Before asking how to steer a quantum system, one can ask whether it can be steered at all. Controllability asks which states or operations are reachable with the available controls. For finite-dimensional systems the answer is algebraic. The reachable set is determined by the dynamical Lie algebra generated by the drift and control Hamiltonians. This is one of the cleanest results of the field, and the starting point for gate design, for the analysis of control landscapes and, more recently, for understanding variational quantum algorithms.

The setting

A closed system with controls entering linearly — a bilinear control system — evolves as

$$\dot U(t) = -i\Big(H_0 + \sum_{k=1}^{K} u_k(t)\,H_k\Big)\,U(t), \qquad U(0) = \mathbb 1 ,$$

with real control functions \(u_k(t)\). The question is which unitaries \(U(T)\) (or which states \(U(T)|\psi_0\rangle\)) can be reached for some choice of controls and some time \(T\).

The Lie-algebra rank condition

Switching quickly between different Hamiltonians generates their commutators: to second order, \(e^{-iA\tau}e^{-iB\tau}e^{iA\tau}e^{iB\tau} \approx e^{-[A,B]\tau^2}\). Repeating the argument, the directions in which the system can move are those of the dynamical Lie algebra

$$\mathfrak L = \mathrm{Lie}\big\{-iH_0,\,-iH_1,\,\dots,\,-iH_K\big\},$$

the smallest real vector space of anti-Hermitian matrices that contains the generators and is closed under commutators. For finite-dimensional systems, the set of reachable unitaries is the connected Lie group \(e^{\mathfrak L}\) generated by this algebra. This rests on the theory of control systems on Lie groups [1], and on the fact that unitary groups are compact, so the drift cannot push the system irreversibly in one direction. In particular:

Complete (operator) controllability

A \(d\)-level system is completely controllable — every unitary can be reached, up to a global phase — if and only if \(\mathfrak L = \mathfrak{su}(d)\) or \(\mathfrak u(d)\) [2].

Computing \(\mathfrak L\) is mechanical. One takes nested commutators of the generators and keeps the linearly independent ones until the dimension stops growing; for full controllability it must reach \(d^2-1\) (or \(d^2\)). For specific classes of systems, sufficient conditions can be read from the structure of the coupling. For example, a single control field driving several transitions at once can still give complete controllability of Morse- and oscillator-like ladders [3].

Different notions of controllability

What is “reachable” depends on what we want [4]:

  • Operator (complete) controllability — every unitary is reachable: \(\mathfrak L \supseteq \mathfrak{su}(d)\). This is what gate synthesis needs.
  • Pure-state controllability — every state can be reached from every other state. This is weaker: it also holds when \(\mathfrak L\) is isomorphic to the symplectic algebra \(\mathfrak{sp}(d/2)\) (with or without \(\mathfrak u(1)\)), which acts transitively on the sphere of states without containing all of \(\mathfrak{su}(d)\).
  • Density-matrix (kinematic) controllability — any two density matrices with the same spectrum are connected. For closed systems it is equivalent to operator controllability.

Symmetries as obstructions

A conserved quantity blocks controllability. If an operator \(S\) commutes with \(H_0\) and with every \(H_k\), then the dynamics never mixes the eigenspaces of \(S\) — for example, total spin or particle number in a symmetric system. The converse is subtler: absence of symmetry gives a convenient necessary condition that is much easier to check than the full Lie-algebra computation. A complete classification of the relevant subalgebras even yields a single necessary and sufficient symmetry criterion [5].

Beyond finite, closed systems

  • Infinite dimensions. For the Schrödinger equation in infinite-dimensional spaces (oscillators, particles in potentials), exact controllability generally fails for bilinear systems [6]. The relevant notion is then approximate controllability — reaching any target to arbitrary precision — studied since the earliest work on quantum controllability [7]. See Bosonic & continuous-variable control.
  • Open systems. With dissipation, the dynamics is no longer unitary and the notions change. Some states may not be reachable at all, while dissipation can also help, for example by driving the system into a target state [8]. See Control of open quantum systems and Reservoir engineering & dissipative preparation.
  • Computability. Whether a given task can be achieved with digitized controls is, in general, an undecidable question for a Turing machine [9]; see QCPOP & polynomial optimization.

Why it matters in practice

  • Universality. In quantum computing, controllability of a multi-qubit system is universality: almost any entangling two-qubit interaction combined with local controls suffices [10].
  • Necessary, not sufficient. Controllability says a target can be reached in some time with some controls. How fast is the subject of Quantum speed limits and Geometric & time-optimal control, and with what resources is the subject of Control complexity & resources.
  • Landscapes. Controllability is one of the assumptions under which control landscapes are free of traps; see Control landscapes.
  • Variational algorithms. The dynamical Lie algebra of a variational ansatz determines the scaling of its gradients, and therefore whether it suffers from barren plateaus [11]; see Variational algorithms & QAOA as control.