Warm Start of Variational Quantum Algorithms for Quadratic Unconstrained Binary Optimization Problems

TL;DR

The paper tackles the trainability challenges of variational quantum algorithms for QUBO problems by introducing a warm-start strategy inspired by imaginary time evolution. It combines a structure-informed ansatz (SIA) with measurement-based (WS-M) and approximation-based (WS-A) parameter initialization to bias the initial state toward low-energy configurations, improving success rates and reducing iteration counts in CVaR-VQE. Through classical simulations on up to 24 qubits, the approach shows robustness to finite-shot errors and partial mitigation of barren plateaus, with WS-M delivering near-infinite-shot performance and WS-A offering a low-cost alternative for small-time evolutions. The work suggests that problem-structured initializations can substantially enhance near-term quantum optimization, and points to extensions to larger systems, hardware experiments, and broader quantum algorithms.

Abstract

Variational Quantum Eigensolver (VQE) is widely used in near-term hardware. However, their performances remain limited by the poor trainability and are dependent on random parameter initialization. In this work, we propose a warm start method inspired by imaginary time evolution, allowing for determining initial parameters that prioritize lower energy states in a resource-efficient way. Using classical simulations, we demonstrate that this warm start method significantly improves the success rate and reduces the number of iterations required for the convergence of VQE. The numerical results also indicate that the warm start approach effectively mitigates statistical errors arising from a finite number of measurements, and to a certain extent alleviates the effect of barren plateaus.

Figures (13)

• Figure 1: Fidelity of the warm start state that mimics ITE with $\tau = 0.2$. The yellow diamonds represent the initial state following the WS-M technique using a state vector simulation, corresponding to an infinite number of measurements when estimating the expectation values. The green dots correspond to the data obtained using 1000 measurements for estimating each of the local Pauli operators. For each system size, the fidelity of 100 random QUBO problems are presented, sorted in ascending order and slightly offset on the $x$-axis for viewer convenience. The dashed line connects the minimum fidelity at each system size to guide the eye. The gray line shows the fidelity between the solution and the uniform superposition.
• Figure 2: (a) Success rate of CVaR-VQE with $\alpha = 0.01$ and (b) the averaged iterations to achieve the fidelity threshold $1\%$, with the errorbar representing the standard error. In both panels, the green diamond represents the data using the WS-M with $\tau = 0.2$. For comparison, the blue dots show the outcomes when the VQE is initialized with a uniform superposition, i.e., all ansatz parameters are set to zero. The number of measurements for the warm start parameter estimation is 1000, and 10000 measurements are used for estimating the CVaR value in each VQE iteration for both cases, with or without the warm start.
• Figure 3: Comparison of warm start fidelity for 20 qubits QUBO problems. The fidelity of the WS-M method introduced in Sec. \ref{['sec: warm-start-general']}, is presented by the yellow diamond for state vector simulation and green dots for simulation with 1000 measurements. The fidelity resulting from the WS-A method is denoted by blue crosses. The gray line represents the fidelity of the uniform superposition, which is $1/2^{20}$.
• Figure 4: Success rate of CVaR-VQE with $\alpha = 0.01$ initiated with different warm start parameters for 20 qubits. Each VQE iteration involved 10,000 measurements for estimating CVaR across all three warm start approaches.
• Figure 5: Scaling of the success rate with number of measurements of the CVaR-VQE with $\alpha=0.01$ (a) and relative standard error (b) for 18 qubits with averaged over 100 QUBO instances. The $x$-axis represents different numbers of measurements taken for determining the cost function in each iteration, ranging from 2000 to 150000. The blue dots represent the data for the VQE with all parameters in SIA-YZ ansatz initialized as zero, the green diamonds the results for initializing the ansatz using the warm start parameters. The grey dashed line in panel (a) indicates a success rate $100\%$.