Stacey Jeffery

Quantum homomorphic encryption for circuits of low T-gate complexity

Fully homomorphic encryption is an encryption method with the property that any computation on the plaintext can be performed by a party having access to the ciphertext only. Here, we formally define and give schemes for quantum homomorphic encryption, which is the encryption of quantum information such that quantum computations can be performed given the ciphertext only. Our schemes allow for arbitrary Clifford group gates, but become inefficient for circuits with large complexity, measured in terms of the non-Clifford portion of the circuit (we use the “\(\pi /8\)” non-Clifford group gate, also known as the \(\mathsf{T}\)-gate).

Nested quantum walks with quantum data structures

We develop a new framework that extends the quantum walk framework of Magniez, Nayak, Roland, and Santha, by utilizing the idea of quantum data structures to construct an efficient method of nesting quantum walks. Surprisingly, only classical data structures were considered before for searching via quantum walks. The recently proposed learning graph framework of Belovs has yielded improved upper bounds for several problems, including triangle finding and more general subgraph detection. We exhibit the power of our framework by giving a simple explicit constructions that reproduce both the O(n35/27) and O(n9/7) learning graph upper bounds (up to logarithmic factors) for triangle finding, and discuss how other known upper bounds in the original learning graph framework can be converted to algorithms in our framework. We hope that the ease of use of this framework will lead to the discovery of new upper bounds.

Improving quantum query complexity of boolean matrix multiplication using graph collision

The quantum query complexity of Boolean matrix multiplication is typically studied as a function of the matrix dimension, n, as well as the number of 1s in the output, l. We prove an upper bound of

Time-Efficient quantum walks for 3-distinctness

\widetilde{\mathrm{O}}(n\sqrt{\ell})

Quantum Algorithms for the Subset-Sum Problem

for all values of l. This is an improvement over previous algorithms for all values of l. On the other hand, we show that for any e<1 and any l≤en2, there is an

Improving Quantum Query Complexity of Boolean Matrix Multiplication Using Graph Collision

\Omega(n\sqrt{\ell})

Circuit Obfuscation Using Braids

lower bound for this problem, showing that our algorithm is essentially tight. We first reduce Boolean matrix multiplication to several instances of graph collision. We then provide an algorithm that takes advantage of the fact that the underlying graph in all of our instances is very dense to find all graph collisions efficiently.

Attacks on the AJPS Mersenne-based cryptosystem

We present two quantum walk algorithms for 3-Distinctness. Both algorithms have time complexity

Quantum Communication Complexity of Distributed Set Joins

\tilde{O}(n^{5/7})

Optimal Parallel Quantum Query Algorithms

, improving the previous