11footnotetext: To reach 99.9% visibility interference for a 1% surface code
threshold Sparrow (2017), resonances must be aligned to within $\sim 3\%$ of
the linewidth.

# On-chip quantum information processing with distinguishable photons

Patrick Yard<EMAIL_ADDRESS>Quantum Engineering and Technology
Laboratories, School of Physics and Department of Electrical and Electronic
Engineering, University of Bristol, Bristol, UK  Alex E. Jones
<EMAIL_ADDRESS>Quantum Engineering and Technology Laboratories, School
of Physics and Department of Electrical and Electronic Engineering, University
of Bristol, Bristol, UK  Stefano Paesani  Quantum Engineering and Technology
Laboratories, School of Physics and Department of Electrical and Electronic
Engineering, University of Bristol, Bristol, UK  Center for Hybrid Quantum
Networks (Hy-Q), Niels Bohr Institute, University of Copenhagen, Blegdamsvej
17, DK-2100 Copenhagen, Denmark  Alexandre Maïnos  Quantum Engineering and
Technology Laboratories, School of Physics and Department of Electrical and
Electronic Engineering, University of Bristol, Bristol, UK  Jacob F. F. Bulmer
Quantum Engineering and Technology Laboratories, School of Physics and
Department of Electrical and Electronic Engineering, University of Bristol,
Bristol, UK  Anthony Laing  Quantum Engineering and Technology Laboratories,
School of Physics and Department of Electrical and Electronic Engineering,
University of Bristol, Bristol, UK

###### Abstract

Multi-photon interference is at the heart of photonic quantum technologies.
Arrays of integrated cavities can support bright sources of single-photons
with high purity and small footprint, but the inevitable spectral
distinguishability between photons generated from non-identical cavities is an
obstacle to scaling. In principle, this problem can be alleviated by measuring
photons with high timing resolution, which erases spectral information through
the time-energy uncertainty relation. Here, we experimentally demonstrate that
detection can be implemented with a temporal resolution sufficient to
interfere photons detuned on the scales necessary for cavity-based integrated
photon sources. By increasing the effective timing resolution of the system
from 200 ps to 20 ps, we observe a $~{}20\%$ increase in the visibility of
quantum interference between independent photons from integrated micro-ring
resonator sources that are detuned by 6.8 GHz. We go on to show how time-
resolved detection of non-ideal photons can be used to improve the fidelity of
an entangling operation and to mitigate the reduction of computational
complexity in boson sampling experiments. These results pave the way for
photonic quantum information processing with many photon sources without the
need for active alignment.

##  I Introduction

Proposed photonic quantum technologies will require large numbers of photons
Wang  _et al._ (2020); Bartolucci  _et al._ (2021a); Li  _et al._ (2015) and
would benefit from cavity based single-photon sources. Integrated cavities can
be used to increase the purity of parametric sources, while reducing their
footprint and power consumption Burridge  _et al._ (2020); Llewellyn  _et al._
(2020), as well as increasing the generation rates from solid state sources
based on two-level systems Gazzano  _et al._ (2013); Somaschi  _et al._
(2016). However, fabrication imperfections produce an inherent misalignment of
emission wavelengths from multiple cavity based sources. While thermal
Silverstone  _et al._ (2015), strain Sun  _et al._ (2013) or electrical
Akopian  _et al._ (2010); Nowak  _et al._ (2014); Zhai  _et al._ (2022a)
tuning techniques can be used to adjust the source emission wavelength post-
fabrication, scaling these techniques to many cavities is impractical; thermal
and electrical cross-talk, for example, make aligning even small numbers of
resonant sources to the required sub-GHz precision a challenge Carolan  _et
al._ (2019); Llewellyn  _et al._ (2020); Paesani  _et al._ (2019); Note (1).

Here, we experimentally address this challenge by demonstrating on-chip
quantum interference of photons with distinguishable emission spectra
generated from integrated cavity sources. Our approach, which does not use
spectral filtering or active tuning, relies on fast photon detection to
directly exploit the conjugate relationship between frequency and time:
provided photons are detected with a high enough timing resolution, their
spectral information can be sufficiently erased to allow quantum interference
between initially distinguishable photons Legero  _et al._ (2003). Previous
works have applied this technique with narrow-bandwidth cavity emission from
atomic systems or parametric processes. Avalanche photodiodes were used to
interfere photons with frequency detunings of tens of MHz Legero  _et al._
(2004); Wang  _et al._ (2018); Zhao  _et al._ (2014), which is insufficient
for the GHz-scale detuning that is typical of integrated photon sources
Pfeifle  _et al._ (2014); Zhai  _et al._ (2022b). Here, we show that
commercial superconducting nanowire single-photon detectors (SNSPDs) enable
the erasure of spectral distinguishability at the GHz scale for photons
generated in standard integrated micro-ring resonator (MRRs) sources. We
demonstrate the viability of this approach for photonic quantum information
processing by performing a range of on-chip experiments using distinguishable
photons, including time-resolved Hong-Ou-Mandel (HOM) interference, error-
mitigated photonic fusion operations, and boson sampling experiments with up
to three interfering (six detected) photons. These results demonstrate that
fast detectors can deliver powerful error mitigation for cavity-based sources,
and can readily be extended to deterministic photon sources in heterogeneous
photonic quantum information processors.

Figure 1: (a) Time-resolved two-photon interference. Distinguishable photons
separated by frequency $\Delta\omega$ input to a balanced beam splitter lead
to interference fringes in the time-domain. Detecting a photon in the first
output projects the other photon into a superposition of coming from either of
the inputs. This results in single-photon interference fringes in the second
photon’s relative arrival time $\tau$, with beat frequency $\Delta\omega$. (b)
Schematic of photonic chip. Pulsed laser light enters the chip through grating
couplers and pumps tuneable MRR sources, with thermo-optic phase shifters
shown in gold. Tuneable filters separate idler and signal photons that are
then interfered or used as heralds, respectively. Photons are detected off-
chip by SNSPDs. (c) Boson sampling using temporal filtering. Low timing
precision means no interference and output probabilities are given by
permanents of positive real matrices. High timing precision and filtering
around $\tau=0$ allow distinguishable photons to interfere, giving an output
distribution described by permanents of the complex transfer matrices. (d)
Boson sampling from the distribution of photon arrival times. Given input
photons with temporal profiles $\zeta_{i}$, the output probability is given by
the permanent of the transfer matrix weighted by the temporal amplitudes (Eq.
1). We depict one possible exchange of photons with its associated amplitude.
The total probability is the squared magnitude of the sum of all possible
amplitudes.

##  II Time-resolved interference theory

It has been shown for HOM interference that provided photons are detected
exactly coincidentally at the output ports of a balanced beam splitter,
bunching can be observed with photons of arbitrary and, in general, different
spectro-temporal profiles Legero  _et al._ (2003). Expressions for time-
resolved interference statistics have since been extended to more photons and
to include spectral impurity Tamma and Laibacher (2015); Shchesnovich and
Bezerra (2020). The probability of detecting $N$ photons with arrival times
$\vec{t}=(t_{1},\ldots,t_{N})$ at the outputs of a linear optical
interferometer described by scattering matrix elements $T_{ij}$ is (see
Supplementary Material)

$P_{\mathrm{coinc}}\left(\vec{t}\right)=|\mathrm{Perm}\left(\Lambda\right)|^{2},$
(1)

where the matrix $\Lambda$ has elements given by
$\Lambda_{ij}=T_{ij}\zeta_{j}(t_{i})$ and $\zeta_{i}$ is the temporal profile
of the $i^{\mathrm{th}}$ input photon. From this we see the complexity of
quantum interference, arising from the calculation of permanents of complex
matrices, is retained, even if the photons have different spectro-temporal
profiles. This effect can be applied in the context of boson sampling Tamma
and Laibacher (2016a, b); Laibacher and Tamma (2018) and also for entanglement
swapping Zhao  _et al._ (2014).

Figure 2: Time-resolved HOM interference fringes. (a) Central panel shows
normalised coincidence counts as the MZI phase $\theta$ is varied, for
different timing windows. The offset in the fringe minimum is due to crosstalk
from AMZI filters. The visibility for each timing window is shown in the
legend (below) and the shaded area around each line shows the 1-$\sigma$
interval, using the error from the fit. Poissonian error bars are smaller than
the symbol size. Inset shows fringe for distinguishable photons. We also show
histograms, at the peaks and dip, and compare to our model. (b) Schematic of
subset of the chip used in this experiment, swept phase is also indicated. (c)
Resonance positions of the two ring resonances relative to the pump pulse
(shown as a dotted line)

The complexity of Eq.1 will be degraded by the imperfect timing resolution of
detectors and time tagging electronics, characterised by the full width at
half maximum (FWHM) of their response functions, often called the ‘jitter’. We
model the overall effect of detection equipment jitter by convolving
interference fringes with a Gaussian distribution whose variance is the sum of
the variances for the detectors and time-tagger channels used.

Figure 3: Time-resolved interference and type-II fusions in a complex
Hadamard. (a) Chip configuration. Tick marks indicate the output pattern used
in (b) and (e). (b) Fringes in $\phi$ for different timing windows showing
that visibility increases with timing resolution. The inset shows the signal
for temporally distinguishable photons generated by consecutive pump pulses:
there is no dependence on $\phi$ or $\delta t$. (c) Type-II fusion gate on
dual-rail photonic qubits, with qubit mode labels shown. The indicated output
click pattern projects onto a Bell state
$\ket{\psi}=\frac{1}{\sqrt{2}}(\ket{+-}+e^{i\phi}\ket{-+})$. (d) Effect of a
type-II fusion gate on entangled resource states. (e) Measured relative
arrival time histograms (purple above) and theoretical model (grey below)
shown for each value of $\phi$.

##  III Experimental setup

A schematic of the integrated photonic device is shown in Fig. 1. A pulsed
pump laser is used to generate pairs of single photons through spontaneous
four-wave mixing (SFWM). The laser is filtered and amplified before being
passively split on-chip by a tree of multi-mode interference couplers,
designed to implement balanced beam-splitters, and pumping up to four silicon
micro-ring resonator sources. The spectra of these sources can be individually
tuned by voltage-controlled thermo-optic phase shifters, and low-power
continuous wave (CW) lasers are used to monitor and align the ring-resonators
to the desired emission wavelengths. Asymmetric Mach-Zehnder interferometer
(AMZI) filters separate signal and idler wavelengths before the signals are
coupled off-chip and the idlers enter a programmable integrated interferometer
(see Fig. 1b).

Photons are detected by high efficiency and low jitter SNSPDs from PhotonSpot,
with an average per-detector jitter measured to be $\sim$75 ps. Timing logic
is performed by a Swabian Ultra II time-tagger, with single channel jitter of
$\sim$15 ps. More detail on the experimental setup is given in the
Supplementary Material.

Figure 4: Scattershot boson sampling with temporal filtering (a)-(d) measured
probability distributions for varying timing window. (e) Theoretical output
distribution for the adversarial model. (f) Corresponding test model
distribution. (g) Spectral alignment of the four ring resonator sources.
Dotted line illustrates the spectral response of the pump pulse. (h) Final
Bayesian probability for larger sample lists (2000 randomly chosen samples,
averaged 300 times). For a timing window of 20 ps we use the entire sample
list.

##  IV Time-resolved two-photon interference

We first show that temporal filtering increases the interference visibility
for two spectrally distinguishable photons, as depicted in Fig. 1c. MRR
resonances are detuned by 55 pm (6.8 GHz) and positioned symmetrically about
the centre wavelength of the pump pulse (see Fig. 2c). We then interfere the
SFWM photons generated in these resonances on a Mach-Zehnder interferometer
(MZI). The output arrival times are recorded as the MZI phase $\theta$ is
varied. The relative arrival times of the interfering photons are calculated
and binned into histograms with 20 ps bin width. In Fig. 2a we plot the
normalised MZI fringes for different effective timing resolutions, obtained by
changing the timing window over which we integrate the depicted coincidence
histograms. The visibility of the resulting fringe in coincidence counts is
defined as
$V=(C_{\mathrm{max}}-C_{\mathrm{min}})/(C_{\mathrm{max}}+C_{\mathrm{min}})$,
where $C_{\mathrm{max}}$ ($C_{\mathrm{min}}$) is the maximum (minimum) fringe
counts within the chosen coincidence window. The fringe visibility for non-
interfering photons is $V=1/3$ Vigliar (2020). For a large integration window
$\delta\tau=200$ ps, the MZI fringe visibility of $0.49\pm 0.01$ reflects some
overlap of the detuned spectra shown in Fig. 2c. As the integration window is
narrowed, the number of counts decreases but the fringe visibility increases
to a maximum of $0.61\pm 0.01$ when $\delta\tau=20$ ps, showing a significant
improvement of the quantum interference via fast detection and temporal
filtering. We find an average statistical fidelity between our model (see
Supplementary Material) and the measured histograms of 0.984 $\pm$ 0.001. The
remaining distinguishability is due to the imperfect timing response of the
detectors and time-tagging electronics used. Moreover, in the inset figure we
also show the fringes for temporally distinguishable photons generated by
consecutive laser pulses (see Supplementary Material). These show no
dependence on the timing window and all fringes have $V=1/3$, as expected.

##  V Fusion gates with time-resolved complex Hadamard interference

We now demonstrate the use of time-resolved techniques to enhance a key
building block for photonic quantum computing: the type-II fusion gate Browne
and Rudolph (2005). Type-II fusion gates perform a partial Bell state
measurement on two qubits, and can be used to fuse two resource states into a
larger entanglement structure (see Fig. 3c,d). They play a central role in the
generation of complex entangled structures for linear optical quantum
computing Bartolucci  _et al._ (2021b), as well as for generating entanglement
in solid state systems Barrett and Kok (2005) and in quantum repeater networks
Duan  _et al._ (2001); Li  _et al._ (2019). The photonic circuitry for type-II
fusion gates can be performed by a four-mode complex Hadamard interferometer.
This type of circuit, depicted in Fig. 3a, exhibits interesting multi-photon
quantum interference features Jones  _et al._ (2020); Laing  _et al._ (2012),
and finds extensive uses in classical encryption Horadam and Udaya (2000);
Hedayat and Wallis (1978), dense coding Werner (2001), and the search for
mutually unbiased bases (MUBs) Durt  _et al._ (2010). We investigate how such
features appear in a time-resolved complex Hadamard interference and analyse
their use for fusion gates.

For dimension $d=4$, all non-equivalent complex Hadamard matrices – those that
are not related by row and column permutations or phase multiplications – are
described by a single parameter $\phi$ in the family $F_{4}$ which, after
normalisation, is given by

$F_{4}\left(\phi\right)=\frac{1}{2}\begin{pmatrix}1&1&1&1\\\
1&e^{i\phi}&-1&-e^{i\phi}\\\ 1&-1&1&-1\\\
1&-e^{i\phi}&-1&e^{i\phi}\end{pmatrix}.$ (2)

In Fig. 3a we show how, with input MZIs set to a balanced splitting ratio, our
programmable integrated interferometer can implement this transfer matrix. The
output statistics for pairs of photons in certain input modes exhibit a
dependence on the internal, controllable phase $\phi$. This is because, while
there are no closed loops inside the interferometer, there is interference of
the pairs of paths that photons can take. The time-resolved output coincidence
probability is

$P(t_{0},t_{0}+\tau)=\frac{1}{16}|\zeta_{1}(t_{0}+\tau)\zeta_{2}(t_{0})+e^{i\phi}\zeta_{1}(t_{0})\zeta_{2}(t_{0}+\tau)|^{2}.$
(3)

When $\phi=\pi$ we recover the same expression as for a balanced beam splitter
Legero  _et al._ (2003). For other values, $\phi$ shifts the relative delays
$\tau$ at which destructive interference suppresses coincidences. This can be
interpreted as adjusting the relative phase for conditioned single-photon
interference, analogous to how applying a variable phase to a path in the
double slit experiment shifts the fringe pattern on a screen. In Fig. 3e, we
see how the complex Hadamard phase shifts the interference fringes as
expected. As before, we can integrate over the measured histograms to obtain
an interference fringe. For indistinguishable photons this fringe should be
given by $P_{\mathrm{indist}}=\frac{1}{2}\left(1+\cos\phi\right)$ and for
distinguishable photons there should be no dependence on $\phi$ Jones  _et
al._ (2020). In Fig. 3b we see that as we change the timing integration
window, we see an increase in the visibility of the fringe with decreasing
timing window. Figure. 3e shows good agreement between our model and measured
histograms with an average statistical fidelity of 0.972 $\pm$ 0.002.

The above measurement probes the interference in the type-II fusion gate
acting on a pair of photonic qubits initialized in the $\ket{00}$ input state.
The value of $\phi$ has the effect of changing the Bell state that is
projected onto the output click patterns. For example, the entangled state
projected onto the output pattern shown in Fig. 3a,c is
$\ket{\psi}=\left[\left(1+e^{i\phi}\right)\ket{\Phi^{-}}+\left(1-e^{i\phi}\right)\ket{\Psi^{-}}\right]/\sqrt{2}$,
where $\ket{\Phi^{\pm}}$ and $\ket{\Psi^{\pm}}$ are the canonical maximally
entangled Bell states. We can map the visibility of the fringe in Fig. 3b to
the fidelity of the fusion gate as $\mathrm{F}=\frac{1+\gamma}{2}$ with
$\gamma=\frac{2V}{1+V}$ Rohde and Ralph (2006). An increase in the visibility
of the fringe due to the temporal resolution thus corresponds to an improved
fidelity of the fusion gate operation. From the experimental fringe in Fig.
3b, we observe a significant improvement of the fusion gate fidelity, from
0.79 $\pm$ 0.02 to 0.875 $\pm$ 0.004, when tuning the timing window. Temporal
filtering allows us to estimate the fusion gate fidelity, in practice,
however, this results in a reduction of the gate success probability. If,
instead, a large timing window is used but the photon arrival times are
measured precisely, then differences in the relative arrival times of the
photons represent a random but heralded phase on the remaining photons and
can, therefore, be accounted for with adaptive measurement Campbell  _et al._
(2007); Zhao  _et al._ (2014)

##  VI Boson sampling with spectrally distinguishable photons

Figure 5: Sampling from the distribution of photon arrival times (a) Two
photon scattershot boson sampling. Probability that a given sample string
arose from our desired model as a function of sample size. Each sample string
is drawn randomly from the total set of samples. We draw each sample size 200
times and plot the average and 1-$\sigma$ error bars. For the final point we
take all measured samples. The inset shows the frequency alignment of the four
ring sources. (b) Dynamic Bayesian updates showing the confidence that a given
sample string arose from our test model.

We now investigate the interference of multiple, spectrally distinguishable
photons in boson sampling tasks. Boson sampling is a model for quantum
computation that, although not universal, can challenge the capabilities of
classical computers, and has been demonstrated as a leading approach to
demonstrating quantum advantage Aaronson and Arkhipov (2011); Zhong  _et al._
(2021, 2020); Madsen  _et al._ (2022). Multiple variants exist and all involve
sampling photon numbers from a quantum state of light. We first consider the
scattershot variant Lund  _et al._ (2014): spontaneous sources are connected
to the inputs of an interferometer and, when pumped simultaneously, a subset
will generate photon pairs. One photon from each pair heralds the input taken
by its partner and so determines the scattering matrix for a particular run of
the sampling experiment. We use four MRR sources to perform two-photon
scattershot boson sampling with distinguishable photons, and investigate using
temporal filtering to recover the complexity of indistinguishable
interference. We align rings 1 and 4 to the same wavelength, and then rings 2
and 3 are aligned to another wavelength detuned by 54 pm (6.7 GHz), see
Fig.4g. The four-mode interferometer is programmed to implement $F_{4}(\pi/2)$
from Eq. 2. We record the time tags for heralded two-photon events and, as
earlier, can apply temporal filtering to coincidence histograms in post-
processing.

We use Bayesian methods to determine whether our experimental samples were
more likely drawn from a distribution involving significant quantum
interference, or from one arising from probabilistic, distinguishable
scattering Paesani  _et al._ (2019). We draw $N$ samples from our experiment,
$x_{i}$, $i=1,\ldots,N$, such that the overall data $D=\\{x_{i}\\}$. We then
calculate the likelihood function

$P\left(M_{test}|D\right)=\frac{P\left(D|M_{test}\right)}{P\left(D|M_{test}\right)+P\left(D|M_{adv}\right)},$
(4)

where $M_{test}$ is a test distribution that is hard to sample from, and
$M_{adv}$ is an adversarial distribution that could describe the experiment
but is easier to sample from. A value of one indicates that the data $D$ were
more likely drawn from $M_{test}$ than $M_{adv}$, and a value of zero
indicates the opposite. We assume samples are uncorrelated, so probabilities
$P(D|M)=\prod_{i}^{N}P_{M}(x_{i})$, where $P_{M}(x_{i})$ is the probability
that sample $x_{i}$ came from distribution $M$.

In our case, $M_{test}$ comprises mainly indistinguishable interference
involving permanents of complex matrices $\sim|\mathrm{Perm}(T_{s})|^{2}$, but
also a small contribution from distinguishable scattering depending on
permanents of real matrices $\sim\mathrm{Perm}|T_{s}|^{2}$. This is to counter
sensitivity of the likelihood test to the observation of photon detection
patterns that are ideally suppressed by the Fourier interferometer employed,
but can happen in our experiment due to imperfections (see Supplementary
Material for details). The adversarial model $M_{adv}$ consists of
distinguishable scattering for the input patterns where the photons are
spectrally distinguishable, and of indistinguishable interference for the
input patterns where the photons are spectrally indistinguishable.

In Figs. 4a–d, we show the measured output distribution as a function of
timing window. These plots include corrections for the input squeezing and
output coupling losses, as well as subtraction of double emissions. We see a
qualitative change, moving from the adversarial distribution in Fig. 4e to the
test distribution in Fig. 4f. Fig. 4h shows the average final Bayesian
probability for sets of 2000 samples, chosen uniformly from the total sample
list, as a function of the timing window used. As we decrease the timing
window, we see a transition in which model best describes the data at between
30 and 60 ps, indicating recovery of quantum interference despite
distinguishability.

Rather than filtering and incurring loss, it is instead possible to use the
precise photon arrival times at the outputs as part of the sample from a boson
sampler. Similarly to the random but known input ports in the scattershot
approach, the arrival times indicate random but known complex factors that
alter the effective scattering matrix (as depicted in Fig. 1d). This approach
was introduced by Tamma et al. Tamma and Laibacher (2015) and has been shown
to be at least as hard as conventional boson sampling Tamma and Laibacher
(2016a, b); Laibacher and Tamma (2015, 2018). We experimentally perform two-
photon scattershot and three-photon standard boson sampling where we
explicitly include the photons’ arrival times. For the two-photon scattershot,
we use the same resonance positions and interferometer as for the temporal
filtering experiment. But for the three-photon experiment, we only pump three
MRRs and reduce the detuning to 5.9 GHz to ensure sufficient squeezing for
six-fold events. We also program a different interferometer as no phase-
dependent three-photon interference appears in $F_{4}$ and both
indistinguishable and distinguishable distributions are uniform for all input
and output combinations (see Supplementary Material).

To verify our experiment we, again, use the Bayesian likelihood function,
given in Eq. 4. The ideal case is now given by Eq. 1, with the corresponding
transfer matrix determined by the measured input and output pattern. We choose
a non-interfering adversarial model with
$P_{\mathrm{adv}}\left(\vec{t}\right)=\mathrm{Perm}\left(|\Lambda|^{2}\right)$.
As above, we include a non-interfering term in the test model. For two-photon,
time-resolved scattershot, we show the averaged Bayesian probability for our
test model for multiple sets containing a given number of samples in Fig. 5a.
For the three-photon case, due to the lower total number of events, we show
the cumulative sample-by-sample probability that the data was drawn from the
desired distribution. This dynamically updated confidence is shown in Fig. 5b,
showing that our samples were more likely drawn from the interfering
distribution.

##  VII Discussion

We have demonstrated photonic quantum information processing experiments with
MRRs detuned up to 6.8 GHz and with photon linewidths of $\sim$3.8 GHz, which
are both two orders of magnitude higher than previous experiments. MRRs are
promising candidates to produce high photon numbers Arrazola  _et al._ (2021)
but experiments have been limited to small numbers of sources due to the
experimental challenge of aligning many sources Carolan  _et al._ (2019). To
perform experiments with no active alignment of MRRs, we require that the
maximum detuning, given by half the resonator free spectral range, is less
than the maximum possible frequency erasure. In this work, the per-detector
jitter, was measured to be $\sim$75 ps, which is already sufficient to observe
quantum interference between photons detuned by approximately twice the photon
linewidth, fulfilling this condition. Further analysis (see Supplementary
Material) shows that using the lowest jitter currently available from
commercial SNSPDs ($15$ ps) Single Quantum (2019), photons detuned up to
$190\text{\,}\mathrm{GHz}$ can result in HOM visibilities above the quantum
limit, while photons detuned up to $5.1\text{\,}\mathrm{GHz}$ can result in
visibilities above $\sim 95\%$, which is sufficient to demonstrate quantum
advantage Renema  _et al._ (2018). Using detectors with the lowest reported
jitter in the literature (3 ps) Korzh  _et al._ (2020), we could expect to
observe non-classical interference from photons detuned by up to
$300\text{\,}\mathrm{GHz}$, and visibilities above the quantum advantage
threshold for photons detuned by $29\text{\,}\mathrm{GHz}$. This work shows
that time-resolved detection can mitigate the demand to actively align
resonant sources and significantly reduce device complexity for large scale
quantum photonics experiments.

##  VIII Acknowledgements

We acknowledge support from the Engineering and Physical Sciences Research
Council (EPSRC) Hub in Quantum Computing and Simulation (EP/T001062/1).
Fellowship support from EPSRC is acknowledged by A.L. (EP/N003470/1). S.P.
acknowledges funding from the Cisco University Research Program Fund nr.
2021-234494. P.Y. would like to thank Emma Foley, Brian Flynn and Naomi
Solomons for useful discussions.

## References

  * Wang  _et al._ (2020) J. Wang, F. Sciarrino, A. Laing, and M. G. Thompson, Nature Photonics 14, 273 (2020).
  * Bartolucci  _et al._ (2021a) S. Bartolucci, P. Birchall, D. Bonneau, H. Cable, M. Gimeno-Segovia, K. Kieling, N. Nickerson, T. Rudolph, and C. Sparrow, arXiv preprint arXiv:2109.13760 (2021a).
  * Li  _et al._ (2015) Y. Li, P. C. Humphreys, G. J. Mendoza, and S. C. Benjamin, Physical Review X 5, 041007 (2015).
  * Burridge  _et al._ (2020) B. M. Burridge, I. I. Faruque, J. G. Rarity, and J. Barreto, Optics Letters 45, 4048 (2020).
  * Llewellyn  _et al._ (2020) D. Llewellyn, Y. Ding, I. I. Faruque, S. Paesani, D. Bacco, R. Santagati, Y.-J. Qian, Y. Li, Y.-F. Xiao, M. Huber, _et al._ , Nature Physics 16, 148 (2020).
  * Gazzano  _et al._ (2013) O. Gazzano, S. Michaelis de Vasconcellos, C. Arnold, A. Nowak, E. Galopin, I. Sagnes, L. Lanco, A. Lemaître, and P. Senellart, Nature communications 4, 1 (2013).
  * Somaschi  _et al._ (2016) N. Somaschi, V. Giesz, L. De Santis, J. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Anton, J. Demory, _et al._ , Nature Photonics 10, 340 (2016).
  * Silverstone  _et al._ (2015) J. W. Silverstone, R. Santagati, D. Bonneau, M. J. Strain, M. Sorel, J. L. O’Brien, and M. G. Thompson, Nature communications 6, 1 (2015).
  * Sun  _et al._ (2013) S. Sun, H. Kim, G. S. Solomon, and E. Waks, Applied Physics Letters 103, 151102 (2013).
  * Akopian  _et al._ (2010) N. Akopian, U. Perinetti, L. Wang, A. Rastelli, O. Schmidt, and V. Zwiller, Applied Physics Letters 97, 082103 (2010).
  * Nowak  _et al._ (2014) A. Nowak, S. Portalupi, V. Giesz, O. Gazzano, C. Dal Savio, P.-F. Braun, K. Karrai, C. Arnold, L. Lanco, I. Sagnes, _et al._ , Nature communications 5, 1 (2014).
  * Zhai  _et al._ (2022a) L. Zhai, G. N. Nguyen, C. Spinnler, J. Ritzmann, M. C. Lobl, A. D. Wieck, A. Ludwig, J. Alisa, and R. J. Warburton, “Quantum interference of identical photons from remote gaas quantum dots,”  (2022a).
  * Carolan  _et al._ (2019) J. Carolan, U. Chakraborty, N. C. Harris, M. Pant, T. Baehr-Jones, M. Hochberg, and D. Englund, Optica 6, 335 (2019).
  * Paesani  _et al._ (2019) S. Paesani, Y. Ding, R. Santagati, L. Chakhmakhchyan, C. Vigliar, K. Rottwitt, L. K. Oxenløwe, J. Wang, M. G. Thompson, and A. Laing, Nature Physics 15, 925 (2019).
  * Note (1) To reach 99.9% visibility interference for a 1% surface code threshold Sparrow (2017), resonances must be aligned to within $\sim 3\%$ of the linewidth.
  * Legero  _et al._ (2003) T. Legero, T. Wilk, A. Kuhn, and G. Rempe, Applied Physics B 77, 797 (2003).
  * Legero  _et al._ (2004) T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, Phys. Rev. Lett. 93, 070503 (2004).
  * Wang  _et al._ (2018) X.-J. Wang, B. Jing, P.-F. Sun, C.-W. Yang, Y. Yu, V. Tamma, X.-H. Bao, and J.-W. Pan, Physical review letters 121, 080501 (2018).
  * Zhao  _et al._ (2014) T.-M. Zhao, H. Zhang, J. Yang, Z.-R. Sang, X. Jiang, X.-H. Bao, and J.-W. Pan, Phys. Rev. Lett. 112, 103602 (2014).
  * Pfeifle  _et al._ (2014) J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, _et al._ , Nature photonics 8, 375 (2014).
  * Zhai  _et al._ (2022b) L. Zhai, G. N. Nguyen, C. Spinnler, J. Ritzmann, M. C. Löbl, A. D. Wieck, A. Ludwig, A. Javadi, and R. J. Warburton, Nature Nanotechnology , 1 (2022b).
  * Tamma and Laibacher (2015) V. Tamma and S. Laibacher, Physical review letters 114, 243601 (2015).
  * Shchesnovich and Bezerra (2020) V. Shchesnovich and M. Bezerra, Physical Review A 101, 053853 (2020).
  * Tamma and Laibacher (2016a) V. Tamma and S. Laibacher, Journal of Modern Optics 63, 41 (2016a).
  * Tamma and Laibacher (2016b) V. Tamma and S. Laibacher, Quantum Information Processing 15, 1241 (2016b).
  * Laibacher and Tamma (2018) S. Laibacher and V. Tamma, arXiv preprint arXiv:1801.03832 (2018).
  * Vigliar (2020) C. Vigliar,  _Graph states in large-scale integrated quantum photonics_ , Ph.D. thesis (2020).
  * Browne and Rudolph (2005) D. E. Browne and T. Rudolph, Physical Review Letters 95, 010501 (2005).
  * Bartolucci  _et al._ (2021b) S. Bartolucci, P. Birchall, H. Bombin, H. Cable, C. Dawson, M. Gimeno-Segovia, E. Johnston, K. Kieling, N. Nickerson, M. Pant, _et al._ , arXiv preprint arXiv:2101.09310 (2021b).
  * Barrett and Kok (2005) S. D. Barrett and P. Kok, Phys. Rev. A 71, 060310 (2005).
  * Duan  _et al._ (2001) L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, Nature 414, 413 (2001).
  * Li  _et al._ (2019) Z.-D. Li, R. Zhang, X.-F. Yin, L.-Z. Liu, Y. Hu, Y.-Q. Fang, Y.-Y. Fei, X. Jiang, J. Zhang, L. Li, _et al._ , Nature photonics 13, 644 (2019).
  * Jones  _et al._ (2020) A. E. Jones, A. J. Menssen, H. M. Chrzanowski, T. A. Wolterink, V. S. Shchesnovich, and I. A. Walmsley, Physical Review Letters 125, 123603 (2020).
  * Laing  _et al._ (2012) A. Laing, T. Lawson, E. M. López, and J. L. O’Brien, Physical review letters 108, 260505 (2012).
  * Horadam and Udaya (2000) K. J. Horadam and P. Udaya, IEEE Transactions on Information Theory 46, 1545 (2000).
  * Hedayat and Wallis (1978) A. Hedayat and W. D. Wallis, The Annals of Statistics , 1184 (1978).
  * Werner (2001) R. F. Werner, Journal of Physics A: Mathematical and General 34, 7081 (2001).
  * Durt  _et al._ (2010) T. Durt, B.-G. Englert, I. Bengtsson, and K. Życzkowski, International Journal of Quantum Information 8, 535 (2010).
  * Rohde and Ralph (2006) P. P. Rohde and T. C. Ralph, Physical Review A 73, 062312 (2006).
  * Campbell  _et al._ (2007) E. T. Campbell, J. Fitzsimons, S. C. Benjamin, and P. Kok, Phys. Rev. A 75, 042303 (2007).
  * Aaronson and Arkhipov (2011) S. Aaronson and A. Arkhipov, in  _Proceedings of the forty-third annual ACM symposium on Theory of computing_ (2011) pp. 333–342.
  * Zhong  _et al._ (2021) H.-S. Zhong, Y.-H. Deng, J. Qin, H. Wang, M.-C. Chen, L.-C. Peng, Y.-H. Luo, D. Wu, S.-Q. Gong, H. Su, Y. Hu, P. Hu, X.-Y. Yang, W.-J. Zhang, H. Li, Y. Li, X. Jiang, L. Gan, G. Yang, L. You, Z. Wang, L. Li, N.-L. Liu, J. Renema, C.-Y. Lu, and J.-W. Pan, “Phase-programmable gaussian boson sampling using stimulated squeezed light,” (2021), arXiv:2106.15534 [quant-ph] .
  * Zhong  _et al._ (2020) H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding, Y. Hu, _et al._ , Science 370, 1460 (2020).
  * Madsen  _et al._ (2022) L. S. Madsen, F. Laudenbach, M. F. Askarani, F. Rortais, T. Vincent, J. F. Bulmer, F. M. Miatto, L. Neuhaus, L. G. Helt, M. J. Collins, _et al._ , Nature 606, 75 (2022).
  * Lund  _et al._ (2014) A. P. Lund, A. Laing, S. Rahimi-Keshari, T. Rudolph, J. L. O’Brien, and T. C. Ralph, Physical review letters 113, 100502 (2014).
  * Laibacher and Tamma (2015) S. Laibacher and V. Tamma, Physical review letters 115, 243605 (2015).
  * Arrazola  _et al._ (2021) J. Arrazola, V. Bergholm, K. Brádler, T. Bromley, M. Collins, I. Dhand, A. Fumagalli, T. Gerrits, A. Goussev, L. Helt, _et al._ , Nature 591, 54 (2021).
  * Single Quantum (2019) Single Quantum,  _Single Quantum Eos Datasheet_ (2019).
  * Renema  _et al._ (2018) J. J. Renema, A. Menssen, W. R. Clements, G. Triginer, W. S. Kolthammer, and I. A. Walmsley, Physical review letters 120, 220502 (2018).
  * Korzh  _et al._ (2020) B. Korzh, Q.-Y. Zhao, J. P. Allmaras, S. Frasca, T. M. Autry, E. A. Bersin, A. D. Beyer, R. M. Briggs, B. Bumble, M. Colangelo, _et al._ , Nature Photonics 14, 250 (2020).
  * Sparrow (2017) C. Sparrow,  _Quantum Interference in Universal Linear Optical Devices for Quantum Computation and Simulation_ , Ph.D. thesis (2017).
  * Scheel (2004) S. Scheel, arXiv preprint quant-ph/0406127 (2004).
  * Vernon  _et al._ (2017) Z. Vernon, M. Menotti, C. Tison, J. Steidle, M. Fanto, P. Thomas, S. Preble, A. Smith, P. Alsing, M. Liscidini, _et al._ , Optics letters 42, 3638 (2017).
  * Swabian Instruments (2022) Swabian Instruments,  _Time tagger series: streaming time-to-digital converters_ (2022).

Supplementary Material

###  VIII.1 N-photon time-resolved interference

In the following analysis we look to derive the probability of detecting $N$
photons at $N$ different times $\vec{t}=(t_{1},t_{2},\dots,t_{N})$ at the
output of an $M$ mode linear optical interferometer, given input photons with
arbitrary temporal spectra, $\zeta_{i}(t)$. For simplicity we assume that our
photons are injected into the first $N$ input modes, and measured in the first
$N$ output modes. Therefore, our measurements at times given by $\vec{t}$
project onto the state

$\ket{\psi_{\text{coinc}}(\vec{t})}=\prod_{i=1}^{N}\hat{b}^{\dagger}_{i}(t_{i})\ket{0}^{\otimes
M}$ (A1)

where $\hat{b}^{\dagger}_{i}(t_{i})$ is the creation operator acting on mode
$i$ at time $t_{i}$. The input state is given by

$\ket{\psi_{\text{in}}}=\prod_{i=1}^{N}\left(\int
dt\zeta_{i}(t)\hat{a}^{\dagger}_{i}(t)\right)\ket{0}^{\otimes M}$ (A2)

which after evolving through an interferometer becomes

$\ket{\psi_{\text{out}}}=\prod_{i=1}^{N}\left(\int
dt\zeta_{i}(t)\left(\sum_{j=1}^{M}T_{ji}\hat{b}^{\dagger}_{j}(t)\right)\right)\ket{0}^{\otimes
M}.$ (A3)

Here, the $T_{ji}$ are the elements of the interferometer scattering matrix
$\mathbf{T}$. We can now calculate the coincidence probability as
$P_{\text{coinc}}(\vec{t})=\left|\braket{\psi_{\text{coinc}}(\vec{t})}{\psi_{\text{out}}}\right|^{2}$.
By inspecting the overlap between $\ket{\psi_{\text{out}}}$ and
$\ket{\psi_{\text{coinc}}}$, we can see that the only the terms which will
survive from $\ket{\psi_{\text{out}}}$ are those which have exactly one
creation operator per mode for the first $N$ modes, and none in the other
modes. This also lets us remove the integrals, as for example

$\int
dt\bra{0}\hat{b}_{k}(t_{k})\zeta_{i}(t)T_{ji}\hat{b}^{\dagger}_{j}(t)\ket{0}=\delta_{kj}\zeta(t_{j})T_{ji}.$
(A4)

As shown, for example in Ref. Scheel (2004), we can therefore write the final
probability as

$\displaystyle P_{\text{coinc}}$ $\displaystyle=\left|\sum_{\sigma\in
S_{N}}\prod_{i=1}^{N}T_{i,\sigma(i)}\zeta_{\sigma(i)}(t_{i})\right|^{2}$ (A5)
$\displaystyle=|\mathrm{Perm}(\Lambda)|^{2},$ (A6)

where $S_{N}$ is the group of $N-$dimensional permutations and $\Lambda$ has
elements given by $\Lambda_{ij}=T_{ij}\zeta_{j}(t_{i})$. We can see that the
amplitude for this state is the permanent of the matrix $\Lambda$ which is
formed from the scattering matrix, weighted by the temporal shape of the
photons. Here, we note that we have recovered the result of Ref. Tamma and
Laibacher (2015).

###  VIII.2 Time-resolved HOM interference with mixed input states

To determine the effect of impure input photons, we detail the derivation of
the coincidence probability of time-resolved HOM interference with photons in
spectrally mixed states. To start we define an arbitrary photon density
operator to be

$\rho_{i}=\sum_{k}\lambda_{k}\ket{\phi_{ik}}\bra{\phi_{ik}},$ (A7)

where

$\ket{\phi_{ik}}=\int dt\;\zeta_{ik}(t)\hat{a}^{\dagger}_{i}(t)\ket{0}_{i}.$
(A8)

We can then write the two photon input state $\ket{11}$ as

$\rho_{\mathrm{in}}=\rho_{1}\otimes\rho_{2}=\sum_{kl}\lambda_{k}\lambda^{\prime}_{l}\ket{\psi_{kl}}\bra{\psi_{kl}},$
(A9)

with

$\ket{\psi_{kl}}=\iint
dtdt^{\prime}\;\zeta_{1k}(t)\zeta_{2l}(t^{\prime})\hat{a}^{\dagger}_{1}(t)\hat{a}^{\dagger}_{2}(t^{\prime})\ket{00}.$
(A10)

Applying the beam-splitter relations to $\rho_{in}$

$\displaystyle\ket{\psi_{kl}}$
$\displaystyle\rightarrow\ket{\psi_{kl}}_{\mathrm{out}}=\iint
dtdt^{\prime}\;\zeta_{1k}(t)\zeta_{2l}(t^{\prime})\left[\hat{a}^{\dagger}_{3}(t)+\hat{a}^{\dagger}_{4}(t)\right]\left[\hat{a}^{\dagger}_{3}(t^{\prime})-\hat{a}^{\dagger}_{4}(t^{\prime})\right]\ket{00}$
(A11) $\displaystyle=\iint
dtdt^{\prime}\;\zeta_{1k}(t)\zeta_{2l}(t^{\prime})\left[\hat{a}^{\dagger}_{3}(t)\hat{a}^{\dagger}_{3}(t^{\prime})+\hat{a}^{\dagger}_{4}(t)\hat{a}^{\dagger}_{3}(t^{\prime})-\hat{a}^{\dagger}_{3}(t)\hat{a}^{\dagger}_{4}(t^{\prime})-\hat{a}^{\dagger}_{4}(t)\hat{a}^{\dagger}_{4}(t^{\prime})\right]\ket{00},$

to give a final density matrix

$\rho_{\mathrm{out}}=\sum_{kl}\lambda_{k}\lambda^{\prime}_{l}\ket{\psi_{kl}}_{\mathrm{out}}\bra{\psi_{kl}}_{\mathrm{out}}.$
(A12)

Using the time-resolved coincidence projector,
$\hat{P}\left(t_{1},t_{2}\right)=\hat{a}^{\dagger}_{3}(t_{1})\hat{a}^{\dagger}_{4}(t_{2})\ket{0}\bra{0}\hat{a}_{4}(t_{2})\hat{a}_{3}(t_{1})$,
we arrive at

$\displaystyle P_{\mathrm{coinc}}\left(t_{1},t_{2}\right)$
$\displaystyle=\mathrm{Tr}\left[\rho_{\mathrm{out}}\times\hat{P}\left(t_{1},t_{2}\right)\right]$
(A13)
$\displaystyle=\sum_{kl}\lambda_{k}\lambda^{\prime}_{l}\;{}_{\mathrm{out}}\bra{\psi_{kl}}\hat{P}\left(t_{1},t_{2}\right)\ket{\psi_{kl}}_{\mathrm{out}}$
$\displaystyle=\sum_{kl}\lambda_{k}\lambda^{\prime}_{l}|\bra{0}\hat{a}_{4}(t_{2})\hat{a}_{3}(t_{1})\ket{\psi_{kl}}_{\mathrm{out}}|^{2}$
$\displaystyle=\sum_{kl}\lambda_{k}\lambda^{\prime}_{l}\bigg{|}\iint
dtdt^{\prime}\zeta_{1k}(t)\zeta_{2l}(t^{\prime})$
$\displaystyle\qquad\qquad\times\big{[}\bra{0}\hat{a}_{4}(t_{2})\hat{a}_{3}(t_{1})\hat{a}^{\dagger}_{4}(t)\hat{a}^{\dagger}_{3}(t^{\prime})\ket{0}-\bra{0}\hat{a}_{4}(t_{2})\hat{a}_{3}(t_{1})\hat{a}^{\dagger}_{3}(t)\hat{a}^{\dagger}_{4}(t^{\prime})\ket{0}\big{]}\bigg{|}^{2}$

To simplify this further, we note that
$\bra{0}\hat{a}_{4}(t_{2})\hat{a}_{3}(t_{1})\hat{a}^{\dagger}_{4}(t^{\prime}_{2})\hat{a}^{\dagger}_{3}(t_{1}^{\prime})\ket{0}=\delta_{t_{1}t^{\prime}_{1}}\delta_{t_{2}t^{\prime}_{2}}$.
This leaves us with only two non-zero terms from the double integral, giving a
final probability of

$P_{\mathrm{coinc}}\left(t_{1},t_{2}\right)=\sum_{kl}\lambda_{k}\lambda^{\prime}_{l}\big{|}\zeta_{1k}(t_{2})\zeta_{2l}(t_{1})-\zeta_{1k}(t_{1})\zeta_{2l}(t_{2})\big{|}^{2}.$
(A14)

The final probability is therefore given by an incoherent sum of pure state
probabilities which for $t_{1}=t_{2}$ returns the indistinguishable
probability, regardless of the number or specific shapes of the Schmidt modes.
For other timing combinations, even with arbitrarily high timing precision,
the incoherent sum masks the quantum interference. We can generalise this
result to $N$ photon interference and arbitrary scattering matrices

$P(\vec{t})=\sum_{k_{1}\cdots
k_{N}}\prod^{N}_{i=1}\lambda_{i,k_{i}}\bigg{|}\mathrm{Perm}\left(\Lambda^{k_{1}\cdots
k_{N}}\right)\bigg{|}^{2},$ (A15)

where the matrix $\Lambda^{k_{1}\cdots k_{N}}$ has elements
$\Lambda^{k_{1}\cdots k_{N}}_{ij}=T_{ij}\zeta_{j,k_{j}}\left(t_{i}\right)$,
$k_{i}$ represents the $k^{\mathrm{th}}$ Schmidt mode of the $i^{\mathrm{th}}$
input photon, and $\zeta_{i,k_{i}}$ is the corresponding temporal profile.

###  VIII.3 Experimental setup

A detailed schematic of the experimental setup is shown in Fig. S1. One high-
powered continuous wave laser from Yenista is used for characterisation of
optical components as well as aligning the ring resonator photon sources to a
primary wavelength. This is used to align the pump resonances of the rings
together. For the experiments where we require aligning of ring sources to two
different wavelengths we employ a second CW Yenista laser.

Figure S1: Detailed experimental schematic along with characterisation of
important components. (a) Typical spectral response of a DWDM channel. (b)
Spectral response at one of the heralding outputs. Combines the ring and AMZI
spectra. (c) Characterisation of a thermo-optic heater. Left panel shows the
transmission of an MZI as a function of applied voltage. Right panel shows the
current-voltage relationship for one heater.(d) Typical coincidence histogram
from one ring source. Also included is the response when using the internal
test signals on two time-tagger channels. This is used to determine the
intrinsic time-tagger jitter.

To ensure that each ring is aligned to the desired wavelength, this second
laser is used to align on resonances $\approx$$9.5\text{\,}\mathrm{nm}$ away
from the pump, this is chosen such that this wavelength is also transmitted
correctly by the on-chip AMZI filters. For generating photons, we employ
pulsed pump lasers from PriTel. For the initial two photon interference
experiments we use a PriTel wavelength tuneable laser with a repetition rate
of $50\text{\,}\mathrm{MHz}$ and for the boson sampling experiments we use a
PriTel ultrafast optical clock with a repetition rate of
$500\text{\,}\mathrm{MHz}$. Both emit pulses with a $2\text{\,}\mathrm{nm}$
bandwidth, however we use a tuneable filter to reduce the bandwidth to
$100\text{\,}\mathrm{pm}$ before amplification by a PriTel erbium doped fibre
amplifier (EDFA). A DWDM is then used to filter any potential unwanted side-
band generation from the amplification. The CW lasers are combined using a
multichannel DWDM filter before being combined with the pump laser on a 50/50
beam-splitter. Polarisation controllers on both CW and pump arms are used to
maximise fibre to chip coupling. The on-chip component uses ring resonator
photon sources (measured parameters given in table 1) to generate two-mode
squeezed vacuum states which are split on chip using AMZI filters which have
free spectral ranges (FSR) of $20\text{\,}\mathrm{nm}$.

Ring | FSR (nm) |  $\Delta\lambda$ (pm) | ER (dB)
---|---|---|---
R1 | 2.27$\pm$0.01 | 29.14$\pm$1.13 | 3.45$\pm$0.02
R2 | 2.27$\pm$0.01 | 29.12$\pm$1.09 | 3.73$\pm$0.02
R3 | 2.27$\pm$0.01 | 32.88$\pm$1.05 | 7.46$\pm$0.03
R4 | 2.27$\pm$0.01 | 34.45$\pm$1.04 | 12.14$\pm$0.05
Table 1: Table showing the measured ring parameters for the 4 sources employed
in this work. FSR is the free spectral range of the resonator. $\Delta\lambda$
is the resonance line-width and ER is the extinction ratio.

Signal photons ($1536.36\text{\,}\mathrm{nm}$, ITU channel 51) are coupled
directly off-chip using grating couplers. The idler photons
($1546.1\text{\,}\mathrm{nm}$, ITU channel 39) enter an integrated
interferometer before being coupled off-chip. DWDM filters are used to reject
the pump and send the single photons to PhotonSpot superconducting nanowire
single photon detectors (SNSPD). Powermeters are used on the output of a 99/1
beamsplitter and each herald output to monitor the input pump power and align
the ring sources, respectively. Coincidence logic is handled by a Swabian
Ultra, used to record coincidences directly as well as saving raw time tags.
To characterise the on-chip thermo-optic phase shifters we first measure the
current-voltage curve of the heater, fitting with a second order polynomial to
account for non-Ohmic effects. This allows us to accurately map a particular
voltage to the dissipated power. To map the power to a phase, we sweep an
interference fringe on the desired heater. The fit of this fringe, along with
the IV fit, allows us to convert the voltage set into a phase. To determine
the system jitter of the experiment we can directly measure the Swabian jitter
using the low-jitter internal test signal. Using this signal on two channels
allows us to measure a coincidence histogram with the FWHM dominated by the
combined electronic jitter of both channels. Assuming, that both channels have
the same jitter, we find that the the single channel FWHM is 15.736 $\pm$
0.002 ps. To determine the combined jitter of detectors and time-tagging
channels, we measure a coincidence histogram between photons generated through
SFWM in a spiral waveguide, with a CW pump. The detector jitter is much larger
than the correlation time between the generated photons and we can therefore
take the coincidence histogram to be the convolution of the detector and
Swabian channels. Modelling each of these jitters as a Gaussian, and assuming
that the detectors each have the same jitter, we find the single detector
channel FWHM jitter to be 75.4 $\pm$ 1.6 ps.

Figure S2: Effect of optimisation protocol for ring alignment. Alignment
wavelength set to $1541.25\text{\,}\mathrm{nm}$. (a) Ring resonances without
optimisation. Here each ring is set separately by setting the voltage that
minimises the transmitted power. Standard deviation of ring positions is
$10.5\text{\,}\mathrm{pm}$ (b) Rings aligned with Nelder-Mead optimisation
algorithm. Starting voltages selected from Gaussian distributions centred at
the voltages used in (a). Standard deviation of ring positions is
$1.6\text{\,}\mathrm{pm}$.

###  VIII.4 Aligning ring resonances

The ability to tune the resonance position thermally also means that thermal
fluctuations in the lab can cause the ring resonances to drift. To combat this
the chip rig is attached to a Peltier and a PID temperature controller to keep
the temperature of the chip stable. We also enclose the chip rig in a box in
order to minimise changes in air temperature from affecting the chip. To
combat cross-talk between thermo-optic phase shifters we use an optimisation
loop to align the ring resonances to the desired wavelength Carolan  _et al._
(2019). By fixing the CW laser at the frequency we wish to align to, we can
define a cost function to be the sum of all powers transmitted by the rings we
wish to align. Using the ring heater voltages as the system parameters an
optimisation loop based on the Nelder-Mead algorithm is used to minimise the
cost function. The starting voltages were sampled from a Gaussian distribution
centred at the voltage found by minimising the transmitted power for each ring
individually. The standard deviation of the distribution was chosen to be wide
enough to allow restarts to escape any local minima but narrow enough to make
sure the algorithm would converge. In Fig. S2, here we compare the situation
where we align each ring to the CW laser independently, by simply choosing the
voltage value that minimises the light transmitted through each ring
individually, and the case where we use the alignment mechanism described
above.

###  VIII.5 Data taking and background subtraction

Figure S3: Use of virtual channels in order to measure background and
distinguishable events. $t$ represents the original channel - i.e. photons
generated in one pulse. $t+\Delta t$ represents a copy of a given channel but
with all time tags delayed by $\Delta t$, when $\Delta t$ is set to the
repetition rate of the laser, we now compare photons generated in the
subsequent pulse. (a) All photons generated in the same pulse. These
correspond to the interference event we wish to observe. (b) Double emissions.
One herald delayed, three photons generated in one pulse - i.e. a double
emission from the source whose herald isn’t delayed. (c) One pair of photons
delayed. Corresponds to interfering photons being generated in subsequent
pulses and therefore are distinguishable in time. (d) Chip configuration for
heralded HOM interference with indistinguishable photons. Fringe acquired by
sweeping angle $\theta$ and measuring four-fold coincidences (two herald and
two signal) (e) Measured fringe with background (green) and without background
(purple) including Poissonian error bars. Shaded area shows the $1\sigma$
interval from the errors on the fitting parameters. Visibility of green curve
with no background subtraction $0.608\pm 0.007$. With background subtraction
this increases to $0.828\pm 0.019$. Inset shows alignment of ring resonators
relative to the pump pulse.

We use the Swabian time tagger to save the time tags of all coincidence events
which we can then post process as desired. We also employ the ability of the
Swabian time tagger to create copies of channels at a hardware level. These
virtual channels can be delayed relative to the original. If the channels are
delayed by a multiple of the repetition rate of the laser we can measure
coincidences from photons generated in different pulses. By changing the
pattern of virtual channels we use we can access the double emissions from
each ring and also the events with no interference. Figure S3 illustrates the
different patterns which are relevant for two photon interference. Figure S3a
shows the case where all four detected photons are created in the same pulse,
the main interference signal used in this work. The double emissions of each
ring are found by shifting one herald channel, as shown in Fig. S3b. This can
be generalised to more than one source by sequentially delaying each herald
channel. The total of all these events consists of $N-1$ times the double
emission rate. We also use the case where interfering photons are generated in
separate pulses and are therefore distinguishable in time and do not
interfere. This is shown in Fig. S3c.

###  VIII.6 Time-resolved fringes

The linear spectral response of a ring resonator should be a function
$l(\omega,\omega_{0},\delta\omega)$, such that
$|l(\omega,\omega_{0},\delta\omega)|^{2}$ gives us a Lorentzian function
peaked at $\omega_{0}$ with a FWHM $\delta\omega$. From Ref Vernon  _et al._
(2017) this is given by

$l(\omega,\omega_{0},\delta\omega)=\sqrt{\frac{\delta\omega}{2\pi}}\times\frac{1}{-i\left(\omega-\omega_{0}\right)+\frac{\delta\omega}{2}}.$
(A16)

The temporal profile, given by the Fourier transform of the spectrum, is a
decaying exponential with a decay constant that is inversely proportional to
resonance width, $\delta\omega$.

Figure S4: Two-photon interference simulations (a) Spectral profile of a ring
resonance for different spectral widths. (b) Corresponding temporal profiles
for (a). Time-resolved fringes as a function of temporal delay between photons
for different detunings of the photon’s central frequency. Fringes beat at the
frequency of the detuning. (d) Total temporal profile acquired by incoherently
summing the pure Schmidt modes. Black line shows temporal profile of a pure
state. (e) Time-resolved fringes for a detuning of $12.6\text{\,}\mathrm{GHz}$
for different photon purities. (f) Modulating envelope for different photon
purities.

To simulate the mixed photons arising from resonant sources, we calculate the
joint spectral amplitude as

$F(\omega_{s},\omega_{i})=\int
d\omega_{p}\alpha(\omega_{p})\alpha(\omega_{s}+\omega_{i}-\omega_{p})l(\omega_{p})l(\omega_{s}+\omega_{i}-\omega_{p})\phi(\omega_{p},\omega_{s},\omega_{i})l(\omega_{s})l(\omega_{i}),$
(A17)

where $\alpha(\omega)$ is the spectral amplitude of the pump and the
phasematching condition $\phi(\omega_{p},\omega_{s},\omega_{i})\approx 1$
$\forall\omega_{p},\omega_{s},\omega_{i}$. The Fourier transform of this
function gives the joint temporal amplitude (JTA) $T(t_{1},t_{2})$. The
temporal Schmidt modes can then be calculated via a singular value
decomposition of the JTA.

As we wish to find fringes in the relative delay between detected photons we
express the array of detection times in Eq. A15 as
$\vec{t}=\\{t_{0},t_{0}+\tau_{1},\cdots,t_{0}+\tau_{N-1}\\}$. Integrating over
$t_{0}$ results in interference fringes that depend on $N-1$ relative delays.
We illustrate these fringes for two photons on a balanced beam-splitter in
Fig. S4c for different detunings of the photons central frequency. We see
sinusoidal fringes, which beat at the frequency detuning of the photons. These
fringes are modulated by an envelope function which for ring resonator photons
is a Lorentzian shape. In Fig. S4d - e we show the effect of reducing the
purity of the input photons. We tune the photon purity by changing the width
of the pump function. A narrower pump pulse results in lower purity photons.

###  VIII.7 Future projections

To analyse the scalability of this technique we use our model to determine HOM
fringe visibilities as a function of photon detuning for a variety of likely
system jitters. We fix the both photon line-widths to be
$30\text{\,}\mathrm{pm}$. Total system jitter results from the convolution of
two detector and two time tagger channels. Assuming Gaussian responses, and
that both detector and time tagger channels have the same jitter, the total
FWHM is given by

$\mathrm{FWHM}_{\mathrm{tot}}=\sqrt{2\mathrm{FWHM}_{\mathrm{det}}+2\mathrm{FWHM}_{\mathrm{tag}}}.$
(A18)

We examine three jitter regimes. First we use detector and time tagger jitter
used in this experiment, corresponding to a total jitter FWHM of
$109\text{\,}\mathrm{ps}$. Secondly, we use the best commercially available
numbers. Detector jitter taken to be $15\text{\,}\mathrm{ps}$ Single Quantum
(2019) and time tagger jitter to be $4.71\text{\,}\mathrm{ps}$ Swabian
Instruments (2022). Finally we use numbers corresponding to the state of the
art, $3\text{\,}\mathrm{ps}$ detector jitter Korzh  _et al._ (2020) and assume
this is the only source of jitter.

Figure S5: Projected HOM fringe visibilities as a function of photon detuning.
We consider three different system jitters. The same as measured in this
experiment, corresponding to a total system jitter of
$109\text{\,}\mathrm{ps}$, the best available commercial equipment
corresponding to a total system jitter of $22.2\text{\,}\mathrm{ps}$ and the
state of the art SNSPD jitter, corresponding to a total system jitter of
$4.2\text{\,}\mathrm{ps}$

###  VIII.8 Boson sampling with indistinguishable photons

To verify that our experimental setup is sufficient for boson sampling we
perform two-photon scattershot and three-photon standard boson sampling with
indistinguishable photons. We avoid three-photon scattershot as the loss in
our system means that photons generated from the fourth source would become
prevalent and would add noise to the data. For two photons, the four fold rate
is high enough that we can reduce the squeezing in the sources and this noise
process is less dominant.

Figure S6: Boson sampling characterisation. (a) - (e) show the results of
background subtraction and loss normalisation. (a) Raw measured counts. (b)
Background subtracted counts. (c) Background subtraction and loss
normalisation. (d) Simulated distribution for test model of indistinguishable
photons plus mixedness. (e) Simulated distribution for adversarial model of
non-interfering photons. (f) Bayesian verification for two-photon scattershot
boson sampling. (g) Simulated distribution for three photon test model. (h)
Simulated distribution for three photon adversarial model. (i) Bayesian
verification for three-photon boson sampling.

For both cases we verify against a test model that is indistinguishable
interfering photons. To combat the 0 probability events corresponding to
suppressed patterns in the Fourier interferometer, which however will occur
due to experimental imperfections, we include a noninterfering term. The test
model therefore becomes
$P_{test}=r|\mathrm{Perm}\left(T_{s}\right)|^{2}+(1-r)\mathrm{Perm}\left(|T_{s}|^{2}\right)$,
where $r$ parameterises the level of interference. We select $r$ to correspond
to a pairwise overlap of 0.8; this value is also used in the main text boson
sampling experiments. We use background subtraction techniques as demonstrated
above and also normalise by the Klyshko efficiency of each output and the
squeezing of the involved sources. For the two-photon interference case, the
interferometer is set to implement a $F_{4}$ matrix with $\phi=\pi/2$. We note
here that due to crosstalk between heaters, there is an offset to the value of
$\phi$, leading to $\phi=$$1.73\text{\,}\mathrm{rad}$. Operating in a
scattershot configuration, we pump all four ring resonator sources and use two
heralding photons to inform the input pattern and two signal photons to inform
the output pattern. The probability is then calculated from the corresponding
submatrix of $F_{4}$. Figure S6(d) shows the input/output probability
distribution for this matrix with indistinguishable photons including
mixedness which operates as our test model. Figure S6(e) shows the
corresponding distribution for our adversarial model of noninterfering
photons. In Fig. S6(a) we show the raw measured counts. Figures. S6(b) and (c)
show the effects of background subtraction and both background subtraction and
Klyshko rescaling, respectively. Figure. S6(f) shows dynamic Bayesian
verification demonstrating that we are likely sampling from our test
distribution. For the three-photon case the distribution of $F_{4}$ for both
indistinguishable and distinguishable photons is uniform. To break this
degeneracy, we tune the input MZIs. We choose the value empirically to be a
value that minimises the indistinguishable/distinguishable distribution
overlap but maximises the non-bunching probabilities to increase the overall
6-fold rate. Figures S6(g) and (h) show the distributions with both input MZI
phases set to 0.9 for the indistinguishable and distinguishable cases,
respectively. Figure S6 shows the Bayesian verification again showing we are
sampling from the desired distribution.

###  VIII.9 Errors

In this section we state some relevant properties with their corresponding
errors.

Experiment | R1 location (nm) | R2 location (nm) | R3 location (nm) | R3 location (nm) |  $\Delta$(pm) |  $\Delta$ (GHz)
---|---|---|---|---|---|---
HOM | 1541.261$\pm$0.005 | 1541.315$\pm$0.002 | N/A | N/A | 54.2 $\pm$ 5.2 | 6.8 $\pm$ 0.7
Fusion | 1541.263$\pm$0.005 | N/A | 1541.316$\pm$0.008 | N/A | 53.5 $\pm$ 9.5 | 6.8 $\pm$ 1.2
2-photon scattershot | 1541.26$\pm$0.015 | 1541.316$\pm$0.005 | 1541.312$\pm$0.005 | 1541.259 $\pm$ 0.03 | (R1,R2) 54.8 $\pm$ 16.6 | 6.9 $\pm$ 2.1
|  |  |  |  | (R1,R3) 50.5 $\pm$ 16.5 | 6.4 $\pm$ 2.1
|  |  |  |  | (R1,R4) 2.1 $\pm$ 16.3 | 0.3 $\pm$ 2.1
|  |  |  |  | (R2,R3) 4.1 $\pm$ 6.6 | 0.5 $\pm$ 0.8
|  |  |  |  | (R2,R4) 57.1 $\pm$ 5.7 | 7.2 $\pm$ 0.7
|  |  |  |  | (R3,R4) 52.9 $\pm$ 5.9 | 6.7 $\pm$ 0.7
3-photon boson sampling | 1541.229$\pm$0.008 | 1541.277$\pm$0.006 | N/A | 1541.23$\pm$0.01 | (R1,R2) 47.8 $\pm$ 10 | 6.04 $\pm$ 1.26
|  |  |  |  | (R1,R4) 1.15 $\pm$ 14 | 0.1 $\pm$ 1.7
|  |  |  |  | (R2,R4) 46.7 $\pm$ 12 | 5.9 $\pm$ 1.5
Table 2: Table showing the measured ring positions and corresponding
detunings, $\Delta$, for the experiments in the main text

In table 2 we show the position of the ring resonators and detunings for each
of the experiments in the main text. We measure the position of the ring
resonator pump resonances by scanning the CW across the resonance and
measuring the transmitted power. This is performed every two hours throughout
each experiment and the values stated in table 2 are the average and standard
deviation of these positions. The average detuning and corresponding error are
determined with Monte-Carlo sampling.

Table 3 shows the visibilities for different timing windows for both the HOM
and fusion experiments. These errors are determined from the error on a
sinusoidal fit of the fringe, including Poissonian error bars. For the fusion
gate we also include the corresponding gate fidelity with accompanying errors
calculated using Monte-Carlo sampling.

Experiment | $\delta\tau=200$ | $\delta\tau=100$ | $\delta\tau=50$ | $\delta\tau=20$
---|---|---|---|---
HOM | 0.49 $\pm$ 0.01 | 0.53 $\pm$ 0.01 | 0.58 $\pm$ 0.01 | 0.61 $\pm$ 0.01
Fusion | 0.40 $\pm$ 0.03 (0.79 $\pm$ 0.02) | 0.46 $\pm$ 0.02(0.81 $\pm$ 0.01) | 0.56 $\pm$ 0.01 (0.859 $\pm$ 0.004) | 0.60 $\pm$ 0.01 (0.875 $\pm$ 0.004)
Table 3: Table showing the visibilities for different timing windows for both
HOM and fusion experiments. For the fusion gate, the corresponding gate
fidelity is shown in parentheses