Computing Ground State Properties with Early Fault-Tolerant Quantum Computers

Abstract

Significant effort in applied quantum computing has been devoted to the problem of ground state energy estimation for molecules and materials. Yet, for many applications of practical value, additional properties of the ground state must be estimated. These include Green's functions used to compute electron transport in materials and the one-particle reduced density matrices used to compute electric dipoles of molecules. In this paper, we propose a quantum-classical hybrid algorithm to efficiently estimate such ground state properties with high accuracy using low-depth quantum circuits. We provide an analysis of various costs (circuit repetitions, maximal evolution time, and expected total runtime) as a function of target accuracy, spectral gap, and initial ground state overlap. This algorithm suggests a concrete approach to using early fault tolerant quantum computers for carrying out industry-relevant molecular and materials calculations.

State Preparation Boosters for Early Fault-Tolerant Quantum Computation

“…Alternatively, one can use the method in Ref. [31] to estimate this quantity without explicitly preparing the ground state. It would be interesting to know which approach is more efficient in practice.…”

Qubit-Efficient Randomized Quantum Algorithms for Linear Algebra

“…Despite having no qubit overhead to implement quantum oracles, the asymptotic complexities of our algorithms can remain comparable with other algorithms in the literature whenever the considered matrices have an amenable structure in the Pauli basis (and when considering equivalent end-to-end problems). Hence, for physically motivated matrices, potential quantum advantages originally requiring QRAM could possibly be similarly obtained in our approach without using a quantum data structure, making them more applicable for the early fault-tolerant regime [26][27][28][29][30][31]. Specifically, given a Fourier series approximation to a function f , and an N × N Hermitian matrix A with known decomposition in the Pauli basis, we give algorithms to sample properties of f (A) using a total of log(N ) + 1 qubits.…”

“…Applications for the early fault-tolerant era of quantum computing have recently begun to be explored, following the motivation to design algorithms that extract practical value out of fault-tolerant quantum algorithms as soon as possible [27][28][29][30][31][32][33]. In this spirit, algorithms have been designed to consume fewer quantum resources for Hamiltonian problems including phase estimation [27][28][29][30], ground state preparation [29], and computing ground state properties [31], by increasing the number of circuit samples required. Until now, these algorithms have predominantly aimed to reduce a proxy for the maximum circuit depth, in the form of the number of calls to a time evolution oracle for a prescribed Hamiltonian in one coherent run of a circuit.…”

“…Different approaches to quantum algorithms for ground state preparation have been previously studied in Refs. [29,31,[53][54][55][56][57]. Recently, Ref.…”

TFermion: A non-Clifford gate cost assessment library of quantum phase estimation algorithms for quantum chemistry

“…While it is known that the general 2-body ground state preparation is QMA-complete, there is hope that the specificity of electronic Hamiltonians will make it easier to solve at least heuristically. In fact, over the years significant effort has been devoted to the formulation of shallow-depth NISQ ansätze [76] to prepare ground states such as the Imaginary Time Evolution ansatz [47] and the Variational Quantum Eigensolver (VQE) with Unitary Coupled-Cluster [54], adaptative [29], and hardware-efficient [32] ansätze.…”