Ecient inference in the transverse eld Ising model E. Dom nguezand H.J. Kappeny Donders Institute for Brain Cognition and Behavior. Radboud University. The Netherlands

2025-05-03 0 0 864.4KB 14 页 10玖币
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Efficient inference in the transverse field Ising model
E. Dom´ınguez∗and H.J. Kappen†
Donders Institute for Brain, Cognition and Behavior. Radboud University. The Netherlands
(Dated: January 30, 2023)
In this paper we introduce an approximate method to solve the quantum cavity equations for
transverse field Ising models. The method relies on a projective approximation of the exact cavity
distributions of imaginary time trajectories (paths). A key feature, novel in the context of similar
algorithms, is the explicit separation of the classical and quantum parts of the distributions. Nu-
merical simulations show accurate results in comparison with the sampled solution of the cavity
equations, the exact diagonalization of the Hamiltonian (when possible) and other approximate
inference methods in the literature. The computational complexity of this new algorithm scales
linearly with the connectivity of the underlying lattice, enabling the study of highly connected
networks, as the ones often encountered in quantum machine learning problems.
I. INTRODUCTION
Many relevant systems in statistical mechanics, combinatorial optimization and quantum information theory can
be mapped to realizations of the so-called transverse field Ising model (TFIM) [1, 2]. Notable examples include Kitaev
1D chains [3, 4], many NP-complete problems such as 3SAT [5] and the foundations of the whole field of adiabatic
quantum computation [6–8]. Given its numerous applications, the study of the properties of this lattice spin model
has attracted considerable attention in the last decades. Thanks to that, much have been learned about the nature
of quantum phase transitions, especially for the 1D case. However, despite all efforts, doing inference (i.e. computing
observables) remains very hard in the general case. The problem stems, as usual, from the exponential increase in size
of the Hilbert space with the number of variables. There are two potential ways around the dimensionality curse. One
is using a quantum computer to directly implement the required Hamiltonian and let Nature deal with the Hilbert
space. Unfortunately, the development of a general purpose quantum computer seems to be at least some decades in
the future and near term devices are challenged by decoherence effects and thermal fluctuations. The other option is
to make use of suitable approximations and make computations in a classical device.
The quantum cavity method (QC), introduced in [9, 10], enables a reduction of the complexity of the inference
tasks for a TFIM. It combines an imaginary-time path integral expansion of the density matrix with the cavity
formalism, well known in the study of classical disordered systems [11]. As in the conventional cavity approach,
QC establishes that all the information of the system is encoded in the set of single variable cavity distributions for
interaction networks lacking short loops. This is not true for general topologies but QC remains a useful mean field
approximation in that scenario.
The set of coupled equations that determine the cavity distributions can be, at least in principle, solved by an
iterative procedure, formally equivalent to the message passing algorithm known as Belief Propagation (BP) [12]. In
practice, the implementation cost is similar to a population dynamics algorithm [13], with the important difference
that in the present case the sampled variables are trajectories and not real values. There are problems for which
the inference calculations must be repeated many times, e.g. for the learning loop of a quantum Boltzmann machine
[14, 15]. Therefore, it is highly desirable to develop time- and memory-efficient (albeit approximated) methods to solve
the QC equations. This is the main purpose of the present work. By focusing of the structure of the QC equations
and the properties of the cavity distributions we will be able to derive a fast, precise and flexible solution to the
inference problem for the TFIM. To put our analysis in context, we will consider the predictions of other previously
developed inference methods. Moreover, we will show how these approximations fit into the QC formalism.
Following the introduction of QC, approximated solution schemes were devised to further reduce its computational
burden. Already in [10], the static approximation was proposed as a cheaper way to obtain qualitatively correct results.
The static numerical predictions, however, are far from ideal. Of special interest to us is the projective procedure
employed in [16], by which the information in the (cavity) distribution is compressed to just a couple effective field
parameters. Remarkably, the projected cavity method (PCM), developed therein, is quite precise in the numerical
estimation of the observables and the critical temperature of sparse ferromagnetic lattices. Two drawbacks of PCM
are the disadvantageous scaling (exponential) of the running time with the system connectivity and that it is specially
∗eduardo.dominguezvazquez@donders.ru.nl
†b.kappen@science.ru.nl
arXiv:2210.11193v2 [cond-mat.dis-nn] 27 Jan 2023
2
fine tuned for longitudinal observables, somewhat at the expense of less transversal precision. Other approximations
considered in [16] are based on more qualitative mean field arguments. This is the case of the so-called naive mean
field and the cavity mean field (CMF) methods. The CMF proposal is simpler and faster than PCM but, as we will
see, it does not reduce to the cavity solution in the classical limit of no tranverse field. This should be a minimal
requirement to all approximate solutions of the quantum cavity equations that aim at being useful for a wide range
of values of the external parameters.
An inference method applicable to the TFIM is the quantum Cluster Variation Method and the related algorithm
called quantum Belief Propagation (qBP) [17, 18]. At first glance, the standard derivation based on the minimization
of a variational Bethe free energy functional is somewhat removed from the QC formalism. However, we can show
that it is possible to establish a connection and consider qBP as an approximate solution scheme of the QC equations.
The rest of the paper is organized as follows. In section II, we give a precise definition of the TFIM and review the
quantum cavity approach. We recast the exact cavity equations in a form that is suitable for subsequent approximate
treatments. In section III we will introduce the main result of this work, an inference algorithm that we call quantum
cavity mean field (qCMF), and discuss the connections with other inference approximations. Numerical simulations
in different scenarios and topologies are discussed in section IV. Finally, in section V we provide an overview of the
results, some concluding remarks and perspectives for future developments.
II. QUANTUM CAVITY SOLUTION OF THE TFIM
In this section we present a summary of the quantum cavity formalism as applied to the transverse field Ising model
on a diluted graph. We will introduce the basic Hamiltonian of the system, the corresponding density matrix and
sketch the mapping that turns the quantum statistical problem into a classical probability distribution. We skip the
detailed derivation and refer the interested reader to the original papers [9, 10].
The TFIM is defined by the following Hamiltonian:
ˆ
H=−X
(ij)
Jij ˆσz
iˆσz
j−X
i
hiˆσz
i−X
i
Biˆσx
i(1)
The transverse fields Biintroduce non-commuting terms to an otherwise classical (diagonal) Hamiltonian. In the
context of the cavity method, the interaction topology specified by the symmetric matrix Jij corresponds to a diluted,
tree-like network. Realizations of such geometries include Cayley trees as well as random regular and Erdos-Renyi
graphs. One can also apply the cavity method in other situations, such as in finite dimensional regular lattices. In
that case, instead of producing an exact solution, it is interpreted as an structured mean field approximation.
The quantum cavity method is based on a Suzuki-Trotter expansion of the density matrix ˆρ=1
Zexp −βˆ
H[19]. This
procedure reveals a mapping between the quantum system defined by (1) and an equivalent classical model with an
extra dimension (the so-called imaginary time coordinate). The classical equivalent has the same interaction topology
as the original, with trajectories (paths) replacing the quantum spins in the nodes of the network structure. The
statistics of the new system will be given by the joint probability distribution ρ(σ) of all trajectories σ= (~σ1, . . . , ~σN).
Each path, denoted ~σi, is a piecewise constant function taking the values ±1 alternatively in the imaginary time
direction, see Fig. (1). A precise definition of ρ(σ) is given in equations (2) to (4):
FIG. 1: The classical variables ~σiare stepwise functions of the (discrete or continuous) imaginary time coordinate.
3
ρ(σ) = 1
Zexp [−βEz(σ)] Y
i
w(~σi, Bi) (2)
Ez(σ) = −X
(ij)
Jij~σi·~σj−X
i
him(~σi) (3)
w(~σi, Bi) =
Ns
Y
τ=1hστ
i|exp βBi
Ns
ˆσx|στ+1
ii(4)
In equation (3) we have defined m(~σi) = 1
NsPτστ
iand ~σi·~σj=1
NsPτστ
iστ
j, where Nsis the number of Suzuki-Trotter
slices used in the expansion. The τindex is the imaginary time coordinate, taking values in the range [1, . . . , Ns].
We will, however, normally work in the continuous time limit Ns→ ∞. Then τbecomes a continuous parameter
and the previous sums become definite integrals in the interval [0,1]. Notice that the energy function Ez(σ) does
not contain interactions between different Trotter slices. Moreover, the external fields hiact homogeneously in the
imaginary time direction. Different slices are coupled only by the w(~σi, Bi) factors. The form of these wfactors imply
that in the classical regime, Bi→0, the trajectories with non-zero probability are only those which are constant in
the imaginary time direction. It is also easy to see in the definition of w(~σi, Bi) that this interaction represents a
one-dimensional ferromagnetic chain in the imaginary time direction.
The quantum-classical mapping is completed once we specify the correspondence of quantum operators representing
physical observables with functions of the spin trajectories: ˆ
F→F(σ). The quantum average hˆ
Fi= Tr[ˆρˆ
F] becomes
a classical average PσF(σ)ρ(σ). For example, to compute mz
i=hˆσz
iione has to average m(~σi) over the marginal
ρi(~σi):
mz
i=X
~σi
m(~σi)ρi(~σi) (5)
See Appendix (A) for the definitions of other observables of interest such as the mx
imagnetization and pairwise
correlations. The important point to note here is that these quantities typically rely on the computation of marginals
of the joint density ρ(σ), that is, on the distributions of small subsets of trajectories. The inference of such marginals
is computationally unfeasible, much like in the original quantum system, where the exponentially growing size of the
Hilbert space forbids any attempts of computing partial traces of the density matrix. At this point is where the cavity
method comes into action.
The cavity approach exploits the structure of the interaction network to simplify the computation of local marginals
such as ρi(~σi) and ρij (~σi, ~σj). The functional form of the resulting local distributions include a Boltzmann-like
term with the interactions appearing in the original Hamiltonian. The effect of the neighbors is included by extra
multiplicative factors. The single spin (trajectory) distribution has the form:
ρi(~σi) = 1
Zi
w(~σi, Bi) exp [βhim(~σi)] Y
k∈∂i
Mk→i(~σi) (6)
here ∂i represents the set of spins that are neighbors of spin i. For interacting pairs we have a similar structure:
ρij (~σi, ~σj) = 1
Zij
exp [βJij ~σi·~σj]µi→j(~σi)µj→i(~σj) (7)
The µand Mfactors above are called cavity distributions. Any given µi→j(~σi) is interpreted as the probability
distribution of the variable ~σiin a system where the interaction with site jis removed. Contrastingly, although
Mk→i(~σi) is positive for all ~σiand can be normalized to a probability distribution, its physical meaning is not so clear
in this quantum scenario. A reader familiar with the classical Belief Propagation will identify (6) and (7) as the belief
distributions and the Mand µfunctions will correspond to messages in a factor graph [12].
The cavity distributions µand Mmust take values such that the consistency of (6) and (7) holds. The probability
function of any pair must be consistent with each single trajectory distribution:
ρi(~σi) = X
~σj
ρij (~σi, ~σj) (8)
The constraints introduced by (8) give the relation between the µand Mdistributions:
µi→j(~σi) = 1
Zµ
i→j
w(~σi, Bi) exp [βhim(~σi)] Y
k∈∂i\j
Mk→i(~σi) (9)
Mk→i(~σi) = 1
ZM
k→iX
~σk
exp [βJik~σi·~σk]µk→i(~σk) (10)
摘要：

Ecientinferenceinthetransverse eldIsingmodelE.DomnguezandH.J.KappenyDondersInstituteforBrain,CognitionandBehavior.RadboudUniversity.TheNetherlands(Dated:January30,2023)Inthispaperweintroduceanapproximatemethodtosolvethequantumcavityequationsfortransverse eldIsingmodels.Themethodreliesonaprojecti...

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