How fast can a quantum system evolve? Quantum speed limits (QSLs) are lower bounds on the time needed to go from one state to another, set by the energy available to the dynamics. In quantum control they appear in two ways: as fundamental limits that no pulse can beat, and, through the closely related minimal control time, as the shortest duration at which a given control problem becomes solvable. Below that time no optimizer can reach unit fidelity, and close to it optimization becomes hard.

The two classic bounds

For a system with time-independent Hamiltonian \(H\), starting in \(|\psi_0\rangle\), let \(\tau_\perp\) be the time to reach an orthogonal state. Two bounds hold:

$$\tau_\perp \;\ge\; \frac{\pi\hbar}{2\,\Delta E} \qquad\text{(Mandelstam–Tamm)}, \qquad\qquad \tau_\perp \;\ge\; \frac{\pi\hbar}{2\,\big(\langle E\rangle - E_0\big)} \qquad\text{(Margolus–Levitin)},$$

where \(\Delta E\) is the energy uncertainty and \(\langle E \rangle - E_0\) the mean energy above the ground state [1, 2]. Both hold at once, so the larger of the two is the relevant bound. This unified bound is tight: for any values of \(\Delta E\) and \(\langle E\rangle - E_0\) there are states that approach it arbitrarily closely [3].

The Mandelstam–Tamm bound has a geometric meaning. The state moves through projective Hilbert space at speed \(\Delta E/\hbar\), as measured by the Fubini–Study metric [4]. Since the distance to a target is the angle \(\arccos|\langle\psi_0|\psi_{\rm tgt}\rangle|\),

$$\tau \;\ge\; \frac{\hbar\,\arccos\big|\langle \psi_0 | \psi_{\rm tgt}\rangle\big|}{\Delta E}$$

for any target, not only orthogonal ones. For time-dependent Hamiltonians, the same argument holds with \(\Delta E\) replaced by its time average along the path.

Generalizations

  • Open systems. Speed limits have been derived for non-unitary dynamics. They are based on the quantum Fisher information [5], on the Lindblad generator in a Mandelstam–Tamm-like form [6], and on operator norms of the generator in a Margolus–Levitin-like form. In the last case, non-Markovian effects can speed up the evolution [7].
  • Other geometries. Every valid measure of distinguishability between quantum states gives a speed limit, and the choice can be made to tighten the bound [8].
  • Classical systems. Speed limits are not uniquely quantum: equivalent bounds exist for classical dynamics in phase space [9].
  • Experiments. The crossover between the Mandelstam–Tamm and Margolus–Levitin regimes has been observed with atoms in a multilevel system, using fast matter-wave interferometry [10].

A comprehensive review groups these results into a geometric approach (distinguishability of states) and a minimal-time approach (control theory) [11].

Speed limits in optimal control

The bounds above refer to a given Hamiltonian. A control problem asks something different: the minimal control time to reach a target when the Hamiltonian can be shaped within constraints. The textbook QSLs usually say little about this time, and bounds adapted to control are needed [12]:

  • Optimal control finds the limit. Scanning the duration \(T\) with an optimizer shows a sharp threshold. Above a minimal time unit fidelity is reached, below it is not, and in paradigmatic cases this threshold coincides with the speed limit allowed by the dynamics [13].
  • Drift and amplitude bounds. For a closed system, the minimal time for a target gate can be bounded from below in terms of the gate, the strength of the drift and the maximal control amplitude. For a single qubit this bound describes the numerical minimal times remarkably well [14].
  • Exact minimal times. For a driven two-level system, the minimal time and the optimal protocol are known exactly [15]. Geometric control gives exact minimal times more generally for small systems.

Many-body systems

In interacting systems, locality matters. Information propagates at a finite speed — the Lieb–Robinson bound [16] — so transferring a state along a chain takes a time that grows with its length. Optimal control can drive an excitation along a spin chain at this maximal propagation rate [17]. When only local controls are available, state preparation obeys a geometric speed limit set by the quantum geometric tensor. In glassy or chaotic systems it can be exponentially larger than the Mandelstam–Tamm time [18]. The quantum speed limit of atom transport has even been explored with a citizen-science game [19].

Why it matters

  • Coherence budgets. Faster operations accumulate less decoherence, so the minimal time is a design target for gates and state preparation; see Quantum gate synthesis and State preparation & transfer.
  • Optimization difficulty. Near the minimal time, the control landscape becomes rugged and even glassy, so optimizers need more restarts or global methods.
  • Resources. How the minimal time depends on bandwidth and on the number of control parameters is the subject of Control complexity & resources.