Quantum Clique Gossiping

This paper establishes a framework of quantum clique gossiping by introducing local clique operations to networks of interconnected qubits. Cliques are local structures in complex networks being complete subgraphs, which can be used to accelerate classical gossip algorithms. Based on cyclic permutations, clique gossiping leads to collective multi-party qubit interactions. We show that at reduced states, these cliques have the same acceleration effects as their roles in accelerating classical gossip algorithms. For randomized selection of cliques, such improved rate of convergence is precisely characterized. On the other hand, the rate of convergence at the coherent states of the overall quantum network is proven to be decided by the spectrum of a mean-square error evolution matrix. Remarkably, the use of larger quantum cliques does not necessarily increase the speed of the network density aggregation, suggesting quantum network dynamics is not entirely decided by its classical topology.

to observe that using larger quantum cliques does not necessarily accelerate the network density aggregation. This shows that the dynamics of a quantum network is not entirely determined by its classical topology.

Main Results
Open Quantum Networks. We consider a group of quantum nodes each holding a quit indexed in the set = … n V {1, 2, , }. Time is slotted for t = 0, 1, 2, …, and at time t the state of qubit i is denoted by a density operator (matrix) ρ ∈ × t ( ) i 2 2  . The network state is denoted by the density operator ρ ∈ ×  t ( ) n n 2 2 . The qubits can be locally connected by a series of environments, which are by themselves quantum systems as well (Please refer to Fig. 1). These local environments induce a classical interaction structure which is described by a generalized graph = G (V, E), where each element in E is a nonempty and non-singleton subset of V. For example, e = {1, 2, 3} is a generalized edge among three nodes 1, 2, and 3. We index the elements in E by e 1 , …e 2 for some c > 0. The quantum generalized interaction graphs is called k-regular if e k j | | = for all j = 1, …, c. Recall that a permutation over a finite alphabet is a bijective mapping over the set onto itself. Particularly, a cyclic permutation is a permutation which maps the elements of certain subset to each other in a cyclic fashion, while mapping each of the other elements to itself. The set of all permutations over the set V is called the n'th permutation group, denoted by P. Associated with any e m , we define a permutation π m over the node set V in the way that = if j e m ≠ for any = … j n 1, , ; (ii) π m is a cyclic permutation.
For example with n = 4, a permutation satisfying π π π π = = = = (1) 2, (2) 3, (3) 1, (4) 4 is a cyclic permutation defined over a generalized edge e = {1, 2, 3}. We note that if the size of e m is k, the number of cyclic permutations over e m is − k ( 1)!. Here for the moment we assume that π m is an arbitrary cyclic permutation to ease the presentation. Note also that any permutation π over V further induces a quantum permutation operator U π over the qubit network by with ψ | 〉 i being any unit vector of the state space of qubit i.
Deterministic Quantum Clique Gossiping. Let σ(·) be a mapping from  0 ≥ to {1, …, c}. Practically, the mapping σ(·) selects a multi-vertex link from the generalized graph E by assigning e σ(t) at time t. It is natural to assume that σ(·) is a periodic signal going through every element in the set E. When e σ(t) is selected, the local environment associated with the qubits in e σ(t) is engineered so that the network density operator evolves along † This defines a deterministic quantum clique-gossiping algorithm. Clearly, when each e σ(t) contains only two nodes, this quantum clique-gossiping becomes a standard quantum gossip algorithm. It should be pointed out that the realization of such a discrete-time quantum algorithm can be made through open quantum systems 19,20 , where the state evolution of the qubits is in continuous time but by switching the dissipative operators the algorithm (1) is achieved in an approximate sense along the switching instants. The following result holds.  (ii) The network density operator where S E is the generating subgroup by the permutations in the set {π 1 , …,π c } associated with the e j c E, 1, , j ∈ = … . This result suggests that as long as all the local environments cover the whole qubit network with sufficient connectivity, the reduced states of the qubits will asymptotically reach an average consensus. However, this condition is not enough for the symmetric-state consensus since the local environment-induced quantum evolution possesses invariant subspaces along the coherent states, which prevents a fully symmetric mixing of the quantum states. Moreover, we would like to point out that the two convergence results illustrated in the above theorem are both at exponential rate, consistent with the results under continuous dynamics 2 . The following result shows the possibility of using k-regular interaction graphs to ensure reduced-state convergence in some finite time steps.

Theorem 2 There exists a k-regular interaction graph G under which a quantum clique-gossiping algorithm (1) can converge to a reduced-state consensus in some finite time steps if and only if n is divisible by k with the same prime factors as k.
Particularly, if there exist factorizations then a fastest k-regular quantum clique-gossiping algorithm drives the qubit states to steps. This result illustrates that cliques have exactly the same acceleration effects at the reduced states in the quantum setting as the classical case 18 . As a matter of fact, the reduced states follow a similar type of evolution where the analysis for classical networks becomes directly applicable.
Random Quantum Clique Gossiping. The cyclic permutations can also be selected in a randomised fashion. Randomization in general will provide the network with the ability of self-organizing node updates. For random scheduling signal, not only the convergence conditions, but also the convergence speed of the quantum states are of interest. There are indeed many possible quantum interaction graphs G = (V, E), and on top of that there are also many choices of the distributions of the random selection among the generalized edges in E. Nonetheless we can benchmark our analysis to the simple case where E is k-regular containing all possible generalized links with k different vertices (this means E has a total of ( ) n k entries), and at each time step each generalized link is selected independently with equal probability. Moreover, of the − k ( 1)! cyclic permutations over the selected k-node generalized set, we also assume it is randomly selected with equal probability. The resulting permutation selection process is denoted as π(t). Let P k contain all permutations that define a k-cyclic permutation over a subset of k nodes and induce an identity mapping over the rest n − k nodes. Then independent with time, π(t) selects a permutation with equal probabilities from P k , and the resulting quantum network state evolution is For the reduced states of the qubits, the following theorem holds.  The case with k = n is trivial since the reduced states of the qubits will reach a consensus in one step deterministically. It is clear from this result that the rate of convergence is precisely improved by n n n k n k n when size-k cliques are used instead of standard gossiping corresponding to k = 2. Therefore, for large n approximately  k n ( / ) is added to the rate of convergence using clique gossiping for the qubits' reduced states. On the other hand, for the network state ρ(t), we establish the following understandings.
Theorem 4 Let n ≥ 5. Along the algorithm (2) the following statements stand.
(i) The quantum network reaches a symmetric-state consensus both almost surely and in the mean-square sense if k is an even number.
both almost surely and in the meansquare sense if k is an odd number, where P even is the subset of P containing all even permutations over V.
In the above result, either k is even or odd the convergence speed is characterized by some v * > 0 in the form of the convergence rate v * can be described by Examples. We now present a few numerical examples to illustrate our findings. We fix the initial state of the qubit network, and average independent sample paths along the considered quantum gossip algorithm so that approximate mean-square error trajectories can be obtained. . The clique selection follows from the random scheduling π(t). In Fig. 2 we plot the function for k = 2, 3, 4, and 5, respectively. Clearly as the size of the cliques increases, the rate of convergence increases at the qubit reduced states.

Network State Convergence.
Apparently it is hard to calculate the mean-square error propagation matrix M through (5). For a network with 5 qubits with initial state ρ(0) = A, in Fig. 3 we plot the function for k = 2, 3, and 4, respectively, where ( ) ρ ∞ area calculated from the results of Theorem 4. As one can see the rate of convergence for the network density operator no longer monotonically depends on the size of the local quantum cliques that are used. This is in contrast to the reduced-state evolution, suggesting nonlinear dependence of the quantum network state dynamics and the classical network structure.

Proof of Theorem 1
(i) Let k  be the two-dimensional Hilbert space associated with the kth qubit. Then clearly This defines a matrix-valued classical averaging consensus algorithm 15  First of all, from vectoring the network density operator ρ(t) we know that the algorithm (1) defines a converging sequence by applying the Perron-Frobenius theory on the state transition matrix 3 . Next, the convergence limit of ρ(t) must be invariant under any permutation U k π associated to any e E k ∈ . This leaves being the consensus limit as the only possibility.
We have now completed the proof of the theorem.

Proof of Theorem 2
The reduced-state representation (6) of the algorithm is a classical clique-gossip algorithm 18 . Therefore, the desired convergence possibility and complexity results follow readily from Theorem 3 of 18 . This is consistent with the physical intuition that the reduced states at each qubit of the quantum network define classical quantities.

Proof of Theorem 3
We stack the reduced state ρ i (t) into Then there holds from the random quantum clique gossip algorithm that Now that E contains all generalized edges with size k and σ t ( ) selects each edge with equal probability independent with time, the distribution of W t can be written as has an eigenvalue − − n k n ( )/( 1) with multiplicity n − 1 and another eigenvalue 1 with multiplicity one. Invoking the analysis for standard gossip algorithm 9 , there holds for This implies that the ρ i (t) converges to reduced-state consensus in the mean-square sense. Since such mean-square convergence is exponential, almost sure convergence is also guaranteed. We have now completed the proof of the desired result.

Proof of Theorem 4
Recall that P k is the set containing all permutations that define a k-cyclic permutation over a subset of k nodes and induce an identity mapping over the rest n − k nodes. From (2) we can vectorize the network density operator by ρ = ∈ X t t ( ) vec( ( )) 4 n and obtain where in the second equality we have used the fact that π τ U t ( ) is a real matrix. Under this vectorization the dynamics of X(t) defines a classical consensus dynamics 2 . Convergence in both mean-square and almost sure sense becomes immediate following the same argument as used in 3 and the proof of Theorem 1. (ii), while the consensus limit would be where S k is the generating subgroup from the set P k . It is easy to verify that S k is a normal subgroup of P. We now investigate two cases, respectively.
• Suppose k is an odd number. Then clearly S k is a subgroup of P even . While it is well known that for n ≥ 5, P even contains no proper normal subgroup. Therefore, S k must be P even .
• Suppose k is an even number. Then ∩ S P k even is a normal subgroup of P even . Thus, ∩ = S P P k even even . As S k contains elements that are not in P even , it must hold that S k = P.
This proves the convergence statements. For the rate of convergence, applying the analysis in the proof of Theorem 3 over the recursion (9) we know that the rate of convergence v * is given by is a random matrix. The form of v * can be made clear when we pick up the distribution of M t from the distribution of π(t), which is exactly the equation of M in (5).

Conclusions
We have established a framework for the acceleration of quantum gossip algorithms by introducing clique operations based on cyclic permutations. It was shown that at reduced states, the cliques have the same acceleration effects as in accelerating classical gossip algorithms, under which finite-time convergence is achievable for suitable network structures. For randomized selection of cliques, it was proven that the rate of convergence is precisely improved by  k n ( / ) at reduced states, where k is the size of the cliques and n is the number of qubits in the network. It remains unanswered regarding how precisely cliques would affect the dynamics of the coherent states of the entire qubit network, where unique quantum features such as entanglements lie in. That would be a natural future direction for the study of distribute quantum algorithms.