# Characterization of Quantum Frequency Processors

Hsuan-Hao Lu<EMAIL_ADDRESS>Quantum Information Science Section, Oak Ridge
National Laboratory, Oak Ridge, Tennessee 37831  Nicholas A. Peters  Quantum
Information Science Section, Oak Ridge National Laboratory, Oak Ridge,
Tennessee 37831  Andrew M. Weiner  School of Electrical and Computer
Engineering and Purdue Quantum Science and Engineering Institute, Purdue
University, West Lafayette, Indiana 47907, USA  Joseph M. Lukens
<EMAIL_ADDRESS>Quantum Information Science Section, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831  Research Technology Office and Quantum
Collaborative, Arizona State University, Tempe, Arizona 85287, USA

###### Abstract

Frequency-bin qubits possess unique synergies with wavelength-multiplexed
lightwave communications, suggesting valuable opportunities for quantum
networking with the existing fiber-optic infrastructure. Although the coherent
manipulation of frequency-bin states requires highly controllable multi-
spectral-mode interference, the quantum frequency processor (QFP) provides a
scalable path for gate synthesis leveraging standard telecom components. Here
we summarize the state of the art in experimental QFP characterization.
Distinguishing between physically motivated “open box” approaches that treat
the QFP as a multiport interferometer, and “black box” approaches that view
the QFP as a general quantum operation, we highlight the assumptions and
results of multiple techniques, including quantum process tomography of a
tunable beamsplitter—to our knowledge the first full process tomography of any
frequency-bin operation. Our findings should inform future characterization
efforts as the QFP increasingly moves beyond proof-of-principle tabletop
demonstrations toward integrated devices and deployed quantum networking
experiments.

##  I Introduction

In modern lightwave communications, the optical and electronic domains work in
concert for the distribution and processing of information Agrell2016 . Light
fields carry data streams with dense multiplexing down low-loss optical fiber,
while electronic circuits perform digital processing for logical computations,
error correction, and packet switching. The inherently ultrawide bandwidth
available to photonic systems—along with the potential for greater speed and
efficiency by avoiding optical-to-electrical conversion—has inspired a
plethora of research into all-optical processing approaches challenging this
status quo and aiming to supplant electronic digital logic with all-optical
interactions Willner2014 . And in the case of photonic _quantum_ information,
all-optical processing is more than just a pathway for improved performance:
instead, it is a prerequisite, since conversion to the digital domain
represents a measurement that irreversibly collapses the quantum state.

Remarkably, Knill, Laflamme, and Milburn (KLM) showed in 2001 that the
photonic interactions necessary for universal quantum information processing
can be realized with multiphoton interference in linear-optical circuits,
combined with detection and feed-forward Knill2001 . Although experimentally
challenging, the KLM scheme is in principle scalable and has stimulated a
large body of research on linear-optical quantum computing Kok2007 , including
important developments in measurement-based computing approaches—both
discrete- Nielsen2004  and continuous-variable Menicucci2006 —that appear
promising for a large-scale photonic quantum computer Rudolph2017 . Beyond
computing alone, KLM concepts form the foundation for all-photonic quantum
repeater designs as well, which could eliminate the need for quantum memories
and two-way communications Azuma2015 ; Munro2015 ; Pant2017 . Although
originally proposed for spatio-polarization encodings, KLM approaches have
been extended into a variety of nontraditional degrees of freedom, including
time bins Humphreys2013  and pulsed modes Brecht2015 .

Mandatory to any KLM-like scheme is the ability to construct arbitrary unitary
transformations. In the case of frequency-bin encoding—intriguing for its
compatibility with wavelength-multiplexing in optical fiber, on-chip photon
sources, and frequency-disparate quantum interconnects—it was unclear whether
arbitrary unitaries could be synthesized with standard fiber-optic components,
until the quantum frequency processor (QFP) was introduced and analyzed in
2017 Lukens2017 . Based on an alternating series of electro-optic phase
modulators (EOMs) and Fourier-transform pulse shapers, the QFP realizes
frequency-bin transformations through what can conceptually be viewed as
successive phase-only filters in the time and frequency domains. In Lukens2017
, explicit designs for a universal gate set (single-qubit phase and Hadamard
$H$, and two-qubit controlled-$Z$) were discovered, supported further by an
argument for linear scaling of resources (EOMs and pulse shapers) with the
number of modes. In the intervening years, many experiments have been
performed demonstrating basic QFP gates, on both single- Lu2018a ; Lu2020b ;
Lu2022b  and two-photon states Lu2018b ; Lu2019a ; Lingaraju2022 . As with any
experimental demonstration, it has been critical to confirm the degree of
agreement between the realized QFP operation and that expected from theory,
for which a variety of probes have been enlisted, including bright frequency
combs, weak coherent states, and entangled photons.

In this article, we present an introduction and overview of the techniques
currently available for QFP characterization, complementing previous high-
level reviews Kues2019 ; Lu2019c  with a much more thorough focus on the
underlying assumptions and information provided by various characterization
approaches. Following background on the theory and implementation in Sec. II,
Secs. III and IV summarize QFP characterization experiments organized in terms
of successively fewer assumptions, including an entirely new experiment
performing Bayesian quantum process tomography (QPT) of the tunable
beamsplitter—the first QPT characterization of a frequency-bin gate.
Throughout, we find it useful to frame the discussion around “open box” and
“black box” techniques, distinguished by their treatment of the QFP’s inner
workings. In addition to furnishing an organized introduction to this
important subfield of time-frequency quantum information, our QPT results open
the door for impartial characterization of emerging QFP designs—e.g., photonic
integrated circuits—comprised of EOMs and pulse shapers whose properties may
deviate strongly from ideal models previously assumed for discrete devices.

##  II QFP Background

Figure 1: QFP characterization concepts. (a) Physically, a QFP consists of an
alternating series of EOMs and pulse shapers, here shown in a three-element
instantiation (EOM/pulse shaper/EOM) flanked by spectral multiplexers. (b) In
the open box viewpoint, the output frequency-bin operators are assumed to
follow from the input through a linear transformation $V$, which can be
characterized by measuring the output spectrum of superposition probe states.
(c) In the black box viewpoint, the QFP is treated as a generic quantum
channel to be analyzed by quantum state or process tomography, returning
output state $\rho$ or process matrix $\chi$, respectively. The examples in
(b) match the mode transformation of a Bell state analyzer Lingaraju2022 ,
while those in (c) correspond to an ideal Hadamard gate Lu2018a .

###  II.1 Theory

As noted above, the distinction between open box and black box
characterization provides a useful framework through which to view previous
and current QFP experiments. The notion of a black box in engineering contexts
is simple to define: a device to which one has access only to the inputs and
outputs, making the system opaque from the perspective of the outside world.
The opposite of such a system has been dubbed, e.g., an “open box” or “white
box” in the literature, defined as device for which at least some of the
internals are known to the tester; in computer science, both open Ostrand2002a
and black box Ostrand2002b  software testing are common. Yet while black box
concepts have a long history in quantum information processing—indeed the
original proposal for QPT used the term explicitly Chuang1997 —open box
terminology has not taken hold in this field, at least in our experience.
Nevertheless, for the QFP the distinction seems particularly apt, given the
gradual increase in complexity of characterization approaches which have been
demonstrated over the past few years. As will be defined mathematically below,
open box QFP approaches are distinguished by assuming the QFP is a true
multiport frequency-bin interferometer, per its design; black box QFP
approaches make no assumptions about the QFP’s insides beyond the minimum for
an arbitrary quantum channel.

Fundamentally, the goal of the QFP is to controllably transform quantum states
of light populating a comb of discrete frequency modes, or bins. Consider $M$
bins centered at frequencies $\omega_{m}=\omega_{0}+m\Omega$
($m\in\\{0,1,...,M-1\\}$), with $\Omega$ a fixed spacing typically in the
radio-frequency (RF) domain. Each of these bins possesses an annihilation
(creation) operator $a_{m}$ ($a_{m}^{\dagger}$) that subtracts (adds) single
photons to mode $m$ when applied to any quantum state. For notational
convenience, we utilize $b_{m}$ ($b_{m}^{\dagger}$) to denote the same
operations but referenced to the output side of the quantum transformation of
interest, so that the input (output) bins are associated with
$a_{m}/a_{m}^{\dagger}$ ($b_{m}/b_{m}^{\dagger}$).

The spectral shape of these bins can in principle be of any form, provided
they are bandlimited to $\omega_{m}\pm\frac{\Omega}{2}$. For the QFP to
function as designed, these bins should possess identical lineshapes and be
easily distinguished within the resolution of the pulse shapers used for line-
by-line phase control Lu2020a . We do note that in the general quantum black
box formalism, these additional assumptions are not strictly required for the
QFP, but only for the system used to measure these bins at the output.
Although typically based on the same technology as the QFP, the measurement
system need not have the same spectral resolution as the QFP under test.
Indeed, emerging QFPs based on integrated photonic circuits Moody2022 ;
SiPhQFP  will likely first be tested with well-characterized discrete fiber-
optic components, so the distinction between QFP and measurement spectral
resolutions is practically as well as theoretically significant.

Each Fock basis state—i.e., a quantum state with a specified number of photons
in each mode—can be described for the input Hilbert space by a length-$M$
vector of photon occupancy numbers
$\mathbf{n}^{\prime}=(n_{0}^{\prime},...,n_{M-1}^{\prime})$, defined as

$\ket{\mathbf{n}^{\prime}}=\prod_{k=0}^{M-1}\frac{(a_{k}^{\dagger})^{n_{k}^{\prime}}}{\sqrt{n_{k}^{\prime}!}}\ket{\mathrm{vac}},$
(1)

where $\ket{\mathrm{vac}}$ denotes the vacuum state over all $M$ modes.
Similarly, the output is spanned by basis states
$\mathbf{n}=(n_{0},...,n_{M-1})$:

$\ket{\mathbf{n}}=\prod_{k=0}^{M-1}\frac{(b_{k}^{\dagger})^{n_{k}}}{\sqrt{n_{k}!}}\ket{\mathrm{vac}}.$
(2)

(As a means to aid in distinguishing between input and output, we use primed
vectors $\textbf{n}^{\prime}$ for the former and n for the latter.)

The QFP transforms between these spaces via an alternating series of (i) pulse
shapers, which apply the phase filters $b_{n}=e^{i\phi_{n}}a_{n}$, and (ii)
EOMs, which apply a temporal phase $\varphi(t)$ periodic at frequency
$\Omega$, leading to the discrete convolution
$b_{n}=\sum_{n^{\prime}}c_{n-n^{\prime}}a_{n^{\prime}}$ where
$e^{i\varphi(t)}=\sum_{k}c_{k}e^{-ik\Omega t}$. Figure 1 offers a schematic of
the physical hardware for a three-element QFP, with input and output
multiplexers to facilitate characterization; three-element QFPs have been the
focus of experiments so far, although deeper QFP circuits remain of interest
for future research. The total effect of an ideal QFP with any number of
elements can be summarized as a multiport frequency-bin interferometer:

$b_{m}=\sum_{m^{\prime}=0}^{M-1}V_{mm^{\prime}}a_{m^{\prime}}.$ (3)

It is important to note that the Fourier series expansion of the phase
modulation in general is unbounded in frequency; thus, even with the input
modes restricted to a finite number $M$, a countably infinite number of output
bins can be populated within and after the interferometer. Indeed, it is over
this infinite set that $V$ is unitary:
$\sum_{k=-\infty}^{\infty}V_{km}^{*}V_{km^{\prime}}=\sum_{k=-\infty}^{\infty}V_{mk}V_{m^{\prime}k}^{*}=\delta_{mm^{\prime}}$.
Although the intrinsically infinite-dimensional space produces interesting
challenges conceptually, it can be accurately approximated by truncating to a
value of $M$ sufficiently large to encompass all nonnegligible probability
amplitudes excited by the input states of interest Lukens2017 . In practice,
the sufficiency of any truncation $M$ can be validated ex post facto by
ensuring the calculated probability amplitudes decay to zero (to within some
desired precision) at the edges of the domain following each EOM in the QFP
circuit.

Because the QFP operates on frequency modes irrespective of the photons
populating them, it can be applied to a variety of information processing
scenarios, including classical lightwave communications Lukens2020a ; Lu2020a
and continuous-variable quantum frequency combs Pizzimenti2021 , which have
emerged as an exciting platform for quantum computing Roslund2014 ;
Pfister2020 . Nonetheless, we focus on discrete-variable encoding in this
paper, the dominant paradigm in QFP characterization experiments so far. In
this case, each logical basis state is associated with a specific Fock state;
under dual-rail encoding, for example, a qubit is encoded by a single photon
populating a superposition of two bins, one defined as $\ket{0}$ and the other
as $\ket{1}$. Experimentally, we always consider the postselected regime,
where all photons of the input are successfully passed through the system.
Therefore any insertion loss (absent in the ideal model above) leads to a
lower throughput without inherently degrading state fidelity.

In the Heisenberg picture, a fixed quantum state $\ket{\psi}$ with $N$ total
photons can be represented in either the input or output basis:
$\ket{\psi}=\sum_{\mathbf{n}}\beta_{\mathbf{n}}\ket{\mathbf{n}}=\sum_{\mathbf{n^{\prime}}}\alpha_{\mathbf{n^{\prime}}}\ket{\mathbf{n^{\prime}}}$,
where each sum considers only terms with
$\sum_{m=0}^{M-1}n_{m}=\sum_{m^{\prime}=0}^{M-1}n_{m}^{\prime}=N$. Solving for
$\beta_{\mathbf{n}}$ using Eqs. (1–3) then leads to the simple vector
expression
$\beta_{\mathbf{n}}=\sum_{\mathbf{n}^{\prime}}W_{\mathbf{n}\mathbf{n}^{\prime}}\alpha_{\mathbf{n}^{\prime}}$,
where

$W_{\mathbf{n}\mathbf{n}^{\prime}}=\frac{1}{\sqrt{\prod_{p=0}^{M-1}n_{p}!n_{p}^{\prime}!}}\sum_{\mathbf{k}^{\prime}\in\mathrm{perms}(\mathbf{m}^{\prime})}V_{m_{1}k_{1}^{\prime}}\cdots
V_{m_{N}k_{N}^{\prime}}.$ (4)

This equation corresponds to the famous “permanents” expression for $N$ bosons
traversing an interferometer Skaar2004 ; Uskov2009 . With each Fock vector
$\mathbf{n}$ [$\mathbf{n^{\prime}}$] we define a length-$N$ mode assignment
vector $\mathbf{m}=(m_{1},...,m_{N})$
[$\mathbf{m}^{\prime}=(m_{1}^{\prime},...,m_{N}^{\prime})$] that lists the
modes occupied by each photon; the summation proceeds over all $N!$
permutations of $\mathbf{m}^{\prime}$. Although this leads to repeated terms
when multiple photons populate the same mode, we have found it useful
numerically, as it treats mode vectors with repeated indices the same as those
without. (Removing repeated terms simply changes the factorial prefactor, as
in the formulation of Skaar2004 .)

The matrix $W$ completely describes the QFP operation, and can be compared to
some target $D\times D^{\prime}$-dimensional matrix $T$ with elements
$T_{\mathbf{n}\mathbf{n}^{\prime}}$, where $D$ ($D^{\prime}$) denotes the
dimension of the output (input) subspace of interest. Both $D,D^{\prime}\leq
D_{\mathrm{max}}$, where $D_{\mathrm{max}}=\begin{pmatrix}N+M-1\\\
M-1\end{pmatrix}$ denotes the Hilbert space dimension of $N$ photons
populating $M$ modes; $D$ and $D^{\prime}$ in practice are often much smaller
than $D_{\mathrm{max}}$ since they can be limited to only the states of
interest in a given problem. $W$ computed for the same set of states can then
be compared to $T$ via matrix fidelity

$\mathcal{F}_{W}=\frac{1}{\mathcal{P}_{W}}\left|\frac{\mathrm{Tr}~{}W^{\dagger}T}{\mathrm{Tr}~{}T^{\dagger}T}\right|^{2},$
(5)

describing the closeness of the operation to the ideal, and success

$\mathcal{P}_{W}=\frac{\mathrm{Tr}~{}W^{\dagger}W}{\mathrm{Tr}~{}T^{\dagger}T},$
(6)

providing the probability that all input photons exit in the bins of interest
Uskov2009 .

At this stage, it is important to observe that the Fock state transformation
matrix $W$—and hence $\mathcal{F}_{W}$ and $\mathcal{P}_{W}$—are completely
determined by the mode transformation matrix $V$ [Eq. (4)]. This assumption
underpins the “open box” approach to QFP characterization, as summarized in
Fig. 1(b); if Eq. (3) holds, one can probe the QFP fully by sending in two-bin
superposition states and measuring the output probabilities. A powerful
technique first introduced in the spatial domain Rahimi2013 , the amplitudes
$|V_{mm^{\prime}}|$ can be recovered from single-bin tests, while
$\arg(V_{mm^{\prime}})$ can be obtained by scanning the relative phase between
two-bin superpositions. Significantly, even classically bright coherent states
can be used, eliminating the need for single-photon detection altogether
Peruzzo2011 ; Spagnolo2013 . This method has been shown to be highly effective
in characterizing linear-optical networks, including large-scale boson
sampling circuits Wang2019 ; Zhong2020 .

This open box viewpoint was adopted to characterize the first experimental QFP
Lu2018a  and has been widely deployed since, with remarkably good agreement
with theory: experimental fidelities up to $\mathcal{F}_{W}=1-10^{-6}$ have
been obtained for the frequency-bin Hadamard Lu2019b . Coupled with our
understanding of the physical processes involved, there is little reason to
question the validity of Eq. (3) in most situations. Nevertheless, the theory
of quantum processes permits much more general models, allowing for QFP
characterization in which its inner mechanisms are completely arbitrary.

Considering any $D^{\prime}\times D^{\prime}$-dimensional mixed input
$\rho^{\prime}$, the $D\times D$ output $\rho$ follows as Kukulski2021

$\rho=\mathcal{E}(\rho^{\prime})=\sum_{k=1}^{DD^{\prime}}A_{k}\rho^{\prime}A_{k}^{\dagger},$
(7)

where for trace preservation the $D\times D^{\prime}$ Kraus operators resolve
the identity $\sum_{k}A_{k}^{\dagger}A_{k}=\mathbbm{1}_{D^{\prime}\times
D^{\prime}}$—a condition following from postselecting $N$ photons at the
output. The open box picture represents the special case of this black box
model when $A_{1}\propto W$ and $A_{k>1}=0$. By preparing known inputs
$\rho^{\prime}$ and performing quantum state tomography (QST) James2001  of
the output $\rho$, insight into the operation $\mathcal{E}(\cdot)$ can be
obtained without any reference to a multiport interferometer. For a single
output $\rho$, agreement with theoretical expectations can be quantified
through the state fidelity

$\mathcal{F}_{\rho}=\left(\mathrm{Tr}~{}\sqrt{\sqrt{\rho_{T}}\rho\sqrt{\rho_{T}}}\right)^{2},$
(8)

where $\rho_{T}\propto T\rho^{\prime}T^{\dagger}$ is the output corresponding
to the ideal target operation. For the entire channel, the process fidelity

$\mathcal{F}_{\Phi}=\left(\mathrm{Tr}~{}\sqrt{\sqrt{\Phi_{T}}\Phi\sqrt{\Phi_{T}}}\right)^{2}$
(9)

serves this purpose, where the $DD^{\prime}\times DD^{\prime}$ density matrix
$\Phi$ ($\Phi_{T}$) is obtained by the Choi–Jamiołkowski isomorphism and
completely specifies the measured (target) channel Gilchrist2005 .

Unlike the open box case, we do not define a success probability
$\mathcal{P}$. For whereas the open box mapping can distinguish between output
states in the targeted subspace (successes) and those outside of this space
(failures), the black box model is intentionally oblivious to all subspaces
apart from the $D$-dimensional output of interest. Considering non-trace-
preserving operations could enable a meaningful success probability
definition, but due to the experimental difficulty of distinguishing between
failure and insertion loss —at least without introducing modes outside of the
quantum channel’s purview—we concentrate on fidelity for the black box
examples below.

Figure 1(c) highlights these concepts for a black box frequency-bin Hadamard.
One state from a set of inputs is sent into the system and measured; in the
example shown, the zero logical input $\rho^{\prime}=\ket{0}\bra{0}$ produces
the superposition output $\rho=\ket{+}\bra{+}$. With many such input/output
tests, the full process can be reconstructed; shown is the ideal Hadamard
process matrix $\chi$ expressed in the Pauli basis
$(\sigma_{0},\sigma_{1},\sigma_{2},\sigma_{3})=(I,X,Y,Z)$; i.e.,
$\mathcal{E}(\rho^{\prime})=\sum_{j=0}^{3}\sum_{k=0}^{3}\chi_{jk}\sigma_{j}\rho^{\prime}\sigma_{k}$.

To conclude this subsection, then, a QFP used to transform Fock states in
frequency-bin encoding can be characterized according to either (i) an open
box viewpoint in which the system is modeled as a multiport interferometer
[Eq. (3)] or (ii) a black box viewpoint in which the QFP is treated as an
arbitrary quantum channel [Eq. (7)]. Both approaches possess advantages and
disadvantages, trading off between physical insight and generality. In the
remainder of the paper, we discuss the findings of QFP experiments applying
each technique.

Figure 2: Photograph of QFP used to demonstrate a frequency-bin tritter
Lu2018a . The setup combines a commercial pulse shaper with two EOMs, each
driven by a superposition of two microwave tones. The profusion of RF cables
and optical fibers highlights the mutual importance of electrical and optical
facets of the QFP. Image credit: Jason Richards, Oak Ridge National
Laboratory.

###  II.2 Experimental Setup

Figure 2 depicts an exemplary tabletop QFP consisting of two EOMs (EOSpace)
and one pulse shaper (Finisar Waveshaper 1000S), which was experimentally
leveraged to realize a balanced frequency tritter—a three-dimensional
extension of the frequency beamsplitter Lu2018a . In addition to the optical
components labeled in the setup, several RF components, including oscillator,
frequency doubler, power splitters, amplifiers, attenuators, phase shifters,
and cables, are employed to synthesize the drive signals for the EOMs. The
spectral resolution of the pulse shaper (typically $\gtrsim$10 GHz) determines
how tight the frequency bins can be packed, while the acceptance bandwidths of
the RF components limit the maximum frequency separation. In this example, the
mode spacing is chosen as $\Omega/2\pi=18.1$ GHz. Our frequency tritter design
is based on both the fundamental tone $\Omega$ and the second harmonic
$2\Omega$ in the EOM drive signals (specific solutions can be found in Lu2018a
; Lu2022b ). To produce the necessary second harmonic, we split a portion of
the 18.1 GHz sinewave from the oscillator and double its frequency to 36.2
GHz. We then use RF amplifiers, attenuators, and phase shifters to set the
correct amplitude and phase for each RF tone before recombining them in an RF
power coupler connected to the EOM.

Applying spectral phase modulation via the commercial pulse shaper is
comparatively simple—in the line-by-line shaping regime Cundiff2010 ;
Weiner2011 , we can program the target phase functions onto the pulse shaper
without causing any distortions in the phase mask. Since EOMs naturally couple
a single input frequency mode to many output modes, the pulse shaper must be
able to address a larger number of modes than the input. For the tritter, the
pulse shaper must shape a total of 12 frequency bins to reach
$\mathcal{F}_{W}\geq 1-10^{-4}$, a small fraction of its total 40 nm (5 THz)
bandwidth. In Lu2022b , we show numerically that the same three-element
(EOM/shaper/EOM) QFP can implement $d$-dimensional discrete Fourier transform
(DFT) gates up to $d=10$ ($d=3$ for the frequency tritter here) with near-
unity fidelity, when $d-1$ RF harmonics drive the EOMs and the pulse shaper
applies phase shifts to a total of $\sim$4$d$ channels. This approach allows
for the scalable construction of high-dimensional frequency-bin quantum
operations that will be valuable for tasks such as entanglement verification
Spengler2012 ; Coles2017  and high-dimensional quantum key distribution
Sheridan2010 ; Islam2017b .

Figure 3: Open box QFP characterization results. Each example shows a mode
matrix $V$ recovered from bright coherent state probes, followed by photon
counting results for quantum inputs. (a) Frequency-bin Hadamard and (b)
tritter. For these two gates, the Hilbert space transformation $W$ and mode
transformation $V$ are equal. (c) Coincidence-basis CNOT. Here the two-photon
transformation $W$ is calculated from the experimentally measured $V$,
revealing a conditional bit flip as designed. For all matrix plots in (a–c),
the color of each circle denotes the amplitude, and the radial line, the
phase, of the relevant element; undefined phases (due to low amplitude) are
depicted as dotted circles. Interferograms in (a,b) reprinted with permission
from Lu2018a . Images in (c) adapted from Lu2019a  under a Creative Commons
Attribution 4.0 International License.

##  III Open Box Characterization

###  III.1 Pure Multiport

Figure 3 illustrates three representative operations realized on the QFP so
far. In the first experimental example [Fig. 3(a)], the QFP was configured to
operate as a Hadamard gate, also known as a balanced frequency beamsplitter
Lu2018a . Mathematically the ideal transformation matrix can be written as

$T_{H}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix}.$ (10)

To characterize the mode matrix $V$ via the open box approach, we first
generated an electro-optic frequency comb at the QFP mode spacing of 25 GHz
and utilized an additional pulse shaper to adjust the amplitude and phase of
each frequency mode. This enabled the preparation of either single-bin or
superposition states. By probing the QFP with a single optical frequency mode
($\ket{1_{\omega_{0}}0_{\omega_{1}}}$ or
$\ket{0_{\omega_{0}}1_{\omega_{1}}}$), the modulus of every matrix element in
$V$ can be obtained by measuring the output optical spectra, as discussed in
Sec. II.1. To determine the unknown phase terms in $V$, we probed the QFP with
equal superpositions of two frequency modes
($\ket{1_{\omega_{0}}0_{\omega_{1}}}+e^{i\phi}\ket{0_{\omega_{0}}1_{\omega_{1}}}$),
scanning $\phi\in[0,2\pi)$ footnote . We extracted the power on specific modes
from a series of optical spectra and observed the resulting interference
patterns as a function of $\phi$. We then used these patterns to determine the
unknown phase terms by fitting them to sinusoidal curves.

Figure 3(a) shows an experimentally obtained mode transformation for the
beamsplitter, which corresponds to $\mathcal{P}_{W}=0.9739$ and
$\mathcal{F}_{W}=0.9999$ when compared to the ideal $T_{H}$. To verify
accuracy at the single-photon level, we attenuated the input state to
$\sim$0.1 photons per detection gate window, and again scanned phase $\phi$.
At each setting, a wavelength-selective switch (WSS) routed the output modes
to an InGaAs single-photon detector. The resulting interference patterns
[Qubit Probe in Fig. 3(a)] show the expected sinuosidal oscillations with
fitted visibilities exceeding 97%, which is primarily limited by the detector
noise.

By incorporating an additional harmonic to the EOM drives, as discussed in
Sec. II.2, we experimentally reconfigured the QFP into a tritter in which the
transformation matrix is a $3d$ DFT given by Lu2018a ; Lu2022b

$T_{\mathrm{tr}}=\frac{1}{\sqrt{3}}\begin{pmatrix}1&1&1\\\ 1&e^{2\pi
i/3}&e^{4\pi i/3}\\\ 1&e^{4\pi i/3}&e^{2\pi i/3}\end{pmatrix}.$ (11)

We again applied the coherent-state-based gate characterization technique, now
with a 18.1 GHz electro-optic frequency comb as the probe, and measured the
$3\times 3$ mode matrix $V$ as depicted in Fig 3(b). The moduli and phases
match those of $T_{\mathrm{tr}}$ well, with an associated success probability
of $\mathcal{P}_{W}=0.9731$ and fidelity of $\mathcal{F}_{W}=0.9992$. Again,
to gain insight into the tritter for single-photon states, we attenuated a
three-mode superposition state
($\ket{1_{\omega_{0}}0_{\omega_{1}}0_{\omega_{2}}}+e^{-i\phi}\ket{0_{\omega_{0}}1_{\omega_{1}}0_{\omega_{2}}}+e^{-2i\phi}\ket{0_{\omega_{0}}0_{\omega_{1}}1_{\omega_{2}}}$)
to the single-photon level and measured the counts in bins 0, 1 and 2 at the
output. The interference patterns, now tracing a sum of two sines with
respective peaks at $\phi\in\\{0,2\pi/3,4\pi/3\\}$, again show excellent
visibilities of over 97%. The reduced flux in bin 1 is primarily due to the
mismatch between the mode spacing (18.1 GHz) and the 12.5 GHz passbands on the
WSS.

The previous two examples are one-photon gates, for which the the mode $V$ and
Hilbert space $W$ transformations are identical; Eq. (4) reduces to one term
with a single factor. Two-qubit gates are also required for universal
computing Nielsen2000  which, following KLM Knill2001 , can be realized via
quantum interference and single-photon detection. Controlled-phase (CPhase)
and controlled-NOT (CNOT) gates were first demonstrated in
polarization/spatial encodings using a network of spatial beamsplitters
Hofmann2002 ; Ralph2002 ; OBrien2003 , where the gate success is postselected
by coincidence events between the desired output ports. Mathematically, the
ideal CNOT can be described in the computational basis as

$T_{\mathrm{CNOT}}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{pmatrix},$ (12)

which flips the state of the target qubit if the control qubit is in state
$|1\rangle$. Similarly, this coincidence-basis CNOT gate can be implemented in
the spectral domain with a QFP Lu2019a . Numerically, we identified a set of
EOM/shaper modulation functions for the three-element QFP that results in
theoretical fidelity $\mathcal{F}_{W}=0.9999$ and success probability
$\mathcal{P}_{W}=0.0445$ Lu2019a , close but slightly below the optimal
success for a coincidence-basis CNOT of $\mathcal{P}_{W}=1/9$ Hofmann2002 ;
Ralph2002  due to our experimentally limited circuit depth. In this solution,
the indices for the logical bins are as follows:
$(C_{0},C_{1},T_{0},T_{1})=(0,6,7,8)$, where we use the notation
$C_{0}~{}(T_{0})$ and $C_{1}~{}(T_{1})$ to denote the frequency bin index $m$
for the logical zero and one mode, respectively, of the control (target)
qubit. Thus the four logical states can be defined in the Fock basis as
$\ket{00}\equiv\ket{1_{C_{0}}0_{C_{1}}1_{T_{0}}0_{T_{1}}}$,
$\ket{01}\equiv\ket{1_{C_{0}}0_{C_{1}}0_{T_{0}}1_{T_{1}}}$,
$\ket{10}\equiv\ket{0_{C_{0}}1_{C_{1}}1_{T_{0}}0_{T_{1}}}$, and
$\ket{11}\equiv\ket{0_{C_{0}}1_{C_{1}}0_{T_{0}}1_{T_{1}}}$.

Standard procedures for characterizing a photonic CNOT gate often involve
preparing a set of input states that span the entire two-photon state space
and then performing state tomography for each of these states at the output
OBrien2004 , which exceeded our resources (number of EOMs and pulse shapers)
available at the time of the experiment. While the use of high-flux coherent
states in the aforementioned characterization procedures cannot reveal the
two-photon interference effects underlying the CNOT gate, it is still useful
in characterizing the mode transformation $V$. Experimentally, we configured
the QFP to realize the frequency-bin CNOT and measured the mode matrix $V$
shown in Fig. 3(c). As expected from the design, coupling between mode $C_{0}$
and $\\{C_{1},T_{0},T_{1}\\}$ was negligible, since $C_{0}$ is spectrally
isolated from the target modes (to preserve the state of the target upon the
presence of a photon in mode $C_{0}$). In contrast, bin $C_{1}$ is close to
both target bins, allowing it to be strongly coupled to both $T_{0}$ and
$T_{1}$ with equal strength. With the mode transformation $V$ at our disposal,
the equivalent two-photon state transformation $W$ was computed from Eq. (4),
which shows all four of the large elements of $W$ to be in-phase and in good
agreement with the theory. We calculated an _inferred_ fidelity of
$\mathcal{F}_{W}^{(\mathrm{inf})}=0.995$ and coincidence-basis success
probability of $\mathcal{P}_{W}^{(\mathrm{inf})}=0.0460$—an indirect estimate
that suggested our frequency-bin CNOT was operating correctly.

To test our gate with truly quantum states, we generated a two-photon
frequency comb—namely, entangled photons spanning discrete pairs of phase-
coherent energy-matched comb lines—by pumping a periodically poled lithium
niobate (PPLN) waveguide with a continuous-wave Ti:sapphire laser and
filtering the broadband emission with an etalon to produce frequency bins. We
then prepared all four computational-basis states by translating the pump
frequency to four different values and using an additional shaper (prior to
QFP) to filter out specific modes. After the gate operation, the output
photons were frequency-demultiplexed and coincidences measured for 600 s for
all 16 combinations of input/output states [plotted in Fig. 3(c)], showing a
total of four distinct, significant peaks as well as a number of smaller bars.
When an input photon is present in mode $C_{0}$, the quantum state is
maintained, but when a photon is present in mode $C_{1}$, the target qubit is
flipped.

###  III.2 Physically Motivated Noise Model

Logical-basis measurements in the case of CNOT characterization, such as the
one presented in Fig. 3(c), can only reveal the amplitude of the state
transformation $W$; to access the phase information in $W$, measurements of
superposition state inputs (for example, $(\ket{0}-\ket{1})\otimes\ket{0}$)
are essential OBrien2003 . However, we have found that by constructing a
physical model that takes into account both single-detector events and
coincidence counts, logical-basis measurements can provide more information
than typically considered. In other words, by augmenting the multiport model
in the open box viewpoint with physically motivated noise processes, it is
possible to estimate the transformation $W$ without reference to the classical
characterization in Fig. 3(c), but rather from the quantum data alone.

Specifically, we considered a model [Fig. 4(a)] where logical input photon
states $\ket{kl}$ ($k,l\in\\{0,1\\}$) are processed by a linear-optical
frequency multiport fully described by a mode matrix $V$ (thus falling under
the open box umbrella), and coincidences between output modes $C_{r}$ and
$T_{s}$ ($r,s\in\\{0,1\\}$) are registered at detectors $A$ and $B$,
respectively. We also accounted for nonideal effects in our model, including
the pathway efficiency $\eta_{A}$ ($\eta_{B}$) from photon-pair generation to
detector $A$ ($B$) and the dark count probability $d_{A}$ ($d_{B}$). For each
set of $\\{k,l,r,s\\}$, we derived the marginal probabilities for clicks on
detector $A$ or $B$, denoted as $p_{A}$ and $p_{B}$, respectively, defined
within a detection frame $\tau$ ($\sim$1.5 ns in our case). The clicks can
occur either due to the arrival of a photon from an entangled pair (with pair
generation probability $\mu$) at the monitored output modes or due to random
background or dark count noise ($d_{A}$ and $d_{B}$); under the assumption
$\mu,\eta_{A},\eta_{B},d_{A},d_{B}\ll 1$, we found the marginal probabilities
to be simply the summation of pair and dark-count contributions (see Lu2019a
for detailed derivation).

The coincidence probability between detectors $A$ and $B$ ($p_{AB}$) also
contains two terms—the correlated coincidences between two photons from the
same entangled photon pair, and the accidental coincidences from two random
clicks within the detection frame $\tau$. The latter term, represented by
$2p_{A}p_{B}$ Eckart1938 ; Pearson2010 , encompasses the noise coincidences
from multipair emission or between a photon and dark count. Together, these
equations show that the probability of detecting a single photon ($p_{A}$ and
$p_{B}$) is determined solely by the magnitude of the elements in the $V$
matrix, whereas the coincidence probability ($p_{AB})$ also depends on the
relative phase of these matrix elements. Consequentially, our goal was to
devise a method to estimate the model parameters of interest
$\beta=\\{V,\mu,\eta_{A},\eta_{B}\\}$ from the measured datasets in Fig. 3(c).

The leading approach in modern quantum tomography, maximum likelihood
estimation (MLE) is typically utilized to find a _single_ set of $\beta$
values that maximizes the likelihood function Hradil1997 ; James2001 .
However, we employed Bayesian inference to sample from a complete posterior
distribution $P(\beta|\bm{\mathcal{D}})$, the probability density of the
parameters $\beta$ given the observed data $\bm{\mathcal{D}}$. While more
computationally intensive than MLE, Bayesian methods have the advantage of
automatically quantifying uncertainties, and the Bayesian mean estimator
attains the minimum squared error on average Robert1999 . Although a detailed
discussion of Bayesian techniques is beyond the scope of the present review,
we direct the interested reader to the proposal and motivation for Bayesian
QST from Blume-Kohout Blume2010 , as well as several papers describing
specific Bayesian tomography models and algorithms DiGuglielmo2009 ; Seah2015
; Granade2016 ; Mai2017 ; Williams2017 ; Lukens2020b ; Chapman2022a ; Lu2022a
; Chapman2022b .

For the considered CNOT experiment, we leveraged the Markov chain Monte Carlo
(MCMC) algorithm known as slice sampling Neal2003 ; Williams2017  to obtain
Bayesian samples numerically, ultimately arriving at the fidelity estimate
$\mathcal{F}_{W}=0.91\pm 0.01$. Figure 4(b) illustrates how this
$\mathcal{F}_{W}$ translates into the output state probabilities for logical-
basis inputs. Significantly, these results show the informative power of the
open box QFP viewpoint: with Eq. (3) combined with a reasonable noise model,
truth table measurements alone [Fig. 3(c)] were in fact sufficient to estimate
even the phases of the matrix $V$.

Figure 4: Incorporating noise into the open box CNOT model. (a) Contributions
to single and coincidence detection probabilities including effects such as
probabilistic pair generation ($\mu$), nonunity efficiency
($\eta_{A},\eta_{B}$), background/dark counts ($d_{A},d_{B}$), and
uncorrelated events ($2p_{A}p_{B}$). (b) Transition probabilities recovered
through Bayesian inference of this model with the two-photon results in Fig.
3(c). Panel (b) reprinted from Lu2019a  under a Creative Commons Attribution
4.0 International License.

##  IV Black Box Characterization

In the previous section, we discussed experimental examples of the open box
approach for characterizing the QFP. With the black box approach, Eqs. (3,4)
are no longer assumed. Instead, one intentionally is given access to the input
and output ports only, without any assumptions regarding the action of the
QFP. By preparing a known input state and then performing measurements of the
output state after the operation, the quantum process can be fully determined
after repeating this procedure for a number of different input states in
multiple bases.

In this section, we will focus on an exemplary class of QFP operations,
namely, tunable frequency-bin beamsplitters. Previously in Lu2018b ; Lu2020b ,
we discovered that the three-element QFP can synthesize tunable frequency
beamsplitters simply by adjusting the depth of the phase shift $\alpha$
imparted by the QFP shaper between frequency bins $\omega_{0}$ and
$\omega_{1}$ while driving the EOMs with $\pi$-phase-shifted sinewaves with a
fixed modulation index $\Theta$. Mathematically, the $2\times 2$
transformation matrix on the qubit Hilbert space is a function of the tunanble
phase jump $\alpha$:

$W=\begin{pmatrix}W_{00}(\alpha)&W_{01}(\alpha)\\\
W_{10}(\alpha)&W_{11}(\alpha)\end{pmatrix},$ (13)

where each of the matrix elements are

$\displaystyle W_{10}(\alpha)$
$\displaystyle=W_{01}(\alpha)=(1-e^{i\alpha})\sum_{k=1}^{\infty}J_{k}(\Theta)J_{k-1}(\Theta)$
(14) $\displaystyle W_{00}(\alpha)$
$\displaystyle=J_{0}^{2}(\Theta)+(1+e^{i\alpha})\frac{1-J_{0}^{2}(\Theta)}{2}$
$\displaystyle W_{11}(\alpha)$
$\displaystyle=e^{i\alpha}J_{0}^{2}(\Theta)+(1+e^{i\alpha})\frac{1-J_{0}^{2}(\Theta)}{2},$

with $J_{k}(\Theta)$ the Bessel function of the first kind. The identity
operation is observed when $\alpha=0$, as the two $\pi$-phase-shifted sinewave
cancel each other out. Furthermore, when $\alpha=\pi$, the reflectivity
$\mathcal{R}\equiv|W_{01}|^{2}=|W_{10}|^{2}$ and the transmissivity
$\mathcal{T}\equiv|W_{00}|^{2}=|W_{11}|^{2}$ are approximately equal; the
elements $\\{W_{00},W_{01},W_{10}\\}$ are all real and positive, while
$W_{11}$ is real and negative. This results in a Hadamard operation with
theoretical fidelity $\mathcal{F}_{W}=1-10^{-7}$. Accordingly, the designed
quantum process transforms any qubit input $\rho^{\prime}$ into the output
$\rho_{T}\propto W\rho^{\prime}W^{\dagger}$. It is important to emphasize that
this formula is not assumed in the characterization experiments below, but
rather is used to compute the ideal output $\rho_{T}$ and process $\Phi_{T}$
against which experimental results are compared.

###  IV.1 Quantum State Tomography (QST)

As the first experimental foray into black box QFP characterization, we
investigated gate performance for converting a fixed input frequency-bin qubit
to a targeted output state, both for arbitrary single-qubit unitaries and the
tunable beamsplitter of interest here Lu2020b . We performed QST of the output
photon and compared the recovered state $\rho$ against the target $\rho_{T}$
via Eq. (8). Specifially, we prepared the fixed single-frequency input
$\ket{0}\equiv\ket{1_{\omega_{0}}0_{\omega_{1}}}$, using the tunable
beamsplitter to convert this state, initially at the north pole of the Bloch
sphere, to a family of states along a predefined trajectory [dashed line in
Fig. 5(a)]; in this implementation, we sampled a total of 21 evenly spaced
$\alpha\in[0,2\pi)$.

To perform QST on a single qubit, we measured projections onto the
eigenvectors of the Pauli basis $\\{X,Y,Z\\}$—namely, $\ket{0}$,
$\ket{1}\equiv\ket{0_{\omega_{0}}1_{\omega_{1}}}$,
$\ket{\pm}\equiv\frac{1}{\sqrt{2}}(\ket{0}\pm\ket{1})$, and $\ket{\pm
i}\equiv\frac{1}{\sqrt{2}}(\ket{0}\pm i\ket{1})$. These Pauli measurements are
equivalent to applying specific quantum gates—$\mathbbm{1}$ for $Z$, $H$ for
$X$, and $HS^{\dagger}$ for $Y$—followed by a measurement in the computational
basis, where $\mathbbm{1}$ is the identity operation, $H$ is the Hadamard, and
$S=\begin{pmatrix}1&0\\\ 0&i\end{pmatrix}$. For the $H$ gate required in both
$X$ and $Y$ measurements, we opted for an inherently probabilistic single-EOM
version Imany2018 ; Seshadri2022  that is simpler than the QFP Hadamard
Lu2018a . This method attains a theoretical fidelity of $\mathcal{F}_{W}=1$
and success of $\mathcal{P}_{W}\approx 0.6$). The $S^{\dagger}$ gate amounts
to a phase shift between two frequency modes, which can be realized by using
an extra pulse shaper, which also blocks any residual photons outside of the
single-qubit space that may occur due to nonunity success of the QFP
operations. As the final step for all three Pauli measurements, we demultiplex
the photons by color with a WSS and record the counts over 1 s with
superconducting nanowire single-photon detectors (SNSPDs), obtaining the full
dataset of counts for all basis states.

As with the CNOT in Sec. III.2, we employed Bayesian inference to estimate the
quantum output state for each beamsplitter setting, this time leveraging
preconditioned Crank–Nicolson MCMC sampling Cotter2013  after the approach of
Lukens2020b . Figure 5(a) plots the resulting states, where each spot consists
of a cluster of 1024 samples from the Bayesian posterior distribution. The
good agreement with the expected trajectory is confirmed quantitatively by the
fidelity calculations in Fig. 5(b), whose means all exceed 0.985.

Figure 5: Black box QST results for the tunable frequency beamsplitter. (a)
Measured density matrices $\rho$ for each beamsplitter setting $\alpha$ with
fixed logical input state $\ket{0}$. Each “cloud” on the Bloch sphere consists
of 1024 discrete samples obtained through Bayesian inference to quantify
uncertainty. (b) State fidelity $\mathcal{F}_{\rho}$ with respect to ideal
output. Reprinted with permission from Lu2020b .

###  IV.2 Quantum Process Tomography (QPT)

The previous QST results above are truly “black box” in that they make no
reference to the inner workings of the QFP. Yet strictly speaking, they
confirm ideal functioning of the tunable beamsplitter for a _single_ input
only—namely $\rho^{\prime}=\ket{0}\bra{0}$. To fully characterize a quantum
process, it is necessary to repeat the aforementioned procedure with a variety
of input states, including frequency-bin superpositions that necessitate an
additional EOM and pulse shaper beyond what was available to us during the
original QST experiments Lu2020b . With a sufficient number of devices now in
hand, however, we proceed with full QPT of the tunable beamsplitter—the first
QPT experiment of a frequency-bin quantum gate.

Figure 6: Experimental QPT setup for tunable frequency-bin beamsplitter.
Single-photon-level inputs traverse the QFP and proceed through a WSS (to
block bins outside of the computational space), an EOM (to interfere frequency
bins), and a final WSS to separate computational bins for detection. All
manual RF phase shifters have been supplanted in favor of electrically
controlled options, both pulse-shaper-induced delays and direct programming of
RF Source 2’s phase with respect to RF Source 1. RF component labels: AMP
(amplifier), ATT (attenuator), PS (power splitter).

In standard QPT, a total of $d^{2}$ states permit an informationally complete
basis for an arbitrary $d$-dimensional quantum process Chuang1997 ;
Nielsen2000 . A typical approach for single-qubit QPT ($d=2$) is to prepare
$\\{\ket{0},\ket{1},\ket{\pm}\\}$ and perform QST on the output state for
each. In our frequency-bin QPT experiment, we consider an overcomplete set of
input states, specifically the Pauli eigenstates defined in Sec. IV.1:
$\\{\ket{0},\ket{1},\ket{\pm},\ket{\pm i}\\}$.

Figure 6 provides a schematic of the experimental setup. The presence of four
EOMs implies the need to precisely set three RF delays, and in contrast to
previous experiments with manual phase shifters, we here control these delays
electronically: two are tuned by applying a linear phase on the first two
pulse shapers, and the third is adjusted by setting the phase of RF Source 2
relative to RF Source 1. To prepare the input states we again utilize
amplitude and phase filtering of a 25 GHz electro-optic frequency comb with a
state preparation pulse shaper. Prior to the QFP, we reduce the input photon
flux to approximately 106 counts per second (roughly one-tenth of the detector
saturation level) with an optical attenuator, mimicking the case of having
true single photons with similar flux at the input.

The central QFP is configured to realize a tunable frequency beamsplitter, in
this case, with reflectivities $\mathcal{R}\in\\{0,0.123,0.373,0.5\\}$
corresponding to $\alpha\in\\{0,\pi/3,2\pi/3,\pi\\}$ in Eq. (14). We continue
the single-EOM approach for tomographic projections, but with two slight
modifications: (i) a dual WSS (Finisar) is utilized to block photons outside
the computational space after the QFP and frequency-demultiplex the photons
before the SNSPDs, and (ii) the $S^{\dagger}$ operation is realized through
precise timing adjustments on the State Tomography EOM, leveraging the
equivalence between linear spectral phase and delay. Finally, we count photons
at the output with 1 s integration time, and do not subtract dark counts. The
full dataset for each QFP setting thus contains 36 numbers: counts for all six
output projections, repeated for each of the six input states.

Figure 7: Experimental QPT results for tunable frequency-bin beasmplitter with
varying reflectivity $\mathcal{R}$. Plotted is the Bayesian mean process
matrix $\chi$ in the Pauli basis. As $\mathcal{R}$ increases, the initially
dominant identity component reduces, while the ideal $X$ and $Z$ contributions
for the Hadamard rise. Process fidelities $\mathcal{F}_{\Phi}$ with the
respect to the designed beasmplitter are (left to right): $0.9925\pm 0.0002$,
$0.9775\pm 0.009$, $0.9878\pm 0.0009$, and $0.9946\pm 0.0002$.

For inference, we once again subscribe to a Bayesian paradigm, following the
QPT procedure introduced in Chapman2022b  which pools all 36 numbers into a
single likelihood and employs a uniform prior based on a recently proposed
Kraus operator parametrization Kukulski2021 . The real and imaginary parts of
the Bayesian mean process matrices $\chi$ follow in Fig. 7, expressed in the
Pauli basis for convenience. The process for $\mathcal{R}=0$ ($\alpha=0$)
matches the identity as expected, with process fidelity
$\mathcal{F}_{\Phi}=0.9925\pm 0.0002$, while that for $\mathcal{R}=0.5$
($\alpha=\pi$) aligns with the Hadamard with $\mathcal{F}_{\Phi}=0.9946\pm
0.0002$. The intermediate cases reveal a clear transition between the two
extremes of a single nonzero component for the identity ($\mathcal{R}=0$) and
four equal components in the $X/Z$ subspace ($\mathcal{R}=0.5$), matching
theoretical expectations with high fidelity: $\mathcal{F}_{\Phi}=0.9775\pm
0.0009$ for $\mathcal{R}=0.123$ and $\mathcal{F}_{\Phi}=0.9878\pm 0.0009$ for
$\mathcal{R}=0.373$.

##  V Discussion

The open box and black box approaches for QFP characterization trade off
advantages in complementary ways, so the method of choice for a given
experiment can vary. From a design perspective, the QFP is the realization of
some desired mode transformation matrix through sequential temporal and
spectral phase modulation. Because this multiport interferometer applies to
any optical input, open box approaches are able to leverage bright coherent
states as probes, enabling extremely precise characterization without the
additional complications of quantum probes sensitive to statistical noise and
background light. Accordingly, open box QFP characterization provides the most
accurate insight into the QFP’s inner workings, and therefore is well suited
to the engineering phase of quantum gate synthesis when the adjustable
parameters of each device are tuned—e.g., EOM voltages and pulse shaper
phases.

On the other hand, the noise effects bypassed by the open box formalism cannot
be ignored in the quantum domain. Thus, after the QFP is tuned and ready for
specific quantum information processing tasks, more general black box
characterization approaches should be preferred, ideally performed at whatever
flux and statistical properties will be used in the application of interest.
QPT incorporates all nonidealities present in the system, consequently
painting the most comprehensive picture of the final QFP operation.

Over the short history of the QFP, open box characterization has proven highly
successful, which can be attributed to the fact that experimental
implementations with commercially available EOMs and pulse shapers have been
near-ideal, to an extent surpassing even our own original expectations. But
this situation will likely change significantly as the QFP enters new
application regimes. For example, fully on-chip QFPs have the potential for
lower loss, tighter bin spacings, and reduced cost compared to tabletop
versions, but on-chip components tend to deviate from the ideal line-by-line
pulse shaper and linear EOM models. As explored in SiPhQFP , microring-based
pulse shapers lead to matrix transformations $V$ that vary depending on
spectral location within a bin, while standard silicon phase modulators suffer
from phase-dependent loss. Similarly, time synchronization of QFPs across
multiple sites—required for applying the QFP to distributed quantum
networking—will experience at least some level of jitter that, depending on
its severity, could cause the transformation $V$ to drift noticeably in time.
It is ultimately the black box viewpoint that possesses sufficient generality
to integrate all such impairments into a single channel description, which in
our opinion points to the increasing importance of QST and QPT in next-
generation QFP experiments.

Finally, although we have focused on the QFP due to the variety of
experimental characterization demonstrations so far, the techniques described
in this paper readily apply to other approaches for frequency-bin quantum
operations. For example, frequency beamsplitters Raymer2010  have been
realized in $\chi^{(2)}$ Kobayashi2016 ; Kobayashi2017  and $\chi^{(3)}$
Clemmen2016 ; Joshi2020  nonlinear materials mediated by classical pump
pulses, as well as in integrated microrings with coupling controlled by
electro-optic modulation Hu2021 . While compatible with single-photon signals,
these experimental frequency-bin gates have yet to be fully characterized in
either the open box multiport or black box quantum process pictures. In the
future, it would therefore prove interesting to invoke the techniques
described here for non-QFP-based frequency-bin operations, facilitating a more
comprehensive and informative understanding of the ever-expanding toolkit for
frequency-bin quantum information.

###### Acknowledgements.

We thank P. Imany, D. E. Leaird, P. Lougovski, O. D. Odele, E. M. Simmerman,
and B. P. Williams for contributions to past publications summarized in this
article Lu2018a ; Lu2019a ; Lu2020b . We thank M. Alshowkan for setting up the
time tagging electronics. This research was performed in part at Oak Ridge
National Laboratory, managed by UT-Battelle, LLC, for the U.S. Department of
Energy under contract no. DE-AC05-00OR22725.

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\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2&\leq \ltwo{\Bbeta_{t}-\Bbeta^*}^2-\frac{1}{12b_0}\eta_t\kl n_l\ltwo{\Bbeta_{t}-\Bbeta^*}^2+\eta_t^2\frac{4}{b_1^2}\ku^2\ltwo{\Bbeta_{t}-\Bbeta^*}^2\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+2\eta_t C_1 b_0 \frac{\ku}{\kl}\frac{d}{\sum_{l=0}^t n_l} + C_2\eta_t^2 \ku \frac{d}{\sum_{l=0}^t n_l}.
\end{align*}

Plug the stepsize $\eta_t\in[c_1,c_2]\cdot\frac{\kl}{\ku^2}\cdot \frac{b_1^2}{b_0}$ into the above equation and then we have

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq \left(1-c\frac{b_1^2}{b_0^2}\cdot\frac{\kl^2}{\ku^2}\right)\cdot\ltwo{\Bbeta_{t}-\Bbeta^*}^2 + C\frac{d}{\ku\sum_{l=0}^t n_l}\cdot b_1^2,
\end{align*}

where $c,C$ are some universal constants. In all, we have

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq \left(1-c\frac{b_1^2}{b_0^2}\cdot\frac{\kl^2}{\ku^2}\right)^{t+1-t_1}\cdot \ltwo{\Bbeta_{t_1}-\Bbeta^*}^2 + C\frac{d}{\ku}\cdot b_1^2\cdot\sum_{s=t_1}^{t}\frac{\left(1-c\frac{b_1^2}{b_0^2}\cdot\frac{\kl^2}{\ku^2}\right)^{t-s} }{\sum_{l=0}^{s} n_l}.
\end{align*}

If the number of arriving samples remains to be $n$, namely, $n_l=n$ for all $l\geq 1$, then equipped with Lemma <ref>, the estimation error rate is upper bounded with

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq \left(1-2c\frac{b_1^2}{b_0^2}\cdot\frac{\kl^2}{\ku^2}\right)^{t+1-t_1}\cdot \ltwo{\Bbeta_{t_1}-\Bbeta^*}^2 + \tilde{C}\frac{\ku}{\kl^2}\cdot \frac{d}{n_0+tn}\cdot \frac{b_0^2}{|\log(1-c\frac{b_1^2}{b_0^2}\cdot\frac{\kl^2}{\ku^2})|},
\end{align*}

where $\tilde{C}$ is some constant irrelevant with $t$. It is worth noting that when $0<x<1$, $|\log(1-x)|\geq x$ holds. Thus the last term of the above equation could be further simplified by

\begin{align*}
\ltwo{\Bbeta_{t+1}-\Bbeta^*}^2\leq \left(1-2c\frac{b_1^2}{b_0^2}\cdot\frac{\kl^2}{\ku^2}\right)^{t+1-t_1}\cdot \ltwo{\Bbeta_{t_1}-\Bbeta^*}^2 + \tilde{C}_1\frac{\ku}{\kl^2}\cdot \frac{d}{n_0+tn}\cdot b_0^2\cdot\left(\frac{b_0}{b_1}\cdot\frac{\ku}{\kl}\right)^2,
\end{align*}

with some constant $\tilde{C}_1$.

§ TECHNICAL LEMMAS

Suppose the loss function is given by the following equation

\begin{align*}
\end{align*}

where $\{\X_i\}_{i=1}^n$ follow Assumption <ref> and $\{(\X_i,Y_i)\}_{i=1}^n$ are independent observations. Then there exist $C_1,C_2$ such that with probability exceeding $1-\exp\left(-C_1d\right)-\exp\left(-\sqrt{n/\log n}\right)$, the following holds for all $\Bbeta_1,\Bbeta_2\in\RR^d$,

\begin{align*}
\left|f(\Bbeta_1)-f(\Bbeta_2)-\EE\left(f(\Bbeta_1)-f(\Bbeta_2) \right)\right|\leq C_2\max\{\tau,1-\tau\}\sqrt{\ku\frac{d}{n}}\ltwo{\Bbeta_1-\Bbeta_2}.
\end{align*}

Suppose the setting is same as Proposition <ref>. Then for any fixed $\Bbeta_1,\Bbeta_2$, with probability exceeding $1-2\exp\left(-C\frac{ns^2}{\ku\ltwo{\Bbeta_1-\Bbeta_2}^2}\right)$ where $C$ is some constant, it has

\begin{align*}
\left|f(\Bbeta_1)-f(\Bbeta_2)-\EE\left[f(\Bbeta_1)-f(\Bbeta_2) \right]\right|\leq s.
\end{align*}

Similarly, with probability exceeding $1-2\exp\left(-c_1\frac{ns^2}{\ku\max\{\tau^2,(1-\tau)^2\}\ltwo{\Bbeta_1-\Bbeta^*}^2}\right)$, it has

\begin{align*}
\frac{1}{n}\left|\sum_{i=1}^{n} \inp{\Bbeta_1-\Bbeta^*}{\X_i}\cdot g_i-\EE \inp{\Bbeta_1-\Bbeta^*}{\X_i}\cdot g_i\right|\leq s,
\end{align*}

where $g_i:= -\tau\mathbbm{1}_{(\xi_i> \inp{\X_i}{\Bbeta_{1}-\Bbeta^*})} + (1-\tau) \mathbbm{1}_{(\xi_i< \inp{\X_i}{\Bbeta_{1}-\Bbeta^*})}+\delta\mathbbm{1}_{(\xi_i= \inp{\X_i}{\Bbeta_{1}-\Bbeta^*})}$ is the sub-gradient with any $\delta\in[-\tau,1-\tau]$.

Note that
\begin{align*}
\left\| \big| |Y_i-\inp{\X_i}{\Bbeta_1}| - |Y_i-\inp{\X_i}{\Bbeta_2}| \big| \right\|_{\Psi_2} \leq \| |\inp{\X_i}{\Bbeta_1-\Bbeta_2}| \|_{\Psi_2}\leq C\sqrt{\ku}\ltwo{\Bbeta_1-\Bbeta_2}.
\end{align*}

Then by sum of independent sub-Gaussians, we obtain the desired inequality. The second consequence could be obtained in a similar way,

\begin{align*}
\left\|  \inp{\Bbeta_1-\Bbeta^*}{\X_i}\cdot g_i\right\|_{\Psi_2}&\leq \max\{\tau,1-\tau\}\left\|  |\inp{\Bbeta_1-\Bbeta^*}{\X_i} |\right\|_{\Psi_2}\leq C\max\{\tau,1-\tau\}\sqrt{\ku}\ltwo{\Bbeta_1-\Bbeta_2}.
\end{align*}

The following lemma provides two types of upper bound for $\ltwo{\g}$ and in the convergence analyses we would choose to use the sharpest one accordingly.

Suppose the predictors and noise term satisfy Assumption <ref> and Assumption <ref>, respectively. And the loss function is based on $n$ independnet observations $f(\Bbeta)=\frac{1}{n}\sum_{i=1}^n\rho_{Q,\tau}(Y_i-\X_i^{\top}\Bbeta)$ with $n\geq Cd$. Suppose $\bcalE:=\{\sup_{\Bbeta_1,\Bbeta_2\in\RR^d} \left|f(\Bbeta_1)-f(\Bbeta_2)-\EE\left[ f(\Bbeta_1)-f(\Bbeta_2)\right]\right|\cdot\ltwo{\Bbeta_1-\Bbeta_2}^{-1}\leq C_1\max\{\tau,1-\tau\}\sqrt{\ku d/n} \}$ holds. Then for all $\Bbeta$ and any sub-gradient $\g\in\partial f(\Bbeta)$, we have

\begin{align*}
\ltwo{\g}\leq\left\{
\begin{array}{l}
\sqrt{\ku}\\
1.5\frac{\ku}{b_1}\ltwo{\Bbeta-\Bbeta^*}+C_1\max\{\tau,1-\tau\}\sqrt{\ku\frac{d}{ n}}
\end{array}
\right. .
\end{align*}

Under the event $\bcalE$ and with sub-gradient definition, for any $\Delta\Bbeta\in\RR^d$, we have

\begin{align}
\inp{\Delta\Bbeta}{\g}\leq f(\Bbeta+\Delta\Bbeta)-f(\Bbeta)&\leq\EE f(\Bbeta+\Delta\Bbeta)-\EE f(\Bbeta)+C_1\max\{\delta,1-\delta\}\sqrt{\ku\frac{d}{n}}\ltwo{\Delta\Bbeta}.
\label{eq1}
\end{align}

We shall provide the two types of upper bounds respectively.

* Note that the quantile function $\rho_{Q,\tau}(\cdot)$ satisfies $\rho_{Q,\tau}(x_1+x_2)\leq \rho_{Q,\tau}(x_1)+\rho_{Q,\tau}(x_2)$ and then for any fixed $\Bbeta,\Delta\Bbeta$, we have

\begin{align*}
\EE f(\Bbeta+\Delta\Bbeta)-\EE f(\Bbeta)&= \frac{1}{n}\sum_{i=1}^n \EE \left[ \rho_{Q,\tau}(\xi_i-\inp{\X_i}{\Bbeta+\Delta\Bbeta-\Bbeta^*})-\rho_{Q,\tau}(\xi_i-\inp{\X_i}{\Bbeta-\Bbeta^*}) \right]\\
&\leq\frac{1}{n}\sum_{i=1}^{n} \EE \rho_{Q,\tau}(-\inp{\X_i}{\Delta\Bbeta}).
\end{align*}

Also notice that $-\inp{\X_i}{\Delta\Bbeta}$ is a mean zero Gaussian variable and then Assumption <ref> implies that $\EE \rho_{Q,\tau}(-\inp{\X_i}{\Delta\Bbeta})\leq \sqrt{\frac{2}{\pi}}\sqrt{\ku}\ltwo{\Bbeta-\Bbeta^*}$. Thus together with $n\geq C_3 \max\{\tau^2,(1-\tau)^2\} d$ we obtain $\EE f(\Bbeta+\Delta\Bbeta)-\EE f(\Bbeta)\leq\sqrt{\ku}\ltwo{\Delta\Bbeta}$. Then (<ref>) becomes
$$\inp{\Delta\Bbeta}{\g}\leq \sqrt{\ku}\ltwo{\Delta\Bbeta},$$

holds for all $\Bbeta,\Delta\Bbeta$. Then insert $\Delta\Bbeta=\g$ into the above equation and we acquire

* According to the first two equations of proof section for Lemma 11 in [Shen et al., 2023], it has

\begin{align*}
\EE f(\Bbeta+\Delta\Bbeta)-\EE f(\Bbeta)\leq\frac{1}{2b_1}\ku\ltwo{\Delta\Bbeta}^2+\frac{1}{b_1}\ku\ltwo{\Delta\Bbeta}\ltwo{\Bbeta-\Bbeta^*}.
\end{align*}

Insert it into (<ref>), take $\Delta\Bbeta=\frac{b_1}{4\ku}\g$ and then we have

\begin{align*}
\ltwo{\g}\leq 1.5\frac{\ku}{b_1}\ltwo{\Bbeta-\Bbeta^*}+C_1\max\{\delta,1-\delta\}\sqrt{\ku\frac{d}{n}}.
\end{align*}

The following lemma is derivable through integral calculations.

Suppose the predictors $\{\X_i\}_{i=1}^{n}$ satisfy Assumption <ref>. Then for any fixed vector $\Bbeta\in\RR^d$, we have
\begin{align*}
n\sqrt{\frac{\kl}{2\pi}} \ltwo{\Bbeta}\leq\EE\sum_{i=1}^{n}\rho_{Q,\tau}(\inp{\Bbeta}{\X_i})\leq n\sqrt{\frac{\ku}{2\pi}} \ltwo{\Bbeta}.
\end{align*}

Suppose the predictors and noise term satisfy Assumption <ref> and Assumption <ref>, respectively. And the loss function is based on $n$ independnet observations $f(\Bbeta)=\frac{1}{n}\sum_{i=1}^n\rho_{Q,\tau}(Y_i-\X_i^{\top}\Bbeta)$ for any $n\geq 1$. Then for any fixed $\Bbeta$, it has

\begin{align*}
\EE\left[f(\Bbeta)-f(\Bbeta^*)\right]\leq \frac{\ku}{b_1}\ltwo{\Bbeta-\Bbeta^*}^2;
\end{align*}

furthermore, if $\ltwo{\Bbeta-\Bbeta^*}\leq8(\ku/\kl)^{1/2}\gamma$ holds, we have

\begin{align*}
\EE\left[f(\Bbeta)-f(\Bbeta^*)\right]\geq \frac{\kl}{12b_0}\ltwo{\Bbeta-\Bbeta^*}^2.
\end{align*}

Under Assumptions <ref> and <ref>, it holds any $\Bbeta\in\RR^d$ that

\begin{align*}
\EE\inp{\Bbeta-\Bbeta^*}{\X}\cdot\partial \rho_{Q,\tau}(\xi-\inp{\X}{\Bbeta-\Bbeta^*})&\leq \frac{1}{b_1}\ku\ltwo{\Bbeta-\Bbeta^*}^2,
\end{align*}

where $\partial \rho_{Q,\tau}(\xi-\inp{\X}{\Bbeta-\Bbeta^*})=-\tau\mathbbm{1}_{(\xi\geq \inp{\X}{\Bbeta-\Bbeta^*})} + (1-\tau) \mathbbm{1}_{(\xi< \inp{\X_i}{\Bbeta-\Bbeta^*})}+\delta\mathbbm{1}_{(\xi_i= \inp{\X_i}{\Bbeta_{1}-\Bbeta^*})}$ is the sub-gradient with any $\delta\in[-\tau,1-\tau]$.

First consider the conditional expectation,

\begin{align*}
\EE\left[\inp{\Bbeta-\Bbeta^*}{\X}\cdot\partial \rho_{Q,\tau}(\xi-\inp{\X}{\Bbeta-\Bbeta^*})\big| \X \right]=\inp{\Bbeta-\Bbeta^*}{\X}\cdot\EE\left[\partial \rho_{Q,\tau}(\xi-\inp{\X}{\Bbeta-\Bbeta^*})\big|\X\right].
\end{align*}

Then the following equation can be obtained via some trivial calculations,

\begin{align*}
&~~~\EE\left[\partial \rho_{Q,\tau}(\xi-\inp{\X}{\Bbeta-\Bbeta^*})\big|\X\right]=\int_{0}^{\inp{\X}{\Bbeta-\Bbeta^*}} h_\xi(x)\; dx.
\end{align*}

Denote $z:=\inp{\X}{\Bbeta-\Bbeta^*}$, which follows mean zero Gaussian distributions with variance $(\Bbeta-\Bbeta^*)^{\top}\bSigma(\Bbeta-\Bbeta^*)\leq\ku\ltwo{\Bbeta-\Bbeta^*}^2$. Let $f_z(\cdot)$ be its density. Thus we obtain

\begin{align*}
&~~~\EE\inp{\Bbeta-\Bbeta^*}{\X}\cdot\partial \rho_{Q,\tau}(\xi-\inp{\X}{\Bbeta-\Bbeta^*})=\EE\left[\inp{\Bbeta-\Bbeta^*}{\X}\cdot\EE\left[\partial \rho_{Q,\tau}(\xi-\inp{\X}{\Bbeta-\Bbeta^*})\big|\X \right]\right]\\
&= \int_{-\infty}^{+\infty}\int_{0}^y y f_z(y) h_\xi(x)\; dx\; dy\leq \frac{1}{b_1}\ku\ltwo{\Bbeta-\Bbeta^*}^2,
\end{align*}

where the last equation uses the upper bound of the density characterized in Assumption <ref>.

Suppose $0<a<1$ is some fixed constant. Define the sequence with $a_n:=\sum_{k=1}^{n}\frac{a^{n-k}}{k+m}$, where $m>0$ is some integer. Then the following holds for each $n$,

\begin{align*}
(m+n)a_n\leq \frac{9}{1-a}\frac{1}{|\log a|}.
\end{align*}

First consider the function $h_a(x):=xa^{x}$ with the domain $x>0$. Its first-order derivative is

\begin{align*}
h_a^{'}(x)=a^{x}-xa^x|\log a|=(1-|\log a|x)a^x.
\end{align*}

It infers that $h_a(x)$ achieves the maximum at $x=\frac{1}{|\log a|}$ and $h_a(x)$ is monotone decreasing in the region $x\in(\frac{1}{|\log a|},+\infty)$. And the $\max_{x>0} h_a(x)=\frac{1}{|\log a|}\exp(\frac{1}{|\log a|}\log a)\leq\frac{3}{|\log a|}$.
In this way we have

\begin{align}
na^{[\frac{n}{2}]}\leq \frac{6}{|\log a|},\quad\text{for all } n\in\ZZ_+.
\label{eq:9-1}
\end{align}

Then consider the sequence $(m+n)a_n$,

\begin{align*}
&= \underbrace{1+\frac{m+n}{m+n-1}a+\cdots+\frac{m+n}{m+n-[\frac{n}{2}]}a^{[\frac{n}{2}]}}_{A_1}+ \underbrace{\frac{m+n}{m+n-[\frac{n}{2}]-1}a^{[\frac{n}{2}]+1}+\cdots +\frac{m+n}{m+1}a^{n-1}}_{A_2}.
\end{align*}

As for term $A_1$, when $k\leq[\frac{n}{2}]$ it has $\frac{m+n}{m+n-k}\leq 2$ and then we obtain

$$A_1\leq 2\left(1+a+\cdots a^{[\frac{n}{2}]}\right)\leq \frac{2}{1-a}.$$
And $A_2$ could be bounded with

\begin{align*}
A_2&\leq \frac{m+n}{m}a^{[\frac{n}{2}]+1}+\cdots+\frac{m+n}{m}a^{n-1}= \frac{m+n}{m}\left(a^{[\frac{n}{2}]+1}+\cdots+a^{n-1}\right)\\
&\leq \frac{1}{1-a}\left(1+\frac{n}{m}\right) a^{[\frac{n}{2}]+1}\leq 7\frac{1}{1-a}\frac{1}{|\log a|},
\end{align*}

where equation (<ref>) is used. Thus altogether, we have $(m+n)a_n\leq \frac{9}{1-a}\frac{1}{|\log a|}$.

The following lemma provides the concentration of Gaussian vectors, an immediate result of Bernstein inequality (Theorem 2.8.1 of [Vershynin, 2018]).

Suppose the random vector $\X \in\RR^d$ satisfies Assumption <ref>. Then with probability exceeding $1-\exp(-cd)$, its $\ell_2$-norm is bounded by $\ltwo{\X}^2\leq 2\ku d$.


§ ACKNOWLEDGMENTS

The authors would like to thank Musa Morena Marcusso Manhães
for her help with setting up the ROS and Gazebo frameworks
as well as Zhe Chen for the help setting up the code to compare
our approach with K-CHSH-RM solver.