# Large Reconfigurable Quantum Circuits with SPAD Arrays and Multimode Fibers

Adrian Makowski  Laboratoire Kastler Brossel, École normale supérieure (ENS) –
Université Paris Sciences & Lettres (PSL), CNRS, Sorbonne Université, Collège
de France, 24 rue Lhomond, Paris 75005, France  Institute of Experimental
Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw,
Poland  Michał Dąbrowski  Laboratoire Kastler Brossel, École normale
supérieure (ENS) – Université Paris Sciences & Lettres (PSL), CNRS, Sorbonne
Université, Collège de France, 24 rue Lhomond, Paris 75005, France
International Centre for Translational Eye Research, Skierniewicka 10A, 01-230
Warsaw, Poland  Department of Physical Chemistry of Biological Systems,
Institute of Physical Chemistry, Polish Academy of Sciences, M. Kasprzaka
44/52, 01-224 Warsaw, Poland  Ivan Michel Antolovic  Advanced Quantum
Architecture Laboratory (AQUA), School of Engineering, École polytechnique
fédérale de Lausanne (EPFL), Rue de la Maladière, Neuchâtel CH-2002,
Switzerland  Pi Imaging Technology SA, 1015 Lausanne, Switzerland  Claudio
Bruschini  Advanced Quantum Architecture Laboratory (AQUA), School of
Engineering, École polytechnique fédérale de Lausanne (EPFL), Rue de la
Maladière, Neuchâtel CH-2002, Switzerland  Hugo Defienne  Sorbonne Université,
CNRS, Institut des NanoSciences de Paris (INSP), Paris F-75005, France
Edoardo Charbon  Advanced Quantum Architecture Laboratory (AQUA), School of
Engineering, École polytechnique fédérale de Lausanne (EPFL), Rue de la
Maladière, Neuchâtel CH-2002, Switzerland  Radek Lapkiewicz  Institute of
Experimental Physics, Faculty of Physics, University of Warsaw, Pasteura 5,
02-093 Warsaw, Poland  Sylvain Gigan<EMAIL_ADDRESS>Laboratoire
Kastler Brossel, École normale supérieure (ENS) – Université Paris Sciences &
Lettres (PSL), CNRS, Sorbonne Université, Collège de France, 24 rue Lhomond,
Paris 75005, France

###### Abstract

Reprogrammable linear optical circuits are essential elements of photonic
quantum technology implementations. Integrated optics provides a natural
platform for tunable photonic circuits, but faces challenges when high
dimensions and high connectivity are involved. Here, we implement high-
dimensional linear transformations on spatial modes of photons using wavefront
shaping together with mode mixing in a multimode fiber, and measure photon
correlations using a time-tagging single-photon avalanche diode (SPAD) array.
In order to prove the suitability of our approach for quantum technologies we
demonstrate two-photon interferences in a tunable complex linear network –- a
generalization of a Hong-Ou-Mandel interference to 22 output ports. We study
the scalability of our approach by quantifying the similarity between the
ideal photon correlations and the correlations obtained experimentally for
various linear transformations. Our results demonstrate the potential of
wavefront shaping in complex media in conjunction with SPAD arrays for
implementing high-dimensional reconfigurable quantum circuits. Specifically,
we achieved $(80.5\pm 6.8)\%$ similarity for indistinguishable photon pairs
and $(84.9\pm 7.0)\%$ similarity for distinguishable photon pairs using 22
detectors and random circuits. These results emphasize the scalability and
reprogrammable nature of our approach.

Quantum optics with indistinguishable photons have emerged as a key resource
in advancing scientific research, particularly in the fields of quantum
information processing[1], communication[2, 3], and metrology[4, 5, 6], owing
to their unique properties such as entanglement[7, 8, 9, 10],
superposition[11, 12, 13], and non-locality[14, 15]. One area of interest is
the study of photonic quantum walks, which explores the behavior of quantum
particles in complex environments[16, 17, 18]. This phenomenon has potential
applications in fields such as quantum algorithms[19], simulations[20], and
metrology[21, 22], as well as quantum computing, communication, and
sensing[23]. Several research groups have made remarkable strides in the
development of quantum walk, including the first experimental realization of
two-dimensional quantum walks on a lattice using single photons[24], and a
quantum walk in a 21-waveguide array[25] or an application of a quantum walk
to the studies of bound states between systems with different topological
properties[26]. However, these experimental setups have stringent limitations
regarding reprogrammability and scalability, which are crucial for scaling the
system to a higher number of modes for practical implementation on near-term
quantum devices[27].

In this letter, we present a reprogrammable and scalable platform for
implementing the quantum walk of a two-photon state using a multimode fiber
(MMF) as a quantum state mixer[28, 29, 30]. Our platform can generate an
arbitrary N-output x 2-input quantum state operations that can be reprogrammed
on demand at a 10 Hz frequency rate (see Fig. 1). This provides significant
advantages over existing experimental setups[31, 32] and makes it a promising
candidate for the future realizations of highly-multimode quantum walk
experiments[33, 34, 35, 36]. The wavefronts of the photons are shaped using a
spatial light modulator (SLM), and then coupled into an MMF. The MMF is a
complex medium that supports around 400 modes at wavelength $\lambda=800$ nm
and has low losses[29] which are the essential features for performing
multidimensional unitary operations on the single photons efficiently [37, 28,
38, 39].

Previous implementations faced a significant limitation in achieving
scalability due to the integration of detection technology[29].This required a
large number of separate avalanche photodiodes, rendering the solution of
problems like boson sampling [40], very impractical. Several experiments have
employed single-outcome projective measurements for sequential analysis of the
output state [41, 18]. However, these approaches suffer from inherent
limitations, including substantial losses (as only two outputs can be detected
simultaneously out of all possibilities) and time-intensive procedures (since
detecting $i$ output modes demands $i^{K}$ measurements, with $K$ representing
the number of photons involved). Consequently, these methods become
impractical for large-scale systems.

To overcome this issue, here we use a 23-single photon avalanche detector
(SPAD23)[42] array to measure the number of counts and coincidences between
photons at the linear network output. The ability to modify the phase pattern
on the SLM allows the MMF to performs a specific linear operation on a two-
photon state. This operation can be easily and reliably adjusted on demand.

Degenerate photon pairs at 810nm are produced by type-II spontaneous
parametric down conversion (SPDC) process in a ppKTP crystal pumped by a 405
nm continuous wave laser. To split the photons with orthogonal polarizations,
we use a polarisation beam splitter (PBS). Our SPDC setup allows us to adjust
the time delay between the photons, and then observe and control Hong-Ou-
Mandel (HOM) interference[43] by changing the photons’ distinguishability. The
measured HOM visibility of photons from our SPDC source is approx. 95%. See
Supplemental Material for more details on the source.

In our experiment, we utilized a detector array consisting of Single-Photon
Avalanche Diodes (SPADs) using CMOS technology[44, 45], specifically the
SPAD23 model from Pi Imaging Technology[42]. This detector offers a sub-ns
temporal resolution (120 ps jitter FWHM and 20 ps for least-significant bit
when using time-to-digital converters as time-taggers), exhibits low dark
noise (less than 100 counts per second at 20oC), has a high pixel fill factor
(80%), and a "dead time" of approximately 50 ns[46]. However, like all SPAD
arrays SPAD23 is prone to cross-talk, which occurs when a photon detected by
one of the array’s detectors is simultaneously counted by a neighboring
detector[47, 44]. While the probability of cross-talk is low (approximately
0.1%), it affects the number of coincidences measured in our experiment but
not the number of single photon counts[44]. To account for cross-talk, we
employed a calibration procedure, which is detailed in the Supplemental
Material.

Figure 1: Reprogrammable and scalable platform for the implementation of
quantum operations on a 2-photon state where wavefronts of two photons,
generated using spontaneous parametric down-conversion (SPDC), are shaped
using two separate parts of the spatial light modulator (SLM) before being
coupled into the multimode fiber (MMF) used as a quantum states mixer[28, 29,
30]. Usage of a single-photon avalanche 23-detector array (SPAD23)[42] enables
for subset $\mathcal{L}^{23}_{2}$ arbitrary 23-output x 2-input quantum state
operations of the general $\mathcal{U}^{23}_{23}$ unitary transformation
performed by MMF.

Figure 2 depicts the experimental setup used in our study. Two separate parts
of the spatial light modulator (SLM), labeled as H and V for orthogonal light
polarizations, were illuminated with two photons created in our SPDC platform.
The SLM shaped the wavefronts of the photons, which were then focused on the
MMF of 50$\mu m$ core. The MMF with SLM induces a specific quantum operation
on a 2-photon state, and modifying the phase pattern on the SLM allows for
easy adjustment of this operation. The resulting speckle image of the light
emerging from the MMF was either imaged on the SPAD23 or the CCD camera after
passing through a polarizer to choose just one particular polarization (for
which the TM was measured). To calibrate the relative position of the MMF and
SPAD23, we used a CCD camera, as depicted in Fig. 2(b). It shows the speckle
image captured by the CCD camera, with the positions of the SPAD23 detectors
marked with red circles. The CCD camera was used just for the calibration
purposes (see Supplemental Material for details). The magnification was
choosen to map one speckle mode of the MMF into one single SPAD detector. The
photon arrival time was measured using SPAD23 detector and then used to
calculate the number of counts $n_{i}$ for each detector $i$ and coincidences
$C_{ij}=\langle n_{i}n_{j}\rangle_{t}$ for each pair $(i,j)$.

Figure 2: (a) Schematic of the experimental setup. A photon pair of two
orthogonal polarizations (H and V) is passed through single-mode fibers. The
photons are collimated, and their wavelengths are shaped by the spatial light
modulator (SLM). The shaped photons are then coupled into a multimode fiber
(MMF). The MMF output is imaged on the SPAD23 or CCD by changing the flip-
mirror position to measure the number of counts and coincidences. By knowing
the MMF’s transmission matrix, we can select a phase pattern on the SLM to
perform an arbitrary operation. (b) Speckle image of the light coming out of
the fiber imaged on the camera, and the localization of the SPAD23 detectors.
Figure 3: (a) Hong-Ou-Mandel (HOM) interference scan for a 4x4 operation,
displaying the number of coincidences as a function of the relative delay
between the two photons. (b)-(e) Example results of a 10x10 operation on the
2-photon state. (b) and (c) show the experimental and theoretical coincidence
counts for indistinguishable photon pairs, respectively. (d) and (e) show the
corresponding experimental and theoretical coincidence counts for
distinguishable photon pairs, acquired over a 100-second measurement period.
The SPAD23 detectors used in the experiment are marked in yellow in Fig. 4.

Our experimental platform establishes a connection between input modes
displayed on the SLM and corresponding output modes measured using 23
detectors described by unitary operator $\mathcal{U}_{23}^{23}$, using the
well-established technique of a transmission matrix (TM) measurement of the
optical system[48]. The measured TM is stable for a few days in normal
laboratory conditions. We measure the TM in a Fourier mode basis by displaying
phase ramps on the SLM with a varying inclination and orientation[49]. This
allows us to scan the different spatial positions at the entrance of the MMF
and as a result to address particular output modes of the MMF after TM
calibration. If the addressed mode is not an eigenvector of the TM, the light
becomes scrambled as it propagates through the MMF. The electric field
amplitude at SPAD23 is linearly dependent on the electric field at the SLM and
can be represented as:

$E^{(k)}_{out}=T^{(1)}_{k}E^{(k)}_{in}\text{, for }k=H,V,$ (1)

where $E^{(k)}_{out}$ is the electric field at SPAD23, $T^{(1)}_{k}$ is a one-
photon transmission matrix for SLM part $k$, and $E^{(k)}_{in}$ is the
electric field on the SLM part $k$, corresponding to the SLM part shaping the
polarisation $k=H,V$. During the TM measurement, the SPAD acquires the number
of photon counts per 10 ns time-window for each SLM mode, and in order to
obtain the electric field on SPAD23 we perform phase-stepping
interferometry[48]. We perform this operation separately for both light
polarization (SLM parts) and then calculate the transmission matrix for the
two-photon state[28]:

$T^{(1)}_{H},T^{(1)}_{V}\rightarrow T^{(2)}.$ (2)

With this information, we can readily calculate the SLM pattern that gives us
the required quantum operation on the two-photon state
$\mathcal{L}\in\mathbb{M}_{2\mathsf{x}N}$, where $N$ is the number of
detectors. The computation of SLM pattern for a given $\mathcal{L}$ takes only
a few seconds. The electric field on SLM can be calculated using the complex
conjugate of the transmission matrix for the two-photon state:

$[E^{(H)}_{in},E^{(V)}_{in}]=T^{\dagger(2)}\mathcal{L}.$ (3)

Knowing the TM of the MMF, one can modify the phase pattern at the speed of
10Hz on the SLM and obtain different N-output x 2-input quantum linear network
operations $\mathcal{L}_{2}^{N}$.

As an example of an all-to-all operator, we emulate a 10x10 Sylvester
operation[50] on the two-photon state generated by the experimental platform
described above. We measured the TM of the MMF in our setup to calculate the
appropriate SLM pattern for performing the 10-dimensional Sylvester operation:

$\mathcal{L}_{S}=\left\\{\begin{array}[]{rcl}1&\mbox{for}&k=V\\\
(-1)^{i}&\mbox{for}&k=H,\end{array}\right.$ (4)

where $i$ denotes the detector index. We measured the number of coincidence
counts for distinguishable and indistinguishable photon pairs.

Figure 4: (a) Sylvester transformation for different numbers of detectors. The
first column exhibits the SPAD23 detector array used in the experiment, with
the detectors used for the measurement with a given number of detectors marked
in yellow. The second column shows the experimental coincidence counts for
various numbers of detectors and indistinguishable photon pairs. The third
column shows the theoretical coincidence counts for the corresponding number
of detectors. (b) A random operation similarity trend for indistinguishable
and distinguishable photon pairs.

The results of this experiment are presented in Fig. 3. Figure 3(a) shows a
HOM interference scan for a 4-dimensional Sylvester operator, which
corresponds to the number of coincidences as a function of the relative delay
between the two photons. The HOM dip in the scan indicates the presence of
interference between the two photons, which is essential for quantum
operations with indistinguishable particles[43]. The HOM visibility $V$ for
different coincidence distributions $C_{i,j}$, ranging from $V=0.74$ to
$V=0.92$ deviates from ideality (the 95% indistinguishably of the source)
mainly because of cross-talk between different SPAD23 detectors (see
Supplemental Material for details) as well as photons spectral dispersion when
propagating through the MMF and non-perfect fidelity of linear network
operator.

Figures 3(b) and 3(d) show the experimental coincidence counts for
indistinguishable and distinguishable photon pairs, respectively over 10
output modes. These counts were acquired for 100 sec by measuring the number
of coincidences between the SPAD23 detectors when the photons were either
indistinguishable or distinguishable. The theoretical coincidence counts for
indistinguishable and distinguishable photon pairs[51, 52] are presented in
Fig. 3(c) and 3(e), respectively.

Comparing the experimental results presented in Figs. 3(b) and 3(d) with the
theoretical predictions in Figs. 3(c) and 3(e), we can see that the presented
results agree well with the theoretical predictions[51, 52]. The experimental
results also demonstrate that the experimental platform can successfully
generate two-photon states for quantum operations, as well as measure and
characterize their properties through coincidence counting.

We finally conducted an investigation into the scalability and reprogrammable
nature of our experimental platform by performing random matrix quantum
operations with varying numbers of detectors. Figure 4(a) presents coincidence
distributions, with the first column showcasing the theoretical coincidence
counts for indistinguishable photon pairs utilizing the Sylvester
operation[50] across a range of detectors. Meanwhile, the second column
demonstrates the corresponding experimental coincidence counts, which we
determined by measuring the number of coincidences between the SPAD detectors.
Because of a high number of dark counts in one of the detectors, we excluded
it from the experiments.

Figure 4(b) shows the difference between the similarity trend for
distinguishable and indistinguishable photon pairs
$\langle\mathcal{S}_{ET}\rangle_{\mathcal{L}_{R}}$ when performing 100 random
$\mathcal{L}_{R}$ operations (random complex numbers from a uniform
distribution) for different number of SPAD detectors. The similarity
$\mathcal{S}_{ET}$ of two coincidence distributions $C^{(E)}_{i,j}$ and
$C^{(T)}_{i,j}$ (corresponding to particular $\mathcal{L}_{R}$ operator),
representing a generalized fidelity for 2-fold coincidences[25], is defined
as:

$\mathcal{S}_{ET}=\frac{\left(\sum_{i,j}\sqrt{C^{(E)}_{i,j}C^{(T)}_{i,j}}\right)^{2}}{\sum_{i,j}C^{(E)}_{i,j}\sum_{i,j}C^{(T)}_{i,j}}.$
(5)

In other words, similarity quantifies the extent to which the experimental
results align with theoretical predictions, with higher values indicating
stronger agreement. We see that the similarity decreases as we increase the
number of detectors from 4 to 22, from $98.3\pm 1.23\%$ ($95.3\pm 5.5$%) to
$84.9\pm 7.0\%$ ($80.5\pm 6.8\%$) for distinguishable (resp.
indistinguishable) photons. The similarity is higher for distinguishable pairs
because of more stringent conditions for 2-photon interference
(distinguishable photons are not affected by phase errors in quantum
interference). For the same reason, the similarity for indistinguishable pairs
of photons is reduced due to the limited accuracy of photon wavefront-shaping
via SLM. Also, for a larger number of detectors, the similarity decreases
because the coincidence distribution becomes more noisy as can be seen in Fig.
4.

To conclude, we present an approach to implementing high-dimensional
reconfigurable quantum circuits using wavefront shaping and mode mixing in the
MMF with a SPAD camera as a detector. It offers advantages in terms of
scalability and flexibility compared to other approaches[32, 53, 54]. We
demonstrate the feasibility of the presented approach by implementing a
complex linear network for two-photon interference. We measure the two-photon
correlations between all the output pairs using a time-tagging SPAD array[47,
55], thus advancing towards scalable detection schemes beyond previously
proposed solutions[28, 29]. The similarity between the ideal photon
correlations and the correlations has been obtained experimentally for up to
22 output modes for various randomly chosen linear transformations for both
indistinguishable and distinguishable photon pairs. The current limitation of
the setup is the number of available photons preventing us from studying high
dimensional linear networks applied to multi-photon states $\mathcal{L}$[56,
57], as well as detector cross-talk[44] that affect coincidences between
detectors located close to each other thus reducing the measured similarity of
the states.

Future work in this area could focus on further optimizing the wavefront
shaping and mode mixing techniques – along with sources delivering more than
two photons enabling the achieving of even higher dimensional
transformations[56]. Additionally, the use of more advanced detectors, either
superconducting nanowire single-photon detectors with near-unity quantum
efficiency[58] or SPAD cameras with more pixels[59, 47] could further improve
the measurement capabilities. The scalability and programmable nature of the
presented approach make it promising for applications in quantum information
processing, such as quantum communication[2, 3] and quantum computing[33, 34,
35, 36], especially in the perspective of using more detectors to test
different boson sampling protocols which can overcome the capabilities of
existing classical information processing schemes[27, 56, 60, 61].

Acknowledgments We would like to thank Saroch Leedumrongwatthanakun for
fruitful discussions and initial guidance with the setup. S.G. acknowledges
funding from the European Research Council ERC Consolidator Grant
(SMARTIES-724473). H.D. acknowledges funding from the European Research
Council ERC Starting Grant (SQIMIC-101039375). A.M acknowledge Scholarship of
French Government - Ph.D. Cotutelle/Codirection and "IV.4.1. A Complex
Programme of Support For UW PhD Students". A.M. and R.L acknowledge the
support by the Foundation for Polish Science under the FIRST TEAM project
‘Spatiotemporal photon correlation measurements for quantum metrology and
super-resolution microscopy’ co-financed by the European Union under the
European Regional Development Fund (POIR.04.04.00-00-3004/17-00). The
International Centre for Translational Eye Research (MAB/2019/12) project is
carried out within the International Research Agendas Program of the
Foundation for Polish Science, co-financed by the European Union under the
European Regional Development Fund.

## References

  * Nielsen and Chuang [2012] M. A. Nielsen and I. L. Chuang,  _Quantum Computation and Quantum Information_ , 10th ed. (Cambridge University Press, Cambridge, 2012).
  * Mattle  _et al._ [1996] K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, Physical Review Letters 76, 4656 (1996).
  * Wehner  _et al._ [2018] S. Wehner, D. Elkouss, and R. Hanson, Science 362 (2018).
  * Giovannetti  _et al._ [2011] V. Giovannetti, S. Lloyd, and L. Maccone, Nature Photonics 5, 222 (2011).
  * Hiekkamäki  _et al._ [2021] M. Hiekkamäki, F. Bouchard, and R. Fickler, Physical Review Letters 127, 263601 (2021).
  * Tenne  _et al._ [2019] R. Tenne, U. Rossman, B. Rephael, Y. Israel, A. Krupinski-Ptaszek, R. Lapkiewicz, Y. Silberberg, and D. Oron, Nature Photonics 13, 116 (2019).
  * Bouwmeester  _et al._ [1997] D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, Nature 390, 575 (1997).
  * Wang  _et al._ [2018a] J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, Science 360, 285 (2018a).
  * Kwiat  _et al._ [1995] P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, Physical Review Letters 75, 4337 (1995).
  * Fickler  _et al._ [2012] R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, Science 338, 640 (2012).
  * Arndt and Hornberger [2014] M. Arndt and K. Hornberger, Nature Physics 10, 271 (2014).
  * Assemat  _et al._ [2019] F. Assemat, D. Grosso, A. Signoles, A. Facon, I. Dotsenko, S. Haroche, J. M. Raimond, M. Brune, and S. Gleyzes, Physical Review Letters 123, 143605 (2019).
  * Hiekkamäki and Fickler [2021] M. Hiekkamäki and R. Fickler, Physical Review Letters 126, 123601 (2021).
  * Aspect  _et al._ [1981] A. Aspect, P. Grangier, and G. Roger, Physical Review Letters 47, 460 (1981).
  * Weihs  _et al._ [1998] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, Physical Review Letters 81, 5039 (1998).
  * Broome  _et al._ [2013] M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, Science 339, 794 (2013).
  * Spring  _et al._ [2013] J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, Science 339, 798 (2013).
  * Valencia  _et al._ [2020] N. H. Valencia, S. Goel, W. McCutcheon, H. Defienne, and M. Malik, Nature Physics 16, 1112 (2020).
  * Childs [2009] A. M. Childs, Physical Review Letters 102, 180501 (2009).
  * Bao  _et al._ [2023] J. Bao, Z. Fu, T. Pramanik, J. Mao, Y. Chi, Y. Cao, C. Zhai, Y. Mao, T. Dai, X. Chen, X. Jia, L. Zhao, Y. Zheng, B. Tang, Z. Li, J. Luo, W. Wang, Y. Yang, Y. Peng, D. Liu, D. Dai, Q. He, A. L. Muthali, L. K. Oxenløwe, C. Vigliar, S. Paesani, H. Hou, R. Santagati, J. W. Silverstone, A. Laing, M. G. Thompson, J. L. O’Brien, Y. Ding, Q. Gong, and J. Wang, Nature Photonics (2023).
  * Brandt  _et al._ [2020] F. Brandt, M. Hiekkamäki, F. Bouchard, M. Huber, and R. Fickler, Optica 7, 98 (2020).
  * Barbieri [2022] M. Barbieri, PRX Quantum 3, 010202 (2022).
  * Aslam  _et al._ [2023] N. Aslam, H. Zhou, E. K. Urbach, M. J. Turner, R. L. Walsworth, M. D. Lukin, and H. Park, Nature Reviews Physics 5, 157 (2023).
  * Schreiber  _et al._ [2012a] A. Schreiber, A. Gábris, P. P. Rohde, K. Laiho, M. Štefaňák, V. Potoček, C. Hamilton, I. Jex, and C. Silberhorn, Science 336, 55 (2012a).
  * Peruzzo  _et al._ [2010] A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, Science 329, 1500 (2010).
  * Kitagawa  _et al._ [2012] T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, Nature Communications 3, 882 (2012).
  * Madsen  _et al._ [2022] L. S. Madsen, F. Laudenbach, M. F. Askarani, F. Rortais, T. Vincent, J. F. F. Bulmer, F. M. Miatto, L. Neuhaus, L. G. Helt, M. J. Collins, A. E. Lita, T. Gerrits, S. W. Nam, V. D. Vaidya, M. Menotti, I. Dhand, Z. Vernon, N. Quesada, and J. Lavoie, Nature 606, 75 (2022).
  * Defienne  _et al._ [2016] H. Defienne, M. Barbieri, I. A. Walmsley, B. J. Smith, and S. Gigan, Science Advances 2 (2016).
  * Leedumrongwatthanakun  _et al._ [2020] S. Leedumrongwatthanakun, L. Innocenti, H. Defienne, T. Juffmann, A. Ferraro, M. Paternostro, and S. Gigan, Nature Photonics 14, 139 (2020).
  * Cavaillès  _et al._ [2022] A. Cavaillès, P. Boucher, L. Daudet, I. Carron, S. Gigan, and K. Müller, Optics Express 30, 30058 (2022).
  * Titchener  _et al._ [2016] J. G. Titchener, A. S. Solntsev, and A. A. Sukhorukov, Optics Letters 41, 4079 (2016).
  * Huisman  _et al._ [2015] S. R. Huisman, T. J. Huisman, T. A. W. Wolterink, A. P. Mosk, and P. W. H. Pinkse, Optics Express 23, 3102 (2015).
  * Schreiber  _et al._ [2012b] A. Schreiber, A. Gábris, P. P. Rohde, K. Laiho, M. Štefaňák, V. Potoček, C. Hamilton, I. Jex, and C. Silberhorn, Science 336, 55 (2012b).
  * Crespi  _et al._ [2013] A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, Nature Photonics 7, 322 (2013).
  * Carolan  _et al._ [2015] J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, Science 349, 711 (2015).
  * Preiss  _et al._ [2015] P. M. Preiss, R. Ma, M. E. Tai, A. Lukin, M. Rispoli, P. Zupancic, Y. Lahini, R. Islam, and M. Greiner, Science 347, 1229 (2015).
  * Li  _et al._ [2021] S. Li, C. Saunders, D. J. Lum, J. Murray-Bruce, V. K. Goyal, T. Čižmár, and D. B. Phillips, Light: Science & Applications 10, 88 (2021).
  * Defienne and Faccio [2020] H. Defienne and D. Faccio, Physical Review A 101, 063830 (2020).
  * Morales-Delgado  _et al._ [2015] E. E. Morales-Delgado, D. Psaltis, and C. Moser, Optics Express 23, 32158 (2015).
  * Brod  _et al._ [2019] D. J. Brod, E. F. Galvão, A. Crespi, R. Osellame, N. Spagnolo, and F. Sciarrino, Advanced Photonics 1, 034001 (2019).
  * Goel  _et al._ [2022] S. Goel, S. Leedumrongwatthanakun, N. H. Valencia, W. McCutcheon, C. Conti, P. W. H. Pinkse, and M. Malik, “Inverse-design of high-dimensional quantum optical circuits in a complex medium,” (2022), arXiv:2204.00578 [quant-ph] .
  * [42] “ _SPAD 23 manual_ ,” https://piimaging.com/datasheet/SPAD23system.pdf, pi Imaging.
  * Hong  _et al._ [1987] C. K. Hong, Z. Y. Ou, and L. Mandel, Physical Review Letters 59, 2044 (1987).
  * Lubin  _et al._ [2019] G. Lubin, R. Tenne, I. M. Antolovic, E. Charbon, C. Bruschini, and D. Oron, Opt. Express 27, 32863 (2019).
  * Ghezzi  _et al._ [2021] A. Ghezzi, A. Farina, A. Bassi, G. Valentini, I. Labanca, G. Acconcia, I. Rech, and C. D’Andrea, Optics Letters 46, 1353 (2021).
  * Antolovic  _et al._ [2018] I. M. Antolovic, C. Bruschini, and E. Charbon, Opt. Express 26, 22234 (2018).
  * Bruschini  _et al._ [2019] C. Bruschini, H. Homulle, I. M. Antolovic, S. Burri, and E. Charbon, Light: Science & Applications 8, 87 (2019).
  * Popoff  _et al._ [2010] S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Physical Review Letters 104, 100601 (2010).
  * Loterie  _et al._ [2015] D. Loterie, S. Farahi, I. Papadopoulos, A. Goy, D. Psaltis, and C. Moser, Opt. Express 23, 23845 (2015).
  * Viggianiello  _et al._ [2018] N. Viggianiello, F. Flamini, L. Innocenti, D. Cozzolino, M. Bentivegna, N. Spagnolo, A. Crespi, D. J. Brod, E. F. Galvão, R. Osellame, and F. Sciarrino, New Journal of Physics 20, 033017 (2018).
  * Vlasov [2017] A. Y. Vlasov, Laser Physics Letters 14, 103001 (2017).
  * Scheel [2004] S. Scheel, “Permanents in linear optical networks,” (2004), arXiv:quant-ph/0406127 [quant-ph] .
  * Wolterink  _et al._ [2016] T. A. W. Wolterink, R. Uppu, G. Ctistis, W. L. Vos, K.-J. Boller, and P. W. H. Pinkse, Physical Review A 93, 053817 (2016).
  * Jang  _et al._ [2017] M. Jang, C. Yang, and I. M. Vellekoop, Physical Review Letters 118, 093902 (2017).
  * Slenders  _et al._ [2021] E. Slenders, M. Castello, M. Buttafava, F. Villa, A. Tosi, L. Lanzanò, S. V. Koho, and G. Vicidomini, Light: Science & Applications 10, 31 (2021).
  * Wang  _et al._ [2019] H. Wang, J. Qin, X. Ding, M.-C. Chen, S. Chen, X. You, Y.-M. He, X. Jiang, L. You, Z. Wang, C. Schneider, J. J. Renema, S. Höfling, C.-Y. Lu, and J.-W. Pan, Physical Review Letters 123, 250503 (2019).
  * Wang  _et al._ [2018b] J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, Science 360, 285 (2018b).
  * Reddy  _et al._ [2020] D. V. Reddy, R. R. Nerem, S. W. Nam, R. P. Mirin, and V. B. Verma, Optica 7, 1649 (2020).
  * Zhang  _et al._ [2018] C. Zhang, S. Lindner, I. M. Antolovic, M. Wolf, and E. Charbon, Sensors 18 (2018), 10.3390/s18114016.
  * Lund  _et al._ [2017] A. P. Lund, M. J. Bremner, and T. C. Ralph, npj Quantum Information 3, 1 (2017).
  * Harrow and Montanaro [2017] A. W. Harrow and A. Montanaro, Nature 549, 203 (2017).