Simplified Quantum Calculations Tackle Complex Molecules

Scientists are continually seeking more efficient methods to model the behaviour of complex chemical systems. Jeremy Canfield from the Department of Physics at Georgetown University, Dominika Zgid from the Department of Chemistry at the University of Michigan, and J K Freericks, also from the Department of Physics at Georgetown University, present a new implementation of the unitary coupled cluster (UCC) algorithm designed to tackle this challenge. Their work details a low-depth UCC approach, reducing computational demands for large systems where traditional methods struggle, particularly those exhibiting strong correlations. This research is significant because it demonstrates a systematic convergence using fewer exact UCC factors than previously possible, enabling more accurate calculations on systems like hydrogen chains and BeH2, and paving the way for modelling increasingly complex molecular structures.

A computational technique to simulate the behaviour of complex molecules with unprecedented accuracy has been developed. This advance tackles a key limitation in quantum chemistry, enabling modelling of larger and more challenging systems than previously possible, and promises to accelerate materials discovery and fundamental chemical understanding.

A new approach to overcome a critical limitation hindering the progress of quantum computing for complex chemical simulations has been demonstrated. The unitary coupled cluster (UCC) algorithm, a promising method for calculating molecular energies on quantum computers, typically requires extremely deep quantum circuits, a major obstacle for current and near-future hardware.

This work introduces a modified UCC technique, termed quadratic UCC (qUCC), that dramatically reduces circuit depth by strategically balancing computational workload between quantum and classical processors. The research focuses on improving the efficiency of UCC calculations for systems exhibiting strong correlation, a phenomenon where electrons interact in complex ways, rendering conventional computational chemistry methods ineffective.

By employing a Taylor series expansion to approximate the effects of numerous UCC factors with small angles, researchers have significantly lessened the demands on the quantum computer’s processing capacity. This allows for the inclusion of a greater number of exact UCC factors, leading to more accurate results, particularly in challenging scenarios where traditional methods fail.

Investigations into hydrogen chains and the BeH2 molecule, systems where the degree of strong correlation can be precisely controlled, demonstrate the systematic convergence of qUCC calculations as more UCC factors are treated exactly. The most difficult regime to converge was found to be the transition from weak to strong coupling, a crucial area for understanding chemical reactivity.

Importantly, the study reveals that only a fraction, typically one-third to one-half, of the total UCC factors need to be calculated directly on the quantum computer to achieve accurate convergence. This advancement effectively shifts the computational burden, enabling the use of classical algorithms for determining the values of the expanded angles while reserving the quantum computer’s resources for generating and storing the complex quantum state.

The team has increased the number of exactly treated UCC factors from approximately 30 to around 300, allowing for rigorous verification of the method’s convergence and accuracy against full configuration interaction (FCI) calculations, the gold standard for quantum chemistry. The result is a more efficient algorithm that prioritizes reduced circuit depth at the expense of increased measurements, a beneficial trade-off for both current and future quantum hardware.

Convergence of quadratic unitary coupled cluster calculations with limited excitation factors

Treating exactly 300 unitary coupled cluster (UCC) factors represents a substantial algorithmic improvement, allowing for detailed convergence testing of the quadratic UCC (qUCC) approach with respect to the number of exact angles included. Calculations demonstrate systematic convergence towards the full UCC result using only a small fraction of all single and double excitation terms treated exactly, typically between one-third and one-half of the total available factors.

This reduction in the number of explicitly calculated UCC factors significantly decreases computational time, exchanging circuit depth for an increased number of measurements, a beneficial trade-off for both near-term and fault-tolerant quantum computers. The study focused on hydrogen chains and the BeH2 molecule, systems chosen for their ability to exhibit varying degrees of strong correlation through geometrical distortions.

Results reveal that the most challenging regime for convergence occurs during the transition from weak to strong coupling, highlighting the sensitivity of the method to changes in electronic correlation. The ability to handle 300 exact UCC factors allows for a more thorough assessment of convergence behaviour than previously possible. This work establishes a systematic method for determining the number of exact UCC factors required to achieve convergence, ensuring accurate determination of the optimised energy and ground state.

The qUCC approach effectively leverages the quantum computer’s memory to store the quantum state generated by the ansatz, restricting the computationally intensive nonlinear optimisation procedure to only the exact angles while determining the remaining angles through linear algebraic methods. This refined algorithm provides a more efficient alternative to full UCC calculations, particularly for systems where traditional coupled cluster methods fail to provide accurate solutions.

Energy expansion and circuit simplification via quadratic unitary coupled cluster theory

A hierarchical approach to the unitary coupled cluster (UCC) algorithm underpinned this work, aiming to reduce the circuit depth inherent in variational quantum eigensolver calculations. The UCC method constructs a quantum state using an exponential ansatz, expressed as the application of an operator σ to a reference state, where σ comprises a sum of excitation operators, singles, doubles, and higher, each associated with a rotation angle θ.

Recognising that many of these angles are small in magnitude, the research team implemented a quadratic UCC (qUCC) approach, performing a Taylor series expansion of the energy around the origin for small angles and optimised values for larger angles. This strategy allows for shallower quantum circuits, substituting increased measurement requirements for reduced computational complexity, a beneficial trade-off for both near-term and fault-tolerant quantum computers.

To facilitate this, the study focused on efficiently handling a significantly larger number of UCC factors than previously possible, increasing the capacity from approximately 30 to around 300 exactly-treated factors. This enhancement enabled a systematic investigation of convergence behaviour as the number of exact angles was varied, carefully assessing the accuracy of the qUCC approximation against full UCC results.

Row reduction techniques were employed to determine the optimal values for the small angles resulting from the Taylor expansion, streamlining the calculation and restricting nonlinear optimisation to only the larger, exactly-treated angles. Investigations centred on the hydrogen chains and beryllium hydride (BeH2) molecule, systems chosen for their ability to exhibit varying degrees of strong correlation due to geometrical distortions.

These molecules allowed for a rigorous examination of the qUCC method’s performance across a range of correlation strengths, particularly focusing on the challenging crossover region between weak and strong coupling. Full configuration interaction (FCI) calculations were performed classically to provide exact reference values for benchmarking the qUCC results, ensuring accurate assessment of the method’s efficacy in cases where conventional coupled cluster methods struggle.

Reducing quantum circuit complexity via Taylor expansion for accurate molecular simulation

Scientists are edging closer to simulating complex molecular interactions with greater efficiency, as demonstrated by advances in the unitary coupled cluster (UCC) algorithm. For years, the promise of accurately modelling molecular behaviour has been hampered by the exponential growth in computational demand as system size increases. Traditional methods, while effective for simpler cases, falter when dealing with strongly correlated systems.

UCC offers a potentially more robust approach, but its implementation on current quantum hardware is limited by the depth of the required circuits. This new work tackles that circuit depth problem head-on, employing a Taylor series expansion to reduce complexity without sacrificing accuracy. The key is a clever trade-off: accepting a greater number of measurements in exchange for a shallower, more manageable quantum circuit.

What’s particularly notable is the systematic convergence observed as more UCC factors are included, even in the challenging regime where weak and strong correlations meet. This suggests a pathway towards tackling previously intractable molecular systems. Looking ahead, the focus will likely shift towards developing adaptive algorithms that intelligently select these factors, and exploring hybrid classical-quantum approaches to further reduce the burden on quantum hardware. The ultimate goal remains a practical quantum simulator capable of designing new materials and catalysts, and this work represents a tangible step in that direction.