Spectral Subspace Extraction Via Incoherent Quantum Phase Estimation Enables Density of States Analysis for Many-body Systems

Quantum phase estimation, a fundamental technique for determining the energy levels of complex systems, traditionally focuses on individual states and demands precise input preparation. Researchers Stefano Scali and Josh Kirsopp, both from Fujitsu Research of Europe Ltd., along with Antonio Márquez Romero and Michał Krompiec, present a new approach that estimates the overall distribution of energy levels, known as the density of states. This advancement, termed DOS-QPE, builds upon previous work and introduces a streamlined circuit design coupled with sophisticated data analysis methods. The team demonstrates that DOS-QPE unlocks access to crucial thermodynamic properties and spectral features, offering a powerful tool for simulating complex systems in areas such as materials science and nuclear physics, and paving the way for more efficient quantum simulations even before fully fault-tolerant quantum computers become available.

Individual eigenstates and their preparation present significant challenges for Quantum Phase Estimation. To address these limitations, researchers adopt an ensemble-based formulation of Quantum Phase Estimation that estimates the density of states of the Hamiltonian governing the evolution. This approach, termed DOS-QPE, builds upon a previously introduced formulation, presenting it as a circuit primitive extended with symmetry-adapted input ensembles and advanced spectrum reconstruction techniques. This variant of Quantum Phase Estimation enables access to thermodynamic properties, symmetry-resolved spectral functions, and features relevant to quantum many-body systems.

Variational and Phase Estimation Algorithms for Nuclear Physics

This text details a rapidly growing field: the application of quantum computing to solve problems in nuclear physics, quantum chemistry, and related areas. The most widely explored algorithm is the Variational Quantum Eigensolver, used to find the ground state energies of complex systems, with ongoing research focused on adaptation and improvement. Quantum Phase Estimation is a powerful algorithm for determining energy eigenvalues, but current research focuses on making it practical for near-term devices by shortening circuits and improving methods. Quantum Subspace Expansion calculates dynamical response functions, potentially useful for nuclear dynamics, while Hamiltonian Simulation is fundamental to many quantum algorithms, with research focused on efficient Hamiltonian representation.

Quantum Machine Learning techniques, such as kernel methods and neural networks, are being explored to accelerate calculations and extract information, and Randomized Measurement Toolbox techniques extract more information from limited quantum measurements. The primary focus of applications in nuclear physics is nuclear structure calculations, where researchers use quantum algorithms to calculate the ground state energies, excited states, and properties of atomic nuclei, potentially solving the Nuclear Shell Model for larger nuclei than currently possible. Scientists are also calculating Nuclear Interactions, determining the forces between nucleons, and exploring Nuclear Excited States, calculating their energies and properties, which are important for understanding nuclear reactions, as well as simulating Nuclear Reactions, modeling Nuclear Dynamics, predicting Neutron Drip Lines, and calculating Nuclear Resonances. Near-term quantum devices have limited coherence times and can only execute relatively shallow circuits, so research focuses on developing algorithms that minimize circuit depth, utilizing Error Mitigation techniques to reduce the impact of errors, and employing Efficient Hamiltonian Representation methods to minimize qubit and gate requirements.

Algorithm Optimization improves efficiency for specific problems, and Hybrid Quantum-Classical Approaches combine the strengths of both computing paradigms, exploiting symmetries to reduce computational complexity. Many of the same quantum algorithms and techniques used in nuclear physics are also applicable to Quantum Chemistry, creating significant cross-fertilization between these fields. Quantum computing is also being used to study strongly correlated electron systems relevant to both Condensed Matter Physics and nuclear physics, and to design new Materials with desired properties. Researchers are using Quantum Machine Learning for Nuclear Physics to accelerate calculations and extract insights from quantum simulations, exploiting Topological Properties to improve algorithm efficiency, developing New Quantum Algorithms tailored to the challenges of nuclear physics, and leveraging Advanced Quantum Hardware, such as increased qubit counts and improved coherence times.

Density of States Estimation for Quantum Simulation

Scientists have developed a novel approach to quantum phase estimation, termed Density of States Quantum Phase Estimation, or DOS-QPE, which estimates the density of states of a Hamiltonian rather than individual quantum states. This work presents DOS-QPE as a fundamental circuit primitive, enhanced with input ensembles adapted to symmetry and advanced spectrum reconstruction techniques, enabling access to thermodynamic properties and spectral functions relevant to complex systems. Experiments utilizing fermionic models and nuclear Hamiltonians demonstrate the potential of DOS-QPE for early fault-tolerant simulations in spectroscopy, electronic structure, and nuclear theory. Achieving a target error requires a minimum number of repetitions, dependent on the dimension of the subspace, and utilizing maximally mixed states allows probing all Hamiltonian eigenvalues with equal probability, estimating spectral resolution and leveraging concepts from garbage state preparation.

To enforce particle-number symmetry, scientists employed Dicke states, specific quantum states with a fixed number of particles, and a fermion-qubit encoding that maps fermionic states to qubits conserving Hamming weight. Preparing Dicke states with deterministic circuits can be achieved with specific depths on devices with all-to-all connectivity, eliminating the need for preparing trial fermionic eigenstates, preserving particle-number symmetry and reducing computational cost. Spectrum reconstruction involves sampling from the normalized density of states and recovering both the continuous eigenvalue locations and their degeneracies from a discretized histogram, with the reconstruction error scaling with the dimension of the subspace and inversely proportional to the number of QPE repetitions, delivering richer information about the spectrum.

Density of States Estimation via Quadratic Programming

Scientists have developed a novel approach to quantum phase estimation, termed Density of States Quantum Phase Estimation, or DOS-QPE, which estimates the density of states of a Hamiltonian rather than individual quantum states. This work presents DOS-QPE as a fundamental circuit primitive, enhanced with input ensembles adapted to symmetry and advanced spectrum reconstruction techniques, enabling access to thermodynamic properties and spectral functions relevant to complex systems. Experiments utilizing fermionic models and nuclear Hamiltonians demonstrate the potential of DOS-QPE for early fault-tolerant simulations in spectroscopy, electronic structure, and nuclear theory, framing the reconstruction of spectral features as a quadratic program solved efficiently using compressed sensing techniques and developing a modular post-processing method to restore physically meaningful integer-value spectra. While the current implementation successfully captures spectral information, future research directions include reducing circuit complexity and resource costs to make the primitive viable for near-term quantum hardware, exploring alternative probe states and spectrum manipulation techniques to enhance signal contrast and mitigate noise, and establishing potential connections to broader algorithmic frameworks, including hybrid optimization, quantum learning, and Green’s function formulations.