Quantum Algorithm Unlocks Accurate Modelling of Unstable, Decaying Systems

Durgesh Pandey and colleagues at Indian Institute of Technology Roorkee present a new approach to the Variational Quantum Eigensolver (VQE) algorithm, addressing a key limitation of current methods. Existing quantum algorithms struggle with non-Hermitian matrices, which are essential for describing open quantum systems exhibiting decay and resonances. The Real Variance-based Variational Quantum Eigensolver (RVVQE) correctly identifies eigenstates for these matrices using only standard, easily implemented Hermitian measurements. The team’s numerical results demonstrate the algorithm’s performance and scalability, enabling more accurate simulations of complex quantum phenomena.

RVVQE algorithm surpasses limitations of Hermitian-based methods in determining complex eigenvalues

Numerical simulations utilising the new Real Variance-based Variational Quantum Eigensolver (RVVQE) achieved a 99.9% convergence rate to true eigenstates. This represents a substantial improvement over existing methods, which typically fail to converge beyond 60% for comparable non-Hermitian matrices. The 99.9% threshold is key, allowing accurate determination of complex eigenvalues previously impossible with standard Variational Quantum Eigensolver (VQE) algorithms designed for Hermitian operators. The significance of this improvement lies in the ability to accurately model systems where energy is not conserved, a common feature in many real-world quantum scenarios.

These algorithms struggle with open quantum systems exhibiting decay and resonances, as they are designed for stable states. Open quantum systems, by definition, interact with their environment, leading to dissipation of energy and the loss of coherence. This interaction is mathematically represented by non-Hermitian operators, where the eigenvalues are complex numbers. The real part of the eigenvalue corresponds to the energy of the state, while the imaginary part represents the decay rate or the width of the resonance. Standard VQE, relying on Hermitian operators, cannot directly handle these complex eigenvalues, leading to inaccurate or non-convergent results. The RVVQE’s success stems from a cost function based on variance minimisation, guaranteeing convergence while utilising only readily measurable Hermitian components, simplifying implementation on current quantum hardware. The RVVQE algorithm addresses the incompatibility of standard quantum algorithms with non-Hermitian operators, enabling the simulation of complex quantum phenomena.

A cost function guaranteeing convergence to true eigenstates was identified, utilising only Hermitian measurements. This cost function is based on minimising the real part of the variance of the non-Hermitian operator. Mathematically, the variance of a non-Hermitian operator M is written as ∆(M) = ⟨M†M⟩−|⟨M⟩|², where ⟨…⟩ denotes the expectation value with respect to a given quantum state and M† is the Hermitian conjugate of M. By minimising this quantity, the algorithm effectively isolates the eigenstates of the non-Hermitian operator. Numerical results demonstrate the algorithm’s performance and scalability on dense non-Hermitian matrices of increasing dimension. Minimising the real part of the variance provides a mathematically sound cost function for identifying eigenstates, regardless of whether the spectrum is real or complex. Requiring only Hermitian measurements makes the method suitable for implementation on current quantum hardware, as these measurements are natively supported by most quantum computing platforms.

A new algorithm successfully recovers complex eigenvalues with accuracy, addressing a long-standing challenge in modelling open quantum systems. Standard quantum algorithms, including the Variational Quantum Eigensolver, are ineffective for non-Hermitian operators describing these systems, as they are designed for stable, unchanging states. Open quantum systems, interacting with their environment and exhibiting behaviours like decay, are central to advances in areas from nuclear physics to drug delivery. Understanding the dynamics of these systems requires accurate knowledge of their complex energy eigenvalues, which dictate the rates of decay and the characteristics of resonances.

The algorithm utilises a Real Variance-based Variational Quantum Eigensolver, employing a cost function that guarantees convergence to the true eigenstates. Its implementation relies on Hermitian measurements, making it suitable for existing quantum hardware. Performance was demonstrated on a series of non-Hermitian matrices, with computational metrics confirming its scalability. Accurate computation of complex eigenvalues is essential for several domains, including quantum chemistry and scattering theory, and the approach defines a hybrid quantum-classical workflow for extracting complex spectra using standard quantum hardware. In quantum chemistry, this allows for the modelling of chemical reactions in realistic environments, including solvent effects and interactions with light. In scattering theory, it enables the calculation of scattering cross-sections, crucial for understanding particle collisions and nuclear reactions.

Guaranteeing convergence for open quantum system eigenstates using readily measurable quantities

This work establishes a new computational approach for simulating open quantum systems, those which interact with their environment and do not conserve energy. Researchers at Indian Institute of Technology Roorkee have overcome a key limitation of existing quantum algorithms unable to accurately model these complex systems by introducing the RVVQE. It guarantees convergence to true eigenstates by utilising only standard, measurable properties, simplifying implementation on quantum computers. The variance of a non-Hermitian operator M is written as ∆(M) = ⟨M†M⟩−|⟨M⟩|². This formulation is crucial because it transforms the problem of finding complex eigenvalues into a problem of minimising a real-valued function, which can be efficiently handled by classical optimisation algorithms within the VQE framework.

The RVVQE algorithm operates within the established paradigm of hybrid quantum-classical computation. A quantum computer is used to prepare trial wavefunctions and measure the expectation value of the Hermitian operator M†M. These measurements are then fed into a classical optimiser, which adjusts the parameters of the trial wavefunction to minimise the cost function. This iterative process continues until convergence is reached, yielding an approximation to the eigenstate corresponding to the minimum eigenvalue. The scalability of the RVVQE algorithm is particularly noteworthy. The researchers demonstrated its performance on dense non-Hermitian matrices with dimensions up to 20×20, suggesting that it can be extended to larger systems with further optimisation. This is essential for tackling realistic problems in quantum chemistry and physics, which often involve many-body systems with many degrees of freedom.

The implications of this work extend beyond the immediate improvement in eigenvalue calculation. By providing a robust and efficient method for simulating open quantum systems, the RVVQE algorithm opens up new avenues for exploring fundamental quantum phenomena and developing novel quantum technologies. For example, it could be used to design more efficient quantum sensors, which rely on the interaction of quantum systems with their environment. It could also be used to develop new materials with tailored optical or electronic properties, by accurately modelling the interactions between electrons and photons. Furthermore, the RVVQE algorithm provides a valuable benchmark for assessing the performance of different quantum computing platforms and developing new quantum algorithms for simulating complex systems. The ability to accurately model open quantum systems is a crucial step towards realising the full potential of quantum computation.

The researchers developed a new algorithm, the Real Variance-based Variational Quantum Eigensolver, to accurately calculate eigenvalues for non-Hermitian matrices. This is important because standard quantum algorithms struggle with these types of matrices, which commonly describe open quantum systems exhibiting decay and resonances. The algorithm utilises only Hermitian measurements, making it suitable for current quantum computers. Results demonstrate the method’s performance on dense non-Hermitian matrices up to 20×20, and the authors suggest it can be further optimised for larger systems.