Homomorphic Encryption of the k=2 Bernstein-Vazirani Algorithm

TL;DR

This paper introduces simplified quantum circuits for the Bernstein-Vazirani problem and applies them to develop efficient, homomorphically encryptable quantum algorithms for delegated quantum computation.

Contribution

It defines a new class of simplified quantum circuits for a specific case of the Bernstein-Vazirani problem and demonstrates their application to quantum homomorphic encryption schemes.

Findings

Circuits with linear gate growth in qubits for the problem.

Application of simplified circuits to efficient quantum homomorphic encryption.

Enhanced security and efficiency in delegated quantum computation.

Abstract

The nonrecursive Bernstein-Vazirani algorithm was the first quantum algorithm to show a superpolynomial improvement over the corresponding best classical algorithm. Here we define a class of circuits that solve a particular case of this problem for second-level recursion. This class of circuits simplifies the number of gates $T$ required to construct the oracle by making it grow linearly with the number of qubits in the problem. We find an application of this scheme to quantum homomorphic encryption (QHE) which is an important cryptographic technology useful for delegated quantum computation. It allows a remote server to perform quantum computations on encrypted quantum data, so that the server cannot know anything about the client's data. Liang developed QHE schemes with perfect security, $\mathcal{F}$-homomorphism, no interaction between server and client, and quasi-compactness bounded by $O(M)$ where M is the number of gates $T$ in the circuit. Precisely these schemes are suitable for circuits with a polynomial number of gates $T/T^{\dagger}$. Following these schemes, the simplified circuits we have constructed can be evaluated homomorphically in an efficient way.