Quantum Simulation

Taylor-Series (Truncated Dyson) Simulation

Building the time-evolution operator by truncating its power series and implementing the sum as a linear combination of unitaries.

The series

The propagator has the Taylor expansion e^(-iHt) = sum over k >= 0 of (-iHt)^k / k!. Truncating at order K gives an approximation whose error is bounded by the tail of the series. For an appropriate K the truncation error falls below any target epsilon, and the truncated sum can be implemented on a quantum computer.

From series to circuit via LCU

Kronos motion — power balance

Write H = sum_l c_l P_l as a sum of unitary Pauli terms. Then H^k is a sum of products of Paulis, each product itself unitary. So the truncated series becomes a large linear combination of unitaries, implementable with the PREPARE/SELECT construction plus amplitude amplification. This is the Berry-Childs-Cleve-Kothari-Somma (2015) algorithm.

The precision advantage

Because the Taylor tail decays factorially, the required truncation order is K = O(log(1/epsilon) / log log(1/epsilon)). This gives error scaling logarithmic in 1/epsilon, exponentially better than product formulas. It was among the first methods to break the polynomial-precision barrier, paving the way for qubitization.

Split total time into short segments so each segment's evolution is well-approximated by a low-order series.
Encode the truncated series as an LCU with PREPARE encoding the 1/k! weights.
Use oblivious amplitude amplification to make the near-unitary segment deterministic.
Chain segments to reach the full time t.

Segmenting time

A single Taylor truncation is accurate only for short times, so the total evolution is divided into r segments of duration t/r, each simulated by its own truncated series. The number of segments scales with the Hamiltonian norm times t, and the per-segment order scales logarithmically with precision, giving overall gate complexity nearly linear in t and logarithmic in 1/epsilon.

Place in the landscape

Taylor-series simulation is the historical bridge between product formulas and qubitization: it introduced the LCU-based, logarithmic-precision paradigm. Qubitization later achieved the same precision scaling with better dependence on the Hamiltonian norm and without amplitude amplification overhead, so it is usually preferred now, but the Taylor/Dyson approach remains important for time-dependent Hamiltonians, where the time-ordered Dyson series generalizes it.