover's search process, i.e. $\ket{\psi_l} := [-G(\pi,\pi)]^l \ket{\psi_0}$, in the invariant subspace $\mathcal{H}_0 = \mathrm{span} \{\ket{r}, \ket{t}\}$, where $\ket{t} := \Pi_{M}\ket{\psi_0}/\lambda$, $\lambda := \left\| \Pi_M \ket{\psi_0} \right\|$ and $\ket{r} := \Pi_{M}^\bot \ket{\psi_0}/\sqrt{1-\lambda^2}$, $\Pi_{M}^\bot := I -\Pi_{M}$. One iteration $-G(\pi,\pi) = (2\ket{\psi_0}\bra{\psi_0}-I) (I-2\ket{t}\bra{t})$ is interpreted as a reflection around $\ket{r}$ followed by a reflection around $\ket{\psi_0}$, and is thus a counter-clockwise rotation by $2\theta := 2\arcsin\lambda$.
• Figure 2: The plot of $P(\lambda) :=\left\|\Pi_M|\psi_l\rangle\right\|$ as a function of $\lambda =\left\|\Pi_M|\psi_0\rangle\right\|$, where we let $w=0.08$, $\delta=0.3$, and $l=\lceil \ln(2/\delta)/(2w) \rceil = 12$ (cf. Theorem \ref{['thm:main']}). It shows that $P(\lambda) \geq \sqrt{1-\delta^2}\approx 0.95$ as long as $\lambda \geq w=0.08$. The explicit formula of $P(\lambda)$ is shown in Eq. \ref{['eq:P_lambda']}.
• Figure 3: A tiling on the $5$-star with positions $\braket{0,1,2,3,4}$. It has one square with weight $2x$ on position $\braket{3}$, one domino with weight $-(1-it_2)(1+it_1)$ on positions $\braket{2,1}$, and one domino with weight $-(1-it_0)(1+it_4)$ on positions $\braket{0,4}$. The weight of this tiling is thus $2x(1-it_2)(1+it_1)(1-it_0)(1+it_4)$.
• Figure 4: A group of type A (see Lemma \ref{['lem:compare_w']} for its definition) tilings $\{T(j)\}_{j=0}^{4}$ on the $5$-star with $n_d=2$ dominos, where $T(j)$ is obtained from shifting all the dominos in $T(0)$ by $j$ positions and updating the weights of shifted dominos correspondingly. The set of products of the two boxed coefficients in each tiling, i.e. $\{(-i)t(0+j)\cdot it(2+j)\}_{j=0}^{4}$, is a $5$-sized set of terms ($5$-terms) appearing in the coefficients of $w^2$ in the total weights of the group $\{T(j)\}_{j=0}^{4}$, and this $L$-terms sum up to $5$ by Eq. \ref{['eq:sum_prod_1']}.

Theorems & Definitions (18)

• theorem 1: fixed_point2014
• lemma 1
• lemma 2
• proof
• lemma 3: Copy of Lemma \ref{['lem:chebyshev']}
• lemma 4
• Remark 1
• proof : Proof of Lemma \ref{['lem:D_L']}
• lemma 5
• lemma 6