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An adiabatic time evolution of a closed quantum system connects eigenspaces of initial and final Hermitian Hamiltonians for slowly driven systems, or, unitary Floquet operators for slowly modulated driven systems. We show that the connection of eigenspaces depends on a topological property of the adiabatic paths for given initial and final points. An example in slowly modulated periodically driven systems is shown. These analysis are based on the topological analysis of the exotic quantum holonomy in adiabatic closed paths.
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We import the tools of Morse theory to study quantum adiabatic evolution, the core mechanism in adiabatic quantum computations (AQC). AQC is computationally equivalent to the (pre-eminent paradigm) of the Gate model but less error-prone, so it is ideally suitable to practically tackle a large number of important applications. AQC remains, however, poorly understood theoretically and its mathematical underpinnings are yet to be satisfactorily identified. Through Morse theory, we bring a novel perspective that we expect will open the door for using such mathematics in the realm of quantum computations, providing a secure foundation for AQC. Here we show that the singular homology of a certain cobordism, which we construct from the given Hamiltonian, defines the adiabatic evolution. Our result is based on E. Wittens construction for Morse homology that was derived in the very different context of supersymmetric quantum mechanics. We investigate how such topological description, in conjunction with Gauss-Bonnet theorem and curvature based reformulation of Morse lemma, can be an obstruction to any computational advantage in AQC. We also explore Conley theory, for the sake of completeness, in advance of any known practical Hamiltonian of interest. We conclude with the instructive case of the ferromagnetic $p-$spin where we show that changing its first order quantum transition (QPT) into a second order QPT, by adding non-stoquastic couplings, amounts to homotopically deform the initial surface accompanied with birth of pairs of critical points. Their number reaches its maximum when the system is fully non-stoquastic. In parallel, the total Gaussian curvature gets redistributed (by the Gauss--Bonnet theorem) around the new neighbouring critical points, which weakens the severity of the QPT.
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.
One of the fundamental physical limits on the speed of time evolution of a quantum state is known in the form of the celebrated Mandelshtam-Tamm inequality. This inequality gives an answer to the question on how fast an isolated quantum system can evolve from its initial state to an orthogonal one. In its turn, the Fleming bound is an extension of the Mandelshtam-Tamm inequality that gives an optimal speed bound for the evolution between non-orthogonal initial and final states. In the present work, we are concerned not with a single state but with a whole (possibly infinite-dimensional) subspace of the system states that are subject to the Schroedinger evolution. By using the concept of maximal angle between subspaces we derive an optimal estimate on the speed of such a subspace evolution that may be viewed as a natural generalization of the Fleming bound.
We describe a general methodology for enhancing the efficiency of adiabatic quantum computations (AQC). It consists of homotopically deforming the original Hamiltonian surface in a way that the redistribution of the Gaussian curvature weakens the effect of the anti-crossing, thus yielding the desired improvement. Our approach is not pertubative but instead is built on our previous global description of AQC in the language of Morse theory. Through the homotopy deformation we witness the birth and death of critical points whilst, in parallel, the Gauss-Bonnet theorem reshuffles the curvature around the changing set of critical points. Therefore, by creating enough critical points around the anti-crossing, the total curvature--which was initially centered at the original anti-crossing--gets redistributed around the new neighbouring critical points, which weakens its severity and so improves the speedup of the AQC. We illustrate this on two examples taken from the literature.
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.
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