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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?

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    $\begingroup$ This question is maybe too broad. What do you mean by "simulating many-body quantum systems"? Do you want an example? $\endgroup$
    – Mauricio
    Nov 10, 2021 at 15:56
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    $\begingroup$ Evidently the OP wants to understand how "QPE can be used for modeling the full time evolution of some quantum systems." I think that is specific enough? But I do not know the answer off-hand. I usually think of it the other way around: one uses time evolution to implement QPE, which provides access to eigenstates and energies. I will read the referenced article some point soon to try and see if it has something else in mind. $\endgroup$
    – jecado
    Nov 10, 2021 at 21:01
  • $\begingroup$ In the first article it is mentioned (see the introductory section) that QPE and quantum amplitude amplification (QAA) can be used so as to do many-body simulation. I do not fully understand the statement given that QPE computes certain eigenvalues. Unless simulation can be reduced to an eigenvalue problem (which I doubt, I think simulation is about mimicking full dynamics and computing quantities such as ground states) I don't really understand yet why QPE is useful there and how their algo is related to QPE it self. $\endgroup$
    – Marion
    Nov 10, 2021 at 21:20
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    $\begingroup$ I will post an answer presently. In the meantime - I didn't see QAA referenced in either article. Just checking to see if you intended to link a different article? $\endgroup$
    – jecado
    Nov 11, 2021 at 15:17
  • $\begingroup$ QAA is briefly mentioned in the Cirac article. $\endgroup$
    – Marion
    Nov 11, 2021 at 20:25

2 Answers 2

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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.

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  • $\begingroup$ That is a good answer but I have to admit I am more puzzled now. Indeed, QPE applies a series of unitaries $U^{j}$ for different durations yielding a big product state out of which (after inverse QFT) we can read out the phase. But, in what sense you can prepare a state using QPE? $\endgroup$
    – Marion
    Nov 11, 2021 at 20:35
  • $\begingroup$ Another comment would be how the algorithm of the 1st paper is similar to QPE. That paper also chops evolution in many pieces (by the way QITE also does this) and applies unitaries and measure expectation values. But I do not know to what extent this is "similar to QPE" as they say. If so, most algos use some sort of Trotterization procedure. $\endgroup$
    – Marion
    Nov 11, 2021 at 20:37
  • $\begingroup$ You know that the phase is obtained in QPE by measuring the control register, after a QFT. That measurement collapses the state register into an eigenstate. The confusion likely comes from the usual presentation of QPE as having a pre-prepared eigenstate as input. But it need not be so! The only catch is that QPE on an arbitrary state, which is composed of many eigenstates, may yield any of the corresponding phases. But, once you've made the measurement, you've collapsed the arbitrary state into one of the eigenstates. $\endgroup$
    – jecado
    Nov 11, 2021 at 20:43
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    $\begingroup$ The eigenstate isn't arbitrary; it's the one corresponding to whichever phase you measured. If you're using QPE as a state preparation algorithm, you have some particular purpose in mind for some particular eigenstate (eg. the ideas presented in my answer). If you measure the wrong phase, you...just try again. :P $\endgroup$
    – jecado
    Nov 11, 2021 at 21:07
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    $\begingroup$ Just added an early reference for QPE for chemistry at the top. I haven't been able to spare the time to read the body of the Cirac paper but my impression is that their reference to QPE is...uh, name-dropping? The term only seems to appear in their introduction, so I don't know whether they meant anything meaningful by the reference other than to contextualize their algorithm as competitive with the most iconic. ^_^ $\endgroup$
    – jecado
    Nov 11, 2021 at 21:11
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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.

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