Abstract
Polynomial systems over the binary field have important applications, especially in symmetric and asymmetric cryptanalysis, multivariate-based post-quantum cryptography, coding theory, and computer algebra. In this work, we study the quantum annealing model for solving Boolean systems of multivariate equations of degree 2, usually referred to as the Multivariate Quadratic problem. We present different methodologies to embed the problem into a Hamiltonian that can be solved by available quantum annealing platforms. In particular, we provide three embedding options, and we highlight their differences in terms of quantum resources. Moreover, we design a machine-agnostic algorithm that adopts an iterative approach to better solve the problem Hamiltonian by repeatedly reducing the search space. Finally, we use D-Wave devices to successfully implement our methodologies on several instances of the Multivariate Quadratic problem.
Highlights
Adiabatic quantum computation is a universal quantum computation scheme [1] where a quantum system is prepared in the ground state of an easy-to-prepare Hamiltonian and evolved towards a Hamiltonian that encodes the solution of a problem in its ground state
We have presented several ways to encode the solution of an multivariate quadratic (MQ) problem into the ground state of a Hamiltonian that can be used for quantum annealing
This is unrealistic because the superconducting chips for quantum annealing provided by D-Wave’s Advantage device support a Pegasus chip architecture [48] and the qubits need to be mapped in accordance with that restriction
Summary
Adiabatic quantum computation is a universal quantum computation scheme [1] where a quantum system is prepared in the ground state of an easy-to-prepare Hamiltonian and evolved towards a Hamiltonian that encodes the solution of a problem in its ground state. In the case of degree 2 polynomials, the problem, usually referred to as the multivariate quadratic (MQ) problem, has important applications in postquantum cryptography, since several postquantum schemes exist basing their security on its difficulty to be solved [17,18] For these reasons, there is a spreading interest in the scientific community to find new algorithms to solve the MQ problem, both in the classical and quantum computation model. We introduce an iterative approach that aids quantum annealing devices in finding the ground state of the Hamiltonian by repeatedly shrinking the search space using information gained in previous executions of the system This method allowed us to successfully solve small instances of the MQ problem using current D-Wave devices. IV and V we present the results of our experiments with the D-Wave quantum annealer and the conclusions of this work, respectively
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