Scalable quantum circuit generation for iterative ground state approximation using Majorana Propagation

Abstract

We introduce the Adaptive Derivative-Assembled Pseudo-Trotter ansatz Variational Majorana Propagation Eigensolver (ADAPT-VMPE), a quantum-inspired classical algorithm that exploits Majorana Propagation (MP) to produce circuits for approximating the ground state of molecular Hamiltonians. Equipped with the theoretical guarantees of MP, which provide controllable bounds on the approximation error, ADAPT-VMPE offers an efficient and scalable approach for iterative ansatz construction. A theoretical analysis of the computational complexity demonstrates that it is polynomial in both the number of qubits and the number of iterations. We present an in-depth analysis of circuit construction strategies, analyzing their impact on convergence and provide practical guidance for efficient ansatz generation. Using ADAPT-VMPE, we construct up to 100-qubit ansätze for a strongly correlated photosensitizer currently undergoing human clinical trials for cancer treatment. Our results demonstrate that constant overlap with the ground state across system sizes can be reached in polynomial time with polynomially sized circuits.

Figures (9)

• Figure 1: Overview of the ADAPT-VMPE algorithm for constructing a fermionic ansatz from a molecular Hamiltonian. The algorithm is initialized with Hartree–Fock state, followed by active rotations. A Fermionic-local operator pool is defined, and the method proceeds iteratively. At each iteration, operators in the pool are evaluated using GGF or gradient selection, and the selected operator is appended to the circuit in either the Schrödinger or Heisenberg representation. Pool trimming may be applied to reduce the number of candidate operators. After each addition, all variational parameters are optimized before the next iteration. The whole pipeline works in polynomial time thanks to using Majorana Propagation in gate selection and parameter optimization.
• Figure 2: ADAPT-VQE simulations comparing the different strategic choices for the iterative ansatz generation procedure. (a) ADAPT-VQE simulations performed with different pools for 16 qubit H-chain molecule. (b) and (c) ADAPT-VQE simulations run with Majorana Propagation simulator performed in different settings for 16 qubit H-chain (b) and 40 qubit Complex A (c). Heisenberg and Schrödinger versions have been considered with and without active rotations. The whole ansatz is optimized at each step except the 'HE w/o AR w/o opt.' where no optimization is performed at all. In all simulations, the GGF selection method is used for gate selection, and the L-BFGS-B algorithm is used for optimization. (d) ADAPT-VQE simulation run with Majorana Propagation simulator for 40 qubit Complex A. The Heisenberg version has been considered with active rotations. The whole ansatz is optimized at each iteration using the L-BFGS-B algorithm.
• Figure 3: Heisenberg ADAPT-VMPE simulations performed for active space sizes ranging between 28 and 100 qubits for Complex A. The MP cutoff is set to 6 for all simulations. A pool reduction strategy is applied, combining GGF selection with faster gradient selection as described in the Methods section. Active rotations are included in the ansatz, and the whole ansatz is optimized after adding each gate using the L-BFGS-B optimizer. In (a), the energy error is plotted with respect to the energy computed by DMRG. In (b), the scaling of computation time with the number of qubits is presented. The cumulative time scales polynomially for all ADAPT iteration counts considered.
• Figure 4: Energy error vs resource cost of ADAPT-VMPE outcomes transpiled for NISQ and FT eras. (a) all-to-all connectivity minimizing 2-qubit gates for the NISQ era, (b) ibm_fez heavy-hexagonal connectivity minimizing 2-qubit gates for the NISQ era, and (c) fault-tolerant scenario minimizing T count.
• Figure 5: Analysis of overlap of the produced states with respect to the ground state for Complex A. Subfigure (a) shows that the lower bounds on the overlap increase monotonically with iterations. For reference, the dashed lines represent the exact overlaps of the Hartree-Fock determinant and the ground state computed with DMRG. In (b), we compare the scaling of the number of Fermionic gates for ADAPT-VMPE to reach 70% and 85% overlap for various active spaces.

Theorems & Definitions (9)

• Theorem 1
• proof
• Theorem 2
• proof
• Definition 1
• Theorem 3
• proof
• Theorem 4
• proof