Short answer: Product formulas approximate the time evolution of a many-body system by a sequence of simpler steps, and they are a central method for simulating quantum dynamics on a quantum computer. The question is how large the error is. A 2021 paper gave tighter error bounds that use the commutativity of the terms, a 2024 paper extended commutator-style bounds to time-dependent Hamiltonians, and another 2024 paper applied them to multi-product formulas. These are theoretical bounds on algorithm cost.
The method and the problem
A 2021 Physical Review X paper explains that the Lie-Trotter formula and its higher-order generalisations decompose the exponential of a sum of operators. Despite significant effort, the authors say the error scaling of such product formulas remained poorly understood. They develop a theory of Trotter error that avoids the limits of earlier approaches based on truncating the Baker-Campbell-Hausdorff expansion. It exploits the commutativity of the operator summands directly, producing tighter error bounds for real- and imaginary-time evolution. Earlier work achieved similar goals for systems with geometric locality or Lie-algebraic structure, while this approach holds in general.
What the 2021 bounds gave
The authors give improved algorithms for digital quantum simulation and quantum Monte Carlo, including second-quantised plane-wave electronic structure, k-local Hamiltonians, power-law interactions, clustered Hamiltonians, the transverse field Ising model and quantum ferromagnets. They say their results nearly match or outperform the best previous results. They show product formulas can preserve locality, so local observables can be simulated with cost independent of system size for power-law interacting systems, which implies a Lieb-Robinson bound as a by-product. Their analysis reproduces known tight bounds for first- and second-order formulas.
Time-dependent Hamiltonians
A 2024 arXiv paper notes that it has been of great interest whether product formula errors can be expressed through commutators among local terms, called commutator scaling, since this substantially suppresses the cost in system size. Recent studies gave such bounds for generic time-independent systems, but time-dependent Hamiltonians remained difficult except for low-order formulas. The authors derive an error bound for generic product formulas on smooth time-dependent Hamiltonians, expressed through nested commutators and their time derivatives. They say it indicates substantial error suppression in system size for generic local Hamiltonians and gives a better estimate of gate counts. The derivation uses Floquet theory, mapping the time-dependent problem to a time-independent one on an infinite-dimensional space, and it covers related cases such as non-unitary dynamics. They also clarify the bound for a time-dependent multi-product formula, which can achieve exponentially better scaling with the target accuracy.
Multi-product formulas
A 2024 Physical Review Research paper describes multi-product formulas as linear combinations of Trotter circuits that give high-quality simulation with fewer Trotter steps. It makes two contributions aimed at near-term use. First, it extends the commutator-scaling theory of the 2021 paper to multi-product formulas, implying a quadratic reduction of Trotter error in the 1-norm on arbitrary time intervals compared with regular product formulas, with no added circuit depth or qubit connectivity and only a constant-factor increase in circuit repetitions. Second, it introduces dynamic multi-product formulas with time-dependent coefficients chosen to minimise an efficiently computable proxy for Trotter error, and uses minimax estimation to make them robust to algorithmic error, sampling and hardware noise, with a rigorous error bound.
What this does not tell you
These are bounds on algorithmic error, and an upper bound can overestimate the true error. They do not report running these simulations on hardware.
FAQ
What is a product formula?
A way to approximate the evolution under a sum of terms by a sequence of evolutions under each term.
What did the 2021 paper improve?
It gave tighter error bounds by using the commutativity of the terms.
What is commutator scaling?
Expressing the error through commutators among local terms, which can cut cost with system size.
Does it cover time-dependent systems?
A 2024 paper derives such a bound for smooth time-dependent Hamiltonians.
What do multi-product formulas offer?
Fewer Trotter steps, and a quadratic reduction in error in the 1-norm according to a 2024 paper.
All field notes