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arXiv:2402.13811 · Quantum Science and Technology

Robustness of diabatic enhancement in quantum annealing

Natasha Feinstein, Ivan Shalashilin, Sougato Bose, Paul A. Warburton

Published Jan 16, 2025• 4 citations
arXivHTMLPDF

Abstract

Abstract In adiabatic quantum annealing, the speed with which an anneal can be run, while still achieving a high final ground state (GS) fidelity, is dictated by the size of the minimum gap that appears between the ground and first excited state in the annealing spectrum. To avoid the exponential slowdown associated with exponentially closing gaps, diabatic transitions to higher energy levels may be exploited in such a way that the system returns to the GS before the end of the anneal. In certain cases, this is facilitated by the original annealing spectrum. However, there are also examples where careful manipulation of the annealing Hamiltonian has been used to alter the spectrum to create a diabatic path to the GS. Since diabatic transitions depend on the evolution rate and the gap sizes in the spectrum, it is important to consider the sensitivity of any potential enhancement to changes in the anneal time as well as any parameters involved in the manipulation of the spectrum. We explore this sensitivity using annealing spectra containing an exponentially closing gap and an additional, tuneable, small gap created by a catalyst. We find that there is a trade-off between the precision needed in the catalyst strength and the anneal time in order to maintain the enhancement to the final GS fidelity.

DOI
10.1088/2058-9565/adab14

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Abstract In adiabatic quantum annealing, the speed with which an anneal can be run, while still achieving a high final ground state (GS) fidelity, is dictated by the size of the minimum gap that appears between the ground and first excited state in the annealing spectrum. To avoid the exponential slowdown associated with exponentially closing gaps, diabatic transitions to higher energy levels may be exploited in such a way that the system returns to the GS before the end of the anneal. In certain cases, this is facilitated by the original annealing spectrum. However, there are also examples where careful manipulation of the annealing Hamiltonian has been used to alter the spectrum to create a diabatic path to the GS. Since diabatic transitions depend on the evolution rate and the gap sizes in the spectrum, it is important to consider the sensitivity of any potential enhancement to changes in the anneal time as well as any parameters involved in the manipulation of the spectrum. We explore this sensitivity using annealing spectra containing an exponentially closing gap and an additional, tuneable, small gap created by a catalyst. We find that there is a trade-off between the precision needed in the catalyst strength and the anneal time in order to maintain the enhancement to the final GS fidelity.
In adiabatic quantum computation the required run-time to reach a given ground-state fidelity is dictated by the size of the minimum gap that appears between the ground and first-excited state in the annealing spectrum. In general the presence of avoided level crossings demands an exponential increase in the annealing time with the system size which has consequences both for the efficiency of the algorithm and the required qubit coherence times. One promising avenue being explored to produce more favorable gap scaling is the introduction of nonstoquastic X X couplings in the form of a catalyst—of particular interest are catalysts which utilize accessible information about the optimization problem in their construction. Here we show extreme sensitivity of the effect of an X X catalyst to subtle changes in the encoding of the optimization problem. In particular, we observe that a targeted catalyst containing a single coupling can significantly reduce the gap closing with system size at an avoided level crossing. For slightly different encodings of the same problems however, these same catalysts result in closing gaps in the annealing spectrum. To understand the origin of these closing gaps, we study how the evolution of the ground-state vector is altered by the presence of the catalyst and find that the negative components of the ground-state vector are key to understanding the response of the gap spectrum. We also consider how and when these closing gaps could be utilized in diabatic quantum annealing protocols—a promising alternative to adiabatic quantum annealing in which transitions to higher energy levels are exploited to reduce the run time of the algorithm. Published by the American Physical Society 2024
Quantum annealing is a promising algorithm for solving combinatorial optimization problems. It searches for the ground state of the Ising model, which corresponds to the optimal solution of a given combinatorial optimization problem. The guiding principle of quantum annealing is the adiabatic theorem in quantum mechanics, which guarantees that a system remains in the ground state of its Hamiltonian if the time evolution is sufficiently slow. According to the adiabatic theorem, the annealing time required for quantum annealing to satisfy the adiabaticity scales inversely proportional to the square of the minimum energy gap between the ground state and the first excited state during time evolution. As a result, finding the ground state becomes significantly more difficult when the energy gap is small, creating a major bottleneck in quantum annealing. Expanding the energy gap is one strategy for improving the performance of quantum annealing; however, its implementation in actual hardware remains difficult. This study proposes a method for efficiently solving instances with small energy gaps by introducing additional local terms to the Hamiltonian and exploiting the diabatic transition remaining in the small energy gap. The proposed method achieves an approximate quadratic speedup of the exponential scaling exponent in time to solution compared to the conventional quantum annealing. In addition, we investigate the transferability of the parameters obtained with the proposed method.
Abstract Quantum annealing (QA) is a continuous-time heuristic quantum algorithm for solving or approximately solving classical optimization problems. The algorithm uses a schedule to interpolate between a driver Hamiltonian with an easy-to-prepare ground state and a problem Hamiltonian whose ground state encodes solutions to an optimization problem. The standard implementation relies on the evolution being adiabatic: keeping the system in the instantaneous ground state with high probability and requiring a time scale inversely related to the minimum energy gap between the instantaneous ground and excited states. However, adiabatic evolution can lead to evolution times that scale exponentially with the system size, even for computationally simple problems. Here, we study whether non-adiabatic evolutions with optimized annealing schedules can bypass this exponential slowdown for one such class of problems called the frustrated ring model. For sufficiently optimized annealing schedules and system sizes of up to 39 qubits, we provide numerical evidence that we can avoid the exponential slowdown. Our work highlights the potential of highly-controllable QA to circumvent bottlenecks associated with the standard implementation of QA.
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.
The potential quantum speedup in solving optimization problems via adiabatic quantum annealing is often hindered by the closing of the energy gap during the anneal, especially when this gap scales exponentially with system size. In this work, we alleviate this bottleneck by demonstrating that for the NP-complete Maximum Weighted Independent Set (MWIS) problem, an informed choice of n -local catalysts (operators involving n qubits) can reopen the gap during the annealing process. By analyzing first-order phase transitions in toy instances of the MWIS problem, we first identify direct-tunneling catalysts that effectively eliminate the transition and provide an analytical discussion on when the sign of the catalyst influences its impact. We then reveal that n -local catalysts exponentially improve gap scaling and in certain scenarios are as effective as direct-tunnel coupling between two minima. Utilizing this understanding, we show that they also increase the efficiency of ground-state preparation via adiabatic quantum annealing in random graphs and analytically demonstrate the necessity of their placement across unfrustrated loops in the graph for effective performance in MWIS problems. Additionally, using a circuit implementation of the n -local catalyst (requiring 2 n nearest-neighbor gates), we demonstrate that both the circuit depth and the total number of gates required to solve the problem are reduced by several orders of magnitude when compared to the discrete-time version of traditional quantum annealing using local drivers. Our analysis suggests that nonlocal quantum fluctuations entangling multiple qubits as a catalyst are key to achieving the desired quantum advantage.
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.