Design and execution of quantum circuits using tens of superconducting qubits and thousands of gates for dense Ising optimization problems

TL;DR

This paper tackles optimization of dense Ising problems on NISQ hardware by introducing Time-Block variants of QAOA and QAMPA that partition the cost Hamiltonian into batched interactions, enabling shallower depth on linearly connected qubit layouts. A novel gate-ordering knob is treated as a variational parameter, providing significant performance gains in experiments on Rigetti’s Aspen-M-3 up to $n=50$ with thousands of gates. Key findings show TB-QAMPA benefits from increased depth and optimized GO, while TB-QAOA shows more modest improvements; adaptive parameter optimization further enhances results, and the method consistently beats random guessing across problem sizes. The work offers practical guidance for co-designing hardware-efficient quantum solvers with current devices and points to future directions in noise modeling, mapping strategies, and integration with classical optimization techniques toward quantum advantage.

Abstract

We develop a hardware-efficient ansatz for variational optimization, derived from existing ansatze in the literature, that parametrizes subsets of all interactions in the Cost Hamiltonian in each layer. We treat gate orderings as a variational parameter and observe that doing so can provide significant performance boosts in experiments. We carried out experimental runs of a compilation-optimized implementation of fully-connected Sherrington-Kirkpatrick Hamiltonians on a 50-qubit linear-chain subsystem of Rigetti Aspen-M-3 transmon processor. Our results indicate that, for the best circuit designs tested, the average performance at optimized angles and gate orderings increases with circuit depth (using more parameters), despite the presence of a high level of noise. We report performance significantly better than using a random guess oracle for circuits involving up to approx 5000 two-qubit and approx 5000 one-qubit native gates. We additionally discuss various takeaways of our results toward more effective utilization of current and future quantum processors for optimization.

Figures (8)

• Figure 1: Left: a subset of $n=20$ qubits with linear connectivity on the physical Aspen-M-3 chip device. The presented subset is an example; the actual linear chains used in our experiments were chosen based on calibration data, see Appendix \ref{['app:expDetails']}. Right: circuits implementing 6-qubit Time-Block (TB) k1-QAOA, k1-QAMPA and k2-QAMPA ansätze. Circuit gates are Hadamards (yellow), Phase-Separator $ZZ_{i,j}^{(J)}$ gates (white) or mixing gates (red, either $X$ rotations or $XY$ operators) and SWAPs. The pictorialized graphs illustrate the progressing scheduling of the interactions as the circuit and the swap network is executed up to the realization of a complete graph (which marks the execution of a full algorithmic round). After $p=1$, the round is repeated but mirrored for compilation efficiency (only the first gates of the next round are drawn).
• Figure 2: Experimental results of $n=50$ QAMPA (left) and QAOA (right) implemented using ansätze introduced in this work. The vertical axis corresponds to the approximation ratio estimated using \ref{['eq:ar_singleshot']}). The horizontal axis shows the depth of the circuit, i.e., physical no. of 2-qubit gates (the fraction of standard QAMPA/QAOA ansatz logical layers, see Remark \ref{['rem:tb_vs_standard']} and discussion in the text). Different colors correspond to different ansatz circuit architectures (see Section \ref{['sec:ansatze_circuits']}). Our ansätze admit optimization over the ordering of circuit gates. The solid lines correspond to optimized gate ordering ("GO" in the plot), chosen as the best out of 10 randomly generated, while dashed lines to the average gate ordering (see Section \ref{['sec:ansatze_permutations']}), both using the random optimizer. The dotted lines correspond to optimization over both angles and gate ordering using adaptive Tree of Parzen Estimators (TPE) optimizer hyperopt2013.
• Figure 3: The left-hand side plot shows an average (over $10$ random Hamiltonian instances) estimated approximation ratio as a function of system size. For each $n$ and $k$, each datapoint is mean over depths corresponding to range $\frac{pk}{n} \in \left[1,2\right]$, i.e., between depth 1 and 2 of a standard ansatz. The dotted lines correspond to the wavefunction simulator. The simulator is run on the same parameters' sets as corresponding experiments (though optimal values may differ). Similarly, the other specifications, including the number of trials and samples, are the same as for experiments. The right-hand side plot shows energy outcome distributions for $k=1,\ n$ with depths corresponding to $\frac{pk}{n}\in\left[1,2\right]$, as compared to random bitstring sampling. The dashed vertical lines correspond to the averages reported in the left-hand side plot.
• Figure 4: Average (over $10$ random Hamiltonian instances) expected value of the renormalized approximation ratio (Eq. \ref{['eq:ar_renormalized']}) calculated from a varying number of the lowest energy samples (Eq. \ref{['eq:ar_quantile']}). Note that due to renormalization, positive values correspond to better-than-random performance. For each $k$, each datapoint is mean over depths corresponding to range $\frac{pk}{n} \in \left[1,2\right]$, i.e., between depth 1 and 2 of a standard ansatz. The rest of the plot's convention is the same as for Fig. \ref{['fig:best_results']}.
• Figure 5: Average (over $10$ random Hamiltonian instances) estimated expected value of the approximation as a function of circuit depth for QAMPA (top) and QAOA (bottom). Solid lines correspond to the best ansatz gate ordering, while dashed lines correspond to the average over 10 random gate orderings. The rest of the plot's convention is the same as for Fig. \ref{['fig:best_results']} (which is zooming in on the rightmost column showing also adaptive parameter optimization results).

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4