Tsallis relative $α$ entropy of coherence dynamics in Grover's search algorithm

Abstract: Quantum coherence plays a central role in Grover's search algorithm. We study the Tsallis relative $α$ entropy of coherence dynamics of the evolved state in Grover's search algorithm. We prove that the Tsallis relative $α$ entropy of coherence decreases with the increase of the success probability, and derive the complementarity relations between the coherence and the success probability. We show that the operator coherence of the first $H^{\otimes n}$ relies on the size of the database $N$, the success probability and the target states. Moreover, we illustrate the relationships between coherence and entanglement of the superposition state of targets, as well as the production and deletion of coherence in Grover iterations.

• The paper establishes how Tsallis relative α entropy quantifies quantum coherence and its depletion as Grover's algorithm success probability increases.
• It decomposes coherence contributions from the Hadamard, oracle, and phase-shift operators, linking analytical and numerical findings.
• The study reveals parameter-sensitive effects of entanglement on coherence dynamics, offering actionable insights for quantum algorithm optimization.

Tsallis Relative $α$ Entropy of Coherence Dynamics in Grover's Search Algorithm

Introduction

Grover's search algorithm (GSA), a fundamental quantum algorithm, is widely recognized for its quadratic speedup in searching unsorted databases relative to classical counterparts. Quantum coherence, manifesting as superposition, is a pivotal resource underpinning quantum computation and algorithmic performance. The paper investigates the dynamics of coherence in GSA utilizing Tsallis relative $α$ entropy, a nonadditive generalization of quantum relative entropy with tunable parameter $\alpha$, which encompasses several coherence quantifiers, including relative entropy and skew information.

Tsallis Relative $α$ Entropy as a Coherence Quantifier

Tsallis relative $α$ entropy, denoted $D_\alpha(\rho || \sigma)$, provides an information-theoretic metric for quantifying the purity and distinguishability between quantum states. The associated coherence quantifier, $C_\alpha(\rho)$, measures a state's deviation from incoherence in a chosen basis. The measure is generalized, interpolating between the relative entropy of coherence ($\alpha\to1$) and skew information ($\alpha=\frac{1}{2}$). For $\alpha\in(0,1)\cup(1,2]$, several monotonicity properties are preserved, although strong monotonicity holds only for suitably modified versions.

Coherence Depletion and Complementarity in Grover's Algorithm

The analysis elucidates the relation between the success probability $α$0 and coherence, showing that, as $α$1 increases during Grover iterations, Tsallis relative $α$2 coherence $α$3 monotonically decreases. Strong numerical results delineate this complementarity:

• For $α$4: $α$5.
• For $α$6: $α$7.

These relations, derived asymptotically for $α$8, formalize the tradeoff: successful search necessitates resource depletion in the form of coherence.

Operator-Level Coherence Dynamics

Unlike prior analyses limited to global algorithmic coherence, this work decomposes Grover iteration ($α$9) to examine the impact of individual basic operators. The Hadamard transform widely influences coherence, dependent on database size $\alpha$0, the success probability $\alpha$1, and target structure. In contrast, the oracle ($\alpha$2) and phase-shift ($\alpha$3) operators are shown to be incoherent, leaving coherence invariant.

Strong numerical results characterize the coherence dynamics post-application of Hadamard ($\alpha$4 and $\alpha$5):

• $\alpha$6 and $\alpha$7 depend on $\alpha$8, structure factor $\alpha$9, and $α$0.
• The coherence production and depletion rates are quantified for consecutive Grover iterations, revealing oscillatory behavior and the existence of turning points where operator-induced coherence transitions occur.

Target State Structure and Entanglement Effects

A comparison is drawn between cases where the target superposition $α$1 is a product state versus an entangled state, focusing on scenarios with few target states ($α$2). For $α$3, Tsallis relative $α$4 entropy of coherence is larger for entangled target superpositions, whereas for $α$5, it is smaller. This dichotomy underscores sensitivity to both entanglement and coherence quantifier parameterization, with implications for resource theory-based algorithm analysis.

The bounds for $α$6 are explicitly derived for product and entangled target scenarios, further contextualizing the interplay between coherence, success probability, and entanglement.

Coherence Production and Depletion: Variational Analysis

The paper rigorously quantifies production and depletion of coherence per operator and per iteration:

• $α$7 and $α$8 correspond to coherence depletion due to $α$9 and $α$0.
• $α$1 signals coherence production by $α$2.
• The relationship $α$3 formalizes operator interdependence.

Turning points arise, delineating regions in iteration space where the sign of coherence production and depletion transitions, encapsulating a non-monotonic, oscillatory behavior.

Comparison with Prior Works

This work extends previous studies on $α$4 norm coherence dynamics in GSA [PMQ], providing generalized parameterized results and exact complementarity relations for Tsallis relative $α$5 entropy. It contrasts the monotonic coherence depletion observed for $α$6 norm with the parameter-dependent depletion and production for Tsallis coherence, offering deeper insight into operator-level contributions. Furthermore, comparison with entanglement dynamics reported in prior literature [MH] highlights the distinctive behavior of coherence versus entanglement during algorithm evolution, including the existence and location of turning points.

Implications and Future Directions

The theoretical analysis reveals practical implications for quantum algorithm design and performance evaluation. The explicit link between success probability and resource depletion, formalized via Tsallis relative $α$7 entropy, offers a foundation for optimizing quantum algorithms in terms of coherence management. The parameter-sensitive response to entanglement structure suggests potential for tailored algorithmic strategies exploiting target superposition properties.

Future research trajectories include quantitative exploration of the intertwining of coherence and entanglement in algorithmic contexts, parametrized resource theory approaches for algorithmic success prediction, and novel quantum algorithm designs leveraging coherence oscillation and depletion dynamics.

Conclusion

The paper delivers a comprehensive characterization of coherence dynamics in Grover's search algorithm via Tsallis relative $α$8 entropy. It establishes complementarity relations between coherence and algorithmic success probability, quantifies operator-induced production and depletion of coherence, and elucidates the dependence on target state entanglement and database structure. The results advance theoretical understanding of quantum resources in algorithmic processes and lay groundwork for resource-centric quantum algorithm optimization.

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References:

"Tsallis relative $α$9 entropy of coherence dynamics in Grover's search algorithm" (2604.13910)