Coherence dynamics in quantum algorithm for linear systems of equations

Abstract: Quantum coherence is a fundamental issue in quantum mechanics and quantum information processing. We explore the coherence dynamics of the evolved states in HHL quantum algorithm for solving the linear system of equation $A\overrightarrow{x}=\overrightarrow{b}$. By using the Tsallis relative $α$ entropy of coherence and the $l_{1,p}$ norm of coherence, we show that the operator coherence of the phase estimation $P$ relies on the coefficients $β{i}$ obtained by decomposing $|b\rangle$ in the eigenbasis of $A$. We prove that the operator coherence of the inverse phase estimation $\widetilde{P}$ relies on the coefficients $β{i}$, eigenvalues of $A$ and the success probability $P_{s}$, and it decreases with the increase of the probability when $α\in(1,2]$. Moreover, the variations of coherence deplete with the increase of the success probability and rely on the eigenvalues of $A$ as well as the success probability.

• The paper demonstrates that coherence can be both produced and depleted in the HHL algorithm, depending on the coherence quantifier applied.
• It employs Tsallis relative α entropy and l₁,p norms to analytically track coherence evolution across phase estimation, conditional rotation, and inverse operations.
• Numerical examples for 2×2 and 4×4 systems validate the predictions, highlighting coherence sensitivity to system parameters and success probability.

Coherence Dynamics in the HHL Quantum Algorithm: Analytical and Numerical Investigations

Introduction

Quantum coherence serves as a foundational resource in quantum information science, underpinning essential quantum phenomena and providing operational advantages for algorithms. The paper "Coherence dynamics in quantum algorithm for linear systems of equations" (2604.14801) investigates the dynamics of quantum coherence within the Harrow-Hassidim-Lloyd (HHL) quantum algorithm, which provides an efficient quantum protocol to solve linear system equations of the form $A\vec{x} = \vec{b}$. The study specifically analyzes the evolution and quantitative transformations of coherence employing two central measures: the Tsallis relative $\alpha$ entropy of coherence and the $l_{1,p}$ norm of coherence, which have distinct mathematical properties and operational interpretations.

This essay provides a technical review of the theoretical developments, analytical results, and numerical examples furnished in the work, and discusses their broader implications for quantum resource theory, quantum algorithm analysis, and potential future directions in quantum computation.

Theoretical Framework: Coherence Measures and the HHL Algorithm

Coherence Quantifiers

The paper adopts two principal coherence quantifiers:

• Tsallis Relative $\alpha$ Entropy of Coherence: Defined as $C_\alpha(\rho)$, this measure interpolates between different entropy-based quantifications of coherence and is parametric in $\alpha \in (0,1) \cup (1,2]$. For $\alpha \to 1$, it converges to the relative entropy of coherence $C_r(\rho)$. For $\alpha=1/2$, it is related to the skew information of coherence, linking coherence to quantum metrology and discord.
• $l_{1,p}$ Norm of Coherence: Denoted $\alpha$0 for $\alpha$1, this measure generalizes the widely used $\alpha$2 norm of coherence (the sum of off-diagonal magnitudes) to higher-order matrix norms. The $\alpha$3 norms capture the structure of off-diagonal coherence in multipartite or higher-dimensional systems.

These quantifiers are applied to the output states at different stages of the HHL algorithm. Their dependency on both system parameters and quantum algorithmic operations is emphasized throughout the analysis.

HHL Algorithm Stages

The HHL algorithm for an $\alpha$4-sparse Hermitian $\alpha$5 matrix $\alpha$6 and normalized vector $\alpha$7 comprises:

1. Phase Estimation ($\alpha$8): Transforms the input state using Hamiltonian simulation and quantum Fourier transform, decomposing $\alpha$9 (where $l_{1,p}$0 are eigenvectors of $l_{1,p}$1 with eigenvalues $l_{1,p}$2).
2. Conditional Rotation ($l_{1,p}$3-$l_{1,p}$4): Introduces ancilla-dependent rotations encoding the $l_{1,p}$5 factors necessary for inverting $l_{1,p}$6.
3. Inverse Phase Estimation ($l_{1,p}$7): Reverses the phase estimation and projects onto the solution state upon successful measurement.

Each step admits exact analytical expressions for quantum coherence, parameterized by the $l_{1,p}$8 (decomposition coefficients), $l_{1,p}$9 (eigenvalues), and the algorithm’s success probability $\alpha$0.

Analytical Results: Coherence Dynamics

Operator Coherence at Each Step

The authors derive closed-form expressions for $\alpha$1 and $\alpha$2 at each stage:

• After phase estimation ($\alpha$3), both coherence measures depend solely on the $\alpha$4 coefficients, reflecting coherence within the eigenbasis of $\alpha$5.
• Following conditional rotation, coherence also depends explicitly on the eigenvalues $\alpha$6 due to the energy-dependent rotation.
• Post-inverse phase estimation and measurement, coherence incorporates the success probability $\alpha$7, which is a function of $\alpha$8 and $\alpha$9. Notably, the Tsallis relative $C_\alpha(\rho)$0 entropy and the $C_\alpha(\rho)$1 norm of coherence generally decrease with increasing $C_\alpha(\rho)$2 for $C_\alpha(\rho)$3, but the former can increase for $C_\alpha(\rho)$4, highlighting nontrivial monotonicity properties.

Strong analytic results delineate the conditions for coherence production or depletion. For instance, the $C_\alpha(\rho)$5 norm of coherence after phase estimation is simply $C_\alpha(\rho)$6.

Coherence Variation and Success Probability

The paper introduces a precise definition for the variation in operator coherence $C_\alpha(\rho)$7 between the initial and final algorithmic steps. The analytical results show:

• Depletion of Coherence: For the $C_\alpha(\rho)$8 norm and Tsallis $C_\alpha(\rho)$9 entropy (with $\alpha \in (0,1) \cup (1,2]$0), $\alpha \in (0,1) \cup (1,2]$1 decreases as $\alpha \in (0,1) \cup (1,2]$2 increases.
• Production of Coherence: For Tsallis $\alpha \in (0,1) \cup (1,2]$3 entropy with $\alpha \in (0,1) \cup (1,2]$4, $\alpha \in (0,1) \cup (1,2]$5 may increase with $\alpha \in (0,1) \cup (1,2]$6.
• The monotonicity and sign of $\alpha \in (0,1) \cup (1,2]$7 depend not only on the quantifier but on the spectral data of $\alpha \in (0,1) \cup (1,2]$8 and expansion coefficients $\alpha \in (0,1) \cup (1,2]$9.

Thus, the coherence resource in HHL is not necessarily monotonic or uniformly depleted, in contrast to the general behavior observed in other quantum algorithms (e.g., Grover’s, Shor’s).

Explicit Coefficient Relations

For $\alpha \to 1$0 systems, the relationship between basis expansion coefficients $\alpha \to 1$1 and the state vector components $\alpha \to 1$2 is given analytically, facilitating exact computation of coherence evolution for arbitrary binary systems.

Numerical Examples

The work provides explicit computations and graphical analyses for selected $\alpha \to 1$3 and $\alpha \to 1$4 linear system instances:

• For a $\alpha \to 1$5 Hermitian case with $\alpha \to 1$6 and $\alpha \to 1$7 specified, the evolution of both coherence quantifiers is tracked rigorously through each algorithmic operation.
• For a $\alpha \to 1$8 diagonal system, the dependence of coherence on both $\alpha \to 1$9 (for Tsallis) and $C_r(\rho)$0 ($C_r(\rho)$1) is illustrated, and transitions between coherence production and depletion regimes are numerically validated.

A notable finding is that for $C_r(\rho)$2 systems, the $C_r(\rho)$3 norm of coherence is independent of $C_r(\rho)$4 at certain steps, whereas for higher dimensions, this is no longer the case. Moreover, the overall algorithmic effect (coherence production vs. depletion) depends critically on the system and input vector.

Implications and Prospects

Practical Implications

The study advances a systematic method to track quantum coherence as a physically meaningful resource across primitive quantum algorithmic steps, providing operational insight for:

• Resource-Aware Quantum Compilation: Quantitative coherence tracking can inform the design of quantum circuits that manage coherence for noise resilience or other resource-theoretic constraints.
• Complexity and Resource Tradeoffs: Understanding coherence dynamics can complement analysis based on entanglement and other resources, offering a more nuanced picture of quantum algorithmic speedups and limitations.
• Quantum Algorithm Design: The nonmonotonicity of coherence is relevant for protocols using coherence as a catalyst or reservoir, and for algorithms where coherence engineering is pivotal.

Theoretical Implications

The results challenge and extend previous indications that quantum algorithmic operation is always associated with coherence consumption. The observed coherence production in HHL for certain quantifiers prompts a re-examination of resource monotonicity principles in the context of quantum algorithms. As the results depend on the basis and coherence measure, future work could address:

• Basis-Independent Resource Synthesis: Development of basis-free or operationally-meaningful quantifiers applicable across diverse quantum algorithms.
• Coherence-Entanglement Relations: Deeper analysis of coherence/entanglement trade-offs, especially in multipartite quantum channels or network algorithms.
• Algorithmic Regimes: Exploration of coherence dynamics in non-Hermitian or noisy variants of HHL, and other quantum linear solver protocols.

Conclusion

This paper offers a rigorous analysis of quantum coherence evolution within the HHL quantum algorithm by leveraging the Tsallis relative $C_r(\rho)$5 entropy and $C_r(\rho)$6 norm quantifiers. The results analytically and numerically demonstrate that coherence can either be produced or depleted depending on algorithmic, spectral, and quantifier parameters—a nontrivial extension beyond the monotonic depletion observed for coherence in many other quantum algorithms. These findings have both practical relevance for resource-theoretic quantum algorithm design and theoretical significance for the foundations of quantum advantage. Future investigations could generalize these techniques to broader algorithmic classes and alternative resource quantifiers, contributing to a more comprehensive theory of quantum computational resources.