Probing nonlinear adiabatic paths with a universal integrator
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چکیده
We apply a flexible numerical integrator to the simulation of adiabatic quantum computation with nonlinear paths. We find that a nonlinear path may significantly improve the performance of adiabatic algorithms versus the conventional straight-line interpolations. The employed integrator is suitable for solving the time-dependent Schrödinger equation for any qubit Hamiltonian. Its flexible storage format significantly reduces cost for storage and matrix-vector multiplication in comparison to common sparse matrix schemes. Simulating quantum systems requires enormous computational resources: Even for a few hundred particles there would be more variables to be stored than atoms exist in the universe [1]. To turn this problem into an advantage, quantum computers may be efficiently used for such simulations, since they are quantum systems themselves [2]. Moreover, quantum algorithms can solve distinct problems like number factoring with exponential speedup compared to classical computers [3]. In the conventional picture, quantum algorithms are implemented as a sequence of unitary operations [4], which implies fast switching of the generating Hamiltonian. In contrast, within the paradigm of adiabatic quantum computation [5], the Hamiltonian is modified slowly from a simple initial Hamiltonian with an easy-to-prepare ground state to a final Hamiltonian which encodes in its ground state the solution to some difficult problem. Most importantly, for a large class of problems, implementation of the final Hamiltonian is possible without knowing the solution of the problem explicitly. The adiabatic theorem implies – provided the evolution is slow enough – that the system will end up near the ground state of the final Hamiltonian, such that the solution to the problem can be obtained by measuring the system. The evolution time is related to the spectral properties of the time-dependent Hamiltonian and thus corresponds to the algorithmic complexity of an adiabatic quantum algorithm (AQA). The conventional circuit picture and the adiabatic approach are known to be polynomially equivalent [6, 7], but exact results for adiabatic algorithms are scarce [8]. It is therefore quite interesting that first numerical simulations of the Schrödinger equation revealed a seemingly polynomial complexity of the adiabatic algorithm for an NP-complete problem [5]. Since then, it has been a strongly debated question whether this scaling would persist for larger problem sizes [9, 10, 11, 12, 13, 14]. Recent findings suggest that the scaling complexity of the conventional straight-line adiabatic interpolation is typically exponential [15, 16]. It may however be conjectured that with modifications of the adiabatic algorithm, its scaling behavior can be considerably improved [17], such that the scaling behavior of adapted algorithms is still an open question. Unfortunately, this question can currently not be settled from the experimental side: Though enormous progress has been made in the last decade, not more than a few quantum bits (qubits) have been entangled so far [18], which currently restricts the execution of quantum algorithms to proof-of-principle demonstrations. As experiments are still neither flexible nor scalable enough to investigate new theoretical models, the demand for classical computer simulations of quantum algorithms is growing. Such simulations are computationally expensive and usually must be
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تاریخ انتشار 2013