Bosonic control steers quantum harmonic oscillators: microwave cavities, the motion of trapped ions, optical and mechanical modes. An oscillator has infinitely many levels. A single long-lived mode can therefore hold an error-correctable qubit — in cat, binomial or grid (GKP) codes — without the many physical qubits other codes need [1, 2]. The catch is that the oscillator’s own evolution is linear. Drives that are linear in the field only displace states, and quadratic Hamiltonians generate only Gaussian operations. Every non-Gaussian state, from a Fock state to a cat state, needs a nonlinearity, and most bosonic control is about using a weak one well.

Control in infinite dimensions

  • Controllability. For bilinear systems in infinite dimensions, exact controllability with bounded controls is ruled out in general [3], so the useful notion is approximate controllability. It has been proven for the Schrödinger equation with a discrete, non-resonant spectrum, using Galerkin (truncated) approximations [4]. See Quantum controllability.
  • Gaussian versus universal. Linear optics and squeezing, generated by Hamiltonians at most quadratic in the field, make up Gaussian quantum information [5, 6]. Adding a single higher-order nonlinearity, such as a Kerr interaction, makes the set universal [7]. The Gaussian part is also the domain of Linear quantum systems & coherent feedback.
  • Truncation. Numerical optimization works in a space cut off at some photon number. A result is trustworthy only if it does not change when the cutoff is raised, and if the controlled states stay well away from the edge.

Oscillators controlled through a qubit

Most experiments couple the oscillator to a nonlinear ancilla, such as a transmon or an ion’s internal levels, and control both.

  • Law–Eberly. Any state of a field mode can be prepared by alternating qubit rotations and qubit–oscillator exchange [8]. In circuit QED this produced Fock states [9] and arbitrary superpositions of them [10].
  • SNAP and displacements. In the strong dispersive regime, the qubit frequency depends on the photon number, so qubit drives can give each Fock state its own phase. This is the selective number-dependent arbitrary phase (SNAP) gate [11]. With displacements, SNAP gates give universal control of the oscillator and prepare \(|n\rangle\) in \(O(\sqrt n)\) steps rather than \(O(n)\) [12]. Optimizing the gate parameters and then the pulse envelopes prepared cat, binomial, GKP and cubic-phase states, robustly against fluctuations of the system parameters [13].
  • Optimal control of the full system. With an accurate Hamiltonian, numerical optimal control of the coupled cavity–transmon system realized decoherence-limited operations (98.5% inferred fidelity) on a qubit encoded in four-component cat states [14].
  • Weak coupling, amplified. Echoed conditional displacements use strong drives to amplify a weak dispersive coupling. Operations run more than an order of magnitude faster than the bare coupling allows, preparing Fock states, squeezed vacuum and binomial and GKP logical states [15].
  • Gates that tolerate ancilla errors. Because the ancilla decoheres much faster than the cavity, gates can be designed so that ancilla errors are corrected in real time rather than passed on to the logical qubit [16].

Engineered nonlinearities and dissipation

  • Cat states. Cat states of about 100 photons were encoded deterministically in a cavity [17]. In a Kerr resonator with a two-photon drive, cat states become eigenstates, can be prepared robustly against single-photon loss, and can be initialized non-adiabatically with transitionless driving [18]. This led to the Kerr-cat qubit [19]. See Shortcuts to adiabaticity.
  • Dissipative cats. Engineered two-photon loss confines the field to the cat manifold [20, 21]. See Reservoir engineering & dissipative preparation.

Grid states and error correction

GKP codes protect a qubit against small shifts in phase space [1]. Grid states were prepared and controlled in the motion of a trapped ion [22] and error-corrected with a feedback protocol in a superconducting cavity [23]. A GKP qubit whose parameters were optimized with model-free reinforcement learning stayed coherent 2.27 times longer than the best of its physical components [24]. See Control for quantum error correction, Measurement-based feedback and Reinforcement & machine learning for control.

Limitations

  • Models must include the small terms that dominate at high photon number: self-Kerr, cross-Kerr, higher-order dispersive shifts and drive-induced frequency shifts.
  • The ancilla is usually the noisiest element, so control speed is a trade-off between ancilla decoherence and the strength of the nonlinearity.
  • Fidelities are hard to verify: Wigner or characteristic-function tomography of large states is slow and depends on the same imperfect ancilla.