Quantum Algorithm Enables Efficient Simulation of Sparse Quartic Hamiltonians for Time Horizons

Understanding how observables change over time under Hamiltonian evolution represents a significant frontier in quantum computing, with the potential to outperform classical algorithms in many scenarios. Giorgio Facelli, Hamza Fawzi, and Omar Fawzi, from the University of Cambridge and Inria, have now demonstrated a setting where this evolution can be efficiently approximated. Their research introduces Majorana Propagation, an algorithm utilising Trotter steps and truncations, and provides the first provable guarantee of its effectiveness in finding a low-degree approximation of the time-evolved observable. This work establishes that Majorana Propagation can efficiently simulate the time dynamics of sparse quartic Hamiltonians, with a runtime scaling favourably with the number of Majorana modes and the interaction strength. The findings formalise the understanding that the algorithm maintains accuracy even when dealing with quadratic Hamiltonians, opening new avenues for practical quantum simulations.

Majorana Propagation for Time-Evolved Observables

The study pioneers a novel approach to simulating quantum systems by focusing on the time dynamics of observables under Hamiltonian evolution, a problem anticipated to challenge classical algorithms. Researchers developed Majorana Propagation (MP), an algorithm combining Trotter steps with degree-based truncations, to efficiently approximate time-evolved observables. This work establishes the first provable performance guarantee for MP in the context of Hamiltonian evolution, demonstrating its ability to find low-degree approximations as soon as they exist.

Scientists engineered a specific truncation strategy, discarding Majorana strings exceeding a defined degree ‘l’ within the observable’s Majorana expansion. The algorithm, denoted as MP(t), iteratively applies a Trotter approximation of the Heisenberg evolution followed by this truncation, producing a degree-l approximation of the time-evolved observable. To quantify approximation accuracy, the team introduced a metric measuring the error between the true time-evolved observable and its best degree-l approximant, using the normalized Frobenius norm as the measure of distance.

Experiments employed a ∆-sparse quartic fermionic Hamiltonian with N Majorana modes, alongside an arbitrary observable A, both normalised to have a Frobenius norm of 1. The runtime and memory complexity of the MP algorithm were determined to be at most (t/δt) · N O(l), where ‘t’ is the time horizon and ‘δt’ is the time step. Crucially, the resulting approximation achieves a Frobenius norm error governed by the quality of the best degree-l approximation when an optimised time step is used.

The research further refined the algorithm’s implementation by partitioning the Hamiltonian into groups with disjoint Majorana strings, enabling tighter control over the degree of the time-evolved observable during the Trotter decomposition. Applying this methodology to weakly interacting Hamiltonians, the study proved that for any observable A, there exists a time limit beyond which the algorithm maintains accuracy, formalising the intuition that MP performs optimally for quadratic Hamiltonians. This advancement provides a significant step towards scalable quantum simulations.

Majorana Propagation Efficiently Models Quantum Dynamics

Scientists achieved a significant breakthrough in understanding the time dynamics of observables under Hamiltonian evolution, demonstrating a novel algorithm called Majorana Propagation (MP). The team measured that this algorithm efficiently approximates the time-evolved observable, finding a low-degree approximation as soon as such an approximation exists, providing the first provable guarantee for its performance. Experiments revealed that MP can efficiently calculate the time dynamics of any sparse quartic Hamiltonian for a time horizon dependent on the interaction strength, a crucial step towards more accurate quantum simulations.

This work establishes a foundational understanding of how to efficiently model complex quantum systems. The research details that for a time horizon of t, the runtime of the algorithm is limited to t/δt * NO(l), where δt represents the time step and NO(l) denotes the computational cost related to the degree truncation parameter l. Measurements confirm that the error, quantified using the normalized Frobenius norm, is bounded by a value dependent on the Hamiltonian sparsity, the degree of truncation, and the initial observable.

Importantly, the team showed that as δt approaches zero, the algorithm’s accuracy improves, formalizing the intuition that it performs optimally when the Hamiltonian is quadratic. Scientists proved that the degree of the approximation is limited by a function of the Hamiltonian, and further, that a greedy algorithm can colour the associated graph using a limited number of colours. The study demonstrates that the algorithm decomposes the Hamiltonian into terms with disjoint support, enabling efficient computation.

Majorana Propagation’s Accuracy and Scalability Demonstrated

Researchers have demonstrated that the Majorana Propagation algorithm can efficiently approximate the time evolution of observables under Hamiltonian dynamics, specifically when a low-degree approximation exists. This work establishes a provable performance guarantee for the algorithm, showing it can accurately calculate time dynamics for sparse quartic Hamiltonians up to a time dependent on the interaction strength. The runtime of the algorithm scales with the number of Majorana modes and the desired error, and importantly, approaches linear time in the limit of small interaction strengths.

The analysis reveals that the error incurred by the Majorana Propagation algorithm is closely related to the error of the best possible low-degree approximation of the time-evolved observable. Specifically, for a ∆-sparse quartic Hamiltonian, the algorithm achieves a specified accuracy with a runtime and memory complexity of (t/δt) · N O(l) , where ‘t’ is the time horizon, ‘δt’ is the time step, ‘N’ is the number of Majorana modes, and ‘l’ is the truncation degree.

The authors acknowledge that the analysis relies on the Hamiltonian being ∆-sparse, a condition broader than assumptions requiring geometric locality. The authors identify that the accuracy of the algorithm is influenced by both the discretization of time and the degree truncation, with an optimal time step identified to minimize error. Future research could explore the application of this algorithm to more complex Hamiltonian systems and investigate alternative truncation strategies to further improve performance. The findings contribute to the development of efficient quantum algorithms by providing a theoretical foundation for understanding the capabilities of Majorana Propagation in simulating quantum dynamics.

Understanding how observables change over time under Hamiltonian evolution represents a significant frontier in quantum computing, with the potential to outperform classical algorithms in many scenarios. This work establishes that Majorana Propagation can efficiently simulate the time dynamics of sparse quartic Hamiltonians, with a runtime scaling favourably with the number of Majorana modes and the interaction strength. The findings formalise the understanding that the algorithm maintains accuracy even when dealing with quadratic Hamiltonians, opening new avenues for practical quantum simulations.