Improving Quantum Approximate Optimization by Noise-Directed Adaptive Remapping

TL;DR

The paper tackles binary optimization on noisy quantum hardware by introducing Noise-Directed Adaptive Remapping (NDAR), which exploits a classical attractor in the noise by adaptively remapping the cost Hamiltonian through bitflip gauges. NDAR iteratively selects the gauge based on prior optimization outcomes, transforming the Hamiltonian to align the attractor with better solutions via $H^{oldsymbol{y}}=P_{oldsymbol{y}}HP_{oldsymbol{y}}$. Applied to $p=1$ QAOA on Sherrington-Kirkpatrick instances with $n=82$, NDAR on Rigetti’s device delivers approximation ratios from $0.9$ to $0.96$, significantly surpassing standard QAOA's $0.34$--$0.51$ under the same cost-function evaluation budget. The work analyzes how noise breaks gauge symmetry and how a greedy outer loop can exploit this to improve optimization, while outlining avenues for generalization to other cost models, ansätze, and non-quantum Ising machines.

Abstract

We present Noise-Directed Adaptive Remapping (NDAR), a heuristic algorithm for approximately solving binary optimization problems by leveraging certain types of noise. We consider access to a noisy quantum processor with dynamics that features a global attractor state. In a standard setting, such noise can be detrimental to the quantum optimization performance. Our algorithm bootstraps the noise attractor state by iteratively gauge-transforming the cost-function Hamiltonian in a way that transforms the noise attractor into higher-quality solutions. The transformation effectively changes the attractor into a higher-quality solution of the Hamiltonian based on the results of the previous step. The end result is that noise aids variational optimization, as opposed to hindering it. We present an improved Quantum Approximate Optimization Algorithm (QAOA) runs in experiments on Rigetti's quantum device. We report approximation ratios $0.9$-$0.96$ for random, fully connected graphs on $n=82$ qubits, using only depth $p=1$ QAOA with NDAR. This compares to $0.34$-$0.51$ for standard $p=1$ QAOA with the same number of function calls.

Figures (5)

• Figure 1: Illustration of a Noise-Directed Adaptive Remapping run for a single Hamiltonian instance. The X-axis represents iterations of the algorithm. The purple dots correspond to the approximation ratio (AR) of the $\ket{0\dots 0}$ (attractor) state, while cyan dots are the best-observed solution (as a result of QAOA runs). In consecutive steps, the Hamiltonian is re-transformed in a way that the best solution from the previous step effectively becomes the (approximate) attractor of the new step, resulting in a stairs-like climb towards higher approximation ratios until convergence. The distributions of ARs are stacked horizontally for each NDAR step, showing how the tail of the distribution from a previous step, becomes a point near the center of the distribution in the next step (in early iterations). The presented data corresponds to a single instance from the experimentally obtained dataset (for $n=82$ qubits) in Fig. \ref{['fig:exp:NDAR']}b. The inset illustrates how binary (spin) variables are associated with the qubit states of a physical QPU system. Qubits are represented by anharmonic oscillators (e.g. transmons) with ground state $\ket{0}$ and first excited state $\ket{1}$. Gates drive coherent transitions (blue arrows) while noise (red arrow) induces asymmetric incoherent transitions.
• Figure 2: Investigation of the effects of applying bitflip gauges to improve an $n=82$-qubit $p=1$ QAOA in experiments on Rigetti's Ankaa-2 superconducting quantum device. a) Correlation between the approximation ratio (AR) of the attractor state $\ket{0\dots0}$ (X-axis) and the AR attained via QAOA (Y-axis) using transformed Hamiltonian. Differently colored regions of the histogram correspond to the AR estimators constructed from different percentiles of the AR distribution. We computed Pearson’s correlation $r_q$ and Spearman’s rank correlation $\rho_q$ coefficients at each percentile $q$. We obtained $r_{0.1\%}=0.87^{+0.03}_{-0.04}$, $r_{10\%}=0.964^{+0.009}_{-0.012}$, and $r_{100\%}=0.984^{+0.004}_{-0.005}$; and $\rho_{0.1\%}=0.86^{+0.04}_{-0.05}$, $\rho_{10\%}=0.96^{+0.01}_{-0.02}$, and $\rho_{100\%}=0.981^{+0.006}_{-0.007}$. Confidence intervals are 95% bias-corrected and accelerated ($BC_{a}$) paired-bootstrap intervals efron1994introductionBootstrap with $10^5$ resamples, calculated using the standard scipy.stats module scipy. Two-sided tests yielded p-values $<10^{-58}$ for all cases. Taken together, these results indicate a strong, nearly linear association between the AR of the $\ket{0\cdots 0}$ state and the quality of optimization. The shaded region corresponds to the $y\leq x$ where the optimized AR is lower than the AR of the attractor state (indicating no performance gain). Data points are aggregated for $10$ Hamiltonian instances. b) Overview of Noise-Directed Adaptive Remapping algorithm (see Algorithm \ref{['alg:NDAR_main']}) performance as a function of iterations (X-axis). The Y-axis shows the AR of the best-found solution (out of $1000*150*(i+1)$, with $i$ being the iteration index). Each dotted line corresponds to a different random Hamiltonian instance, solid lines indicate the maximal and minimal performance across $10$ instances. Besides best-found solutions obtained via NDAR applied to QAOA (orange), we plot standard QAOA optimization (light blue), and a baseline that corresponds to uniform random sampling (grey). Since in the case of QAOA and random baseline the protocols are not iterative, the iteration index corresponds to increasing the number of cost function evaluations/gathered samples to match those of NDAR.
• Figure 3: Evolution of the samples' distributions properties as a function of NDAR iteration. The Y-axis is NDAR iteration, and the Z-axis represents a (normalized) number of occurrences. The X-axis corresponds to the a) Approximation Ratio, b) the physically recorded, raw Hamming weight, and c) the effective Hamming weight attained due to NDAR bitstrings relabeling. First and last iterations are colored red and blue, respectively; while intermediate iterations are graduated green. The plots show aggregated data for $10$ Hamiltonians. For each Hamiltonian, there is a total of $1000$ samples shown for each iteration (taken from the optimized trials from the same data as in Fig. \ref{['fig:exp:NDAR']}b).
• Figure 4: Investigation of the effects of applying bitflip gauges to improve an $n=16$-qubit variational optimization with $p=1$ QAOA in simulations. The data presented is of the same type as Fig. \ref{['fig:exp:NDAR']} in the main text, see description of that figure for details.
• Figure 5: Schematics of the Rigetti's Ankaa-2 superconducting transmon chip topology with respect to 2-qubit iSWAP operation. Missing edges not shown. Source: rigetti.com.

Theorems & Definitions (3)

• Lemma 1
• proof
• Remark 1