Implementing the Grover Algorithm in Homomorphic Encryption Schemes

TL;DR

This paper demonstrates how to implement Grover's quantum search algorithm within homomorphic encryption schemes, enabling secure quantum computations on encrypted data with efficiency improvements for specific cases.

Contribution

It introduces a method to evaluate Grover's algorithm homomorphically using quantum homomorphic encryption schemes suitable for circuits with many $T/T^{\

Findings

Successfully simulated a 3-qubit Grover circuit using QHE.

Showed that any Grover circuit can be evaluated homomorphically with polynomial complexity.

Proposed an efficient decryption method for the case with a single marked item.

Abstract

We apply quantum homomorphic encryption (QHE) schemes suitable for circuits with a polynomial number of $T/T^{\dagger}$-gates to Grover's algorithm, performing a simulation in Qiskit of a Grover circuit that contains 3 qubits. The $T/T^{\dagger}$ gate complexity of Grover's algorithm is also analysed in order to show that any Grover circuit can be evaluated homomorphically in an efficient manner. We discuss how to apply these QHE schemes to allow for the efficient homomorphic evaluation of any Grover circuit composed of $n$ qubits using $n-2$ extra ancilla qubits. We also show how the homomorphic evaluation of the special case where there is only one marked item can be implemented using an algorithm that makes the decryption process more efficient compared to the standard Grover algorithm.