Off-Grid Compressed Sensing Enables Heisenberg-Limited Estimation of Multiple Eigenvalues

Precise phase estimation lies at the heart of many quantum simulation algorithms, and researchers are continually seeking ways to improve its efficiency and accuracy. Davide Castaldo, from the University of Padova and ETH Zurich, along with Stefano Corni and colleagues at the University of Padova’s Department of Chemical Sciences, Padua Quantum Technologies Research Center, and the CNR Institute of Nanosciences, now demonstrate a new approach to simultaneously estimating multiple eigenvalues of a unitary matrix. Their work introduces an ‘off-grid’ compressed sensing protocol, combined with advanced signal classification, that achieves the Heisenberg limit , a fundamental benchmark in precision measurement , while analysing only a small fraction of the necessary data. This breakthrough not only significantly reduces the computational resources required for accurate phase estimation, but also offers improved resilience to imperfections in the initial quantum state, paving the way for more robust and efficient quantum simulations.

Reducing Quantum Simulation Resource Requirements

Quantum computing promises revolutionary advances in fields like materials science and drug discovery, but realizing this potential requires overcoming significant hurdles. A central challenge lies in the sheer computational power needed to simulate quantum systems accurately. Researchers are actively exploring ways to reduce these resource requirements, focusing on algorithms that minimize circuit complexity and overall circuit depth. A promising approach draws a parallel between quantum simulation and signal recovery, a technique used in classical signal processing. This connection allows scientists to frame the problem of estimating the energy levels of a quantum system as reconstructing a signal from limited data.

By exploiting the natural sparsity often found in the spectral decomposition of quantum states, researchers aim to significantly reduce the computational burden. This work builds upon recent advances in compressed sensing, a signal processing technique that allows accurate reconstruction of sparse signals using far fewer samples than traditionally required. The team has developed a novel algorithm that achieves performance approaching the Heisenberg limit, while requiring remarkably short computation times. Furthermore, the researchers have introduced an innovative “off-grid” recovery algorithm that is both computationally efficient and robust to noise. Importantly, the team also demonstrates the algorithm’s resilience to imperfections in the initial preparation of the quantum state, a crucial factor for building reliable and scalable quantum simulations. This combination of improvements brings the prospect of practical quantum advantage significantly closer to reality.

Quantum Phase Estimation via Compressed Sensing

Researchers have developed a novel approach to quantum phase estimation, a crucial algorithm for simulating quantum systems on future computers. This method tackles the challenge of resource-intensive calculations by drawing a parallel between quantum computation and signal recovery techniques used in classical signal processing. The team recognised that accurately determining the energy levels of a quantum system is akin to reconstructing a signal, and they leveraged this insight to dramatically reduce the computational demands. The core of this innovation lies in the application of compressed sensing, a technique that allows accurate reconstruction of signals from far fewer samples than traditionally required.

By identifying and focusing on crucial elements, the researchers can reconstruct the signal, and therefore estimate the quantum system’s energy levels, with significantly fewer measurements. This is particularly effective in quantum systems where the distribution of energy levels often exhibits natural sparsity. This new algorithm extends previous work by enabling the simultaneous estimation of multiple energy levels, rather than just a single one. The team developed an efficient protocol that not only achieves this multi-level estimation but also saturates the Heisenberg limit. Furthermore, the method demonstrates resilience to imperfections in the initial quantum state, suggesting it could be more robust in practical applications.

This approach moves beyond simply reducing the number of measurements; it also aims to minimise the complexity of the quantum circuits required. By reducing both the width and depth of these circuits, the researchers hope to bring the prospect of practical quantum advantage closer to reality. The algorithm’s efficiency stems from a carefully designed protocol that exploits the inherent structure of quantum systems and applies advanced signal processing techniques.

Efficient Energy Estimation via Signal Recovery

Researchers have developed a new method for estimating the energies of quantum systems, particularly relevant for simulating molecules and materials. This approach reframes the problem of energy estimation as a signal recovery task, allowing for more efficient algorithms. The core innovation lies in combining compressed sensing with a sophisticated signal classification method to extract multiple energy levels simultaneously. The process begins by measuring the system’s autocorrelation function using a quantum processor. However, instead of requiring a vast amount of data, this new method achieves accurate estimations with remarkably few samples, a significant advantage for current and near-future quantum computers.

The team demonstrates that their algorithm operates at the Heisenberg limit, even in complex scenarios with strong interactions between particles. Importantly, the algorithm exhibits resilience to imperfections in the initial state preparation, a practical concern for real-world quantum computations. The researchers show that even with approximate initial states, the method can still accurately estimate energies, enhancing its feasibility for complex systems. Furthermore, the algorithm’s efficiency stems from an “off-grid” approach to signal recovery, avoiding computational bottlenecks. The team’s results indicate that the algorithm outperforms standard quantum phase estimation in terms of computational scaling, achieving comparable accuracy with significantly reduced time requirements. They demonstrate the ability to simultaneously recover both ground and excited state energies, a crucial capability for understanding molecular behavior. This advancement opens new possibilities for simulating complex quantum systems and designing novel materials.

Eigenvalue Estimation Nears Heisenberg Limit

The research presents a novel algorithm for estimating multiple eigenvalues of a quantum system, crucial for simulating complex phenomena in fields like quantum chemistry. The core of this method lies in combining an ‘off-grid’ compressed sensing protocol with a state-of-the-art signal classification technique. This approach allows for the simultaneous estimation of multiple eigenvalues using a relatively small number of measurements. Numerical results demonstrate that the algorithm achieves performance approaching the Heisenberg limit, both in strongly and weakly correlated systems, and requires short evolution times to achieve accurate estimates.

Furthermore, the team developed an improved ‘off-grid’ protocol that incorporates prior knowledge of the underlying signal, enhancing both the speed and accuracy of the recovery process. The study also investigated the algorithm’s resilience to imperfections in the initial input state, a critical consideration for practical implementation. The algorithm’s efficiency stems from an ‘off-grid’ approach to signal recovery, avoiding computational bottlenecks. Future work could focus on further improvements to the algorithm and exploring its application to specific problems in quantum chemistry, potentially demonstrating a quantum advantage for simulating molecular excitation energies.