Dual-VQE: A quantum algorithm to lower bound the ground-state energy

TL;DR

This work introduces dual-VQE, a quantum algorithm that yields a lower bound on the ground-state energy by leveraging SDP duality and a penalty-based reformulation of the optimization problem. Paired with VQE, dual-VQE provides a sandwich of bounds that can serve as a quality check for near-term quantum computations, without requiring exact classical solutions. The approach uses a parameterized mixed-state $\omega(\theta)$, with either a purification or convex-combination ansatz, and estimates key terms via destructive swap tests and sampling; moreover, matrix-product-state pretraining is proposed to mitigate barren plateaus and effectively warm-start the optimization. Numerical experiments on transverse-field Ising models show that dual-VQE can approach the true energy with errors on the order of $10^{-2}$ without pretraining and around $0.5\%$ relative error with MPS pretraining, highlighting its potential as a practical lower-bound verifier for quantum hardware results.

Abstract

The variational quantum eigensolver (VQE) is a hybrid quantum-classical variational algorithm that produces an upper-bound estimate of the ground-state energy of a Hamiltonian. As quantum computers become more powerful and go beyond the reach of classical brute-force simulation, it is important to assess the quality of solutions produced by them. Here we propose a dual variational quantum eigensolver (dual-VQE) that produces a lower-bound estimate of the ground-state energy. As such, VQE and dual-VQE can serve as quality checks on their solutions; in the ideal case, the VQE upper bound and the dual-VQE lower bound form an interval containing the true optimal value of the ground-state energy. The idea behind dual-VQE is to employ semi-definite programming duality to rewrite the ground-state optimization problem as a constrained maximization problem, which itself can be bounded from below by an unconstrained optimization problem to be solved by a variational quantum algorithm. When using a convex combination ansatz in conjunction with a classical generative model, the quantum computational resources needed to evaluate the objective function of dual-VQE are no greater than those needed for that of VQE. We also show that the problem is well suited for classical pretraining using matrix product states and these methods help warm-start the optimization. We simulated the performance of dual-VQE on the transverse-field Ising model with and without pretraining and found that, for the example considered, while dual-VQE training is slower and noisier than VQE, it approaches the true value with an error of order $10^{-2}$.

Figures (8)

• Figure 1: (a) Realizing mixed states using MPSs. The red tensors are the reference qubits, and the blue tensors are the system qubits. (b) Tensor network for estimating $\operatorname{Tr}[H\omega]$. The tensors in yellow represent the Pauli strings in the decomposition of $H$. (c) Tensor network for estimating $\operatorname{Tr}[\omega^2]$.
• Figure 2: Convergence of dual-VQE and comparison to standard VQE for a two-qubit example problem instance. The solid line shows the median value of the estimate, the shaded region represents the interquartile range, while the ground truth is marked by the dashed line. We were able to achieve an error of $10^{-2}$ after $20,000$ iterations of training.
• Figure 3: Objective function and penalty value for the Ising model problem using dual-VQE and MPS pretraining. After 2,000 iterations of pretraining followed by 25,000 iterations of quantum training, the true solution is reached when $\chi_{\max} = 8$, with a relative error of 0.5%.
• Figure 4: Fidelity of translation between the final MPS state after pretraining and the translated quantum circuit.
• Figure 5: (a) Representation of a pure state in tensor-network notation. (b) Alternate representation of a pure state in tensor-network notation.

Theorems & Definitions (2)

• Theorem 1: Dual-VQE lower bound
• Remark 2