Metaphotonics and Metamaterials. KAF acknowledges support from the Lu Stanford
Graduate Fellowship and the National Defense Science and Engineering Graduate
Fellowship. RT acknowledges support from Kailath Stanford Graduate Fellowship.

The authors kindly thank Emily Davis, Leigh Martin, and Tatsuhiro Onodera for
helpful discussions. The authors are also grateful to Joshua Combes for
carefully reading the paper and providing thoughtful feedback.

##  Appendix A Photon flux and interpretation of temporal modes

We can now briefly comment on the reason for calling the modes ‘temporal’ by
examining the photon flux operator [57]. The expectation of the normally-
ordered intensity operator at $\vec{r}=0$ in the Schrödinger picture is

$\displaystyle\braket{:I(\vec{r}=0):}$ $\displaystyle\propto$
$\displaystyle\braket{:E(\vec{r}=0)^{2}:}$ (192a) $\displaystyle\propto$
$\displaystyle\braket{\int_{0}^{\infty}\mathop{\textrm{d}\omega_{1}}\int_{0}^{\infty}\mathop{\textrm{d}\omega_{2}}\sqrt{\omega_{1}\omega_{2}}\,b_{\omega_{1}}^{\dagger}b_{\omega_{2}}},$
(192b)

which would be measured by an experimental photon detector ($:I:$ denotes the
normal ordering of $I$). Making the same approximation in Sec. IIC that the
frequency content of the state is narrowband around $\omega_{0}$

$\braket{:I(0):}\propto\omega_{0}\braket{\int_{-\infty}^{\infty}\mathop{\textrm{d}\omega_{1}}\int_{-\infty}^{\infty}\mathop{\textrm{d}\omega_{2}}\,b_{\omega_{1}}^{\dagger}b_{\omega_{2}}}.$
(193)

We label this operator on the right and transform into the temporal mode basis

$\displaystyle F_{\vec{r}=0}(0)$ $\displaystyle\equiv$
$\displaystyle\int_{-\infty}^{\infty}\mathop{\textrm{d}\omega_{1}}\int_{-\infty}^{\infty}\mathop{\textrm{d}\omega_{2}}\,b_{\omega_{1}}^{\dagger}b_{\omega_{2}}$
(194a) $\displaystyle=$ $\displaystyle b_{\tau=0}^{\dagger}(0)b_{\tau=0}(0).$
(194b)

In the interaction picture, $F_{0}$ then acquires the time dependence

$F_{0}(t)=b_{0}^{\dagger}(t)b_{0}(t),$ (195)

and hence $\braket{:I(0,t):}\propto\braket{F_{0}(t)}$ with the free evolution
of the waveguide lumped in with the operator rather than the state.

Let’s evaluate the expectation of $F_{0}(t)$ for an arbitrary state in the
temporal mode basis, where we ignore any low-dimensional system’s interaction
with the waveguide $\gamma\rightarrow 0$

$\braket{F_{0}(t)}=\braket{\psi_{\textrm{B,I}}}{b_{0}^{\dagger}(t)b_{0}(t)}{\psi_{\textrm{B,I}}}.$
(196)

(We note this correlation is also a special case of Eq. 1.) Expanding this
expectation by inserting resolutions of the identity in the temporal mode
basis

$\braket{F_{0}(t)}=\sum_{m=0}^{\infty}\sum_{m^{\prime}=0}^{\infty}\int\mathop{\textrm{d}\vec{\bm{\tau}}^{(m)}}\int\mathop{\textrm{d}\vec{\bm{\tau}}^{\prime(m^{\prime})}}\braket{\psi_{\textrm{B,I}}}{\vec{\bm{\tau}}^{(m)}}\braket{\vec{\bm{\tau}}^{(m)}}{b_{0}^{\dagger}(t)b_{0}(t)}{\vec{\bm{\tau}}^{\prime(m^{\prime})}}\braket{\vec{\bm{\tau}}^{\prime(m^{\prime})}}{\psi_{\textrm{B,I}}}.$
(197)

Now, we evaluate

$\braket{\vec{\bm{\tau}}^{(m)}}{b_{0}^{\dagger}(t)b_{0}(t)}{\vec{\bm{\tau}}^{\prime(m^{\prime})}}=\delta_{mm^{\prime}}\delta(\vec{\bm{\tau}}^{(m)}-\vec{\bm{\tau}}^{\prime(m)})\sum_{q=1}^{m}\delta(t-\tau_{q})\,[m>0],$
(198)

given the explicit ordering of the indices $\tau_{1}<\cdots<\tau_{m}$. Then,

$\displaystyle\braket{F_{0}(t)}$ $\displaystyle=$
$\displaystyle\sum_{m=1}^{\infty}\int\mathop{\textrm{d}\vec{\bm{\tau}}^{(m)}}\int\mathop{\textrm{d}\vec{\bm{\tau}}^{\prime(m)}}\braket{\psi_{\textrm{B,I}}}{\vec{\bm{\tau}}^{(m)}}\braket{\vec{\bm{\tau}}^{\prime(m)}}{\psi_{\textrm{B,I}}}\delta(\vec{\bm{\tau}}^{(m)}-\vec{\bm{\tau}}^{\prime(m)})\sum_{q=1}^{m}\delta(t-\tau_{q})$
(199a) $\displaystyle=$
$\displaystyle\sum_{m=1}^{\infty}\sum_{q=1}^{m}\int\mathop{\textrm{d}\vec{\bm{\tau}}^{(m)}}\left|\braket{\vec{\bm{\tau}}^{(m)}}{\psi_{\textrm{B,I}}}\right|^{2}\delta(t-\tau_{q}).$
(199b)

Experimentally, an ideal photon detector is likely to detect a photon with
probability proportional to $\braket{F_{0}(t)}$ [57]. This expression then has
the interesting form that it represents a summation over all possible ways for
the state to yield a single photon detection event at time $t$. Each
$m$-photon wavepacket contributes independently.

For instance, $\braket{F_{0}(t)}$ picks out just the occupation in the
$t^{\textrm{th}}$ temporal mode from the single-photon wavepacket

$\left|\braket{t}{\psi_{\textrm{B,I}}}\right|^{2}.$ (200)

Then, for the two-photon wavepacket, there are two possible ways that a
single-photon may be detected at time $t$: a photon was either detected before
or after. Hence,

$\displaystyle\int_{0}^{\infty}\mathop{\textrm{d}\tau_{1}}\int_{\tau_{1}}^{\infty}\mathop{\textrm{d}\tau_{2}}\left|\braket{\tau_{1},\tau_{2}}{\psi_{\textrm{B,I}}}\right|^{2}\left(\delta(t-\tau_{1})+\delta(t-\tau_{2})\right)$
$\displaystyle\hskip
50.0pt=\int_{t}^{\infty}\mathop{\textrm{d}\tau_{2}}\left|\braket{t,\tau_{2}}{\psi_{\textrm{B,I}}}\right|^{2}+\int_{0}^{t}\mathop{\textrm{d}\tau_{1}}\left|\braket{\tau_{1},t}{\psi_{\textrm{B,I}}}\right|^{2}.$
(201)

We also show the case for a three-photon wavepacket, where there are three
possible ways that a single-photon may be detected at time $t$: two after, one
before and one after, and two before

$\displaystyle\int_{0}^{\infty}\mathop{\textrm{d}\tau_{1}}\int_{\tau_{1}}^{\infty}\mathop{\textrm{d}\tau_{2}}\int_{\tau_{2}}^{\infty}\mathop{\textrm{d}\tau_{3}}\left|\braket{\tau_{1},\tau_{2},\tau_{3}}{\psi_{\textrm{B,I}}}\right|^{2}\left(\delta(t-\tau_{1})+\delta(t-\tau_{2})+\delta(t-\tau_{3})\right)=$
(202)
$\displaystyle\int_{t}^{\infty}\mathop{\textrm{d}\tau_{2}}\int_{\tau_{2}}^{\infty}\mathop{\textrm{d}\tau_{3}}\left|\braket{t,\tau_{2},\tau_{3}}{\psi_{\textrm{B,I}}}\right|^{2}+\int_{0}^{t}\mathop{\textrm{d}\tau_{1}}\int_{t}^{\infty}\mathop{\textrm{d}\tau_{3}}\left|\braket{\tau_{1},t,\tau_{3}}{\psi_{\textrm{B,I}}}\right|^{2}+\int_{0}^{t}\mathop{\textrm{d}\tau_{1}}\int_{\tau_{1}}^{t}\mathop{\textrm{d}\tau_{2}}\left|\braket{\tau_{1},\tau_{2},t}{\psi_{\textrm{B,I}}}\right|^{2}.\quad$

The pattern continues rather intuitively, where Eq. 199b is just a shorthand
for counting all possible ways each $m$-photon wavepacket can contribute to a
single excitation at time $t$. Notably, these expectations depend only on the
magnitude of projections onto the $\tau$-modes, i.e. on
$\left|\braket{\vec{\bm{\tau}}^{(m)}}{\psi_{\textrm{B,I}}}\right|^{2}$. This
corresponds to what we physically expect because intensity detectors do not
respond to the quantum phase of wavefunctions [57]. Hence, we can construct
classical probability density functions

$\mathbb{P}(\vec{\bm{\tau}}^{(m)})\equiv
p_{m}(\vec{\bm{\tau}}^{(m)})=\left|\braket{\vec{\bm{\tau}}^{(m)}}{\psi_{\textrm{B,I}}}\right|^{2}$
(203)

that correspond to the density of detecting $m$ photons at times
$t_{1},\dots,t_{m}$ (like we used in Fischer et al. [88])—for a given
scattered state, this is also a restatement of Eq. 136. Then, the
probabilities to detect a given number of photons after the entire pulse has
interacted with an ideal detector are given by the photocount distribution

$P_{m}=\int\mathop{\textrm{d}\vec{\bm{\tau}}^{(m)}}p_{m}(\vec{\bm{\tau}}^{(m)}).$
(204)

From these results come the phrase ‘temporal mode’, where an occupation in a
given temporal mode indexed by $t$ can trigger a detection at time $t$ by a
detector.

##  Appendix B Second-order coherence with temporal modes

We also note that a similar process can be repeated for the second-order
coherence

$\displaystyle G^{(2)}(t_{1},t_{2})$ $\displaystyle=$
$\displaystyle\braket{\psi_{\text{B,I}}}{b_{0}^{\dagger}(t_{1})b_{0}^{\dagger}(t_{2})b_{0}(t_{2})b_{0}(t_{1})}{\psi_{\text{B,I}}},$
(205)

as in Appendix A. After inserting resolutions of the identity, we instead need
to use

$\displaystyle\braket{\vec{\bm{\tau}}^{(m)}}{b_{0}^{\dagger}(t_{1})b_{0}^{\dagger}(t_{2})b_{0}(t_{2})b_{0}(t_{1})}{\vec{\bm{\tau}}^{\prime(m^{\prime})}}=$
(206)
$\displaystyle\delta_{mm^{\prime}}\delta(\vec{\bm{\tau}}^{(m)}-\vec{\bm{\tau}}^{\prime(m)})\sum_{q_{1}=1}^{m}\sum_{q_{2}=1}^{m}\delta(t_{1}-\tau_{q_{1}})\delta(t_{2}-\tau_{q_{2}})\,[(m>1)\land(q_{1}\neq
q_{2})]$

and then

$\displaystyle
G^{(2)}(t_{1},t_{2})=\sum_{m=2}^{\infty}\sum_{q_{1}=1}^{m}\sum_{q_{2}=1}^{m}\int\mathop{\textrm{d}\vec{\bm{\tau}}^{(m)}}\left|\braket{\vec{\bm{\tau}}^{(m)}}{\psi_{\textrm{B,I}}}\right|^{2}\delta(t_{1}-\tau_{q_{1}})\delta(t_{2}-\tau_{q_{2}})\,[q_{1}\neq
q_{2}].$ (207)

Intuitively, this expectation is used to count all possible ways to detect
two-photons from an $m$-photon wavepacket. For instance, the three-photon
wavepacket’s contribution to $G^{(2)}(t_{1},t_{2})$ for $t_{1}<t_{2}$ is

$\displaystyle\int_{t_{2}}^{\infty}\mathop{\textrm{d}\tau_{3}}\left|\braket{t_{1},t_{2},\tau_{3}}{\psi_{\textrm{B,I}}}\right|^{2}+\int_{t_{1}}^{t_{2}}\mathop{\textrm{d}\tau_{2}}\left|\braket{t_{1},\tau_{2},t_{2}}{\psi_{\textrm{B,I}}}\right|^{2}+\int_{0}^{t_{1}}\mathop{\textrm{d}\tau_{1}}\left|\braket{\tau_{1},t_{1},t_{2}}{\psi_{\textrm{B,I}}}\right|^{2}$
(208)

or

$\int_{t_{2}}^{\infty}\mathop{\textrm{d}\tau_{3}}p_{3}(t_{1},t_{2},\tau_{3})+\int_{t_{1}}^{t_{2}}\mathop{\textrm{d}\tau_{2}}p_{3}(t_{1},\tau_{2},t_{2})+\int_{0}^{t_{1}}\mathop{\textrm{d}\tau_{1}}p_{3}(\tau_{1},t_{1},t_{2}).$
(209)

(If $t_{1}>t_{2}$ then exchange those indices in Eqs. 208 or 209.)

From this point, it should be clear that all $m$-photon wavepackets contribute
to the coherence functions when $m$ is greater than the order of the coherence
function. Thus, although every correlation from Eq. 1 technically captures the
entire state of the field, transforming between the correlations and a state
vector is quite difficult, which of course also requires the assumption that
the state is pure. Further, in the case of scattering from energy-
nonconserving systems, it’s often hard to know before doing a calculation how
many correlation orders will contribute to a given $m$-photon wavepacket.

It is easy to extrapolate the pattern to the $R^{\text{th}}$ order
correlation, where

$\displaystyle G^{(R)}(t_{1},\dots,t_{R})$ (210)
$\displaystyle=\sum_{m=R}^{\infty}\sum_{q_{1}=1}^{m}\cdots\sum_{q_{R}=1}^{m}\int\mathop{\textrm{d}\vec{\bm{\tau}}^{(m)}}\left|\braket{\vec{\bm{\tau}}^{(m)}}{\psi_{\textrm{B,I}}}\right|^{2}\delta(t_{1}-\tau_{q_{1}})\delta(t_{2}-\tau_{q_{2}})\cdots\delta(t_{R}-\tau_{q_{R}})\,[q_{l}\neq
q_{k}],$

where by $[q_{l}\neq q_{k}]$ we mean that we can only have one delta function
at a given time.

##  Appendix C Mollow transformation

Figure 7: The Mollow transformation is a unitary operation that removes a
coherent state from a waveguide coupled to a system. It captures the effects
of a coherent state drive in the system Hamiltonian itself. In the new frame,
however, energy is not conserved.

Often, a classical laser pulse is incident on a quantum-mechanical system,
causing it to undergo transitions between its various levels and then re-emit
the absorbed energy. Typically the effect of the coherent pulse is lumped into
the system Hamiltonian as a time-dependent operator (acting on
$\mathcal{H}_{\textrm{S}}$) that has the time-dependence of the pulse’s phase.
Many often assume that this Hamiltonian is semi-classical, but Mollow
identified it is actually exact, provided the loss channel where the coherent
state originated is retained in the calculation [5]. He did this by finding a
unitary transformation that removes the coherent state from the waveguide
(Fig. 7). We now detail this calculation.

Consider a system with Hamiltonian $H_{0\text{S}}\\{a_{i}\\}$ coupled to a
single chiral waveguide described by a continuum of modes $b_{\omega}$. The
complete system can be modeled by

$H=H_{0\text{S}}\\{a_{i}\\}+\int_{-\infty}^{\infty}\mathop{\textrm{d}\omega}\omega\,b_{\omega}^{\dagger}b_{\omega}+\xi\int_{-\infty}^{\infty}\mathop{\textrm{d}\omega}\left(a_{0}b_{\omega}^{\dagger}+b_{\omega}a_{0}^{\dagger}\right),$
(211)

where $\xi=\sqrt{\gamma_{0}/2\pi}$ is the coupling constant between the
waveguide and the local system, with $\kappa$ being the decay rate induced by
the loss channel in the local system. The system is driven by a coherent state
through the waveguide, which corresponds to the following state at $t=0$

$|\psi(t=0)\rangle=\mathcal{D}\\{\alpha(\omega)\\}|0\rangle\otimes|\psi_{\text{S}}(0)\rangle,$
(212)

where
$\mathcal{D}\\{\alpha(\omega)\\}=\exp\left(\int_{-\infty}^{\infty}\mathop{\textrm{d}\omega}\,(\alpha(\omega)b_{\omega}^{\dagger}-\alpha^{*}(\omega)b_{\omega})\right)$
is the displacement operator creating a coherent state in the waveguide. We
define a transformed state $|\tilde{\psi}(t)\rangle$ via

$|\tilde{\psi}(t)\rangle=\mathcal{D}_{t}^{\dagger}\\{\alpha(\omega)\\}|\psi(t)\rangle,$
(213)

where

$\mathcal{D}_{t}\\{\alpha(\omega)\\}=\exp\left(\int_{-\infty}^{\infty}\mathop{\textrm{d}\omega}\left(\alpha(\omega)\textrm{e}^{-\textrm{i}\omega
t}b_{\omega}^{\dagger}-\alpha^{*}(\omega)\textrm{e}^{\textrm{i}\omega
t}b_{\omega}\right)\right).$ (214)

The time evolution of the state $|\tilde{\psi}(t)\rangle$ can be computed by
differentiating Eq. 213 to obtain an effective Hamiltonian

$\tilde{H}(t)=\mathcal{D}_{t}^{\dagger}\\{\alpha(\omega)\\}H\mathcal{D}_{t}\\{\alpha(\omega)\\}+\text{i}\frac{\mathop{\textrm{d}}}{\mathop{\textrm{d}t}}\mathcal{D}^{\dagger}_{t}\\{\alpha(\omega)\\}\mathcal{D}_{t}\\{\alpha(\omega)\\}\quad\textrm{with}\quad\textrm{i}\frac{\mathop{\textrm{d}}}{\mathop{\textrm{d}t}}|\tilde{\psi}(t)\rangle=\tilde{H}(t)|\tilde{\psi}(t)\rangle.$
(215)

We next compute the effective Hamiltonian $\tilde{H}(t)$ using Eq. 211. In
particular, it follows from the identities
$\mathcal{D}^{\dagger}\\{\alpha(\omega)\\}b_{\omega}\mathcal{D}\\{\alpha(\omega)\\}=b_{\omega}+\alpha(\omega)$
and
$\mathcal{D}^{\dagger}\\{\alpha(\omega)\\}b_{\omega}^{\dagger}\mathcal{D}\\{\alpha(\omega)\\}=b_{\omega}^{\dagger}+\alpha^{*}(\omega)$
that

$\displaystyle\mathcal{D}_{t}^{\dagger}\\{\alpha(\omega)\\}H\mathcal{D}_{t}\\{\alpha(\omega)\\}=H_{0\text{S}}\\{a_{i}\\}+\sqrt{\gamma_{0}}\left(\alpha(t)a_{0}^{\dagger}+\alpha^{*}(t)a_{0}\right)+\int_{-\infty}^{\infty}\mathop{\textrm{d}\omega}\omega
b_{\omega}^{\dagger}b_{\omega}+\int_{-\infty}^{\infty}\mathop{\textrm{d}\omega}\omega|\alpha(\omega)|^{2}+$
$\displaystyle\int_{-\infty}^{\infty}\mathop{\textrm{d}\omega}\omega\left(b_{\omega}\alpha^{*}(\omega)\textrm{e}^{\textrm{i}\omega
t}+b_{\omega}^{\dagger}\alpha(\omega)\textrm{e}^{-\textrm{i}\omega
t}\right)+\xi\int_{-\infty}^{\infty}\mathop{\textrm{d}\omega}\left(a_{0}^{\dagger}b_{\omega}+b_{\omega}^{\dagger}a_{0}\right),$
(216)

where
$\alpha(t)=\int_{-\infty}^{\infty}\frac{\mathop{\textrm{d}\omega}}{\sqrt{2\pi}}\textrm{e}^{-\textrm{i}\omega
t}\alpha(\omega)$.

Now, we need to compute
$\mathop{\textrm{d}\mathcal{D}_{t}^{\dagger}}\\{\alpha(\omega)\\}/\mathop{\textrm{d}t}$.
Consider the problem of computing the derivative of an exponential of a time
dependent operator $A(t)=\exp(\phi(t))$, with the property that
$[\phi(t),\text{d}\phi(t)/\text{d}t]=\zeta(t)$, with $\zeta(t)$ being a
scalar. The derivative of $A(t)$ can be computed as

$\frac{\text{d}A(t)}{\text{d}t}=\lim_{\delta t\to 0}\frac{\exp(\phi(t+\delta
t))-\exp(\phi(t))}{\delta t}.$ (217)

Using the Baker–Campbell–Hausdorff equality it follows that

$\exp(\phi(t+\delta t))\approx\exp\bigg{(}\phi(t)+\delta
t\frac{d\phi(t)}{dt}\bigg{)}=\exp(\phi(t))\exp\bigg{(}\delta
t\frac{\text{d}\phi(t)}{\text{d}t}\bigg{)}\exp(-\delta t\,\zeta(t)/2)$ (218)

resulting in

$\frac{\text{d}A(t)}{\text{d}t}=\exp(\phi(t))\bigg{(}\frac{\text{d}\phi(t)}{\text{d}t}-\frac{\zeta(t)}{2}\bigg{)}.$
(219)

Specializing this result to:
$A(t)=\mathcal{D}^{\dagger}_{t}\\{\alpha(\omega)\\}$ with
$\phi(t)=\int_{-\infty}^{\infty}\mathop{\textrm{d}\omega}\left(\alpha^{*}(\omega)\textrm{e}^{\textrm{i}\omega
t}b_{\omega}-\alpha(\omega)\textrm{e}^{-\textrm{i}\omega
t}b_{\omega}^{\dagger}\right)$ and
$\zeta(t)=2\text{i}\int_{-\infty}^{\infty}\mathop{\textrm{d}\omega}\omega|\alpha(\omega)|^{2}$
then

$\frac{\mathop{\textrm{d}\mathcal{D}_{t}^{\dagger}\\{\alpha(\omega)\\}}}{\mathop{\textrm{d}t}}=\textrm{i}\mathcal{D}_{t}^{\dagger}\\{\alpha(\omega)\\}\bigg{(}\int_{-\infty}^{\infty}\mathop{\textrm{d}\omega}\omega\left(\alpha^{*}(\omega)b_{\omega}\textrm{e}^{\textrm{i}\omega
t}+\alpha(\omega)b_{\omega}^{\dagger}\textrm{e}^{-\textrm{i}\omega
t}\right)-\int_{-\infty}^{\infty}\mathop{\textrm{d}\omega}\omega|\alpha(\omega)|^{2}\bigg{)},$
(220)

with which we obtain

$\frac{\mathop{\textrm{d}\mathcal{D}_{t}^{\dagger}\\{\alpha(\omega)\\}}}{\mathop{\textrm{d}t}}\mathcal{D}_{t}\\{\alpha(\omega)\\}=\textrm{i}\int_{-\infty}^{\infty}\mathop{\textrm{d}\omega}\omega\left(\alpha^{*}(\omega)b_{\omega}\textrm{e}^{\textrm{i}\omega
t}+\alpha(\omega)b_{\omega}^{\dagger}\textrm{e}^{-\textrm{i}\omega
t}\right)+\int_{-\infty}^{\infty}\mathop{\textrm{d}\omega}\omega|\alpha(\omega)|^{2}.$
(221)

Putting together Eqs. 215, C and 221

$\tilde{H}(t)=H_{\text{S}}(t)+\int_{-\infty}^{\infty}\mathop{\textrm{d}\omega}\omega\,b_{\omega}^{\dagger}b_{\omega}+\xi\int_{-\infty}^{\infty}\mathop{\textrm{d}\omega}\left(a_{0}^{\dagger}b_{\omega}+a_{0}b_{\omega}^{\dagger}\right),$
(222)

where

$H_{\text{S}}(t)=H_{0\text{S}}+H_{1\text{S}}(t)\quad\text{and}\quad
H_{1\text{S}}(t)=\sqrt{\kappa}\left(\alpha(t)a_{0}^{\dagger}+\alpha^{*}(t)a_{0}\right)$
(223)

is the system Hamiltonian with an effective classical driving field added to
it. (Here we drop the tilde in $H_{\text{S}}$ due to the appearance of the
explicit time-dependence.) In this frame, energy is no longer conserved due to
the time-dependent operator $H_{1\text{S}}(t)$ added to $H_{0\text{S}}$.

This transformation caries through with any arbitrary number of waveguides
also coupled to the system Hamiltonian as depicted in Fig. 7, though we did
not want to clutter the calculation with the extra notation. In experiment,
the coherent state sometimes has a very large photon number but is only weakly
coupled to the system through $\gamma_{0}$, while other waveguides act as the
primary decay channels for the system. In this case
$\gamma_{0}\ll\gamma_{1},\dots,\gamma_{M}$, and in the limit that
$\gamma_{0}\rightarrow 0$ and $|\alpha(t)|\rightarrow\infty$ while keeping
$\sqrt{\gamma_{0}}|\alpha(t)|$ fixed, the state of the system factorizes with
the $0^{\textrm{th}}$ waveguide at all times while still pumping energy into
$\mathcal{H}_{\text{S}}$ at a finite rate—this is the semi-classical coherent
driving limit.

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# ML4CO-KIDA: Knowledge Inheritance in Dataset Aggregation

Zixuan Cao
Peking University
Megvii Research
<EMAIL_ADDRESS>
&Yang Xu
Peking University
Megvii Research
<EMAIL_ADDRESS>
&Zhewei Huang
Megvii Research
<EMAIL_ADDRESS>
&Shuchang Zhou
Megvii Research
<EMAIL_ADDRESS>

###### Abstract

The Machine Learning for Combinatorial Optimization (ML4CO) NeurIPS 2021
competition aims to improve state-of-the-art combinatorial optimization
solvers by replacing key heuristic components with machine learning models. On
the dual task, we design models to make branching decisions to promote the
dual bound increase faster. We propose a knowledge inheritance method to
generalize knowledge of different models from the dataset aggregation process,
named KIDA. Our improvement overcomes some defects of the baseline graph-
neural-networks-based methods. Further, we won the $1$st Place on the dual
task. We hope this report can provide useful experience for developers and
researchers. The code is available at https://github.com/megvii-
research/NeurIPS2021-ML4CO-KIDA.

In the dual task of ML4CO NeurIPS $2021$ competition, as shown in Table 1, our
team (Nuri) ranks $1$st, $6$th, and $1$st in three benchmarks of Balanced Item
Placement, Workload Apportionment, and Anonymous Problem separately. Our
report is organized as follows: Section 1 introduces the basic information of
ML4CO; Section 2 introduces issues we observe when following the baseline
method; Section 3 explains in detail the main method we develop to improve the
quality of branching decisions further; Section 4 shows the performance of
different models in each benchmark; Section 5 discusses the failure of Strong
Branching [1] which is regarded as expert knowledge; We conclude our report in
Section 6.

Table 1: Leaderboard of the Dual Task in ML4CO NeurIPS 2021 competition

Team | Item Placement | Load Balancing | Anonymous | Score $\downarrow$
---|---|---|---|---
Cum. Reward $\uparrow$ | Cum. Reward $\uparrow$ | Cum. Reward $\uparrow$
Nuri (Ours) | 6684.00 | 630787.18 | 27810782.42 | $\bm{6}~{}(1\times 6\times 1)$
EI-OROAS | 6670.30 | 631744.31 | 27158442.74 | $8~{}(2\times 1\times 4)$
EFPP | 6487.53 | 631365.02 | 26340264.47 | $117~{}(3\times 3\times 13)$
KAIST_OSI | 6196.56 | 631410.58 | 26626410.86 | $126~{}(7\times 2\times 9)$
qqy | 6377.23 | 630557.31 | 27221499.03 | $132~{}(6\times 11\times 2)$

##  1 Preliminary

We will introduce the background knowledge needed in the dual task and the
official baseline based on graph neural network (GNN) [19, 13].

###  1.1 Dual Task

For mixed-integer linear programs (MILPs), a well-known solving method is
branch-and-bound (B&B) [14]. One of the key issues of B&B is how to select
variables for Branching. In ML4CO competition, the goal of the dual task is to
promote the dual bound to increase faster by selecting proper branching
variables. The final metric is the dual integral over time, shown in Figure 1.

###  1.2 Branching Algorithms

Although making branching decisions has received little theoretical
understanding to this day [16], there are many heuristic algorithms. Strong
Branching involves testing which of the candidate variable gives the best
improvement to the objective function before actually branching on them.
Strong Branching is a high-quality but computationally expensive method. In
practice, modern B&B solvers rely on hybrid-branching [15] and reliability
branching [1], which only use Strong Branching at certain nodes of the search
tree and use pseudo-cost branching [3] at other nodes.

Figure 1: Evaluation Metric of the Dual Task. Our goal is to minimize the dual
integral over time

###  1.3 Official Baseline

The baseline method [8] regards the decision of Strong Branching as expert
knowledge and train models by imitating the expert knowledge. Specifically, we
convert the combinatorial optimization problems to bipartite graphs,
illustrated in Figure 2. Then we use the GNN to obtain the embedding of each
variable. These embeddings are used to generate branching strategies for the
current state. We train this GNN by minimizing the cross-entropy loss between
its output and the expert decision.

$\mathbf{v}_{1}$$\mathbf{v}_{2}$$\mathbf{v}_{3}$$\mathbf{c}_{1}$$\mathbf{c}_{2}$$\mathbf{e}_{1,1}$$\mathbf{e}_{1,3}$$\mathbf{e}_{1,2}$$\mathbf{e}_{2,3}$

$\mathbf{C}$$\mathbf{E}$$\mathbf{V}$$\mathbf{C}^{1}$$\mathbf{V}^{1}$$\mathbf{C}^{2}$$\mathbf{V}^{2}$$\pi(\mathbf{x})$$n\times
d$$m\times n\times e$$m\times c$$n\times 64$$m\times 64$$m\times 64$$n\times
64$$n\times
1$initialembedding$\mathcal{C}$-sideconvolution$\mathcal{V}$-sideconvolutionfinalembedding\+
softmax

Figure 2: Copy from [8]. Left: the bipartite state representation
$\mathbf{s}_{t}=(\mathcal{G},\mathbf{C},\mathbf{E},\mathbf{V})$ with $n=3$
variables and $m=2$ constraints. Right: the bipartite GCNN architecture for
parametrizing baseline policy $\pi_{\theta}(\mathbf{a}\mid\mathbf{s}_{t})$

##  2 Bad Generalization Issue

Following the method in [8], we summarize two issues that may reduce the final
performance.

First, there is a major gap between offline training and the online deployment
phase. In offline training, the data is collected beforehand. The GNN model
achieves an accuracy of around $0.80$ (validation). However, the accuracy
shows a sharp decline when deploying a trained model. As shown in Figure 3,
when making branching decisions for a random online instance, the averaged
accuracy of different steps is about $0.1$.

Figure 3: Accuracy when Interacting with SCIP Solver. During the deployment
phase, the accuracy drops a lot

Second, the higher accuracy does not necessarily lead to a higher reward. As
shown in Table 2, comparing models in different epochs, the accuracy is not
strongly related to the final reward.

Table 2: Performance Comparison of Different Epochs. The imitation accuracy is not directly related to model performance Epoch | Top 1 Acc. | Top 3 Acc. | Top 5 Acc. | Cum. Reward
---|---|---|---|---
1 | 0.780 | 0.932 | 0.971 | 5202.6
5 | 0.803 | 0.948 | 0.981 | 5545.7
10 | 0.808 | 0.946 | 0.978 | 5131.9
20 | 0.810 | 0.947 | 0.979 | 5038.8

##  3 Method

###  3.1 DAgger

The major performance gap between training and deployment can be explained by
the defects of Behavior Cloning [2], which forces agents to learn expert
behavior. There is a common problem that the distribution of collected data
and the data the model encounters in the real scene may be inconsistent [4,
17]. For example, a driving-related dataset is rich in driver’s operation data
during normal driving. But when the car is about to hit the wall, there may be
a lack of data to deal with the contingency. To make a more completely
collected dataset, we need to constantly explore the unseen data and label
these data with expert knowledge.

Dataset Aggregation (DAgger) [18] is such a method to reduce the gap between
training and depolyment. Based on DAgger, we collect data iteratively and
train new models constantly. Details are described in Algorithm 1.

Algorithm 1 DAgger in Dual Task

1:Initialize a random model $\pi_{0}$ and an empty dataset $D$

2:for each $i\in[1,50]$ do

3: Interact with the solver with $0.95$ probability of using model $\pi_{i-1}$
and 0.05 probability of using Strong Branching

4: Collect data obtained by Strong Branching as $D_{i}$

5: $D=D\cup D_{i}$

6: if $i~{}mod~{}10==0$ then

7: Train $\pi_{i}$ with $100$ epochs

8: else

9: Train $\pi_{i}$ with $10$ epochs

10: end if

11:end for

###  3.2 KIDA

As shown in Section 2, in the Item Placement benchmark, higher accuracy does
not lead to higher rewards. Experiments show that the performance of different
models obtained in DAgger is not stable. Therefore, we use model ensemble [5,
12] to improve the performance of our model further. A noticeable difficulty
is that the practicality of the model is time-sensitive. Averaging the output
of different models is time-consuming. We consider averaging the weights of
different models [21]. Here, we define Knowledge Inheritance in Dataset
Aggregation (KIDA) as building a new model $\pi_{avg}$ by averaging the
weights of trained models during the dataset aggregation process. The
framework is shown in Figure 4. The pipeline of KIDA is similar to Born-Again
Neural Networks [6]. In KIDA, the training of the current model depends on the
generation using the last model. The final model is the ensemble of trained
models.

Formally, for models obtained from dataset aggregation with parameters
$(\theta_{0},\theta_{1},...,\theta_{n-1})$, the parameters of $\pi_{avg}$ are
obtained by: $\theta_{avg}=\sum_{i=0}^{n-1}\theta_{i}/n.$

Figure 4: Framework of KIDA. We train each model based on the data generation
using the last model. Finally, we average the parameter of trained models

Figure 5: KIDA vs. Epoch Weight Average

Figure 5 shows the difference between KIDA and the popular epoch weight
average method. KIDA focuses on models trained by different data from
different iteration rounds of DAgger. Although the training data of models are
different, these data are related. Common epoch weight average such as
Snapshot ensembles [10] and SWA [11] focuses on models that are obtained in
the same training process of different epochs.

##  4 Experiment

In ML4CO competition, there are three benchmarks, including Balanced Item
Placement, Workload Apportionment, and Anonymous Problem 111Details about each
benchmark can refer to https://www.ecole.ai/2021/ml4co-competition/ which need
to be evaluated separately. In this section, we discuss the performance of
different models in each benchmark.

###  4.1 Anonymous Problem

We show the performance of models with different settings in the validation
set in Figure 6. With the same data size, the model using DAgger performs
better. With the increase of data size, the performance increases. The model
we finally submitted is a new DAgger model with Dropout [20] layers and a
longer solver time when collecting data.

Figure 6: The Cumulative Reward of Anonymous in Validation Dataset. The
performance of DAgger models is related to the number of iteration rounds

###  4.2 Balanced Item Placement

Overall Performance. The performance of DAgger at different iteration rounds
is shown in Figure 7. Unlike in the Anonymous benchmark, the performance of
DAgger is unstable. But when we apply KIDA in this benchmark, the performance
has significantly improved.

Figure 7: The Cumulative Reward of Item Placement in Validation set. KIDA has
significant advantages over just applying DAgger

Detailed Evaluation. To further explore our method, we randomly process a
problem from the validation set using Strong Branching and compare the output
of different models with Strong Branching labels. The accuracy of different
models is shown in Figure 8. DAgger model still has the highest accuracy, and
the accuracy of the baseline model is the worst.

Figure 8: Accuracy when Interacting with the Solver. Overall accuracy of DAgger 20, KIDA and the baseline are 0.635, 0.182, 0.071 respectively. Table 3: Performance Comparison between DAgger models and the KIDA model. KIDA suffers from an accuracy drop but has a lower loss Model | Top 1 Acc. | Top 3 Acc. | Top 5 Acc. | Loss | Cum. Reward
---|---|---|---|---|---
Model 0 | 0.850 | 0.957 | 0.981 | 7.38 | 5304.9
Model 1 | 0.797 | 0.917 | 0.966 | 9.50 | 5319.8
Model 2 | 0.795 | 0.916 | 0.961 | 6.18 | 5237.5
KIDA | 0.721 | 0.822 | 0.870 | 2.96 | 7561.6

Models used in KIDA are the top $3$ performance models of DAgger. Table 3
shows the comparison between these models and the KIDA model. Although KIDA
has lower accuracy in collected validation data, it has a lower loss value and
higher reward.

###  4.3 Workload Apportionment

In Workload Apportionment, baselines are trained and compared with random
policy. From the table, we can see that the top $1$ accuracy of the baseline
model is 45.6%. The accuracy of a classification task with more than one
hundred labels shows that the model has learned much expert knowledge.
However, random strategies that do not rely on prior knowledge can obtain
higher cumulative rewards.

Table 4: Performance Comparison between Baseline and Random Policy. The baseline model gets great accuracy but fails to achieve higher rewards | Top 1 Acc. | Top 3 Acc. | Top 5 Acc. | Cum. Reward
---|---|---|---|---
Baseline | 0.456 | 0.729 | 0.820 | 624043.6
Random | 0.013 | 0.034 | 0.052 | 624928.9

##  5 Discussion

Overall, we show that Strong Branching can not produce completely reliable
labels. To further explore the performance of Strong Branching, we compare the
variation of the dual bound using different methods.

(a) Item Placement

(b) Load Balancing

(c) Anonymous

Figure 9: Dual Bound Improvements with Steps. Strong Branching may still fall
behind GNN-based policy or even random policy

As shown in Figure 9, we select a random instance for each benchmark, the
variation of the dual bound using different strategies. Even if we ignore the
expensive time cost of Strong Branching, the dual bound improvement may still
fall behind GNN models. This observation may be explained by dual degeneracy
[9]. If there is a high dual degeneracy in LP solution, the product score of
Strong Branching that SCIP uses to combine the improvements of the two-child
nodes would be close to zero [7]. In that case, Strong Branching may fail, and
the expert knowledge needs to be redesigned based on other information
accumulated in the problem-solving process.

##  6 Conclusion

In this paper, we devise a knowledge inheritance method in the dataset
aggregation process for the dual task of ML4CO competition. By dataset
aggregation, the inconsistency between training and deployment is reduced. By
averaging the weights of different models from the dataset aggregation
process, learned knowledge is generalized to get better results. Our model
gets great improvements on Item Placement and Anonymous. Further, our
experiments show models closer to expert knowledge do not necessarily achieve
better results, which indicates Strong Branching fails in some cases. More
heuristic branching algorithms need to be taken into account to build more
reasonable expert knowledge to facilitate learning to branch.

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