Quantum Algorithm Estimates Eigenstate Properties with Reduced Error and Complexity.

A new hybrid classical-quantum algorithm estimates observable expectation values in eigenstates using a single quantum state copy. Random time evolution creates a mixed state virtually purified with exponential efficacy, achieving error suppression dependent on the Hamiltonian’s energy gap. Simulations, including a 100-qubit example, validate its applicability and inspire a tensor network-based classical approach.

Accurately determining the properties of quantum systems, particularly the expectation values of observables within specific energy eigenstates, remains a significant challenge in quantum computation. Researchers are continually seeking methods to mitigate errors and enhance the efficiency of these calculations, especially in the context of near-term and early fault-tolerant quantum devices. A new algorithm, detailed in a recent publication, offers a novel approach to this problem by employing a technique conceptually related to purification-based error mitigation, but achieving exponential error suppression with a single quantum state. This work, titled ‘Exponential distillation of dominant eigenproperties’, is the result of collaborative efforts from Bence Bakó, Tenzan Araki, and Bálint Koczor, all affiliated with the Mathematical Institute at the University of Oxford.

Hybrid Quantum Algorithm Improves Eigenstate Property Estimation

Researchers have developed a new hybrid classical-quantum algorithm to estimate the expectation values of observables – measurable physical quantities – within the eigenstates of quantum systems. This addresses a key challenge in utilising early fault-tolerant quantum computers for practical applications. The algorithm efficiently determines the average value of an observable when the quantum system is in a specific energy eigenstate.

The method begins with an initial quantum state that shares significant overlap with the target eigenstate, but may also exhibit overlap with other eigenstates. It builds upon established purification-based error mitigation techniques – methods to reduce the impact of errors in quantum computations – but distinguishes itself by achieving exponential suppression of algorithmic errors using only a single copy of the quantum state. Many existing strategies require multiple copies, increasing computational cost.

The core innovation centres on the application of random time evolution. This process generates an average mixed state – a statistical ensemble of quantum states – which then undergoes ‘virtual purification’. This mathematical procedure demonstrably enhances the accuracy of the expectation value estimation. Rigorous performance guarantees underpin the approach, establishing a quantifiable relationship between the algorithm’s computational complexity and the energy gap – the difference in energy between eigenstates – within the problem’s Hamiltonian (the operator describing the total energy of the system). This positions the algorithm competitively alongside other established hybrid quantum-classical techniques.

Extensive numerical simulations validate the algorithm’s applicability in both near-term (current, limited-scale) and early fault-tolerant quantum computing scenarios. A particularly compelling demonstration involved a 100-qubit system. In this case, direct classical simulation, utilising tensor network techniques – a method for efficiently representing and manipulating many-body quantum states – facilitated the prediction of both ground and excited state properties. This effectively mirrored the algorithm’s principles within a classical computing framework, fostering a synergistic relationship between quantum and classical computation.

Researchers are currently investigating extensions to the algorithm’s capabilities, including application to more complex quantum systems and the development of more efficient implementations. Further work focuses on improving robustness to noise and errors, and exploring potential combinations with other quantum algorithms to enhance computational power.

This development represents a notable advancement in quantum simulation, offering a promising route towards solving complex scientific and engineering problems. The work underscores the importance of interdisciplinary collaboration and continued investment in quantum research.

Recent research, particularly from 2018 to 2023, demonstrates a concentrated effort on developing and refining quantum algorithms, specifically for simulation and optimisation, driving advancements in the field.