Commuting Hamiltonians and Fast-Forwarding
Special Hamiltonians whose structure lets their evolution be simulated in time independent of the duration, escaping the no-fast-forwarding bound.
The exception to the rule
The no-fast-forwarding theorem forbids simulating generic Hamiltonian evolution in time sublinear in t. But it applies only to generic H. Certain structured Hamiltonians can be fast-forwarded: their evolution for arbitrarily long time t costs the same as for short time, because their spectrum can be computed and applied directly.
Commuting Hamiltonians
If H = sum_j H_j where all H_j mutually commute, then e^(-iHt) = product of e^(-i H_j t) exactly, with no Trotter error, and each factor is a simple gate. More strongly, commuting Hamiltonians can be simultaneously diagonalized, so evolution is just applying phases in the shared eigenbasis, a cost independent of t.
Quadratic (free) systems
- Free fermions: quadratic in creation/annihilation operators, diagonalized by a Bogoliubov transformation.
- Free bosons and harmonic systems: diagonalized by normal-mode transformations.
- Both allow evolution via a basis change, phases, and inverse change, all t-independent in circuit structure.
- Classical simulation is also efficient here, so no quantum advantage, but the circuits are exact and cheap.
Diagonalizable-in-known-basis Hamiltonians
Any Hamiltonian for which an efficient quantum circuit diagonalizes it can be fast-forwarded: change to the eigenbasis, apply the phase e^(-i E t) for each eigenvalue, change back. The time t enters only as parameters of phase gates, not as circuit depth. Free-fermion systems and certain integrable models fall in this class.
Why it matters
Recognizing fast-forwardable structure can turn an apparently expensive long-time simulation into a cheap one. In quantum chemistry, the one-body (quadratic) part of a Hamiltonian is fast-forwardable and can be handled exactly, leaving only the two-body interaction to Trotterize, an optimization used in low-depth chemistry circuits. Fast-forwarding also sharpens the picture of the no-fast-forwarding theorem: hardness comes specifically from non-commuting, non-integrable interactions, exactly the ingredients that make many-body physics interesting and classically hard.