# Mod2 VQLS: A Variational Quantum Algorithm for Solving Systems of Linear Equations Modulo 2
## 7 Applications of Mod2 VQLS and Related Techniques  
Mod2 VQLS was motivated by the use of modulo 2 linear algebra in integer factorization [2]. In this use-case, the goal is to find relations of the form \(a^{2}\equiv b^{2}\mod n\), by multiplying various equations with known factors to obtain suitable \(a^{2}\) and \(b^{2}\). This requires that exponents of all the factors in the prime factorizations of both \(a^{2}\) and \(b^{2}\) are even; thus, assembling such a combination by multiplying several equations becomes translated into finding a suitable binary vector \(x\) so that \(Ax\equiv 0\in\mathbb{F}_{2}^{n}\). The surprising importance of this 'playing with numbers' game is explained by [PERSON] [1].  
In the integer factorization application, the dimension \(n\) is determined by the number of (small) prime factors used in the 'factor basis' used to search for nearby composite numbers. The number of such relations determines the number of rows of \(A\) and the dimension \(m\) of the target space. This number is unknown at the beginning of the process; ideally, using the smallest number of relations possible reduces the burden on the'search phase' of the algorithm, but this number is often large in practice. Hence it was a requirement for Mod2 VQLS to work with a range of dimensions, and in particular, without the square matrix constraint that the number of equations must be equal to the number of unknowns.  
The are other applications for binary linear solvers as well as integer factoring. The solution of some Boolean satisfiability problems can be optimized using Gaussian elimination with binary arithmetic, especially for cases with XOR constraints [31]. (Quantum approaches to more general Boolean satisfiability problems are surveyed by [PERSON] et al. [32].) Applications in cryptography and cryptanalysis, such as finding inverses of elements in the finite field \(GF(2^{m})\), have encouraged research on optimizing solutions of binary linear systems [33, 34]. A component that solves linear systems modulo 2 is therefore required in various applications. Choosing an optimal approach typically depends on features including the dimension, number of equations, and sparsity; research on this problem is ongoing [35]. While it is unlikely that Mod2 VQLS would have a significant enough advantage over block Wiedermann to motivate the expense of developing quantum computers for this purpose, it is much more likely that large-scale quantum computers will become commercially available for a range of applications, and that Mod2 VQLS will have particular niches in this environment.  
More generally, computations over the field \(\mathbb{F}_{2}\) are fundamental in information and coding theory, and the data structures used sometimes overlap with work in quantum computing. For example, low-density parity-check (LDPC) codes work by comparing values between \"value nodes\" and \"check nodes\" in a bipartite graph [36, 37]. Quantum graph learning is also a current field of research [38, 39], and there are various proposed methods for developing quantum LDPC codes [40, 41]. These methods involve various operations over \(\mathbb{F}_{2}\), most of which do not involve solving complete sets of equations. An extra consideration is that some quantum processes can be parallelized in ways that enhance the security of distributed information [42], so there may be incentives for using quantum computers for specific system processes, beyond the traditional goals of reducing computational complexity and speeding-up classically intractable computations. As described in Section 3 above, the simple circuit design for computing the product of a matrix and a vector over \(\mathbb{F}_{2}\) is the heart of the Mod2 VQLS system, and components like this may end up being put to many different uses as the quantum computing ecosystem develops.