# Building a Personalized Dialogue System with Prompt-Tuning

Tomohito Kasahara1, Daisuke Kawahara1,
Nguyen Tung2, Shengzhe Li2, Kenta Shinzato2, Toshinori Sato2
${}^{\text{1}}$Waseda University, ${}^{\text{2}}$LINE Corporation
{tomo_k<EMAIL_ADDRESS>
<EMAIL_ADDRESS>

###### Abstract

Dialogue systems without consistent responses are not fascinating. In this
study, we build a dialogue system that can respond based on a given character
setting (persona) to bring consistency. Considering the trend of the rapidly
increasing scale of language models, we propose an approach that uses prompt-
tuning, which has low learning costs, on pre-trained large-scale language
models. The results of automatic and manual evaluations in English and
Japanese show that it is possible to build a dialogue system with more natural
and personalized responses using less computational resources than fine-
tuning.

##  1 Introduction

Large dialogue corpora used to train dialogue systems using neural network
models contain utterances from various speakers. This has the disadvantage
that the trained system is often inconsistent in the generated utterances (Li
et al., 2016b). For example, after the system says, “I am from Tokyo,” it
might say, “I am from Kyoto.”

We aim to build a dialogue system that can respond based on a persona to avoid
inconsistent utterances. A simple method of giving a persona to a model can be
to concatenate the persona to the model’s input in natural language (Zhang et
al., 2018). However, this method is not suitable because the more persona
information is added, the longer the input text becomes. Therefore, we propose
to freeze all parameters of a pre-trained language model and add a new fixed-
length prompt before the input token sequence to embed the persona
information. Specifically, only the embedding vectors of the added prompt are
optimized using a dialogue corpus in which utterances are made based on the
persona.

We conduct experiments on two languages: English and Japanese. Automatic and
manual evaluations show that our method can build a dialogue system capable of
natural responses based on a persona. Since our approach does not update the
parameters of the pre-trained model, it can reduce the computational cost
required for training. We also show that it is possible to build a
personalized dialogue system with even a small dataset consisting of hundreds
to thousands of utterance-response pairs.

##  2 Related Work

###  2.1 Prompt-Tuning

With the advent of pre-trained models such as BERT (Devlin et al., 2019) and
T5 (Raffel et al., 2020), a method that adapts a pre-trained model to a target
task by fine-tuning has become mainstream. However, as the scale of models
grows and the cost of fine-tuning increases, methods for adapting a pre-
trained model to a target task without updating their parameters are gaining
attention.

Brown et al. (2020) proposed a zero/few-shot learning method based on language
models with manually created task descriptions and zero/a few task examples
(collectively called prompt). Although there are some studies on improving
this method (Reynolds and McDonell, 2021; Zhao et al., 2021), they are
inferior to fine-tuning in terms of accuracy.

Prompt-tuning is a method for automatically optimizing a prompt without
creating it by manual labor. There are two kinds of methods in prompt-tuning:
one is to select the best words from a discrete vocabulary (Shin et al.,
2020), and the other is to optimize continuous embedding vectors (Qin and
Eisner, 2021; Li and Liang, 2021; Lester et al., 2021; Liu et al., 2021; Vu et
al., 2021). Prefix-tuning (Lester et al., 2021; Li and Liang, 2021) adds a
sequence of tokens, called prefix tokens, to the beginning of the input and
optimizes only their embedding vectors. There is also a study on multimodal
prompt-tuning for images and natural language (Tsimpoukelli et al., 2021).

###  2.2 Persona-Based Dialogue Systems

According to Roller et al. (2021), for dialogue systems to interact more
naturally with humans, it is essential to consider three perspectives: having
a consistent personality, having knowledge, and having emotions and empathy
for the interlocutor. Among these three perspectives, we focus on personality
because we believe that it is the most important to generate consistent
responses.

The Persona-Chat dataset (Zhang et al., 2018) is a dataset created with the
goal of adding personality to a dialogue system. It consists of multi-turn
dialogues between two crowdworkers, each of whom is given approximately five
persona sentences, which describe their character settings. There are 1,155
personas in the Persona-Chat dataset. There are two types of persona sentences
per persona: original, which the worker used in the dialogue, and revised,
which is a paraphrased version of the original. In the experiments conducted
by Zhang et al. (2018), models were trained using all the data in the Persona-
Chat dataset, which contains utterances based on various personas. On the
other hand, our method uses dialogue data uttered based on only one persona to
train models. There is also a Japanese version of the Persona-Chat dataset,
JPersonaChat (Sugiyama et al., 2021). Other dialogue corpora that contain
speaker persona information include PersonalDialog (Zheng et al., 2019) and a
corpus of dialogue data from Reddit (Mazaré et al., 2018). Zheng et al. (2019)
proposed a method to add encoded persona information to the input before it is
fed into a seq2seq model.

##  3 Method

This section describes our proposed method. The detailed setup for our
experiments is described in Sections 4.1 and 4.2.

###  3.1 Proposed Model

We propose a Transformer-based model with an additional embedding layer for
tokens that embed persona information. We refer to these tokens as persona
info tokens. The architecture and input-output relation of the proposed model
are shown in Figure 1.

###  3.2 Datasets

Conversations in daily life are not always related to personal information
(Song et al., 2021). To allow the model to generate not only utterances that
are related to the persona but also utterances that are not related to the
persona, we make a dialogue dataset that consists of two types of datasets.
The first is a dialogue dataset where each utterance is based on the persona,
and the second is a dialogue dataset that is not related to the persona.

###  3.3 Training

The newly added embedding layer embeds persona info tokens, and the embedding
layer of the pre-trained language model embeds each pair of utterance and
response (which consists of tokens already generated during training). These
embedding vectors are combined and then input into the model. During training,
the cross-entropy loss is calculated for the output tokens of the response
sentence, and only the parameters of the embedding layer for the persona info
tokens are updated.

The embedding layer for the persona info tokens is initialized with the
persona sentences included in the Persona-Chat dataset. These sentences are
embedded into vectors by the embedding layer of the pre-trained language model
and then used for initialization. If the number of the tokens of the persona
sentences is less than the length of the persona info tokens, the persona
sentences are repeatedly arranged until the number is satisfied.

Figure 1: Architecture and input-output relation of the proposed model. All
parameters of the pre-trained language model and its embedding layer are
frozen. Only the newly added embedding layer for persona info tokens is tuned.

##  4 Experiments

Based on the method in Section 3, we build a personalized dialogue system. We
used Hugging Face’s Transformers to build the system and the NVIDIA A100 SXM4
GPU with a GPU memory size of 40 GB. The main experiments are conducted in
English, and the results of additional experiments in Japanese are included at
the end of this section.

###  4.1 Datasets Setup

We use the Persona-Chat
dataset111https://github.com/facebookresearch/ParlAI/tree/main/parlai/tasks/personachat
and DailyDialog (Li et al., 2017)222https://aclanthology.org/I17-1099/ for our
experiments in English.

####  4.1.1 Training Datasets

First, the multi-turn dialogues in the Persona-Chat dataset are divided into
two utterances of one round trip. We refer to this pair of two utterances as a
dialogue pair. The dialogue pairs are aggregated according to the persona type
given to the responder. There are 1,155 personas in the Persona-Chat dataset,
but we use the three personas with the most dialogue pairs in our experiments.
The reason for this is that we intend to experiment with a relatively large
number of dialogue pairs even in the small dataset. The number of dialogue
pairs based on these three personas is 185, 167, and 166, respectively. Three
models corresponding to the three personas are trained and evaluated for each
experimental setup. The aggregated dialogue pairs are divided into training
and evaluation pairs in a ratio of 9:1.

The Persona-Chat dataset does not contain many short utterances or utterances
unrelated to persona. To add utterances that are short and not related to
persona to the dataset, we also use dialogue pairs contained in DailyDialog
whose topic is Relationship,333Each dialogue is assigned a topic. There are
ten topics: Attitude & Emotion, Culture & Education, Finance, Health, Ordinary
Life, Politics, Relationship, School Life, Tourism, and Work. which contains
many such utterances. Among them, dialogue pairs in which the lengths of both
the utterance and the response are less than 50 characters are mixed into the
training datasets in a certain ratio. Based on the results of preliminary
experiments, we determined the ratio of dialogue pairs added from DailyDialog
to the number of those obtained from the Persona-Chat dataset as 1:1. We call
this the ratio of the training datasets.

####  4.1.2 Evaluation Datasets

We made two datasets for evaluation: the persona eval dataset and the general
eval dataset. The persona eval dataset is 10% of the 9:1 dataset described in
Section 4.1.1. The general eval dataset consists of dialogue pairs obtained
from DailyDialog under the same conditions as in Section 4.1.1, but not used
for training.

Training Method | Model | Dist-1 | Dist-2
---|---|---|---
Fine-Tuning (added) | GPT2-XL | 0.199 | 0.526
Fine-Tuning (none) | 0.210 | 0.568
Prompt-Tuning | 0.177 | 0.494
GPT-J-6B | 0.213 | 0.595
Table 1: Results of automatic evaluation by distinct-1, 2. The prompt-tuned GPT-J-6B model generates the most diverse responses. “Added” and “none” mean whether the persona sentences are added to the input sentence or not. Eval Dataset | Training Method | Model | Fluency | Engagingness | Relevance
---|---|---|---|---|---
Persona Eval | Fine-Tuning (none) | GPT2-XL | 3.52 (1.26) | 3.70 (1.22) | 3.30 (1.27)
Prompt-Tuning | 3.82 (1.06) | 3.74 (1.17) | 3.62 (1.02)
GPT-J-6B |  3.90 (0.90) |  3.98 (0.95) |  3.82 (0.96)
General Eval | Fine-Tuning (none) | GPT2-XL | 3.93 (1.19) |  3.82 (1.20) | 3.77 (1.16)
Prompt-Tuning |  4.04 (1.01) | 3.81 (1.19) |  3.96 (1.13)
GPT-J-6B | 3.98 (1.03) | 3.80 (1.01) | 3.89 (1.05)
Human | 4.31 (1.07) | 4.25 (1.06) | 4.36 (0.92)
Table 2: We evaluated the generated responses on a 5-point scale for fluency, engagingness, and relevance. We asked five workers to answer each question, and the averages of all answers and standard deviations (in parentheses) are shown. The prompt-tuned GPT-J-6B model scored highest in all aspects in the persona eval dataset. No significant differences were found in the general eval dataset. Eval Dataset | Training Method | Model | [1,2) | [2,3) | [3,4) | [4,5]
---|---|---|---|---|---|---
Persona Eval | Fine-Tuning (none) | GPT2-XL | 0 | 5 | 33 | 12
Prompt-Tuning | 0 | 7 | 41 | 2
GPT-J-6B | 0 | 2 | 29 | 19
General Eval | Fine-Tuning (none) | GPT2-XL | 0 | 11 | 105 | 34
Prompt-Tuning | 0 | 8 | 75 | 67
GPT-J-6B | 0 | 1 | 91 | 58
Table 3: The generated responses were rated on a 5-point scale for persona consideration, and their distribution is shown. 1 is inconsistent with the persona, 3 is irrelevant to the persona, and 5 is in line with the persona. $[1,2)$ means the number of sentences scored between 1 and 2, including 1. In each setting, the number of samples from the persona eval dataset is 50 and that from the general eval dataset is 150. Utterance | Generated Response
---|---
| are you tired ?
---
| i am a little tired.
---
| cool , i am eating donuts . what do you do ?
---
| i am retired, i am a gym teacher. i used to teach gym.
---
| how about the korean restaurant next to it ?
---
| i heard it’s very good.
---
| i used to cook paella for a kitchen in spain .
---
| i love paella. that sounds great.
---
i bet it is a hard job to cook it right?
| the tv news is reporting a bank robbery .
---
| i heard it on the radio. what a shock.
---
| what do you like to do in your spare time ?
---
| i volunteer at a local soup kitchen,
---
helping people in need.
| hi , i like to keep fit and work out 5 times a week
---
| i like to eat cheeseburgers
---
and watch war documentaries
| stop producing cigarettes .
---
| i like the idea, but how can we do that?
---
Table 4: Response examples generated by the prompt-tuned GPT-J-6B model, which
got the highest score in the manual evaluation. We can see that both of
responses based on the persona (e.g., the sixth response) and responses
unrelated to the persona (e.g., the first one) are generated. Persona Sentence
---
i am a retired gym teacher.
i volunteer at a soup kitchen.
i was poor growing up.
cheeseburgers are my favorite food.
i like watching war documentaries.
Table 5: The persona used in the generated response examples in Table 4. Training Method | Model | Dist-1 | Dist-2
---|---|---|---
Fine-Tuning (none) | GPT2-XL | 0.134 | 0.379
Prompt-Tuning | 0.118 | 0.336
HyperCLOVA | 0.106 | 0.322
Table 6: Results of automatic evaluation by distinct-1, 2 in experiments in Japanese.  Eval Dataset | Training Method | Model | Fluency | Engagingness | Relevance
---|---|---|---|---|---
Persona Eval | Fine-Tuning (none) | GPT2-XL | 3.81 (1.12) | 3.63 (1.00) | 3.81 (1.06)
Prompt-Tuning | 3.68 (1.23) | 3.67 (1.13) | 3.71 (1.17)
HyperCLOVA |  3.87 (1.11) |  3.92 (0.98) |  3.90 (1.08)
General Eval | Fine-Tuning (none) | GPT2-XL | 4.01 (0.96) | 3.82 (0.89) | 3.82 (1.00)
Prompt-Tuning | 3.99 (1.09) | 3.68 (1.03) | 3.92 (1.08)
HyperCLOVA |  4.07 (1.01) |  3.86 (0.95) |  4.06 (0.99)
Human | 4.31 (1.07) | 4.25 (1.06) | 4.36 (0.92)
Table 7: Results of manual evaluation of fluency, engagingness, and relevance for the generated responses in the Japanese experiments. We asked five workers to answer each question, and the averages of all answers and standard deviations (in parentheses) are shown. Prompt-tuned HyperCLOVA scored highest in all aspects on both datasets. Eval Dataset | Training Method | Model | [1,2) | [2,3) | [3,4) | [4,5]
---|---|---|---|---|---|---
Persona Eval | Fine-Tuning (none) | GPT2-XL | 0 | 5 | 105 | 40
Prompt-Tuning | 1 | 14 | 84 | 51
HyperCLOVA | 0 | 18 | 77 | 55
General Eval | Fine-Tuning (none) | GPT2-XL | 0 | 8 | 122 | 20
Prompt-Tuning | 0 | 14 | 115 | 21
HyperCLOVA | 0 | 19 | 125 | 6
Table 8: Distribution of manually evaluated persona consideration in Japanese.
In each setting, the number of samples is 150 for both persona eval and
general eval datasets.

###  4.2 Model Setup

To compare our prompt-tuning model with fine-tuning, we use the datasets in
Section 4.1 and tune the pre-trained models of GPT series. We use two model
sizes: GPT2-XL (1.5B parameters) and GPT-J-6B (Wang and Komatsuzaki, 2021).
Fine-tuning of the GPT-J-6B model is not tested due to the lack of GPU memory.

The hyperparameters for prompt-tuning are based on the settings of (Lester et
al., 2021). The length of the persona info tokens was set to 200 based on the
results of preliminary experiments. The strategy for generating the response
sentences is the greedy search. The number of epochs was set to a value such
that the loss during learning converges. For fine-tuning, we experimented with
two methods: one is to input only dialogue pairs, and the other is to add
persona sentences before the dialogue pair’s utterance and then input it into
the model. Other hyperparameter values are given in Appendix B.

###  4.3 Results

We input the utterances of dialogue pairs from the evaluation datasets into
the trained models. We automatically evaluate the diversity of the generated
responses and manually assess whether the responses are natural and based on
the persona.

####  4.3.1 Automatic Evaluation

We evaluate the diversity of the generated responses by distinct-N (Li et al.,
2016a). The values of distinct-1 and distinct-2 are shown in Table 1. The
evaluation values are the average of all the generation results of the
persona, general eval datasets from each model corresponding to the three
types of personas. The results show that the GPT-J-6B model trained by prompt-
tuning generates the most diverse responses. In fine-tuning, we also find that
the results are better when persona sentences are not added to the input,
similar to the experimental results using the seq2seq model in the experiments
by Zhang et al. (2018).

####  4.3.2 Manual Evaluation

We use Amazon Mechanical Turk to manually evaluate whether the generated
responses are natural and persona-based. Following the method of Zhang et al.
(2018), the responses are rated on a 5-point scale on four aspects: fluency,
engagingness, relevance, and persona consideration. We ask five workers to
answer each question. In each setting, the number of samples from the persona
eval dataset is 50 and that from the general eval dataset is 150. An example
of tasks given to workers is shown in Appendix C.

The results of the first three aspects are shown in Table 2. The human scores
are taken from the experiments by Zhang et al. (2018). In fine-tuning, when
persona sentences are added to the input, the automatic evaluation results are
worse than when they are not, and thus we only evaluate the models without
persona sentences. From Table 2, the manual evaluation results using the
persona eval dataset show that the prompt-tuned GPT-J-6B model achieves the
best scores in all aspects. This can be attributed to the fact that the larger
the model size was, the more knowledge was stored in the model through pre-
training, and the more natural responses were generated by using this
knowledge. Manual evaluation using the general eval dataset did not produce a
significant difference. This is probably because most of the utterances in the
general eval dataset are short and general, such as greetings, and the
responses are also short and simple sentences.

For persona consideration, the distribution of the evaluation results is shown
in Table 3, where 1 is inconsistent with the persona, 3 is irrelevant to the
persona, and 5 is in line with the persona. The average score of persona
consideration is calculated for each generated response. Table 3 shows that
the majority of the generated responses are based on personas.

Comparing fine-tuning and prompt-tuning with the largest models that can be
trained with a fixed GPU memory size, we can say that prompt-tuning can build
a dialogue system with more natural responses based on the persona.

Table 4 shows response examples generated by the prompt-tuned GPT-J-6B model,
which got the highest score in the manual evaluation. These responses are
generated from the model trained with the dialogue pairs based on persona
sentences shown in Table 5. We can see that training with small training
datasets of only a few hundred pairs can produce a response with a natural and
consistent personality, as shown in Table 4.

###  4.4 Experiments in Japanese

For our Japanese experiments, we use two datasets: JPersonaChat and
JEmpatheticDialogues (Sugiyama et al.,
2021).444https://github.com/nttcslab/japanese-dialog-transformers As in the
English experiments, three personas are used, and the number of dialogue pairs
from JPersonaChat are 527, 525 and 525, respectively. To create training
datasets, the same process as in the English experiments is used. Since most
of the utterances in JEmpatheticDialogues are shorter and more general than
those in JPersonaChat, we did not set any conditions for adding the utterances
from JEmpatheticDialogues to the training datasets. The ratio of the training
datasets is set to 1:10 based on the results of preliminary experiments. For
the models, we use GPT2-XL555https://huggingface.co/rinna/japanese-gpt-1b with
1.3B parameters and HyperCLOVA (Kim et al., 2021), a GPT3-like model with 6.9B
parameters.

In the automatic evaluation results shown in Table 6, in contrast to the
English experiments, HyperCLOVA, which has a higher number of parameters,
tends to score lower. This can be attributed to the fact that there were many
instances in which HyperCLOVA begins its response with back-channeling.

Table 7 shows the average scores for the three aspects within the manual
evaluation results. For both the persona eval dataset and general eval
dataset, the HyperCLOVA model with prompt-tuning scored the highest. The
distribution of persona consideration is shown in Table 8. As in the English
experiments, many responses are based on the persona and few are inconsistent
with the persona. Generated response examples are shown in Appendix A.

##  5 Conclusion

We proposed a method for prompt-tuning a pre-trained language model using
dialogue data based on a single persona. Automatic and manual evaluations
showed that we could construct a dialogue system that can respond more
naturally and persona-based, with less computational resources than fine-
tuning. Compared to the generated responses in English, those in Japanese look
natural due to the larger persona dataset. In the future, this method can be
used not only to add personality to a dialogue system but also to build a
dialogue system to generate responses with emotions by making a prompt for
each emotion.

## Acknowledgements

This work was supported by a joint research grant from LINE Corporation.

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  * Zhang et al. (2018) Saizheng Zhang, Emily Dinan, Jack Urbanek, Arthur Szlam, Douwe Kiela, and Jason Weston. 2018.  Personalizing dialogue agents: I have a dog, do you have pets too? In _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 2204–2213, Melbourne, Australia. Association for Computational Linguistics.
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##  Appendix A Generated response examples in Japanese

Response examples that generated by prompt-tuned HyperCLOVA are shown in Table
9. These responses are generated from the model trained with the dialogue
pairs based on persona sentences shown in Table 10.

Utterance | Generated Response
---|---
| こんにちは。最近パーマをかけて
---
イメチェンしたから今度みてほしいな。
(Hello. I recently got a perm and a makeover,
so I hope you’ll take a look at it sometime.)
| いいよ。いつがいい?わたしは家政婦の仕事が
---
あるから、夕方以降なら大丈夫だよ。
(Okay. When is a good time? I have to work
as a housekeeper, so I can come after the evening.)
| こんにちは!ウサギ飼ってるんですね、羨ましい。
---
(Hello! I see you have a rabbit, I’m jealous.)
| はい、飼ってます。ペットショップで
---
一目惚れして、そのまま連れて帰りました。
(Yes, I have one. I fell in love with her
at first sight at a pet store and took her home with me.)
| 絵の勉強をヨーロッパでやるとしたら、
---
どの国がいいんですか?
(If you were to study painting in Europe,
which country would you prefer?)
| ゴッホが好きなら、
---
オランダがいいんじゃないですか?
(If you like Van Gogh, why not Holland?)
| 大丈夫、ありがとう。短かったんだけど、
---
突然真っ暗になったから本当に怖い思いをした。
(No problem, thanks. It was short, but I was really
scared because all of a sudden it went dark.)
| それは怖いね。無事でよかった。
---
それにしても、停電って本当に困るよね。
(That’s scary. I’m glad you’re safe.
Still, power outages are really annoying, aren’t they?)
| パイン飴ってそんなに昔からあるの?
---
(Has candy pineapple been around that long?)
| うん、子どもの頃から食べてたよ。
---
(Yeah, I’ve been eating it since I was a kid.)
Table 9: Response examples generated by prompt-tuned HyperCLOVA. Persona
Sentence
---
私は家政婦です。 (I am a housekeeper.)
私は姉と暮らしています。 (I live with my sister.)
私のペットはウサギです。 (My pet is a rabbit.)
私が尊敬する人は、画家のゴッホです。 (The person I admire is the painter Van Gogh.)
私は美術部に入っていました。 (I was in the art club.)
Table 10: The persona used in the generated response examples in Table 9.

##  Appendix B Hyperparameter

Table 11 shows hyperparameters during model training in our experiment.

Hyperparameter | Fine-Tuning (En) | Prompt-Tuning (En) | Fine-Tuning (Ja) | Prompt-Tuning (Ja)
---|---|---|---|---
Optimizer | Adam | Adam | Adam | Adam
Learning Rate | 5e-5 | 1e-3 | 1e-5 | 1e-3
Table 11: Hyperparameters during model training in our experiment.

##  Appendix C An example of tasks used in crowdsourcing

Figure 2 shows an example of tasks used in crowdsourcing.

Figure 2: An example of tasks given to workers on Amazon Mechanical Turk for
the manual evaluation.


# Efficient preparation of the AKLT State with Measurement-based Imaginary
Time Evolution

Tianqi Chen<EMAIL_ADDRESS>School of Physical and Mathematical Sciences,
Nanyang Technological University, Singapore 639798  New York University
Shanghai; NYU-ECNU Institute of Physics at NYU Shanghai; Shanghai Frontiers
Science Center of Artificial Intelligence and Deep Learning, 567 West Yangsi
Road, Pudong New District, Shanghai 200126, China  Department of Physics,
National University of Singapore, Singapore 117551  Tim Byrnes
<EMAIL_ADDRESS>New York University Shanghai; NYU-ECNU Institute of Physics
at NYU Shanghai; Shanghai Frontiers Science Center of Artificial Intelligence
and Deep Learning, 567 West Yangsi Road, Pudong New District, Shanghai 200126,
China  State Key Laboratory of Precision Spectroscopy, School of Physical and
Material Sciences, East China Normal University, Shanghai 200062, China
Center for Quantum and Topological Systems (CQTS), NYUAD Research Institute,
New York University Abu Dhabi, UAE  Department of Physics, New York
University, New York, NY 10003, USA

###### Abstract

Quantum state preparation plays a crucial role in several areas of quantum
information science, in applications such as quantum simulation, quantum
metrology and quantum computing. However, typically state preparation requires
resources that scale exponentially with the problem size, due to their
probabilistic nature or otherwise, making studying such models challenging. In
this article, we propose a method to prepare the ground state of the Affleck-
Lieb-Kennedy-Tasaki (AKLT) model deterministically using an measurement-based
imaginary time evolution (MITE) approach. By taking advantage of the special
properties of the AKLT state, we show that it can be prepared efficiently
using the MITE approach. Estimates based on the convergence of a sequence of
local projections, as well as direct evolution of the MITE algorithm suggest a
constant scaling with respect to the number of AKLT sites, which is an
exponential improvement over the naive estimate for convergence. We show that
the procedure is compatible with qubit-based simulators, and show that using a
variational quantum algorithm for circuit recompilation, the measurement
operator required for MITE can be well approximated by a circuit with a much
shallower circuit depth compared with the one obtained using the default
Qiskit method.

##  I Introduction

The last few decades have witnessed tremendous progress in the study of
quantum simulation of many-body quantum systems [1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17, 18], one of the earliest proposed practical
applications of a quantum computer as suggested by Feynman [19, 20]. Recently,
with the emergence of noisy intermediate scale quantum (NISQ) era devices
[21], together with improvements in algorithms and hardware, there has been an
increasing focus on utilizing such devices to facilitate certain tasks which
were beyond the scope of numerical approaches on classical computers. Examples
of models that have been examined are strongly correlated lattice models [22,
23, 24, 25], quantum spin liquids [26, 27], fractional quantum Hall effect
[28], and models involving Majorana fermions [29]. The NISQ-era devices have
also been applied in characterizing non-equilibrium phase transition [30] as
well as dissipative quantum systems involving non-unitary operations [31, 32].

In order to examine the properties of a physical model, the task of state
preparation is a crucial step for simulating quantum many-body systems, and
has attracted much attention recently [33, 34, 35]. The aim of quantum
simulation is to investigate the properties of a particular system, which
usually amounts to obtaining the low-lying energy eigenstates of a specified
Hamiltonian [36]. Several methods have been proposed for realizing this.
First, in an analogue quantum simulation approach [5, 37, 38], the system is
physically cooled down, and the Hamiltonian of interest is realized using
experimental techniques such as optical lattices and optical tweezers [39, 40,
41]. In a digital quantum simulation approach, the quantum phase estimation
[42] algorithm, or other Fourier transform methods [43], can be used to read
out of the energy eigenstates of the Hamiltonian. However, in order to study
the low energy eigenstates, one must be able to prepare a state with a
suitably high overlap with the low energy eigenstates. One approach to prepare
this is to use an adiabatic procedure, where one slowly changes the
Hamiltonian from a known one to the target Hamiltonian [44]. Another approach
that has gained recent popularity are variation approaches, such as the
variational quantum eigensolver (VQE) [45, 46] and the quantum approximate
optimization algorithm (QAOA) [47]. In these approaches, a measurement-
feedback approach is to used to obtain a low-energy state of a parameterized
circuit. Finally, several methods that mimic imaginary time evolution (ITE)
have been developed recently. The variational imaginary time evolution (VITE)
[48] method works in a similar way to the measurement-feedback approaches
above, and uses a parameterized circuit to find a unitary evolution that best
approximates the imaginary time evolution. Another approach is measurement-
based imaginary time evolution (MITE), which uses measurements combined with
adaptive unitary operations to result in a stochastic, yet deterministic
procedure to prepare eigenstates of Hamiltonians [49, 50]. Other recent
approaches which deal with non-unitary time evolution either approximate the
target non-unitary operators as unitary ones [51], or transform them into
unitary operators with additional ancilla qubits [52].

In all the techniques listed above, finding the ground state of a given
Hamiltonian is typically a difficult problem. For example, Bittel and Kliesch
showed that training VQE is NP-hard [53]. Methods involving postselection tend
to scale badly for large-scale systems due to the low probability of obtaining
the desired result [54, 55, 35]. Both adiabatic methods and MITE are known to
have an exponential convergence time in the typical case, due to an
exponentially small gap [44, 49]. Hence performing state preparation in a
large-scale system with a non-trivial wavefunction is an outstanding problem
due to the exponential resources required. This is also true not only for
quantum simulation but also for related subfields such as quantum metrology
[56, 57, 58] and quantum computing [59, 60, 61], as these approaches require
preparation of a large-scale entangled quantum state.

Recently, there has been some interest in preparing the ground state of the
Afflect, Kennedy, Lieb and Tasaki (AKLT) model [62, 63]. The AKLT model is a
fundamental model in condensed matter physics which has served to understand
the physics of fractional excitations at its boundaries [62, 64, 65], as well
as the strongly correlated symmetry protected topological (SPT) phase with a
Haldane gap [63], due to its exactly solvable nature. It is also interesting
from the point of view of quantum computing since it was shown to be
realizable experimentally with photonics systems using cluster states [59, 66,
67], which is the primary resource of measurement-based quantum computing [68,
69, 70, 71]. Several works have discussed creating the state on NISQ-era
quantum processors [72, 73, 74], including tensor networks-based approaches
[75, 76, 77], and approaches involving dissipation on a qutrit superconducting
array [78]. However, those methods either require mid-circuit measurements
[73], adiabatic preparation of the state [75], or measurement-based post
selection [72, 74]. Other methods dealing with various non-unitary operators
include matrix decompositions [79] and diagonal operators [80]. Postselection
makes the methods inefficient for longer chains since the probability of a
successful preparation reduces drastically with chain length.

Hence, approaches that work without the necessity of postselection or complex
operations are in demand, such that they are practically implementable for
today’s NISQ-era quantum processors.

In this work, we propose an efficient, deterministic (i.e. no postselection)
method to prepare the ground state of the AKLT model. Our approach is based
upon the MITE methods introduced in Refs. [49, 50], where the ground state of
a given Hamiltonian can be prepared deterministically, using measurements and
adaptive unitary operations. In addition to the fundamental interest of the
AKLT state, the model is interesting to consider in the context of the MITE
algorithm due to the very high preparation efficiency that can be attained.
For a general Hamiltonian the ground state preparation for MITE requires an
exponential time overhead with system size [49]. Remarkably, in the case of
AKLT ground state, this can be reduced to constant time with respect to the
chain length $N$. This exponential speedup is thanks to the specific
properties of the AKLT model that makes this possible, despite the large
Hilbert space and the complexity of the quantum many-body wavefunction. We
also show that our approach can be implemented not only on qudit-based devices
directly, but also on qubit-based devices. The key procedure of MITE, which
involves a measurement in the energy eigenbasis of a given Hamiltonian, can be
implemented efficiently using a variational optimization algorithm for circuit
recompilation.

This paper is structured as follows: in Sec. II, we present the AKLT model
with periodic boundary conditions (PBCs) and define the key properties of the
ground state that we aim to prepare. In Sec. III we give a brief review of
MITE, and our particular procedure to prepare ground state. We then show the
performance for the preparation of the target AKLT state in the spin-$1$
representation in Sec. IV. In Sec. V, we show how the procedure can be
performed on a qubit based quantum computer, by mapping the AKLT model onto a
spin-$1/2$ Hamiltonian, and discussing the implementation of the measurement
operator required for MITE. Finally, we draw our conclusions and provide an
outlook in Sec. VI.

##  II The AKLT model

Figure 1: (a) The $N$-site AKLT model. Each spin-$1$ site is represented by an
oval, which may be either a spin-$1$ particle or two coupled spin-$1/2$
particles. The lines between the sites represent the interactions, indicating
the PBC geometry. The AKLT ground state is the common zero eigenstate [Eq.
(1)] of the $S=2$ projectors $\hat{\mathcal{P}}_{j,j+1}^{S=2}$ on adjacent
sites, indicated by the branched arrows. (b) State preparation of the AKLT
state using MITE for $N=6$. The MITE subroutine, denoted by
$\text{MITE}_{[m,n]}$ (forked arrows) is executed simultaneously on
neighboring pairs of sites, firstly for the odd bonds (upper panel) and
subsequently for the even bonds (lower panel). The procedure repeats until
convergence.

We consider a one-dimensional AKLT model, which consists of a linear chain of
$N$ spin-$1$ particles. The Hamiltonian for such an AKLT model is

$\displaystyle\hat{\mathcal{H}}$
$\displaystyle=\sum_{j=1}^{N}\hat{\mathcal{P}}_{j,j+1}^{S=2}$ (1)

where periodic boundary conditions (PBC) is used throughout this work, and

$\displaystyle\hat{\mathcal{P}}_{j,j+1}^{S=2}=\frac{1}{24}\left(\mathbf{S}_{j}+\mathbf{S}_{j+1}\right)^{2}\left[\left(\mathbf{S}_{j}+\mathbf{S}_{j+1}\right)^{2}-2\hat{I}\right]$
(2)
$\displaystyle=\frac{1}{2}\left[\mathbf{S}_{j}\cdot\mathbf{S}_{j+1}+\frac{1}{3}\left(\mathbf{S}_{j}\cdot\mathbf{S}_{j+1}\right)^{2}+\frac{2}{3}\hat{{I}}\right],$

is the projector on the $S=2$ subspace, where we used
$\mathbf{S}_{j}\cdot\mathbf{S}_{j}=2\hat{I}$. Here we defined
$\mathbf{S}_{j}=(S_{j}^{x},S_{j}^{y},S_{j}^{z})$ for $j=[1,N]$, where

$\displaystyle S_{j}^{x}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&1&0\\\ 1&0&1\\\
0&1&0\end{pmatrix},S_{j}^{y}=\frac{i}{\sqrt{2}}\begin{pmatrix}0&-1&0\\\
1&0&-1\\\ 0&1&0\end{pmatrix},S_{j}^{z}=\begin{pmatrix}1&0&0\\\ 0&0&0\\\
0&0&-1\end{pmatrix}.$ (3)

Throughout this work, we consider periodic boundary conditions (PBC), such
that the site $j=N+1$ is equivalent to site $j=1$.

The ground state of the one-dimensional AKLT model with PBC has no degeneracy
111For the one-dimensional AKLT model with open boundary conditions, it has a
four-fold degeneracy because of the two separate spin-$1/2$’s located at each
boundary.. The ground state has an eigenvalue equal to zero exactly

$\displaystyle\hat{\mathcal{H}}|\text{AKLT}\rangle=0.$ (4)

We shall henceforth call this non-degenerate ground state of the AKLT model
(1) the “AKLT state”. The ground state also satisfies for all $j$ [67]

$\displaystyle\hat{\mathcal{P}}_{j,j+1}^{S=2}|\text{AKLT}\rangle=0,$ (5)

which will be important in our case. This implies that for each neighboring
sites, the total net spin should consist of $S=0$ and $S=1$ states. Note that
the projectors on adjacent pairs of sites do not commute,
$[\hat{\mathcal{P}}_{j-1,j}^{S=2},\hat{\mathcal{P}}_{j,j+1}^{S=2}]\neq 0$,
even though the ground state is a common eigenstate of all the projection
operators. We note that since an AKLT state can be written exactly as a matrix
product state (MPS) [82], this also implies that for each pair of sites, the
local reduced density matrix has a minimum eigenvalue of $0$.

Instead of using genuine $S=1$ particles, one may also create the AKLT state
by using two $S=1/2$ particle per site, as shown in Fig. 1. While we will not
directly use this in our procedure, we summarize some of these properties in
this formulation. The AKLT state can be prepared by first preparing singlet
states between the sites

$\displaystyle\bigotimes_{j=1}^{N}\left(|0\rangle_{j,a}|1\rangle_{j+1,b}-|1\rangle_{j,a}|0\rangle_{j+1,b}\right),$
(6)

where $l\in\\{a,b\\}$ label the two subspins on each site. Then applying a
projection operator for all $j\in[1,N]$

$\displaystyle\hat{\mathcal{P}}_{j}^{S=1}=\frac{4}{3}\bm{\sigma}_{j,a}\cdot\bm{\sigma}_{j,b}$
(7)

where

$\displaystyle\bm{\sigma}_{j,l}=(\sigma_{j,l}^{x},\sigma_{j,l}^{y},\sigma_{j,l}^{z})$
(8)

for $j=[1,N]$ and

$\displaystyle\sigma_{j,l}^{x}=\frac{1}{2}\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\sigma_{j,l}^{y}=\frac{i}{2}\begin{pmatrix}0&-1\\\
1&0\end{pmatrix},\sigma_{j,l}^{z}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},$
(9)

one achieves the ground state of the Hamiltonian from Eq. (1). In Eq. (6), the
states are defined as the eigenstates of the Pauli-$z$ operator
$\sigma_{j,l}^{z}|n\rangle_{j,l}=(-1)^{n}|n\rangle_{j,l}$ for $n\in\\{0,1\\}$.
The projective (i.e. measurement) nature of Eq. (7) suggests that it might be
compatible with MITE, but in fact the random nature of quantum measurements,
and the specific initial state from Eq. (6) that is required makes a reliable
preparation more difficult. We explain an approach that is more robust and
efficient in the following section.

##  III Measurement-based imaginary time evolution algorithm

###  III.1 Overall scheme

We now describe a deterministic procedure to generate the AKLT state using
measurements and unitary operations. To this end, we use the measurement-based
imaginary time evolution (MITE) approach [49], adapted towards the specific
state of interest in this case. The MITE algorithm uses a combination of
quantum measurements with adaptive unitary transformations to
deterministically prepare an eigenstate of a given Hamiltonian. In the case of
the AKLT state, since it is the common eigenvalue of the
$\hat{\mathcal{P}}_{j,j+1}^{S=2}$ operators as shown in Eq. (5), this allows
for a more efficient procedure than simply applying the MITE algorithm to the
Hamiltonian from Eq. (1).

In imaginary time evolution, the goal is to amplify the ground state, such
that we obtain the AKLT state from an arbitrary initial state
$|\psi_{0}\rangle$

$\displaystyle\lim_{\tau\rightarrow\infty}e^{-\hat{\mathcal{H}}\tau}|\psi_{0}\rangle=|\text{AKLT}\rangle.$
(10)

Due to the fact that the Hamiltonian is the sum of all local projectors on the
$S=2$ subspace, the property from Eq. (5) can therefore be applied to the
Hamiltonian, and we observe that we also have

$\displaystyle\lim_{\tau\rightarrow\infty}\prod_{j=1}^{N}e^{-\tau\hat{\mathcal{P}}_{j,j+1}^{S=2}}|\psi_{0}\rangle=|\text{AKLT}\rangle.$
(11)

Note that this is a specific relation for the AKLT state since
$\hat{\mathcal{P}}_{j,j+1}^{S=2}$ do not all commute. Our approach to MITE
will therefore be to perform a MITE subroutine on each pair of sites $j$ and
$j+1$ locally, and sweep over all pair of sites for many iterations until the
fidelity is converged to one with the target state. The purpose of the sweeps
is to converge towards a consistent state such as to satisfy Eq. (5) for all
$j$. We illustrate the sequence for the preparation of the AKLT state in Fig.
1(b). The MITE subroutine is simultaneously applied to each pair of sites with
the same $j$ parity. For example, for the $N=6$ case the
$(j,j+1)=(1,2),(3,4),(5,6)$ are applied at the same time. This is followed by
$(j,j+1)=(2,3),(4,5),(6,1)$. This parallelization is possible because these
operations all commute, as they do not share common spins.

###  III.2 MITE subroutine on a pair of sites

We now describe how the MITE subroutine works on a pair of sites $(j,j+1)$.
One of the key ingredients of MITE is to perform a measurement in the
eigenbasis of the desired Hamiltonian [49]. In the standard MITE algorithm,
this would be the eigenstates of the Hamiltonian from Eq. (1). For a system
with a large Hilbert space, this may require long convergence times, since in
general the convergence scales with the Hilbert space dimension [49]. Here, we
propose a more efficient method, taking advantage of the property as given in
Eq. (5).

We instead form the measurement operators in the eigenbasis of
$\hat{\mathcal{P}}_{j,j+1}^{S=2}$, which are

$\displaystyle M_{j,j+1}^{q}=$ (12)
$\displaystyle\frac{1}{2}\sum_{S=0}^{2}\sum_{m=-S}^{m=S}\left(\cos\epsilon
E_{S,m}-(-1)^{q}\sin\epsilon E_{S,m}\right)|S,m\rangle_{j,j+1}\langle
S,m|_{j,j+1}$

where $q\in\\{0,1\\}$ labels the measurement outcomes. In Ref. [49] it is
described how to realize a measurement of this form using an ancilla qubit
prepared in the state $|+\rangle_{a}=(|0\rangle_{a}+|1\rangle_{a})/\sqrt{2}$.
Applying a unitary evolution
$\exp(-i\epsilon\hat{\mathcal{P}}^{S=2}_{j,j+1}\otimes\sigma_{a}^{y})$ and
measuring the ancilla qubit in the $z$-basis produces the measurement in Eq.
(12). Here, $|S,m\rangle_{j,j+1}$ are angular momentum eigenstates with total
spin $S$ and projection $m$ and are eigenstates of
$\hat{\mathcal{P}}^{S=2}_{j,j+1}$. The eigenstates consists of two coupled
$S=1$ particles on sites $j$ and $j+1$. The eigenvalues of the projection
operator are given by

$\displaystyle E_{S,m}=\delta_{S,2},$ (13)

i.e. it is 1 if $S=2$ and 0 otherwise.

In a single MITE subroutine, a sequence of measurements given by Eq. (12) are
performed, where the outcomes $q\in\\{0,1\\}$ occur randomly according to Born
probabilities. For a particular sequence, if there are a total of $k_{0}$
counts of $q=0$ and $k_{1}$ counts of $q=1$, the resultant state is

$\displaystyle(M_{j,j+1}^{0})^{k_{0}}(M_{j,j+1}^{1})^{k_{1}}|\psi_{0}\rangle=\sum_{S,m}A_{k_{0}k_{1}}\left(\epsilon
E_{S,m}\right)\langle E_{S,m}|\psi_{0}\rangle|E_{S,m}\rangle$ (14)

where

$\displaystyle
A_{k_{0}k_{1}}(\chi)=\cos^{k_{0}}(\chi+\pi/4)\sin^{k_{0}}(\chi-\pi/4)$ (15)

is an amplitude function which has a Gaussian form for a large number of
measurements. The sequence of measurements therefore causes a collapse of the
state in energy space, peaked at the value

$\displaystyle E_{k_{0}k_{1}}^{\rm
peak}=\frac{1}{2\epsilon}\arcsin{\frac{k_{1}-k_{0}}{k_{0}+k_{1}}}.$ (16)

Since in our case, the only energy eigenstates are $E=0,1$ according to Eq.
(13), there will be an amplitude gain of the desired $E=0$ states as long as
$E_{k_{0}k_{1}}^{\rm peak}<1/2$, the midway point between the energy levels.

Applying the measurement sequence [Eq. (14)] randomly converges to either the
target $E=0$ states (with $S=0,1$), or the $E=1$ states (with $S=2$). In order
to make a deterministic procedure to produce the AKLT state, we require an
adaptive operation, where we introduce a conditional unitary operator applied
to the measurement outcomes. By monitoring the location of the Gaussian peak
[Eq. (16)], if its value is higher than a pre-chosen energy threshold $E_{\rm
th}$ where $E_{\text{th}}=\epsilon/\eta$, $\epsilon$ is the interaction time,
and $\eta$ is an adjustable modification factor, then we apply a corrective
unitary to the state. If $E^{\rm peak}_{k_{0}k_{1}}$ falls into the desired
range $E^{\rm peak}_{k_{0}k_{1}}<E_{\rm th}$, then no further operation is
executed. To make this concrete, we define the adaptive unitary operation as

$\displaystyle U_{j,j+1}=\begin{cases}\hat{I}&\quad\text{if}\,\,E^{\rm
peak}_{k_{0}k_{1}}<E_{\rm th}\\\
U^{C}_{j,j+1}&\quad\text{otherwise},\end{cases}$ (17)

where $\hat{I}$ is the identity operator. Combining the adaptive unitary with
the measurement, we hence perform the sequence after the $T$th measurement as

$\displaystyle|\psi_{T}\rangle\rightarrow|\psi_{T+1}\rangle=\frac{U_{j,j+1}M_{j,j+1}^{q}|\psi_{T}\rangle}{\sqrt{\langle\psi_{T}|M_{j,j+1}^{q,\dagger}M_{j,j+1}^{q}|\psi_{T}\rangle}}$
(18)

where the choice of which unitary that is made in Eq. (17) is made by keeping
track of the counts $k_{0}$ and $k_{1}$. Once the operator $U^{C}$ is applied,
the counts are reset to zero $k_{0}=k_{1}=0$.

The particular choice of adaptive unitary that is made has a large amount of
freedom, and can be any operator that creates a transition between the $E=0$
and $E=1$ energy states. We choose a form that rotates the spins randomly on
each of the sites

$\displaystyle U_{j,j+1}^{C}=$ $\displaystyle\exp\left[2\pi
i\left(a_{j}^{x}S_{j}^{x}+a_{j}^{y}S_{j}^{y}+a_{j}^{z}S_{j}^{z}\right)\right]$
$\displaystyle\otimes\exp\left[2\pi
i\left(a_{j+1}^{x}S_{j+1}^{x}+a_{j+1}^{y}S_{k}^{y}+a_{j+1}^{z}S_{k}^{z}\right)\right]$
(19)

where the $a_{j}^{\alpha}\in[0,1]$ for $\alpha\in\\{x,y,z\\}$ a randomly
generated real number.

For each pair of sites $(j,j+1)$, the procedure as described in Eq. (18) is
repeated up to $N_{\text{iter}}$ times, or until convergence on the MITE
subroutine is attained. Convergence is attained in a sequence of measurements
when $E^{\rm peak}_{k_{0}k_{1}}<E_{\rm th}$ for many iterations. The MITE
subroutine on separate spins can be straightforwardly parallelized, since the
updates from Eq. (18) can be performed completely locally. For example, in the
$N=6$ case shown in Fig. 1(b), the MITE subroutine is applied to odd and even
$j$ for the pairs of sites involved in the measurement shown in Eq. (12).
Iterating the MITE subroutines on the odd and even $j$ pairs of sites, the
state converges to the AKLT state.

##  IV The AKLT state preparation with spin-$1$ particles

In this section, we show numerical results of the performance for the AKLT
state preparation with spin-$1$ particles. This is a direct preparation of the
AKLT state in the spin-$1$ representation of Fig. 1. This can be physically
implemented using qutrits [83, 84] on various platorms such as trapped ions
[85], superconducting quantum processors [86], silicon carbide [87], as well
as silicon-photonic integrated circuits [88].

The initial state for the evolution is chosen to be:

$\displaystyle|\psi_{0}\rangle=\bigotimes_{j=1}^{N}|S=1,m=1\rangle_{j}$ (20)

where $|S=1,m\rangle$ are the eigenstates of $S^{z}$. We choose this state as
an easily preparable initial state experimentally, although the MITE routine
is generally not sensitive on the initial conditions.

To measure the performance, we use the state fidelity with respect to the AKLT
state

$\displaystyle F_{\rm tot}=|\langle\text{AKLT}|\psi_{T}\rangle|^{2}.$ (21)

We also compute the partial fidelities for each pair of sites $(j,j+1)$,
defined as

$\displaystyle
F_{j,j+1}=\langle\psi_{T}|(\hat{I}-\hat{\mathcal{P}}^{S=2}_{j,j+1})|\psi_{T}\rangle.$
(22)

This calculates the fidelity of successfully converging to the zero eigenvalue
state for a single projector $\hat{\mathcal{P}}^{S=2}_{j,j+1}$. Here,
$|\psi_{T}\rangle$ is the state with the update rule as shown in Eq. (18)
during the MITE subroutine, after $T$ measurements are made.

Figure 2: Partial fidelities using the spin-$1$ representation with $N=4$
between (a) odd pair of sites ($F_{\rm 3,4}$ as an example) and (b) even pair
of sites ($F_{\rm 1,4}$ as an example). The three curves are three different
runs of the MITE subroutine. The data point is recorded for each measurement
step $T$ within each MITE subroutine up to $T=100$. For all panels, the
interaction time is chosen to be $\epsilon=0.5$. The factor $\eta$ is chosen
to be $\eta=4$ such that $E_{\text{th}}=1/8$.

The partial fidelities are shown in Fig. 2. We observe that there is a fast
convergence of the partial fidelities to unity with typically $T\sim 20$
measurements, for all chain lengths that we have tried. Both odd and even
parity site pairs converge with a similar number of MITE steps. The trajectory
for each particular run is different, due to the stochastic nature of quantum
measurements, but every run converges to a state with $F_{j,j+1}=1$. We expect
that the convergence occurs at similar timescales regardless of the chain
length $N$ because the dimension of the MITE subroutine for a pair is constant
in each case. Specifically, there are two sites involved with a dimension of
$3^{2}=9$. Due to the key property of Eq. (5), the convergence of each MITE
subroutine is independent of the chain length.

Figure 3: AKLT state preparation using the spin-$1$ representation with $N=4$.
(a) the total state fidelity $F_{\rm tot}$ as a function of MITE subroutine
round $r$. For each $r$, MITE routines are performed throughout the whole
system. (b) The peak value of the amplitude function as a function of time. We
perform $100$ trials of simulations, and randomly choose three of them for the
illustration in this figure. The data points are recorded within the iteration
for each MITE subroutine round $r$ throughout the whole system up to $r=100$.
For all panels, we have used $N_{\rm iter}=30$, and the interaction time is
chosen to be $\epsilon=0.5$. The factor $\eta$ is chosen to be $\eta=4$ such
that $E_{\text{th}}=1/8$.

The fidelities with respect to the AKLT state are shown in Fig. 3(a)(c)(d).
Here we plot the total fidelity after $r$ complete rounds of applications of
MITE subroutines applied to the whole chain, as shown in Fig. 1(b). We see
that the AKLT state can be prepared in typically $r\sim 20$ rounds.
Remarkably, there is little dependence of the number of rounds on different
chain lengths, despite the exponential difference in Hilbert space dimensions.

We interpret this in the following way. Consider a MITE subroutine acting on
sites $j$ and $j+1$. While the MITE evolution deterministically evolves the
state towards the eigenstate given in Eq. (5), it may disrupt the eigenvalue
relation of its nearby operators, such as
$\hat{\mathcal{P}}_{j-1,j}^{S=2},\hat{\mathcal{P}}_{j+1,j+2}^{S=2}$. The
disruption is however limited to the vicinity of the sites $j,j+1$ due to the
short-ranged correlations of the AKLT state. Due to the limited range of the
disruption due to the MITE operations, the chain length is not the relevant
factor determining the convergence. Different parts of the chain effectively
act independently, and convergence is attained once the state independently
finds the state that is a common zero eigenvalue of all the projectors in Eq.
(5).

To test this hypothesis, we plot the convergence of the fidelity of the
following state

$\displaystyle\left[\left(\prod_{j\in\text{even}}(\hat{I}-\hat{\mathcal{P}}_{j,j+1}^{S=2})\right)\left(\prod_{j\in\text{odd}}(\hat{I}-\hat{\mathcal{P}}_{j,j+1}^{S=2})\right)\right]^{r}|\psi_{0}\rangle$
(23)

with the AKLT state. Here, $|\psi_{0}\rangle$ is the state as in Eq. (20) but
the eigenstates of $S^{x}$. This may be considered to by a simplified version
of our MITE-based procedure, where we remove the stochastic aspect of the MITE
subroutines and focus on the convergence of the common projections. The
smaller numerical overhead allows us to go to larger $N$. As can be seen in
Fig. 4(a), for different system sizes, the state converges to the target AKLT
state within $8$ rounds. In Fig. 4(b) we plot the number of rounds required to
attain a fidelity of 0.9. We see that the number of rounds remains
approximately constant for all chain lengths $N$. The particular case $N=7$
shows a higher number of rounds due to the slow increase in fidelity in the
early rounds, as see in Fig. 4(a). For this case, the initial state appears to
be a particularly poor choice, requiring a longer time to convergence. The
convergence confirms our hypothesis and suggests that in our procedure, the
total number of rounds $r$ for the convergence of the target state should
remain constant with the system size $N$.

In Fig. 3(b), we also plot the peak value computed using Eq. (16) which is
recorded at the end each round. The dashed horizontal lines correspond to the
energy eigenvalues given in Eq. (13) multiplied by $\epsilon$. We see that the
evolutions eventually converge to the target zero energy values after an
initially chaotic regime. For an interaction time $\epsilon=0.5$ the sequence
shows good convergence but for smaller values we find the convergence is
slower. This is as expected since a shorter interaction time corresponds to a
weaker measurement, and the time taken to collapse takes longer.

Figure 4: Target AKLT state convergence using direct applications from Eq.
(23): (a) the total fidelity $F_{\rm tot}$ as a function of direct application
rounds $r$ up to $r=15$. The grey dashed line represents $F_{\rm tot}=0.9$.
Lighter to darker colors indicate different system size from $N=3$ to $N=9$.
(b) the critical value of rounds $r_{c}$ when $F-{\rm tot}=0.9$ as a function
of system size $N$. All critical values of rounds $r_{c}$ were obtained via
interpolation of the data from panel (a) by numerically solving $F_{\rm
tot}(r)=0.9$.

##  V The AKLT state preparation with qubits

While the standard form of the AKLT Hamiltonian is written with respect to
spin-1 particles, typical quantum computers work with qubits, which may be
considered to be spin-$1/2$ particles. In this section, we demonstrate that
our methods are compatible with qubit-based quantum devices, such as transmon
qubit superconducting quantum computers [89] using IBM Q Qiskit [90] or Google
Cirq [91]. To show the compatibility with qubits, we first show how to map our
scheme onto spin-$1/2$ particles. Second, we show that the key component of
MITE, the measurement operators as given in Eq. (12), can be implemented using
a circuit recompilation of the corresponding unitary operator.

###  V.1 Mapping onto qubits

We perform the mapping onto qubits via the transformation of the spin-$1$
operators as

$\displaystyle\mathbf{S}_{j}\rightarrow\bm{\sigma}_{j,a}+\bm{\sigma}_{j,b}$
(24)

where the Pauli operators $\bm{\sigma}_{j,l}$ is defined in Eq. (8). The terms
in Eq. (2) can be written in terms of Pauli matrices as

$\displaystyle\mathbf{S}_{j}\cdot\mathbf{S}_{j+1}\rightarrow\left(\bm{\sigma}_{j,a}+\bm{\sigma}_{j,b}\right)\cdot\left(\bm{\sigma}_{j+1,a}+\bm{\sigma}_{j+1,b}\right)$
(25)

Since each physical spin-$1$ site has been transformed to two spin-$1/2$
sites, the Hilbert space of the total chain has also been enlarged from
$3^{N}$ to $2^{2N}=4^{N}$. By the usual rules of angular momentum coupling,
the two spin-$1/2$ can couple to form a spin-$0$ and spin-$1$. The states that
are symmetric under interchange of the $a$ and $b$ spins form a spin-$1$
triplet, while the states that are antisymmetric form a spin-$0$ singlet.
Since the Hamiltonian is symmetric under $a\leftrightarrow b$ interchange, and
we will start in an initial state that is also symmetric

$\displaystyle|\psi_{0}\rangle=\bigotimes_{j=1}^{N}|0\rangle_{j,a}|0\rangle_{j,b},$
(26)

the evolution remains in the symmetric subspace of dimension $3^{N}$, and the
antisymmetric sector remains unpopulated throughout the subsequent evolution.

Our numerical results for the MITE preparation of the AKLT state using in the
qubit framework are shown in Fig. 5. Similarly to the spin-$1$ case, a high
fidelity AKLT state can be obtained typically after $r\sim 10$ rounds of MITE
subroutines performed throughout the whole system. Interestingly, the state
appears to converge with a smaller number of rounds and the peak value
$\chi_{\rm max}$ of the Gaussian function reaches the desired value faster
than the spin-$1$ case, despite the mathematical equivalence of the models. We
note that although the Hilbert space for the spin-$1/2$ representation is
larger than the case with spin-$1$ representation, due to the symmetry obeyed
by the Hamiltonian and initial state, the evolution strictly remains in the
symmetric sector and the additional Hilbert space available never affects the
convergence.

As with the $S=1$ case, for smaller $\epsilon$ we find that the convergence is
slower, and the peak value $\chi_{\rm max}$ fluctuates significantly before
converges to the desired values, respectively.

Figure 5: The AKLT state preparation in the spin-$1/2$ representation for
$N=3$. (a) the total state fidelity $F_{\rm tot}$ as a function of MITE
subroutine round $r$. For each $r$, MITE routines are performed throughout the
whole system. (b) the peak value of the amplitude function as a function of
time. We perform $100$ trials of simulations, and randomly choose three of
them for the illustration in this figure. The data points are recorded within
the iteration for each MITE subroutine round $r$ performed throughout the
whole system up to $r=100$. For all panels, we have used $N_{\rm iter}=30$,
and the interaction time is chosen to be $\epsilon=0.5$. The factor $\eta$ is
chosen to be $\eta=2$ such that $E_{\text{th}}=1/4$.

###  V.2 Implementation of the measurement operator

The most straightforward implementation of the measurement operator in Eq.
(12) involves applying a Hamiltonian of the form

$\displaystyle\hat{H}_{M}=\hat{\mathcal{P}}^{S=2}_{j,j+1}\otimes\sigma_{a}^{y}.$
(27)

In the qubit language, the projector involves the square of the operator given
in Eq. (25), which involve fourth order spin interactions. Together with the
ancilla qubit, this is a fifth order qubit interaction, which is likely to be
difficult to implement using operations available on NISQ devices. However,
using the variational optimization algorithm for the quantum circuit
recompilation [92, 93, 94, 95, 96, 97, 53, 98, 99], the measurement operator
from Eq. (12) can be converted to a more readily implementable form.

In Fig. 6, we substitute the original unitary operation for the measurement
$\hat{U}=\exp(-i\epsilon\hat{H}_{M})$ into a variational parameterized circuit
$\hat{V}$. This involves 5 qubits since the measurement is on two sites, which
consist of two qubits each, and there is one ancilla qubit. The parameterized
circuit consists of an initial layer of $U_{3}$ gates, followed by alternate
“odd” and “even” layers. Here the $U_{3}$ gate is defined in the same way as
in Qiskit [90]: $U_{3}=U_{3}(\theta,\phi,\lambda)$. Each layer is composed of
one layer of CNOT gates followed by a single layer of $U_{3}$ gates. The total
depth of the parameterized circuit is $n_{L}$. We use the Quimb package [93]
to optimize the parametrized circuit $\hat{V}$ through a limited memory
Broyden-Fletcher-Goldfarb-Shanno algorithm with box constraints (L-BFGSB) [92,
100, 101], with a basin-hopping method [102, 103, 104, 105]. The loss function
is the same as that given in Ref. [93].

The optimization results are shown in Fig. 7, where we show both the averaged
fidelity $\bar{\mathcal{F}}(U,V)$ as well as the maximum fidelity
$\mathcal{F}_{\rm max}(U,V)$ between the original operator $\hat{U}$ and the
parametrized circuit operator $\hat{V}$ of $100$ repetitions to characterize
the performance of the optimization procedure. It is found that when the
circuit layer depth is larger than $4$, of all the $100$ repititions, the
maximum fidelity between $\hat{U}$ and $\hat{V}$ already reaches the value
larger than $99.99\%$ [Fig. 7(a)]. When the depth is larger than $6$, although
$\mathcal{F}_{\rm max}(U,V)$ remains above $99.99\%$, the average fidelity
$\bar{\mathcal{F}}(U,V)$ starts to decrease due to overparametrization and
other basin-hopping parameters chosen.

Hence in summary, we are able to approximate the unitary $\hat{U}$ with only a
single-digit number of layers of $U_{3}$ and CNOT gates, where the number of
CNOT gates is of the magnitude of $\sim 10$, and the fidelity between the
original target weak measurement operator and the ansatz circuit is larger
than $99.99\%$. Using the default transpile function from Qiskit [90] for the
same measurement operator, the number of CNOT gates is 499.

Figure 6: Recompiled quantum circuit for the measurement operator (12). (left
panel) The target unitary evolution operator $\hat{U}$; (right panel) the
ansatz $\hat{V}$ and its quantum circuit: an initial layer composed of $U_{3}$
gates followed by an odd layer and an even layer. The odd (even) layer has
$CX$ gates on odd (even) bonds followed by $U_{3}$ gates placed on the
corresponding physical sites. Figure 7: Circuit recompilation of the
measurement operator (12). (a) the averaged fidelity between target operator
$U$ and the ansatz circuit $V$. The average value and the error bar are
calculated over $100$ repetitions. (b) the maximum of the fidelity between
target operator $U$ and the ansatz circuit $V$ within every $100$ repetitions.
Other parameters are: $N_{\rm iter}=100$, and $N_{\rm hop}=5$ from a basin-
hopping method.

##  VI Conclusions

We have proposed an efficient quantum algorithm based on the measurement-based
imaginary time evolution (MITE) algorithm for preparation of the AKLT state.
Our results show a fast convergence to the target state typically with
approximately $T\sim 20$ time steps per MITE subroutine. The total number of
rounds is typically $r\sim 20$. For the case of spin-$1/2$ representation, by
using the variational optimization quantum circuit recompilation algorithm,
the five-qubit weak measurement operator can be transformed into a circuit
with $\sim 10$ CNOT gates, which allows for a direct implementation on qubit-
based quantum computers. The speed of the overall convergence benefits from a
optimized interaction time parameter $\epsilon$, if this is too small the
convergence tends to occur much more slowly.

For a completely generic Hamiltonian, the complexity of convergence of the
MITE algorithm typically scales with the dimension of the Hilbert space [49].
However in the case of the AKLT state, due to the key property given in Eq.
(5), this allowed us to parallelize all MITE evolutions corresponding to even
and odd $j$ projectors $\hat{\mathcal{P}}^{S=2}_{j,j+1}$. Apart from the
$O(N)$ improvement in scaling, more importantly, since each projector only
involves a convergence on a Hilbert space dimension of 2 sites (Hilbert space
dimension of $3^{2}=9$), the convergence of each MITE subroutine is
independent of the chain length. Remarkably, finding the the common eigenstate
for all $j$ in Eq. (5) is independent of the chain length, due to the short-
ranged correlations in the AKLT state. This was directly observed in our MITE
simulations, as well as the prototypical case examined in Fig. 4. This means
that the whole convergence is attained in approximately constant time, which
is an exponential speedup compared to the naive estimate of convergence
related to the full Hilbert space dimension $3^{N}$.

We note that the efficiency of preparation observed here is a special case
thanks to the properties of the AKLT state. There are many other interesting
states that could be also prepared using the MITE approach, including exotic
dissipative many-body physics [106, 107, 108, 109, 110, 111, 112, 113, 114,
115, 18], as well as the simulation of quantum spin Hall effects beyond the
solid-state systems [116, 117]. It is however an open question of whether
there are other models that can be prepared with the exceptional efficiency of
the AKLT state, and exactly what properties the Hamiltonian must obey to
attain this efficiency. It was shown in Ref. [50] that cluster states could
also be prepared efficiently due to the commuting stabilizers that define the
state. Even in the generic case, the MITE approach does not require any
postselection, and is therefore deterministic: one could obtain the target
AKLT state for every run of the sequence. Furthermore, no knowledge of the
ground state wavefunction is required for the convergence. Hence it is
potentially useful as a generic procedure in a quantum simulation context,
where eigenstates of Hamiltonians require preparation.

##  VII acknowledgements

T. B. is supported by the National Natural Science Foundation of China
(62071301); NYU-ECNU Institute of Physics at NYU Shanghai; Shanghai Frontiers
Science Center of Artificial Intelligence and Deep Learning; the Joint Physics
Research Institute Challenge Grant; the Science and Technology Commission of
Shanghai Municipality (19XD1423000,22ZR1444600); the NYU Shanghai Boost Fund;
the China Foreign Experts Program (G2021013002L); the NYU Shanghai Major-
Grants Seed Fund; Tamkeen under the NYU Abu Dhabi Research Institute grant
CG008; and the SMEC Scientific Research Innovation Project (2023ZKZD55). T. C.
acknowledges support from the Singapore National Research Foundation (NRF)
under NRF fellowship award NRF-NRFF12-2020-0005. The computational work for
this article was partially performed on resources of the National
Supercomputing Centre, Singapore (NSCC) (https://www.nscc.sg), and on
resources of the National University of Singapore (NUS)’s high-performance
computing facilities, partially supported by NUS IT’s Research Computing
group.

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