Emergent-Coupling-Based Ansatz Evaluated on a Superconducting Quantum Processor

Abstract

The performance of the variational quantum eigensolver depends critically on the choice of ansatz. In this work, we experimentally evaluate the emergent-coupling-based ansatz (ECBA), a physically motivated variational ansatz for disordered systems. The ECBA is based on a renormalization (semi-)group approach to determine the dominant effective couplings, resulting in shallow circuits that capture the essential long-range entanglement structure while balancing local correlations. We implement the ECBA on superconducting quantum processors and benchmark it on disordered Heisenberg chain models. Using classically pre-optimized parameters and error mitigation techniques, we study systems of up to 30 qubits and observe an experimental relative energy accuracy of 96.47% for the largest system. Furthermore, we find that the ECBA can be efficiently embedded on hardware with two-dimensional square-lattice connectivity. We compare to commonly used hardware efficient ansätze and observe that the ECBA achieves significantly higher accuracy at a similar gate count.

Figures (5)

• Figure 1: (a) Schematic illustration of the rainbow model for a system size of six qubits. The upper part shows the qubit chain, the lower part a mapping to a qubit ladder structure. (b) Example of an embedding of the rainbow chain for the CBA onto the IQM Garnet device. Blue connection lines indicate nearest-neighbor gates, green dashed lines denote long-range gates, and red numbers label the mapping of the $i$-th qubit of the chain to the corresponding physical qubit. (c) Hardware-efficient ansatz (HEA) consisting exclusively of single-qubit and nearest-neighbor gates, shown as an example for six qubits. (d) Coupling-based ansatz (CBA) composed of alternating layers of single-qubit gates combined with long-range two-qubit gates reflecting the coupling structure of the model and layers of single-qubit gates combined with nearest-neighbor gates. (e) Relative accuracy of the simulated (noise-free) data and the experimental data obtained for 10 qubits on the IQM Garnet device, each compared to the exact value, for both the HEA and the CBA with and without error mitigation. Error mitigation is applied only to the experimental data.
• Figure 2: (a) Schematic illustration of the random quantum critical point model. The upper part depicts the RG decimation procedure: if $J_i$ is the strongest coupling in the chain, sites $i$ and $i+1$ form a singlet (dashed line). Quantum fluctuations generate an effective interaction between sites $i-1$ and $i+2$ (dotted line). The lower part shows one possible configuration illustratively for six qubits, where singlets may connect arbitrarily distant sites. (b) HEA for the random chain, shown for six qubits: The initial state is constructed from singlets between neighboring qubits (dashed lines), with solid lines indicating the $J$ couplings, followed by a layer of $U_\alpha := R_{xx} R_{yy} R_{zz}$ gates entangling the qubits in a brick wall structure. (c) ECBA for the random chain, shown for six qubits and a single realization of the RG flow. The initial state is constructed from singlets corresponding to the RG singlet pairs. The variational circuit first applies $U_\alpha$ gates to qubit pairs associated with the strongest couplings not forming RG singlets, followed by $U_\alpha$ gates acting on the RG singlet pairs. (d) Embedding of the 20 qubit configuration with disorder parameter $\delta=2$, used for the results shown in Figs. \ref{['fig:results_rg']}, \ref{['fig:heatmap']} and \ref{['fig:heatmap2']}, on the IQM Emerald device. Green dashed lines indicate RG singlet pairs, blue lines denote the strongest remaining couplings not associated with RG singlets, and red numbers label the mapping from logical qubit indices to physical qubits. The long-range singlet pairs are given by (5, 8) and (0, 17). The two isolated singlets (18,19) emerge because the influence of the long-range couplings outweighs that of the couplings between the neighboring qubits (17,18). (e) Relative energy accuracy of experimental results with EM and of ideal simulated results for different system sizes and disorder parameters, each referenced to the exact ground-state energy.
• Figure 3: Logarithmic value of the relative error between the respective VQE simulation and the exact result as function of the system size. The inset shows the average variance of the gradient sampled over various initial parameters. The error bars indicate the variance across the different gradient components.
• Figure 4: Correlation matrix $C_{ij}$ for 20 qubits and $\delta=2$ for the ECBA. Comparison between ideal simulated results of the VQE circuit (lower left half) and experimental results (upper right half).
• Figure 5: Correlation matrix $1 - C_{ij}$ for 20 qubits and $\delta=2$ for the ECBA shown on a logarithmic scale. Comparison between exact (lower left half) and experimental results (upper right half).