arXiv:2604.16190 · Europhysics Letters (EPL)

Coherence dynamics in Simon's quantum algorithm

Linlin Ye, Zhaoqi Wu, Shao-Ming Fei

quant-phSubmitted Apr 17, 2026• Announced Apr 20, 2026• 6 citations

Abstract

Quantum coherence plays a pivotal role in quantum algorithms. We study the coherence dynamics of the evolved states in Simon's quantum algorithm based on Tsallis relative $α$ entropy and $l_{1,p}$ norm. We prove that the coherences of the first register and the second register both rely on the dimension $N$ of the state spaces of the $n$ qubit systems, and increase with the increase of $N$. We show that the oracle operator $O$ does not change the coherence. Moreover, we study the coherence dynamics in the Simon's quantum algorithm and prove that in overall the coherence is in production when $N>4$ and in depletion when $N<4$.

Comments: 10 pages

Related papers

Coherence dynamics in quantum algorithm for linear systems of equations

Linlin Ye, Zhaoqi Wu, Shao-Ming Fei

quant-pharXiv:2604.14801Physica ScriptaApr 17, 20267 citations

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

Linlin Ye, Zhaoqi Wu, Shao-Ming Fei

quant-pharXiv:2604.13910Communications in Theoretical PhysicsApr 16, 202611 citations

Coherence and entanglement dynamics in Shor's algorithm

Linlin Ye, Zhaoqi Wu, Shao-Ming Fei

quant-pharXiv:2604.06639Communications in Theoretical PhysicsApr 9, 20263 citations

Operator coherence dynamics in Grover's quantum search algorithm

Minghua Pan, Daowen Qiu

Physical Review AJul 31, 201925 citations

Coherence and Entanglement Dynamics in Training Variational Quantum Perceptron

Min Namkung, Younghun Kwon

EntropyNov 11, 20205 citations

Parameterized coherence measure

Meng‐Li Guo, Zhi‐Xiang Jin, Jin‐Min Liang +2 more

arXiv:2306.11973Results in PhysicsJun 7, 20232 citations

arXiv:2604.16190 · Europhysics Letters (EPL)

Coherence dynamics in Simon's quantum algorithm

Linlin Ye, Zhaoqi Wu, Shao-Ming Fei

quant-phSubmitted Apr 17, 2026• Announced Apr 20, 2026• 6 citations

arXiv HTML PDF

Abstract

Quantum coherence plays a pivotal role in quantum algorithms. We study the coherence dynamics of the evolved states in Simon's quantum algorithm based on Tsallis relative

α

entropy and

l_{1,p}

norm. We prove that the coherences of the first register and the second register both rely on the dimension

N

of the state spaces of the

n

qubit systems, and increase with the increase of

N

. We show that the oracle operator

O

does not change the coherence. Moreover, we study the coherence dynamics in the Simon's quantum algorithm and prove that in overall the coherence is in production when

N>4

and in depletion when

N<4

Comments: 10 pages

Related papers

Coherence dynamics in quantum algorithm for linear systems of equations

Linlin Ye, Zhaoqi Wu, Shao-Ming Fei

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.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.

quant-pharXiv:2604.14801Physica ScriptaApr 17, 20267 citations

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

Linlin Ye, Zhaoqi Wu, Shao-Ming Fei

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.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.

quant-pharXiv:2604.13910Communications in Theoretical PhysicsApr 16, 202611 citations

Coherence and entanglement dynamics in Shor's algorithm

Linlin Ye, Zhaoqi Wu, Shao-Ming Fei

Shor's algorithm outperforms its classical counterpart in efficient prime factorization. We explore the coherence and entanglement dynamics of the evolved states within Shor's algorithm, showing that the coherence in each step relies on the dimension of register or the order, and discuss the relations between geometric coherence and geometric entanglement. We investigate how unitary operators induce variations in coherence and entanglement, and analyze the variations of coherence and entanglement within the entire algorithm, demonstrating that the overall effect of Shor's algorithm tends to deplete coherence and produce entanglement. Our research not only deepens the understanding of this algorithm but also provides methodological references for studying resource dynamics in other quantum algorithms.Shor's algorithm outperforms its classical counterpart in efficient prime factorization. We explore the coherence and entanglement dynamics of the evolved states within Shor's algorithm, showing that the coherence in each step relies on the dimension of register or the order, and discuss the relations between geometric coherence and geometric entanglement. We investigate how unitary operators induce variations in coherence and entanglement, and analyze the variations of coherence and entanglement within the entire algorithm, demonstrating that the overall effect of Shor's algorithm tends to deplete coherence and produce entanglement. Our research not only deepens the understanding of this algorithm but also provides methodological references for studying resource dynamics in other quantum algorithms.

quant-pharXiv:2604.06639Communications in Theoretical PhysicsApr 9, 20263 citations

Operator coherence dynamics in Grover's quantum search algorithm

Minghua Pan, Daowen Qiu

Coherence is one of the most basic concepts and resources in quantum information. To clear the role coherence plays on the essential operator level in Grover's search algorithm, here we discuss the coherence dynamics of the state after each basic operator is applyied. As it is known, Grover's search algorithm repeats the application of Grover operator $G$ , which can be decomposed into $G={H}^{\ensuremath{\bigotimes}n}P{H}^{\ensuremath{\bigotimes}n}O$, where $H$ is Hadamard operator, $P$ is the condition phase-shift operator, and $O$ is the oracle operator. First, we show that $O$ and $P$ are incoherent operators while ${H}^{\ensuremath{\bigotimes}n}$ is coherent. Second, we prove that the amount of the operator coherence of the first ${H}^{\ensuremath{\bigotimes}n}$ and the operator coherence produced or depleted by ${H}^{\ensuremath{\bigotimes}n}$ depends not only on the size of the database and the success probability, but also on target states. Moreover, the amount of operator coherence is larger when the superposition state of targets is entangled rather than product. Third, we show that the two ${H}^{\ensuremath{\bigotimes}n}$ have different effects on coherence that one produces coherence and the other depletes coherence, and the depletion plays a major role. Therefore, the coherence is vibrating during the search process and the overall effect is that coherence is in depletion.Coherence is one of the most basic concepts and resources in quantum information. To clear the role coherence plays on the essential operator level in Grover's search algorithm, here we discuss the coherence dynamics of the state after each basic operator is applyied. As it is known, Grover's search algorithm repeats the application of Grover operator $G$, which can be decomposed into $G={H}^{\ensuremath{\bigotimes}n}P{H}^{\ensuremath{\bigotimes}n}O$, where $H$ is Hadamard operator, $P$ is the condition phase-shift operator, and $O$ is the oracle operator. First, we show that $O$ and $P$ are incoherent operators while ${H}^{\ensuremath{\bigotimes}n}$ is coherent. Second, we prove that the amount of the operator coherence of the first ${H}^{\ensuremath{\bigotimes}n}$ and the operator coherence produced or depleted by ${H}^{\ensuremath{\bigotimes}n}$ depends not only on the size of the database and the success probability, but also on target states. Moreover, the amount of operator coherence is larger when the superposition state of targets is entangled rather than product. Third, we show that the two ${H}^{\ensuremath{\bigotimes}n}$ have different effects on coherence that one produces coherence and the other depletes coherence, and the depletion plays a major role. Therefore, the coherence is vibrating during the search process and the overall effect is that coherence is in depletion.

Physical Review AJul 31, 201925 citations

Coherence and Entanglement Dynamics in Training Variational Quantum Perceptron

Min Namkung, Younghun Kwon

In quantum computation, what contributes supremacy of quantum computation? One of the candidates is known to be a quantum coherence because it is a resource used in the various quantum algorithms. We reveal that quantum coherence contributes to the training of variational quantum perceptron proposed by Y. Du et al., arXiv:1809.06056 (2018). In detail, we show that in the first part of the training of the variational quantum perceptron, the quantum coherence of the total system is concentrated in the index register and in the second part, the Grover algorithm consumes the quantum coherence in the index register. This implies that the quantum coherence distribution and the quantum coherence depletion are required in the training of variational quantum perceptron. In addition, we investigate the behavior of entanglement during the training of variational quantum perceptron. We show that the bipartite concurrence between feature and index register decreases since Grover operation is only performed on the index register. Also, we reveal that the concurrence between the two qubits of index register increases as the variational quantum perceptron is trained.In quantum computation, what contributes supremacy of quantum computation? One of the candidates is known to be a quantum coherence because it is a resource used in the various quantum algorithms. We reveal that quantum coherence contributes to the training of variational quantum perceptron proposed by Y. Du et al., arXiv:1809.06056 (2018). In detail, we show that in the first part of the training of the variational quantum perceptron, the quantum coherence of the total system is concentrated in the index register and in the second part, the Grover algorithm consumes the quantum coherence in the index register. This implies that the quantum coherence distribution and the quantum coherence depletion are required in the training of variational quantum perceptron. In addition, we investigate the behavior of entanglement during the training of variational quantum perceptron. We show that the bipartite concurrence between feature and index register decreases since Grover operation is only performed on the index register. Also, we reveal that the concurrence between the two qubits of index register increases as the variational quantum perceptron is trained.

EntropyNov 11, 20205 citations

Parameterized coherence measure

Meng‐Li Guo, Zhi‐Xiang Jin, Jin‐Min Liang +2 more

Quantifying coherence is an essential endeavor for both quantum mechanical foundations and quantum technologies. We present a bona fide measure of quantum coherence by utilizing the Tsallis relative operator (α,β)-entropy. We first prove that the proposed coherence measure fulfills all the criteria of a well defined coherence measure, including the strong monotonicity in the resource theories of quantum coherence. We then study the ordering of the Tsallis relative operator (α,β)-entropy of coherence, Tsallis relative α-entropies of coherence, Rényi α-entropy of coherence and l1 norm of coherence for both pure and mixed qubit states. This provides a new method for defining new coherence measure and entanglement measure, and also provides a new idea for further study of quantum coherence.Quantifying coherence is an essential endeavor for both quantum mechanical foundations and quantum technologies. We present a bona fide measure of quantum coherence by utilizing the Tsallis relative operator (α,β)-entropy. We first prove that the proposed coherence measure fulfills all the criteria of a well defined coherence measure, including the strong monotonicity in the resource theories of quantum coherence. We then study the ordering of the Tsallis relative operator (α,β)-entropy of coherence, Tsallis relative α-entropies of coherence, Rényi α-entropy of coherence and l1 norm of coherence for both pure and mixed qubit states. This provides a new method for defining new coherence measure and entanglement measure, and also provides a new idea for further study of quantum coherence.

arXiv:2306.11973Results in PhysicsJun 7, 20232 citations

Quantum coherence plays a pivotal role in quantum algorithms. We study the coherence dynamics of the evolved states in Simon's quantum algorithm based on Tsallis relative

α

entropy and

l_{1,p}

norm. We prove that the coherences of the first register and the second register both rely on the dimension

N

of the state spaces of the

n

qubit systems, and increase with the increase of

N

. We show that the oracle operator

O

does not change the coherence. Moreover, we study the coherence dynamics in the Simon's quantum algorithm and prove that in overall the coherence is in production when

N>4

and in depletion when

N<4

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.

Tsallis relative

α

entropy of coherence dynamics in Grover's search algorithm

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.

G

, which can be decomposed into $G={H}^{\ensuremath{\bigotimes}n}P{H}^{\ensuremath{\bigotimes}n}O$, where

H

is Hadamard operator,

P

is the condition phase-shift operator, and

O

is the oracle operator. First, we show that

O

and

P

are incoherent operators while ${H}^{\ensuremath{\bigotimes}n}$ is coherent. Second, we prove that the amount of the operator coherence of the first ${H}^{\ensuremath{\bigotimes}n}$ and the operator coherence produced or depleted by ${H}^{\ensuremath{\bigotimes}n}$ depends not only on the size of the database and the success probability, but also on target states. Moreover, the amount of operator coherence is larger when the superposition state of targets is entangled rather than product. Third, we show that the two ${H}^{\ensuremath{\bigotimes}n}$ have different effects on coherence that one produces coherence and the other depletes coherence, and the depletion plays a major role. Therefore, the coherence is vibrating during the search process and the overall effect is that coherence is in depletion.