Finding the Ground State of Quantum Many-Body Systems

 By Ieva Čepaitė, University of Strathclyde, Glasgow

Ground states of many-body quantum systems (especially systems made of spins) are cool – and I don’t just mean their temperature. Finding a system's ground state is often equivalent to solving a very hard optimization 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 do it? One way is counterdiabatic driving, and it's a lot like running with a glass of water on a tray. Let me explain.

We think of quantum states in terms of their Hamiltonian, an operator that represents a system's potential and kinetic energy as an equation. In order to represent the ground state, we begin with a pared-down, time-dependent version of the Hamiltonian called H(0). This state is easy to prepare, but isn't particularly useful—it's basically just a snapshot of the system. However, we can then evolve this Hamiltonian for a time T to Hamiltonian H(T)—like combining snapshots into a movie—whose ground state encodes something interesting (say, the solution to the aforementioned optimization 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 point in time (assuming all ground states are non-degenerate, meaning they represent unique energies 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 [1], which tells us, simply, that functions describing states evolving slowly will adapt to the slowly-changing conditions. However, if the time T is too short, the system has a high chance of transitioning out of the ground state into an excited one. Preparing ground states in this way, or adiabatically, is difficult; 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 is to use something called counterdiabatic driving (CD) [2], which involves counteracting potential excitations by applying an external drive. The concept of CD can be made clear using a very popular analogy (see Fig. 1):

This may LOOK like a waiter's hand carrying a tray with a glass of water (or whatever suits your fancy), but really it's a quantum system in (A) its ground state at t = 0, (B) evolving via time-dependent Hamiltonian H(t) and (C) evolving via H(t) with an added counterdiabatic drive.

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 the waiter 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 "wobble" too much and 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, we want to counteract the possible excitations that varying the Hamiltonian with time 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 exactly derive the form of these possible excitations throughout the state evolution, then we would know the CD exactly, too. This would allow us to perform something called transitionless driving, meaning that no excitations would even be possible no matter how quickly our Hamiltonian changed. This is a consequence of work done by Michael Berry [3], who showed that there can be a Hamiltonian which doesn’t allow for transitions to other states regardless of speed, although this Hamiltonian is hard to find. 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 [4], 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 guess for the form of the CD based on the system we’re working with and can optimize it by treating it as a sort of perturbation theory. These approximate drives – extra terms in the Hamiltonian like an additional magnetic field gradient or electric field pulse – 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 [5]). It looks like a promising new direction with many improvements still to be made, applications in many-body physics, and even potential benefits for optimization problems.

[1] A. Childs, LECTURE 18: The quantum adiabatic theorem, University of Waterloo Quantum Algorithms course, 2008
[2] S. A. Rice, M. Demirplak, Adiabatic Population Transfer with Control Fields, J. Phys. Chem. A 107, 46, 9937–9945, 2003
[3] M.V. Berry, Transitionless quantum driving, J. Phys. A: Math. Theor. 42 365303, 2009
[4] D. Sels, A. Polkovnikov, Minimizing irreversible losses in quantum systems by local counterdiabatic driving, 2 PNAS  114 (20) E3909-E3916, 2017
[5] 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