Performant near-term quantum combinatorial optimization

TL;DR

The paper tackles near-term quantum combinatorial optimization by introducing a variational algorithm with linear-depth circuits. It uses a Hamiltonian-generator-based ansatz and a modified quantum imaginary time evolution to optimize parameters, achieving convergence to optimal solutions in MAXCUT instances. Adaptive spectrum transformations and a diagonalized cost Hamiltonian enable stable, efficient convergence, with numerical simulations and trapped-ion hardware demonstrations up to 32 qubits showing high-fidelity optima without error mitigation. The results suggest a promising, resource-minimal route toward quantum advantages in combinatorial optimization on near-term devices.

Abstract

Combinatorial optimization is a promising application for near-term quantum computers, however, identifying performant algorithms suited to noisy quantum hardware remains as an important goal to potentially realizing quantum computational advantages. To address this we present a variational quantum algorithm for solving combinatorial optimization problems with linear-depth circuits. Our algorithm uses an ansatz composed of Hamiltonian generators designed to control each term in the target combinatorial function, along with parameter updates following a modified version of quantum imaginary time evolution. We evaluate this ansatz in numerical simulations that target solutions to the MAXCUT problem. The state evolution is shown to closely mimic imaginary time evolution, and its optimal-solution convergence is further improved using adaptive transformations of the classical Hamiltonian spectrum. With these innovations, the algorithm consistently converges to optimal solutions, with interesting highly-entangled dynamics along the way. We further demonstrate the success of this approach by performing optimization of a truncated version of our ansatz with up to 32 qubits in a trapped-ion quantum computer, using measurements from the quantum computer to train the ansatz and prepare optimal solutions with high fidelity in the majority of cases we consider. The success of these large-scale quantum circuit optimizations, in the presence of realistic hardware constraints and without error mitigation, mark significant progress on the path towards solving combinatorial problems using quantum computers. We conclude our performant and resource-minimal approach is a promising candidate for potential quantum computational advantages.

Figures (5)

• Figure 1: Variational imaginary time evolution, following (\ref{['eq:ansatz_form']}) and (\ref{['eq:varit']}), closely matches exact imaginary time evolution for an example 8 vertex, fully connected graph with edge weights sampled from U(0,1) and timestep $\Delta\tau = 0.1$.
• Figure 2: Convergence of 100 random instances of 3 regular (left) and NWS graph types averaged, for a 16 vertex graphs. Also plotted is the average performance of 8 layer QAOA, taken from dupont2024.
• Figure 3: Iterations required to reach convergence of random 3 regular, and NWS graphs with vertex size 8-16
• Figure 4: Demonstration of the effect of excluding rotations below a given $\epsilon$ on a) maximum ansatz depth in terms of equivalent layers of QAOA, and b) iterations required to reach an approximation ratio of 1.
• Figure 5: (a) Entanglement entropy $S$ of a random bipartition of $N/2$ qubits during training, details in text. (b) Average maximum entropy fit by $S(N) = aN + b$, with $a=0.181 \pm 0.004$ and $b=0.35 \pm 0.05$.