Approximate Quadratization of High-Order Hamiltonians for Combinatorial Quantum Optimization

TL;DR

This work tackles the difficulty of solving high-order combinatorial optimization problems with quantum hardware by introducing approximate quadratization methods that convert HUBOs into qubit-efficient QUBOs without adding qubits. It presents two strategies—the hypergraph clique expansion and a jointly optimized quadratization $H_2'(\boldsymbol{\theta})$—showing that depth-1 and depth-2 QAOA with these approximations can achieve substantial performance gains on LABS-type problems, though with trade-offs in convergence guarantees. The paper also develops hardware-friendly Ansätze using SWAP-strategy truncation to reduce circuit depth, demonstrating that under noise there exists an optimal balance between SWAP layers and QAOA depth, and that CVaR post-processing can recover near-noiseless performance. Overall, the results suggest a practical path to improve quantum optimization on near-term devices by sacrificing some exactness for shallower, more robust circuits and hardware-aware transpilation. The approach has implications for applying QAOA to dense HUBOs and motivates hybrid strategies that combine approximate quadratization with exact implementations for selected terms.

Abstract

Combinatorial optimization problems have wide-ranging applications in industry and academia. Quantum computers may help solve them by sampling from carefully prepared Ansatz quantum circuits. However, current quantum computers are limited by their qubit count, connectivity, and noise. This is particularly restrictive when considering optimization problems beyond the quadratic order. Here, we introduce Ansatze based on an approximate quadratization of high-order Hamiltonians which do not incur a qubit overhead. The price paid is a loss in the quality of the noiseless solution. Crucially, this approximation yields shallower Ansatze which are more robust to noise than the standard QAOA one. We show this through simulations of systems of 8 to 16 qubits with variable noise strengths. Furthermore, we also propose a noise-aware Ansatz design method for quadratic optimization problems. This method implements only part of the corresponding Hamiltonian by limiting the number of layers of SWAP gates in the Ansatz. We find that for both problem types, under noise, our approximate implementation of the full problem structure can significantly enhance the solution quality. Our work opens a path to enhance the solution quality that approximate quantum optimization achieves on noisy hardware.

Figures (7)

• Figure 1: Circuit complexity of $e^{-i\gamma H_C}$ for $H_4^{\text{full}}$ (teal) and LABS (blue) compared to a fully-connected QUBO (purple). Panel (a) shows the number of $CZ$ gates as a function of problem size. Panel (b) shows the $CZ$ gate depth of the transpiled circuits. The continuous lines connect raw data points. The horizontal dashed line corresponds to the number of $CZ$ gates that fit within the median qubit $T_1$ time on ibm_fez.
• Figure 2: Construction of shallow Ansatze. (a) and (b) show how a four-local gate $\exp(-i\gamma ZZZZ)$ and a two-local gate $\exp(-i\gamma ZZ)$, respectively, decompose into $R_Z$ and $CX$ gates. (c) The hypergraph clique expansion generates a 2D graph that is easier to implement on quantum hardware. The graph is generated such that each pair of nodes in the original hypergraph has a corresponding edge. Here, the edge $(0,1)$ has weight $w_c = (w_a + w_b) / 2$, whereas the other edges preserve the weights of the single hyperedge they correspondingly belong to. (d) Circuit implementation of the resulting graph with layers of SWAP gates. To limit the effect of hardware noise, we skip implementing the last two SWAP layers, at the cost losing some correlation with the original problem.
• Figure 3: Convergence of the quadratizations as function of COBYLA iterations. The clique expansion, i.e., the minimization of Eq. (\ref{['eq:clique_proj']}), is shown as the dotted pink curve on the right $y$-axis. The solid blue and violet curves show the minimization of Eq. (\ref{['eqn:qaoa_proj']}) for QAOA depths one and two, respectively.
• Figure 4: Cumulative distribution function of the LABS Hamiltonian energy over $2^{14}$ samples obtained from different quantum circuits. The vertical solid line shows the minimum energy found by direct diagonalization. The dotted vertical line shows the best solution sampled from the $H_2'(\boldsymbol{\theta})$ quadratization. The full energy range of the 12 qubits LABS problem is $[-28, 110]$.
• Figure 5: Evolution of expected performance of depth-two QAOA for LABS versus strength of depolarizing noise for a system of 12 qubits, using the standard Ansatz transpiled on hardware with full connectivity (pink) and with line connectivity (blue) compared to using the quadratized Ansatz transpiled on a line of qubits (purple). The circuit corresponding to the pink line has 600 two-qubit gates, the blue 2094, and the purple 374. Each data point is the average of 10 circuit runs and the error bars represent shot noise over 10 000 samples.