Kernpiler: Compiler Optimization for Quantum Hamiltonian Simulation with Partial Trotterization

TL;DR

This paper introduces Kernpiler, a compiler framework for quantum Hamiltonian simulation that leverages partial Trotterization to reduce error per Trotter step by partitioning non-commuting terms into dense, small-unityary blocks. It combines dense-term partitioning, commuting-group reordering with edge-focused Trotter steps, and Monte Carlo Tree Search to synthesize grouped exponentials into optimized gate sequences, achieving significant depth and gate-count reductions compared with state-of-the-art baselines. Theoretical analysis shows commutator-based error cancellation scales with partition size, and empirical results across spin, fermionic, and molecular Hamiltonians demonstrate substantial improvements in both gate counts and circuit depth, especially for first-order Trotterization. The approach also integrates randomized compilation concepts to convert coherent errors into stochastic noise, increasing robustness and scalability. Overall, Kernpiler offers a practical route to more efficient quantum simulations on near- and mid-term devices by exploiting fine-grained error structure and learned synthesis patterns.

Abstract

Quantum computing promises transformative impacts in simulating Hamiltonian dynamics, essential for studying physical systems inaccessible by classical computing. However, existing compilation techniques for Hamiltonian simulation, in particular the commonly used Trotter formulas struggle to provide gate counts feasible on current quantum computers for beyond-classical simulations. We propose partial Trotterization, where sets of non-commuting Hamiltonian terms are directly compiled allowing for less error per Trotter step and therefore a reduction of Trotter steps overall. Furthermore, a suite of novel optimizations are introduced which complement the new partial Trotterization technique, including reinforcement learning for complex unitary decompositions and high level Hamiltonian analysis for unitary reduction. We demonstrate with numerical simulations across spin and fermionic Hamiltonians that compared to state of the art methods such as Qiskit's Rustiq and Qiskit's Paulievolutiongate, our novel compiler presents up to 10x gate and depth count reductions.

Figures (9)

• Figure 1: Conventional compilation flow vs the proposed Kernpiler compiler. b) Pipeline for reducing gates through error term reduction. First we group into partial Trotter steps which act on a subset of N qubits, in our case N=3. Then we perform an efficient numerical rewrite of the partial Trotter unitaries. Next step, group into commuting subsets of unitaries placing the largest two groups of unitaries on the edges of the Trotter step. Finally, we use a partially symmetric Trotter step to cancel error terms in the expansion by alternating every other Trotter steps order. Commuting unitaries then merge back together naturally allowing for a unitary reduction with no additional error. The compilation finishes at circuit-level.
• Figure 2: The four stages of the Monte Carlo search tree. 1. Selection of a node for expansion and evaluation. 2) Expansion: choosing a new action and state combination that has not been explored. 3) Simulation: Randomly traversing states and actions to a terminal state and evaluating the outcome. 4) Backpropagation: updating tree metadata on outcomes learned through simulation
• Figure 3: 1) Create Groups: A conflict graph is constructed showing commutation relations of Hamiltonian terms. A vertex indicates a unitary of the Trotter step. An edge indicates that two unitaries do not commute. Independent sets are created about the graph which are used to group unitaries with other pairwise commuting unitaries. 2) Order Full Groups: The groups created are ordered in the Trotter step for cancellation with other groups. The two largest groups are placed on edges of the Trotter step. At the neighboring Trotter steps, the groups placed at the edges swap places such that identical groups are neighboring each other. Unitaries are then merged via commutation equivalences. 3) Shuffling Group Term Order: The order of terms within each group is shuffled to invoke stochastic noise over coherent noise.
• Figure 4: Unitary decomposition method 1) Selection: Select a node in the search tree which represents a partially synthesized circuit which has unexplored child actions. 2) Expansion: Select a CNOT gate among choices from the gateset to append to the circuit. 3) Simulation: Starting from the newly expanded state, append CNOTs until we reach a terminal circuit length. After, interleave a fixed number of single qubit gates at random in between the CNOT gates. Optimize parameters with the Gauss-Newton method. 4) Backpropagation: Update values of nodes in the tree based on the result of the simulation stage to identify if the newly explored state was valuable.
• Figure 5: Absolute data comparisons with no error reduction considered. Hamiltonians are compiled to a fixed number of Trotter steps.