Digital Discovery of 100 diverse Quantum Experiments with PyTheus

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Digital Discovery of 100 diverse Quantum Experiments with

PyTheus

Carlos Ruiz-Gonzalez§1, Sören Arlt§1, Jan Petermann1, Sharareh Sayyad1, Tareq Jaouni2,

Ebrahim Karimi1,2, Nora Tischler3, Xuemei Gu1, and Mario Krenn1

1Max Planck Institute for the Science of Light, Erlangen, Germany.

2Nexus for Quantum Technologies, University of Ottawa, K1N 6N5, ON, Ottawa, Canada.

3Centre for Quantum Computation and Communication Technology (Australian Research Council), Centre for Quantum Dy-

namics, Griﬃth University, Brisbane, Australia.

Photons are the physical system of

choice for performing experimental tests

of the foundations of quantum mechan-

ics. Furthermore, photonic quantum tech-

nology is a main player in the second

quantum revolution, promising the devel-

opment of better sensors, secure com-

munications, and quantum-enhanced com-

putation. These endeavors require gen-

erating speciﬁc quantum states or eﬃ-

ciently performing quantum tasks. The

design of the corresponding optical exper-

iments was historically powered by hu-

man creativity but is recently being auto-

mated with advanced computer algorithms

and artiﬁcial intelligence. While sev-

eral computer-designed experiments have

been experimentally realized, this ap-

proach has not yet been widely adopted

by the broader photonic quantum optics

community. The main roadblocks con-

sist of most systems being closed-source,

ineﬃcient, or targeted to very speciﬁc

use-cases that are diﬃcult to generalize.

Here, we overcome these problems with

a highly-eﬃcient, open-source digital dis-

covery framework PyTheus, which can em-

ploy a wide range of experimental devices

from modern quantum labs to solve var-

ious tasks. This includes the discovery of

highly entangled quantum states, quantum

measurement schemes, quantum commu-

nication protocols, multi-particle quantum

Carlos Ruiz-Gonzalez§:cruizgo@proton.me

Sören Arlt§:soeren.arlt@mpl.mpg.de

Mario Krenn: mario.krenn@mpl.mpg.de

gates, as well as the optimization of contin-

uous and discrete properties of quantum

experiments or quantum states. PyTheus

produces interpretable designs for com-

plex experimental problems which human

researchers can often readily conceptual-

ize. PyTheus is an example of a powerful

framework that can lead to scientiﬁc dis-

coveries – one of the core goals of artiﬁ-

cial intelligence in science. We hope it will

help accelerate the development of quan-

tum optics and provide new ideas in quan-

tum hardware and technology.

Contents

1 Introduction 2

1.1 Related Work ............ 3

2 Graphs and Quantum Experiments 4

2.1 Quantum State Generation .... 5

2.1.1 Probabilistic Photon-Pair

Sources ........... 5

2.1.2 Deterministic Single-

Photon Sources ....... 7

2.1.3 Mixed States ........ 8

2.1.4 States Entangled in the

Photon-Number Basis ... 9

2.2 Quantum Communication ..... 10

2.3 Quantum Measurements ...... 11

2.4 Quantum Computation ....... 11

3 The PyTheus Library 12

§These authors contributed equally to this work.

The order was decided by a coin ﬂip.

Accepted in Quantum 2023-12-03, click title to verify. Published under CC-BY 4.0. 1

arXiv:2210.09980v2 [quant-ph] 8 Dec 2023

4 Hundred Experiments 14

4.1 Generation of Entangled States . . 14

4.2 Maximizing Entanglement ..... 21

4.3 Generation of Mixed States . . . . 26

4.4 Generation of Entanglement in the

Photon-Number Basis ....... 27

4.5 Towards Quantum Simulation . . . 30

4.6 Quantum Communication ..... 38

4.7 Quantum Measurements ...... 39

4.8 Quantum Gates ........... 41

4.9 Combinatorial Measures ...... 42

5 Outlook 43

References 44

1 Introduction

Photons, the individual particles of light, have

long been used as the core player for fundamen-

tal experiments and applications in quantum in-

formation science [1]. Photons do not easily inter-

act with their environments; therefore, they can

be distributed over large distances – which makes

them a key resource for long-distance quantum

communication [2,3] and experiments that re-

quire strict Einstein locality conditions [4–6]. Us-

ing advanced measurement-based quantum com-

puting schemes, photons are among the most

promising candidates for future quantum com-

puters [7]. Entanglement between two or more

photons can be produced without a vacuum or

cooling, and therefore many advanced experimen-

tal results can be achieved directly with table-top

setups. Furthermore, the bosonic nature of pho-

tons allows for the generation of complex entan-

gled quantum states of indistinguishable photons

that are a key resource for quantum-enhanced

measurements [8]. These potential applications

have lead to enormous technological advances

in integrated chips for fast and precise control

of photonic quantum states [9–12], high-quality

single-photon sources [13–16], novel photon-pair

sources [17], photon number resolving detectors

[18,19], and advanced high-quality multi-photon

interference [20–23].

One question now is how to utilize these tech-

nologies to build up exciting new experiments for

the foundations of quantum physics and practical

quantum hardware.

Historically, the design of quantum experi-

ments strongly relied on the intuition and cre-

ativity of human experts who leverage their ex-

perience and come up with blueprints of exper-

iments. However, due to the unintuitive phe-

nomena and enormous combinatorial space of the

potential designs, it becomes extremely diﬃcult

for human researchers to discover more complex

quantum setups. It might be possible that there

are high-quality solutions to experimental design

questions far outside of the region where humans’

intuition fails. How could we possibly ﬁnd such

extraordinary solutions?

This question has sparked a strong interest

in the automated discovery of quantum experi-

ments with computers, overviewed in [24]. The

invention of these tools for quantum optics ex-

periments [25] have indeed overcome experimen-

tal limitations and allowed for new avenues in

laboratories for entanglement research [26–29].

One crucial question is whether we can also learn

something about physics from these tools. And

indeed, several new concepts have been pub-

lished that were purely discovered through au-

tomated design [25], such as a new general idea

of entanglement structure [30], and generalized

constructions of photonic quantum gates [31].

Those concepts were discovered by tedious anal-

ysis of the computer’s solutions, which was time-

consuming. The problem was that the algorithms

were powerful enough to ﬁnd unknown solutions

but had no incentive to present a simple, human-

understandable form of it.

This was solved by the invention of Theseus

[32], an eﬃcient algorithm for the discovery of

new quantum experiments that can readily be

interpreted by humans. The key insight was

a shift in the representation. Rather than de-

scribing quantum experiments as quantum opti-

cal components on an optical table, experiments

are described as graphs of correlations between

photons. This representation, which has been a

derivative of a computer-discovered concept itself,

was developed in [33–35] – and allows working

with a much more natural representation, which

can be translated back at any point to an exper-

iment consisting of optical elements. (It should

be noted that the representation is independent

of photonic graph states for measurement-based

quantum computing [36–38], and it is so far un-

known how to translate among them.)

Accepted in Quantum 2023-12-03, click title to verify. Published under CC-BY 4.0. 2

In this paper, we introduce PyTheus2, a

highly-eﬃcient, open-source, automated design

and discovery framework for quantum optics ex-

periments. At the core, PyTheus uses a much

extended graph-based representation of quantum

optics, which allows us not only to represent en-

tanglement and quantum gates, but lets us de-

sign quantum measurements, quantum commu-

nication protocols, optimize experimental prop-

erties, and discover quantum systems that in-

volve single-photon sources, mixed states, and

states entangled in the photon-number basis. Be-

sides the advances of the scientiﬁc scope, we note

that PyTheus is written in Python, and there-

fore can readily be combined with machine learn-

ing frameworks such as TensorFlow and Py-

Torch, and allows for immediate parallelization

in computer clusters.

To showcase the applicability of PyTheus,

we demonstrate the discovery of 100 previously

unknown or advanced implementations of quan-

tum optics experiments, ranging from exciting

new systems for entanglement research to quan-

tum states from condensed matter physics that

are interesting for quantum simulation purposes,

new ways of performing quantum communication

tasks such as entanglement swapping, new quan-

tum state measurements, and quantum gates.

The experiments can involve both probabilistic

photon sources and deterministic single-photon

sources, and many of them are readily imple-

mentable in today’s modern quantum optics labs.

In the GitHub repository, we present the instruc-

tions for PyTheus that discover each of the ex-

amples. We hope that PyTheus’s eﬃciency, gen-

erality, and low entry barrier kick-starts the ap-

plication of computer-discovered quantum setups

in experimental laboratories worldwide, and in-

spires new exciting computer-inspired ideas and

directions for fundamentals and applications of

photonic quantum physics research.

While the goal of this paper was to demon-

strate the discovery capability of PyTheus, in

several cases, it was impossible not to see clear

generalizations and reasons why the solutions

work. We show this in some cases below. One

of the exceptionally interesting concepts we dis-

2GitHub:

https://github.com/artificial-scientist-lab/

PyTheus

covered was a new quantum multiphoton interfer-

ence eﬀect that can simulate probabilistic multi-

pair sources just with photon pairs. We describe

this new physics concept and its application in a

parallel paper [39].

The article is structured in the following way:

In section 2, we introduce the graph-based rep-

resentation of quantum optics, which lies at the

heart of PyTheus. In section 3, we introduce the

idea of the computational PyTheus framework,

which we then apply to the discovery of 100 new

quantum experiments in section 4. In section 5,

we explain some future(istic) ideas that might lie

ahead of us.

1.1 Related Work

The ﬁrst automated and artiﬁcial-intelligence-

driven design methods for new quantum exper-

iments were introduced in 2016 (for a more de-

tailed review on the topic see [24,40]). One

of them, Melvin , was focused on speciﬁc pho-

tonic quantum information tasks such as quan-

tum state generation and quantum transforma-

tions, using discrete learning techniques [25]. The

second one, Tachikoma, focused on the discovery

of new experimental setups for quantum metrol-

ogy tasks and used genetic algorithms for discrete

optimization [41]. Tachikoma has been expanded

to incorporate neural network surrogate models

to speed up the search process for new quantum-

enhanced measurements [42,43]. At the same

time, the ideas of Melvin have led to numerous

implementations of experiments in various labo-

ratories [26–29] and the extraction of new ideas

and concepts in quantum physics [30,31]. Auto-

mated design tools have helped to build new ways

to perform quantum information tasks such as

quantum cloning [44]. Compared to these tools,

PyTheus does not work on the discrete search

space. Discrete spaces are very challenging to

navigate as gradients cannot be used. Rather,

PyTheus uses domain knowledge in the form of

a new physics-inspired representation that is en-

tirely continuous.

These ideas have later been expanded by using

reinforcement-learning [45,46], for quantum com-

munication [47,48], recurrent neural networks

[49], and deep generative models such as varia-

tional autoencoders [50] or logical AI [51]. Com-

pared to these tools, PyTheus uses direct opti-

mization on the outputs of a physical simulator

Accepted in Quantum 2023-12-03, click title to verify. Published under CC-BY 4.0. 3

and does not rely on learned simulators or strate-

gies. That makes it signiﬁcantly faster.

Various quantum physics groups and compa-

nies have also developed simulators and optimiz-

ers since. A main focus there is on the design

of experimental settings for photonic quantum

computing [52–55]. One remarkable simulator is

Strawberryﬁelds [53], which is focused on design

and optimization tasks for continuous-variable

photonic quantum computing and quantum ma-

chine learning tasks. Recent updates include

auto-diﬀerentiation which signiﬁcantly speeds up

the rate of optimization. A related software pack-

age is [56], which allows for fast computation of

tasks related to Gaussian boson sampling. Com-

pared to these tools, the focus of PyTheus is

diﬀerent. PyTheus is built for discrete-variable

quantum optics, and not targeted to photonic

quantum computing (or boson sampling tasks).

Furthermore, one main motivation is the inter-

pretability of the discovered results, which is

achieved via a topological optimization on the

graph-based representation.

An alternative methodology that focuses not

only on the design question but also on un-

derstanding the underlying physical concepts is

Theseus [32]. There, the algorithm employs a

graph-based representation to describe photonic

experiments, and the ﬁnal results are topologi-

cally simpliﬁed graphs that can be interpreted

and conceptualized in a much more straightfor-

ward way than representations that work directly

on the optimization of optical elements. Com-

pared to Theseus,PyTheus expands this idea

and applies it to many new situations, inac-

cessible before, such as the design of quantum

measurement and communication setups via the

Choi–Jamiołkowski isomorphism.

Related work has shown how quantum com-

puters could overcome the enormous computa-

tional of designing quantum optical hardware

[57]. PyTheus relies, for now, on classical com-

puters, but recent hardware advances might en-

able the execution of tools like PyTheus on

quantum hardware [58].

2 Graphs and Quantum Experiments

The connection between quantum optical exper-

iments and graph theory was discovered a few

years ago [33–35] and has been further developed

as a design algorithm Theseus for new quan-

tum experiments [32]. In the graph-experiment

representation, each colored weighted graph cor-

responds to a quantum experimental setup, and

vice versa. Each edge and each vertex of the

graph represent a correlated photon pair and a

photon path, respectively. Its complex weight de-

notes the amplitude of the photon pair, and the

edge color represents a photon’s internal mode

number for a given path, which corresponds to

the photon’s degree of freedom such as polar-

ization [59], path [10,60,12], transverse spa-

tial modes [61–64], time-bin [65] or frequency

[66,17]. This abstract graph representation al-

lows us to have the full information of quantum

optical experiments and has been used for discov-

ering quantum states and transformations [32].

At ﬁrst glance, it might seem that there is fun-

damentally no diﬀerence between the path de-

gree of freedom (which is encoded as vertices)

and the internal degrees of freedom of photons

(which are encoded as colors of edges). However,

this is only true information theoretically. Phys-

ically, the path degree of freedom is exceptional,

because it allows to add spatially separation be-

tween photons and thereby perform non-locality

experiments.

Here we signiﬁcantly extend the bridge be-

Graph theory Experiment

color weighted graph quantum experiment

vertex

path to photodetector

heralded optical path*

single photon source

incoming photon

ancillary photodetector

number resolving detector

environment interaction

edge

color internal mode number

weight ∈Camplitude**

negative amplitude

correlated photon pair

single photon path

Table 1: The correspondence between graph theory and

quantum experiments. *For some experiments one must

known the total amount of photons crossing a group of

optical paths (see section 4.4). This is clariﬁed in the

graph ﬁgures with a gray envelop. **Unless the contrary

is speciﬁed, all weights are real values.

Accepted in Quantum 2023-12-03, click title to verify. Published under CC-BY 4.0. 4

tween graphs and experiments, which allows us to

perform design and discovery tasks for quantum

state generation (for pure and mixed states and

on the photon number basis), quantum measure-

ments, quantum communication protocols, and

gates for quantum computing. In Table. 1, we

show the correspondence between graph theory

and quantum experiments. The graph represen-

tation can be directly translated into diﬀerent

experimental implementations. In the remain-

ing part of this section, we explain how these

graphs encode quantum states, and how to trans-

late them to experimental setups.

2.1 Quantum State Generation

2.1.1 Probabilistic Photon-Pair Sources

Probabilistic photon-pair sources, which are typ-

ically based on nonlinear processes such as spon-

taneous parametric down conversion (SPDC) and

four-wave mixing (FWM) [67], are one of the

most widespread resources to generate entangled

and correlated pairs of photons. A range of pho-

tonic quantum experiments using probabilistic

sources can be interpreted as a weighted colored

graph [32–35]. There, each vertex represents an

optical path to a detector and each edge refers to

the correlated photon pair produced by a prob-

abilistic photon-pair source. The edge weight is

the amplitude associated with the photons, and

the edge color describes the photon’s internal

mode number (i.e., the degree of freedom of a

photon). The connection between the graph and

the corresponding quantum state is given by the

weight function [32]

Φ(ω) = X

m!

X

e∈E(G)

ω(e)x†(e)y†(e) + h.c.



(1)

where E(G)is the set of edges of the graph. The

quantum state is obtained by applying the weight

function to the vacuum, i.e. |ψ⟩= Φ(ω)|vac⟩.

The term h.c. stands for hermitian conjugate,

which includes annihilation operators. As an ex-

ample of states using four path (i.e., a,b,c, and

d) with two-dimensional internal modes (i.e., 0

and 1) in Fig. 1, the Φ(ω)is given as

Φ(ω)≈X

N!(ω0,0

a,b a†

0b†

0+ω1,1

b,d b†

1d†

+ω0,0

c,d c†

0d†

0+ω1,1

a,c c†

1a†

1+h.c.)N,(2)

where ω= (ω0,0

a,b , ω1,1

b,d , ω0,0

c,d , ω1,1

a,c )is a list of edge

weights ωi,j

x,y ∈Cand |ωi,j

x,y|2<1, the superscript

and subscript represent the mode number and the

optical path, respectively. x†

kis the creation oper-

ator of a photon in path xwith mode k. The pair-

emission process is up to the N-order, and the

probability of occurrence for lower-order events is

higher than that of higher-order ones. In princi-

ple, the hermitian conjugate terms inﬂuence the

ﬁnal state. However, for the low pump regime,

the eﬀect of the annihilation terms is negligible

in many cases. For example this is the case when

the ﬁnal state is conditioned on having a photon

in every detector. In all of the examples that we

present here, it is safe to neglect the annihilation

operators.

An alternative method to compute these sys-

tems in a non-approximate way is to follow

the Takagi decomposition (see [68], speciﬁcally

Eq.27) , which leads to reliable solutions also in

the strong-pump regime. Naturally, this method

is more expensive to compute. In our manuscript,

we do not need to use this more involved method,

as the low-pump approximation holds with high

accuracy. Experimentally, a common way to ob-

tain a quantum state is to condition the experi-

mental results on a simultaneous detection event

in each detector, which is also called the n-fold co-

incidence detection. In our graph representation,

this only happens when a subset of the edges con-

tains each of the nvertices exactly once (see the

subset of blue edges or red edges in Fig. 1), which

is the so-called perfect matching of a graph. For

the example in Fig. 1, we neglect the empty mode

and higher-order terms N > 2to post-select the

quantum state. There are two perfect matchings

(two blue edges and two red edges) in Fig. 1,

which contribute to two quantum terms |0000⟩

and |1111⟩. The weight of a perfect matching is

the product of all its edge weights. For each term

in a quantum state, the weight is given by the sum

of all weights of the perfect matchings that con-

tribute to it. Therefore, the weights for quantum

terms |0000⟩and |1111⟩are ω0,0

a,b ω0,0

c,d and ω1,1

a,c ω1,1

b,d ,

respectively. In the end, a coherent superposition

of the two perfect matchings in the graph leads

to the post-selected quantum state, which is

|ψ⟩ ≈ ω0,0

a,b ω0,0

c,d |0000⟩+ω1,1

a,c ω1,1

b,d |1111⟩.(3)

If we set all weights the same and normalize

the state, we can then reach a four-particle

Greenberger-Horne-Zeilinger (GHZ) state.

Accepted in Quantum 2023-12-03, click title to verify. Published under CC-BY 4.0. 5

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DigitalDiscoveryof100diverseQuantumExperimentswithPyTheusCarlosRuiz-Gonzalez§1,SörenArlt§1,JanPetermann1,ShararehSayyad1,TareqJaouni2,EbrahimKarimi1,2,NoraTischler3,XuemeiGu1,andMarioKrenn11MaxPlanckInstitutefortheScienceofLight,Erlangen,Germany.2NexusforQuantumTechnologies,UniversityofOttawa,K1N6N5...

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