Digital Discovery of a Scientic Concept at the Core of Experimental Quantum Optics S oren Arlt1Carlos Ruiz-Gonzalez1and Mario Krenn1y 1Max Planck Institute for the Science of Light Erlangen Germany

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Digital Discovery of a Scientiﬁc Concept at the Core of Experimental Quantum Optics

S¨oren Arlt,1, ∗Carlos Ruiz-Gonzalez,1and Mario Krenn1, †

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

(Dated: October 19, 2022)

Entanglement is a crucial resource for quantum technologies ranging from quantum communica-

tion to quantum-enhanced measurements and computation. Finding experimental setups for these

tasks is a conceptual challenge for human scientists due to the counterintuitive behavior of multipar-

ticle interference and the enormously large combinatorial search space. Recently, new possibilities

have been opened by artiﬁcial discovery where artiﬁcial intelligence proposes experimental setups

for the creation and manipulation of high-dimensional multi-particle entanglement. While digitally

discovered experiments go beyond what has been conceived by human experts, a crucial goal is to

understand the underlying concepts which enable these new useful experimental blueprints. Here,

we present Halo (Hyperedge Assembly by Linear Optics), a new form of multiphoton quantum

interference with surprising properties. Halos were used by our digital discovery framework to

solve previously open questions. We – the human part of this collaboration – were then able to

conceptualize the idea behind the computer discovery and describe them in terms of eﬀective proba-

bilistic multi-photon emitters. We then demonstrate its usefulness as a core of new experiments for

highly entangled states, communication in quantum networks, and photonic quantum gates. Our

manuscript has two conclusions. First, we introduce and explain the physics of a new practically

useful multi-photon interference phenomenon that can readily be realized in advanced setups such

as integrated photonic circuits. Second, our manuscript demonstrates how artiﬁcial intelligence can

act as a source of inspiration for the scientiﬁc discoveries of new actionable concepts in physics.

A central element of physics (and science in general)

is the formulation of concepts. A range of related ob-

jects, processes or properties are uniﬁed under one name

(such as atom,scattering,charge), simplifying our de-

scription of the world. Using artiﬁcial intelligence (AI)

to inspire and ultimately perform conceptualization could

accelerate scientiﬁc progress greatly. AI has been applied

in physics to rediscover hidden symmetries [1], orbital

mechanics [2] [3], models for gravitational-wave popula-

tions [4], conserved quantities [5] and phase diagrams [6].

These works show the great potential that lies in ap-

plying machine learning to the discovery of underlying

concepts in physics. However, their results focus mostly

on rediscovery and it is not per se clear whether and how

to extend these results to the discovery of new concepts

and ideas.

Here we go beyond rediscovery, and use AI to discover

a hidden concept that lies at the heart of quantum optics

– a ﬁeld in which automated discovery and design have

been strongly employed recently [7–14] (see a review in

[15]). We obtain these results using pytheus– a highly

eﬃcient discovery framework that provides interpretable

results, which is presented in a parallel article [16] and is

available as open-source software 1. The concept, which

we call Halo (Hyperedge Assembly by Linear Optics) is

a multiphoton interference eﬀect with a surprising prop-

erty: It acts as a probabilistic source of multi-photon

pairs, while being constructed from only pair sources un-

der common experimental post-selection conditions. As

∗soeren.arlt@mpl.mpg.de

†mario.krenn@mpl.mpg.de

1https://github.com/artificial-scientist-lab/Theseus

Figure 1. Structure of this article. The design principle Halo

is formulated after analysis of the results found by pytheus

algorithm. The concept is applied in three diﬀerent areas of

quantum optics, where solutions discovered by the computer

are extended by hand.

we show, this resource can now be applied to construct

by hand experimental setups for creating new forms of

multi-particle entangled states, entanglement swapping,

and quantum gates, which were previously beyond our in-

tuition. Furthermore, we see that a rudimentary form of

Halos can be found in several of the pioneering quan-

tum information experiments with photons, which can

now be understood as special cases of the much more

general design principle Halo.

At a broader view, we demonstrate how computer al-

gorithms can act as a source of inspiration to increase

our scientiﬁc understanding [17]. We illustrate the con-

nections between the parts of this article in Fig. 1.

Digital Discovery – The digital discovery framework

pytheus relies on a representation of quantum experi-

ments using graphs [18]. This representation was ﬁrst

arXiv:2210.09981v1 [quant-ph] 18 Oct 2022

developed for entanglement by path identity [19,20]

but also extends to bulk optics or integrated photonics

[12,21]. Integrated photonic chips have made impressive

technological progress recently and oﬀer great potential

for the experimental realization of setups discovered by

pytheus [22–27]. The search for an experiment creating

a target state is formulated as an optimization that max-

imizes the ﬁdelity of a graph. When a solution is found,

the corresponding graph can directly be trannslated to an

experimetnal setup consisting of standard optical compo-

nents. A detailed explanation of digital discovery using

pytheus is provided in the accompanying paper [16].

Here we focus on the scientiﬁc consequences of the new

quantum optics concept Halo, the underlying physics,

its connection to modern quantum optics experiments,

and how it can be productively used.

A prominent class of examples for genuine multi-particle

entanglement are the GHZ states [28] [29].

|GHZid

n=1

√d

i=0 |ii⊗n(1)

where nis the number of particles and dis the dimen-

sion of the GHZ state. |GHZid

nis a generalization of

the original |GHZi2

3state [30]. Fig. 2shows a correla-

tion network for the state |GHZi3

4. An edge of the graph

corresponds to a two-particle correlation as it is intro-

duced by probabilistic photon pair sources, for example

by spontaneous parametric down-conversion (SPDC). A

vertex corresponds to a path to a detector at the end of

the setup. We condition the ﬁnal quantum state on the

detection of one photon in each of the detectors, which

motivates the usage of perfect matchings in graphs. A

perfect matching is a set of edges by which each vertex

of a graph is covered exactly once. In an event where all

crystals corresponding to the edges in a perfect matching

ﬁre, exactly one photon will enter each detector. Fig. 2

also shows that each perfect matching of the graph can

be understood as a contribution to the created state.

Fig. 2also shows that it is not straightforward to extend

the construction to |GHZi4

4. It is impossible to create

|GHZi4

4using linear optics without the use of additional

resources in the form of ancillary photons [18,21]. This

is a physical limitation that can be explained by graph

theory as follows. There are at most three disjoint per-

fect matchings in a four-vertex graph. This makes it

impossible to create four diﬀerent GHZ terms without

introducing cross terms with a four-vertex graph.

Including additional resources allows us to go beyond

this limitation. We can use pytheus to search for a

graph corresponding to the state

|ψi=|GHZi4

4⊗ |0000i,(2)

which is the four-dimensional four-particle GHZ state in

a product state with four ancillary particles. For this tar-

get, pytheus discovers minimal solution in Fig. 3(b).

The solution shown has 12 perfect matchings. Four pairs

Figure 2. (a) A graph corresponding to the experimental

setup for the creation of the |GHZi3

4state. A perfect matching

is a set of edges by which each vertex of a graph is covered ex-

actly once. The graph has three perfect matchings (blue, red,

and green). In this case, each perfect matching corresponds

to one term in the target state. (b) shows an incorrect at-

tempt at drawing a graph corresponding to the state |GHZi4

The two additional edges create the wanted fourth term at

the cost of two unwanted cross-terms.

interfere constructively creating the four GHZ terms.

The remaining two pairs each correspond to cross-terms

but interfere destructively.

The physics of HALO – By closer inspection of the

solution for |GHZi4

4(Fig. 3(b)), we see that the graph for

|GHZi3

4(shown in Fig. 2(a)) is included as a subgraph.

The experimental setup for |GHZi3

4can be seen as a basic

setup that is extended by the components corresponding

to the remaining edges of the graph |GHZi4

4. When all

detectors click, the additional components either produce

four correlated particles or none, with all cross-terms de-

structively interfering, imitating a four-particle emitter.

The physical interpretation of the ancillary subgraph is

shown in Fig. 4. This is achieved by an interference

pattern that can be interpreted as an extension of frus-

trated multiphoton interference, an eﬀect described in

[21] and experimentally observed in [31,32]). Building

probabilistic multi-particle emitters is an active area of

research [33,34]. Halo oﬀers a way of emulating cor-

related multi-particle emitters with pair sources in post-

selected experiments, which is a complementary experi-

ment route that can employ physically well-understood

technologies.

Concept Extraction – We give the concept that ap-

pears in the solution for |GHZi4

4a name:

AHalo (Hyperedge Assembly by Linear Optics) is a

subsystem of a linear optics setup, which eﬀectively acts

as a probabilistic multi-photon source.

This deﬁnition is not used by the algorithm to produce

solutions, rather we abstract it from the solutions. In the

abstract graph representation, a multi-particle emitter

can be described by a hyperedge (shown in Fig. 3(c)).

A hyperedge is drawn as a shape enclosing the n > 2

vertices it connects. Thus, a Halo-subgraph constructed

from regular edges can also be represented by hyperedges.

Hypergraphs describing quantum experiments involving

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DigitalDiscoveryofaScienticConceptattheCoreofExperimentalQuantumOpticsSorenArlt,1,CarlosRuiz-Gonzalez,1andMarioKrenn1,y1MaxPlanckInstitutefortheScienceofLight,Erlangen,Germany(Dated:October19,2022)Entanglementisacrucialresourceforquantumtechnologiesrangingfromquantumcommunica-tiontoquantum-enhanc...

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Digital Discovery of a Scientic Concept at the Core of Experimental Quantum Optics S oren Arlt1Carlos Ruiz-Gonzalez1and Mario Krenn1y 1Max Planck Institute for the Science of Light Erlangen Germany

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