Distributed Quantum Computation via Entanglement Forging and Teleportation
Tian-Ren Jin Kai Xu Heng Fan


    

    
        
        Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
        School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

    
        
        Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
        School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
        Beijing Academy of Quantum Information Sciences, Beijing 100193, China
        Hefei National Laboratory, Hefei 230088, China
        Songshan Lake Materials Laboratory, Dongguan 523808, China
        CAS Center for Excellence in Topological Quantum Computation, UCAS, Beijing 100190, China

    
        hfan@iphy.ac.cn
        Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
        School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
        Beijing Academy of Quantum Information Sciences, Beijing 100193, China
        Hefei National Laboratory, Hefei 230088, China
        Songshan Lake Materials Laboratory, Dongguan 523808, China
        CAS Center for Excellence in Topological Quantum Computation, UCAS, Beijing 100190, China

    § ABSTRACT 
        Distributed quantum computation is a practical method for large-scale quantum computation on quantum processors with limited size.
        It can be realized by direct quantum channels in flying qubits.
        Moreover, the pre-established quantum entanglements can also play the role of quantum channels with local operations and classical channels.
        However, without quantum correlations like quantum channels and entanglements, the entanglement forging technique allows us to classically forge the entangled states with local operations and classical channels only.
        In this paper, we demonstrate the methods to implement a nonlocal quantum circuit on two quantum processors without any quantum correlations, which is based on the fact that teleportation with classically forged Bell states is equivalent to quantum state tomography.
        In compensation, the overhead of single-shot measurement will increase, and several auxiliary qubits are required.
        Our results extend the possibility of integrating quantum processors.
        We expect that our methods will complement the toolbox of distributed quantum computation, and facilitate the extension of the scale of quantum computations. 
    
    Distributed Quantum Computation via Entanglement Forging and Teleportation
    Heng Fan
     
==========================================================================



    

§ INTRODUCTION


    Quantum computation has attracted widespread attention in recent years, since it takes advantage when simulating large quantum systems, or solving specific problems with efficient quantum algorithms.
    Practical quantum computation requires millions of qubits in a relatively low level of noise, which may obstruct the applications of quantum computation.
    Quantum error correction and other techniques provide a systematic way to deal with the noises in quantum operations <cit.>.
    Recently, the remarkable progress in quantum error correction codes has declared that the level of noise in the state-of-art experimental techniques is close to the threshold of fault-tolerant quantum computation <cit.>. 

    However, fabricating such many qubits on an individual quantum chip is challenging for state-of-the-art experimental techniques.
    Quantum internet and distributed quantum computation provide a feasible way to extend the size of quantum processors <cit.>.
    On many integrated quantum processors, the scales of implementable quantum tasks can be much larger than those on an individual quantum processor.

    To integrate two spatially separate quantum processors, it can be directly correlated with flying qubits to transmit quantum states, which have been realized by both photons <cit.> and phonons <cit.>.
    The famous quantum teleportation protocol also shows that quantum channels can be undertaken by entangled states with classical channels <cit.>. 
    In addition, the gate teleportation protocol teleports the nonlocal controlled-unitary quantum gates with the assistance of only a single Bell pair <cit.>.
    Therefore, nonlocal circuits can be implemented on separate quantum processors with quantum correlations.

    However, when two quantum processors are separated with only local operations and classical channels (LOCCs), the nonlocal quantum circuits can still be implemented.
    Recently, a technique called entanglement forging has been used to double the size of quantum simulators <cit.>.
    In this technique, entangled states can be forged by separable states classically.
    Entanglement forging has been used only to double the size of quantum processors in some specially designed circuits <cit.>.
    With the entanglement forging and the teleportation protocol, any nonlocal quantum circuits can be forged on many separate quantum processors with only classical channels.
    
    This paper is organized as follows. 
    In Sec. <ref>, we review the entanglement forging and the gate teleportation protocol.
    In Sec. <ref>, we show that the quantum state teleportation with entanglement forging is equivalent to quantum state tomography, and demonstrate the methods to forge a nonlocal quantum circuit on two separated quantum processors with LOCCs. 
    In Sec. <ref>, we discuss the suppression of measurement noise in our methods with measurement error mitigation.
    The conclusion and discussion are given in Sec. <ref>.

    

§ PRELIMILARIES
 

    

 §.§ Entanglement Forging


        Entanglement forging employs the Schmidt decomposition of a bipartite entangled state
        
    |ψ⟩ = (Û⊗V̂) ∑_iλ_i |i⟩⊗|i⟩,
 
        where |i⟩ is the computational basis of local Hilbert space, and Û and V̂ are unitaries, and the Schmidt coefficients λ_i are positive.
        In terms of the density matrix, it can be written as
        
    ρ_ψ    = (Û⊗V̂) ∑_i (λ_i^2 |i⟩⟨i|⊗|i⟩⟨i| + ∑_j<i. 
           .×∑_p ∈ℤ_4 (-1)^p |ij_p⟩⟨ij_p|⊗|ij⟩⟨ij_p|) (Û^†⊗V̂^†),
 
        where |ij_p⟩ = 1/√(2)(|i⟩ + i^p |j⟩), with p ∈{1,2,3,4}.

        In this decomposition, the entangled state |ψ⟩ is not the classical probabilistic mixture of separable states, where all the coefficients of the decomposition are positive, due to the factor (-1)^p.
        This kind of decomposition is called local pesudomixture <cit.> in the investigation of the robustness measure of quantum entanglement.
        In general, we can decompose a state ρ_AB of system AB into the product states of AB
        
    ρ_AB = ∑_i x_i ρ_A^i ⊗ρ_B^i,
 
        where the coefficients x_i can be both positive and negative.
        With the normalization of states, the coefficients satisfy ∑_i x_i = 1, thus this decomposition is called the quasiprobability decomposition. 

        If all the coefficients are positive, this state is a separable state by definition.
        Therefore, for an entangled state, there exist coefficients x_i <0.
        The decomposition of the entangled state can be rewritten as 
        
    ρ_AB = Z ∑_i sgn(x_i) q_i ρ_A^i ⊗ρ_B^i,

        where Z = ∑_i |x_i|, and q_i = |x_i|/Z is a probabilistic distribution.
        Therefore, the expectation of observable Ô in ρ_AB is
        
    ⟨Ô|_⟩ρ_AB = Z ∑_i q_i (sgn(x_i) ⟨Ô|_⟩ρ_A^i ⊗ρ_B^i),

        the probabilistic mixture of the expectations in separable state ρ_A^i ⊗ρ_B^i with signature sgn(x_i).
        The entangled state can be simulated by a separable state in the sense of expectations.
        A similar technique is also used in the probabilistic error cancellation method of quantum error mitigation <cit.>.

        The cost of separable states in the simulation of an entangled state is evaluated as the factor Z, and the minimal cost over all possible decomposition is called the implementability of the state ρ_AB with respect to separable states <cit.>
        
    p_𝒬(ρ_AB) = min{∑_i |x_i|| ρ_AB = ∑_i x_i ρ_A^i ⊗ρ_B^i}.


    

 §.§ Gate Teleportation


        

        Gate teleportation protocol is put forward to nonlocally implement the quantum CNOT gate with the assistance of a pair of qubits in the maximal entangled state.
        It can also be extended to teleport any N-controlled unitary gate.
        Given a controlled-unitary gate 𝒰_C, there are universal starting and ending processes <cit.>, which allows for nonlocal application of the gate 𝒰_C with the consumption of only a single Bell pair.
        On the contrary, for the nonlocal implementation of a general two-bit gate, with the teleportation of state, it consumes two Bell pairs. 

        The gate teleportation protocol of a controlled-unitary gate 𝒰_C is shown in Fig. <ref>(c).
        The starting and ending process is part of the teleportation of state, as shown in Fig. <ref>(d), which contains the Bell measurement and the feedback.   
        Assume the input state of the starting process is |ψ_in⟩ = α|0⟩ + β|1⟩, which is the state of a qubit called the control qubit in the following. 
        Then the output state of the starting process is  
        
    |ψ_start⟩ = α|00⟩ + β|11⟩,

        where the second qubit is an auxiliary qubit.
        The ending process can take the state |ψ_start⟩ back to |ψ_in⟩, without the cost of nonlocal quantum operations or quantum state.
        
        If a two-qubit quantum gate Û is acted before the ending process, the action of the output state will generally not be the two-qubit quantum gate Û 
        Expand the quantum gate Û into the Pauli basis on the auxiliary qubit after starting the process as
        
    Û = ∑_i=0,x,y,zσ̂_i ⊗Û_i.

        The action of this gate is 
        
    Î⊗Û|ψ_start⟩⊗|ψ_t⟩
        = ∑_i (α|0⟩⊗σ_i |0⟩ + β|1⟩⊗σ_i |1⟩)⊗Û_i |ψ_t⟩,
    
        where |ψ_t⟩ is the state of qubit not involved in the starting and ending process, which is called target qubit in the following.
        After the ending process of the measurement in x direction on auxiliary qubit and feedback, the state on control and target qubit is 
        
    |ψ_out^+⟩    = [Î⊗(Û_0 + Û_x) + σ̂_z ⊗(Û_z + iÛ_y )]|Ψ_in⟩ , 
    |ψ_out^-⟩    = [Î⊗(Û_0 - Û_x) + σ̂_z ⊗(Û_z -iÛ_y)]|Ψ_in⟩ ,

        where |Ψ_in⟩ = |ψ_in⟩⊗|ψ_t⟩.
        To realize the gate teleportation, it requires that
        
    |ψ_out^+⟩ = |ψ_out^-⟩ = Û|Ψ_in⟩,

        which gives that 
        
    Û_x = Û_y = 0.

        This means that the quantum gates, which can be teleported with the starting and ending process, are in the form
        
    Û = Î⊗Û_0 + Û_cσ̂_z Û_c^†⊗Û_z,

        where Û_c is arbitrary unitary transformation acting on the control qubit.
        In this protocol, the gate teleportation costs one pair of qubits in the Bell state.
        For the general two-qubit quantum gate, the gate teleportation is realized with the state teleportation twice, which costs two pairs of Bell state.  

    

§ DISTRIBUTED QUANTUM COMPUTATION WITH LOCAL OPERATIONS AND CLASSICAL CHANNELS
 

    

 §.§ Teleportation Forged by Separable States

    
    The teleportation protocol teleports qubits with classical channels in the consumption of Bell state <cit.>, so the entangled states are resources for quantum communication. 
    The entangled states cannot be prepared by local operations and classical channels (LOCC), which is the free set of the quantum entanglement theory <cit.>.
    However, we can simulate the entangled states ρ^E by the quasiprobability decomposition.

    Here is an example of the simulation of the Bell state.
    
    |B^+⟩⟨B^+| =   1/2(|00⟩⟨00| + |00⟩⟨11|. 
        + . |11⟩⟨00| + |11⟩⟨11|) 
    
            =   1/4(Î + Ẑ_1 Ẑ_2 + X̂_1 X̂_2 - Ŷ_1 Ŷ_2 ) 
    
            =   1/4(Î + Ẑ_1 Ẑ_2) + 1/4(Î + X̂_1 X̂_2) 
        - 1/4(Î + Ŷ_1 Ŷ_2),
 
    where
    
    1/4(Î + Ẑ_1 Ẑ_2) =    1/2(|0_z0_z⟩⟨0_z0_z| + |1_z1_z⟩⟨1_z1_z|) ,
    1/4(Î + X̂_1 X̂_2) =    1/2(|0_x0_x⟩⟨0_x0_x| + |1_x1_x⟩⟨1_x1_x|) ,
    1/4(Î + Ŷ_1 Ŷ_2) =    1/2(|0_y0_y⟩⟨0_y0_y| + |1_y1_y⟩⟨1_y1_y|) .


    The teleportation protocol is implemented by appending a state ρ_C and performing the Bell state measurements Π̂_s, s=0,1,2,3 on the system CA.
    If the measurement outcome is s_0, then the state is 
    
    Π̂_s_0ρ_C ⊗ρ_ABΠ̂_s_0    = ∑_i x_i Π̂_s_0ρ_C ⊗ρ_Ai⊗ρ_BiΠ̂_s_0
        = Π̂_s_0⊗∑_i x_i p(s_0|i) ρ_Bi,

    where p(s|i) = Tr(Π̂_sρ_C ⊗ρ_Ai) = Tr(ρ_Ai^Tσ_sρ_C σ_s).
    The transported state ρ_C is obtained by 
    
    ρ_C ∝∑_i x_i p(s_0|i) 𝒰_s_0(ρ_Bi) = ∑_s ∑_i x_i p(s|i) 𝒰_s(ρ_Bi).

    Thus, with decomposition Eq. (<ref>), the classical information of p(s|i) for a fixed s is sufficient to recover the state ρ_C.
    Therefore, the teleportation protocol can be simulated by separable states completely, which means that the quantum information of Alice is converted to Bob completely by classical channels.

    For simplicity, we consider σ_s=0 = I, then p(0|i) = Tr(ρ_Ai^Tρ_C ).
    In the decomposition Eq. (<ref>), ρ_Ai = |i_α⟩⟨i_α|, where i = 0,1 and α = x, y, z, so as ρ_Ai^T.
    Therefore, the information of p(0|i) is the same as the required information of standard quantum state tomography (QST) on single qubit <cit.>, so the teleportation with the Bell state simulated in this decomposition is equivalent to the standard QST. 
    Because the dimension of the single qubit state space is d = 3, which is equal to the number of quantities measured by standard QST, it is reasonable to believe that the implementability is attained by this decomposition, p_𝒮(ρ_AB) = 3.

    When simulating many Bell pairs, let the system of Bell states ρ_AB be AB = ⊗_a A_a B_a, where A_a, B_a are single qubit.
    If the state that can be freely used is the separable state in the systems A_a and B_a,  ℱ_𝒮 = conv[⊗_a𝒮(A_aB_a)], it can be shown that (Proposition 4 in Ref. <cit.>)
    
    p_ℱ_𝒮(ρ_AB) = ∏_a p_𝒮(ρ_A_a B_a) = 3^L,

    where L is the number of simulated Bell pairs.
    The teleportation with Bell pairs in this simulation is equivalent to the local tomography <cit.>.
    Equation (<ref>) implies that the standard QST is the most efficient scheme in local tomography.

    In the contrary, if the entangled states between system A_a (or system B_a) can be freely used, ℱ = 𝒮(AB) = conv[𝒬(A) ⊗𝒬(B)], it can be show that (Proposition 5 and Proposition 11 in Ref. <cit.>)
    
    2^L ≤ p_𝒮(ρ_AB) ≤ 3^L.

    The investigations in classical teleportation also show that the standard QST is not the most efficient scheme in global tomography <cit.>.

    With the classically forged Bell states and the teleportation protocols, a quantum computational task can be performed on separate quantum processors with only classical communications.
    In compensation, there are exponentially increasing overhead of repeated measurements.
    For an arbitrary quantum gate, its teleportation requires two pairs of qubits in the Bell state.
    In the decomposition of Eq. (<ref>), the overhead for classical teleportation of one nonlocal quantum gate is 9 times more than the performing on one chip, and 4 auxiliary qubit.
    However, to teleport a general two-qubit gate, it requires two pairs of qubits in Bell states.
    In this case, the entanglement forging in Eq. (<ref>) is not optimal, and the optimal one has overhead larger than 4 times.
    Since the classical teleportations of nonlocal quantum gates are independent, assume that the quantum circuit has L layers, where each layer contains one nonlocal gate between two processors, the p^2L times overhead and the 4 L auxiliary qubits are required, where 2 ≤ p ≤ 3.

    

 §.§ Identical operations Forged by Projective Measurements


    

        Since the distributed quantum computation is aimed at scaling the size of the quantum computer, the increasing auxiliary qubit will confine its usage.
        In the following, we illustrate a method with less number of auxiliary qubits.

        The state teleportation can be viewed as an identical operation between two qubits.
        In the state teleportation with entanglement forging, one auxiliary qubit is needed to realize the identical operations.
        Therefore, we hope to classically simulate this operation directly.

        In terms of the Choi-Jamiołkowsky (CJ) isomorphism <cit.>, the Bell state is dual to the identical operations
        
    |B^+⟩⟨B^+| =   1/2(|00⟩⟨00| + |00⟩⟨11|. 
        + . |11⟩⟨00| + |11⟩⟨11|) 
    ↦ℐ(·) =    |0⟩⟨0|(·) |0⟩⟨0| + |0⟩⟨0|(·) |1⟩⟨1|
        + |1⟩⟨1|(·) |0⟩⟨0| + |1⟩⟨1|(·) |1⟩⟨1|.
 
        This inspires us to construct the identical operations from the classical forging of Bell state, Eq. (<ref>), by transposing the “bra” and “ket” in the middle of the states.
        This construction gives 
        
    ℐ(·)     =  Π̂_0z(·)Π̂_0z + Π̂_1z(·)Π̂_1z + Π̂_0x(·)Π̂_0x
            + Π̂_1x(·)Π̂_1x- Π̂_01y(·)Π̂_10y + Π̂_10y(·)Π̂_01y ,

        where Π̂_ijα = |i_α⟩⟨j_α| and Π̂_iα≡Π̂_iiα.
        It is the mixture of the projective measurements in x and z directions with the operations [Π̂_01y(·)Π̂_10y + Π̂_10y(·)Π̂_01y] removed.
        The removed operation is not the projective measurement in y direction, since 
        
    |0_y⟩^T = ⟨1_y|,   |1_y⟩^T = ⟨0_y|.


        Although, the operations [Π̂_01y(·)Π̂_10y + Π̂_10y(·)Π̂_01y] is not a projective measurement.
        It is equivalent to projective measurement up to a local unitary transformation
        
    Π̂_01y(·)Π̂_10y + Π̂_10y(·)Π̂_01y = 𝒫_z ∘ M_y(·),

        where 𝒫_z is the Pauli z operation.
        Therefore, the identical operation is 
        
    ℐ(·) = ℳ_z(·) + ℳ_x(·) - 𝒫_z ∘ℳ_y(·),

        where
        
    ℳ_α(·) = Π̂_0α(·)Π̂_0α + Π̂_1α(·)Π̂_1α

        is projective measurement in x,y, and z direction.
        This allows us to simulate the identical channels classically.

        Assume Alice (A) and Bob (B) want to perform a nonlocal circuit 𝒰.
        This circuit can be represented as 
        
    𝒰 = (𝒱_2A⊗𝒱_2B) ∘𝒱∘ (𝒱_1A⊗𝒱_1B),

        where 𝒱 is a nonlocal gate acting on one qubit q_a of Alice and one qubit q_b of Bob.
        Let the initial state be ρ_A⊗ρ_B. 
        Typically, it can be selected as |0⟩_A^⊗ m⊗|0⟩_B^⊗ n in practice.
        The output state is
        
    ρ_out = 𝒰(ρ_A⊗ρ_B).


        Insert ℐ before and after the gate 𝒱 on the qubit of Alice q_a (or Bob q_b), where the gate 𝒱 acts on.
        Then, the state can be realized by 
        
    ρ_out    = ∑_i,j,k,l∑_α, β, μ, ν M_jβ,iα M_lν,kμ
           ×ρ(A|i_α,l_ν) ⊗ρ(B|k_μ,j_β),

        where the unnormalized states ρ(A|i_α,l_ν), ρ(B|k_μ,j_β) are
        
    ρ(A|i_α,l_ν)     = Tr_q_a [Π̂_iα^(q_a)𝒱_2A^(q_c→ q_a)
           ∘𝒱_1A^(A)(ρ_A⊗Π̂_lν^(q_c))], 
    ρ(B|k_μ,j_β)     = Tr_q_d [Π̂_kμ^(q_d)𝒱_2B^(B)∘𝒱^(q_d → q_a)
           ∘𝒱_1B^(B)(ρ_B⊗Π̂_jβ^(q_d))],

        and the transition matrix is 
        
    M_jβ,iα = (-1)^δ_α,yδ_α,β [δ_i,j(1-δ_α,y) + (1-δ_i,j)δ_α,y].

        Here, q_c and q_d are two auxiliary qubits, and the notation [·]^(q_c,d→ q_a) represents that the operation [·] is acted by exchanging q_a and q_c,d
        
    [·]^(q_c,d→ q_a) = [·] ∘𝒮_q_c,d↔ q_a.


        This method splits the nonlocal circuit into two separate circuits with two auxiliary qubits.
        The diagram is shown in Fig. <ref>. 
        The auxiliary qubits q_c, q_d are prepared randomly in |i⟩_α, and the qubits q_a and q_d are measured randomly in x,y and z direction.
        After the preparations, evolutions, and measurements in q_a and q_d, they obtain the states ρ(A|i_α,l_ν) and ρ(B|k_μ,j_β), which are labeled by the prepared states |i_α⟩, |k_μ⟩, and the outcome |j_β⟩, |l_ν⟩ of measurements.
        In the post-processing, they pairs the state ρ(A|i_α,l_ν) with ρ(B|k_μ,j_β) and signature ϵ_αϵ_ν.
        The index |i_α⟩ is related to ⟨j_β|, and |k_μ⟩ is related to |l_ν⟩, in the way that for a pair (|i_α⟩, ⟨j_β|), i = j if α = β = x,z and i ≠  j if α = β = y.
        The signature ϵ_α = (-1)^δ_α,y is negative when α = y.
        This construction is from the equivalence between the teleportation forged by separable states and the standard QST shown in the previous.

        For a circuit with more nonlocal gates between Alice and Bob, the circuit can be split into separated circuits with the same method.
        The auxiliary qubits are 2 L for splitting L nonlocal gates.
        Moreover, if the qubit q_a can be reset, the qubit q_c can reuse q_a after the measurement, thus only L auxiliary qubits are necessary.
        However, in compensation, the overhead is 3^2L times, which is optimal since the preparation in state |i_α⟩ (and |k_μ⟩) and measurement in ⟨l_ν| (and ⟨j_β|) on Alice's (and Bob's) side have no quantum correlation.

    

§ MITIGATION OF THE NOISE IN MEASUREMENTS
 
    
        The perfect realizations of both the methods discussed in the previous section depend on perfect preparations of state and measurements.
        In practice, there always exist errors in preparations and measurements.
        These errors can be canceled by the measurement error mitigation <cit.>.
        With the post-processing of measurement error mitigation, the overhead will increase additionally.
        In the following, we consider the cancellation of measurement errors in detail.

        Assume the expectation of observables of interest can be calculated from the projective measurement in basis |s_A⟩⊗|s_B⟩, where s_A and s_B are bit strings.
        Then, the probabilities are 
        
    P(s_A, s_B)     = ∑_i,j,k,l∑_α, β, μ, ν M_jβ,iα M_lν,kμ
           × P(s_A|i_α,l_ν) P(s_B|k_μ,j_β),

        where 
        
    P(s_A|i_α,l_ν)     = P(s_A, i_α, l_ν)/∑_s_A P(s_A, i_α, l_ν), 
    
                P(s_B|k_μ,j_β)     = P(s_B, k_μ,j_β)/∑_s_B P(s_B, k_μ,j_β),

        where P(s_A, i_α, l_ν) and P(s_B, k_μ,j_β) can be counted from measurement data of experiments.

        In measurement error mitigation, there is an assignment matrix Â, which depicts the relation between the probabilities of ideal outcome and noisy outcome
        
    P_error(s) = ∑_s' A(s,s') P(s'),

        where P and P_error are the ideal and noisy probabilities of outcome.
        This matrix can be measured from experiments.
        The error is mitigated by 
        
    P(s) = ∑_s' A^-1(s,s') P_error(s').

        In this manner, with the noisy probabilities P_error(s_A, i_α, l_ν) and P_error(s_B, k_μ,j_β) from experiments, the ideal probabilities is calculated by 
        
        
    P(s_A|i_α,l_ν)     = ∑_s_A', i_α', l_ν' A^-1(s_A, i_α, l_ν;s_A', i_α', l_ν') P_error(s_A', i_α', l_ν')/∑_s_A∑_s_A', i_α', l_ν' A^-1(s_A, i_α, l_ν;s_A', i_α', l_ν') P_error(s_A', i_α', l_ν'), 
    
                P(s_B|k_μ,j_β)     = ∑_s_B', k_μ',j_β' A^-1(s_B, k_μ,j_β;s_B', k_μ',j_β') P_error(s_B', k_μ',j_β')/∑_s_B∑_s_B', k_μ',j_β' A^-1(s_B, k_μ,j_β;s_B', k_μ',j_β') P_error(s_B', k_μ',j_β').
 
        
    
    

§ CONCLUSION
 

        In this paper, we demonstrate two methods for distributed quantum computation on separated quantum processors with local operations and classical channels.
        There is a simple method that is based on the teleportation protocol with classical forged entanglement.
        It requires 4 auxiliary qubits and 9 times overhead for classically teleporting per nonlocal gate. 
        However, the overhead of this construction may not be optimal.

        Moreover, we construct another method that requires 2 auxiliary qubits per gate, which is less than the first one.
        This method is based on the identical operations forged by projective measurements.
        The demonstrated construction requires also 9 times overhead, however, which is optimal.
        We also show that the measurement error mitigation technique can cancel the error of the protocol in noisy cases.

        Our results demonstrate the methods to implement a nonlocal quantum circuit on two separate quantum processors only with local operations and classical channels, which extend the possibility of combining two quantum processors.
        We expect that our methods will complement the toolbox of distributed quantum computation, and facilitate the extension of the scale of quantum computations. 

        
            This work was supported by the Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0301800), National Natural Science Foundation of China (Grants Nos. T2121001, 92265207, 12122504). We also acknowlege the supported from the Synergetic Extreme Condition User Facility (SECUF) in Beijing, China.
        
    
    
    







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