What on Earth is counterdiabatic driving? (Or: the waiter and the glass of wine (water) example that’s been done to death)

Ground states of quantum systems are cool – and I don’t just mean their temperature. Finding the ground state of a particular Hamiltonian is often equivalent to solving a very hard optimisation problem. Being able to prepare and manipulate such ground states is also incredibly important in understanding various quantum phenomena, such as high-temperature superfluidity and superconductivity as well as, say, topological quantum computing.

So how do we prepare these quantum states? One popular approach is to start with the ground state of a time-dependent Hamiltonian H(0), which we know how to prepare easily but might not be particularly useful to us. We can then evolve this for a time T to Hamiltonian to H(T) whose ground state encodes something interesting (say, the solution to the aforementioned optimisation problem or some novel phase of matter). If we perform this evolution slowly enough, we can keep the quantum system in the instantaneous ground state for every time t ∈ [0, T]. This means that at every point in time, the quantum system is in its ground state of H(t) (assuming all ground states are non-degenerate throughout the evolution).

Importantly, the change from H(0) to H(T) has to be slow enough. This is because of something called the Quantum Adiabatic Theorem [Childs08], which tells us that the speed at which you can evolve a time-dependent Hamiltonian depends on the energy gap ∆ϵ between the ground state and first excited state of the system. If the time T is too short, the system has a high chance of transitioning out of the ground state into an excited one. This is, unfortunately, a pretty big problem when trying to prepare ground states adiabatically: evolving a quantum system for very long times makes it difficult to maintain coherence, but evolving it too quickly excites it out of the required state. The question then becomes how to speed up these Hamiltonian dynamics without exciting unwanted transitions.

One way of achieving this exact thing is to use something called counterdiabatic driving (CD) [Demirplak03], which involves counteracting the excitations by applying an external drive. The concept of CD can be made clear using a very popular analogy (see Fig. 1):

Imagine a waiter carrying a glass of water on a tray from the bar to a customer at a table. When the waiter starts moving they accelerate and induce a ‘force’ on the glass, making it wobble and splash around. The faster the waiter moves, the stronger the force, so the only way to keep the glass from spilling is to move very slowly. One way they can counteract this force and walk quicker is by tilting the tray with the glass so as to keep the water from spilling, modifying the tilt as they weave around the restaurant to account for the shifting direction and magnitude of the force. Note that we don’t care what position the glass is while it’s being carried, only that it is upright, still and full of water at the beginning and end of its journey.

If we now imagine the glass of water and tray to be the quantum system, then its ground state at time t=0 is standing still and upright on the bar and its ground state at t=T is standing perfectly still and upright on the customer’s table. The time-dependent Hamiltonian is, therefore, the path of the waiter from the bar to the customer. When the waiter moves too fast, the glass wobbles – equally, when the Hamiltonian changes too quickly in time, the quantum system may get excited from the ground state. The trick in the quantum case is to try and imitate what the waiter does and tilt the tray.

As you may have already guessed, this tilting is a stand-in for the idea of CD. As in the case of the waiter, what we want to do as we vary our Hamiltonian with time is to counteract the possible excitations it induces in the quantum state and we can do this by applying an external ‘drive’ (tilt of the tray) on top of the original Hamiltonian. In essence, if we could derive the form of these excitations throughout the state evolution exactly we would know the CD exactly too and we could perform something called transitionless driving, meaning that no excitations would even be possible no matter how quickly our Hamiltonian changed. Sounds fantastic, right?

Well, the issue here is that deriving the exact form of the CD required to have lossless Hamiltonian evolution is very very hard and we’d need to have exact knowledge of the entire energy spectrum of the system at every point in time to be able to do this. Even for systems of only a few spins (or qubits, should you prefer) and very simple dynamics this becomes almost impossible. In practice, we need to try something a little different: approximate CD.

A good example of this is the approach in a paper by D. Sels and A. Polkovnikov [Sels17], where they take inspiration from the waiter example in a more practical way. When the waiter tilts the glass as in Fig. 1, they don’t actually know the exact microscopic movements of each molecule of water in the glass that they need to counteract, only the approximate direction and magnitude of the tilt to keep the glass from tipping over. In this vein, Sels and Polkovnikov propose an approximate CD protocol where we can make a decent Ansatz for the form of the CD based on the system we’re working with and can optimise it by treating it as a sort of perturbation theory. These approximate drives can be incredibly effective even to first-order, as in the case of the waiter and they can be applied to essentially any closed system with a time-dependent Hamiltonian (though not always as efficiently depending on the application at hand).

Several approaches now exist that are inspired by approximate CD in one way or another for various applications (see, for example, this paper where a two-parameter CD drive is applied to investigating quantum phase transitions in for the p-spin model [Prielinger21]). It looks like a promising new direction with many improvements still to be made and many applications in many-body physics and optimisation problems to be exploited.

References

[Childs08] A. Childs, LECTURE 18: The quantum adiabatic theorem, University of Waterloo Quantum Algorithms course, 2008

[Demirplak03] S. A. Rice, M. Demirplak, Adiabatic Population Transfer with Control Fields, J. Phys. Chem. A 107, 46, 9937–9945, 2003

[Berry09] M.V. Berry, Transitionless quantum driving, J. Phys. A: Math. Theor. 42 365303, 2009

[Sels17] D. Sels, A. Polkovnikov, Minimizing irreversible losses in quantum systems by local counterdiabatic driving, 2 PNAS  114 (20) E3909-E3916, 2017

[Prielinger21] L. Prielinger, A. Hartmann, Y. Yamashiro, K. Nishimura, W. Lechner, H. Nishimori, Two-parameter counter-diabatic driving in quantum annealing, Phys. Rev. Research 3, 013227, 2021