Client-friendly continuous-variable blind and verifiable quantum computing

We present a verifiable and blind protocol for assisted universal quantum computing on continuous-variable (CV) platforms. This protocol is experimentally friendly to the client, as it only requires Gaussian-operation capabilities from the latter. Moreover, the server does not require universal quantum-computational power either, its only function being to supply the client with copies of a single-mode non-Gaussian state. Universality is attained based on state injection of the server's non-Gaussian supplies. The protocol is automatically blind because the non-Gaussian resource requested to the server is always the same, regardless of the specific computation. Verification, in turn, is possible thanks to an efficient non-Gaussian state fidelity test where we assume identical state preparation by the server. It is based on Gaussian measurements by the client on the injected states, which is potentially interesting on its own. The division of quantum hardware between client and server assumed here is in agreement with the experimental constraints expected in realistic schemes for CV cloud quantum computing.

Figure 1

Universal CV quantum computation. To implement an arbitrary CV computation $U|{\mathrm{\Psi }}_{\text{in}}\rangle$, one requires only Gaussian operations and at least one non-Gaussian operation. The non-Gaussian operation can be implemented by Alice when she uses Bob's non-Gaussian state resource $\rho$ and applies unitary Gaussian operations $\mathcal{G}$ and Gaussian measurements $\mathcal{M}$. See Fig. 2 and the text for more details.

Figure 2

Alice's circuit to implement a cubic phase gate. Alice implements an approximation to the cubic phase gate ${\mathbb{1}}_k\otimes C\left(\gamma \right)\otimes {\mathbb{1}}_{m-k-1}$ acting on the state $|{\mathrm{\Psi }}_{\text{in}}\rangle$ by using the resource state $|\tilde{\gamma }{\rangle }_s$ given by Bob. Here $C\left(\gamma \right)$ acts on the ${(k+1)}^{\text{th}}$ mode of $|{\mathrm{\Psi }}_{\text{in}}\rangle$ and the resource state is on the ${(m+1)}^{\text{th}}$ mode. Here $S\left(r\right)$ is the single-mode squeezing operator with $r={(\gamma /\tilde{\gamma })}^{1/3}$, where the value $r$ is only known to Alice. The initial gate $S_1\equiv {\mathbb{1}}_k\otimes S\left(r\right)\otimes {\mathbb{1}}_{m-k-1}$ and final gate $S_2\equiv {\mathbb{1}}_k\otimes S^{†}\left(r\right)G^{-1}\left(y\right)\otimes {\mathbb{1}}_{m-k-1}$ acting on the top register are used to hide the value of $\gamma$ from Bob. Here $G^{-1}\left(y\right)=e^{-i\tilde{\gamma }{y}^3}{e}^{-3i\tilde{\gamma }y\widehat{x}(\widehat{x}+y)}$ is a Gaussian operator and $y$ is the measurement outcome on the lower register of the operator $\widehat{x}$.