**J. C. Lemm and J. Uhlig
Institut für Theoretische Physik I
Universität Münster, 48149 Münster, Germany
**

A new method is presented to reconstruct
the potential of a quantum mechanical many-body system
from observational data,
combining a nonparametric Bayesian approach
with a Hartree-Fock approximation.
*A priori* information
is implemented
as a stochastic process,
defined on the space of potentials.
The method is computationally feasible
and provides a general framework to treat
inverse problems for quantum mechanical many-body systems.

The reconstruction of inter-particle forces from observational data is of key importance for any application of quantum mechanics to real world systems. Such inverse problems have been studied intensively in inverse scattering theory and in inverse spectral theory for one-body systems in one and, later, in three dimensions [1,2]. In this Letter we now outline a method, designed to deal with inverse problems for many-body systems.

Inverse problems are notoriously ill-posed
[3].
It is well known that for ill-posed problems
additional *a priori* information is required
to obtain a unique and stable solution.
In this Letter we refer to a nonparametric Bayesian framework
[4,5]
where *a priori* information
is implemented explicitly, i.e.,
in form of stochastic processes over potentials
[6].

Calculating an exact solution is typically not feasible for inverse many-body problems. As an example of a possible approximation scheme we will study in the following an `Inverse Hartree-Fock Approximation' (IHFA). For situations where a Hartree-Fock (HF) ansatz is not sufficient, the inverse problem would have to be solved on top of other approximation schemes. A Random Phase Approximation or a full Time-Dependent Hartree-Fock Approximation [7,8,9,10], for example, would go beyond HF.

Bayesian methods
can easily be adapted to different learning situations
and have therefore been applied
to a variety of empirical learning problems,
including classification, regression, density estimation
[11,12,13],
and, recently, to quantum statistics [6].
In particular, using a Bayesian approach for quantum systems
it is straightforward
to deal with measurements of arbitrary quantum mechanical observables,
to include classical noise processes,
and to implement *a priori* information
explicitly in terms of the potential.

Computationally, on the other hand, working with stochastic processes, or discretized versions thereof, is much more demanding than, for example, fitting a small number of parameters. This holds especially for applications to quantum mechanics where one cannot take full advantage of the convenient analytical features of Gaussian processes. Due to increasing computational resources, however, the corresponding learning algorithms become now numerically feasible.

To define the problem
let us consider many-fermion systems with Hamiltonians,
= , consisting of a one-body part
(e.g., in coordinate space representation ,
with Laplacian ,
mass , = 1),
and a two-body potential .
Introducing fermionic creation and annihilation operators
, , corresponding to a complete
single particle basis
,
such Hamiltonians can be written,

(1) |

In order to apply the Bayesian framework we need two model inputs:
firstly, the probability
of measuring the observational data given a potential
(for fixed also known as *likelihood* of ),
and, secondly,
a *prior* probability for .
The probability for given data ,
also called the *posterior* probability for ,
is then obtained according to Bayes' rule,

As the first step,
we have to identify
the likelihood of for observational data .
In order to be able to obtain information about the potential,
the system has to be prepared in a -dependent state.
Such a state can be a stationary statistical state,
e.g. a canonical ensemble,
or a time-dependent state evolving according to the Hamiltonian
of the system.
In the following we will discuss many-body systems
prepared in their ground state .
The (normalized) -particle ground state wave function
depends on and is antisymmetrized for fermions.
In particular,
we will study two kinds of observational data =
:
(A) simultaneous measurements of the coordinates
of all particles,
(B) measurements of the coordinates of a single particle.
In case ,
the th measurement results in a vector = ,
consisting of components ,
each representing the coordinates of a single particle
(which may also form a vector, e.g., a three dimensional one).
According to the axioms of quantum mechanics,
the probability of measuring
the coordinate vector , given , is,

Similarly, it is for example straightforward to obtain the likelihoods for measuring inter-particle distances , the particle number in a certain region, or particle momenta.

In contrast to an ideal measurement of a classical system,
the state of a quantum system is typically
changed by the measurement process.
In particular, its state is projected
in the space of eigenfunctions of the measured observable
with eigenvalue equal to the measurement result.
Hence, if we want to deal with
independent, identically distributed data,
the system must
be prepared in the same state before each measurement.
Under that condition the total likelihood factorizes

As the second step, we have to choose a prior probability .
A common and convenient choice are Gaussian prior probabilities
(or, for functions, Gaussian processes),
for which

Having defined
a likelihood for many-body quantum systems
and a prior probability
the next step is to solve the stationarity equation
for the posterior probability (2)

denoting the gradient with respect to , having components . To calculate the derivatives of the likelihoods (3) and (4) we need to know . It is straightforward to show, by taking the derivative of = with respect to that, for a nondegenerate ground state,

with a complete basis of eigenstates , energies , and requiring = 0 to fix norm and phase. Furthermore, from = directly follows

(10) |

being the -lowest orthonormalized eigenstates of , i.e.,

The corresponding normalized -particle Hartree-Fock ground state = is the Slater determinant made of the -lowest single particle orbitals . The Hartree-Fock likelihood replacing (3) becomes,

defining the overlap matrix with matrix elements = . Analogously, the likelihood (4) becomes

From (13) follows = + , so that expanding the determinant = in terms of the cofactors, = , yields

(15) |

Again, we proceed by taking the derivative of
Eq. (12)
and obtain after standard manipulations
(for nondegenerate
and
= 0),

Therefore, the derivative of the orbitals is found by inserting Eq. (17) into Eq. (16)

That ``inverse HF equation'' (18) can quite effectively be solved by iteration, starting for example with initial guess = 0. Because depends on and , Eq. (18), being the central equation of the IHFA, has the dimension of a two-body equation for the lowest orbitals. Introducing, analogously to , the matrix with = (for , , , ), it is straightforward to check that the derivatives of the logarithm of the HF likelihoods (13) and (14) are given by

The stationarity equation (7) can now be solved by iteration, for example,

choosing a positive step width and starting from an initial guess .

In conclusion,
reconstructing a potential from data by IHFA is based on
the definition of a *prior probability*
for
and requires the iterative solution of

1. the *stationarity equation for the potential* (7),
needing as input for each iteration step (21)

2. the *derivatives of the likelihoods* (19),
obtained by solving the (two-body-like) equation (18)
for given

3. *single particle orbitals*, defined in (12) as solutions of
the direct (one-body) Hartree-Fock equation.

We tested the numerical feasibility of the IHFA for a Hamiltonian = including a local two-body potential to be reconstructed and a local one-body potential , with diagonal elements = 0 for and = elsewhere, to confine the particles and to break translational symmetry.

To check the validity of the IHFA we must be able to sample artificial data from the exact true many-body likelihood (3) for a given true potential . Because this requires to solve the corresponding many-body problem exactly, we have chosen a two-particle system with one-dimensional (on a grid with 21 points, = 10) for which the true ground state can be calculated by diagonalizing numerically. We want to stress, however, that, in the case of real data, application of the IHFA to systems with particles is straightforward and only requires to solve Eq. (18) for instead for two orbitals.

We selected a local true two-body potential with diagonal elements = and parameter values ( = 1, = 10, = = 21 for mass = ) for which the iteration of the HF equation (12) converges. (For two-body systems the HF iteration leads easily to oscillations.) Calculating the true ground state for and the corresponding true likelihoods (3) and (4) we then sampled: (A) = 100 two-particle data from the true likelihood (3) and (B) = 200 single particle data from the true likelihood (4). (See Figs. 1a and 2a.)

The calculations have been done for a Gaussian prior probability as in (6) with (with identity , = ) as inverse covariance , and a reference potential of the form of , but with = 1 (so it becomes nearly linear in the shown interval.) Furthermore, we have set all potentials to zero at the origin and constant beyond the right boundary. The reconstructed potential has then been obtained by iterating according to Eq. (21) and solving Eqs. (12) and (18) within each iteration step.

The resulting IHFA likelihoods (case A, Fig. 1a) or (case B, Fig. 2a) did indeed fit well the true likelihoods or , respectively. (Fig. 1a shows instead of the two-dimensional for vectors the one-dimensional = for the inter-particle distance .) In particular, in case A the reconstructed likelihood is over the whole range an improvement over the reference likelihood , while in case B the IHFA solution is nearly exactly the same as the true likelihood . That perfect result for case B is due to the fact that reconstructing the likelihood for single particle data is a much simpler task than reconstructing the full .

The situation is more complex
for potentials (Figs. 1b and 2b).
Firstly, one sees that the correlation information
contained in the two-particle data of case A
yields a better reconstruction for
than the less informative single particle data of case B.
In both cases, however, the true potential
is only well approximated at medium inter-particle distances.
For large and small distances, on the other hand,
the IHFA solution is still dominated by
the reference potential
of the prior probability .
This effect is a consequence of the lack of empirical data
in those regions:
The probability to find particles at large distances is small,
because the true potential has its maximum at large distances.
Also, measuring small distances is unlikely,
because antisymmetry
forbids two fermions to be at the same place.
In such low data regions one must therefore
rely on *a priori* information.

We are grateful to A. Weiguny for stimulating discussions.