Embedding of Tree Tensor Networks into Shallow Quantum Circuits
Shota Sugawara
1 a
, Kazuki Inomata
1 b
, Tsuyoshi Okubo
2 c
and Synge Todo
1,2,3 d
1
Department of Physics, The University of Tokyo, Tokyo, 113-0033, Japan
2
Institute for Physics of Intelligence, The University of Tokyo, Tokyo, 113-0033, Japan
3
Institute for Solid State Physics, The University of Tokyo, Kashiwa, 277-8581, Japan
Keywords:
Tree Tensor Networks, Quantum Circuits, Variational Quantum Algorithms.
Abstract:
Variational Quantum Algorithms (VQAs) are being highlighted as key quantum algorithms for demonstrating
quantum advantage on Noisy Intermediate-Scale Quantum (NISQ) devices, which are limited to executing
shallow quantum circuits because of noise. However, the barren plateau problem, where the gradient of the
loss function becomes exponentially small with system size, hinders this goal. Recent studies suggest that
embedding tensor networks into quantum circuits and initializing the parameters can avoid the barren plateau.
Yet, embedding tensor networks into quantum circuits is generally difficult, and methods have been limited
to the simplest structure, Matrix Product States (MPSs). This study proposes a method to embed Tree Tensor
Networks (TTNs), characterized by their hierarchical structure, into shallow quantum circuits. TTNs are
suitable for representing two-dimensional systems and systems with long-range correlations, which MPSs are
inadequate for representing. Our numerical results show that embedding TTNs provides better initial quantum
circuits than MPS. Additionally, our method has a practical computational complexity, making it applicable to
a wide range of TTNs. This study is expected to extend the application of VQAs to two-dimensional systems
and those with long-range correlations, which have been challenging to utilize.
1 INTRODUCTION
Variational Quantum Algorithms (VQAs) represent
the foremost approach for achieving quantum advan-
tage with the current generation of quantum com-
puting technologies. Quantum computers are antici-
pated to outperform classical ones, with some algo-
rithms already proving more efficient (Shor, 1994;
Grover, 1996). However, the currently available
Noisy Intermediate-Scale Quantum (NISQ) devices
are incapable of executing most algorithms due to
their limited number of qubits and susceptibility to
noise (Preskill, 2018).
VQAs are hybrid methods where classical com-
puters optimize parameters of quantum circuits’
ansatz to minimize the cost function evaluated by
quantum computers. VQAs require only shallow cir-
cuits, making them notable as algorithms that can
be executed on NISQ devices (Cerezo et al., 2021a).
VQAs come in many forms, such as Quantum Ma-
a
https://orcid.org/0009-0005-8758-2940
b
https://orcid.org/0009-0008-4030-5518
c
https://orcid.org/0000-0003-4334-7293
d
https://orcid.org/0000-0001-9338-0548
chine Learning (QML) (Biamonte et al., 2017; Mi-
tarai et al., 2018; Schuld and Killoran, 2019), Vari-
ational Quantum Eigensolver (VQE) (Abrams and
Lloyd, 1999; Peruzzo et al., 2005; Peruzzo et al.,
2014), and Quantum Approximate Optimization Al-
gorithm (QAOA) (Farhi et al., 2014; Wang et al.,
2018; Zhou et al., 2020), with potential applications
across diverse industries and fields.
However, a significant challenge known as the
barren plateau stands in the way of realizing quan-
tum advantage (McClean et al., 2018; Holmes et al.,
2022). The barren plateau phenomenon refers to
the challenge in VQAs where the gradient of the
cost function decreases exponentially as the system
size increases. This phenomenon occurs regard-
less of whether the optimization method is gradient-
based (Cerezo and Coles, 2021) or gradient-free (Ar-
rasmith et al., 2021), and it has been observed even
in shallow quantum circuits (Cerezo et al., 2021b).
Furthermore, it has been confirmed that this phe-
nomenon also occurs in practical tasks using real-
world data (Holmes et al., 2021; Sharma et al., 2022;
Ortiz Marrero et al., 2021). Avoiding the barren
plateau is a critical challenge in demonstrating the
Sugawara, S., Inomata, K., Okubo, T. and Todo, S.
Embedding of Tree Tensor Networks into Shallow Quantum Circuits.
DOI: 10.5220/0013403200003890
Paper published under CC license (CC BY-NC-ND 4.0)
In Proceedings of the 17th International Conference on Agents and Artificial Intelligence (ICAART 2025) - Volume 1, pages 793-803
ISBN: 978-989-758-737-5; ISSN: 2184-433X
Proceedings Copyright © 2025 by SCITEPRESS – Science and Technology Publications, Lda.
793

superiority of quantum algorithms using NISQ de-
vices. To avoid the barren plateau, appropriate pa-
rameter initialization in VQAs is crucial since ran-
domly initializing the parameters can result in the al-
gorithm starting far from the solution or near a local
minimum (Zhou et al., 2020). Although various ini-
tialization methods have been considered (Wiersema
et al., 2020; Grant et al., 2019; Friedrich and Maziero,
2022), using tensor networks is natural due to their
compatibility with quantum circuits.
Tensor networks are originally developed to ef-
ficiently represent quantum many-body wave func-
tions. Any quantum circuit can be naturally regarded
as a tensor network (Markov and Shi, 2008), and it
is sometimes possible to simulate quantum comput-
ers with a practical amount of time using tensor net-
works (Liu et al., 2021). Moreover, in recent years,
their utility has been recognized and applied across
various fields such as machine learning (Levine et al.,
2019) and language models (Gallego and Orus, 2022)
In this study, we focus particularly on Ma-
trix Product States (MPSs) and Tree Tensor Net-
works (TTNs) among various structures of ten-
sor networks. MPSs are one-dimensional arrays
of tensors. Its simplest and easiest-to-use struc-
ture, along with the presence of advanced al-
gorithms such as Density Matrix Renormalization
Group (DMRG) (White, 1992) and Time-evolving
block decimation (TEBD) (Vidal, 2003), has led to
its application across a wide range of fields. Recently,
these excellent algorithms have been applied to the
field of machine learning, continuing the exploration
of new possibilities (Stoudenmire and Schwab, 2016;
Han et al., 2018). TTNs are tree-like structures of
tensors. The procedures used in DMRG and TEBD
have been applied to TTNs (Shi et al., 2006; Silvi
et al., 2010). Additionally, the distance between any
pair of leaf nodes scales in logarithmic order in TTNs
while in linear order in MPSs. As connected corre-
lation functions generally decay exponentially with
path length within a tensor network, TTNs are better
suited to capture longer-range correlations and rep-
resent two-dimensional objects than MPSs. TTNs
are utilized in various fields such as chemical prob-
lems (Murg et al., 2010; Gunst et al., 2019), con-
densed matter physics (Nagaj et al., 2008; Taglia-
cozzo et al., 2009; Lin et al., 2017) and machine learn-
ing (Liu et al., 2019; Cheng et al., 2019).
Rudolph et al. have successfully avoided the
barren plateau problem by utilizing tensor net-
works (Rudolph et al., 2023b). They proposed a
method of first optimizing tensor networks, then map-
ping the optimized tensor networks to quantum cir-
cuits, and finally executing VQAs. While numeri-
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Embedding
Figure 1: A schematic diagram illustrating the embedding
of a TTN into a shallow quantum circuit composed solely
of two-qubit gates. The aim is to ensure that the quantum
state output by the quantum circuit closely approximates the
quantum state represented by the TTN. The diagram depicts
a case with three layers, and the approximation accuracy
improves as the number of quantum circuit layers increases.
cal results have shown that this method can indeed
avoid the barren plateau, there is a general challenge
in mapping tensor network states into shallow quan-
tum circuits. The case where the tensor network struc-
ture is an MPS has already been well-studied (Ran,
2020; Rudolph et al., 2023a; Shirakawa et al., 2024;
Malz et al., 2024). However, effective methods for
mapping tensor networks other than MPSs to shallow
quantum circuits have not yet been devised.
In this paper, we propose a method to embed
TTNs into shallow quantum circuits composed of
only two-qubit gates as shown in Figure 1. The pri-
mary obstacle to embedding TTNs into shallow quan-
tum circuits has been the complexity of contractions
arising from the intricate structure of TTNs. In gen-
eral, the contraction of tensor networks requires an
exponentially large memory footprint relative to the
number of qubits if performed naively. Furthermore,
determining the optimal contraction order for ten-
sor networks is an NP-complete problem (Chi-Chung
et al., 1997). Through the innovative design of a con-
traction method that balances minimal approximation
error and computational efficiency, we successfully
extend the embedding method of MPSs (Rudolph
et al., 2023a) to TTNs. Additionally, by applying the
proposed method to practical problems, we success-
fully prepare better initial quantum circuits for VQAs
than those provided by MPSs with a practical compu-
tational complexity. This study is expected to extend
the application of VQAs to two-dimensional systems
and those with long-range correlations, which have
previously been challenging to utilize.
2 BACKGROUND
2.1 Tensor Networks
Tensor networks are powerful mathematical frame-
works that efficiently represent and manipulate high-
dimensional data by decomposing tensors into inter-
connected lower-dimensional components (Bridge-
man and Chubb, 2017; Or
´
us, 2014). They have
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794

been widely applied in various fields, including quan-
tum many-body physics for approximating ground
states of complex systems (White, 1992; Vidal,
2003; Xiang, 2023) and machine learning for real
data (Stoudenmire and Schwab, 2016; Han et al.,
2018). Recently, increasing attention has been di-
rected towards their compatibility with quantum com-
puting, as their structure aligns well with quantum
circuits and facilitates hybrid quantum-classical algo-
rithms (Markov and Shi, 2008).
The widespread application of tensor networks
across various fields can be attributed to their simple
and comprehensible notation. Tensor network nota-
tion provides a clear and intuitive way to represent
complex tensor contractions and operations. In tensor
network notation, a rank-r tensor is depicted as a geo-
metric shape with r legs. The geometric shape and the
direction of the legs are determined by the properties
of the tensor and its indices. When representing quan-
tum states, the direction of the legs often indicates
whether the vectors are in the Hilbert space for kets or
its dual space. When two tensors share a single leg,
the leg is referred to as the bond, and the dimension
of that leg is referred to as the bond dimension. By
adjusting the bond dimension, the expressive power
of the tensor network can be controlled. Reducing the
bond dimension to decrease memory requirements is
referred to as truncation. One of the most commonly
used operations is contraction, which combines mul-
tiple tensors into a single tensor. In tensor network
diagrams, contraction corresponds to connecting ten-
sors with lines. The contraction of A’s x-th index and
B’s y-th index is defined as
C
i
1
,...,i
x−1
,i
x+1
,...,i
r
, j
1
, j
y−1
, j
y+1
,..., j
s
=
∑
α
A
i
1
,...,i
x−1
,α,i
x+1
,i
r
B
j
1
,..., j
y−1
,α, j
y+1
,..., j
s
,
(1)
where C is the result of the contraction. Operations
such as the inner product of vectors, matrix-vector
multiplications, and matrix-matrix multiplications are
specific examples of contractions. A contraction net-
work forms a tensor network, allowing for the consid-
eration of various networks depending on the objec-
tive.
2.2 MPSs and TTNs
Let |ψ⟩ be a tensor network in the form of either an
MPS or a TTN. Number the tensors in |ψ⟩ from left
to right for MPS, and in breadth-first search (BFS)
order from the root node for TTN, denoting the i-th
tensor as A
(i)
(i = 1,... ,N). By adding a leg with bond
dimension one, a two-legged tensor can be converted
into a three-legged tensor, thus all tensors in |ψ⟩ can
be considered three-legged. |ψ⟩ is in canonical form
if there exists a node A
(i)
, called the canonical center,
such that for all other nodes A
(i
′
)
the following holds
∑
l,m
A
(i
′
)
l,m,n
A
(i
′
)∗
l,m,n
′
= I
n,n
′
, (2)
where the leg denoted by index n is the unique leg of
A
(i
′
)
pointing towards A
(i)
. A tensor that satisfies this
equation is referred to as an isometric tensor. Any |ψ⟩
can be transformed into this canonical form and the
position of the canonical center can be freely moved
without changing the quantum state.
The canonical form offers numerous advantages.
In this paper, the key benefit is that at the canon-
ical center, the local Singular Value Decomposi-
tion (SVD) matches the global SVD, allowing for
precise bond dimension reduction through truncation
while appropriately moving the canonical center. Ad-
ditionally, in the canonical form, each tensor is an
isometry, facilitating easy conversion to unitary form
and embedding into quantum circuits. Unless other-
wise specified, this paper assumes that any MPS and
TTN are converted to the canonical form with A
(0)
as
the canonical center.
2.3 Embedding of MPSs
Although embedding general tensor networks into
shallow quantum circuits is challenging, several
workable methods have been proposed for MPSs. In
this subsection, we overview the technique for seam-
lessly embedding MPSs into shallow quantum cir-
cuits.
Ran introduced a systematic decomposition
method for an MPS into several layers of two-qubit
gates with a linear next-neighbor topology (Ran,
2020). First, the MPS |ψ
(k)
⟩ with bond dimension
χ is truncated to a bond dimension two MPS |ψ
(k)
χ=2
⟩.
Next, the isometric tensors in |ψ
(k)
χ=2
⟩ are converted
into unitary tensors. The resulting set of unitary ten-
sors, L[U]
(k)
, is referred to as a layer and can be em-
bedded into a quantum circuit composed of two-qubit
gates. Additionally, since
|ψ
(k)
χ=2
⟩ = L[U]
(k)
|0⟩ (3)
and
L[U]
(k)†
|ψ
(k)
χ=2
⟩ = |0⟩ (4)
are hold, L[U]
(k)†
can be considered a disentan-
gler, transforming the quantum state into a product
state. Finally, a new MPS |ψ
(k+1)
⟩ = L[U]
(k)†
|ψ
(k)
⟩
can be obtained. Since L[U]
(k)†
acts as a disen-
tangler, |ψ
(k+1)
⟩ should have reduced entanglement
Embedding of Tree Tensor Networks into Shallow Quantum Circuits
795

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(b)
(c)
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U
p
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o
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a
b
c
d
a
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c
d
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d
c
a
a
b
c
d
Figure 2: A schematic diagram illustrating the systematic decomposition for a TTN. (a) The TTN is truncated to a bond
dimension of two using SVD. This simplified TTN is then embedded into a quantum circuit by converting tensors into
unitaries via Gram-Schmidt orthogonalization. These unitaries act as disentanglers and are applied to the original TTN. By
repeating these steps, a multi-layer quantum circuit is formed. (b) A diagram of a penetration algorithm. It contracts two
connected tensors, reorders axes, and separates them using SVD, making it seem like their positions have swapped. (c) A
schematic diagram explaining the transformation from a complex tensor network into a TTN. We number the tensors based
on their positions in the network. We first use SVD to split it into an upper and a lower tensor. Then, we apply the penetration
algorithm iteratively until the upper tensor is connected with the corresponding tensor. This process is repeated for the lower
tensor as well. Finally, we contract the upper and lower tensors with the corresponding tensor to form a new tensor in the
TTN. This process is performed sequentially from the highest-numbered component of the disentangler.
compared to |ψ
(k)
⟩. By repeating this process start-
ing from the original MPS |ψ
(0)
⟩, a quantum circuit
∏
K
k=1
L[U]
(k)
|0⟩ with multiple layers can be gener-
ated.
Optimization-based methods are employed as an
alternative approach to embedding tensor networks
into quantum circuits. This method sequentially op-
timizes the unitary operators within the quantum cir-
cuit to maximize the magnitude of the inner product
between the quantum circuit and the tensor network.
Evenbly and Vidal proposed an iterative optimization
technique that utilizes the calculation of environment
tensors and SVD (Evenbly and Vidal, 2009). Simi-
larly, Shirakawa et al. employed this iterative opti-
mization method to embed quantum states into quan-
tum circuits (Shirakawa et al., 2024). An environ-
ment tensor is calculated to update a unitary by re-
moving the unitary from the circuit and contracting
the remaining tensor network. We perform the SVD
of the environment tensor and utilize the fact that the
resulting unitary matrix serves as the optimal opera-
tor to increase the fidelity between the original tensor
network and the constructed quantum circuit. For a
more detailed explanation of the optimization algo-
rithm, please refer to (Rudolph et al., 2023a; Shi-
rakawa et al., 2024) or the chapter on optimization
algorithms for TTNs described later.
Rudolph et al. investigated the optimal sequence
for combining these two methods to achieve the high-
est accuracy (Rudolph et al., 2023a). The study con-
cludes that the best results are obtained by adding
new layers through a systematic decomposition step
and optimizing all layers with each addition. This
method has achieved exceptional accuracy in embed-
ding a wide range of MPSs in fields such as physics,
machine learning, and random systems. Therefore, it
can be considered one of the best current techniques
for embedding MPSs into shallow circuits. However,
this method does not support embedding tensor net-
works other than MPSs. Consequently, embedding
tensor networks such as TTNs and other non-MPS
structures remains an unresolved issue.
3 PROPOSED METHOD
We propose a method for embedding TTNs into shal-
low quantum circuits. Any shape of TTN can be
converted into a binary tree using SVD (Hikihara
et al., 2023), so we assume a binary tree for sim-
plicity. However, this method can be generalized to
embed TTNs of any shape. The fundamental ap-
proach of the proposed method is similar to that of
MPSs (Rudolph et al., 2023a). By repeatedly adding
layers using the systematic decomposition algorithm
and optimizing the entire circuit, we generate highly
accurate embedded quantum circuits. This section
first introduces the systematic decomposition algo-
rithm for TTNs. Due to the increased structural com-
plexity of TTNs compared to MPSs, the contraction
methods become non-trivial. We propose a method
that balances approximation error and computational
cost. Next, we describe the optimization algorithm for
TTNs and integrate it with the systematic decomposi-
tion. Finally, we discuss the computational cost and
demonstrate that embedding TTNs is feasible within
QAIO 2025 - Workshop on Quantum Artificial Intelligence and Optimization 2025
796

practical computational limits.
3.1 Systematic Decomposition
Our systematic decomposition algorithm for TTNs is
an extension of the algorithm for MPSs introduced in
(Ran, 2020). We denote the original TTNs as |ψ
0
⟩, the
k-th layer of the quantum circuit as L[U ]
(k)
, the num-
ber of layers in resulting quantum circuit as K, and the
resulting quantum circuit as
∏
K
k=1
L[U]
(k)
|0⟩. Algo-
rithm 1 details the systematic decomposition process
for TTNs, as depicted in Figure 2(a).
Input: TTN|ψ
0
⟩,Maximum layers K
Output: Quantum Circuit
∏
K
k=1
L[U]
(k)
|0⟩
|ψ
(1)
⟩ ← |ψ
0
⟩;
for k = 1 to K do
Truncate |ψ
(k)
⟩ to |ψ
(k)
χ=2
⟩ via SVD;
Convert |ψ
(k)
χ=2
⟩ to L[U ]
(k)
;
|ψ
(k+1)
⟩ ← L[U ]
(k)†
|ψ
(k)
⟩;
end
Algorithm 1: Systematic decomposition.
In this algorithm, a copy of the original TTN is
truncated to a lower dimension using SVD. Truncat-
ing bond dimension χ TTNs to bond dimension two
TTNs can be done accurately, similar to MPSs, by
shifting the canonical center so that the local SVD
matches the global SVD. This truncated TTN is then
transformed into a single layer of two-qubit gates by
converting the isometric tensors in the layer into uni-
tary tensors using the Gram-Schmidt orthogonaliza-
tion process. The inverse of this layer is applied to
the original TTN, resulting in a partially disentangled
state with potentially reduced entanglement and bond
dimensions. This process can be iteratively repeated
to generate multiple layers, which are indexed in re-
verse order to form a circuit that approximates the tar-
get TTN. Notably, the final layer of the disentangling
circuit is used as the initial layer of the quantum cir-
cuit for approximation.
While this algorithm appears to function effec-
tively at first glance, the computation of |ψ
(k+1)
⟩ ←
L[U]
(k)†
|ψ
(k)
⟩ is exceedingly challenging from
the perspective of tensor networks. The struc-
ture of |ψ
(k+1)
⟩ is TTN, whereas tensor network
L[U]
(k)†
|ψ
(k)
⟩ has a more complex structure, neces-
sitating its transformation into the shape of TTN. As
previously mentioned, naively transforming a general
tensor network can require exponentially large mem-
ory relative to the number of qubits. Additionally, al-
though limiting the bond dimension during the trans-
formation can facilitate the process, the sequence of
transformations can lead to substantial approximation
errors.
To address the issue of tensor network transforma-
tions arising from the complexity of TTN structures,
we employ a penetration algorithm as a submodule.
As illustrated in Figure 2(b), the penetration algo-
rithm operates by contracting two tensors connected
by a single edge along the connected axis, appropri-
ately reordering the axes, and then separating them
using SVD. This makes it appear as if the positions of
the two tensors have swapped. Additionally, by ad-
justing the number of singular values retained during
the SVD, we can balance approximation accuracy and
computational cost.
Input: L[U ]
(k)†
|ψ
(k)
⟩
Output: TTN |ψ
(k+1)
⟩
for i = N − 1 to 1 do
Split L[U]
(k)†
i
into U
p
and L
o
via SVD;
while U
p
is not connected to A
(i)
do
A ← U
p
’s left tensor;
Make U
p
penetrate A;
end
while L
o
is not conntected to A
(i)
do
A ← L
o
’s left tensor;
Make L
o
penetrate A;
end
A
(i)
← Contract A
(i)
, U
p
, and L
o
;
end
Algorithm 2: Transformation of tensor networks using the
penetration algorithm.
Algorithm 2 details the transformation of ten-
sor networks using the penetration algorithm, as de-
picted in Figure 2(c). We assign numbers to |ψ
(k)
⟩
in breadth-first search (BFS) order from the root. We
denote i-th tensor of |ψ
(k)
⟩ as A
(i)
. We also assign
numbers to L[U]
(k)†
based on its origin in the TTN,
denoting the tensor with number i as L[U]
(k)†
i
. For
each tensor L[U ]
(k)†
i
in L[U]
(k)†
, first use SVD to split
it into upper tensor U
p
and lower tensor Lo to reduce
the computational complexity in the penetration al-
gorithm. Then apply the penetration algorithm iter-
atively until the upper tensor is connected with A
(i)
.
Repeat this for the lower tensor too. Finally, con-
tract the upper and lower tensors with A
(i)
to form a
new TTN’s i-th tensor. Perform this process sequen-
tially from the highest-numbered tensor in L[U]
(k)†
.
This algorithm allows contraction with each tensor
in L[U]
(k)†
without changing the TTN structure of
|ψ
(k)
⟩. Consequently, despite minor approximation
errors during penetration, L[U]
(k)†
|ψ
(k)
⟩ can be trans-
Embedding of Tree Tensor Networks into Shallow Quantum Circuits
797

formed into a TTN structure.
3.2 Integration with Optimization
As demonstrated in (Rudolph et al., 2023a), the sys-
tematic decomposition algorithm achieves the highest
embedding accuracy when appropriately combined
with the optimization algorithm. In this study, we
also integrate systematic decomposition with the op-
timization algorithm. The fundamental concept of the
optimization algorithm for quantum circuits embed-
ded with TTNs is analogous to that for quantum cir-
cuits embedded with MPS. The primary difference
lies in the ansatz of the quantum circuits; however,
the optimization algorithm can be executed in a simi-
lar manner.
Input: Quantum Circuit
∏
K
k=1
L[U]
(k)
|0⟩,
number of sweeps T , learning rate
r ∈ [0, 1]
Output: Optimized Quantum Circuit
∏
K
k=1
L[U]
(k)
|0⟩
for t = 1 to T do
for k = 1 to K do
for i = 1 to N − 1 do
U
old
← L[U ]
(k)
i
;
Calculate environment tensor E;
SVD E = U SV
†
;
U
new
← UV
†
;
L[U]
(k)
i
← U
old
(U
†
old
U
new
)
r
;
end
end
end
Algorithm 2: Optimization.
Algorithm 2 details the optimization process for
TTNs, as depicted in Figure 3. The environment ten-
sor E is obtained by contracting all tensors except for
the one of interest. In this algorithm, the tensor of
interest is L[U]
(k)
i
and E becomes a four-dimensional
tensor with two legs on the left and two on the right.
We compute the SVD of E and utilize the fact that the
product UV
†
is the unitary matrix that maximizes the
magnitude of the inner product between the original
TTN and the generated quantum circuit (Shirakawa
et al., 2024). Given the strength of this local up-
date, we introduce a learning rate r, which modifies
the unitary update rule via U
old
(U
†
old
U
new
)
r
. Replac-
ing L[U]
(k)
i
with the unitary operator calculated in this
manner for all operators constitutes one step, and re-
peating this process for T steps completes the opti-
mization algorithm.
The integration of the systematic decomposition
and optimization algorithms involves a method where
a new layer is added using the systematic decompo-
sition algorithm, followed by optimizing the entire
quantum circuit with the optimization algorithm. This
process is repeated iteratively. Rudolph et al. re-
fer to this method as Iter[D
i
, O
all
], confirming that
it achieves the highest accuracy regardless of the type
of MPSs (Rudolph et al., 2023a). When creating the
k + 1-th layer using the systematic decomposition al-
gorithm, the layers up to k are first absorbed into the
original TTN using the penetration algorithm before
executing the systematic decomposition algorithm. It
should be noted that since we are using the penetra-
tion algorithm, the |ψ⟩ that has absorbed L[U]
j†
re-
tains its TTN structure. The integrated algorithm is
presented as Algorithm 3.
Input: TTNψ
0
,Maximum layers K
Output: Quantum Circuit
∏
K
k=1
L[U]
(k)
|0⟩
|ψ⟩ ← |ψ
0
⟩;
for k = 1 to K do
Truncate |ψ⟩ to |ψ
χ=2
⟩ via SVD;
Convert |ψ
χ=2
⟩ to L[U ]
(k)
;
Optimize
∏
k
k
′
=1
L[U]
(k
′
)
;
for j = 1 to k do
Absorb L[U]
j†
into |ψ⟩;
end
end
Algorithm 3: Proposed method.
3.3 Computational Complexity
For a TTN with a maximal bond dimension χ, the
memory requirements scale O(Nχ
3
). However, the
computational complexity of transforming the TTN
into its canonical form scale O(Nχ
4
). Most of the al-
gorithms related to TTNs require transformation into
canonical form. Therefore, it is reasonable to assume
that we set χ such that computations of O(Nχ
4
) can
be performed efficiently.
In Algorithm 3, truncating |ψ⟩ to |ψ
χ=2
⟩ requires
O(Nχ
3
). Optimizing the entire circuit is significantly
more challenging compared to MPS due to the com-
plexity of the TTN structure. Generally, exponen-
tial memory is required with respect to N, but by us-
ing an appropriate contraction order and caching, the
computation can be performed in O(Nχ
3
4
K
), which
is linear in N. Although the computational complex-
ity increases exponentially with the number of layers,
this algorithm is designed for embedding into shallow
quantum circuits, and it operates efficiently for K ≈ 7,
as used in our experiments. Furthermore, when em-
bedding TTNs into a larger number of layers, it is
QAIO 2025 - Workshop on Quantum Artificial Intelligence and Optimization 2025
798

|0>
|0>
|0>
|0>
|0>
|0>
|0>
|0>
|0>
|0>
|0>
|0>
|0>
|0>
|0>
|0>
|0>
|0>
|0>
|0>
|0>
|0>
|0>
|0>
E U
S
†
VUV
†
Figure 3: The environment tensor E is obtained by contracting all tensors except the one of interest. We compute its SVD
E = U SV
†
and calculate UV
†
to find the unitary operator closest to the environment tensor. The generated unitary operator is
positioned at the location of the removed unitary operator.
possible to achieve O(N logNχ
4
KT ), where T is the
number of sweeps, linear computational complexity
with respect to the number of layers, bond dimension
by introducing approximations that prevent the bond
dimension from exceeding χ during the contraction
of environment tensors, as used in the embedding of
MPS (Rudolph et al., 2023a).
The computational complexity of absorbing K
layers into the original TTN is O(N logNχ
4
K). In
the penetration algorithm, the SVD of a matrix with
row dimension χ
2
and column dimension χ has
a complexity of O(χ
4
). Given that the depth of
the TTN is O(logN), the penetration operation is
performed O(log N) times per tensor. Since each
layer contains O(N) tensors, the complexity of ab-
sorbing one layer is O(N logNχ
4
), leading to a to-
tal complexity of O(N logNχ
4
K) for K layers. In
summary, the computational complexity to gener-
ate one layer is O(max(N log Nχ
4
K,Nχ
3
4
K
)), and
it is O(max(N logNχ
4
K
2
,Nχ
3
4
K
)) for K layers. It
scales with the number of qubits. Additionally, the
bond dimension is constrained to χ
4
, making it suit-
able for embedding TTNs with large bond dimen-
sions. Furthermore, by allowing approximations
in optimization, the complexity can be reduced to
O(N log Nχ
4
K
2
T ) for K layers.
4 EXPERIMENTS
We conducted experiments using two distinct state
vectors from the fields of machine learning and
physics. The first state vector represents a uniform
superposition over the binary data samples in the
4 × 4 bars and stripes (BAS) dataset (MacKay, 2003),
which has become a canonical benchmark for gen-
erative modeling tasks in quantum machine learning.
The second state vector represents the ground state of
the J
1
-J
2
Heisenberg model, a model that character-
izes competing interactions in quantum spin systems.
The Hamiltonian for this model is given by the fol-
lowing equation,
H = J
1
∑
<i, j>
S
i
· S
j
+ J
2
∑
<<i, j>>
S
i
· S
j
, (5)
where J
1
(J
2
) represents the nearest (next-nearest)
neighbor interactions. By varying the ratio of the first
and second nearest-neighbor interactions, the J
1
-J
2
Heisenberg model generates complex quantum many-
body phenomena and has been widely studied. In this
paper, we utilized J
2
/J
1
= 0.5. Both state vectors
are two-dimensional systems with long-range corre-
lations, making them more suitably represented by
TTNs rather than MPSs.
The state vector of the BAS was manually pre-
pared, while the ground state of the J
1
-J
2
Heisenberg
model was generated using a numerical solver pack-
age H Φ (Kawamura et al., 2017), which is designed
for a wide range of quantum lattice models. MPSs and
TTNs were constructed by iteratively applying SVD
to the state vector from the edges.
The conversion from MPSs and TTNs to quantum
circuits employed the Iter[D
i
, O
all
] method, which
is also utilized in the proposed method. Addition-
ally, to compare with the proposed method, we also
used D
all
, O
all
, and Iter[I
i
, O
all
] from (Rudolph et al.,
2023a). The D
all
method generates circuits solely
through systematic decomposition. The O
all
method
optimizes circuits starting from an initial state com-
posed only of identity gates. The Iter[I
i
, O
all
] method
sequentially adds identity layers, optimizing the en-
tire circuit at each step. The number of sweeps in the
optimization process was set to 1000. Experiments
were conducted using various learning rates ranging
from 0.5 to 0.7, and the rate that demonstrated the
highest convergence accuracy was selected.
To measure the accuracy of embedding into quan-
tum circuits, we utilized infidelity as the evaluation
metric. The infidelity I
f
between two quantum states,
|Ψ⟩ and |Φ⟩, is expressed by
I
f
= 1 − |⟨Φ|Ψ⟩ |, (6)
and a smaller infidelity indicates that the two quantum
states are closer. In this study, infidelity quantifies the
success of our transformations.
Figure 4 illustrates the infidelity between the orig-
inal state vector and the quantum circuits embedded
with either TTN or MPS. Despite the greater diffi-
culty in embedding TTNs into shallow quantum cir-
cuits due to their hierarchical structure, the infidelities
Embedding of Tree Tensor Networks into Shallow Quantum Circuits
799

Figure 4: Infidelities between original state vectors and generated quantum circuits. The bond dimensions of tensor networks
are eight and the number of optimization steps is 1000. The learning rate for optimization is set to 0.65 for the BAS Dataset
and 0.6 for the J
1
-J
2
Heisenberg model. MPS is represented by the blue line, while TTN is depicted by the orange line.
Additionally, the infidelity between the tensor network and the state vector is indicated by the dotted line.
Figure 5: Infidelities between original TTNs and generated quantum circuits. The bond dimensions of TTNs are 16 and the
number of optimization steps is 1000. The learning rate for optimization is set to 0.6.
between original state vectors and generated quantum
circuits from TTNs are sufficiently low, indicating the
successful embedding of the TTNs into shallow quan-
tum circuits. Notably, in the BAS dataset, the TTN’s
superior representational capacity enables more accu-
rate embeddings than MPS. The result of the J
1
-J
2
Heisenberg model reveals an intriguing pattern: TTN
outperforms in both shallow and deep quantum cir-
cuits, while MPS excels in the intermediate region.
The superior performance of TTN in shallow circuits
is due to the minimal loss from the penetration algo-
rithm, enabling high-precision systematic decomposi-
tion. In deep circuits, TTN’s higher representational
capacity leads to better convergence accuracy, as in-
dicated by the dotted lines. Thus, the J
1
-J
2
model
results enhance our understanding of the differences
between MPS and TTN embeddings.
Figure 5 highlights the importance of the system-
atic decomposition algorithm for embedding TTNs
into quantum circuits. As demonstrated in previous
research on MPS (Rudolph et al., 2023a), methods
such as Iter[D
i
, O
all
] and Iter[I
i
, O
all
], which sequen-
tially add layers and optimize all circuits, achieve
high accuracy. Furthermore, as suggested in previous
studies, it was confirmed that Iter[I
i
, O
all
] struggles
with the BAS dataset even in TTN embeddings, where
QAIO 2025 - Workshop on Quantum Artificial Intelligence and Optimization 2025
800

the singular values at the bond become discontinuous.
This indicates that Iter[D
i
, O
all
] is the most effective
embedding method, achieving high accuracy in mul-
tiple fields, including machine learning and physics,
thus affirming the critical importance of the system-
atic decomposition algorithm. This result also verifies
the effective performance of the proposed systematic
decomposition method.
5 CONCLUSION
To avoid the barren plateau issue in VQAs, there is
increasing interest in using tensor networks to ini-
tialize quantum circuits. However, embedding tensor
networks into shallow quantum circuits is generally
difficult, and prior research has been limited to em-
bedding MPSs. In this study, we propose a method
for embedding TTNs, which have a more complex
structure than MPSs and can efficiently represent two-
dimensional systems and systems with long-range
correlations, into shallow quantum circuits composed
solely of two-qubit gates. We applied our proposed
method to various types of TTNs and confirmed that it
prepares quantum circuit parameters with better accu-
racy than embedding MPSs. Additionally, the compu-
tational complexity is O(max(N log Nχ
4
K
2
,Nχ
3
4
K
)),
or O(N log Nχ
4
K
2
T ) with approximation, making it
applicable to practical problems. This study will serve
as an important bridge for implementing hybrid algo-
rithms combining tree tensor networks and quantum
computing.
ACKNOWLEDGEMENTS
This work was supported by the Center of Innovation
for Sustainable Quantum AI, JST Grant Number JP-
MJPF2221, and by Japan Society for the Promotion of
Science KAKENHI, Grant Numbers 22K18682 and
23H03818. We acknowledge the use of Copilot (Mi-
crosoft, https://copilot.microsoft.com/) for the trans-
lation and proofreading of certain sentences within
our paper.
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