A quantum system of \(n\) qubits has a Hilbert space of dimension \(2^n\). Naively, controlling it would need an exponential number of control parameters and exponentially long pulses. Yet optimal control routinely prepares many-body states with a few dozen parameters. Control complexity explains why, and asks what resources a control task really needs: how many parameters, how much bandwidth, how much time, and how much classical information.

The illusion of Hilbert space

Physical dynamics cannot explore all of Hilbert space. States that local, time-dependent Hamiltonians can generate in a time polynomial in the system size occupy an exponentially small volume of it. The overwhelming majority of states can only be produced after an exponentially long time [1]. Control problems of physical interest therefore live on a much smaller manifold than the full space.

How many parameters are needed?

The size of the space of control parameters needed to solve a control problem — for pure states, mixed states or unitaries — is polynomially bounded by the size of the set of states reachable in polynomial time [2]. Two consequences:

  • Few parameters can be enough. If the dynamics has an efficient classical description — for example one-dimensional, slightly entangled dynamics — the control problem can be solved efficiently to finite precision [2]. This underlies the success of compact parametrizations such as CRAB and of tensor-network-based Many-body optimal control.
  • The dimension of the dynamics sets the cost. The control complexity of a many-body evolution is bounded from below by the dimension of the manifold that supports it. This makes the role of integrability explicit: conserved quantities confine the dynamics to smaller manifolds [3]. With partial control of the Hamiltonian, arbitrary many-body evolutions can even be reversed, in a way much more robust to noise than standard time-reversal procedures [3].

Simulations of constrained problems tell the same story from the optimizer’s side. The minimum number of controls that ensures a high yield depends on the number of accessible energy levels [4]. With too few control variables the landscape develops traps [5]; see Control landscapes.

Bandwidth and time

Time and bandwidth trade off against each other. Lloyd and Montangero give a bound on the minimal time of a control process for a given control bandwidth, a continuous analogue of the Solovay–Kitaev theorem [2]. Minimal times scale with the inverse of the bandwidth, a scaling that optimization in a basis of Slepian functions reaches for entangling two-qubit gates [6]. See Quantum speed limits and Pulse parametrization & constraints.

Noise and information

Control fields carry information, and noise degrades it. Information–time bounds quantify how well a system can be controlled under arbitrary noise sources. They agree with standard decoherence formulas and predict how the achievable accuracy scales with the noise, which has been tested numerically with dCRAB [7]. See Robust & ensemble optimal control and Control of open quantum systems.

  • Geometric (circuit) complexity. The number of gates needed for a unitary can be cast as a geodesic length in a suitably weighted geometry on the unitary group [8]; see Geometric & time-optimal control.
  • Computational complexity. Deciding whether a control task is possible at all is undecidable for digitized controls on a Turing machine [9]. In the real-number model of computation, the polynomial formulation of QCPOP makes optimal control computable [10].
  • Algebraic complexity. The dimension of the dynamical Lie algebra determines whether gradients of variational circuits vanish exponentially (barren plateaus) [11]. It is a measure of how “large” the controlled dynamics is; see Variational algorithms & QAOA as control.

Practical lessons

  • Start with a compact, physically motivated parametrization and add parameters only until fidelity saturates.
  • If a problem needs an exponential number of parameters, it is probably asking for a state that the dynamics cannot reach in reasonable time.
  • Low entanglement in the controlled dynamics is a resource: it makes both simulation and control efficient.