Quantum Hamiltonian Algorithms for Maximum Independent Sets

TL;DR

Intriguingly, this equivalence unveils that the PXP model, recently prominent in quantum dynamics research, can be viewed as quantum diffusion over the median graph of all independent sets governed by the non-Abelian gauge matrix.

Abstract

With qubits encoded into atomic ground and Rydberg states and situated on the vertexes of a graph, the conditional quantum dynamics of Rydberg blockade, which inhibits simultaneous excitation of nearby atoms, has been employed recently to find maximum independent sets following an adiabatic evolution algorithm hereafter denoted by HV [Science 376, 1209 (2022)]. An alternative algorithm, short named the PK algorithm, reveals that the independent sets diffuse over a media graph governed by a non-abelian gauge matrix of an emergent PXP model. This work shows the above two algorithms are mathematically equivalent, despite of their seemingly different physical implementations. More importantly, we demonstrated that although the two are mathematically equivalent, the PK algorithm exhibits more efficient and resource-saving performance. Within the same range of experimental parameters, our numerical studies suggest that the PK algorithm performs at least 25% better on average and saves at least $6\times10^6$ measurements ($\sim 900$ hours of continuous operation) for each graph when compared to the HV algorithm. We further consider the measurement error and point out that this may cause the oscillations in the performance of the HV's optimization process.

Figures (6)

• Figure 1: A graph with 8 vertices and 12 edges. The circles stands for vertices and the lines stands for edges. The red circles form one of its maximum independent sets.
• Figure 2: The unoptimized path of variational quantum adiabatic algorithm is represented by the solid lines, the same as the path given in FIG. S8. of Lukin2022. The adiabatic path of the PK algorithm, given by Eqs. (\ref{['path0']}) and (\ref{['path2']}), is drawn as the dashed lines with $\omega_{\theta}=\pi/T$ and $\omega_{\phi}/\omega_{\theta}=-11$.
• Figure 3: $P_\text{MIS}$ and $P_\text{IS}$ denote respectively the probabilities for finding MIS and IS. $N_\text{HV}$ and $N_\text{PK}$ denote the numbers of graphs employed for running HV algorithm and PK algorithms respectively. The graphs are 1000 unit disk graphs with 7 vertices. The parameters adopted are from the experiment Lukin2022, i.e. $V_{\text{NN}}/h=107$ MHz, $V_{\text{NNN}}/h=13$ MHz, and $T=1.5$$\mu$s. (a) The average success rate using the unoptimized path of HV algorithm is 45% with standard deviation of 41.2%. Using the adiabatic path of the PK algorithm increases the success rate to 97% with standard deviation of 2.2%. (b) The average rate of finding independent sets by HV algorithm is 46% with standard deviation of 42.2%, which means in most unsuccessful cases, the HV algorithm finds non-independent sets.
• Figure 4: $S$ denotes the optimization steps required for the HV algorithm to reach the success rate $\min\ (99\% \ {\rm or}\ P_{\text{MIS}} \text{ of the PK algorithm})$ and takes a maximum value of $500$. $N_\text{SGD}$ and $N_\text{Adam}$ denote respectively the numbers of graphs using SGD or Adam optimizer. $P_\text{MIS}$ denotes the probabilities for finding MIS for the graphs. $N_\text{SGD-max}$ and $N_\text{Adam-max}$ denote respectively the numbers of graphs optimized for 500 steps using SGD or Adam optimizer. A total of 500 unit disk graphs with 7 vertices are included for each optimizer. (a) The average success rate and average optimization steps are 72% and 262, 70% and 287, respectively from using SGD or Adam optimizer. The corresponding percentages of graphs optimized for 500 steps are 48.8% and 54.6% respectively. (b) For the graphs that reach the maximum optimization steps, the average maximum success rates are 47% with standard deviation of 19.1% and 46% with standard deviation of 17.8%, respectively from using SGD and Adam optimizer.
• Figure 5: A graph with 5 vertices and 6 edges.