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Register a new userGeneralized Ramsey numbers through adiabatic quantum optimization
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Ramsey theory is an active research area in combinatorics whose central theme is the emergence of order in large disordered structures, with Ramsey numbers marking the threshold at which this order first appears. For generalized Ramsey numbers $r(G,H)$, the emergent order is characterized by graphs $G$ and $H$. In this paper we: (i) present a quantum algorithm for computing generalized Ramsey numbers by reformulating the computation as a combinatorial optimization problem which is solved using adiabatic quantum optimization; and (ii) determine the Ramsey numbers $r(mathcal{T}_{m},mathcal{T}_{n})$ for trees of order $m,n = 6,7,8$, most of which were previously unknown.
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The graph-theoretic Ramsey numbers are notoriously difficult to calculate. In fact, for the two-color Ramsey numbers $R(m,n)$ with $m,ngeq 3$, only nine are currently known. We present a quantum algorithm for the computation of the Ramsey numbers $R(m,n)$. We show how the computation of $R(m,n)$ can be mapped to a combinatorial optimization problem whose solution can be found using adiabatic quantum evolution. We numerically simulate this adiabatic quantum algorithm and show that it correctly determines the Ramsey numbers R(3,3) and R(2,s) for $5leq sleq 7$. We then discuss the algorithms experimental implementation, and close by showing that Ramsey number computation belongs to the quantum complexity class QMA.
We prove an infinite Ramsey theorem for noncommutative graphs realized as unital self-adjoint subspaces of linear operators acting on an infinite dimensional Hilbert space. Specifically, we prove that if V is such a subspace, then provided there is no obvious obstruction, there is an infinite rank projection P with the property that the compression PVP is either maximal or minimal in a certain natural sense.
We present generalized adiabatic theorems for closed and open quantum systems that can be applied to slow modulations of rapidly varying fields, such as oscillatory fields that occur in optical experiments and light induced processes. The generalized adiabatic theorems show that a sufficiently slow modulation conserves the dynamical modes of time dependent reference Hamiltonians. In the limiting case of modulations of static fields, the standard adiabatic theorems are recovered. Applying these results to periodic fields shows that they remain in Floquet states rather than in energy eigenstates. More generally, these adiabatic theorems can be applied to transformations of arbitrary time-dependent fields, by accounting for the rapidly varying part of the field through the dynamical normal modes, and treating the slow modulation adiabatically. As examples, we apply the generalized theorem to (a) predict the dynamics of a two level system driven by a frequency modulated resonant oscillation, a pathological situation beyond the applicability of earlier results, and (b) to show that open quantum systems driven by slowly turned-on incoherent light, such as biomolecules under natural illumination conditions, can only display coherences that survive in the steady state.
We generalize Katos adiabatic theorem to nonunitary dynamics with an isospectral generator. This enables us to unify two strong-coupling limits: one driven by fast oscillations under a Hamiltonian, and the other driven by strong damping under a Lindbladian. We discuss the case where both mechanisms are present and provide nonperturbative error bounds. We also analyze the links with the quantum Zeno effect and dynamics.
Ramsey theory is a highly active research area in mathematics that studies the emergence of order in large disordered structures. Ramsey numbers mark the threshold at which order first appears and are extremely difficult to calculate due to their explosive rate of growth. Recently, an algorithm that can be implemented using adiabatic quantum evolution has been proposed that calculates the two-color Ramsey numbers $R(m,n)$. Here we present results of an experimental implementation of this algorithm and show that it correctly determines the Ramsey numbers R(3,3) and $R(m,2)$ for $4leq mleq 8$. The R(8,2) computation used 84 qubits of which 28 were computational qubits. This computation is the largest experimental implementation of a scientifically meaningful adiabatic evolution algorithm that has been done to date.
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