Optimizing the convergence rate of the quantum consensus: A discrete-time model

Jafarizadeh, Saber

doi:10.1016/j.automatica.2016.07.029

Cited by 13 publications

(19 citation statements)

References 39 publications

(103 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hence, instead of solving the Fastest Quantum Gossip problem , we solve . Formulation of the optimization problem is similar to that of the fastest discrete‐time quantum consensus problem …”

Section: Quantum Gossip Algorithmmentioning

confidence: 99%

“…The second partition of interest is feasible for N ≤ d 2 and, in the work of Jafarizadeh, it has been shown that the Laplacian corresponding to this partition includes the corresponding spectrum of all other partitions. Furthermore, in the work of the aforementioned author, by generalizing the Aldous' conjecture to all partitions, it is shown that the second smallest eigenvalues of the Laplacian matrices of all partitions are equal. Therefore, for all relative values of N and d , λ 2 ( W Q , α ) and

λ_{2} false({\overset{bold-italicW}{true}}_{Q}, \overset{α}{true} false)

can be obtained from the Laplacian matrix corresponding to partition ( N −1,1) where its induced graph is identical to the underlying graph of the quantum network.…”

Section: Quantum Gossip Algorithmmentioning

confidence: 99%

“…Therefore, for all relative values of N and d , λ 2 ( W Q , α ) and

λ_{2} false({\overset{bold-italicW}{true}}_{Q}, \overset{α}{true} false)

can be obtained from the Laplacian matrix corresponding to partition ( N −1,1) where its induced graph is identical to the underlying graph of the quantum network. Unlike the second smallest eigenvalues, the greatest eigenvalue of the induced graphs corresponding to different partitions are not the same . As a result, determining the value of

λ_{d^{2 N}} false({bold-italicW}_{Q}, α false)

and

λ_{d^{2 N}} false({\overset{bold-italicW}{true}}_{Q}, \overset{α}{true} false)

depends on the relative value of N and d , and solution of the optimization problem can be categorized into the following cases.…”

Section: Quantum Gossip Algorithmmentioning

confidence: 99%

“…I. For N ≤ d 2 , as shown in the work of Jafarizadeh,

false| λ_{d^{2 N}} false({bold-italicW}_{Q}, α false) false| = false| 1 - 2 α false|

and

false| λ_{d^{2 N}} false({\overset{bold-italicW}{true}}_{Q}, \overset{α}{true} false) false| = false| 1 - 2 \overset{α}{true} false|

. Therefore, the objective function of the optimization problem can be written as

\max false{λ_{2} false({bold-italicW}_{Q}, α false), false| 1 - 2 α false|, λ_{2} false({\overset{bold-italicW}{true}}_{Q}, \overset{α}{true} false), false| 1 - 2 \overset{α}{true} false| false}

.…”

Section: Quantum Gossip Algorithmmentioning

confidence: 99%

“…II. For N ≥ d 2 +1, it is shown that SLEM( W Q )= λ 2 ( W Q , α ) and

SLEM false({\overset{bold-italicW}{true}}_{Q} false) = λ_{2} false({\overset{bold-italicW}{true}}_{Q}, \overset{α}{true} false)

, since

λ_{2} false({bold-italicW}_{Q}, α false) < - λ_{d^{2 N}} false({bold-italicW}_{Q}, α false)

results in α >1, which is not acceptable. For

α \leq \frac{1}{2}

(where

\overset{α}{true} \geq α

), we have

λ_{2} false({bold-italicW}_{Q}, α false) \geq λ_{2} false({\overset{bold-italicW}{true}}_{Q}, \overset{α}{true} false)

.…”

Section: Quantum Gossip Algorithmmentioning

confidence: 99%

See 4 more Smart Citations

Gossip algorithm with nonuniform clock distribution: Optimization over classical and quantum networks

Jafarizadeh

2019

Optim Control Appl Methods

Self Cite

View full text Add to dashboard Cite

Summary Distributed gossip algorithm has been studied for practical implementation of fundamental algorithms employed for collaborative processing. This paper focuses on optimizing the convergence rate of the gossip algorithm for both classical and quantum networks. By modifying the rate of the Poisson process, a new model of the gossip algorithm with nonuniform clock distribution is proposed which can reach the optimal convergence rate of the continuous‐time consensus algorithm. For quantum gossip algorithm, the evolution equation of the quantum gossip algorithm is transformed to the state update equation of the classical gossip algorithm. Defining the quantum gossip operator, the original optimization problem is formulated as a convex optimization problem, where the analytical answer is provided for a series of topologies. It is shown that the optimal results obtained for uniform clock distribution are suboptimal compared to those of the nonuniform one and for nonuniform distribution the optimal answer is not unique. Based on the optimal continuous‐time consensus algorithm and the detailed balance property, an effective method of obtaining one of these optimal answers is proposed.

show abstract

Section: Quantum Gossip Algorithmmentioning

confidence: 99%

λ_{2} false({\overset{bold-italicW}{true}}_{Q}, \overset{α}{true} false)

can be obtained from the Laplacian matrix corresponding to partition ( N −1,1) where its induced graph is identical to the underlying graph of the quantum network.…”

Section: Quantum Gossip Algorithmmentioning

confidence: 99%

“…Therefore, for all relative values of N and d , λ 2 ( W Q , α ) and

λ_{2} false({\overset{bold-italicW}{true}}_{Q}, \overset{α}{true} false)

λ_{d^{2 N}} false({bold-italicW}_{Q}, α false)

and

λ_{d^{2 N}} false({\overset{bold-italicW}{true}}_{Q}, \overset{α}{true} false)

depends on the relative value of N and d , and solution of the optimization problem can be categorized into the following cases.…”

Section: Quantum Gossip Algorithmmentioning

confidence: 99%

“…I. For N ≤ d 2 , as shown in the work of Jafarizadeh,

false| λ_{d^{2 N}} false({bold-italicW}_{Q}, α false) false| = false| 1 - 2 α false|

and

false| λ_{d^{2 N}} false({\overset{bold-italicW}{true}}_{Q}, \overset{α}{true} false) false| = false| 1 - 2 \overset{α}{true} false|

. Therefore, the objective function of the optimization problem can be written as

\max false{λ_{2} false({bold-italicW}_{Q}, α false), false| 1 - 2 α false|, λ_{2} false({\overset{bold-italicW}{true}}_{Q}, \overset{α}{true} false), false| 1 - 2 \overset{α}{true} false| false}

.…”

Section: Quantum Gossip Algorithmmentioning

confidence: 99%

“…II. For N ≥ d 2 +1, it is shown that SLEM( W Q )= λ 2 ( W Q , α ) and

SLEM false({\overset{bold-italicW}{true}}_{Q} false) = λ_{2} false({\overset{bold-italicW}{true}}_{Q}, \overset{α}{true} false)

, since

λ_{2} false({bold-italicW}_{Q}, α false) < - λ_{d^{2 N}} false({bold-italicW}_{Q}, α false)

results in α >1, which is not acceptable. For

α \leq \frac{1}{2}

(where

\overset{α}{true} \geq α

), we have

λ_{2} false({bold-italicW}_{Q}, α false) \geq λ_{2} false({\overset{bold-italicW}{true}}_{Q}, \overset{α}{true} false)

.…”

Section: Quantum Gossip Algorithmmentioning

confidence: 99%

See 3 more Smart Citations

Gossip algorithm with nonuniform clock distribution: Optimization over classical and quantum networks

Jafarizadeh

2019

Optim Control Appl Methods

Self Cite

View full text Add to dashboard Cite

show abstract