# Teleportation-based quantum homomorphic encryption scheme with quasi-compactness and perfect security

## TL;DR

This paper introduces a teleportation-based quantum homomorphic encryption scheme that achieves perfect security and quasi-compactness, optimized for circuits with low T/T†-gate complexity, advancing secure quantum computation.

## Contribution

It presents two new non-interactive, perfectly secure QHE schemes, exttt{GT} and exttt{VGT}, with bounded quasi-compactness proportional to T/T†-gates, improving efficiency for low gate complexity circuits.

## Key findings

- Schemes are exttt{F}-homomorphic and quasi-compact with complexity O(M).
- exttt{VGT} is proven optimal with M-quasi-compactness.
- Suitable for polynomial-size quantum circuits with low T/T†-gate count.

## Abstract

This article defines encrypted gate, which is denoted by $EG[U]:|\alpha\rangle\rightarrow\left((a,b),Enc_{a,b}(U|\alpha\rangle)\right)$. We present a gate-teleportation-based two-party computation scheme for $EG[U]$, where one party gives arbitrary quantum state $|\alpha\rangle$ as input and obtains the encrypted $U$-computing result $Enc_{a,b}(U|\alpha\rangle)$, and the other party obtains the random bits $a,b$. Based on $EG[P^x](x\in\{0,1\})$, we propose a method to remove the $P$-error generated in the homomorphic evaluation of $T/T^\dagger$-gate. Using this method, we design two non-interactive and perfectly secure QHE schemes named \texttt{GT} and \texttt{VGT}. Both of them are $\mathcal{F}$-homomorphic and quasi-compact (the decryption complexity depends on the $T/T^\dagger$-gate complexity). Assume $\mathcal{F}$-homomorphism, non-interaction and perfect security are necessary property, the quasi-compactness is proved to be bounded by $O(M)$, where $M$ is the total number of $T/T^\dagger$-gates in the evaluated circuit. \texttt{VGT} is proved to be optimal and has $M$-quasi-compactness. According to our QHE schemes, the decryption would be inefficient if the evaluated circuit contains exponential number of $T/T^\dagger$-gates. Thus our schemes are suitable for homomorphic evaluation of any quantum circuit with low $T/T^\dagger$-gate complexity, such as any polynomial-size quantum circuit or any quantum circuit with polynomial number of $T/T^\dagger$-gates.