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arXiv:1003.2172 · Physical Review A

Optimal time schedule for adiabatic evolution

J. E. Avron, Martin Fraas, Gian Michele Graf, Philip Grech

Published Oct 22, 2010• 23 citations
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Abstract

We show that, provided dephasing is taken into account, there is a unique timetable which maximizes the fidelity with a target state in adiabatic evolutions. The optimum has constant tunneling rate along the path. Application to quantum search algorithms recovers the Grover result for appropriate scaling of the dephasing with the size of the database. Moreover, the Grover bound imposes constraints on the dephasing rates of systems coupled to a universal Markovian bath.

DOI
10.1103/physreva.82.040304

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We show that, provided dephasing is taken into account, there is a unique timetable which maximizes the fidelity with a target state in adiabatic evolutions. The optimum has constant tunneling rate along the path. Application to quantum search algorithms recovers the Grover result for appropriate scaling of the dephasing with the size of the database. Moreover, the Grover bound imposes constraints on the dephasing rates of systems coupled to a universal Markovian bath.
Adiabatic quantum algorithms must evolve slowly enough to suppress non-adiabatic transitions while remaining fast enough to be practical. In open systems, this trade-off is reshaped by decoherence. For Hamiltonians subject to dephasing Lindbladians, Avron et al. [Phys. Rev. A 82, 040304(R) (2010)] showed that a unique timetable exists that maximizes the fidelity with a target state. This optimal schedule is characterized by a constant tunneling rate along the adiabatic path. In this work, we revisit their analysis and apply it to the adiabatic Grover search framework, obtaining closed-form expressions for the optimal evolution schedule, the minimum runtime, and the resulting achievable fidelity. Moreover, by invoking an energy-time uncertainty argument, we identify a critical dephasing threshold beyond which further noise-assisted acceleration is prohibited, showing that the apparent advantage of strong dephasing is a Zeno-type stabilization mechanism rather than a conventional algorithmic speedup over unitary Grover search.
The adiabatic theorem has been recently used to design quantum algorithms of a new kind, where the quantum computer evolves slowly enough so that it remains near its instantaneous ground state, which tends to the solution. We apply this time-dependent Hamiltonian approach to Grover's problem, i.e., searching a marked item in an unstructured database. We find that by adjusting the evolution rate of the Hamiltonian so as to keep the evolution adiabatic on each infinitesimal time interval, the total running time is of order N,\sqrt{N},N​, where N is the number of items in the database. We thus recover the advantage of Grover's standard algorithm as compared to a classical search, scaling as N. This is in contrast with the constant-rate adiabatic approach of Farhi et al. (e-print quant-ph/0001106), where the requirement of adiabaticity is expressed only globally, resulting in a time of order N.
We present an optimized adiabatic quantum schedule for unstructured search building on the original approach of Roland and Cerf [Phys. Rev. A 65, 042308 (2002)]. Our schedule adiabatically varies the Hamiltonian even more rapidly at the endpoints of its evolution, preserving Grover's well-known quadratic quantum speedup. In the errorless adiabatic limit, the probability of successfully obtaining the marked state from a measurement increases directly proportional to time, suggesting efficient parallelization. Numerical simulations of an appropriate reduced two-dimensional Schrödinger system confirm adiabaticity while demonstrating superior performance in terms of probability compared to existing adiabatic algorithms and Grover's algorithm, benefiting applications with possible premature termination. We introduce a protocol that ensures a marked-state probability at least ppp in time of order N(1+p/ε)\sqrt{N}(1+p/\varepsilon)N​(1+p/ε), and analyze its implications for realistic bounded-resource scenarios. Our findings suggest that quantum advantage may still be achievable under constrained coherence times (where other algorithms fail), provided the hardware allows for them to be sufficiently long.
Grover's algorithm is one of the most important quantum algorithms, which performs the task of searching an unsorted database without a priori probability. Recently the adiabatic evolution has been used to design and reproduce quantum algorithms, including Grover's algorithm. In this paper, we show that quantum search algorithm by adiabatic evolution has two properties that conventional quantum search algorithm doesn't have. Firstly, we show that in the initial state of the algorithm only the amplitude of the basis state corresponding to the solution affects the running time of the algorithm, while other amplitudes do not. Using this property, if we know a priori probability about the location of the solution before search, we can modify the adiabatic evolution to make the algorithm faster. Secondly, we show that by a factor for the initial and finial Hamiltonians we can reduce the running time of the algorithm arbitrarily. Especially, we can reduce the running time of adiabatic search algorithm to a constant time independent of the size of the database. The second property can be extended to other adiabatic algorithms.
The efficiency of adiabatic quantum evolution is governed by the evolution time T , which typically scales as O ( Δ − 2 ) with the minimum energy gap Δ . However, the rigorous lower bound is O ( L Δ − 1 ) , where L is the adiabatic path length. Although L is formally upper bounded by O ( Δ − 1 ) , such a bound is often too loose in practice, and L can be bounded independently of Δ . This indicates the potential for a quadratic speedup through adiabatic schedule construction. Here, we introduce the constant geometric speed (CGS) schedule, which traverses the adiabatic path at a uniform rate. We show that this approach reduces the scaling of the evolution time by a factor of Δ − 1 , provided L remains bounded independently of Δ . We propose a segmented CGS protocol where path segment lengths are computed from eigenstate overlaps on the fly, reducing the prior spectral-knowledge requirement from the full gap function Δ ( s ) to just a global lower bound on the energy gap. Numerical tests on adiabatic unstructured search, N 2 , and a [2Fe-2S] cluster demonstrate the optimal Δ − 1 scaling, confirming a quadratic speedup over the standard linear schedule.
We report on a detailed analysis of generalization of the local adiabatic search algorithm. Instead of evolving directly from an initial ground state Hamiltonian to a solution Hamiltonian a different evolution path is introduced and is shown that the time required to find an item in a database of size NNN can be made to be independent of the size of the database by modifying the Hamiltonian used to evolve the system.