How is Quantum Phase Estimation useful for simulating dynamics of a many-body system?

Question

I am quite aware of the Quantum Fourier Transform (QFT) as well as the very closely related topic of Quantum Phase Estimation (QPE). The latter is usually motivated as follows:

Given a unitary $U$ and a state $|\psi \rangle$ that is promised to be an eigenstate of the unitary, $U|\psi \rangle = \lambda |\psi \rangle$, compute the global phase $\lambda = \mathrm{e}^{i\vartheta}$. The circuit implementation is quite straight forward to me, with the inverse QFT needed in the end so as to obtain $\vartheta$.

Reading this article the authors claim, by quoting papers such as this Lloyd's paper, that QPE can be used for modelling the full time evolution of some quantum systems.

Although I could not really understand Lloyd's paper in detail, I could not find how the QPE is implemented or used.

My question is how one can use QPE for simulating maby-body quantum systems?

Answer 1

I generally think of it the other way around. Simulating dynamics (ie. evolving a system in time) is used in Quantum Phase Estimation (QPE). That is, the $U$ appearing in QPE is the time-evolution operator $U=e^{-iHt}$, where $H$ is the Hamiltonian of the system. Part of the QPE protocol requires implementing $U^k$ for increasing powers $k$. This reduces to simply evolving the system for longer and longer times $t$.

Here's a good reference: https://arxiv.org/abs/quant-ph/0604193

(Published version: https://www.science.org/doi/10.1126/science.1113479)

Now, why bother doing QPE for a physical system?

The phases $\lambda$ given by QPE are the eigenvalues of the time evolution operator $U$. These are relatively easily mapped back onto the eigenenergies of the Hamiltonian $H$. These eigenenergies are usually the thing chemists are trying to find, as they alone are sufficient to deduce a host of chemical properties like ionization potential and equilibrium constants.

Better yet, QPE reads out an eigenphase classically, but it also prepares the corresponding eigen-state. This makes it a useful starting point for calculating other useful observables besides energy, or perhaps simply preparing a reference state to obtain the ground-state energy for a more precise Hamiltonian. It seems like this is what the first article linked in the question is referring to.

Moreover, once you have all the eigenstates and eigenenergies in a region of interest, you have enough to very easily understand how a (closed) system develops in time - you simply decompose your initial state into a sum of the eigenstates and develop each eigenstate according to the phase you found in QPE. The final state is an interferometric sum of all components.

Do note that QPE is not the only algorithm available for doing all these things, though it is the most iconic. To name just a few, the Variational Quantum Eigensolver (VQE) is also quite popular, Quantum Imaginary Time Evolution (QITE) strikes me as promising (but I'm no expert), and the first article linked in the question itself presents two more.

I'm happy to provide more detail if requested.

Answer 2

After some research I found an illustrating example on how QPE is used for simulation of many-body systems. The idea is the following:

1. Start with a (possibly time dependent) Hamiltonian $H$ of a many-body system, e.g. a molecular system. The evolution operator then reads: $$ U = \mathrm{e}^{-{\rm i}Ht}. $$
2. Use Trotter decomposition so as to break this unitary to a sum/product of many unitaries: $$ U \approx ( {\rm e}^{-{\rm i}\, h_1\,\delta t}\ldots {\rm e}^{-{\rm i}\, h_n\,\delta t})^{1/\delta t} $$ for Hermitian matrices $h_i$. $U$ can be viewed as the propagator.
3. Use the Jordan-Wigner mapping to convert the propagator above into a sequence of quantum gates.
4. Knowing that for a stationary initial state $|\psi_0\rangle$ it holds that the action $\mathrm{e}^{-{\rm i}Ht}|\psi_0\rangle$ will produce an eigenvalue $ \mathrm{e}^{-{\rm i}E_0t}$, use QPE to compute ${\rm e}^{2\pi {\rm i}(\phi+k)}$

Specifically for the QPE algorithm we read:

The key idea is to Fourier transform the oscillating phase, $\langle\psi(0)|\psi(t)\rangle = \exp(−{\rm i}Et)$, to obtain the electronic energy. The eigenenergy is converted into relative phases. The relative phase can then be measured using the quantum phase estimation algorithm. As the phase is measured the input state partially collapses to the set of states consistent with the measurements obtained up to that point.

Details can be founds in this paper by James D. Whitfield, Jacob Biamonte, Alán Aspuru-Guzik.