In recent years, there have been significant developments in the field of quantum algorithms, especially in applications involving quantum chemistry, and representations of molecular Hamiltonians. These include variational quantum algorithms that aim to maximize the algorithmic performance despite the limited coherence times of currently available hardware. Unfortunately, that comes at the cost of introducing heuristic aspects, making it difficult to obtain rigorous performance guarantees. In contrast, quantum phase estimation approaches provide a route to accurately calculate eigenstates to small specifiable error, assuming sufficiently high overlap with the true eigenstates during the preparation of approximate eigenstates. Also, in fault-tolerant quantum computation, required resources will depend on the ability to bound errors in the algorithm which means that tighter error estimates will lead to fewer resource requirements.
Estimation of the resources required for phase estimation can be done using product-formula decomposition, also known as Trotterization. Previous works have shown that the number of fermions in the system can be used to offset the dependence of the error on the number of orbitals that can be applied to chemical systems in realistically sized basis sets. In this work, the authors present three factorized decompositions and use these in conjunction with the fermionic seminorm to obtain tighter Trotter error bounds in practice and apply it to the particular case of simulating a uniform electron gas (Jellium). Another objective of this approach is reducing the classical runtime and the gate complexity of quantum algorithms.
Each of the three factorized decompositions used in the work; namely spectral decomposition, cosine decomposition, and Cholesky decomposition, exhibits its own advantage. The spectral decomposition is generally applicable and extends beyond the plane wave dual basis. The cosine decomposition best exploits fermion number information and thus can be the most effective in the low-filling fraction regime. The Cholesky decomposition has the smallest constant factor overhead and therefore performs best in the medium and half-filling regimes.
The approach is applied to simulation of a uniform electron gas, finding substantial (over 100×) improvements in Trotter error for low-filling fraction and pushing towards much higher numbers of orbitals than is possible with existing methods. The approach was benchmarked against three prior art bounds: the analytic SHC bound, the fermionic commutator approach, and a similar Pauli commutator approach. A substantial classical runtime advantage for the calculation of the bounds was observed. In the case of fermionic and Pauli commutator approaches, the calculation of large spin-orbital number N>200 becomes intractable without access to > 100 GB of RAM. In contrast, the new bounds on a 512 spin-orbital can be calculated in less than 6 hours using < 16 GB of RAM. The work also calculates the T-bound for phase estimation of the ground state energy for Jellium, with significant improvements in gate complexity of over 10× as compared to the existing Trotter-based approaches. The gate counts in the results demonstrate that the trotter approach can comparatively perform better in specific regimes of interest than post-Trotter methods like qubitization. This holds true even while using fewer gates than qubitization for a large enough Wigner-Seitz radius.
The work focuses on second-order Trotter in the plane wave dual basis, but the techniques can be further generalized to higher-order trotter bounds. While the spectral and Cholesky decompositions are applicable in case of compact basis sets, an interesting potential lies in the cosine decomposition to obtain an even tighter bound in the low-filling fraction regime. Also, calculation of the higher-order trotter bounds would require time (scaling as O(N^7)), making it a costlier run. A potential alternative would be the post-Trotter methods, which exhibit superior asymptotic performance with respect to target error as well as being cost-effective due to the use of Hamiltonian factorizations. However, trotter methods possess good constant pre-factors in the runtime and few additional ancilla qubits requirements compared to post-Trotter methods, suggesting that trotter methods could perform comparatively better at specific tasks in the pre-asymptotic regime. It would be interesting to explore whether second quantized post-Trotter methods can similarly exploit low-filling fractions, where Trotter methods perform better.