Trotter error compensation with polylogarithmic precision and nested-commutator scaling without ancillas

Product formulas (Trotter methods) are a cornerstone of quantum Hamiltonian simulation due to their simplicity and low resource requirements. However, they traditionally suffer from a polynomial dependence on the target precision, making high-precision simulations computationally expensive. This paper introduces the High-order Nested-Commutator Compensation (HNCC) algorithm, which overcomes this limitation while maintaining the desirable properties of standard product formulas, such as the absence of ancilla qubits and the exploitation of nested-commutator error bounds.

The HNCC algorithm addresses the Trotter remainder—the difference between the ideal unitary evolution and the product formula approximation—by representing it as a linear combination of quantum channels (LCQC). The authors use a truncated Baker-Campbell-Hausdorff (BCH) expansion to decompose the Trotter error into nested commutators. By converting these commutators into a sum of Pauli-rotation channels using a parameter-shift identity, the algorithm allows for the compensation of Trotter errors at the superoperator level. This approach avoids the need for Hadamard tests or ancillary qubits, which are typically required in unitary-level compensation methods.

HNCC achieves polylogarithmic precision dependence in the circuit size while maintaining the standard O(ε⁻²) sampling cost. For a K-th order product formula, the algorithm effectively matches the time dependence of a (2K+1)-th order formula. Numerical estimates for the periodic Heisenberg chain demonstrate that HNCC significantly reduces the number of CNOT and T-gates per circuit compared to uncompensated product formulas. The method is particularly well-suited for devices with limited connectivity, as it avoids the complex controlled operations often required by other compensation techniques.