State preparation drives a quantum system from a known initial state into a target state: a superposition, an entangled state of many qubits, the ground state of a many-body Hamiltonian, a Fock or cat state of an oscillator. State transfer moves an unknown state from one place to another, along a chain or between distant nodes. These were the first problems of quantum control, in the steering of molecular wave packets to select chemical products [1, 2, 3]. They remain the most common ones outside gate design.

The problem

For pure states, the objective is the overlap \(F = |\langle\psi_\mathrm{target}|\psi(T)\rangle|^2\). It involves one state rather than a full unitary, so it is cheaper to compute and needs fewer control resources. Its landscape is generically free of traps when the system is controllable. For mixed or open systems, the objective becomes a distance between density matrices or the expectation of a projector. See Problem formulation & figures of merit. The minimal duration is set by the speed limit, and an optimizer scanning the duration finds it as the time below which the target can no longer be reached [4].

Routes to a target state

  • Adiabatic following. Slowly changing the Hamiltonian keeps the system in an eigenstate that connects the initial and target states. STIRAP and rapid adiabatic passage transfer population completely and robustly [5, 6]. Adiabatic quantum computation prepares ground states of problem Hamiltonians the same way [7]. See Adiabatic passage.
  • Shortcuts. Shortcuts to adiabaticity reach the same final state in a finite time [8], for example cooling atoms in harmonic traps fast and without final excitation [9]. See Shortcuts to adiabaticity.
  • Optimal control. Numerical optimization handles the cases without analytic solutions, from single qubits to many-body states [10, 11]. See GRAPE, CRAB & dCRAB and Many-body optimal control.
  • Constructive sequences. For an oscillator coupled to a qubit, any state can be built step by step [12], as done for arbitrary superpositions of photon numbers [13]. See Bosonic & continuous-variable control.
  • Circuits. Entangling gates build GHZ states. In 14 trapped ions, their coherence decayed with the square of the number of qubits, as expected for correlated phase noise [14].
  • Dissipation. Any pure multipartite state can be the unique steady state of an engineered dissipative process, reached from any initial state [15, 16]. See Reservoir engineering & dissipative preparation.
  • Measurement and feedback. Real-time feedback prepared and stabilized photon-number states in a cavity [17]. See Measurement-based feedback.

State transfer

  • Along spin chains. Even an unmodulated Heisenberg chain transmits a qubit with better-than-classical fidelity over tens of spins [18]. With engineered but fixed couplings, the transfer is perfect over any length of chain [19]. With time-dependent control, the transfer can run at the speed limit [20], and control at one end of the chain can be enough [21].
  • Between distant nodes. In a quantum network, a laser pulse can map an atom’s state onto a time-symmetric photon that a second atom absorbs with unit probability [22]. The microwave version transferred states deterministically between superconducting qubits on separate chips [23].

Cooling and initialization

Preparing a pure state from a thermal one requires removing entropy, so it needs dissipation or measurement. Optimal control has been used to design laser cooling of internal molecular degrees of freedom [24]. The third law limits how fast this can go: for general cooling processes, the attainable temperature scales as an inverse power of the cooling time [25]. Fast, unconditional reset of qubits is part of Control for quantum error correction [26]; see also Control in quantum thermodynamics.

In experiments

  • Cold atoms. Closed-loop optimization sped up atom loading into optical lattices and the crossing of the superfluid–Mott-insulator transition more than threefold [27]. With condensates in optical lattices, both a protocol at the speed limit and a robust superadiabatic (transitionless) protocol were realized [28]. See Ultracold atoms & BECs.
  • Rydberg arrays. Optimal control prepared GHZ states of up to 20 atoms [11]. See Neutral atoms & Rydberg arrays.
  • Citizen science. In a game based on moving atoms with optical tweezers, players found good solutions in regimes where purely numerical optimization struggled [29].

Limitations

  • Verifying a large entangled state is hard: tomography scales exponentially, and witnesses or fidelity estimates replace it.
  • A state-to-state solution works only for the initial state it was designed for, while a gate works for all inputs. Errors in the initial state propagate.
  • Many optimal solutions exist for the same target, and the choice between them should be based on robustness rather than nominal fidelity alone.