Quantum Fast-Forwarding in Dynamics

• Quantum fast-forwarding is a set of methods that simulate long-time quantum evolution with sublinear resource scaling by exploiting specific structural properties.
• It integrates techniques from Hamiltonian simulation, reversible Markov chains, and stable differential equations to circumvent standard no-fast-forwarding limits.
• Practical applications on NISQ devices and control-theoretic implementations demonstrate its potential to accelerate quantum simulations and enhance property testing.

Quantum fast-forwarding (QFF) denotes families of methods that realize long-time evolution, or compute long-time dynamical properties, with resources scaling sublinearly in the evolution time by exploiting special structure rather than generic black-box simulation. In the algorithmic literature, QFF typically means implementing $e^{-iHt}$, $P^t$, or related propagators with gate, query, or depth complexity that is asymptotically smaller than linear in $t$; in the quantum-control literature, “fast-forward” often refers to shortcut-to-adiabaticity-style driving that reproduces a target evolution in shorter physical time by modifying the Hamiltonian controls (Gu et al., 2021, Villazon et al., 2019). The term is therefore not uniform across subfields. A common thread is that generic systems remain constrained by no-fast-forwarding barriers, whereas reversible Markov chains, compact Lie-algebraic representations, stable or dissipative differential equations, and carefully chosen NISQ subspaces admit nontrivial speedups (Apers et al., 2018, An et al., 2024).

1. Definitions and conceptual scope

Algorithmic QFF is formalized relative to a no-fast-forwarding baseline. For Hamiltonians, one formulation asks whether a family $\{H_n\}$ admits circuits implementing $e^{-itH_n}$ on a specified subspace with gate complexity $G(n)$ that asymptotically beats the worst-case simulation cost $l(n)$ for an ambient Hamiltonian class; this accommodates both exponential and polynomial fast-forwarding (Gu et al., 2021). A complementary formulation characterizes exponential fast-forwarding through a measurement-theoretic equivalence: a normalized Hamiltonian belongs to $\mathrm{FF}_{\exp}$ if and only if it belongs to $\mathrm{SEEM}_{\exp}$, where $\mathrm{SEEM}_{\exp}$ denotes super-efficient energy measurement with exponentially small $P^t$0 and polynomial computational resources (Atia et al., 2016).

In differential-equation settings, QFF is defined operationally by the scaling of the complexity required to prepare either a history state or a final-time state. For linear dissipative ordinary differential equations, fast-forwarding means that history-state preparation scales polylogarithmically in $P^t$1, or that final-state preparation scales as $P^t$2 rather than $P^t$3 (An et al., 2024). For stable linear systems more generally, the history state can always be output with complexity $P^t$4, while quantum Hamiltonian dynamics appears as a boundary case that does not admit this stability-induced mechanism (Jennings et al., 2023).

A recurring misconception is that QFF is a generic property of quantum dynamics. The literature instead shows the opposite. General Hamiltonian simulation obeys an $P^t$5 lower bound, and reversible Markov-chain QFF relies on Hermitian discriminants with spectrum in $P^t$6 (Apers et al., 2018, An et al., 2024). Likewise, purely dissipative Lindbladian fast-forwarding in logarithmic depth requires additional structure, such as block-diagonal Pauli jump operators, and exact Chebyshev-based QFF fails for the nonreversible $P^t$7-perturbed $P^t$8-cycle once the spectrum leaves $P^t$9 (Shang et al., 11 Sep 2025, Banerjee et al., 25 Jun 2026).

2. Reversible Markov chains and polynomial spectral fast-forwarding

The modern algorithmic notion of QFF was crystallized for reversible Markov chains. Let $t$0 be a reversible transition matrix and $t$1 its discriminant matrix. Using a Szegedy-type walk $t$2 and the projector $t$3 onto the flat subspace, the projected walk satisfies the exact Chebyshev identity

$t$4

where $t$5 is the Chebyshev polynomial of the first kind (Apers et al., 2018). This identity converts the problem of producing $t$6 into a polynomial transformation problem. The scalar relation

$t$7

with $t$8 for a simple symmetric random walk implies that truncating at degree $t$9 suffices in the reversible case, yielding $\{H_n\}$0 walk complexity (Apers et al., 2018, Banerjee et al., 25 Jun 2026).

This construction was used to quadratically accelerate transient random-walk behavior, not merely asymptotic mixing. One can prepare a state $\{H_n\}$1-close to $\{H_n\}$2 using

$\{H_n\}$3

expected quantum walk steps and $\{H_n\}$4 expected reflections about $\{H_n\}$5 (Apers et al., 2018). The same mechanism underpins quantum property testers for graph expansion and clusterability, and later expansion-testing work improves the dependence on graph size by first growing a localized seed set via an evolving set process and then applying QFF from the resulting superposition (Apers, 2019).

The reversible construction does not extend verbatim to nonreversible chains. For the $\{H_n\}$6-perturbed $\{H_n\}$7-cycle, the eigenvalues become

$\{H_n\}$8

so for $\{H_n\}$9 the spectrum leaves $e^{-itH_n}$0 and $e^{-itH_n}$1 is no longer uniformly bounded (Banerjee et al., 25 Jun 2026). The paper proves that for any $e^{-itH_n}$2 and $e^{-itH_n}$3 there is no unitary compression $e^{-itH_n}$4 for all $e^{-itH_n}$5. A finite-time approximation survives through truncated Chebyshev and LCU methods, but the required degree becomes

$e^{-itH_n}$6

so the reversible $e^{-itH_n}$7 behavior is recovered only in the nearly reversible regime $e^{-itH_n}$8 (Banerjee et al., 25 Jun 2026).

3. Hamiltonian fast-forwarding, precise energy measurement, and algebraic structure

For Hamiltonians, fast-forwarding is tightly linked to structure in the dynamical Lie algebra, spectral decomposition, or solvable normal forms. The equivalence between exponential fast-forwarding and exponentially precise energy measurement is one of the foundational results in the area: a Hamiltonian is exponentially fast-forwardable if and only if its eigenvalues can be measured with exponentially small uncertainty by polynomial-size circuits (Atia et al., 2016). This perspective explains why Shor-type modular multiplication Hamiltonians, commuting local Hamiltonians, and quadratic fermionic Hamiltonians admit exponential violations of the computational time-energy uncertainty principle and therefore exponential fast-forwarding (Atia et al., 2016).

A broader framework extends the definition beyond exponential speedups. Local spin systems that can be taken into block diagonal form using an efficient quantum circuit, such as permutation-invariant Hamiltonians, can be exponentially fast-forwarded; frustration-free positive semidefinite local spin systems can be polynomially fast-forwarded on sufficiently low-energy subspaces; and all quadratic fermionic systems together with number-conserving quadratic bosonic systems can be exponentially fast-forwarded in appropriate gate models (Gu et al., 2021). The same work develops a correspondence between fast-forwarding and precise energy measurements that also accommodates polynomial improvements (Gu et al., 2021).

Several recent results sharpen this structural theme. In the oscillator representation of $e^{-itH_n}$9, QFF is realized by factorizing

$G(n)$0

with $G(n)$1 and $G(n)$2, and implementing the resulting diagonal or Fourier-diagonal operators through the Quantum Hermite Transform (Guan et al., 7 Jun 2026). Applied to the quantum Kravchuk transform, this yields gate complexity

$G(n)$3

polylogarithmic in both the dimension and the inverse precision (Guan et al., 7 Jun 2026).

For fermion-boson systems, the polaron transform identifies a fast-forwardable interaction-picture free part. In the Hubbard–Holstein model, the conditional displacement diagonalizes $G(n)$4 so that $G(n)$5 in first quantization, with no linear dependence on $G(n)$6, and the overall interaction-picture simulation reduces the bosonic-cutoff dependence from $G(n)$7 or $G(n)$8 to polylogarithmic factors (Apel et al., 25 Jan 2026). For purely dissipative Lindbladians with unitary jump operators, additive query complexity

$G(n)$9

is achieved in general, and when the jumps are block-diagonal Paulis the circuit depth becomes

$l(n)$0

an exponential fast-forwarding in circuit depth while preserving the same linear-in-$l(n)$1 query complexity (Shang et al., 11 Sep 2025).

An algebraic variant was realized experimentally for the two-site Anderson impurity model relevant to dynamical mean-field theory. There, a Cartan decomposition produces a fixed-depth factorization

$l(n)$2

so only a small set of commuting rotation angles scales with $l(n)$3 (Steckmann et al., 2021). After optimization, the full Green’s-function experiment required 77 CNOTs, independent of the simulated time, and this enabled extraction of the quasiparticle resonance near the Mott transition on noisy cloud hardware (Steckmann et al., 2021).

4. Linear ODEs, stability, and dissipative fast-forwarding

Quantum algorithms for linear differential equations reveal a distinct fast-forwarding mechanism based on stability and dissipativity rather than Lie-algebraic compilation. For homogeneous and inhomogeneous linear ODEs,

$l(n)$4

generic lower bounds depend on two forms of “non-quantumness”: the real-part gap

$l(n)$5

and the non-normality measure

$l(n)$6

Real-part gaps induce exponential overheads and non-normality induces linear overheads in worst-case state-preparation complexity (An et al., 2022). Positive results then exploit special cases: inhomogeneous systems with negative definite $l(n)$7 admit $l(n)$8 dependence, while normal matrices with efficiently implementable eigensystems can yield complexities independent of $l(n)$9 and $\mathrm{FF}_{\exp}$0 in favorable regimes (An et al., 2022).

The 2023 resource analysis made this stability mechanism explicit and non-asymptotic. Instead of fast-forwarding $\mathrm{FF}_{\exp}$1 directly, the algorithm embeds the ODE into a structured linear system $\mathrm{FF}_{\exp}$2, where the solution vector coherently encodes the trajectory or final state. For Lyapunov-stable systems, the condition number of $\mathrm{FF}_{\exp}$3 scales only as $\mathrm{FF}_{\exp}$4, and the history state can always be output with complexity $\mathrm{FF}_{\exp}$5 (Jennings et al., 2023). The paper emphasizes that quantum Hamiltonian dynamics, with $\mathrm{FF}_{\exp}$6, is a boundary case that does not allow this form of stability-induced fast-forwarding (Jennings et al., 2023).

A stronger dissipative result appears in the 2024 truncated-Dyson analysis of linear dissipative ODEs. For

$\mathrm{FF}_{\exp}$7

the homogeneous propagator is contractive,

$\mathrm{FF}_{\exp}$8

and this yields a refined condition-number estimate for the all-at-once linear system: the condition number scales as $\mathrm{FF}_{\exp}$9 instead of the usual $\mathrm{SEEM}_{\exp}$0 (An et al., 2024). With a truncated Dyson series, one can prepare dissipative ODE history states up to time $\mathrm{SEEM}_{\exp}$1 with cost

$\mathrm{SEEM}_{\exp}$2

and final states at time $\mathrm{SEEM}_{\exp}$3 with cost

$\mathrm{SEEM}_{\exp}$4

an exponential and polynomial improvement in $\mathrm{SEEM}_{\exp}$5, respectively (An et al., 2024). Even lower-order methods such as forward Euler and the trapezoidal rule retain $\mathrm{SEEM}_{\exp}$6 scaling for final-state preparation, and the framework applies to dissipative non-Hermitian dynamics and heat processes (An et al., 2024).

A subtle limitation is that homogeneous dissipative final-state preparation is not fast-forwardable in general: the ratio $\mathrm{SEEM}_{\exp}$7 can be exponentially large because the signal decays exponentially (An et al., 2024). This sharpens the distinction between history-state and final-state tasks and prevents overgeneralizing the dissipative speedup.

5. NISQ, subspace, and cloud-based fast-forwarding

A separate strand of QFF targets finite-coherence hardware by learning or measuring a reduced dynamical representation and then performing long-time propagation classically or with fixed-depth circuits. Quantum Krylov fast-forwarding (QKFF) constructs a real-time Krylov subspace

$\mathrm{SEEM}_{\exp}$8

and solves the projected Schrödinger equation

$\mathrm{SEEM}_{\exp}$9

with the subspace matrices $\mathrm{SEEM}_{\exp}$0 and $\mathrm{SEEM}_{\exp}$1 measured on hardware (Cortes et al., 2022). The multi-reference variant improves long-time fidelity without increasing circuit depth, and numerical experiments on $\mathrm{SEEM}_{\exp}$2, $\mathrm{SEEM}_{\exp}$3, and linear $\mathrm{SEEM}_{\exp}$4 report fidelities $\mathrm{SEEM}_{\exp}$5 over longer times as the number of references increases (Cortes et al., 2022).

Fixed-state variational fast-forwarding (fsVFF) restricts diagonalization to the subspace relevant to a chosen initial state. In the XY chain, the method learns an approximate diagonalization of a short-time propagator and then reuses the diagonal form for arbitrarily many steps, so the long-time circuit depth is essentially independent of the simulated time (Gibbs et al., 2021). On IBM and Rigetti hardware, fsVFF maintained a fidelity of at least $\mathrm{SEEM}_{\exp}$6 for over 600 time steps and a fidelity of at least $\mathrm{SEEM}_{\exp}$7 for over 1275 time steps in a two-qubit XY simulation, a factor of 150 longer than is possible using the iterated Trotter method (Gibbs et al., 2021). The same paper emphasizes that the method only diagonalizes on the subspace spanned by the initial state, rather than on the total Hilbert space (Gibbs et al., 2021).

Classical Quantum Fast Forwarding (CQFF) removes the variational feedback loop entirely. It chooses a cumulative $\mathrm{SEEM}_{\exp}$8-moment basis, measures overlap matrices

$\mathrm{SEEM}_{\exp}$9

solves the generalized eigenproblem $P^t$00 classically, and then evaluates

$P^t$01

without any time-dependent circuit growth (Lim et al., 2021). For Pauli Hamiltonians this reduces to sampling in Pauli-rotated bases. On IBM hardware, the method achieved fidelity essentially 1 up to 2,500,000 steps for the XY dimer with $P^t$02, while first-order Trotterization broke down at about 25 steps; the paper describes this as about a $P^t$03 extension in effective simulation time and a $P^t$04 improvement over the previous record (Lim et al., 2021).

The DMFT impurity-solver implementation provides a different NISQ realization. By combining Cartan-based algebraic fast-forwarding, a symmetry-preserving one-parameter ground-state ansatz, and a noise-resilient spectral analysis based on two sampling grids and post-selection, the method mapped the metal-insulator phase diagram of the two-site Hubbard model on noisy cloud hardware (Steckmann et al., 2021). Near the Mott transition, the quasiparticle resonance frequency is extremely close to zero, and the fixed-depth fast-forwarded circuit maintains accuracy where Trotter error would otherwise dominate because of the long evolution times required for spectral resolution (Steckmann et al., 2021).

6. Fast-forward driving in quantum control and open systems

In quantum control, “fast-forward” typically means physical acceleration or deceleration of a target trajectory by reshaping the Hamiltonian. In finite-dimensional Hilbert spaces, fast-forward scaling introduces a time magnification factor $P^t$05 and a scaled time

$P^t$06

together with a unitary gauge transformation $P^t$07 so that

$P^t$08

satisfies a driven Schrödinger equation under a fast-forward Hamiltonian

$P^t$09

(Takahashi, 2014). For two-level systems this yields explicit diagonal acceleration potentials, and when applied to transitionless driving the fast-forward potential can be interpreted as a counterdiabatic term (Takahashi, 2014).

This control-theoretic notion has been applied directly to tunneling. For a charged particle, one can choose electromagnetic potentials so that the fast-forwarded state equals $P^t$10 exactly, with both amplitude and phase accelerated throughout the protocol. The vector potential is

$P^t$11

the magnetic field vanishes, and the current obeys

$P^t$12

The time-averaged tunneling rate therefore satisfies

$P^t$13

(Khujakulov et al., 2016). In the high-barrier $P^t$14-barrier regime, choosing $P^t$15 restores a recognizable tunneling current from the suppressed $P^t$16 scale (Khujakulov et al., 2016).

Open-system and resonance-control variants extend the same logic. In a locally coupled oscillator-bath model, fast-forward driving uses only accessible controls—the system frequency and the system-bath coupling—to reproduce the outcome of an adiabatic heat-exchange process in finite time (Villazon et al., 2019). The rotating-wave fast-forward protocol modifies

$P^t$17

so as to induce a resonant state exchange with a bath mode, while a Floquet-engineered protocol extends the construction beyond the rotating-wave regime (Villazon et al., 2019). For two linearly coupled qubits, inter-trajectory travel supplements fast-forward scaling when no continuous speed-controlled trajectory exists: a virtual phase trajectory $P^t$18 is constructed to keep a mismatch measure $P^t$19 small, allowing both acceleration and deceleration with high fidelity using only detuning control (Masuda et al., 2021).

These control-theoretic works clarify an important terminological issue. Here fast-forwarding is not a statement about query complexity or circuit depth; it is a statement about reproducing a target state or adiabatic outcome in shorter physical time by adding accessible control fields (Villazon et al., 2019, Masuda et al., 2021). This differs categorically from algorithmic QFF, even though both involve structurally enabled shortcuts.

The resulting landscape is therefore heterogeneous but coherent. Reversible Markov chains, structured Hamiltonians, stable or dissipative ODEs, and several NISQ subspace methods admit genuine algorithmic speedups; strict dissipativity can even produce $P^t$20 history-state preparation (An et al., 2024). By contrast, generic Hamiltonian simulation, generic nonreversible Markov dynamics, and marginally stable unitary evolution remain excluded by no-fast-forwarding barriers or boundary-case arguments (Banerjee et al., 25 Jun 2026, Jennings et al., 2023). A plausible implication is that QFF is best understood not as a universal capability, but as a collection of structure-sensitive mechanisms whose power is determined by reversibility, compact algebraic closure, diagonalisability, contractivity, or experimentally accessible gauge freedom.