We use the HHL algorithm to retrieve a quantum state holding the algebraic normal form (ANF) of a Boolean function. Unlike the standard HHL applications, we do not describe the cipher as an exponentially big system of equations. Rather, we perform a set of small matrix inversions which correspond to the Boolean Möbius transform. This creates a superposition holding information about the ANF in the form |Af⟩=1C∑I=02n−1cI|I⟩, where cI is the coefficient of the ANF and C is a scaling factor. The procedure has a time complexity of O~(n) for a Boolean function with n-bit input. We also propose two approaches by which some information about the ANF can be extracted from such a state. Next, we use a similar approach, the Dual Boolean Möbius transform, to compute the preimage under the algebraic transition matrix. We show that such a matrix is well-suited for the HHL algorithm when the attacker gets oracle access in the Q2 setting to the Boolean function.We use the HHL algorithm to retrieve a quantum state holding the algebraic normal form (ANF) of a Boolean function. Unlike the standard HHL applications, we do not describe the cipher as an exponentially big system of equations. Rather, we perform a set of small matrix inversions which correspond to the Boolean Möbius transform. This creates a superposition holding information about the ANF in the form |Af⟩=1C∑I=02n−1cI|I⟩, where cI is the coefficient of the ANF and C is a scaling factor. The procedure has a time complexity of O~(n) for a Boolean function with n-bit input. We also propose two approaches by which some information about the ANF can be extracted from such a state. Next, we use a similar approach, the Dual Boolean Möbius transform, to compute the preimage under the algebraic transition matrix. We show that such a matrix is well-suited for the HHL algorithm when the attacker gets oracle access in the Q2 setting to the Boolean function.

Quantum ReportsAug 19, 20251 citation

A quantum algorithm for approximating the influences of Boolean functions and its applications

Hongwei Li, Li Yang

arXiv:1409.1416Quantum Information ProcessingMar 4, 201531 citations

Tunable Quantum Neural Networks for Boolean Functions

Viet Pham Ngoc, Herbert Wiklicky

In this paper we propose a new approach to quantum neural networks. Our multi-layer architecture avoids the use of measurements that usually emulate the non-linear activation functions which are characteristic of the classical neural networks. Despite this, our proposed architecture is still able to learn any Boolean function. This ability arises from the correspondence that exists between a Boolean function and a particular quantum circuit made out of multi-controlled NOT gates. This correspondence is built via a polynomial representation of the function called the algebraic normal form. We use this construction to introduce the idea of a generic quantum circuit whose gates can be tuned to learn any Boolean functions. In order to perform the learning task, we have devised an algorithm that leverages the absence of measurements. When presented with a superposition of all the binary inputs of length $n$ , the network can learn the target function in at most $n+1$ updates.In this paper we propose a new approach to quantum neural networks. Our multi-layer architecture avoids the use of measurements that usually emulate the non-linear activation functions which are characteristic of the classical neural networks. Despite this, our proposed architecture is still able to learn any Boolean function. This ability arises from the correspondence that exists between a Boolean function and a particular quantum circuit made out of multi-controlled NOT gates. This correspondence is built via a polynomial representation of the function called the algebraic normal form. We use this construction to introduce the idea of a generic quantum circuit whose gates can be tuned to learn any Boolean functions. In order to perform the learning task, we have devised an algorithm that leverages the absence of measurements. When presented with a superposition of all the binary inputs of length $n$, the network can learn the target function in at most $n+1$ updates.

arXiv:2003.14122Mar 31, 20204 citations

n

, the network can learn the target function in at most

n+1

updates.