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arXiv:2301.13861 · Physical Review Research

Bounding first-order quantum phase transitions in adiabatic quantum computing

Matthias Werner, Artur García-Sáez, Marta P. Estarellas

Published Dec 13, 2023• 7 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevresearch.5.043236

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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.
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.
We present a perturbative method to estimate the spectral gap for adiabatic quantum optimization, based on the structure of the energy levels in the problem Hamiltonian. We show that, for problems that have an exponentially large number of local minima close to the global minimum, the gap becomes exponentially small making the computation time exponentially long. The quantum advantage of adiabatic quantum computation may then be accessed only via the local adiabatic evolution, which requires phase coherence throughout the evolution and knowledge of the spectrum. Such problems, therefore, are not suitable for adiabatic quantum computation.
This paper explores the phenomenon of avoided level crossings in quantum annealing, a promising framework for quantum computing that may provide a quantum advantage for certain tasks. Quantum annealing (QA) involves letting a quantum system evolve according to the Schr\"odinger equation, with the goal of obtaining the optimal solution to an optimization problem through measurements of the final state. However, the continuous nature of quantum annealing makes analytical analysis challenging, particularly with regard to the instantaneous eigenenergies. The adiabatic theorem provides a theoretical result for the annealing time required to obtain the optimal solution with high probability, which is inversely proportional to the square of the minimum spectral gap. Avoided level crossings can create exponentially closing gaps, which can lead to exponentially long running times for optimization problems. In this paper, we use a perturbative expansion to derive a condition for the occurrence of an avoided level crossing during the annealing process. We then apply this condition to the MaxCut problem on bipartite graphs. We show that no exponentially small gaps arise for regular bipartite graphs, implying that QA can efficiently solve MaxCut in that case. On the other hand, we show that irregularities in the vertex degrees can lead to the satisfaction of the avoided level crossing occurrence condition. We provide numerical evidence to support this theoretical development, and discuss the relation between the presence of exponentially closing gaps and the failure of quantum annealing.
Towards better understanding of how to design efficient adiabatic quantum algorithms, we study how the adiabatic gap depends on the spectra of the initial and final Hamiltonians in a natural family of test-bed examples. We show that perhaps counter-intuitively, changing the energy in the initial and final Hamiltonians of only highly excited states (we do this by assigning all eigenstates above a certain cutoff the same value), can turn the adiabatic algorithm from being successful to failing. Interestingly, our system exhibits a phase transition; when the cutoff in the spectrum becomes smaller than roughly n/2n/2n/2, nnn being the number of qubits, the behavior transitions from a successful adiabatic process to a failed one. To analyze this behavior, and provide an upper bound on both the minimal gap as well as the success of the adiabatic algorithm, we introduce the notion of "escape rate", which quantifies the rate by which the system escapes the initial ground state (a related notion was also used by Ilin and Lychkovskiy in arXiv:1805.04083). Our results indicate a phenomenon that is interesting on its own right: an adiabatic evolution may be robust to bounded-rank perturbations, even when the latter closes the gap or makes it exponentially small.
The quantum adiabatic theorem ensures that a slowly changing system, initially prepared in its ground state, will evolve to its final ground state with arbitrary precision. As a first result this thesis extends the original theorem to projection operators keeping the statement valid for Hamiltonians with degenerate ground spaces. Yet the main focus of this work lies in studying the efficiency of quantum circuit simulations by adabatic quantum computation. The standard Hamiltonian construction by Kitaev is based on a path graph reflecting the LLL computation steps and influencing the scaling of the necessary evolution time by its spectral gap of O(1L2)\mathcal{O}\left(\frac{1}{L^2}\right)O(L21​). Aspiring to an improved running time we generalize Kitaev's Hamiltonian to so-called standard graph Hamiltonians based on graph families with a different spectral gap. In this generalized construction the first two time derivatives of the Hamiltonian and the fraction of initial vertices appear as additional parameters of running time. In a first step the time derivatives can be proven to be constant. Expansion results from spectral graph theory however impose a trade-off between the spectral gap and the fraction of initial vertices as well as the fraction of final vertices which corresponds to the probability for obtaining the correct computational output. Graphs with spectral gap O(1Lk)\mathcal{O}\left(\frac{1}{L^k}\right)O(Lk1​), k<2k<2k<2, turn out to contradict very likely, graphs with k<1k<1k<1 even for sure at least one of the efficiency criterias for running time, output probability or Hamiltonian implementation. The above results may also be obstacles for a possible quantum PCP-theorem in complexity theory claiming the local Hamiltonian problem with constant gap to be QMA-complete since the very same Kitaev Hamiltonian is constructed in the QMA-hardness proof for the Local Hamiltonian problem.
Quantum annealing (QA) is a method for solving combinatorial optimization problems. We can estimate the computational time for QA using what is referred to as the adiabatic condition derived from the adiabatic theorem. The adiabatic condition consists of two parts: an energy gap and a transition matrix. Most past studies have focused on the relationship between the energy gap and computational time. The success probability of QA is considered to decrease exponentially owing to the exponentially decreasing energy gap at the first-order phase-transition point. In this study, through a detailed analysis of the relationship between the energy gap, transition matrix, and computational cost during QA, we propose a general method for constructing counterintuitive models wherein QA with a constant annealing time fails despite a constant energy gap, based on polynomial energy. We assume that the energy of the total Hamiltonian is at most $\mathrm{\ensuremath{\Theta}}(L)$, where LLL is the number of qubits. In our formalism, we choose a known model that exhibits an exponentially small energy gap during QA, and modify the model by adding a specific penalty term to the Hamiltonian. In the modified model, the transition matrix in the adiabatic condition becomes exponentially large as the number of qubits increases, while the energy gap remains constant. Moreover, we achieve a quadratic speedup, for which the upper bound for improvement in the adiabatic condition is determined by the polynomial energy. For concrete examples, we consider the adiabatic Grover search and the ferromagnetic ppp-spin model. In these cases, with the addition of the penalty term, although the success probability of QA on the modified models becomes exponentially small despite a constant energy gap, we are able to achieve a success probability considerably higher than that of conventional QA. Moreover, in concrete examples, we numerically show the scaling of the computational cost is quadratically improved compared to the conventional QA. Our findings pave the way for a better understanding of QA performance.