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arXiv:1010.1220

Avoid First Order Quantum Phase Transition by Changing Problem Hamiltonians

Vicky Choi

Published Oct 6, 2010• 6 citations
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Abstract

In Amin and Choi \cite{AC09}, we show that an adiabatic quantum algorithm for the NP-hard maximum independent set (MIS) problem on a set of special family of graphs in which there are exponentially many local maxima would have the exponentially small minimum spectral gap and thus would require the exponential time, due to the first order quantum phase transition (FQPT). The problem Hamiltonian of the adiabatic quantum algorithm for MIS is based on the reduction to the Ising problem and has flexible parameters. In this paper, we show numerically on the 15-vertex graph that by choosing the parameters appropriately in the problem Hamiltonian (without changing the problem to be solved) for MIS, we can prevent the FQPT and drastically increase the minimum spectral gap. The result is further supported by visualization from the Decomposed State Evolution Visualization (\desev) --- a visualization tool we introduced. Furthermore, our result also serves to concretely clarify that it is not sufficient to consider one specific problem Hamiltonian for proving the failure of adiabatic quantum optimization for a problem. We also raise the basic question about what the appropriate formulation of adiabatic running time should be.

DOI
10.48550/arxiv.1010.1220

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In Amin and Choi \cite{AC09}, we show that an adiabatic quantum algorithm for the NP-hard maximum independent set (MIS) problem on a set of special family of graphs in which there are exponentially many local maxima would have the exponentially small minimum spectral gap and thus would require the exponential time, due to the first order quantum phase transition (FQPT). The problem Hamiltonian of the adiabatic quantum algorithm for MIS is based on the reduction to the Ising problem and has flexible parameters. In this paper, we show numerically on the 15-vertex graph that by choosing the parameters appropriately in the problem Hamiltonian (without changing the problem to be solved) for MIS, we can prevent the FQPT and drastically increase the minimum spectral gap. The result is further supported by visualization from the Decomposed State Evolution Visualization (\desev) --- a visualization tool we introduced. Furthermore, our result also serves to concretely clarify that it is not sufficient to consider one specific problem Hamiltonian for proving the failure of adiabatic quantum optimization for a problem. We also raise the basic question about what the appropriate formulation of adiabatic running time should be.
The problem Hamiltonian of the adiabatic quantum algorithm for the maximum-weight independent set problem (MIS) that is based on the reduction to the Ising problem (as described in [Choi08]) has flexible parameters. We show that by choosing the parameters appropriately in the problem Hamiltonian (without changing the problem to be solved) for MIS on CK graphs, we can prevent the first order quantum phase transition and significantly change the minimum spectral gap. We raise the basic question about what the appropriate formulation of adiabatic running time should be. We also describe adiabatic quantum algorithms for Exact Cover and 3SAT in which the problem Hamiltonians are based on the reduction to MIS. We point out that the argument in Altshuler et al.(arXiv:0908.2782 [quant-ph]) that their adiabatic quantum algorithm failed with high probability for randomly generated instances of Exact Cover does not carry over to this new algorithm.
The problem Hamiltonian of the adiabatic quantum algorithm for the maximum-weight independent set problem (MIS) that is based on the reduction to the Ising problem (as described in [Choi08]) has flexible parameters. We show that by choosing the parameters appropriately in the problem Hamiltonian (without changing the problem to be solved) for MIS on CK graphs, we can prevent the first order quantum phase transition and significantly change the minimum spectral gap. We raise the basic question about what the appropriate formulation of adiabatic running time should be. We also describe adiabatic quantum algorithms for Exact Cover and 3SAT in which the problem Hamiltonians are based on the reduction to MIS. We point out that the argument in Altshuler et al.(arXiv:0908.2782 [quant-ph]) that their adiabatic quantum algorithm failed with high probability for randomly generated instances of Exact Cover does not carry over to this new algorithm.
It has been recently argued that adiabatic quantum optimization would fail in solving NP-complete problems because of the occurrence of exponentially small gaps due to crossing of local minima of the final Hamiltonian with its global minimum near the end of the adiabatic evolution. Using perturbation expansion, we analytically show that for the NP-hard problem of maximum independent set there always exist adiabatic paths along which no such crossings occur. Therefore, in order to prove that adiabatic quantum optimization fails for any NP-complete problem, one must prove that it is impossible to find any such path in polynomial time.
In the context of adiabatic quantum computation (AQC), it has been argued that first-order quantum phase transitions (QPTs) due to localization phenomena cause AQC to fail by exponentially decreasing the minimal spectral gap of the Hamiltonian along the annealing path as a function of the qubit number. The vanishing of the spectral gap is often linked to the localization of the ground state in a local minimum, requiring the system to tunnel into the global minimum at a later stage of the annealing. Recent methods have been proposed to avoid this phenomenon by carefully designing the involved Hamiltonians. However, it remains a challenge to formulate a comprehensive theory of the effect of the various parameters and the conditions under which QPTs make the AQC algorithm fail. Equipped with concepts from graph theory, in this work we link graph quantities associated with the Hamiltonians along the annealing path with the occurrence of QPTs. These links allow us to derive bounds on the location of the minimal spectral gap along the annealing path, augmenting the toolbox for the analysis of strategies to improve the runtime of AQC algorithms.
We present a quantum adiabatic algorithm for solving MIS with Rydberg atoms, which uses graph degrees to engineer the Hamiltonian for adiabatic evolution. Our approach extends the algorithm of Pichler et al. for MIS state preparation, which uses time-dependent global controls to excite atoms while the Rydberg blockade mechanism suppresses adjacent excitations. To accelerate adiabatic evolution, we introduce local light shifts that favor excitation of low-degree nodes and constrain the final Hamiltonian so that energy scales inversely with the number of Rydberg excitations, preventing convergence to non-MIS states. Benchmarking our degree-dependent MIS algorithm with the traditional adiabatic algorithm on a set of 1000 MIS instances, we demonstrate numerically that our algorithm inflates the minimal spectral gap and, consequently, accelerates the convergence. By fitting power-law relations for both the time-to-solution and success probability metrics as a function of the hardness parameter, we demonstrate a superior scaling exponent for our algorithm over the traditional global protocol. Furthermore, the classical Hamiltonian engineering step required to construct degree-dependent controls introduces only polynomial overhead. These results indicate that our algorithm can be used to extend the size of graphs for which MIS can be found on physical hardware. Moreover, our approach extends beyond MIS to additional graph problems such as MAX-CUT.
We investigate the connection between local minima in the problem Hamiltonian and first-order quantum phase transitions during adiabatic quantum computation. We demonstrate how some properties of the local minima can lead to an extremely small gap that is exponentially sensitive to the Hamiltonian parameters. Using perturbation expansion, we derive an analytical formula that cannot only predict the behavior of the gap, but also provide insight on how to controllably vary the gap size by changing the parameters. We show agreement with numerical calculations for a weighted maximum independent set problem instance.