Entropy Density Benchmarking of Near-Term Quantum Circuits

TL;DR

This work introduces entropy-density benchmarking as a bridge between circuit- and application-level benchmarking for NISQ devices. It develops a simple global depolarizing heuristic to model entropy accumulation in variational quantum circuits, validates it against classical-simulation and experimental data, and refines it with T1-relaxation effects. By combining this entropy-accumulation model with an application-level quantum-advantage framework, the authors derive tighter circuit-size bounds that delineate where quantum advantage is still feasible, demonstrating practical implications for MAX-CUT on superconducting hardware. The study highlights both the promise of entropy-based benchmarks and the need for more sophisticated noise models and error-mitigation strategies to tighten predictions for larger systems and diverse platforms.

Abstract

Understanding the limitations imposed by noise on current and next-generation quantum devices is a crucial step towards demonstrating practical quantum advantage. In this work, we investigate the accumulation of entropy density as a benchmark to monitor the performance of quantum processing units. We provide a proof-of-principle demonstration of our novel methodology which entails developing simple heuristic models of how entropy accumulates, testing them against real QPU experiments, and finally using these models to determine a circuit volume threshold above which quantum advantage is unattainable. Monitoring entropy density not only offers a novel approach that complements existing circuit-level benchmarking techniques, but more importantly, it bridges the gap between circuit-level and application-level benchmarking protocols. In particular, our heuristic model of entropy accumulation allows us to outperform existing techniques that bound the circuit size threshold for quantum advantage.

Figures (9)

• Figure 1: First two circuit layers of a 5-qubit hardware-efficient parameterized quantum circuit $U(\Vec{\theta})$. Each circuit layer is composed of a single-qubit $R_X(\theta)$ rotation followed by a $R_Y(\theta)$ rotation and a final layer of $CZ$ gates between nearest neighbours. The angles $\theta_i$s are drawn from the uniform distribution over $[0,2\pi)$.
• Figure 2: Numerical simulation and heuristic model. We simulate a single VQA circuit represented in Fig. \ref{['fig:ideal_hardware_efficient_param_circuit']}, with randomly selected parameters (random seed np.random.seed(837)) affected by local depolarizing noise ($p_1=0.008$ and $p_2=0.054$ from calibration data of Rigetti's QPU, Aspen-M-3). The dots characterise the Renyi-2 entropy density evolution of the quantum register after each circuit layer for different number of qubits $n$ (see colour label). The black horizontal dash-dotted line corresponds to the entropy density of the maximally mixed state $\sigma_0\coloneqq I/2^n$. The solid lines correspond to interpolations using a global depolarizing heuristic model from Eq. \ref{['eq:purity-model-from-class-sim']} with fitting parameters $\alpha_1$ and $\alpha_2$ ($\alpha_i\approx p_i$), as detailed in subsection \ref{['subsubsec:analytical-model']}.
• Figure 3: Classical shadows protocol. Each $V_i$ is a single-qubit Clifford gate such that its combination with a computational basis measurement corresponds to a random measurement in the $X$, $Y$, or $Z$ basis.
• Figure 4: Experimental results and validation of our heuristic model. We consider a VQA circuit as in Fig. \ref{['fig:ideal_hardware_efficient_param_circuit']} applied to $n=3$ qubits, with fixed circuit parameters (fixed random seed np.random.seed(837)). Golden error bars give the purity and Renyi-2 entropy evolution as functions of circuit depth on Rigetti's QPU Aspen-M-3 using two different methods. Each error bar was obtained by running the classical shadows protocol $3$ times, and computing the average estimate and the standard deviation over those $3$ samples. Results obtained in the same setting but using a local depolarizing noise model ($p_1=0.008$ and $p_2=0.054$ from calibration data of Aspen-M-3) and without measurement errors correspond to the light blue error bars. The dark blue error bars in Fig. \ref{['subfig:QPU_run_n=3-R2d']} correspond to the same simulation with added measurement errors from calibration data (measurement circuit gate errors $p_1=0.008$ and errors in the detectors $P(0|1)=0.03$ and $P(1|0)=0.02$). The dark blue error bars in Fig. \ref{['subfig:QPU_run_n=3-R2d-T1']} also include $T_1$ relaxation errors both at the gate level ($\gamma_1, \gamma_2 = 0.003, 0.015$) right after depolarization ($p_1, p_2 = 0.006, 0.038$) and in the measurement stage ($\gamma_{meas}=0.05$) just before readout errors ($P(0|1)=P(1|0)=0.03$) based on calibration data. The black solid lines show the purity and Renyi-2 entropy density evolution obtained via density matrix simulation under our local depolarizing noise model without measurement errors (similar to light blue error bars). The black horizontal dash-dotted lines correspond to the purity or entropy density of the maximally mixed state $\sigma_0\coloneqq I/2^n$. Fit of our global depolarizing heuristic model, where we have imposed that $\alpha_2/\alpha_1=p_2/p_1$, and where $\beta$ models readout errors and noise in the measurement circuit, corresponds to the solid golden and blue lines.
• Figure 5: Quantum advantage benchmarking framework from stilck_franca_limitations_2021. For an optimization task of the form $\min_{\rho} Tr[H\rho]$, the authors derive a lower bound (blue solid line) on the quantum device's solution to the energy density ($\mathop{\mathrm{Tr}}\nolimits[H\rho_{out}]/n$) as a function of the accumulated entropy density ($S(\rho_{out})/n$). The state-of-the-art classical solvers' solution to this optimization problem (red horizontal line) defines an entropy density threshold $c$ such that if the quantum device's candidate solution $\rho_{out}$ satisfies $\frac{S(\rho_{out})}{n} \geq c$ then quantum advantage is out of reach for this device and for this optimization problem.