Decoding Quantum Search Advantage: The Critical Role of State Properties in Random Walks

TL;DR

The paper investigates how intrinsic quantum-state properties govern the performance of random-walk-based quantum search. It introduces three SKW variants (SKW-1, SKW-2, SKW-3) and an optimized version (OSKW) and derives exact success-probability forms tied to state resources: the coherence fraction $f_c$, Groverian entanglement $E_g$, and the coherence measure $C_f$. The main results show $P_{\text{max-1}}=\tfrac{1}{2} f_c + O(\tfrac{1}{\sqrt{N}})$ for SKW-1, $P_{\text{max-2}}=\tfrac{1}{2}(1-E_g^2)$ and $P_{\text{max-2}}^{\text{opt}}=1-E_g^2$ for SKW-2, and $P_{\text{max-3}}=\tfrac{1}{2}(1-C_f^2)$ for SKW-3, with $P_{\text{max-1}}^{\text{opt}}=f_c+O(\tfrac{1}{\sqrt{N}})$. These results reveal that increasing coherence can enhance SKW-1 performance while entanglement and coherence can diminish SKW-2/3 performance, highlighting a nuanced, state-property-driven mechanism underlying quantum search advantages and guiding algorithm design for quantum-friendly AI applications.

Abstract

Quantum algorithms have demonstrated provable speedups over classical counterparts, yet establishing a comprehensive theoretical framework to understand the quantum advantage remains a core challenge. In this work, we decode the quantum search advantage by investigating the critical role of quantum state properties in random-walk-based algorithms. We propose three distinct variants of quantum random-walk search algorithms and derive exact analytical expressions for their success probabilities. These probabilities are fundamentally determined by specific initial state properties: the coherence fraction governs the first algorithm's performance, while entanglement and coherence dominate the outcomes of the second and third algorithms, respectively. We show that increased coherence fraction enhances success probability, but greater entanglement and coherence reduce it in the latter two cases. These findings reveal fundamental insights into harnessing quantum properties for advantage and guide algorithm design. Our searches achieve Grover-like speedups and show significant potential for quantum-enhanced machine learning.

Figures (5)

• Figure 1: Diagrammatic sketch of the quantum properties of the initial state in SKW algorithm. Three modified versions of the SKW algorithm give connections between the algorithm's success probabilities and the coherence fraction, entanglement and coherence, respectively.
• Figure 2: Quantum circuit for SKW-1 algorithm. Applying the Hadamard operation $H^{\otimes m}$ ($m^2=n$) to the input state $|0^m\rangle$ on the direction space to obtain $|S^c\rangle=\frac{1}{\sqrt{n}}\sum_{d=1}^{n}|d\rangle$. While an arbitrary unitary quantum gate $\mathcal{U}$ is applied to the input state $|\vec{0^{n}}\rangle$ of the register. The initial state is $|\psi\rangle=\mathcal{U}|\vec{0^{n}}\rangle=\sum_{x=0}^{N-1}a_x|\vec{x}\rangle$, where $a_x$ is the amplitude of $|\vec{x}\rangle$. The resulting state is $|S^c\rangle\otimes|\psi\rangle$. Subsequently, the perturbed evolution operator $\mathcal{V}=\mathcal{S}\mathcal{C}$ is applied $\tau$ times, and the final state is then measured.
• Figure 3: Quantum circuit for SKW-2 algorithm. Applying the Hadamard operation $H^{\otimes m}$ ($m^2=n$) to the input state $|0^m\rangle$ on the direction space to obtained $|S^c\rangle=\frac{1}{\sqrt{n}}\sum_{d=1}^{n}|d\rangle$. While an arbitrary unitary quantum gate $\mathcal{U}$ is applied to the input state $|\vec{0^{n}}\rangle$ of the register. The initial state is $|\psi\rangle=\mathcal{U}|\vec{0^{n}}\rangle=\sum_{x=0}^{N-1}a_x|\vec{x}\rangle$, where $a_x$ is the amplitude of $|\vec{x}\rangle$. The resulting state is $|S^c\rangle|\psi\rangle$. Perform a product of arbitrary local operations $U_{1}\otimes U_{2}\otimes \cdots \otimes U_{n}$ on the register, where $U_{j}$ is an arbitrary local unitary gate acting on the $j$th qubit. Subsequently, the perturbed evolution operator $\mathcal{V}=\mathcal{S}\mathcal{C}$ is applied $\tau$ times, and the final state is measured on the computational basis.
• Figure 4: Quantum circuit for SKW-3 algorithm. Applying the Hadamard operation $H^{\otimes m}$ ($m^2=n$) to the input state $|0^m\rangle$ on the direction space to obtained $|S^c\rangle=\frac{1}{\sqrt{n}}\sum_{d=1}^{n}|d\rangle$. While an arbitrary unitary quantum gate $\mathcal{U}$ is applied to the input state $|\vec{0^{n}}\rangle$ of the register. The initial state is $|\psi\rangle=\mathcal{U}|\vec{0^{n}}\rangle=\sum_{x=0}^{N-1}a_x|\vec{x}\rangle$, where $a_x$ is the amplitude of $|\vec{x}\rangle$. The resulting state is $|S^c\rangle|\psi\rangle$. Apply a product of arbitrary local operations $V_{1}\otimes V_{2}\otimes \cdots \otimes V_{n}$ on the initial state $|\psi\rangle$, where $V_{j}$ is chosen from the three single-qubit Pauli gates (X, Y, and Z) acting on the $j$th qubit. Then perform a Hadamard gate $H$ to each qubit in the register. Subsequently, the perturbed evolution operator $\mathcal{V}=\mathcal{S}\mathcal{C}$ is applied $\tau$ times, and the final state is measured on the computational basis.
• Figure 5: The relationship among the success probabilities of the three modified SKW algorithms and the properties of the initial states. With respect to Eq. (\ref{['max-1']}), (\ref{['max-2']}) and (\ref{['max-3']}), the blue, green and pink lines represent the relationships between the success probabilities of the SKW-1, SKW-2 and SKW-3 algorithms and the coherence fraction, entanglement and coherence of the initial state, respectively.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Theorem 3
• Proof
• Proof
• Proof