New annealing schedules for quantum annealing are proposed based on the adiabatic theorem. These schedules exhibit faster decrease of the excitation probability than a linear schedule. To derive this conclusion, the asymptotic form of the excitation probability for quantum annealing is explicitly obtained in the limit of long annealing time. Its first-order term, which is inversely proportional to the square of the annealing time, is shown to be determined only by the information at the initial and final times. Our annealing schedules make it possible to drop this term, thus leading to a higher order (smaller) excitation probability. We verify these results by solving numerically the time-dependent Schrodinger equation for small size systemsNew annealing schedules for quantum annealing are proposed based on the adiabatic theorem. These schedules exhibit faster decrease of the excitation probability than a linear schedule. To derive this conclusion, the asymptotic form of the excitation probability for quantum annealing is explicitly obtained in the limit of long annealing time. Its first-order term, which is inversely proportional to the square of the annealing time, is shown to be determined only by the information at the initial and final times. Our annealing schedules make it possible to drop this term, thus leading to a higher order (smaller) excitation probability. We verify these results by solving numerically the time-dependent Schrodinger equation for small size systems

Mar 7, 20077 citations

Difference between Quantum Annealing by Imaginary-Time and Real-Time Schrödinger Equations of Grover’s Search

Shuntaro Okada, Masayuki Ohzeki, Kazuyuki Tanaka

We confirmed the annealing time of Grover's search which is required to obtain desired success probability for quantum annealing by the imaginary-time and the real-time Schrödinger equation with two kinds of schedulings; one linearly decreases the quantum fluctuation and the other tunes the evolution rate of the Hamiltonian based on the adiabatic condition. With linear scheduling, the required annealing time for quantum annealing by the imaginary-time Schrödinger equation is of order $\log N$ , which is very different from $O(N)$ required for the quantum annealing by the real-time Schrödinger equation. With the scheduling based on the adiabatic condition, the required annealing time is of order $\sqrt{N}$ , which is identical to the annealing time for quantum annealing by the real-time Schrödinger equation. Although the scheduling based on the adiabatic condition is optimal for the quantum annealing by the real-time Schrödinger equation, it is inefficient for the quantum annealing by the imaginary-time Schrödinger equation. This result implies that the optimal scheduling for the quantum annealing by the imaginary-time and the real-time Schrödinger equation is very different, and the efficient scheduling considered with the quantum Monte Carlo methods, which is based on imaginary-time Schrödinger equation, is not necessarily effective to improve the performance of quantum annealing by the real-time Schrödinger equation. We discuss the efficient scheduling for quantum annealing by the imaginary-time Schrödinger equation with respect to the exponential decay of excited states.We confirmed the annealing time of Grover's search which is required to obtain desired success probability for quantum annealing by the imaginary-time and the real-time Schrödinger equation with two kinds of schedulings; one linearly decreases the quantum fluctuation and the other tunes the evolution rate of the Hamiltonian based on the adiabatic condition. With linear scheduling, the required annealing time for quantum annealing by the imaginary-time Schrödinger equation is of order $\log N$, which is very different from $O(N)$ required for the quantum annealing by the real-time Schrödinger equation. With the scheduling based on the adiabatic condition, the required annealing time is of order $\sqrt{N}$, which is identical to the annealing time for quantum annealing by the real-time Schrödinger equation. Although the scheduling based on the adiabatic condition is optimal for the quantum annealing by the real-time Schrödinger equation, it is inefficient for the quantum annealing by the imaginary-time Schrödinger equation. This result implies that the optimal scheduling for the quantum annealing by the imaginary-time and the real-time Schrödinger equation is very different, and the efficient scheduling considered with the quantum Monte Carlo methods, which is based on imaginary-time Schrödinger equation, is not necessarily effective to improve the performance of quantum annealing by the real-time Schrödinger equation. We discuss the efficient scheduling for quantum annealing by the imaginary-time Schrödinger equation with respect to the exponential decay of excited states.

arXiv:1801.06297Journal of the Physical Society of JapanJan 16, 20195 citations

Lower Bounds on Quantum Annealing Times

Luis Pedro García-Pintos, Lucas T. Brady, Jacob Bringewatt +1 more

The adiabatic theorem provides sufficient conditions for the time needed to prepare a target ground state. While it is possible to prepare a target state much faster with more general quantum annealing protocols, rigorous results beyond the adiabatic regime are rare. Here, we provide such a result, deriving lower bounds on the time needed to successfully perform quantum annealing. The bounds are asymptotically saturated by three toy models where fast annealing schedules are known: the Roland and Cerf unstructured search model, the Hamming spike problem, and the ferromagnetic p-spin model. Our bounds demonstrate that these schedules have optimal scaling. Our results also show that rapid annealing requires coherent superpositions of energy eigenstates, singling out quantum coherence as a computational resource.The adiabatic theorem provides sufficient conditions for the time needed to prepare a target ground state. While it is possible to prepare a target state much faster with more general quantum annealing protocols, rigorous results beyond the adiabatic regime are rare. Here, we provide such a result, deriving lower bounds on the time needed to successfully perform quantum annealing. The bounds are asymptotically saturated by three toy models where fast annealing schedules are known: the Roland and Cerf unstructured search model, the Hamming spike problem, and the ferromagnetic p-spin model. Our bounds demonstrate that these schedules have optimal scaling. Our results also show that rapid annealing requires coherent superpositions of energy eigenstates, singling out quantum coherence as a computational resource.

arXiv:2210.15687Physical Review LettersApr 5, 202317 citations

Steered quantum annealing: improving time efficiency with partial information

Ana Palacios de Luis, Artur García-Sáez, Marta P. Estarellas

In the computational model of quantum annealing, the size of the minimum gap between the ground state and the first excited state of the system is of particular importance, since it is inversely proportional to the running time of the algorithm. Thus, it is desirable to keep the gap as large as possible during the annealing process, since it allows the computation to remain under the protection of the adiabatic theorem while staying efficient. We propose steered quantum annealing as a new method to enlarge the gap throughout the process, in the case of diagonal final Hamiltonians, based on the exploitation of some assumptions we can make about the particular problem instance. In order to introduce this information, we propose beginning the anneal from a biased Hamiltonian that incorporates reliable assumptions about the final ground state. Our simulations show that this method yields a larger average gap throughout the whole computation, which results in an increased robustness of the overall annealing process.In the computational model of quantum annealing, the size of the minimum gap between the ground state and the first excited state of the system is of particular importance, since it is inversely proportional to the running time of the algorithm. Thus, it is desirable to keep the gap as large as possible during the annealing process, since it allows the computation to remain under the protection of the adiabatic theorem while staying efficient. We propose steered quantum annealing as a new method to enlarge the gap throughout the process, in the case of diagonal final Hamiltonians, based on the exploitation of some assumptions we can make about the particular problem instance. In order to introduce this information, we propose beginning the anneal from a biased Hamiltonian that incorporates reliable assumptions about the final ground state. Our simulations show that this method yields a larger average gap throughout the whole computation, which results in an increased robustness of the overall annealing process.

arXiv:2206.07646Jun 15, 20220 citations

Rigorous convergence condition for quantum annealing

Yusuke Kimura, Hidetoshi Nishimori

Abstract We derive a generic bound on the rate of decrease of transverse field for quantum annealing to converge to the ground state of a generic Ising model when quantum annealing is formulated as an infinite-time process. Our theorem is based on a rigorous upper bound on the excitation probability in the infinite-time limit and is a mathematically rigorous counterpart of a previously known result derived only from the leading-order term of the asymptotic expansion of adiabatic condition. Since our theorem gives a sufficient condition of convergence for a generic transverse-field Ising model, any specific problem may allow a better, faster, control of the coefficient.Abstract We derive a generic bound on the rate of decrease of transverse field for quantum annealing to converge to the ground state of a generic Ising model when quantum annealing is formulated as an infinite-time process. Our theorem is based on a rigorous upper bound on the excitation probability in the infinite-time limit and is a mathematically rigorous counterpart of a previously known result derived only from the leading-order term of the asymptotic expansion of adiabatic condition. Since our theorem gives a sufficient condition of convergence for a generic transverse-field Ising model, any specific problem may allow a better, faster, control of the coefficient.

arXiv:2207.12096Journal of Physics A Mathematical and TheoreticalOct 27, 20226 citations

Convergence of Quantum Annealing with Real-Time Schrödinger Dynamics

Satoshi Morita, Hidetoshi Nishimori

Convergence conditions for quantum annealing are derived for optimization problems represented by the Ising model of a general form. Quantum fluctuations are introduced as a transverse field and/or transverse ferromagnetic interactions, and the time evolution follows the real-time Schrodinger equation. It is shown that the system stays arbitrarily close to the instantaneous ground state, finally reaching the target optimal state, if the strength of quantum fluctuations decreases sufficiently slowly, in particular inversely proportionally to the power of time in the asymptotic region. This is the same condition as the other implementations of quantum annealing, quantum Monte Carlo and Green's function Monte Carlo simulations, in spite of the essential difference in the type of dynamics. The method of analysis is an application of the adiabatic theorem in conjunction with an estimate of a lower bound of the energy gap based on the recently proposed idea of Somma et. al. for the analysis of classical simulated annealing using a classical-quantum correspondence.Convergence conditions for quantum annealing are derived for optimization problems represented by the Ising model of a general form. Quantum fluctuations are introduced as a transverse field and/or transverse ferromagnetic interactions, and the time evolution follows the real-time Schrodinger equation. It is shown that the system stays arbitrarily close to the instantaneous ground state, finally reaching the target optimal state, if the strength of quantum fluctuations decreases sufficiently slowly, in particular inversely proportionally to the power of time in the asymptotic region. This is the same condition as the other implementations of quantum annealing, quantum Monte Carlo and Green's function Monte Carlo simulations, in spite of the essential difference in the type of dynamics. The method of analysis is an application of the adiabatic theorem in conjunction with an estimate of a lower bound of the energy gap based on the recently proposed idea of Somma et. al. for the analysis of classical simulated annealing using a classical-quantum correspondence.

arXiv:quant-ph/0702252Journal of the Physical Society of JapanMay 25, 200741 citations

\log N

, which is very different from

O(N)

required for the quantum annealing by the real-time Schrödinger equation. With the scheduling based on the adiabatic condition, the required annealing time is of order

\sqrt{N}

, which is identical to the annealing time for quantum annealing by the real-time Schrödinger equation. Although the scheduling based on the adiabatic condition is optimal for the quantum annealing by the real-time Schrödinger equation, it is inefficient for the quantum annealing by the imaginary-time Schrödinger equation. This result implies that the optimal scheduling for the quantum annealing by the imaginary-time and the real-time Schrödinger equation is very different, and the efficient scheduling considered with the quantum Monte Carlo methods, which is based on imaginary-time Schrödinger equation, is not necessarily effective to improve the performance of quantum annealing by the real-time Schrödinger equation. We discuss the efficient scheduling for quantum annealing by the imaginary-time Schrödinger equation with respect to the exponential decay of excited states.