Abstract
This paper describes adiabatic-inspired algorithms to solve the approximate shortest vector problem. Numerical simulations varying algorithm sweep-time suggest the existence of a Goldilocks zone for maximizing likelihood of success.
Highlights
The advent of quantum computers heralds an age of new computational possibilities
Even though time complexity is generally difficult to estimate for this class of algorithms, they seem suitable for attacks on LBC for two reasons: first, because lattice problems can be formulated as optimization problems, as we will demonstrate for a quantum setting; second, while a major drawback of adiabatic quantum computation (AQC) is the prohibitive time cost of achieving adiabaticity, this may not be a problem here as, up to a threshold, approximate solutions are admissible
Little is known analytically about the time scaling for adiabatic quantum algorithms, beyond a worst case energy gap dependence of 1/ 3 [29]—with denoting the minimum energy gap between ground state and second lowest eigenstate—whereas with quantum gate algorithms neat closed form scalings are known for a handful of algorithms, for example Shor’s exponential speedup for integer factorization and discrete logarithm computation [4] and Grover’s quadratic speedup for searching unsorted lists [30]
Summary
The advent of quantum computers heralds an age of new computational possibilities. Two paradigms of quantum computing are gate model and adiabatic quantum computation (AQC): the gate model closely resembles current computing architecture, replacing bits with qubits and retaining control over the smallest building-blocks of the system, and in AQC the solution for the problem to be solved is encoded into the ground state of a Hamiltonian [1,2]. One cannot prepare this ground state directly, otherwise the problem would be straightforward to solve. The adiabatic theorem guarantees that a sufficiently slow change from this initial Hamiltonian to the problem Hamiltonian HP lets the system evolve into the ground state of the latter. Both paradigms have been demonstrated to be equivalent [3], though there is not a general way of mapping from one paradigm to the other. The most impactful quantum algorithm discovered far is that of Shor for integer factorization and discrete logarithm computation [4].
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