Coherence Fraction in Grover Search Algorithm

TL;DR

This work identifies coherence fraction as the central resource governing the success probability of Grover-type search, independent of entanglement or conventional coherence in the initial state. By introducing the generalized Grover search algorithm (GGSA) with an arbitrary unitary before the Grover operator, the authors derive exact expressions showing the average success probability is determined by the coherence fraction $f_c$ and the oracle-query count, with a tight upper bound achieved when the initial state approaches the equal superposition $|\eta\rangle$. They extend the framework to mixed states and connect the coherence fraction to a quantum minimization paradigm (GQMA), showing that the optimal performance again corresponds to the coherence fraction. The results clarify the origins of quantum advantage in Grover-like tasks, relate the coherence fraction to known teleportation metrics via the fully entangled fraction analog, and suggest design principles for new Grover-based algorithms and quantum machine-learning subroutines. Overall, the paper reframes quantum speedups in terms of fidelity to maximum coherence and provides a concrete, testable target for algorithm development.

Abstract

The question of which resources drive the advantages in quantum algorithms has long been a fundamental challenge. While entanglement and coherence are critical to many quantum algorithms, our results indicate that they do not fully explain the quantum advantage achieved by the Grover search algorithm. By introducing a generalized Grover search algorithm, we demonstrate that the success probability depends not only on the querying number of oracles but also on the coherence fraction, which quantifies the fidelity between an arbitrary initial quantum state and the equal superposition state. Additionally, we explore the role of the coherence fraction in the quantum minimization algorithm, which offers a framework for solving complex problems in quantum machine learning. These findings offer insights into the origins of quantum advantage and open pathways for the development of new quantum algorithms.

Figures (6)

• Figure 1: Diagrammatic sketch of coherence fraction in Grover search algorithm. The upper bound of the average success probability for the generalized Grover search algorithm depends solely on the coherence fraction, rather than quantum entanglement or coherence.
• Figure 2: Quantum Circuit for GGSA. An arbitrary unitary quantum gate $\mathcal{U}$ is applied to the input state $|0^{n}\rangle$ of the register, while the Hadamard gate $H$ is applied to the ancilla qubit $|1\rangle_q$. The initial state is $|\psi\rangle:=\mathcal{U}|0^{n}\rangle=\sum_{x=0}^{N-1}a_x|x\rangle$, where $a_x$ is the amplitude of $|x\rangle$. The resulting state is $|\psi\rangle|-\rangle$. Subsequently, the Grover operator $\mathcal{G}$ is applied $\tau$ times, and the final state is measured on the computational basis.
• Figure 3: Grover operator $\mathcal{G}_{\mathcal{M}_{i}}$. The algorithm is depicted in three outer boxes, each representing a different step. The first box initializes the state as $|\psi\rangle$. In the second box, the Grover operator $\mathcal{G}_{\mathcal{M}_{i}}$ is applied, resulting in the final state $|\psi_{\mathcal{M}_{i}}\rangle$. The small boxes inside signify the selections of a specific marked state each time the algorithm proceeds. The final box represents the measurement.
• Figure 4: The optimal average success probability as a function of the coherence fraction. We set $N=2^5$ and the plot follows the Eq. (\ref{['P_opt']}).
• Figure 5: The link between parameters ($\alpha$, $\beta$, $\theta$) and the optimal average success probability $P_{\text{ave}}^{\mathrm{opt}}(|\psi(\alpha, \beta, \theta)\rangle)$.Example (a). Fixed the parameter $\theta=\pi/4$, the success probability $P_{\text{ave}}^{\mathrm{opt}}(|\psi(\alpha, \beta, \pi/4)\rangle)=1$ when $\alpha=\beta+2k\pi$, $(k=0, \pm 1, \cdots)$. Here is the case of $n=2$. Example (b). Fixed parameters $\alpha=\beta=0$, the upper bound of $P_{\text{ave}}^{\mathrm{opt}}(|\psi(0, 0, \theta)\rangle)=1$ is achieved when $\theta=\pi/4$. Three lines in the figure demonstrate 2, 3, and 4 qubits from top to bottom.

Theorems & Definitions (1)

• Theorem 1