Bayesian autotuning of Hubbard model quantum simulators Ludmila Szulakowska Jun Dai Department of Chemistry University of British Columbia Vancouver B.C. V6T 1Z1 Canada

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Bayesian autotuning of Hubbard model quantum simulators

Ludmila Szulakowska, Jun Dai

Department of Chemistry, University of British Columbia, Vancouver, B.C. V6T 1Z1, Canada

Stewart Blusson Quantum Matter Institute, Vancouver, B.C. V6T 1Z4, Canada

(Dated: October 7, 2022)

Spins in gated semiconductor quantum dots (QDs) are a promising platform for Hubbard model

simulation inaccessible to computation. Precise control of the tunnel couplings by tuning voltages on

metallic gates is vital for a successful QD-based simulator. However, the number of tunable voltages

and the complexity of the relationships between gate voltages and the parameters of the resulting

Hubbard models quickly increase with the number of quantum dots. As a consequence, it is not

known if and how a particular gate geometry yields a target Hubbard model. To solve this problem,

we propose a hybrid machine-learning approach using a combination of support vector machines

(SVMs) and Bayesian optimization (BO) to identify combinations of voltages that realize a desired

Hubbard model. SVM constrains the space of voltages by rejecting voltage combinations producing

potentials unsuitable for tight-binding (TB) approximation. The target voltage combinations are

then identiﬁed by BO in the constrained subdomain. For large QD arrays, we propose a scalable

eﬃcient iterative procedure using our SVM-BO approach, which optimises voltage subsets and

utilises a two-QD SVM model for large systems. Our results use experimental gate lithography

images and accurate integrals calculated with linear combinations of harmonic orbitals to train the

machine learning algorithms.

I. INTRODUCTION

Recent progress in manipulating spins in gated semi-

conductor quantum dots (QDs) [1–3] oﬀers an oppor-

tunity for realizing scalable quantum systems for ap-

plications in quantum computing [4–9] and simulation,

such as Kitaev chain or Hubbard model simulation [10–

14]. Hubbard models with many sites, inhomogeneous or

time-varying parameters remain largely unexplored due

to computational complexity, but can be studied exper-

imentally with new generations of gated QD arrays. A

successful QD-based device appropriate for this goal must

provide means to control the charge occupation, chemi-

cal potential of QDs and tunnel couplings with consider-

able precision in order to prepare, manipulate and read

out many-particle quantum states [4, 5, 15, 16]. There-

fore, developing the tools to identify the charge states and

tune their properties to desired operating regimes is vital

for scaling up the complexity of quantum simulators and

discovering new physical phenomena, such as topological

phases [11, 17–20] and strongly-correlated ground states

[10, 12–14, 16, 21]. In particular, the inter-dot tunneling

amplitude requires special attention as it determines the

exchange interaction of spins in QDs [16, 22], which af-

fects all their applications, ranging from qubit design to

parameters of interacting electron models under study.

Design of QD-based simulator relies on well-established

techniques of trapping electrons in electrostatic potential

wells, i.e. QDs, created at the interface of semiconductor

devices build with silicon [1, 2, 23], germanium [3, 24],

III-V materials [13, 25–27] or 2D semiconductors [7, 28–

34]. This conﬁning potential landscape is determined by

a set of voltages applied to lithographically fabricated

metallic gates placed on top of the nanomaterial. Con-

tacts are reservoirs of electrons placed at the edges of the

structure to allow for electron tunneling into the con-

ﬁning potential wells, where they can be manipulated.

Barrier gates are used to control tunneling between QDs,

while plunger gates are designed to alter the depth of each

potential well. Changing the voltages on all those gates

allows for realizing a vast range of electron states for

various applications. Characterization of charge states

achieved with diﬀerent voltage combinations is usually

performed by repeatedly measuring the charge stability

diagram – an image of transport features as a function

of gate voltages, essential for experimentally tuning the

system to a desired regime [26, 35].

This approach to control the gate voltages limits the

scalability of QD-based simulators. For systems with

many quantum dots, the number of gates is large and

the design complexity makes the voltage calibration pro-

cess impractical for manual tuning. Moreover, in dense

devices, the relative proximity of gates produces substan-

tial cross-talk, which further complicates the independent

QD control [12, 36–38]. An additional obstacle for prac-

tical QD simulators is the presence of charge impurities

unavoidable in the fabrication process, which alter the

potential landscape and lead to non-uniform device per-

formance [12, 36, 37]. These challenges, combined with

variations of gate geometry, and a wide range of material

parameters and screening eﬀects impede the development

of practical tools for experimental control of QD-based

simulators.

Machine learning (ML) has emerged as a promis-

ing tool for some of the experimental challenges with

QD control [37]. Deep neural networks [39–46], im-

age recognition [38, 40, 42, 47–50] and supervised clas-

siﬁcation [40, 46, 51] have been demonstrated to aid

charge state characterization [41, 42, 48, 49, 51], cou-

pling parameter tuning [37] and gate voltage optimiza-

tion [36, 41, 43, 46, 49, 52] in a single QD [43, 49, 51],

double QDs [36–38, 42–44, 51], triple QDs and arrays

arXiv:2210.03077v1 [cond-mat.mes-hall] 6 Oct 2022

of QDs [36, 37, 41, 45, 48]. Unsupervised statistical

methods [52, 53] and deterministic algorithms [36, 49–

51] have also been used for double-QD tuning. ML also

proven useful for compensating for cross-capacitance in

devices [45], calibration of virtual gates in place of real

ones [45, 48] and in the analysis and parameter-extraction

from charge stability diagrams [48].

A vast majority of these approaches relied on experi-

mentally obtained data as input [38, 48] or intermediate

step in a feedback protocol [36, 37, 42, 49, 51, 52], which

required numerous measurements or involved readjust-

ments and recapturing procedures [36, 37, 42, 43, 49, 51–

53]. Although scarcity of experimental data has been

addressed in Refs [43, 44] with synthetic data, many ML

solutions for QD simulators suﬀer from crude theoretical

assumptions. This includes the Thomas-Fermi approxi-

mation for electron density [44, 54], the use of exponen-

tial ﬁts to tunneling couplings [48] or constant interac-

tion model with weak coupling and absent barrier gates

[50], which limits their applicability to a wider range of

designs and materials. Another limitation of the opti-

mization techniques used in Refs [38, 45] is the need for

obtaining the gradients of gate voltages in the parameter

search, which may be prone to vanishing gradient prob-

lem [55]. The relationship between gate voltages and the

parameters of the resulting quantum simulator is further

complicated by the scarcity of the physical simulation

domain: the majority of the voltage combinations pro-

duce unphysical potentials. Thus, an automated ﬁrst-

principle design of quantum simulators must be able to

recognize the physical subdomain of experimentally tun-

able parameters.

To address this problem, we propose a hybrid machine-

learning approach using a combination of support vector

machines (SVMs) and Bayesian optimization to identify

combinations of voltages that realize a desired Hubbard

model. SVM constrains the space of voltages by rejecting

voltage combinations producing potentials unsuitable for

tight-binding (TB) approximation. The target voltage

combinations are then identiﬁed by Bayesian optimiza-

tion (BO) in the constrained subdomain. We perform

BO of gate voltages to produce a double QD system with

tailored tunneling parameter tand on-site Hubbard en-

ergy U, using experimental gate lithography images as

input for realistic calculations of tand Uwith the linear

combination of harmonic orbitals method (LCHO) [56].

This approach allows us to predict tand Ufor variable

electrode design and with ﬂexible material parameters

and custom heterostructures. Our BO procedure oper-

ates without gradients or input from experimental charge

stability diagrams, which are tedious to measure for large

systems. BO is also suitable for problems with multiple

local optima and noisy data. We also develop an itera-

tive, scalable SVM-BO approach for multiple-site arrays,

which is able to reach optimal solution by only optimis-

ing subsets of voltages at a time and uses only the two-

site SVM. We predict the optimal voltage combination

needed in experiment to prepare an on-demand double

QD Hubbard model within 1.5% error for model gates

and 6% for experimental gates, as well as for three-site

system with 10% error. This procedure can be combined

with existing methods of preparing a charged state within

quantum dots [36, 41–43, 46, 48, 49, 51, 52].

This paper is organized as follows: Section I describes

the numerical calculations of the electrostatic potentials

from the metallic gate input image by solving Poisson’s

equation. Section II presents the SVM model developed

to classify voltage combinations and demonstrates its ex-

cellent performance for rejecting undesired voltage com-

binations for two gate designs: an ideal simple-shaped

gate set and a realistic experimental gate set obtained

from lithography images. Section III describes the LCHO

method for the calculation of the Hubbard model param-

eters and presents the results for possible U/t ratios that

can be achieved with various gate designs. The adapta-

tion of BO for the title problem is described in Section IV.

We present our results on optimal voltage combinations

for the model and experimental sets of gates in Section

II. ELECTROSTATIC POTENTIAL FROM

METALLIC GATES

We begin by computing the electrostatic conﬁning po-

tential produced by two sets of metallic gates: the model

basic-shape gates in Fig. 1 and a realistic gate image

from experiment [57] in Fig. 1. Both sets of metallic

gates are assumed to be 10 nm tall and are placed within

a heterostructure inside a material with = 10 (Fig. 1).

The pattern of the gates serves as input to a ﬁnite diﬀer-

ence numerical method solving the Poisson’s equation:

∇ · (r)∇V(r)=−ρ(r)

0

,(1)

where r= (x, y, z)is the position vector in 3D space, ρ

is the charge density, (r)is the dielectric constant and

0is the permittivity of free space.

We use ﬁnite diﬀerence method with the varied dielec-

tric constant and two types of boundary conditions: the

Dirichlet boundary condition at the top and bottom of

computational box as well as inside the box, where the

gate is placed, the Neumann boundary condition at the

sides of the computational box, in xand ydirection. We

use a grid of 150×150×127 points in x×y×zdirections.

The sample is modeled with 30nm of vaccum above the

heterostructure and 30nm wetting layer below the metal-

lic gate layer. The successive over-relaxation technique

[58] is used to speed up the numerical procedure.

Fig 1 shows the resulting 2D electrostatic potential (for

experimental gates) for an sample set of voltages, consist-

ing of two conﬁning wells (quantum dots) which act as

sites in a 2-site Hubbard model. The red line marks a 1D

cut of the potential passing through the minima of the

two wells. The 1D example potential cuts for model as

well as experimental gates are shown in Fig. 1.

[eV]

vaccum

silicon

ab c

def

FIG. 1: a) Model gate design with 2 conﬁning plunger

gates and 3 barrier gates controlling the tunnelling. b)

Image of experimental gate design with 2 plunger gates,

5 barrier gates and 4 accumulation gates. Red box corre-

sponds to the 2D potential landscape shown in c). Two

potential wells are visible (dark blue) in c) Red line corre-

sponds to crosssection potential in f). d) Semiconductor

heterostructure design crosssection along z axis used for

numerical Poisson equation solution. Metallic gates are

placed inside the heterostructure with silicon substrate.

Colors denote dielectric constant regions. Red line cor-

responds to 2DEG subject to conﬁnement. e) Conﬁn-

ing 2QD potential for model gate deisgn (cut along x).

f) Conﬁning 1QD potential for experimental gate design

(cut along x).

III. HUBBARD MODEL SIMULATION

DOMAIN

In a typical experiment with gated QDs, each QD is

tuned by 3electrodes. As the number of QDs increases

as needed for many-site Hubbard models, the space of

gate voltages becomes high-dimensional. More impor-

tantly, the vast majority of the gate voltage combinations

results in electrostatic potentials that are unsuitable for

quantum simulation of Hubbard models and must, there-

fore, be discarded as unphysical. Identifying the target

gate voltages thus amounts to optimization in a highly

constrained subdomain.

In order to identify the subdomain of gate voltages pro-

ducing suitable potentials, we develop an SVM ﬁlter of

gate voltages. We use SVM to solve a classiﬁcation prob-

lem as implemented in the scikit-learn python library [59]

with a radial basis function (RBF) kernel. The SVM

models are trained by the results of single-particle (SP)

quantum calculation. For a given combination of volt-

ages, we obtain the electrostatic potential as described

in Section II and solve the Schrödiger equation to obtain

the particle density pi∈[1,N0]for N≤N0lowest-energy

eigenstates.

We adopt the following criteria for the classiﬁcation

problem. The gate voltages are accepted as suitable for

quantum simulation of the Hubbard models provided: a)

a signiﬁcant portion p=p1·p2> p0of the particle

density in well 1 or 2 (p1or p2) is enclosed within a given

radius Rfrom the centres of the quantum dots (and the

charge density within any single well does not vanish); b)

a chemical potential µis set and all N < N0SP energy

levels Eiare populated, i.e. EN0≤µ. For the present

calculations, we use N0= 6, p0= 0.05, R = 50 nm, µ=

−0.15 eV. SVM is trained over 5·104training points and

achieves approximately 99% overall success and above

98% rejection success.

e f g

FIG. 2: a) (b) Potentials classiﬁed as rejected (accepted)

during SVM testing, in agreement with training labels.

Grey lines represent all potentials included in classiﬁca-

tion. The potentials in b) follow a uniﬁed trend, while

those in a) do not. c) (d) Potentials classiﬁed as rejected

(accepted) during SVM testing, contrary to training la-

bels. c) and d) are borderline examples and results de-

pend on training label parameters. It is apparent that

the SVM classiﬁcation accepts a good selection of poten-

tials for Hubbard model. e) (f) Classiﬁcation of poten-

tials in space of voltages for cuts along plunger-plunger

(plunger-barrier) voltage plane. Percent of accepted po-

tentials during test as a function of the size of the train-

ing sample (model gates). The accepted portion is of the

order of 2% for large enough sample size.

The classiﬁcation problem we consider here is imbal-

anced, i.e. around 2% of the potentials are suitable for

Hubbard model. In practice, it is important to tune this

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BayesianautotuningofHubbardmodelquantumsimulatorsLudmilaSzulakowska,JunDaiDepartmentofChemistry,UniversityofBritishColumbia,Vancouver,B.C.V6T1Z1,CanadaStewartBlussonQuantumMatterInstitute,Vancouver,B.C.V6T1Z4,Canada(Dated:October7,2022)Spinsingatedsemiconductorquantumdots(QDs)areapromisingplatformfo...

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Bayesian autotuning of Hubbard model quantum simulators Ludmila Szulakowska Jun Dai Department of Chemistry University of British Columbia Vancouver B.C. V6T 1Z1 Canada

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