Measurement-Only Verifiable Universal Blind Quantum Computation

The example protocol achieves the functionality of Secure Verifiable Delegated Quantum Computation which is the task of assigning quantum computation to an untrusted device while maintaining privacy of the computation. This protocol allows the client to verify the correctness of the blind delegated quantum computing with high probability. Here, the server prepares and sends a universal resource quantum state to the client, and the client performs measurements on the resource state to carry out the quantum computation. Using this method, it is easy to verify with high probability whether the server is honest. The computation here remains perfectly private from the server and this protocol can implement any quantum computation.

Tags Two Party,Universal Task, Quantum Functionality, Quantum Online communication, Classical Online communication, Measurement Based Quantum Computation (MBQC)

Assumptions

• The protocol assumes perfect state preparation, transmissions and measurements.
• There is no communication channel from the client to the server.
• The quantum channel from the server to the client should not be too lossy.

Outline

This protocol is based on MBQC and is mainly derived from Measurement-Only Universal Blind Quantum Computation. In this protocol, the server prepares and sends the resource state to the client, whereas the client performs measurements on the received states. The server is considered to be a general adversary and any computational deviations can be detected in this method by the client.

By performing measurements, the client creates the final state to be a mixture of resource state and trap qubits, in a random distribution. If the measurement of all the trap qubits matches the expected outcome, then it shows with high probability that the server is honest and has not deviated from the protocol.

This protocol is dived into two stages: Servers' preparation and Client's measurement.

• Server's preparation: Server prepares the resource graph state required for MBQC by the Client. The server sends each qubit of this state to the Client, one by one.
• Client's Measurement
  • The client receives the qubits from the server and measures them in order to finally create an n-qubit state with the server, following the principle of MBQC. This n-qubit state comprises of randomly distributed resource state of measurement-based quantum computation and trap qubits. Based on no-signaling principle, the server can never find out about the random distribution pattern.
  • On receiving the final n-qubit state, the client performs the measurement on these states with certain corrections. If the measurement of all the traps matches the expected outcome, the results are accepted otherwise the protocol is aborted.

Notation

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_{m\times n}} : Resource state
• $m\times n$: Resource state size
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle_P} : $P(|R\rangle \otimes |+\rangle ^{\otimes {\frac {N}{3}}}\otimes |0\rangle ^{\otimes {\frac {N}{3}}})$, this is the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -qubit state left with the server which contains the trap qubits (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0\rangle} , $|+\rangle$) and resource state.
• $|R\rangle$: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{n}{3}} -qubit resource state
• $P$: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -qubit permutation, which keeps the order of qubits in $|R\rangle$
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |+\rangle} : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{\sqrt{2}} (|0\rangle + |1\rangle)}
• q: $(x_{1},...,x_{n},z_{1},...,z_{n})\in \{0,1\}^{2n}$
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_q} : $\bigotimes _{j=1}^{n}X_{j}^{x_{j}}Z_{j}^{z_{j}}$

Hardware Requirements

• Quantum computation resources for the server.
• A quantum channel from the server to the client to transfer the quantum states.
• Measurement device for the client.
• No channel is required from client to the server.

Properties

• This protocol detects a cheating server with high probability.
• Universality: This protocol is universal in nature. The resource state used is universal and thus can implement any quantum computation.
• Correctness: The correctness of the protocol is implied from the measurement based quantum computing used.
• Blindness: As there exists no quantum channel from the client to the server, the no-signaling theorem ensures that no information about the states is sent to the server using just the measurements.
• The security of this protocol is device independent, which means client does not need to trust their measurement device in order to guarantee the security.
• One advantage of this protocol is that no random number generators are required.

Pseudocode

Stage 1: Server's preparation
Input: Dimensions of the resource state.
Output: Client: receives all qubits

• Server creates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_{m\times n}} and sends each qubit to Client

Stage 2: Client's measurement
Input: Resource state qubits
Output: Final outcome

• For $i=1,2,...m-1$:
  • For $j=1,2,...n$:
    • Client receives resource state qubit $|\psi \rangle _{i,j,0}$ from server.
    • The client performs the measurement on the qubit according to the measurement pattern.
  • Through the measurements, Client creates the state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_q|\psi\rangle_P} in the server's possession, where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle_P = P(|R\rangle\otimes |+\rangle^{\otimes \frac{N}{3}} \otimes |0\rangle^{\otimes \frac{N}{3}}) }

• For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = 1, 2, ...n} : (for every qubit of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_q|\psi\rangle_P} )
  • Client receives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_q{|\psi\rangle_P}_i} from the server.
  • Measurement is done after applying the correction $\sigma _{q}^{\dagger }$ on the qubit received.
  • If result obtained is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |1\rangle} (for trap qubit $|0\rangle$) or $|-\rangle$(for trap qubit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |+\rangle} ):
    • Protocol is aborted.
  • else:
    • Protocol is continued and accepted if all trap qubits show expected outcome ($|0\rangle$ or $|+\rangle$).

Further Information

A second protocol exists which uses the property of the topological code, and does not use any trap qubits. Here, after the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_q|\psi\rangle_P} state is prepared, client does topological measurement-based quantum computation with a correcting factor. If any error is detected, the protocol is aborted.

*contributed by Rhea Parekh