Scalable distributed gate-model quantum computers

A scalable model for a distributed quantum computation is a challenging problem due to the complexity of the problem space provided by the diversity of possible quantum systems, from small-scale quantum devices to large-scale quantum computers. Here, we define a model of scalable distributed gate-model quantum computation in near-term quantum systems of the NISQ (noisy intermediate scale quantum) technology era. We prove that the proposed architecture can maximize an objective function of a computational problem in a distributed manner. We study the impacts of decoherence on distributed objective function evaluation.

As the development of quantum computers evolve extensively  , the power of quantum computations has become more interpretable for efficient problem-solving. However, while experimental quantum computers are currently under development, smaller quantum devices and quantum terminals are currently available in practice.
The problem of scalable quantum computation in a distributed quantum system is a challenge because of the complexity of the problem space provided by the diversity of possible quantum systems. The distributed quantum computational model has to include arbitrarily scaled quantum systems, from smaller quantum devices to large-scale quantum computers and the quantum Internet. As a corollary, the definition and parameterization of a scalable model for a distributed gate-model quantum computation is a hard problem, and no general solution is currently available.
Here, we study the problem of scalable quantum processing in distributed near-term quantum systems. We define a scalable distributed model of gate-model quantum computation and conceive the scaling attributes and unitaries of a distributed quantum information processing for problem-solving. The proposed scalable distributed quantum network integrates distributed quantum processing in arbitrarily scaled quantum systems.
In our context, an arbitrarily scaled quantum system can identify small, medium, or large-scale distributed quantum systems. The system model consists of an arbitrary number of quantum nodes connected by different levels of entangled connections (level of entanglement refers to the number of spanned nodes between a source and target node). The quantum system can refer to a quantum device, a quantum computer, or an arbitrary quantum Internet setting in which several quantum computers (quantum nodes) share entanglement to perform distributed quantum computations. The quantum nodes have to achieve the objective function maximization in a distributed way such that each node is allowed to apply local unitaries and connected via an arbitrary level of entanglement. In a small-scale system, the quantum nodes are connected by one-level entanglement while for a medium-or large-scale system, the level of entanglement between quantum nodes can be arbitrarily large. The local unitaries of the nodes are defined in a way that allows the distributed quantum system to implement a gate-model quantum computation in a distributed way.
We characterize a system model of a scalable distributed quantum system that allows for the performance of distributed gate-model quantum computation in a scalable manner. We define the scalable attributes of the system model and the gate parameters of the local unitaries of the quantum nodes for the objective function where β j ∈ [0, π] is the gate parameter of the unitary, while X is the Pauli σ x operator, and on qubits j and k in nodes V x and V y , β k ∈ [0, π].
The node pair is also allowed to realize a distributed unitary on qubits j and k using the l-level entangled connection E j = jk , where γ jk ∈ [0, 2π] is the gate parameter of the distributed unitary 12,14 , defined as where γ j , γ k ∈ [0, π] are the local gate parameters applied on qubits j and k, Z is the Pauli σ z operator, while Thus, setting the result in (8) can be evaluated as A node V x can also apply an U C x local coupling unitary to connect qubits i and j from entangled connections �(i − 1)(i)� and jk in V x , as where H (i,j) is a Hamiltonian, and also in V y on the qubits k and k + 1 of entangled connections jk and �(k + 1)(k + 2)� , as www.nature.com/scientificreports/ where H (k,k+1) is a Hamiltonian, to connect qubits k and k + 1 , and remote entangled connections. Therefore, the U xy unitary associated to a given node pair V x , V y connected by an l-level entanglement E j in the distributed quantum system N is defined as where U x is the unitary of a node V x , x = 1, . . . , L , defined as while U y is the unitary of its neighbor node V y , as Since unitaries (14) and (15) allows us to realize a gate-model quantum computation 14,29 , it follows that the V x , V y node pairs of the distributed quantum system N can implement quantum computation using their entangled connections in a distributed manner.

Methods. Proposition 2
To model multipartite entanglement in a particular node V x , qubit j has entangled connection with k to formulate jk , and also with Ŵ j remote qubits, n 1 , . . . , n Ŵ j , which are not neighbors of qubit k (These Ŵ j qubits have no connections with qubit k.). The total number of qubits that are neighbor of j but not neighbor of k is Ŵ j + 1.
Proof Each entangled connection E j has a contribution ζ E j to an F P(A→B) target function of a computational path P(A → B) (will be proven in Sect. 3) where |ϕ * � is the output state of P(A → B) , defined as where |+� = 1 √ 2 (|0� + |1�) , while U P(A→B) is defined as a unitary sequence associated to P(A → B) , as where jk ∈ P(A → B) refers to an E j entangled connection between qubits j and k on the computational path P(A → B).
The n-qubit length input system |s� of the distributed system N, is defined as a product of σ x eigenstates 12,14 , as where |z� is a computational basis state, z is an n-length string, where z i identifies an i-th bit, z i ∈ {−1, 1} , and |+� i is the input system of an i-th computational path (As N is a quantum computer system or a quantum device with quantum registers, then |s� refers to a quantum register in the superposition of n qubits, while a given node A i identifies the i-th source qubit, |+� i , of the n-length quantum register. In the current system model, the input system fed into the distributed system can also refer to a quantum register, physically not distributed between distant parties.) P(A i → B i ).
The nodes of the distributed system also can perform M[m b ] local measurements in a base m b ∈ {m 0 , m 1 } (see (35), (36)) to realize an L U upload 76,142 and an L D download 76,143 procedure. The L U upload procedure is an information delocalization method 76,142 , in which a source system is uploaded by a source node onto the network state formulated by the entangled connections of the intermediate nodes of the distributed system. The L D download procedure is an information localization procedure 76,143 , in which the uploaded and transformed information (transformed by the local unitaries of the quantum nodes in our setting) is localized into a particular target node from the network state of intermediate nodes. Since the distributed quantum system evolves with (13)  www.nature.com/scientificreports/ time, the timing of a local measurement also represents a scalable attribute in the distributed system (see (27) and (37)).

Proposition 3
In a source node A i , the L U (|+� i ) uploading is realized by a M B Bell measurement 76 applied on input system |+� i and the first particle of chain | � i that identifies the | � i network state of computational path P(A i → B i ).

Proof
The | � i network state is defined as where sub-index 1 identifies the first particle of | � i of P(A i → B i ) maximally entangled with the remaining 2(L − 1) qubits of the chain of P( The L U (|s�) operation therefore results in that yields the output system of N distributed between the n receiver nodes B 1 , . . . , B n . Thus. the outputs of the n paths, U(N)|s� can be localized onto the n receivers in the downloading procedure 76,143 .
To verify (22) and (27), we recall the formalisms of 144,145 . The input system |+� i of a given node A i can be rewritten as and let | � i be as given in (21), then where indices 0 and 1 identify the input system |+� i and the first qubit of the first EPR pair of chain | � i that serves as an |aux� auxiliary qubit system, H aux = C 2 , maximally entangled with the 2(L − 1)-qubit length system , while {|m s �} is an orthogonal basis 76,144,145 .
Then, in node A i an M B Bell measurement is applied on subsystems 0 and 1, that yields a projection onto while the ′ i post-measurement network state is evaluated as which coincidences with (22). Extending the derivations to n computational paths such that the paths realize the n-qubit unitary U(N) , each A i apply a Bell measurement M B (A i ) , thus the post-measurement network state ′ n 1 is as defined on H L = C 2 ⊗n2(L−1) since the entangled network structure of the distributed system is formulated via n2(L − 1) entangled states over the n computational paths, while n auxiliary qubit systems, |aux� 1 |aux� 2 . . . |aux� n , are measured via the Bell measurements in the n source nodes, H aux 1...n = C 2 ⊗n , that confirms the result in (27).
The L D downloading process 76,143 for receiver node B i results in To obtain (34) (Assuming that the entangled connections between the nodes are maximally entangled, ς = π , and ς < π otherwise. This parameter is also referred to as entanglement factor, see also 76 ). Then, it can be verified 76,143 that by applying M[m b ] local measurements in the L − 2 intermediate nodes between A i and B i as defined by (35) and (36), Bob B i obtains the Therefore, applying the measurement procedure in the intermediate nodes of the n computational paths, results in (28) at the receiver side in a distributed manner, as with probability over the n paths. Thus, if the network is maximally entangled it yields a deterministic download at the receiver with Pr (|φ * �) = 1. Ethics statement. This work did not involve any active collection of human data.

entangled connection between qubits j and k.
Proof Let N be the physical distributed quantum system, with a particular objective function C of a computational problem subject of a maximization. To simplify the discussion in the following section, allow us to focus on a single computational path P(A → B) , thus we set n = 1 , and N = P(A → B) with |s� = |+� ; however, the derivations and results are not restricted to this case.
Let U( θ) be the unitary realized via the computational path P(A → B) , as where i = 1, . . . , 2L , L is the number of nodes of N (number of distributed subsystems), 2L is the total number of unitaries in the L nodes (each node is defined via 2 unitaries) θ i is a gate parameter associated with U i , i.e., θ i = β i or θ i = γ i , and θ is the gate parameter vector defined as (|0� + |1�) , |s� = 1 √ 2 n z |z� , while |z� is an n-qubit length computational basis state. The aim of the distributed network system N is to maximize a C objective function of a computational problem in a distributed manner. The distributed system realizes the distributed unitary U(N) and outputs a distributed system |φ * � = U(N)|s� . The M distributed measurements are performed in the n receiver nodes B 1 , . . . , B n to produce the string z that allows the nodes to evaluate C(z) in a distributed way. The , where index 1 identifies the first particle of computational path P(A i → B i ) , formulated via the results of the unitaries of the n computational paths, where an i-th path Applying L U and L D for all source and receiver nodes, results in L D (L U (|+� 1 . . . |+� n )) = U(N)|s� at the receiver in a distributed manner.  Alice applies an M B Bell measurement on the input system |+� and on the first particle of the chain to achieve the L U (|+�) uploading procedure. A node pair V xy = V x , V y with a shared l-level entangled connection E j , j = 1, . . . , L − 1 ( l = 1 for a small-scale system while l ≥ 1 for a medium-and large-scale system by a convention) is allowed to (1) apply a local coupling unitary U C x = exp −itH (i,j) and U C y = exp −itH (k,k+1) to connect qubits i (connected to V x−1 ) to and j in V x , and qubits k and k + 1 (connected to V x+1 ), (2) to perform a local single-qubit unitaries U X j , β j and U(X k , β k ) on qubits j and k in V x and V y , (3) to realize a distributed two-qubit unitary U Z j Z k , γ jk on qubits j and k using the l-level entangled connection E j , and (4) to In a given V x , qubit j formulates a multipartite entanglement: j has an entangled connection with qubit k in V y , and j is also entangled with Ŵ j other neighbor qubits, n 1 , . . . , n Ŵ j , called remote entangled connections of j (not neighbors of qubit k), and the total number of qubits that are neighbors of j but not neighbors of k is Ŵ j + 1 . Each entangled connection E j has a contribution ζ E j to the expected target function value F P(A→B) = 1 Scientific Reports | (2021) 11:5172 | https://doi.org/10.1038/s41598-020-76728-5 www.nature.com/scientificreports/ where jk ∈ N is an l-level, l ≥ 1 , entangled connection between qubits j and k, with gate parameter vector and where Z j Z k = σ j z σ k z . At a particular physical entangled connection topology in N, the objective function C can be written as where C jk (z) is the objective function component 12,14 evaluated for entangled connection jk ∈ N , as where z is an n-length input bitstring, For a given z, a |z� computational basis state is defined as and |ϕ� output system of N at a single path at input (52) is defined as (For a level-p circuit, a set of p β and γ gate parameter vectors are used as β (1) , . . . , β (p) , and γ (1) , . . . , γ (p) . For simplicity, here we assume p = 1 , however the results can be extended for arbitrary p 14 . For further details, see 14 .) Then, let |s� be an n-qubit length input system of N, defined as in (19), thus for n = 1, and the output system |ϕ * � is evaluated as given in (17). The maximization of objective function C is identified via a target function F, as and for a particular entangled connection jk of N, the aim is the maximization of target function F jk , as where ϕ * N,jk is a target state defined as For the total system N, the objective function values of all entangled connections are summed, thus C(z) is as given in (49). For all connected qubits, the target function is set as www.nature.com/scientificreports/ where ϕ * N,jk is given in (57). Then, assuming that N consists of n computational paths, and |s� is an n-qubit length input as defined in (19), the result in (58) can be extended as that concludes the proof. (|0� + |1�) , while |A� and |B� are d = 2 dimensional vectors that represent the input and output systems (boundary conditions in the extended correlation space).
The system state of (60) can be rewritten as By recalling Observation 2 from 145 , allows us to the define δ i via the ς ∈ [0, π] measurement coefficient used in the definition of measurement operators (35) and (36), as where ω i identifies computational bases b ω i ∈ 0 ω i , 1 ω i , as Using ω i along with ς , a diagonal matrix D(ω i , ς) can be defined as where δ j is as given in (65) . . . W n S δ n,L W n S δ n,L−1 . . . W n S δ n,1 , www.nature.com/scientificreports/ where ζ E j is the contribution of an l-level E j entangled connection between qubits j and k in target function F P(A→B) , defined as where Ŵ j is the number of remote neighbor entangled qubits of j such that not neighbors of qubit k, while β j , β k and γ jk are the gate parameters of unitaries of U xy in (13) (The evaluation of (79) utilizes an abstraction. The structure of the distributed system is mapped onto a grid such that the vertices of the grid represent the qubits in the nodes, while an edge between the qubits identifies an l-level E j entangled connection in the distributed system. Since all connections between the qubits are entangled, the vertices on the grid are separated only by the particular edge that directly connects the qubits, thus the distance between the qubits on the grid is set to unit for all connections 14 .) Assuming that γ jk is set to the same value for all k, k = 1, . . . , Ŵ j + 1 , at β j = β k the result in (79) exp iβ j X j Z j exp −iβ j X j = cos β j I + i sin β j X j Z j cos β j I − i sin β j X j = cos β j Z j + i sin β j Z j X j cos β j I − i sin β j X j = cos β j Z j + i sin β j Y j cos β j I − i sin β j X j = cos 2 β j Z j − i cos β j sin β j Z j X j + i cos β j sin β j Y j − i 2 sin 2 β j Y j X j = cos 2 β j Z j + i cos β j sin β j X j Z j + i cos β j sin β j Y j − i 2 sin 2 β j Y j X j = cos 2 β j Z j + i cos β j sin β j Y j + i cos β j sin β j Y j + i 2 sin 2 β j Z j = cos 2 β j Z j + i 2 sin 2 β j Z j + 2i cos β j sin β j Y j U N, γ jk Z j U † N, γ jk = cos γ jk I − i sin γ jk Z j Z k Z j cos γ jk I + i sin γ jk Z j Z k = Z j cos γ jk I − iZ j sin γ jk Z j Z k cos γ jk I + i sin γ jk Z j Z k = cos γ jk Z j − i sin γ jk Z k cos γ jk I + i sin γ jk Z j Z k = cos 2 γ jk Z j + i cos γ jk sin γ jk Z k − i cos γ jk sin γ jk Z k − i 2 sin 2 γ jk Z j = cos 2 γ jk Z j − i 2 sin 2 γ jk Z j = cos 2 γ jk Z j + sin 2 γ jk Z j = 1 2 1 + cos 2γ jk Z j + 1 2 1 − cos 2γ jk Z j = 1 2 Z j + cos 2γ jk Z j + 1 2 Z j − cos 2γ jk Z j = Z j ,  www.nature.com/scientificreports/ Then, by utilizing the fact that input system |+� , and therefore also |s� , is an eigenstate of each X with eigenvalue 1 14 (see also (93) and (94)), the terms containing Y and Z vanish from (98), while X can be replaced as X = 1 . As follows, (98) can be rewritten as Further assuming that β j = β k = β holds, (99) can be simplified as (98) cos γ jk − X j X k sin γ jk sin 2β j sin 2β k cos γ jk + X j X k sin γ jk sin 2β j sin 2β k cos γ jk + X j X k sin γ jk sin 2β j sin 2β k cos γ jk + X j cos 2β k sin γ jk sin 2β j Assuming that (80) holds, (103) is simplified as If for each node the same β j , γ jk and Ŵ j values are set, (105) can be rewritten as After some calculations, the gate-parameter values β j and γ jk that maximize ζ E j (and therefore C P(A→B) ) are at and that yields gate parameter values and (100) χ jk = −2 cos 2β sin γ jk sin 2β sin 4β j sin γ jk cos (Ŵj+1) γ jk = 1 4 + 1 2 L−1 j=1 1 2 Ŵ j + 2 + sin 4β j sin γ jk cos (Ŵj+1) γ jk .
(107) 4 cos 4β j = 0 (108) cos (Ŵj+2) γ jk − Ŵ j + 1 cos (Ŵj) γ jk sin 2 γ jk = 0, The maximized C P(A→B) objective function value of (105) for a given computational path is therefore and the maximized value of (106) is as Figure 3. The values of ζ E j in function of gate parameters β j and γ j = 1 2 γ jk , for different Ŵ j values ( γ jk is set for the same value for all k, k = 1, . . . , www.nature.com/scientificreports/ The proof is concluded here. The values of ζ E j in function of gate parameters β j and γ j = 1 2 γ jk , for different Ŵ j values ( γ jk is set for the same value for all k, k = 1, . . . , Ŵ j + 1 ) are depicted in Fig. 3.
The objective function values (106) for a computational path P(A → B) in function of gate parameters β j and γ j = 1 2 γ jk , at different L node number and Ŵ j values ( β j and γ j are set as the same for all j, j = 1, . . . , L − 1 and k, k = 1, . . . , Ŵ j + 1 ) are depicted in Fig. 4.
The gate parameter values β j and γ jk for the maximization of ζ E j are depicted in Fig. 5.
. Proof Let assume that the total number of entangled connections of N is D = n(L − 1) . Then, let t ij (N) be a vector of initialization time parameters of the target states of the entangled connections, defined as For the survival amplitudes of the system states associated to the entangled connections at a given t, we also define a A N (t) vector of survival amplitudes associated to the D target states, as where A (j) (t) is the survival amplitude of ϕ * N,jk (t) , while j is the decay rate belongs to A (j) (t) defined via Û t, t Using (118), for a given qubit j, we define the µ j (t) cumulated target state intensity which is dynamic term to model the interaction within the entangled network structure, as follows. Term µ j (t) is defined as the sum of weighted target state decoherence terms (weighted target state intensities) of existing neighboring entangled connections and the actual weighted target state intensity at a local decoherence (local target function intensity), as where term � jk t, t (j) 0 is defined as the intensity of a target state ϕ * N,jk (t) , is the initial target function value at t (j) 0 , while jl refer to the neighboring entangled connections of j, l = 1, . . . , Ŵ j + 2 − 1, l � = k , while G l (s) is a control parameter 148 , defined as where s ≤ T.
Using (124), the µ N (t) = (µ 1 (t), . . . , µ D (t)) T cumulated target state intensity of N can be defined as As follows, the µ N t, τ (j) cumulated target state intensity of the global entangled structure can be decomposed into a sum of target state intensities before measurement M τ (j) [m b ] in the intermediate nodes, and after measurement in the target node. As a corollary of the M τ (j) [m b ] measurement on j, for any t ≤ τ (j) , the target state intensities of connections entangled with j vanish from the cumulated target state intensity in the intermediate nodes.
As the measurement on j is performed in the intermediate node, we focus to Bob B i , to evaluate the target state intensity on his localized system state. As follows, at Bob B i , the target state intensity of the localized system is as  The proof is concluded here.
The scaled A j t, t (j) 0 survival amplitude of the � B i (t) target function intensity of a given jk at different τ (j) measurement delays and j decay rates are depicted in Fig. 6.
Scaled computational cost. Lemma 2 (Cost of target function evaluation). The f C F jk computational cost associated to a given F jk is the total application time of the local unitaries. The cost function is scalable via Ŵ j in a multipartite entanglement system. Proof Let P(A → B) be a computational path in N with L nodes and (L − 1) entangled connections. Then, for a given jk ∈ N , let β * j , β * k and γ * jk refer to the gate parameters set to maximize the target function F jk , set via (109) and (110).
The f C F P(A→B) computational cost of the maximization of target function F P(A→B) is defined as where f C F jk is the computational cost associated to a given F jk of an entangled connection jk , as that measures the computational cost as the total application time of the local unitaries. As follows, (141) depends only on Ŵ j , thus the scaling coefficient of the computational cost is Ŵ j .
The S R f C F jk series representation of (142) for Ŵ j 1 + Ŵ j < 1 , is while the S E f C F jk series expansion of (142) at Ŵ j = ∞ is as The f C (N) total computational cost of N at n computational paths, is therefore  www.nature.com/scientificreports/ In Fig. 7, the scaled f C F jk cost function of F jk is depicted.

Conclusions
Here, we defined a scalable model of distributed gate-model quantum computation in near-term quantum systems. We evaluated the scaling attributes and the unitaries of a distributed system for solving optimization problems. We showed that the computational model is an extended correlation space. We studied how decoherence affects the distributed computational model and characterized a cost function. The proposed results are applicable in different scenarios of experimental gate-model quantum computations.