Editing Measurement-Only Universal Blind Quantum Computation

The [https://journals.aps.org/pra/abstract/10.1103/PhysRevA.87.050301 example protocol] achieves the functionality of [[Secure Client- Server Delegated Computation]] by assigning quantum computation to an untrusted device while maintaining privacy of the input, output and computation of the client. The client requires to be able to prepare and send quantum states while the server requires to possess a device with quantum memory, measurement and entanglement generation technology. Following description deals with a method which involves quantum online and classical online communication, called Blind Quantum Computation. It means the protocol needs continuous quantum and classical communication between the parties, throughout the execution. It comes with the properties of [[Secure Client- Server Delegated Quantum Computation#Properties|correctness]], [[Secure Client- Server Delegated Quantum Computation#Properties|blindness]] and [[Secure Client- Server Delegated Quantum Computation#Properties|universality]].</br></br>
'''Tags:''' [[Category: Two Party Protocols]] [[:Category: Two Party Protocols|Two Party]], [[Category: Universal Task]][[:Category: Universal Task|Universal Task]], [[Category: Quantum Functionality]] [[:Category: Quantum Functionality|Quantum Functionality]], Quantum Online communication, Classical Online communication, [[Supplementary Information#Measurement Based Quantum Computation|Measurement Based Quantum Computation (MBQC)]],
[[Prepare and Send-Universal Blind Quantum Computation|Prepare and Send-UBQC]], [[Measurement-Only Verifiable Universal Blind Quantum Computation|Measurement Only Verifiable UBQC]], [[Quantum Key Distribution|QKD]], [[Quantum Teleportation]].

==Assumptions==
* This protocol is secure against honest but curious adversary setting
*Client should have the classical means to compute the measurement pattern
*Client should have measurement devices.
*Protocol 1a assumes that quantum channel is not too lossy.
*No unwanted leakage from Client is assumed, i.e. Server cannot bug Client’s laboratory, a fundamental assumption in QKD.

==Outline==
The following Universal Blind Quantum Computation (UBQC) protocol uses the unique feature of [[Supplementary Information#Measurement Based Quantum Computation|Measurement Based Quantum Computation (MBQC)]] that separates the classical and quantum parts of a computation. Based on its counterpart Prepare and Send UBQC, this protocol requires Client to possess only a measurement device in order to perform blind quantum computation, hence the name 'Measurement Only UBQC'. The motivation behind this protocol lies in the fact that for several experimental setups like optical systems, measurement of a state is much easier than the generation of a state. Presented below are two versions of the protocol. The first protocol needs only quantum communication throughout the protocol while the second needs both quantum and classical throughout the communication. These protocols are designed for classical input and output. It can be extended to quantum input/output by modifying the measurement angles of the Client according to Prepare and Send UBQC in order to hide her quantum output from the Server. Like all the other delegated quantum computing protocols, this protocol is also divided into two stages, Preparation and Computation.
===Protocol 1a: Device Independent===
*''Server’s preparation'': Server prepares the resource graph state required for MBQC by the Client.
*'' Interaction and Client’s Computation'': Server sends single qubits of the prepared resource state to the Client who measures it in the basis required to carry out the quantum computation according to the measurement pattern in her mind. She records the outcomes and at the end of the computation stage, gets the result of her computation. This protocol is not tolerant to channel losses.

===Protocol 1b: Tolerant to high channel losses===
*''Server’s preparation'': This step remains the same as protocol 1a
*'' Interaction and Client’s Computation'': Server prepares a Bell pair and sends one half of the Bell Pair to the Client. The Client informs the Server if she receives it or else if she doesn’t, Client asks Server to send it again. The client measures her share of entangled pair in a certain measurement basis depending on her MBQC pattern. The Server then entangles his share of Bell pair and qubit of the resource state using CZ gate which transfers the gate/ measurement operated by Client to the resource qubit. Then he measures the resource qubit in X basis and communicates his classical measurement outcome to the Client. Client records it and uses it to compute her final outcome.

==Requirements==
*'''Network Stage:''' [[:Category:Quantum Memory Network Stage|Quantum Memory]] [[Category:Quantum Memory Network Stage]]
*'''Required Network Parameters:''' 
**'''<math>\epsilon_j</math>''', which measures the error due to noisy operations.
**Number of communication rounds
**Circuit depth
**Number of physical qubits used
*Client should have measurement devices
*Quantum offline channel
*Classical online channel
*Server should be able to generate and store large network of entangled quantum states.

==Knowledge Graph==

{{graph}}

==Properties==
*Universality - Any model of quantum computation based on MBQC can be changed made blind using these protocols, thus, the universality of the protocol is implied by the universality of the resource state used.
*Correctness - Correctness for both protocols is implied from MBQC implementing the quantum computation successfully.
*Blindness - Blindness for protocol 1a is implied by no-signalling theorem as Client does not send any information to Server by measuring her states.
*Security of protocol 1a is device independent i.e. Client does not need to trust her measurement device in order to guarantee privacy.
*Protocol 1a can cope with Client’s measurement device inefficiency.
*Protocol 1b can cope with high channel losses but is no longer a no-signalling protocol. In order to make it no-signalling Client needs to discard measurement device after one use or use a random number generator to indicate if the particle was received or not.
*Both protocols follow the following definition of blindness: A protocol is blind if,
**The conditional probability distribution of Alice’s computational angles, given all the classical information Bob can obtain during the protocol, and given the measurement results of any POVMs which Bob may perform on his system at any stage of the protocol, is equal to the a priori probability distribution of Alice’s computational angles, and
**The conditional probability distribution of the final output of Alice’s algorithm, given all the classical information Bob can obtain during the protocol, and given the measurement results of any POVMs which Bob may perform on his system at any stage of the protocol, is equal to the a priori probability distribution of the final output of Alice’s algorithm.

==Notations==
*(m,n,o) dimensions of cluster state. It could be 2D or 3D.
* <math>G_{\text{mxnxo}}</math>: Graph state/Resource state created by the Server, as required by Client
* <math>|\psi\rangle_{i,j,k}\rangle</math>: A qubit of the resource state at position (i,j,k)
* <math>\Phi_{1,2}</math>: [[Glossary#Bell States|Bell pair]]
* <math>|\phi_2\rangle</math>: Client's half of the Bell pair
* <math>|\phi_1\rangle</math>: Server's half of the Bell pair
* <math>\theta</math>: Measurement angle as determined by Client's MBQC pattern. <math>\theta \epsilon\{0,\pi /2\}</math> in case of Clifford gates while <math>\theta \epsilon\{\pi /4\}</math>  in case of non-Clifford gates.

==Pseudocode== 
*Unless given specific mention in [.], following steps apply to both protocols
*'''Input:''' Server: Dimensions of Resource State (m,n,o)
*'''Output:''' Client: Final Outcome
#Server’s preparation
##Server creates a resource state <math>G_{\text{mxnxo}}</math>
#Interaction and Computation
##For i= 1,2,...m, j= 1,2,...n, k= 1,2,...o
##[Protocol 1a]
###Server sends <math>|\psi\rangle_{i,j,k}\rangle</math> to Client
###Client measures <math>|\psi\rangle_{i,j,k}\rangle</math> in the required measurement basis according to her measurement pattern
##[Protocol 1b]
###Server creates <math>\Phi_{1,2}=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)</math>
###Server sends to Client (<math>|\phi_2\rangle</math>) and waits for Client's response
###Client checks if she received and tells the Server as Client.Response()
###'''If Client.Response()=No''', Server repeats the previous two steps 
###'''Else''' Client measures (<math>|\phi_2\rangle</math>) in measurement basis {<math>|0\rangle \pm e^{i\theta}|1\rangle</math>
###'''Server's Computation: [[Glossary#Gate Teleportation|Gate Teleportation]]'''
####He entangles <math>|\phi_2\rangle</math> with <math>|\psi\rangle_{i,j,k}</math> by performing [[Glossary#Unitary Operations|C-Z]]
####He measures <math>|\psi\rangle_{i,j,k}</math> in X basis ({<math>|+\rangle,|-\rangle</math>}) 
####Server's applies correction on the classical outcome using Gate Teleporation
###Server communicates the corrected outcome
####Client records Server’s outcome and uses it when computing the final result or measurement angles for further qubits

*Interaction and Computation steps are repeated until all the qubits of resource state are measured.

==Further Information==

<div style='text-align: right;'>''*contributed by Shraddha Singh''</div>