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Quantum inverse iteration algorithm for programmable quantum simulators

Oleksandr Kyriienko

Published Aug 1, 2025• 23 citations
OpenAlex

Abstract

We propose a quantum inverse iteration algorithm which can be used to estimate the ground state properties of a programmable quantum device. The method relies on the inverse power iteration technique, where the sequential application of the Hamiltonian inverse to an initial state prepares an approximate groundstate. To apply the inverse Hamiltonian operation, we write it as a sum of unitary evolution operators using the Fourier approximation approach. This allows to reformulate the protocol as separate measurements for the overlap of initial and propagated wavefunction. The algorithm thus crucially depends on the ability to run Hamiltonian dynamics with an available quantum device. We benchmark the performance using paradigmatic examples of quantum chemistry, corresponding to molecular hydrogen and beryllium hydride. Finally, we show its use for studying the ground state properties of relevant material science models which can be simulated with existing devices, considering an example of the Bose-Hubbard atomic simulator.

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arXiv:1204.3600•The Journal of Chemical Physics•Dec 21, 2012•30 citations
We propose a quantum inverse iteration algorithm which can be used to estimate the ground state properties of a programmable quantum device. The method relies on the inverse power iteration technique, where the sequential application of the Hamiltonian inverse to an initial state prepares an approximate groundstate. To apply the inverse Hamiltonian operation, we write it as a sum of unitary evolution operators using the Fourier approximation approach. This allows to reformulate the protocol as separate measurements for the overlap of initial and propagated wavefunction. The algorithm thus crucially depends on the ability to run Hamiltonian dynamics with an available quantum device. We benchmark the performance using paradigmatic examples of quantum chemistry, corresponding to molecular hydrogen and beryllium hydride. Finally, we show its use for studying the ground state properties of relevant material science models which can be simulated with existing devices, considering an example of the Bose-Hubbard atomic simulator.
Abstract We propose a quantum inverse iteration algorithm, which can be used to estimate ground state properties of a programmable quantum device. The method relies on the inverse power iteration technique, where the sequential application of the Hamiltonian inverse to an initial state prepares the approximate ground state. To apply the inverse Hamiltonian operation, we write it as a sum of unitary evolution operators using the Fourier approximation approach. This allows to reformulate the protocol as separate measurements for the overlap of initial and propagated wavefunction. The algorithm thus crucially depends on the ability to run Hamiltonian dynamics with an available quantum device, and can be used for analog quantum simulators. We benchmark the performance using paradigmatic examples of quantum chemistry, corresponding to molecular hydrogen and beryllium hydride. Finally, we show its use for studying the ground state properties of relevant material science models, which can be simulated with existing devices, considering an example of the Bose-Hubbard atomic simulator.
Quantum simulation provides a powerful route for exploring many-body phenomena beyond the capabilities of classical computation. Existing approaches typically proceed in the forward direction: a model Hamiltonian is specified, implemented on a programmable quantum platform, and its phase diagram and properties are explored. Here we present a quantum algorithmic framework for inverse quantum simulation, enabling quantum material design with desired properties. Target material characteristics are encoded as a cost function, which is minimized on quantum hardware to prepare a many-body state with the desired properties in quantum memory. Hamiltonian learning is then used to reconstruct a low-energy Hamiltonian for which this state is an approximate ground state, yielding a physically interpretable model that can guide experimental synthesis. As illustrative applications, we outline how the method can be used to search for high-temperature superconductors within the fermionic Hubbard model, enhancing ddd-wave correlations over a broad range of dopings and temperatures, design quantum phases by stabilizing a topological order through continuous Hamiltonian modifications, and optimize dynamical properties relevant for photochemistry and frequency- and momentum-resolved condensed-matter data. These results extend the scope of quantum simulators from exploring quantum many-body systems to designing and discovering new quantum materials.
Accurately determining ground-state properties of quantum many-body systems remains one of the major challenges of quantum simulation. In this work, we present a protocol for estimating the ground-state energy using only global time evolution under a target Hamiltonian. This avoids the need for controlled operations that are typically required in conventional quantum phase estimation and extends the algorithm applicability to analog simulators. Our method extracts energy differences from measurements of the Loschmidt echo over an initial ground-state approximation, combines them with direct energy measurements, and solves a set of equations to infer the individual eigenenergies. We benchmark this protocol on free-fermion systems, showing orders-of-magnitude precision gains over direct energy measurements on the initial state, with accuracy improving rapidly with initial-state fidelity and persisting for hundreds of modes. We further demonstrate applicability to the 2D Ising and Fermi-Hubbard models and show that the approach extends naturally to other observables such as order parameters. Finally, we analyze the effect of experimental imperfections and propose error-mitigation strategies. These results establish a practical route to compute physically relevant quantities with high precision using globally controlled quantum simulators.
Abstract In this chapter, we present quantum algorithms to approximate Hamiltonian eigenstates. We start discussing the quantum phase estimation algorithms, an important subroutine that allows us to approximate eigenpairs of a Hamiltonian operator given an initial guess and a quantum circuit implementing the time evolution operator. We then present the adiabatic state preparation, a technique to transform the ground state of a Hamiltonian into the ground state of another Hamiltonian. We describe two important heuristic methods, the quantum approximate optimization algorithm, and the variational quantum eigensolver. The second part of the chapter discusses concrete applications in the field of quantum chemistry. After introducing the Born-Oppenheimer Hamiltonian in second quantization and the Hartree-Fock method, we illustrate how a chemical problem can be mapped onto a quantum computer. We conclude the chapter with the simulation of the hydrogen molecule in a minimal basis with the iterative phase estimation algorithm.
Abstract Imaginary-time evolution plays an important role in many areas of quantum physics and has been widely applied to the ground-state determination of various Hamiltonian in the quantum computation field. In this work, we propose an iteration-free quantum algorithm in a full gate-based frame using the approximate unitary expansion to simulate the imaginary-time evolution operator, avoiding the resource overhead caused by repeated measurement for state reconstruction or complex pre-calculations in the classical computers. We detail the algorithm and analyze the complexity and related characteristics including a lower bound for ancillary qubits at a given success probability. Then an application demonstration of the algorithm in quantum chemistry with hydrogen molecule under noiseless and noisy conditions is offered. In addition, we present another imaginary-time evolution simulation method based on similar construction schemes. Our algorithms can serve as the alternative proposals for the imaginary-time evolution realization in the future fault-tolerant quantum computers.
Unlike fixed designs, programmable circuit designs support an infinite number of operators. The functionality of a programmable circuit can be altered by simply changing the angle values of the rotation gates in the circuit. Here, we present a new quantum circuit design technique resulting in two general programmable circuit schemes. The circuit schemes can be used to simulate any given operator by setting the angle values in the circuit. This provides a fixed circuit design whose angles are determined from the elements of the given matrix-which can be non-unitary-in an efficient way. We also give both the classical and quantum complexity analysis for these circuits and show that the circuits require a few classical computations. For the electronic structure simulation on a quantum computer, one has to perform the following steps: prepare the initial wave function of the system; present the evolution operator U = e(-iHt) for a given atomic and molecular Hamiltonian H in terms of quantum gates array and apply the phase estimation algorithm to find the energy eigenvalues. Thus, in the circuit model of quantum computing for quantum chemistry, a crucial step is presenting the evolution operator for the atomic and molecular Hamiltonians in terms of quantum gate arrays. Since the presented circuit designs are independent from the matrix decomposition techniques and the global optimization processes used to find quantum circuits for a given operator, high accuracy simulations can be done for the unitary propagators of molecular Hamiltonians on quantum computers. As an example, we show how to build the circuit design for the hydrogen molecule.