Recently the study of distributed quantum consensus algorithms drew attention in the research community [1, 2, 3, 4], where the goal is to build the analogue of classical consensus algorithms [5, 6, 7, 8] towards distributed and scalable control and computation means for quantum networks. In classical networks, nodes holding real values can achieve a common state by self-organized communication and local computations . In networks of qubits, consensus can be defined over a set of different notions, but using the idea of classical gossip algorithms quantum consensus algorithms can indeed be developed with conceptual consistency. For both open quantum networks[2, 3] and hybrid quantum networks, quantum consensus can in fact be conveniently studied by building the bridge to its classical counterpart.
The prospects of carrying out quantum computation and quantum communication via networks of quantum subsystems have already been noted in the past few years [10, 13, 14, 11, 12]. The development of quantum consensus and synchronization algorithms[1, 2, 3, 4]
marks a continuing flow of this line of research as simple but fundamental blocks of network computation and information dissemination. The expectation is that more advanced algorithms may be developed for a variety of control, computation, and estimation tasks over complex quantum networks on top of such foundations, as witnessed in the engineering of classical networks in the past decades.
One critical performance metric of distributed algorithms is their rates of convergence. As a foundational block for distributed algorithms, classical gossiping in randomized form achieve asymptotical convergence, whose speed of convergence is governed by the underlying network structure. Even finite-time convergence is possible for classical gossiping under selected network topologies . In order to improve the rate of convergence of distributed algorithms, either classical or quantum, an immediate thought was to respect the network structure while adjusting the weights of the links representing strength of interactions. With the ability of designing the network structure, optimization is not an easy problem due to the arising combinatorial obstacles. However, utilizing certain micro-structures such as cliques , i.e., local complete subgraphs of a network, one can resolve previously impossible convergence requirements or significantly improve convergence speed.
This paper aims to establish a framework for the acceleration of quantum gossip algorithms by introducing clique operations to networks of qubits. For a local complete subgraph over such networks, cyclic permutations are used to define their collective interactions which can be physically realized by a series of local environments. The focus is then the convergence conditions and convergence rates with deterministic or random scheduling of the cliques. We first show that at reduced states, these cliques have the same acceleration effects, which can even enable finite-time convergence for suitable network structures. Next, we show that for random selection of cliques, the rate of convergence is improved by at reduced states, where is the size of the cliques and is the number of qubits in the network. The rate of convergence of the network coherent states is established via the spectrum of a mean-square error evolution matrix. Explicit calculation of such matrix seems to be extremely difficult, however, the effect of cliques on the coherent states can be seen via numerical examples. It is surprising 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.
2 Main Results
2.1 Open Quantum Networks
We consider a group of quantum nodes each holding a quit indexed in the set . Time is slotted for , and at time the state of qubit is denoted by a density operator (matrix) . The network state is denoted by the density operator . The qubits can be locally connected by a series of environments, which are by themselves quantum systems as well. These local environments induce a classical interaction structure which is described by a generalized graph , where each element in is a nonempty and non-singleton subset of . For example, is a generalized edge among three nodes , , and . We index the elements in by for some . The quantum generalized interaction graphs is called -regular if for all .
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 is called the ’th permutation group, denoted by . Associated with any , we define a permutation over the node set in the way that
if for any ;
is a cyclic permutation.
For example with , a permutation satisfying
is a cyclic permutation defined over a generalized edge . We note that if the size of is , the number of cyclic permutations over is
. Here for the moment we assume thatis an arbitrary cyclic permutation to ease the presentation. Note also that any permutation over further induces a quantum permutation operator over the qubit network by
being any unit vector of the state space of qubit.
2.2 Deterministic Quantum Clique Gossiping
Let be a mapping from to . Practically, the mapping selects a multi-vertex link from the generalized graph by assigning at time . It is natural to assume that is a periodic signal going through every element in the set . When is selected, the local environment associated with the qubits in is engineered so that the network density operator evolves along
This defines a deterministic quantum clique-gossiping algorithm. Clearly, when each 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.
Along the deterministic quantum clique gossip algorithm (1), the following statements hold.
(i) The network achieves reduced-state consensus in the sense that
if and only if and .
(ii) The network density operator satisfies
where is the generating subgroup by the permutations in the set associated with the .
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. The following result shows the possibility of using -regular interaction graphs to ensure reduced-state convergence in some finite time steps.
There exists a -regular interaction graph under which a quantum clique-gossiping algorithm (1) can converge to a reduced-state consensus in some finite time steps if and only if is divisible by with the same prime factors as .
Particularly, if there exist factorizations and with being prime numbers and for all , then a fastest -regular quantum clique-gossiping algorithm drives the qubit states to
in steps. This result illustrates that cliques have exactly the same acceleration effects at the reduced states in the quantum setting as the classical case. As a matter of fact, the reduced states follow a similar type of evolution where the analysis for classical networks becomes directly applicable.
2.3 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 , and on top of that there are also many choices of the distributions of the random selection among the generalized edges in . Nonetheless we can benchmark our analysis to the simple case where is -regular containing all possible generalized links with different vertices (this means has a total of
entries), and at each time step each generalized link is selected independently with equal probability. Moreover, of thecyclic permutations over the selected -node generalized set, we also assume it is randomly selected with equal probability. The resulting permutation selection process is denoted as . Let contain all permutations that define a -cyclic permutation over a subset of nodes and induce an identity mapping over the rest nodes. Then independent with time, selects a permutation with equal probabilities from , and the resulting quantum network state evolution is described by
For the reduced states of the qubits, the following theorem holds.
Let . Along the algorithm (2) the reduced states converge to a consensus both almost surely and in the mean-square sense. Particularly, the convergence speed is characterized by in the sense that
The case with 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
when size- cliques are used instead of standard gossiping corresponding to . Therefore, for large approximately is added to the rate of convergence using clique gossiping for the qubits’ reduced states. On the other hand, for the network state , we establish the following understandings.
Let . 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 is an even number.
(ii) The quantum network reaches
both almost surely and in the mean-square sense if is an odd number, where
is an odd number, whereis the subset of containing all even permutations over .
In the above result, either is even or odd the convergence speed is characterized by some in the form of
the convergence rate can be described by
where denotes the spectrum of the matrix . An explicit calculation of is rather challenging due to the complexity of .
2.4.1 Reduced State Convergence
We consider a network of qubits. Let the initial network state be with
where and . The clique selection follows from the random scheduling . In Figure 2 we plot the function
for , , , and , respectively. Clearly as the size of the cliques increases, the rate of convergence increases at the qubit reduced states.
2.4.2 Network State Convergence
Apparently it is hard to calculate the mean-square error propagation matrix through (5). For a network with qubits with initial state in Figure 3 we plot the function
for , , and , respectively, where 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.
3.1 Proof of Theorem 1
(i) Let be the two-dimensional Hilbert space associated with the th qubit. Then clearly as the reduced state of qubit after tracing out the rest of the qubits. From the algorithm (1) it is clear that at the reduced states there holds
This defines a matrix-valued classical averaging consensus algorithm, where the updates happen along each entry of the matrices independently. From the results of we know that the desired convergence holds if and only if the edge set
forms a connected undirected graph over the node set , which is in turn equivalent to the conditions that and .
(ii) This part of the conclusions is analogous to the results for continuous-time quantum consensus dynamics. First of all, from vectoring the network density operator we know that the algorithm (1) defines a converging sequence by applying the Perron-Frobenius theory on the state transition matrix . Next, the convergence limit of must be invariant under any permutation associated to any . This leaves being the consensus limit as the only possibility.
We have now completed the proof of the theorem.
3.2 Proof of Theorem 2
The reduced-state representation (6) of the algorithm is a classical clique-gossip algorithm. Therefore, the desired convergence possibility and complexity results follow readily from Theorem 3 of . This is consistent with the physical intuition that the reduced states at each qubit of the quantum network define classical quantities.
3.3 Proof of Theorem 3
We stack the reduced state into
Then there holds from the random quantum clique gossip algorithm that
is a random matrix andis the identity matrix.
Let be the unit -dimensional vector with the th entry being . Introduce
for any with the being pairwise distinct. We can verify by direct calculation that each satisfies
Now that contains all generalized edges with size and selects each edge with equal probability independent with time, the distribution of can be written as
for any . Consequently, we obtain
which implies that
has an eigenvaluewith multiplicity and another eigenvalue with multiplicity one. Invoking the analysis for standard gossip algorithm, there holds for
This implies that the 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.
3.4 Proof of Theorem 4
Recall that is the set containing all permutations that define a -cyclic permutation over a subset of nodes and induce an identity mapping over the rest nodes. From (2) we can vectorize the network density operator by and obtain
where in the second equality we have used the fact that is a real matrix. Under this vectorization the dynamics of defines a classical consensus dynamics. Convergence in both mean-square and almost sure sense becomes immediate following the same argument as used in  and the proof of Theorem 1.(ii), while the consensus limit would be
where is the generating subgroup from the set . It is easy to verify that is a normal subgroup of . We now investigate two cases, respectively.
Suppose is an odd number. Then clearly is a subgroup of . While it is well known that for , contains no proper normal subgroup. Therefore, must be .
Suppose is an even number. Then is a normal subgroup of . Thus, . As contains elements that are not in , it must hold that .
where is a random matrix. The form of can be made clear when we pick up the distribution of from the distribution of , which is exactly the equation of in (5).
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 at reduced states, where is the size of the cliques and 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.
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