Many-body optimal control deals with systems of many interacting particles: spin chains, atoms in optical lattices, Rydberg arrays, Bose–Einstein condensates. The tasks are the same as elsewhere — prepare a state, cross a phase transition quickly, transfer an excitation, implement an evolution — but two features change the problem. The Hilbert space grows exponentially with the number of particles, so the dynamics can rarely be simulated exactly. And the controls are usually few, global fields (a lattice depth, a laser detuning, a Rabi frequency) acting on all particles at once.

What makes it hard

  • Simulation. Exact propagation, and with it GRAPE-type gradients, is limited to a few tens of spins. Larger systems need approximate dynamics: tensor networks in one dimension — matrix product states, the time-evolving block decimation [1] and time-dependent DMRG [2] — or mean-field descriptions, such as the Gross–Pitaevskii equation for condensates [3, 4]. See Numerical propagation & simulation.
  • Reachability. Global controls respect the symmetries of the system, which restrict what can be reached [5]. Conversely, acting on a small part can be enough. Heisenberg chains, for instance, are fully controllable by operating on one spin at their end [6]. See Quantum controllability.
  • Time. Correlations spread at a finite speed in lattice systems [7], and transfers along chains have a minimal time [8]. Crossing a quantum phase transition without creating defects takes at least a time of order \(\pi/\Delta\), where \(\Delta\) is the minimal spectral gap. The optimal dynamics reaching this bound is not adiabatic [9]. See Quantum speed limits.

How few parameters suffice

States that local Hamiltonians can reach in polynomial time fill an exponentially small part of Hilbert space. The number of control parameters a problem needs is set by the dimension of that reachable set, not of the full space. When the dynamics has an efficient classical description, such as slightly entangled one-dimensional dynamics, the control problem can be solved efficiently [10]. Conserved quantities shrink the manifold further [11]. See Control complexity & resources.

This is why compact parametrizations work so well here. CRAB was introduced precisely to control a many-body system simulated with time-dependent DMRG [12]. Optimizing a handful of coefficients keeps the number of expensive simulations small, and dCRAB removes the traps that a truncated basis introduces [13, 14].

Methods in use

  • Open-loop optimization on approximate models: CRAB/dCRAB with tensor networks [12], and GRAPE or Krotov with the Gross–Pitaevskii equation [4].
  • Closed-loop optimization on the experiment itself, which removes the need to simulate the dynamics at all [15, 16]. See Closed-loop & in-situ optimization.
  • Counterdiabatic and variational driving, which approximates the adiabatic gauge potential with local terms [17]. See Shortcuts to adiabaticity.
  • Reinforcement learning, which also showed that many-body state preparation has distinct “phases” of control, set by the protocol duration [18], including a glassy phase with many near-optimal protocols [19]. See Reinforcement & machine learning for control.

In experiments

  • Optical lattices. Closed-loop optimization of atom loading and of the superfluid–Mott-insulator crossing improved the result and sped up the process more than threefold compared with adiabatic ramps [15]. Optimal transformations were computed and verified experimentally for the motional states of a condensate and for a phase-transition crossing in small lattice systems, and they proved robust to temperature and atom-number fluctuations [20].
  • Rydberg arrays. Optimal control protocols were proposed for ordered many-body ground states and GHZ states of Rydberg lattice gases [21]. Experimentally, GHZ states of up to 20 atoms were created by engineering the energy spectrum and using optimal control of the many-body system [22]. See Neutral atoms & Rydberg arrays.

Connections

Many-body control meets quantum computing in Variational algorithms & QAOA as control, where QAOA and quantum annealing are control protocols for spin Hamiltonians. It is also the natural setting for State preparation & transfer on cold-atom and Rydberg simulators.

Limitations

  • Optimal controls are only as good as the approximate model used to find them: truncation errors in a tensor network or the breakdown of mean-field theory can be mistaken for good performance.
  • Tensor-network methods are efficient only while entanglement stays low, which fast, strongly driven protocols can violate. Two-dimensional systems remain hard.
  • Global controls cannot correct local imperfections, such as disorder or inhomogeneous couplings, without help.