Adiabatic quantum annealing is a paradigm of analog quantum computation, where a given computational job is converted to the task of finding the global minimum of some classical potential energy function and the search for the global potential minimum is performed by employing external kinetic quantum fluctuations and subsequent slow reduction (annealing) of them. In this method, the entire potential energy landscape (PEL) may be accessed simultaneously through a delocalized wave-function, in contrast to a classical search, where the searcher has to visit different points in the landscape (i.e., individual classical configurations) sequentially. Thus in such searches, the role of the potential energy might be significantly different in the two cases. Here we discuss this in the context of searching of a single isolated hole (potential minimum) in a golf-course type gradient free PEL. We show, that the quantum particle would be able to locate the hole faster if the hole is deeper, while the classical particle of course would have no scope to exploit the depth of the hole. We also discuss the effect of the underlying quantum phase transition on the adiabatic dynamics.
Hamiltonian H, which may be a physical Hamiltonian with many degrees of freedom, or a suitable mathematical function depending on many variables, and the task is to determine its global minimum. In order to introduce the quantum fluctuations necessary for the AQA of such a Hamiltonian, one adds a quantum kinetic part H ′ (t) to it, such that H ′ (t) and H do not commute. Initially, one keeps |H ′ (t = 0)| ≫ |H| so that the total Hamiltonian H tot (t) = H ′ (t) + H is well approximated by the kinetic part only (H tot (0) ≈ H ′ (0)). If the system is initially prepared to be in the ground state of H ′ (0) (one chooses H ′ (0) to have a easily realizable ground state) and H ′ (t) is reduced slowly enough, then according to the adiabatic theorem of quantum mechanics, the overlap | ψ(t)|E -(t) |, between the instantaneous lowest-eigenvalue state |E -(t) and the instantaneous state of |ψ(t) of the evolving system, will always stay near its initial value (which is close to unity, since H tot (0) ≈ H ′ (0)). Hence at the end of such an evolution, when H ′ (t) is reduced to zero at t = τ (the annealing time), the system will be found in a state |ψ(τ ) with | ψ(τ )|E -(τ ) | ≈ 1, where |E -(τ ) is the ground state of H tot (τ ), which is nothing but the surviving classical part H.
Thus at the end of an adiabatic annealing the system is found in the ground state of the classical Hamiltonian with a high probability. Based on this principle, algorithms can be framed to anneal complex physical systems like spin glasses as well as the objective functions of hard combinatorial optimization problems (like the Traveling Salesman Problem), towards their ground (optimal) states [2,6,8,11].
In order to ensure adiabaticity, the evolution should be such that
where
|E + (t) being the instantaneous first excited state of the total Hamiltonian H tot (t), ∆(t)
being the instantaneous gap between the ground state and the first excited state energies and α being the adiabatic factor (for a simple proof see [14]).
One key feature, believed to be behind the success of AQA over the classical ones [2,6,8] in glass-like rugged PEL, is the ability of the quantum systems to tunnel easily through potential energy barriers even if they are very high, provided they are narrow enough, in contrast to the the classical ones, which always has to scale the barrier height with its kinetic energy (temperature) irrespective of the width [5,7,10,13,15]. Here we show that there is another aspect which makes a quantum mechanical searcher more advantagous over the classical ones-it can utilize the depth of the potential energy minimum in locating it in absence of any potential gradient which a classical searcher cannot.
We consider a lattice with N sites, |i denoting the state of a particle localized at the i-th site. At each site, there is a potential, which is zero at all the sites i = w, and is -χ
, where w is chosen randomly. Thus the PEL is essentially a flat one without any gradient, with a single hole (minimum) at i = w with a depth χ. This is precisely some kind of analog version of Grover’s algorithm for searching a particular entry in an unstructured database [16] - [18]. But in those studies, the possibility of utilizing the depth of the hole in favor of faster search was not considered, and the gain over the classical ones is limited by the optimal Grover’s bound of O( √ N ) speed-up [18,19].
Let us consider that the lattice points are connected to each other by an infinite range hopping term Γ between any two sites. The question is how fast a particle can locate the hole starting from a state which does not assume any knowledge of the position of the hole, by reducing its kinetic energy Γ, and tuning the hole depth χ. The Hamiltonian for a particle on such a lattice will be given by-
In order to anneal the particle to the hole, one has to reduce Γ from a very high value to a very low final value and tune χ in the opposite manner, so that Γ(t = 0) ≫ χ(t = 0) and Γ(t = τ ) ≪ χ(t = τ ), where τ is the annealing time. The evolution should satisfy the adiabatic condition (1). The eigen-spectrum of H tot (t) consists of a ground state |E -(t)
and first excited state |E + (t) (in the order of increasing eigen values) with energies
respectively, all the time dependencies being implicit, through the time dependence of Γ and χ. The instantaneous gap is thus given by
The instantaneous first and second excited states |E ± (t) are given by -
where
The second excited state is (N -2)-fold degenerate, with eigenvalue -Γ, and the time evolving Hamiltonian never mixes the first two eigenstate with any of the second excited states. This can be easily argued noting that a state of the form
for all t. For all allowed combinations of i and j we get (N -2) such linearly-independent eigenstates. Form these (N -2) linearly-independent eigenstates we can construct (N -2) mutually orthogonal eigenstates, each of which will obviously satisfy the above non-
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