Simultaneous estimation of multiple eigenvalues with short-depth quantum circuit on early fault-tolerant quantum computers

Zhiyan Ding1 and Lin Lin1,2,3

1Department of Mathematics, University of California, Berkeley, CA 94720, USA
2Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
3Challenge Institute of Quantum Computation, University of California, Berkeley, CA 94720, USA

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We introduce a multi-modal, multi-level quantum complex exponential least squares (MM-QCELS) method to simultaneously estimate multiple eigenvalues of a quantum Hamiltonian on early fault-tolerant quantum computers. Our theoretical analysis demonstrates that the algorithm exhibits Heisenberg-limited scaling in terms of circuit depth and total cost. Notably, the proposed quantum circuit utilizes just one ancilla qubit, and with appropriate initial state conditions, it achieves significantly shorter circuit depths compared to circuits based on quantum phase estimation (QPE). Numerical results suggest that compared to QPE, the circuit depth can be reduced by around two orders of magnitude under several settings for estimating ground-state and excited-state energies of certain quantum systems.

Phase estimation is one of the most important quantum primitives. This paper focuses on designing phase estimation algorithms that can simultaneously estimate ground-and excited-state energies of a Hamiltonian, which are essential for understanding the optical and electronic properties of materials.

In our paper, we introduce the multi-modal, multi-level quantum complex exponential least squares (MM-QCELS) method for estimating multiple eigenvalues of a quantum Hamiltonian. Our approach employs a simple quantum circuit with only one ancilla qubit. We prove that the circuit depth and total cost of our method satisfies the Heisenberg-limited scaling. Furthermore, with suitable initial-state conditions, our circuit depth can be significantly shorter than that of quantum phase estimation (QPE) circuits. Therefore, this method is especially suitable for early fault-tolerant quantum computers.

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