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Editing Measurement-Only Universal Blind Quantum Computation

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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>
==Functionality Description==
'''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)]],
Delegated Computation is the task of assigning quantum computation to an untrusted device while maintaining privacy of the computation. It can be done via classical online/offline and quantum online/offline communication. Following description deals with a method which involves quantum online and classical online communication, called Blind Quantum Computation. It comes with the properties of correctness i.e. if both parties follow the protocol the final outcome is correct, blindness i.e. the Client to have Server carry out a quantum computation for her (Client) such that the Client’s inputs, outputs and circuit used for computation remain perfectly private from the Server and Universality i.e. the following protocol can implement any quantum computation.
[[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]].
'''Tags''' [[Two Party Protocols|Two Party]], [[Universal Task|Universal Task]], [[Quantum Functionality|Quantum Functionality]], [[Secure Delegated Quantum Computation|Secure Delegated Quantum Computation]], Quantum Online communication, Classical Online communication, [[Supplementary Information#Measurement Based Quantum Computation|Measurement Based Quantum Computation (MBQC)]].


==Assumptions==
==See also==
[[Prepare and Send-Universal Blind Quantum Computation|Prepare and Send-UBQC]], [[Measurement Only Verifiable Universal Blind Quantum Computation|Measurement Only Verifiable UBQC]], [[QKD|QKD]], [[Supplementary Information|Teleportation]].
 
==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 setup like optical systems, measurement of a state is much easier than generation of a state. Presented below are two versions of the protocol. The first protocol needs only quantum communication throughout the protocol while second needs both quantum and classical throughout 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 in the end of 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. Client informs the Server if she receives the it or else if she doesn’t, Client asks Server to send it again. Client measures her share of entangled pair in a certain measurement basis depending on her MBQC pattern. 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.
==Figure==
==Properties==
===Adversarial Assumption===
* This protocol is secure against honest but curious adversary setting
* This protocol is secure against honest but curious adversary setting
===Setup Assumptions===
*Client should have the classical means to compute the measurement pattern
*Client should have the classical means to compute the measurement pattern
*Client should have measurement devices.
*Client should have measurement devices.
*Protocol 1a assumes that quantum channel is not too lossy.
*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.
*No unwanted leakage from Client is assumed, i.e. Server cannot bug Client’s laboratory, a fundamental assumption in QKD.
 
===Parameters===
==Outline==
*(m,n,o) dimensions of cluster state. It could be 2D or 3D.
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.
===Security Claim/ Theorems===
===Protocol 1a: Device Independent===
*Universality Any model of quantum computation based on MBQC can be changed made blind using these protocols, thus, universality of protocol is implied by universality of the resource state used.
*''Server’s preparation'': Server prepares the resource graph state required for MBQC by the Client.
*CorrectnessCorrectness for both protocols is implied from MBQC implementing the quantum computation successfully.
*'' 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.
*Blindness Blindness for protocol 1a is implied by no-signalling theorem as Client does not send any information to Server by measuring her states.
 
===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.
*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 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.
*Protocol 1b can cope with high channel losses but is no longer a no-signalling protocol. In order to make it no-signallig Client needs to discard measurement device after one use or use a random number generator to indicate if particle was received or not.
*Both protocols follow the following definition of blindness: A protocol is blind if,
*Both protocols follow the follwing 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 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.
**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.
 
==Pseudocode==  
==Notations==
*Unless given specific mention in [.], following steps apply to both protcols
*(m,n,o) dimensions of cluster state. It could be 2D or 3D.
*'''Input:''' Server: Dimeonsions of Resource State (m,n,o)
* <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.
 
==Protocol Description==  
*Unless given specific mention in [.], following steps apply to both protocols
*'''Input:''' Server: Dimensions of Resource State (m,n,o)
*'''Output:''' Client: Final Outcome
*'''Output:''' Client: Final Outcome
#Server’s preparation
#Server’s preparation
##Server creates a resource state <math>G_{\text{mxnxo}}</math>
##Server creates a resource state Gmxnxo
#Interaction and Computation
#Interaction and Computation For i= 1 →m, j= 1 →n, k= 1 →o
##For i= 1,2,...m, j= 1,2,...n, k= 1,2,...o
##[Protocol 1a]
##[Protocol 1a]
###Server sends <math>|\psi\rangle_{i,j,k}\rangle</math> to Client
###Server directly sends the qubit |ψi,j,ki to Client
###Client measures <math>|\psi\rangle_{i,j,k}\rangle</math> in the required measurement basis according to her measurement pattern
###Client measures his qubit in the measurement basis according to the measurement pattern
##[Protocol 1b]
##[Protocol 1b]
###Server creates <math>\Phi_{1,2}=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)</math>
###Server creates Bell pair
###Server sends to Client (<math>|\phi_2\rangle</math>) and waits for Client's response
###Server sends one half (|Φ2i) of the Bell pair to Client
###Client checks if she received and tells the Server as Client.Response()
###Client tells her response to Server if she received the sent qubit or not iv. If she didn’t, Server repeats the previous two processes, otherwise
###'''If Client.Response()=No''', Server repeats the previous two steps
###Client measures her share of entangled qubit (|Φ2i) in measurement basis {|0i ± eθ |1i} determined by measurement pattern.  in case of Clifford gates while {π/4} in case of non-Clifford gates.
###'''Else''' Client measures (<math>|\phi_2\rangle</math>) in measurement basis {<math>|0\rangle \pm e^{i\theta}|1\rangle</math>
###Server uses gate teleportation to apply this unknown gate on the qubit of resource state as follows
###'''Server's Computation: [[Glossary#Gate Teleportation|Gate Teleportation]]'''
####He entangles his share of Bell pair with the qubit of the resource state |ψi,j,ki by performing CZ
####He entangles <math>|\phi_2\rangle</math> with <math>|\psi\rangle_{i,j,k}</math> by performing [[Glossary#Unitary Operations|C-Z]]
####He measures the qubit in the register, |ψi,j,ki in X basis ({|+i,|−i}) and communicates the outcome to the Client. This applies the required measurement on the qubit of the resource state with some correction depending on the outcome
####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
####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.
*Interaction and Computation steps are repeated until all the qubits of resource state are measured.


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