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arXiv:1007.3294 · New Journal of Physics

Testing quantum adiabaticity with quench echo

H T Quan, W H Zurek

Published Sep 15, 2010• 35 citations
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Abstract

Adiabaticity of quantum evolution is important in many settings; one example is adiabatic quantum computation (AQC). Nevertheless, to date, there is no effective method available for testing the adiabaticity of the evolution for the case where the eigenenergies of the driven Hamiltonian are not known. We propose a simple method for checking the adiabaticity of a quantum process for an arbitrary quantum system. We further propose an operational method for finding more efficient protocols that approximate adiabaticity, and suggest a 'uniformly adiabatic' quench scheme based on the Kibble–Zurek mechanism for the case where the initial and final Hamiltonians are given. This method should help in implementing AQC and other tasks where preserving the system in the ground state of a time-dependent Hamiltonian is desired.

DOI
10.1088/1367-2630/12/9/093025

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Adiabaticity of quantum evolution is important in many settings; one example is adiabatic quantum computation (AQC). Nevertheless, to date, there is no effective method available for testing the adiabaticity of the evolution for the case where the eigenenergies of the driven Hamiltonian are not known. We propose a simple method for checking the adiabaticity of a quantum process for an arbitrary quantum system. We further propose an operational method for finding more efficient protocols that approximate adiabaticity, and suggest a 'uniformly adiabatic' quench scheme based on the Kibble–Zurek mechanism for the case where the initial and final Hamiltonians are given. This method should help in implementing AQC and other tasks where preserving the system in the ground state of a time-dependent Hamiltonian is desired.
Adiabatic quantum computation (AQC) is a universal model for quantum computation which seeks to transform the initial ground state of a quantum system into a final ground state encoding the answer to a computational problem. AQC initial Hamiltonians conventionally have a uniform superposition as ground state. We diverge from this practice by introducing a simple form of heuristics: the ability to start the quantum evolution with a state which is a guess to the solution of the problem. With this goal in mind, we explain the viability of this approach and the needed modifications to the conventional AQC (CAQC) algorithm. By performing a numerical study on hard-to-satisfy 6 and 7 bit random instances of the satisfiability problem (3-SAT), we show how this heuristic approach is possible and we identify that the performance of the particular algorithm proposed is largely determined by the Hamming distance of the chosen initial guess state with respect to the solution. Besides the possibility of introducing educated guesses as initial states, the new strategy allows for the possibility of restarting a failed adiabatic process from the measured excited state as opposed to restarting from the full superposition of states as in CAQC. The outcome of the measurement can be used as a more refined guess state to restart the adiabatic evolution. This concatenated restart process is another heuristic that the CAQC strategy cannot capture.
The adiabatic quantum computation (AQC) has been proven to be equivalent to the standard circuit model. Conventionally, AQC evolves from the initial Hamiltonian which has a uniform equal superposition of the computational basis to the final Hamiltonian whose ground state encodes the solution to a computation problem. We propose an alternative approach to construct the initial Hamiltonian of the AQC which has an unequal superposition of the possible solutions to the problem and show that an educated guess can improve the performance of AQC.
Ideally, in an adiabatic quantum computation or quantum state preparation, a simple ground state of an initial Hamiltonian is evolved adiabatically to an interesting ground state of a final Hamiltonian. Unfortunately, the simple and the interesting are often different enough to be separated by a quantum phase transition. Due to a vanishing energy gap between the ground state and excited states at the critical point, near the transition the adiabaticity is bound to fail. This failure is quantified by a quantum version of the Kibble-Zurek mechanism (KZM). In these notes I introduce KZM in its general textbook form, based on adiabatic/impulse approximation, and then support the picture by an exact solution of the integrable transverse field quantum Ising chain and an approximate one of the non-integrable Bose-Hubbard model. The last model illustrates typical problems with adiabatic quantum state preparation that are encoutered in atomic quantum simulators.
We give a quantum algorithm for solving instances of the satisfiability problem, based on adiabatic evolution. The evolution of the quantum state is governed by a time-dependent Hamiltonian that interpolates between an initial Hamiltonian, whose ground state is easy to construct, and a final Hamiltonian, whose ground state encodes the satisfying assignment. To ensure that the system evolves to the desired final ground state, the evolution time must be big enough. The time required depends on the minimum energy difference between the two lowest states of the interpolating Hamiltonian. We are unable to estimate this gap in general. We give some special symmetric cases of the satisfiability problem where the symmetry allows us to estimate the gap and we show that, in these cases, our algorithm runs in polynomial time.
Farhi et al. have proposed an adiabatic quantum computation (AQC), which can be applied to NP-problems if one can know an appropriate Hamiltonian for a target problem. We have proposed a neuromorphic adiabatic quantum computation (NAQC) as the AQC with energy dissipation and an efficient method for designing a final Hamiltonian in consideration of the analogy with a neural network. The NAQC can be applied to optimization problems if its cost function can be expressed in a quadratic form. And successful operations have been confirmed by numerical simulations. In addition, local adiabatic evolution for quantum search have proposed by Roland et al. in order to speed-up the calculation time. In this paper, we show preliminary results for NAQC with local adiabatic evolution by numerical simulations.
We give a quantum algorithm for solving instances of the satisfiability problem, based on adiabatic evolution. The evolution of the quantum state is governed by a time-dependent Hamiltonian that interpolates between an initial Hamiltonian, whose ground state is easy to construct, and a final Hamiltonian, whose ground state encodes the satisfying assignment. To ensure that the system evolves to the desired final ground state, the evolution time must be big enough. The time required depends on the minimum energy difference between the two lowest states of the interpolating Hamiltonian. We are unable to estimate this gap in general. We give some special symmetric cases of the satisfiability problem where the symmetry allows us to estimate the gap and we show that, in these cases, our algorithm runs in polynomial time.