This paper introduces HNCC, a new algorithm to reduce error in Hamiltonian simulation without using extra qubits. The problem they address is that practical “product formula” or Trotter methods for simulating quantum dynamics have simple circuits and error bounds that exploit nested commutators. But their circuit size grows as a polynomial when you ask for higher precision. HNCC aims to keep the advantages of product formulas while making the circuit size grow only polylogarithmically with the required precision.

The authors build HNCC (high-order nested-commutator compensation) by writing the short-time Trotter error in terms of nested commutators using a truncated Baker–Campbell–Hausdorff (BCH) expansion. They then approximate the corrective operation as a linear combination of quantum channels (LCQC). Each channel in that combination is implemented by sequences of Pauli-rotation channels and sampled at random. Because the correction is applied at the channel, or “superoperator,” level, their scheme avoids Hadamard-test measurements and needs no ancillary qubits. The randomized implementation recovers the ideal evolution in expectation, and the classical bookkeeping only records sampled signs.

What this delivers in complexity terms is a notable trade. For a K-th order product formula applied to a k-local Hamiltonian on N qubits with Γ Pauli terms and local interaction strength g0, HNCC estimates an observable to additive precision ε with the usual O(ε−2) number of circuit repetitions. The maximum gate count per circuit scales as O(N^{2/(2K+1)} (k g0 t log(1/ε))^{1+1/(2K+1)} k(Γ+log(1/ε))). In plain language: the method preserves the nested-commutator (system-size) scaling of product formulas, achieves a time dependence like a product formula of effective order 2K+1, and replaces the previously polynomial dependence on 1/ε in circuit size by a polylogarithmic one.