Solving the electronic structure problem via unitary evolution of the electronic Hamiltonian is one of the promising applications of digital quantum computers. One of the practical strategies to implement the unitary evolution is via Trotterization, where a sequence of short-time evolutions of fast-forwardable (i.e. efficiently diagonalizable) Hamiltonian fragments is used. Given multiple choices of possible Hamiltonian decompositions to fast-forwardable fragments, the accuracy of the Hamiltonian evolution depends on the choice of the fragments. We assess efficiency of multiple Hamiltonian partitioning techniques using fermionic and qubit algebras for the Trotterization. Use of symmetries of the electronic Hamiltonian and its fragments significantly reduces the Trotter error. This reduction makes fermionic-based partitioning Trotter errors lower compared to those in qubit-based techniques. However, from the simulation-cost standpoint, fermionic methods tend to introduce quantum circuits with a greater number of T-gates at each Trotter step and thus are more computationally expensive compared to their qubit counterparts.
Solving the electronic structure problem via unitary evolution of the electronic Hamiltonian is one of the promising applications of digital quantum computers. One of the practical strategies to implement the unitary evolution is via Trotterization, where a sequence of short-time evolutions of fast-forwardable (i.e. efficiently diagonalizable) Hamiltonian fragments is used. Given multiple choices of possible Hamiltonian decompositions to fast-forwardable fragments, the accuracy of the Hamiltonian evolution depends on the choice of the fragments. We assess efficiency of multiple Hamiltonian partitioning techniques using fermionic and qubit algebras for the Trotterization. Use of symmetries of the electronic Hamiltonian and its fragments significantly reduces the Trotter error. This reduction makes fermionic-based partitioning Trotter errors lower compared to those in qubit-based techniques. However, from the simulation-cost standpoint, fermionic methods tend to introduce quantum circuits with a greater number of T-gates at each Trotter step and thus are more computationally expensive compared to their qubit counterparts.
A key application of quantum computing is solving the electronic problem using Quantum Phase Estimation (QPE). The Trotter Approximation (TA) is one of the techniques used in QPE for encoding the Hamiltonian on a quantum computer. In TA, a Hamiltonian is partitioned into easily diagonalizable fragments. However, there is an error introduced in the unitary evolution operator when TA is applied. Different partitionings of a Hamiltonian and the ordering of the fragments yield different TA errors. In this work, we propose a novel two-step algorithm to obtain a near-optimal ordering for a given partitioning of a Hamiltonian. We find that when we use the Qubit-Wise Commuting partitioning method for the electronic Hamiltonian of moleculeH2, our algorithm is approximately 6 times faster than evaluating all possible orderings exhaustively.
Achieving an accurate description of fermionic systems typically requires considerably many more orbitals than fermions. Previous resource analyses of quantum chemistry simulation often failed to exploit this low fermionic number information in the implementation of Trotter-based approaches and overestimated the quantum-computer runtime as a result. They also depended on numerical procedures that are computationally too expensive to scale up to large systems of practical interest. Here we propose techniques that solve both problems by using various factorized decompositions of the electronic structure Hamiltonian. We showcase our techniques for the uniform electron gas, finding substantial (over $100\ifmmode\times\else\texttimes\fi{}$) improvements in Trotter error for low-filling fraction and pushing to much higher numbers of orbitals than is possible with existing methods. Finally, we calculate the T-count to perform phase estimation on Jellium. In the low-filling regime, we observe improvements in gate complexity of over $10\ifmmode\times\else\texttimes\fi{}$ compared to the best Trotter-based approach reported to date. We also report gate counts competitive with qubitization-based approaches for Wigner-Seitz values of physical interest.
Trotter–Suzuki decompositions are frequently used in the quantum simulation of quantum chemistry. They transform the evolution operator into a form implementable on a quantum device, while incurring an error—the Trotter error. The Trotter error can be made arbitrarily small by increasing the Trotter number. However, this increases the length of the quantum circuits required, which may be impractical. It is therefore desirable to find methods of reducing the Trotter error through alternate means. The Trotter error is dependent on the order in which individual term unitaries are applied. Due to the factorial growth in the number of possible orderings with respect to the number of terms, finding an optimal strategy for ordering Trotter sequences is difficult. In this paper, we propose three ordering strategies, and assess their impact on the Trotter error incurred. Initially, we exhaustively examine the possible orderings for molecular hydrogen in a STO-3G basis. We demonstrate how the optimal ordering scheme depends on the compatibility graph of the Hamiltonian, and show how it varies with increasing bond length. We then use 44 molecular Hamiltonians to evaluate two strategies based on coloring their incompatibility graphs, while considering the properties of the obtained colorings. We find that the Trotter error for most systems involving heavy atoms, using a reference magnitude ordering, is less than 1 kcal/mol. Relative to this, the difference between ordering schemes can be substantial, being approximately on the order of millihartrees. The coloring-based ordering schemes are reasonably promising—particularly for systems involving heavy atoms—however further work is required to increase dependence on the magnitude of terms. Finally, we consider ordering strategies based on the norm of the Trotter error operator, including an iterative method for generating the new error operator terms added upon insertion of a term into an ordered Hamiltonian.
Trotter–Suzuki decompositions are frequently used in the quantum simulation of quantum chemistry. They transform the evolution operator into a form implementable on a quantum device, while incurring an error—the Trotter error. The Trotter error can be made arbitrarily small by increasing the Trotter number. However, this increases the length of the quantum circuits required, which may be impractical. It is therefore desirable to find methods of reducing the Trotter error through alternate means. The Trotter error is dependent on the order in which individual term unitaries are applied. Due to the factorial growth in the number of possible orderings with respect to the number of terms, finding an optimal strategy for ordering Trotter sequences is difficult. In this paper, we propose three ordering strategies, and assess their impact on the Trotter error incurred. Initially, we exhaustively examine the possible orderings for molecular hydrogen in a STO-3G basis. We demonstrate how the optimal ordering scheme depends on the compatibility graph of the Hamiltonian, and show how it varies with increasing bond length. We then use 44 molecular Hamiltonians to evaluate two strategies based on coloring their incompatibility graphs, while considering the properties of the obtained colorings. We find that the Trotter error for most systems involving heavy atoms, using a reference magnitude ordering, is less than 1 kcal/mol. Relative to this, the difference between ordering schemes can be substantial, being approximately on the order of millihartrees. The coloring-based ordering schemes are reasonably promising—particularly for systems involving heavy atoms—however further work is required to increase dependence on the magnitude of terms. Finally, we consider ordering strategies based on the norm of the Trotter error operator, including an iterative method for generating the new error operator terms added upon insertion of a term into an ordered Hamiltonian.
Simulating vibrational dynamics is essential for understanding molecular structure, unlocking useful applications such as vibrational spectroscopy for high-fidelity chemical detection. Quantum algorithms for vibrational dynamics are emerging as a promising alternative to resource-demanding classical approaches, but this domain is largely underdeveloped compared with quantum simulations of electronic structure. In this work, we describe in detail three distinct forms of the vibrational Hamiltonian: canonical bosonic quantization, real space representation, and the Christiansen second-quantized form. Leveraging the Lie algebraic properties of each, we develop efficient fragmentation schemes to enable the use of Trotter product formulas for simulating time evolution. We introduce circuits required to implement time evolution in each form and highlight factors that contribute to the simulation cost, including the choice of vibrational coordinates. Using a perturbative approach for the Trotter error, we obtain tight estimates of the T gate cost for the simulation of time evolution in each form, enabling their quantitative comparison. Combining tight Trotter error estimates and efficient fragmentation schemes, we find that for the CH 4 molecule with 9 vibrational modes, time evolution for approximately 1.8 ps may be simulated using as few as 36 qubits and approximately 3 × 10 8 T gates─an order-of-magnitude speedup over state-of-the-art algorithms. Finally, we present calculations of vibrational spectra using each form to demonstrate the fidelity of our algorithms. This work presents a unified and highly optimized framework that makes simulating vibrational dynamics an attractive use case for quantum computers.
We propose a variational alternative to the Trotter-Suzuki decomposition that provides greater control over errors while preserving the unitary structure of time evolution. The variational parameters in our ansatz are derived from a global action principle, where Euler-Lagrange equations govern their optimal dynamics. Unlike conventional wavefunction-based variational methods, our approach specifically targets the time evolution operation and this allows a single set of optimized parameters to be applied to any initial state for a fixed Hamiltonian avoiding costly optimization procedures. Our method outperforms the standard Trotter-Suzuki formulas, typically achieving higher accuracy than higher-order Suzuki schemes. This translates directly to quantum computing applications, where it enables the design of quantum circuits with fewer gates which reduces noise and improves precision. Although we focus on quantum dynamics, the method is broadly applicable to problems involving general time-evolution operators. Applied to various model Hamiltonians, our approach reduces errors by factors of 2 to 5 compared to Trotter-Suzuki decompositions, demonstrating its promise for accurate quantum simulation with improved efficiency. In certain cases, the variational ansatz achieves higher accuracy than more complex higher-order Suzuki formulas while reducing the gate count by nearly half within a single circuit layer. Furthermore, we derive approximate analytical expressions for the variational parameters up to cubic order in time, valid for generic Hamiltonians. These approximations enable long-time quantum simulations with improved accuracy over equivalent Suzuki decompositions, providing ready-to-use evolution formulas that match Suzuki's gate complexity while delivering better performance.