Nested commutator expansion for the commutation function

Xinzhao Wang, Shuo Zhou, Ziruo Wang +5 more

Product formulas are among the most practical approaches to Hamiltonian simulation, requiring no ancillary qubits and exhibiting error bounds governed by nested commutators rather than only by Hamiltonian norms. Their circuit size, however, scales polynomially with the inverse precision. We develop a high-order nested-commutator compensation (HNCC) algorithm that preserves the main advantages of product formulas while achieving polylogarithmic precision dependence in the circuit size and the standard $\mathcal{O}(\varepsilon^{-2})$ sampling cost. HNCC uses a truncated Baker--Campbell--Hausdorff expansion to represent high-order Trotter errors by products of nested commutators and compensates these errors at the channel level through randomly sampled Pauli-rotation channels, avoiding Hadamard tests and ancillary qubits. For a fixed $K$-th order product formula applied to a $k$-local Hamiltonian on $N$ qubits with $Γ$ Pauli terms and local interaction strength $g_0$, HNCC estimates $\operatorname{tr}[Oe^{-i tH}ρe^{i tH}]$ to additive precision $\varepsilon\|O\|$ using $\mathcal{O}(\varepsilon^{-2})$ repetitions. Its maximum gate count per circuit is $\mathcal{O}\bigl( kN^{\frac{1}{2K+1}} Γ^{1-\frac{1}{2K+1}} \max\{Γ,N\log(1/\varepsilon)\}^{\frac{1}{2K+1}} (kg_0t\log(1/\varepsilon))^{1+\frac{1}{2K+1}} \bigr)$. Finite-size resource estimates for the periodic Heisenberg chain indicate that HNCC has the lowest estimated $T$-gate count per circuit among the product-formula-based methods considered.

Peter Fong

Kaoru Mizuta

A multi-product formula (MPF) is a promising approach for Hamiltonian simulation efficiently both in the system size N and the inverse allowable error 1 / ε by combining Trotterization and the linear combination of unitaries (LCU). It achieves poly-logarithmic cost in 1 / ε like LCU [G. H. Low, V. Kliuchnikov, N. Wiebe, (2019)]. The efficiency in N is expected to come from the commutator scaling in Trotterization, and this appears to be confirmed by the error bound of MPF expressed by nested commutators [J. Aftab, D. An, K. Trivisa, (2024)]. However, we point out that the efficiency of MPF in the system size N is not exactly resolved yet in that the present error bound expressed by nested commutators is incompatible with the size-efficient complexity reflecting the commutator scaling. The problem is that q -fold nested commutators with arbitrarily large q are involved in their requirement and error bound. The benefit of commutator scaling by locality is absent, and the cost efficient in N becomes prohibited in general. In this paper, we show an alternative commutator-scaling error of MPF and derive its size-efficient cost properly inheriting the advantage in Trotterization. The requirement and the error bound in our analysis, derived by techniques from the Floquet-Magnus expansion, have a certain truncation order in the nested commutators and can fully exploit the locality. We prove that Hamiltonian simulation by MPF certainly achieves the cost whose system-size dependence is as large as Trotterization while keeping the p o l y l o g ( 1 / ε ) -scaling like the LCU. Our results will provide improved or accurate error and cost also for various algorithms using interpolation or extrapolation of Trotterization.

Oleg Lychkovskiy, Viacheslav Khrushchev, Ilya Shirokov

We present QCommute, a software tool implemented in C++ for symbolic computation of nested commutators between a Hamiltonian and local observables in quantum many-body spin-1/2 systems on one-, two-, and three-dimensional hypercubic lattices. The computation is performed algebraically directly in the thermodynamic limit, and the Hamiltonian parameters are kept symbolic. Importantly, this way the entire parameter space is covered in a single run. The implementation supports extensive parallelization to achieve high computational performance. QCommute can serve as a computational backend for Heisenberg-picture approaches to quantum dynamics in strongly correlated regimes, ranging from direct Taylor expansion in time to advanced techniques such as the recursion method.

Freeman J. Dyson

Jialiang Tang, Xi Chen, Zhi-Yuan Wei

Counterdiabatic (CD) driving enables efficient quantum state preparation, but it requires implementing highly nonlocal adiabatic gauge potentials (AGP) that are impractical to compute and realize in large many-body systems. We introduce a \textit{weighted nested-commutator} (WNC) ansatz to approximate AGP using local operators. The WNC ansatz generalizes the standard nested-commutator ansatz by assigning independent variational weights to commutators of local Hamiltonian terms, thereby enlarging the variational space while preserving a fixed operator range. We show that the WNC ansatz can be efficiently optimized using a local optimization scheme. Moreover, it systematically outperforms the nested-commutator ansatz in preparing one-dimensional matrix product states (MPS) and the ground state of a nonintegrable quantum Ising model. We then numerically demonstrate that CD driving based on the WNC ansatz significantly accelerates the preparation of 1D MPS for system sizes up to $N = 1000$ qubits, as well as the two-dimensional Affleck-Kennedy-Lieb-Tasaki state on a hexagonal lattice with up to $N = 3 \times 10$ sites.