Adiabatic processes are robust: if a Hamiltonian changes slowly enough, a system follows its instantaneous eigenstate and ends up exactly where intended. But “slowly enough” can be too slow for the available coherence time. Shortcuts to adiabaticity (STA) reach the same final state as a perfectly adiabatic process in a finite, short time, by designing the Hamiltonian so that the excitations created along the way are cancelled or undone [1, 2]. Most shortcuts are analytic or semi-analytic, which makes them a natural partner and starting point for numerical optimal control.
The adiabatic starting point
For a Hamiltonian \(H_0(\lambda(t))\) with instantaneous eigenstates \(|n(t)\rangle\), the adiabatic theorem [3, 4] guarantees that a system starting in \(|n(0)\rangle\) stays in \(|n(t)\rangle\) (up to a phase) if the rate of change is small compared with the energy gaps. Shortcuts remove the need for slowness in three main ways.
Counterdiabatic (transitionless) driving
Add a term that exactly cancels the non-adiabatic couplings. With
$$H(t) = H_0(t) + H_1(t), \qquad H_1(t) = i\hbar \sum_n \Big( |\partial_t n\rangle\langle n| - \langle n|\partial_t n\rangle\, |n\rangle\langle n| \Big),$$
the system follows the eigenstates of \(H_0\) exactly, at any speed [5, 6]. Equivalently \(H_1 = \dot\lambda\,\mathcal A_\lambda\), where \(\mathcal A_\lambda\) is the adiabatic gauge potential [7].
For a two-level system \(H_0 = \tfrac{\hbar}{2}\big(\Delta(t)\,\sigma_z + \Omega(t)\,\sigma_x\big)\) — a Landau–Zener sweep — the counterdiabatic term is a field along \(\sigma_y\), \(H_1 = \tfrac{\hbar}{2}\,\Omega_a(t)\,\sigma_y\), with \(\Omega_a = (\dot\Omega\Delta - \Omega\dot\Delta)/(\Omega^2 + \Delta^2)\) (up to sign conventions). It has been realized with Bose–Einstein condensates in optical lattices [8] and with a single NV-centre spin [9].
The difficulty is that \(H_1\) requires knowledge of all eigenstates and, in many-body systems, typically involves non-local, many-body terms that cannot be implemented. Two ideas address this:
- Experimentally realizable protocols for single-particle, many-body and non-linear systems that do not need the spectrum as input, applied for example to fast decompression of condensates [10];
- Local counterdiabatic driving, which approximates the gauge potential variationally with the terms that are actually available [11], or as a series of nested commutators realized by periodic driving [12]. See Floquet engineering & periodic driving.
Invariant-based inverse engineering
A Lewis–Riesenfeld invariant \(I(t)\) satisfies \(\partial_t I + \tfrac{1}{i\hbar}[I, H] = 0\), and the eigenstates of \(I(t)\), with suitable phases, solve the Schrödinger equation exactly [13]. One designs \(I(t)\) to commute with \(H\) at the initial and final times, then inverts it to find the Hamiltonian that realizes it. Applied to expanding a harmonic trap, this gives frictionless cooling in a time much shorter than adiabatic expansion. The trap may transiently become expulsive, and the cooling times are shorter than those of bang-bang optimal control restricted to real trap frequencies [14]. Formulated as a time-optimal control problem, the minimal-time solution is itself bang-bang [15]; see Geometric & time-optimal control.
Fast-forward
The fast-forward approach derives driving potentials that reproduce, in any desired short time, the result of a slow adiabatic process — transport, splitting or squeezing of a wavefunction — up to a global phase [16].
In quantum computing
Counterdiabatic terms can be added to digitized adiabatic algorithms, improving fidelity with shallow circuits on current hardware [17]. Adding non-stoquastic counterdiabatic terms to adiabatic optimization for Ising spin glasses has been proposed to give a polynomial enhancement over standard adiabatic optimization [18]. See Variational algorithms & QAOA as control.
Costs and limits
Shortcuts are not free. There is a rigorous trade-off between speed and the energetic cost of the counterdiabatic field. Instantaneous driving would need infinite cost, and the spectral gap governs both the speed limit and the cost [19]; see Quantum speed limits. Different shortcuts to the same target also differ in their sensitivity to noise and systematic errors, and the most robust one can be selected [20]; see Robust & ensemble optimal control.
Shortcuts and optimal control
The two approaches are complementary. Shortcuts provide interpretable, analytic protocols and good initial guesses. Optimal control can then add constraints (bounded amplitudes, limited couplings), robustness, or push towards the minimal time. In thermodynamics, shortcuts suppress the “quantum friction” of finite-time cycles; see Control in quantum thermodynamics. Applications span cold-atom transport and cooling, population transfer, and State preparation & transfer.