Enhancement of quantum annealing via n-local catalysts

TL;DR

This work tackles the bottleneck of exponentially small energy gaps in adiabatic quantum annealing for NP-hard MWIS problems. It introduces n-local catalysts, including a universal product catalyst and locality-graded multi-qubit terms, to transform first-order phase transitions into crossovers and dramatically improve gap scaling. Through toy models and random-graph MWIS instances, it demonstrates that carefully placed 2- to 3- and higher-local couplings can open the gap and enhance ground-state preparation, with stoquastic catalysts offering systematic improvements in many cases. Gate-based implementations show that these catalysts can be realized with polynomially bounded circuit depth and substantially fewer gates than traditional discrete-time drivers, implying practical resource reductions for quantum annealing. The findings point to non-local quantum fluctuations as a key ingredient for achieving quantum advantage in optimization tasks and provide a scalable, adaptive framework for catalyst design.

Abstract

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 re-open 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 $2n$ nearest-neighbour 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 non-local quantum fluctuations entangling multiple qubits as a catalyst are key to achieving the desired quantum advantage.

Figures (18)

• Figure 1: Schematic diagram of two simple examples of energy gap closures at first order phase transition during quantum annealing. (a) The para-ferro transition where the ground state changes from paramagnetic $\ket{P}$, to ferromagnetic $\ket{F}$ during the annealing through a first order transition. This occurs for example, in $p-$spin models for odd $p\ge 3$. For even $p$, $\ket{F}$ has a degeneracy. (b) Example of a first order quantum phase transition that can occur deep inside the ordered phase, the 'perturbative crossing'. Here the states involved are perturbations around computational states with $O(L)$ Hamming distance between them; an example is provided in the figure. There is a subtle difference in the mechanism of gap closure in the two cases, which is described in detail in the text.
• Figure 2: Bipartite toy model, $A$ and $B$ are two subsystems of the bipartite system. We represent the $L=7$ model in the diagram where $A$ has spins $1,2,3,4$ and $B$ has spins $5,6,7$. Partition $A[B]$ has a total weight of $W_1[W_2]$ which is equally divided among the four(three) spins. Thus $w_{1,2,3,4}=W_1/4$ and $w_{5,6,7}=W_2/3$. $J$ is a constant coupling between the spins across the bipartition. We typically choose $W_1=W$ and $W_2=W+\delta W$
• Figure 3: (a) The variation of energy gap $\Delta$ with the anneal parameter $s$ for a system of size $L=7$ with $4$ spins in subsystem $A$ and $3$ spins in $B$. The different colours indicate the three different cases. Blue ($H_c=0$) indicates the case when we do not add a catalyst, Eq. \ref{['eq:anneal']}. The others correspond to the anneal according to Eq. \ref{['eq:annealcat']}: red indicates the case where an $XX$ interaction is added on all the $ZZ$ bonds ($H_c=H_{cXX}$) and black represents the case when the product catalyst ($H_c=H_{cp}$) is used. (b) Variation of the order parameter with $s$ showing the presence and absence of a transition in different cases.
• Figure 4: Scaling of minimum energy gap $\Delta_{\rm min}$ with system size $L$ under different catalysts. The dashed line indicates the best fit $\Delta_{\rm min}=A e^{-b L}$. For no catalyst $(H_c=0)$, $b\approx1.54$, and for the multiple XX catalyst $(H_{cXX})$$b\approx1$.
• Figure 5: Examples of $n-$local catalysts with n$=2,3,4$. The simplest is 2-local, which corresponds to the buliding blocks of $H_{cXX}$ in Fig. \ref{['fig:7site']}. The product catalyst is the extreme case of $n-$locality where all the sites are connected.