Abstract
How to reduce the quantum cost of the client is a hot topic in blind quantum computation. However, making full use of the server’s quantum ability to reduce the client’s quantum cost is rarely acknowledged. In this research, we show that twice measurements on a three-qubits cluster state can be used to realize an indeterminate single-qubit gates by converting each qubit of the cluster state into another form in measurement-based quantum computation. In blind quantum computation, combining with this approach and adding different circuit-based quantum computation capabilities to the server, we present two models that the client can send fewer qubits to the server to implement a deterministic single-qubit gate. Finally, to keep the entire calculation process blind to the server, we substitute some bricks that realize a tensor of two single-qubit gates to the above models in the brickwork state. As a result, two types of the brickwork states with fewer particles were proposed, and we also give an upper bound for the number of replaced bricks. This reduces the quantum cost of the client.
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This work is supported by the National Natural Science Foundation of China (Grant No.11671284), Sichuan Science Foundation and Technology Program (Grant No.2020YFG0290).
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Appendix
Appendix
1.1 I: The concrete process from Eq.(1) to Eq.(3)
In this subsection, we will explain how Eq.(1) is evolved into Eq.(3). In the first place, we introduce a conclusion, from Ref. [19], that the result of measuring tangled state \(CZ_{12}|\psi \rangle _1|+\rangle _2\) based on \(M_1(\theta )\) is \(X^{s_1}HR_z(\theta )|\psi \rangle \), where \(\psi = \alpha |0\rangle + \beta |1\rangle \) and \(\alpha ,\beta \) satisfy \(|\alpha |^2+|\beta |^2 = 1\). To be specific, we can obtain that
For Eq.(1), we execute \(M_1(\delta _1)\) on qubit 1, then qubit 2 collapses into
from the result of Eq.(14), Eq.(15) can be evolved into
Next, we express qubit 2 as \(\alpha _2|0\rangle +\beta _2|1\rangle \) and carry out \(CZ_{23}\) between qubit 2 and 3. Then the total state can be shown as
after that we measure qubit 2 based on \(M_2(\delta _2)\), which is similar to the process of transmitting Eq.(1) into Eq.(16), the qubit 3 from Eq.(17) collapses into
from Eq.(14), Eq.(18) can evolve into
Then, substituting Eq.(16) for \(\alpha _2|0\rangle +\beta _2|1\rangle \) and utilizing \(R_z(\delta )X^s = X^sR_z((-1)^s\delta )\) and \(R_z(\delta )Z^s = Z^sR_z(\delta ) \) for adjusting order, the qubit 3 will be expressed as Eq.(3) ultimately.
1.2 II: The concrete process from Eq.(4) to Eq.(6)
In this subsection, we make clear the detail of process from Eq.(4) to Eq.(6). After replacing \(\delta _i\;(i=1,2)\) in Eq.(4) for \(\theta _i+\phi _i+r_i\pi \), we get that
on the other hand
where \(s,r \in \{0,1\}\). So Eq.(20) can be transmitted to
when a global phase factor is ignored.
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Yang, Z., Bai, MQ. & Mo, ZW. The brickwork state with fewer qubits in blind quantum computation. Quantum Inf Process 21, 125 (2022). https://doi.org/10.1007/s11128-022-03473-1
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DOI: https://doi.org/10.1007/s11128-022-03473-1