Local implementations of non-local quantum gates in linear entangled channel

In this paper, we demonstrate n-party controlled unitary gate implementations locally on arbitrary remote state through linear entangled channel where control parties share entanglement with the adjacent control parties and only one of them shares entanglement with the target party. In such a network, we describe the protocol of simultaneous implementation of controlled-Hermitian gate starting from three party scenario. We also explicate the implementation of three party controlled-Unitary gate, a generalized form of Toffoli gate and subsequently generalize the protocol for n-party using minimal cost.

Simultaneous Implementation of Controlled-Hermitian Gate: Consider three remote parties Alice, Bob and Charlie possess qubits 1,4 and 7 respectively of the arbitrary state with Σ 7 i=0 |d i | 2 = 1. Now Alice and Bob want to implement Controlled-Hermitian Gate (as well as unitary) on Charlie's system simultaneously where the target qubit is common for both the control parties. To achieve this task, Alice and Bob share a Bell state between their respective qubits 2 and 3; Bob and Charlie share a Bell state between their respective qubits 5 and 6 : Here Alice shares entanglement with another control party and Bob shares entanglement with the target party, which makes a linear entanglement connection (shown in Fig. 1). The combined state of all the qubits possessed by Alice, Bob and Charlie is given by, The details protocol of simultaneous C H implementation through linear network is described below, Step 1: Alice first applies controlled-NOT gate C N 12 on her qubits 1 and 2.
Step 2: Alice measures on qubit 2 in computational basis and Bob applies local operations according to the outcomes of the measurements as follows, (here σ i x , σ i z , σ i y are Pauli operators, with superscript 'i' indicating the qubit

Outcomes of measurements
Local operations Combined state after on 2 after measurements measurement and operations operand; C U mn denotes controlled-Unitary gate, where 'm' is the control bit and 'n' is target bit) Step 3: Bob measures qubit 5 in computational basis and Charlie applies local operations according to the outcomes of the measurements as follows,

Outcomes of measurements
Local operations Combined state after on 5 after measurements measurement and operations Step 4: Finally qubit 3 and 6 are measured in Hadamard basis by Bob and Charlie and corresponding Alice applies unitary operations to obtain the desired state,

Outcomes of measurements
Local operations Combined state of qubits 1,4,7 on 3 and 6 after measurements after measurement and operations | + + 36 Figure 1: Simultaneous C H implementation through linear network. The pictorial representation of local unitary operations, measurements and classical communications of this protocol has been depicted in Fig. 1. The simultaneous remote implementation of controlled-Hermitian gate from two parties to one consumes 2 ebits and total 5 cbits to communicate the measurement outcomes. The generalized protocol for n-party of simultaneous C H gate implementation described in Fig. 2, is an extension of the above protocol. For n-party, the communication cost is (n − 1) ebits and (n 2 + n − 2)/2 cbits.
From the above protocol, it can be inferred that the Unitary as well as Hermitian operators have significance in linear entangled network. This operator has the additional property of involution (i.e., the operator is same as its inverse), which is responsible for making this protocol deterministic. Most of the important gates like controlled-Pauli gates, controlled-Hadamard gate etc., belong to this category, making this implementation powerful.
Multiparty Controlled Unitary Gate Implementation: It has been shown that a more general form of Toffoli gate, i.e., controlled-controlled-Unitary gate can be deterministically implemented using two Bell pairs (2 ebits of entanglement) and 4 cbits to communicate the measurement outcomes [10]. Here we demonstrate the implementation of this non-local gate with the same communication cost using linear entangled channel (shown in Fig. 3). For illustration, we consider the same qubit distribution shared by Alice, Bob and Charlie described in Eq. 3 : where we want to implement controlled-controlled-Unitary gate, C U 147 (here qubit 1 and 4 are control bits and qubit 7 is target bit) on |ψ 147 . The details of the protocol is illustrated below, Step 1: Alice first applies controlled-NOT gate C N 12 , on her two qubits.
Step 2: Then she measures qubit 2 in computational basis and Bob applies unitary gates as follow,

Outcomes of Local
Combined state obtained measurements operations after measurements and operations on qubit 2 by Bob Step 3: Bob measures on qubit 5 in computational basis and accordingly Charlie performs unitary gates, (here U|ψ is denoted as|ψ )

Outcomes of
Local Combined state obtained measurements operations after measurements and operations on qubit 5 by Bob Step 4: After that Charlie measures qubit 6 in Hadamard basis and Bob applies local operations depending on the outcomes,

Outcomes of
Local Combined state obtained measurements operations after measurements and operations on qubit 6 by Bob Step 5: Finally qubit 3 is measured in Hadamard basis and Bob performs unitary gates to get the desired state which is shared by three parties,

Outcomes of
Local Combined state obtained measurements operations after measurements and operations on qubit 3 by Bob The above procedure can be generalized to implement a n-qubit gate, where (n − 1) qubits are controls and the unitary operator acts on the target qubit, only if, all the control qubits are |1 s. The protocol is illustrated in Fig. 4 and for n-party gate the communication cost is (n − 1) ebits and 2(n − 1) cbits which is optimal as shown in Ref. [10].
Discussion: In conclusion, we have described non-local gate implementation protocols in linear entanglement network by LOCC. Although the classical communication cost for implementing simultaneous controlled-gate is more as compared to the Eisert et. al. [10] scenario, the linear network is advantageous for large n, as each party shares only two entangled states and the target as well as the first control party share one entangled state. The fact that, our network comprises of Bell states, which are realized in laboratory conditions, makes our protocol experimentally achievable [3]. Optimal protocol for the simultaneous controlled-Unitary and other non-local gate implementations in linear entangled channels can be further investigated.