Shot- and Noise-Aware Pauli Correlation Encoding for Max-Cut Experiments on IBM Heron Quantum Processor

Abstract

Pauli correlation encoding (PCE) encodes $m$ binary variables in the signs of $k$-body Pauli correlators of $n=O(m^{1/k})$ qubits, and trains a parameterized circuit on a relaxed loss of sharpness $α$. The strongest published schedules sharpen this loss aggressively and are benchmarked with exact expectation values, whereas a quantum computer estimates correlators from a finite number of shots. We show that the sharpness is a hidden shot budget. Since the expected sign of a correlator estimated from $S$ shots, $\mathrm{erf}(c\sqrt{S/2})$, agrees with $\tanh(0.851\sqrt{S}\,c)$ to within $0.019$, the edge term of the relaxed loss at $α=0.851\sqrt{S}$ approximately equals, up to a constant, minus twice the cut expected from decoding $S$ shots. This explains why schedules tuned with exact expectation values lose their advantage at realistic budgets, and yields a shot-aware schedule whose final sharpness is set by the shot budget. On five G-set graphs with $800$ variables on $13$ qubits, this schedule outperforms every published schedule at its target budget in all $150$ paired comparisons of the decoded circuit output, owing to the sharpness at $1.6\times10^4$ shots and mostly to switching off the regularization at $10^3$; after a classical bit-swap search, it remains the best on average at $1.6\times10^4$ shots. Device noise damps the correlators by a factor $s$, which amounts to a smaller shot budget $s^2S$. On the IBM Heron processor ibm_fez, an all-to-all ansatz of the type used for the best published G-set results compiles to $835$ CZ gates and decodes to a random cut, while our $96$-CZ hardware-native circuit, trained in simulation and sharpened to $α\simeq0.65\,s\sqrt{S}$, retains up to $0.919$ of the best-known cut. For new circuits, a damping and an offset calibrated on other circuits of the same graph predicted the decoded cuts at $10^3$ shots to within $0.002$.