Neural quantum states
for supersymmetric quantum gauge theories

Xizhi Han
Kavli Institute for Theoretical Physics
Santa Barbara, CA, US
hanxzh@ucsb.edu
Enrico Rinaldi
University of Michigan, Ann Arbor, MI, US
and Theoretical Quantum Physics Laboratory, RIKEN, Japanand Center for Quantum Computing, RIKEN, Japanerinaldi@umich.edu

Abstract
Supersymmetric quantum gauge theories are important mathematical tools in highenergy physics. As an example, supersymmetric matrix models can be used asa holographic description of quantum black holes. The wave function of suchsupersymmetric gauge theories is not known and it is challenging to obtain withtraditional techniques. We employ a neural quantum state ansatz for the wave
function of a supersymmetric matrix model and use a variational quantum Monte
Carlo approach to discover the ground state of the system. We discuss the difficultyof including bosonic particles and fermionic particles, as well as gauge degrees of
freedom.

1

Introduction

The powerful conjecture of gauge/gravity duality [1, 2] translates difficult (or intractable) problems inquantum gravity to well-defined problems in non-gravitational quantum theories. Quantum mechanics
of matrices (or matrix models in the following) can provide us with a nonperturbative formulation ofsuperstring/M-theory [2–5]: at strong coupling, and in the limit of large N × N matrices, weaklycoupled gravity with small stringy corrections can be described by matrix models of Yang–Mills-type(with gauge degrees of freedom). A bound state of D0-branes with open strings can be interpreted
as a black hole and described by a supersymmetric quantum field theory with bosonic, fermionic,
and gauge degrees of freedom. On the other hand, such bound state of N D0-branes has a dynamics
accurately described by quantum mechanics of large N × N traceless Hermitean matrices.
The ground state wave function of complicated matrix models, such as the aforementioned onesarising from dimensional reduction of supersymmetric SU(N ) Yang–Mills theories, is not known.
However, inspecting the wave function’s structure is of paramount importance to understand quantum
information concepts and one interesting application is the emergence of space-time geometry fromthe entanglement of quantum information in a holographic approach [6]. Unfortunately, traditional
methods used to solve matrix models, such as Lattice Monte Carlo simulations of the Euclidean
path integral [7], can not access the wave function, and are challenging for systems with a sign
problem, where the Boltzmann weight for each path can not be interpreted as a positive probability
measure [8]. Other numerical approaches to quantum states include quantum computing and tensor
network methods. The former is currently limited by the small number of physical qubits and the
Fourth Workshop on Machine Learning and the Physical Sciences (NeurIPS 2021).

latter are unlikely to yield an advantage in this case because matrix models lack the spatial geometriclocality which is present in lattice spin models.
Deep artificial neural network architectures, routinely used in the context of density estimation, canbe used to efficiently approximate quantum states, without imposing constraining structures on the
wave function or limiting their representation power by focusing on simple distributions. Recently,
different numerical approaches to matrix model wave functions have been compared in [9].

2Related works

There has been tremendous work on studying neural network wave function ansatz for quantum
many-body systems [10, 11]. Compared to spin models, neural quantum states of bosons and fermionsare less well understood, and previous research mostly focused on representing the state in the particle
number (Fock) basis [12–14]. In the Fock basis, the number of bosons or fermions in each mode isa discrete variable, and hence can be processed in similar ways as a quantum spin. An interesting
proposal of continuous variable RBM states was discussed in Ref. [15]. In our work, we employ
generative flows as variational wave functions without truncating the bosonic modes. Bosons are
represented in their physical infinite-dimensional Hilbert space. We observe both lower variationalenergies and better symmetry properties compared to the Fock basis methods.
The generative variational wave functions in this work are accompanied by generative flow Monte
Carlo samplers for improved efficiency. The energy of the variational wave function is often evaluatedwith Monte Carlo methods, for example with a Gibbs sampler. Recently, neural generative models
have shown advantages over the conventional methods, both in spin [16–19] and field systems [20–23]. We use the Block Neural Autoregressive Flow (BNAF) [24] as the Monte Carlo sampler, which
generates uncorrelated bosonic and fermionic configurations.
Preservation of gauge symmetries is essential for physical interpretations of the resulting variational
state. Recent success in designing symmetry-invariant or equivariant neural states [23, 25–29]
has shed light on this difficult task. Research for equivariant variational states with non-Abelian
symmetries, or even supersymmetries, however, is still incomplete. In this work the gauge symmetry
is not imposed exactly. Instead, we present two general penalty and projection methods for finding
gauge-invariant states, and observe satisfactory results. In the supersymmetric case supersymmetries(or more generally, the BPS conditions [30]) are also preserved to high accuracy.

3

Methods

3.1

Variational Quantum Monte Carlo

The variational quantum Monte Carlo method consists of three components: a wave function network,a sampler network, and an optimizer. The wave function network is a complex function ψθ (X)parametrized by θ. Here X is the input vector denoting a basis state, for example a binary vector ofspins ups and downs, or a real vector of coordinates of the particles. The sampler is a generativemodel capable of evaluating and sampling from a probability distribution pη (X) (η is the vector ofparameters), where the samples can be fed into the wave function network. The optimizer updates
the parameters θ and η in the networks to lower the variational energy.
The variational energy is computed from Monte Carlo samples: for a Hamiltonian operator Ĥ, thevariational energy Eθ isZθ (X)|ψθ (X)|22 hX|Ĥ|ψθ i
Eθ ≡ hψθ |Ĥ|ψθ i = dX|ψθ (X)| ·= EX∼pη.
(1)ψθ (X)pη (X)
Here θ (X) is defined as hX|Ĥ|ψθ i/ψθ (X), and the expectation value is estimated as the mean of
the samples.
During training, the parameters θ are updated to minimize the energy Eθ and the sampler parameters
η are updated to minimize the Kullback–Leibler divergence between distributions pη and |ψθ |2 :pη (X)DKL (pη k|ψθ |2 ) = EX∼pη log.
(2)|ψθ (X)|22

In particular, during each step we assign θ ← θ − β∇θ Eθ , and η ← η − β∇η DKL (pη k|ψθ |2 ), with alearning rate β = 10−3 . The gradients are∇θ (θ (X)|ψθ (X)|2 )∇θ Eθ = EX∼pη
,(3)pη (X)pη (X),(4)∇η DKL (pη k|ψθ |2 ) = EX∼pη ∇η (log pη ) log|ψθ (X)|2
PK
and the expectation values E[f (X)] are estimated as Monte Carlo sample averages i=1 f (Xi )/K
in stochastic gradient descents. In (4), ∇η (log pη ) comes from a gradient of the sampling distribution
X ∼ pη , and we have also noted that EX∼pη [∇η log pη (X)] = 0. The (locally) minimal value forEθ found is a variational upper bound for the ground state energy.
We have chosen to minimize DKL (pη k|ψθ |2 ) to reduce the variance of the Monte Carlo estimate for(1). This choice is guided by the following consideration. In the limit where both the wavefunction andthe sampler networks have strong representation power, pη = |ψθ |2 minimizes the KL divergence (2)and ψθ = ψ0 minimizes the energy Eθ , where Ĥ|ψ0 i = E0 |ψ0 i is the true ground state wavefunction.
In this limit, the quantity in the expectation value of (1) is a constant E0 , and hence its Monte Carloestimate has zero variance. Hence in practice we expect and see small and controlled sample variancefor the estimate of (1), showing the effectiveness of this procedure.
3.2

Wave function and sampler

Generative flows are essential in both our wave function and sampler networks. The basic idea offlow models is to generate complex distributions by applying nonlinear but reversible transformations
to simpler ones. If x is a sample from the (continuous) distribution pX (x), and y = F (x), by achange of variable we know that pY (y) = pX (F −1 (y))|detDF |−1 . The resulting pY (y) can be very
complicated if F is highly nonlinear. We can then use feed-forward networks as the transformation
F , but we should guarantee that F is reversible and the Jacobian is efficient to compute.
In the BNAFs, reversibility is guaranteed by imposing some autoregressive structure. Basically ify and x are vectors, yi depends only on x1 to xi , and ∂yi /∂xi > 0, theQJacobian would be lower
triangular with positive diagonals, and hence invertible. Its determinant i ∂yi /∂xi is also easy tocompute.
Let pθ (X) be a probability distribution constructedfrom the BNAF. In the bosonic case, our variap
tional wave function ansatz is then ψθ (X) = pθ (X) exp(iFθ (X)), where Fθ is a fully-connected
feed-forward network generating the phase. The sampler could just be pθ , sharing the parameters, ora separate BNAF pη .
For supersymmetric models, the quantum wave function ψ(X, ξ) depends on bosonic matrices X
and fermionic matrices ξ. For ξ we choose the fermion number basis as the basis for quantum states:
the coordinatesp (X, ξ) for the wave function contain real numbers for X and binaries for ξ. In thiscase we use pθ (X)(fξ (X) + igξ (X)) as the variational wave function, where pθ (X) is given by a
BNAF and fξ and gξ are real functions given by fully-connected neural networks. The weights andbiases in fξ and gξ depend on the binary vector ξ, via additional fully-connected networks. Becauseof the use of fξ and gξ , the sampler is a separate BNAF, and in general will be different from pθ .
In more detail, we implement the same BNAF architecture as described in Ref. [24], with default
hyperparameter choices as follows. For each BNAF network we have one hidden layer. The activation
function is SinhArcsinh with trainable skewness and tailweight parameters. The number of hidden
units is α times the number of input (or output) units. Results for different α’s are compared in Tables
1 and 2. We have also experimented with other generative flow architectures [25], and BNAF shows
the best accuracy in the models we consider.
3.3

Preservation of gauge symmetries

In this work gauge invariance is imposed by either adding a gauge Casimir penalty, or by projecting
the state onto the gauge invariant subspace, during or after training. A gauge Casimir penalty can beP
added as a perturbation to the Hamiltonian, Ĥ 0 = Ĥ + c α Ĝ2α , where c > 0 is a hyperparameter3

and Ĝα are operators generating the gauge symmetry. The projection
P̂ onto the singlet sector can beRwritten as an average over gauge transformations P̂ |ψi = dU Û |ψi, where the integration is overthe gauge group. For any gauge invariant observable Ô, its expectation value in the projected state is
hÔisinglet = hψ|P̂ Ô|ψi/hψ|P̂ |ψi, and can be estimated with Monte Carlo samples of U and X.

4

Results

We consider two matrix models to benchmark our method: a boson-only matrix model with 2 matrices
and SU(2) gauge group (hereafter, the bosonic model), and a supersymmetric matrix model with
2 bosonic matrices and 1 fermionic matrix with SU(2) gauge group (hereafter, the minimal BMN
model [5]). Both models can be defined with local Hamiltonians described in details by Ref. [9] andthe sought-after physical states are those invariant under SU(2) gauge transformations.
Our benchmark focuses on the expectation value of relevant operators (energy, gauge Casimir) for
various strengths of the gauge coupling λ = g 2 N = 2g 2 . For these SU(2) gauge models, where
the number of degrees of freedom is limited, we can compare directly with results from exact
diagonalization of a truncated Hamiltonian (using the Fock basis) in the limit where the truncation
effects are negligible [9]. Moreover, for the supersymmetric model we know that the energy of theground state is exactly zero at each coupling due to supersymmetry.
All results in this section are obtained by training the parameters (θ, η) on a multi-core CPU laptop inless than 24 hours, with 200 samples in each mini-batch, 1000 batches in each epoch, and about 150
training epochs for each coupling. In training, the Adam optimizer is used with an initial learning
rate β = 10−3 and the rate is reduced when the loss function plateaus. The training is stopped when
the learning rate drops below 10−5 . The final expectation value of observables is evaluated from 106
Monte Carlo samples.
Table 1: Neural variational quantum Monte Carlo ground state energy for the bosonic model. The
exact result of the Hamiltonian truncation (HT) is reported in the last column.
α

1

2

5

10

20

50

HT (exact)

λ = 0.2
λ = 0.5
λ = 1.0
λ = 2.0

3.137(2)3.313(2)3.544(3)3.914(3)3.137(2)3.312(2)3.544(2)3.910(3)3.140(2)3.308(2)3.541(3)3.892(3)3.138(2)3.307(2)3.528(2)3.872(3)3.137(2)3.302(2)3.519(2)3.857(3)3.135(2)3.305(2)3.520(2)3.859(3)

3.134
3.297
3.516
3.854

For the SU(2) bosonic model, as a warm up test for our neural ansatz, we assess the accuracy of thevariational energies by increasing the ratio α of the number of hidden units to features in the BNAFψθ (X). With more parameters (larger α) we expect that the ansatz is more flexible and accurate, andindeed convergence to exact results for the energy of the ground state E0 is observed in Table 1 forα ≥ 20. More parameters are needed to represent the ground state accurately at larger gauge couplingstrengths (for reference, the zero coupling limit of this model is just a system with 6 independent
harmonic oscillators). We have also tested our neural ansatz on the case of larger SU(N ) algebras,
which amounts to having a larger number of continuous variables as input to the neural network,without any appreciable degradation of the accuracy of the results, or an inflation of the training time.
In the supersymmetric model we see vanishing variational energies as we increase α, as P
expected. In
Table 2 we also report the convergence of other observables: the gauge Casimir G2 = α G2α , theSO(2) angular momentum M , and the fermion number F . When evaluated on the variational state,
they also converge to correct values for α ≥ 20. For example, the ground-state energy of bosonic andfermionic oscillators are ±(N 2 − 1) = ±3, and they cancel out. The results at α = 20 means thecancellation is reproduced up to one-percent-order error.
Results at α = 20 for other couplings are summarized in Table 3 and we can see that they become lessprecise as the system becomes more strongly coupled (note change in F for different λ is expected).
4

Table 2: Neural variational quantum Monte Carlo ground state observables for the SU(2) minimal
BMN model, at coupling λ = g 2 N = 1.0 and µ = 1. The exact result of the Hamiltonian truncation(HT) is reported in the last column.
α

1

5

10

20

50

HT (exact)

H
G2
M
F

0.058(6)0.007(8)-0.0003(3)0.1844(6)0.041(6)0.014(8)-0.0001(4)0.1895(6)0.031(6)0.007(9)0.0001(4)0.1922(6)0.014(6)0.022(9)-0.0003(5)0.1946(7)0.005(6)0.012(9)-0.0001(4)0.1935(7)

0.000
0.000
0.00000.2034

Table 3: Neural variational quantum Monte Carlo ground state observables for the SU(2) minimal
BMN model at various couplings λ.

5

λ

0.5

1.0

2.0

H
G2
M
F

0.009(5)0.010(6)-0.0002(3)0.1224(4)0.014(6)0.022(9)-0.0003(5)0.1946(7)0.034(7)0.038(14)0.0006(7)0.2729(9)Conclusions

In this work we have shown how effective a BNAF variational quantum state is in a supersymmetricmodel with bosonic, fermionic, and gauge degrees of freedom. The ground state properties arecorrectly reproduced when enough parameters are included in the neural ansatz using continuous
variables to represent bosons. We note how this ansatz is more limited when the system is driven
to stronger coupling where the number of fermions and bosons can increase and the wave function
becomes more complicated.
As a remark, we think that these supersymmetric matrix models could become challenging benchmark
tasks for developing new neural ansatz architectures and algorithms. On such benchmark tasks one
could tune the difficulty by choosing the number of bosons (and even add fermions) and the gauge
group algebra, and some tasks also have exact results or analytical predictions (e.g. at very large N ).
At large N and strong coupling, the wave function should contain details of quantum gravity, andhence an efficient numerical method would be both powerful and exciting.

Acknowledgments and Disclosure of Funding
We thank Masanori Hanada, Michael McGuigan, Franco Nori, Mohammad Hassan, and Yuan Feng forcollaboration on a related work. E.R. is supported by Nippon Telegraph and Telephone Corporation(NTT) Research. We acknowledge the use of the Supercomputer HOKUSAI BigWaterfall at RIKEN.

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