# Enhanced-sensitivity interferometry with phase-sensitive unbiased multiports

Christopher R. Schwarze<EMAIL_ADDRESS>Department of Electrical and Computer
Engineering & Photonics Center, Boston University, 8 Saint Mary’s St., Boston,
Massachusetts 02215, USA  David S. Simon<EMAIL_ADDRESS>Department of
Electrical and Computer Engineering & Photonics Center, Boston University, 8
Saint Mary’s St., Boston, Massachusetts 02215, USA  Department of Physics and
Astronomy, Stonehill College, 320 Washington Street, Easton, Massachusetts
02357, USA  Alexander V. Sergienko<EMAIL_ADDRESS>Department of Electrical
and Computer Engineering & Photonics Center, Boston University, 8 Saint Mary’s
St., Boston, Massachusetts 02215, USA  Department of Physics, Boston
University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA

###### Abstract

Here we introduce interferometric devices by combining optical feedback
(cavities) with unbiased multiports, which unlike traditional beam dividers,
allow light to reflect back out of the port from which it originated. By
replacing the traditional, directionally-biased beam-splitter in a Michelson
interferometer with an unbiased multiport,  the functional dependence of the
scattering amplitudes changes. As a result, the derivative of transmittance
with respect to an external phase perturbation can be made substantially
large. This significantly enhances the resolution of phase measurement, and
allows the phase response curves to be altered in real time by tuning an
externally-controllable phase shift.

##  I Introduction

Beam dividing is a fundamental manipulation of light, forming the basis of
applications such as interferometry [1] and optical information processing
[2].  Many optical apparatuses rely on a variant of the standard beam-splitter
to separate and combine light. This might be a cube beam-splitter, a pellicle
one, or perhaps a fiber coupler. In any case, these devices share the
following characteristic: the light entering a given port cannot exit that
same port. This directionally-biased behavior effectively lowers the
dimensionality of the device: an output state is unable to populate as many
modes as there are ports due to the lack of a coupling between the input port
and the mode counter-propagating from that port. Sometimes this feed-forward
nature is desirable, for instance, to prevent back-reflections from re-
entering a laser cavity. However, the use of strictly feed-forward scattering
devices constrains the output states that can be generated, and consequently,
limits the capabilities of the systems that use them.

A generalization to the class of feed-forward linear-optical scatterers was
introduced in [3]. These new devices are directionally-unbiased, allowing
light to reflect out the port that it entered. In addition to reducing the
number of optical elements needed to enact certain operations, this property
has driven a number of theoretical developments in entangled-state information
processing [4], Hamiltonian simulation of topologically nontrivial systems
[5][6], and optical sensing with the Sagnac effect[7].

In this article, we use a particular directionally-unbiased counterpart of the
beam-splitter called the Grover “$N$-sided coin”, which will be introduced in
greater detail later. With the four-port Grover coin, we form a generalization
of the Michelson interferometer which can be configured to have an arbitrarily
large sensitivity to a phase perturbation. This behavior stems from the
optical cavities which get formed by replacing the beam-splitter with the
unbiased Grover coin. Intentionally leaky cavities of this sort formed the
basis of the free-space realization of the 3-port Grover coin [8]. In this
case, we analyze how a controllable phase shift acquired during each round
trip in the cavity can be used to create a lower-dimensional device with a
tunable scattering matrix.

The outline for the remainder of the article is as follows. In the next
section, we review linear-optical scattering theory for monochromatic
radiation interacting with devices such as the beam-splitter or Grover coin.
In Section III, we compare various unbiased and tunable scattering devices
formed from either a single beamsplitter or a 4-port Grover coin. All devices
considered are tuned by changing optical phase elements; that is, for new
values of each phase shift, a new scattering matrix is produced which is
periodic in $2\pi$. In Section IV, we focus on a two-cavity configuration
which resembles the Michelson interferometer. Unlike the standard Michelson
interferometer, this Grover-based version can be tuned to have a transmittance
with an arbitrarily steep slope, which can be used to obtain enhanced sensing.
Conclusions are drawn in Section V.

##  II Linear-optical scattering theory

We will mathematically represent a beam-splitter as a particular linear,
spatially-coherent scattering transformation. Any optical device that can be
expressed this way we will call a “multiport”. The action of a multiport is
often expressed in the ideal case as a unitary transformation, which operates
on the probability amplitudes of an optical state. It is common to express
such a state in a Fock basis of spatial modes.

Figure 1: (left) One particular realization of a directionally-unbiased
optical multiport, formed by linking four 50:50 beam-splitters to form a leaky
cavity. The distance between beam-splitters is twice that between a mirror and
neighboring beam-splitter. By fixing $\phi=-3\pi/4$, so that the total phase
per mirror-unit is $-\pi/2$, one obtains the four-port Grover coin [6]. We
abstract this four-port device into the circuit diagram (right), which will be
used throughout the rest of the article.

If we consider this basis and additionally assume all excitations of interest
are confined to four  ports, a general superposition of arbitrarily polarized,
monochromatic radiation takes the form
$|\psi\rangle=(c_{1}a_{1}^{\dagger}+c_{2}a_{2}^{\dagger}+c_{3}a_{3}^{\dagger}+c_{4}a_{4}^{\dagger})|0\rangle$.
The subscripts on the creation operators indicate the port numbers of a given
four-port device, such as those depicted in Fig. 1. The field modes could be
fiber modes or free space plane waves, depending on the physical
implementation of the system. Next we identify each  photon creation operator
$a_{j}^{\dagger}$ with the standard basis vector $e_{j}$, which equals $1$ at
its $j$th element and $0$ elsewhere.  In our notation, ingoing and outgoing
modes of the same port will be associated with the same standard basis vector.

In this formalism, the state $|\psi\rangle$ may be expressed as a column
vector

$|\psi\rangle=\begin{pmatrix}c_{1}\\\ c_{2}\\\ c_{3}\\\ c_{4}\end{pmatrix}.$
(1)

while a 50:50 beam-splitter would have a scattering matrix given by

$B=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&1&1\\\ 0&0&1&-1\\\ 1&1&0&0\\\
1&-1&0&0\end{pmatrix},$ (2)

and the output state of $|\psi\rangle$ interacting with the beam-splitter is
then given by $B|\psi\rangle$ [9][10].

The transformation (2), however common in practice, is relatively sparse:
modes 1 and 2 are only routed to modes 3 and 4 and vice versa; no energy
returns to the mode from which it came. In this sense, the device is feed-
forward or directionally-biased. Hence the scattering matrix above is often
depicted by the $2\times 2$ matrix

$H=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix}$ (3)

and the feed-forward nature is assumed implicitly.

When static scattering devices such as the beam-splitter are combined with
optical phase shift elements, the resulting device may be viewed as a
realization of a tunable scattering matrix. Non-trivial device behavior is
often obtained with interference, such as when excitations in multiple spatial
modes are superimposed after a tunable delay is introduced between them. To
obtain exotic forms of interference, an increasing number of optical elements
and/or degrees of freedom are typically required. However, as we will discuss
later, interference across an infinite number of cavity paths can give rise to
novel effects without substantially increasing either.

An important instance of an unbiased multiport is the Grover coin, which
originates from a matrix that appeared in Grover’s search algorithm [11] and
has since been adopted to studies of quantum walks [12]. The $d$-port Grover
coin (where $d\geq 3$) is defined to have a reflection amplitude of $(2/d-1)$
and transmission amplitude of $2/d$ at each output port, for all input ports.
Therefore in addition to being unitary, it is also real and symmetric. Grover
coins have been experimentally realized for the case of $d=3$ [8][13] and
fabrication of an integrated version of the $d=4$ case is underway.

In the $d=4$ case, the scattering matrix $G$ is given by

$G=\frac{1}{2}\begin{pmatrix}-1&1&1&1\\\ 1&-1&1&1\\\ 1&1&-1&1\\\ 1&1&1&-1\\\
\end{pmatrix}.$ (4)

As in Eq. (2), the output probabilities are equal at each port.  The free-
space realization of the device, first presented in [6], is shown next to its
abstract circuit symbol in Fig. 1. In order for this realization to possess
the scattering matrix (4), the phase imparted in each arm $\phi$ must equal
$-\pi/2$. Moreover, the source of radiation must be coherent with respect to
the length scales of the beam-splitter arrangement. This prevents various
internal routes that a photon could propagate along from being distinguished
by the photon exit times, causing the amplitudes corresponding to these paths
to interfere. These coherence relations are further discussed in [3].

The permutation-symmetry of the matrix (4) manifests as a rotational symmetry
in the device in Fig. 1 (left). Since this Grover coin and the beam-splitter
both have four ports, it is interesting to consider what results when
traditional beam-splitters are replaced with Grover 4-ports in conventional
interferometers. It has recently been shown that in the Mach-Zehnder topology,
the Grover coin enables simultaneous measurements of different phase shifts
[7].

##  III Grover Optical Cavities

Figure 2: (left) A simple optical cavity coupled to a Grover coin (G). The
spatial mode at the lower port is converted to a cavity mode by sealing the
port with a mirror. Light in this mode oscillates back and forth, depicted
schematically with the blue arrow. As the light bounces within the cavity,
some energy leaks into the output ports, interfering with the radiation in
these ports. As $\phi$ changes, the interference in the output ports changes,
modulating the final scattering matrix. (right) The same topology but with a
traditional beam-splitter instead of a Grover coin. Because the beam-splitter
is directionally-biased, this configuration does not form an optical cavity.
All energy will leave the sealed port mode after a single round trip.

Consider the configuration with one simple cavity affixed to a 4-port Grover
coin, such as the sealed 4-port Grover coin pictured in Fig. 2. Light entering
the cavity strikes the mirror, accumulating phase $\pi$ from the mirror in
addition to a round-trip phase $\phi$. Since the device has three open ports,
the scattering matrix will be $3\times 3$ and its elements will vary
$2\pi$-periodically with $\phi$. After each pass through the cavity, 3/4 of
the energy of the optical state is transferred from the cavity mode uniformly
into the spatial modes of the open ports.

When computing the scattering matrix, one can avoid making redundant
calculations using the permutation symmetry of the Grover coin; after
computing the output amplitudes for input at one port, the output state for
input at a different port is the same but with the corresponding amplitudes
permuted. In this example, the cavity mode amplitudes map for each round trip
into a simple recursive form which can be converted to a geometric series.
Ultimately, the reflected amplitude is of the form $r=(e^{i\phi}-2)^{-1}$ and
the transmitted amplitude is $t=1+r$. One may readily verify these amplitudes
are normalized: $|r|^{2}+2|t|^{2}=1$. For $\phi=0$, the device behaves as a
$3$-sided mirror.

More interestingly, for $\phi=\pi$, the reflection probability reaches a
minimum and the device behaves as a Grover $3$-port. It can be shown this
property generalizes to Grover coins of arbitrary dimension $d$: a single seal
cavity will create a device that interpolates between a $(d-1)$-port mirror
and $(d-1)$-port Grover coin. Conversely, if one coherently couples an
$n$-port Grover coin to a $d$-port Grover coin with a net 0 phase shift
(modulo $2\pi$) in the cavity, a $(d+n-2)$-port Grover coin is formed.
Applications of these facts will be explored elsewhere, and in the remainder
of this article we focus on applications with low-dimensional devices.

Figure 3: (left) Traditional Michelson interferometer. (right) A cavity-
coupled beam-splitter device. The scattering coefficients for both of these
devices can be expressed as a function of the quantity $(\phi_{1}+\phi_{2})$
or $(\phi_{1}-\phi_{2})$.

The cavity formed in the previous example will not exist if a beam-splitter is
used instead of a Grover coin. It will have a partially-unbiased scattering
matrix: input at two ports will be directionally-unbiased while input at the
third does not interact with the mirror and is consequently feed-forward. In
fact, this device formed the essential building block of the original free-
space realization of the Grover multiport [3][8].

Placing a pair of mirrors on adjacent sides of a beam-splitter will not form a
cavity either but doing so will form another device of great practical use:
the Michelson interferometer, shown in Fig. 3 (left). We can view this
interferometer as a tunable, unbiased, two-port scattering device. Its
reflection and transmission probabilities are given by

$\displaystyle R$
$\displaystyle=\cos^{2}\bigg{(}\frac{\phi_{1}-\phi_{2}}{2}\bigg{)}$ (5a)
$\displaystyle T$
$\displaystyle=1-R=\sin^{2}\bigg{(}\frac{\phi_{1}-\phi_{2}}{2}\bigg{)}$ (5b)

where $\phi_{1}$ is the phase in the first arm of the interferometer, and
$\phi_{2}$ is that of the second.

If two mirrors are alternatively arranged to lie on opposite sides of the
beam-splitter, as shown in Fig. 3 (right), an optical cavity is formed. The
scattering matrix elements are found by coherently summing the probability
amplitudes for all paths light can take between a given input port and a given
output port. This device is also tunable, with the reflection and transmission
probabilities given by

$\displaystyle R$ $\displaystyle=\frac{1}{5-4\cos(\phi_{1}+\phi_{2})},$ (6a)
$\displaystyle T$ $\displaystyle=1-R.$ (6b)

In the case of using a conventional beam-splitter, the cavity and Michelson
device probabilities may be expressed as functions of either
$(\phi_{1}+\phi_{2})$ or $(\phi_{1}-\phi_{2})$. If $\phi_{2}$ is fixed at some
value, the probabilities can be viewed as a $2\pi$-periodic curve
parameterized by $\phi_{1}$, or vice versa. Altering $\phi_{2}$ then produces
a new curve which is function of $\phi_{1}$. The points on any particular
curve of fixed $\phi_{2}$ are prescribed for each value of $\phi_{1}$.  Due to
the functional dependence of the form $(\phi_{1}\pm\phi_{2})$, when the fixed
phase $\phi_{2}$ is allowed to vary, the entire transmission curve is merely
translated along the horizontal axis without making changes to its structure.
Hence an entire degree of freedom is redundant. In addition to this, the
cavity-based beam-splitter device cannot be tuned to obtain all reflection and
transmission probabilities in the range $[0,1]$; this can be seen since the
minimum in Eq. (6a) occurs when the denominator is maximized, which in turn
occurs when the cosine term equals its extreme value of -1, so the minimum
reflection probability obtainable is 1/9.

##  IV Enhanced sensitivity Grover-Michelson interferometer

Figure 4: Grover-Michelson interferometer, formed by replacing the beam-
splitter in a conventional Michelson interferometer with a Grover coin. By
attaching a source and detector to the open ports, an enhanced phase sensing
device is formed; the interference of cavity amplitudes leads to a nonlinear,
continuously modified phase response, allowing the slope of the output
probabilities with respect to a phase perturbation be made as steep or flat as
desired.

A novel two-port device can be created from the 4-port Grover coin by forming
separate cavities at two ports. The device topologies in Fig. 3 coalesce into
this “Grover-Michelson” device when the beam-splitter is replaced by a Grover
coin, as the latter device is permutation symmetric. The resulting
configuration is shown schematically in Fig. 4. Light from a coherent source
enters a port and the transmission probability is measured at the other open
port. Some of the incident energy is coupled into two cavities, reflecting
from a mirror and re-entering the Grover coin after accumulating a phase shift
$\phi_{j}$ in cavity $j$.

Assume the cavities are coupled to ports 3 and 4. If one studies the round-
trip propagation of $(a_{3}^{\dagger}+a_{4}^{\dagger})$ and
$(a_{3}^{\dagger}-a_{4}^{\dagger})$ rather than $a_{3}^{\dagger}$ and
$a_{4}^{\dagger}$ individually, the transformation from one round-trip to the
next forms a recurrence relation which may be unrolled into a geometric series
and then explicitly summed (see Appendix B). Thus these linear combinations
somewhat emulate the role of coupled-cavity supermodes. In the appendix we
show the output state corresponding to
$|\psi_{0}\rangle=a_{1}^{\dagger}|0\rangle$ is given by

$\displaystyle|\psi_{\text{out}}\rangle=\bigg{[}\bigg{(}\frac{C(\phi_{1},\phi_{2})^{2}}{2B(\phi_{1},\phi_{2})-2}-\frac{B(\phi_{1},\phi_{2})}{2}-\frac{1}{2}\bigg{)}a_{1}^{\dagger}$
$\displaystyle+\bigg{(}\frac{C(\phi_{1},\phi_{2})^{2}}{2B(\phi_{1},\phi_{2})-2}-\frac{B(\phi_{1},\phi_{2})}{2}+\frac{1}{2}\bigg{)}a_{2}^{\dagger}$
$\displaystyle\bigg{]}|0\rangle,$ (7)

where

$\displaystyle B(\phi_{1},\phi_{2})$
$\displaystyle\coloneqq\frac{1}{2}(e^{i\phi_{1}}+e^{i\phi_{2}})$ (8a)
$\displaystyle C(\phi_{1},\phi_{2})$
$\displaystyle\coloneqq\frac{1}{2}(e^{i\phi_{1}}-e^{i\phi_{2}}).$ (8b)

In fact, we show in the appendix that up to a $\pi$ phase shift, $B$ and $C$
are respectively the $r$ and $t$ for the standard Michelson interferometer.
Therefore the use of a Grover coin instead of a beam-splitter results in a
direct nonlinear transformation of the device’s scattering coefficients.

Over the parameters $\phi_{1}$ and $\phi_{2}$, both the Michelson and Grover-
Michelson interferometers span the line $R+T=1$ with $R,T\geq 0$. This line
forms the state space of classical scattering transformations derived from a
tunable $U(2)$ device. Despite sharing this space entirely, the Grover-
Michelson carries a significant advantage in the way its dependence on
$\phi_{1}$ and $\phi_{2}$ covers the line $R+T=1$. To see this, consider how
its reflection and transmission probability curves $R_{\phi_{2}}(\phi_{1})$
and $T_{\phi_{2}}(\phi_{1})$ vary with $\phi_{2}$, as in Fig. 5.  In the case
of the regular Michelson interferometer, we recall from the previous section
the scattering probabilities were functions of the quantity
$(\phi_{1}-\phi_{2})$. This meant that as one phase was varied, the
probability curve was merely translated.

However, in the Grover-Michelson device, the dependence on $\phi_{1}$ and
$\phi_{2}$ cannot be expressed in this way. The geometric series summation of
cavity amplitudes places phase dependence in the denominator in the output
amplitudes, resulting in nonlinear behavior in $R$ and $T$. Examples of
reflection probability curves are plotted in Fig. 5. The deformation of each
probability curve constrained on the domain $[0,2\pi]$ is over a family of
continuous paths with fixed endpoints, which is an instance of homotopy. At
$\phi_{2}=\pi$ the tuning curves are symmetric about $\phi_{1}=\pi$.  However,
as $\phi_{2}$ increases in distance from $\pi$, the curves are increasingly
skewed toward the periodic endpoint, increasing the maximum slope obtainable
on a given curve.

Figure 5: Transmission probability curve in one period $0\leq\phi_{1}\leq
2\pi$ for the Grover-Michelson interferometer at various values of $\phi_{2}$.
At $\phi_{2}=\pi$, $T$ is symmetric, but as $\phi_{2}$’s distance from $\pi$
is increased, the curve becomes increasingly skewed, leaving the periodic
endpoints fixed. The homotopy originates from the fact that $R$ and $T$ are
given by composition of nonlinear functions of $(\phi_{1},\phi_{2})$, which
itself stems from the cavity-coupled nature of the device. Given suitable
control of the phases $\phi_{1}$ and $\phi_{2}$, perturbations in the
$\phi_{1}$ can be made to have a small or large response in $R$ and $T$, while
in the standard Michelson the maximum slope is always 1. The gray boxes
illustrate how this difference in modulation $\Delta T$ for the same phase
variation $\Delta\phi$ can be made very large.

An advantage of the Grover-Michelson configuration over the traditional one is
that the sensitivity of the output state can be made arbitrarily large or
small by fixing $\phi_{2}$ at some value near an integer multiple of $2\pi$.
Sensitivity here is quantified by the magnitude of the slope of the
probability $R$ or $T$ vs. $\phi_{1}$ curve, such as $|\partial
T/\partial\phi_{1}|$. A comparison of the maximum sensitivity as a function of
the curve index $\phi_{2}$ is shown in Fig. 6. The phase sensitivity can be
seen to grow arbitrarily large as $\phi_{2}$ approaches integral multiples of
$2\pi$.

Figure 6: Maximum sensitivity of the Michelson and Grover-Michelson
interferometers versus $\phi_{2}$, shown from $10^{-5}$ to $2\pi-10^{-5}$. The
curves are periodic in $2\pi$. The sensitivity is given by the absolute value
of the slope of either the reflection or transmission probability $P$,
$|\partial P_{(}\phi_{1},\phi_{2})/\partial\phi_{1}|$. The maximum is taken
over $\phi_{1}$. The Grover-Michelson dominates the Michelson for all values
of $\phi_{2}$ and becomes arbitrarily large as $\phi_{2}$ approaches integral
multiples of $2\pi$.

From a metrological standpoint, the controllable sensitivity behavior can be
used to substantially increase the resolution of the phase readout. Assuming
the phase shift $\phi_{2}$ can be stably controlled, then this sharper
response can be useful if the phase $\phi_{1}$ is perturbed about the
sensitive region by a small, unknown amount. If the interferometer is
calibrated with a controllable phase $\phi_{1}$ to bias the system about the
point of 50:50 power splitting, say in the case $\phi_{2}=\pi/8$ in Fig. 5,
then a small unknown variation in phase $\delta$ brings the effective cavity
round-trip phase $\phi_{1}$ to $({\color[rgb]{0,0,0}\phi_{1}}+\delta)$.
Accordingly, the reflectance and transmittance will see a substantially
greater degree of modulation in comparison to this same variation affecting a
standard Michelson interferometer. This is illustrated with the shaded gray
boxes in Fig. 5; the boxes illustrate the same value of unknown phase
disturbance leads to a substantially larger modulation in transmittance for
the Grover-Michelson.

Using a phase-shifting element in the second arm to control $\phi_{2}$, the
slope at a given bias point can be changed, allowing the system to be field-
programmed to accommodate varying perturbation strengths, even if the
strengths themselves are unknown. If the slope is large enough for a given
perturbation, the output transmission will jump past the sensitive region and
saturate into the flat-sloped, low-transmission region of the next period, at
which readout of the phase becomes difficult to resolve accurately. This
exemplifies the trade-off between sensitivity and dynamic range intrinsic to
metrology. The larger the slope, the smaller $\Delta\phi$ must be in order to
be extracted unambiguously. Assuming the same perturbation can be reapplied at
will, one may repeatedly re-calibrate the bias point at different slopes and
then reapply the perturbation here until the result is non-saturating. For
large perturbations, this may mean operating in a low-curvature bias point to
obtain a larger dynamic range. For instance, in Fig. 5, biasing at the
inflection point $\phi_{1}=\pi,\phi_{2}=\pi/8$ in the Grover-Michelson
interferometer would be the preferred readout location in comparison to the
$\phi_{1}=\pi$ zero-sloped region of high curvature in the traditional device.

As a coherent device, conditions must be met to guarantee the device behaves
as predicted by the assumption that amplitudes corresponding to
indistinguishable paths interfere. As stated in Section II, the source must be
coherent with respect to the length scales of the Grover coin in use. This
allows the physical Grover coin such as the one in Fig. 1 to behave as such.
The conditions pertaining to that physical realization are discussed in Ref.
[3]. The coherence length of the source must also be larger than the length of
a single cavity arm so that amplitudes exiting after a different number of
cavity round-trips can interfere. Otherwise, the result would be a statistical
mixture of the output amplitudes corresponding to interference over an equal
number of round-trips only. Fortunately, the geometric series that determines
the output ampltiudes converges quickly, so interference over only a few
round-trips would likely be sufficient in practice.

Nonetheless, if the source coherence may be an issue, the condition may be
improved by bringing the arms closer to the coin; however, if the arms become
too close to the coin, they will couple to the internal cavity modes of the
coin, thereby changing the coin to a new device of its own. Instead, one might
ensure the coin acts instantaneously on each round-trip by using a detector
with a slower response then characteristic lifetime of the coin’s internal
cavity. Then the time spent within the coin becomes negligible in relation to
the time spent inside a cavity arm.

Other configurations have been known to impart a large, possibly nonlinear
phase due to the interference of light in a cavity. For instance, a Fabry-
Perot could provide a collinear cavity to be placed in the arm of a Michelson
interferometer. If the back surface is fully reflective, a Gires-Tournois
interferometer is formed [14][15]. While such a device can be designed to
produce a sharp response to a phase perturbation, the curve it produces stems
from fixed parameters such as surface reflectivity, thickness, and refractive
index. This means the device can only produce one curve for each wavelength.
One substantial advantage of the tunable response curves of the Grover-
Michelson device is that any fixed wavelength can be used with the device, so
long as the unbiased coin continues to act according to Eq. (4). At the new
wavelength, one obtains the same family of curves, by adjusting values of
$\phi_{2}$.

##  V Conclusion

In this work we have shown that replacing the beam-splitter in a standard
Michelson interferometer with a Grover coin generates a tunable-sensitivity
device. The new device behavior is the result of field interference over an
infinite number of cavity round-trip paths, which generates a nonlinear phase
dependence in the scattering matrix amplitudes. As a result, the transmittance
can be made as sensitive or insensitive as desired by tuning an external phase
and can operate at any fixed wavelength. This would provide an enhanced-
resolution measurement of phase disturbances in any physical situation. Higher
dimensional extensions of this work will be explored in the future.

## Acknowledgments

This research was supported by the Air Force Office of Scientific Research
MURI award number FA9550-22-1-0312.

## Appendix

Here we derive the scattering amplitudes for the standard Michelson and
Grover-Michelson interferometers. We work in the Heisenberg picture, in which
the scattering matrix acts on the  photon creation operators. This is
equivalent to the Schrödinger picture in which the probability amplitudes of
each operator are transformed under the same mappings. Indeed, if a
monochromatic but otherwise general optical scattering state is given by

$|\psi\rangle=\sum_{j}c_{j}a_{j}^{\dagger}|0\rangle=\sum_{j,k}c_{j}a^{\dagger}_{k}\delta_{jk}|0\rangle,$

then applying the linear scattering transformation $A$ on $|\psi\rangle$ gives

$A|\psi\rangle=\sum_{j,k,\ell}(A_{kj}c_{j})a^{\dagger}_{\ell}\delta_{j\ell}|0\rangle=\sum_{j,k,\ell}c_{j}(A_{k\ell}a^{\dagger}_{\ell})\delta_{j\ell}|0\rangle.$

###  V.1 Standard Michelson Interferometer

We use the beam-splitter scattering matrix in Eq. (2). Because the beam-
splitter is biased there is no cavity summation. Thus the initial excitation
is split into each arm, hits the mirror ($M$) and phase shifts ($\Phi)$ once,
and then overlaps at the beam-splitter again before exiting. In accordance
with this, we see

$\displaystyle a_{1}^{\dagger}$
$\displaystyle\xrightarrow{B}\frac{1}{\sqrt{2}}(a_{3}^{\dagger}+a_{4}^{\dagger})$
$\displaystyle\xrightarrow{\Phi,M}-\frac{1}{\sqrt{2}}(e^{i\phi_{1}}a_{3}^{\dagger}+e^{i\phi_{2}}a_{4}^{\dagger})$
$\displaystyle\xrightarrow{B}-\frac{1}{2}\bigg{(}(e^{i\phi_{1}}(a_{1}^{\dagger}+a_{2}^{\dagger})+e^{i\phi_{2}}(a_{1}^{\dagger}-a_{2}^{\dagger}))\bigg{)}$
$\displaystyle=-\frac{1}{2}\bigg{(}a_{1}^{\dagger}(e^{i\phi_{1}}+e^{i\phi_{2}})+a_{2}^{\dagger}(e^{i\phi_{1}}-e^{i\phi_{2}})\bigg{)}$

Reading off the scattering amplitudes,

$\displaystyle r$ $\displaystyle=-\frac{1}{2}(e^{i\phi_{1}}+e^{i\phi_{2}}),$
$\displaystyle t$ $\displaystyle=-\frac{1}{2}(e^{i\phi_{1}}-e^{i\phi_{2}}).$

The square-modulus of these leads to the scattering probabilities of eqs. (5a)
and (5b). A similar calculation starting with $a_{2}^{\dagger}$ results in the
same output amplitudes.

###  V.2 Grover-Michelson Interferometer

In a Grover coin $G$, we will seal port 3 with a phase shift $\phi_{1}$ and
mirror and do the same for port 4 with a phase shift $\phi_{2}$ and mirror.
Collectively the linear phase transformations these devices enact during each
round trip will be denoted $\Phi$ and $M$. To simplify the calculation, we
will introduce some new variables and first show how a single round-trip
affects $(a_{3}^{\dagger}+a_{4}^{\dagger})$ and
$(a_{3}^{\dagger}-a_{4}^{\dagger})$. We will find that these linear
combinations of cavity modes are mapped recursively into themselves, somewhat
emulating the role of coupled-cavity supermodes.

To that end, define the following:

$\displaystyle A$
$\displaystyle\coloneqq\frac{1}{2}(-a_{1}^{\dagger}+a_{2}^{\dagger}),$
$\displaystyle B$
$\displaystyle\coloneqq\frac{1}{2}(e^{i\phi_{1}}+e^{i\phi_{2}}),$
$\displaystyle C$
$\displaystyle\coloneqq\frac{1}{2}(e^{i\phi_{1}}-e^{i\phi_{2}}).$

Next, for an excitation $(a_{3}^{\dagger}+a_{4}^{\dagger})$ making a single
round-trip,

$\displaystyle(a_{3}^{\dagger}+a_{4}^{\dagger})$
$\displaystyle\xrightarrow{M,\Phi}-e^{i\phi_{1}}a_{3}^{\dagger}-e^{i\phi_{2}}a_{4}^{\dagger}$
(9a)
$\displaystyle\xrightarrow{G}-\frac{1}{2}(e^{i\phi_{1}}(a_{1}^{\dagger}+a_{2}^{\dagger}-a_{3}^{\dagger}+a_{4}^{\dagger})$
$\displaystyle\
+e^{i\phi_{2}}(a_{1}^{\dagger}+a_{2}^{\dagger}+a_{3}^{\dagger}-a_{4}^{\dagger}))$
(9b)
$\displaystyle=-\frac{1}{2}(e^{i\phi_{1}}+e^{i\phi_{2}})(a_{1}^{\dagger}+a_{2}^{\dagger})$
$\displaystyle\
+\frac{1}{2}(e^{i\phi_{1}}-e^{i\phi_{2}})(a_{3}^{\dagger}-a_{4}^{\dagger})$
(9c)
$\displaystyle=-B(a_{1}^{\dagger}+a_{2}^{\dagger})+C(a_{3}^{\dagger}-a_{4}^{\dagger})$
(9d)

and similarly

$\displaystyle(a_{3}^{\dagger}-a_{4}^{\dagger})$
$\displaystyle\xrightarrow{M,\Phi}-e^{i\phi_{1}}a_{3}^{\dagger}+e^{i\phi_{2}}a_{4}^{\dagger}$
(10a)
$\displaystyle\xrightarrow{G}-\frac{1}{2}(e^{i\phi_{1}}(a_{1}^{\dagger}+a_{2}^{\dagger}-a_{3}^{\dagger}+a_{4}^{\dagger})$
$\displaystyle-e^{i\phi_{2}}(a_{1}^{\dagger}+a_{2}^{\dagger}+a_{3}^{\dagger}-a_{4}^{\dagger}))$
(10b)
$\displaystyle=-\frac{1}{2}(e^{i\phi_{1}}-e^{i\phi_{2}})(a_{1}^{\dagger}+a_{2}^{\dagger})$
$\displaystyle\
+\frac{1}{2}(e^{i\phi_{1}}+e^{i\phi_{2}})(a_{3}^{\dagger}-a_{4}^{\dagger})$
(10c)
$\displaystyle=-C(a_{1}^{\dagger}+a_{2}^{\dagger})+B(a_{3}^{\dagger}-a_{4}^{\dagger}).$
(10d)

We see $(a_{3}^{\dagger}-a_{4}^{\dagger})$ maps directly into itself after
each round trip. This recursion can be explicitly unrolled into a geometric
series and summed like so

$\displaystyle(a_{3}^{\dagger}-a_{4}^{\dagger})$
$\displaystyle\xrightarrow{N}-C(a_{1}^{\dagger}+a_{2}^{\dagger})\sum_{n=0}^{N}B^{n}+B^{N+1}(a_{3}^{\dagger}-a_{4}^{\dagger})$
(11a)
$\displaystyle\xrightarrow{N\rightarrow\infty}-C(a_{1}^{\dagger}+a_{2}^{\dagger})\sum_{n=0}^{\infty}B^{n}$
(11b)
$\displaystyle=\bigg{(}\frac{C}{B-1}\bigg{)}(a_{1}^{\dagger}+a_{2}^{\dagger}).$
(11c)

Now we use the above to derive the S-matrix. The Grover coin maps a photon
incident on the first port to

$\displaystyle a_{1}|0\rangle$
$\displaystyle\rightarrow\frac{1}{2}(-a_{1}^{\dagger}+a_{2}^{\dagger}+a_{3}^{\dagger}+a_{4}^{\dagger})|0\rangle=(A+\frac{1}{2}(a_{3}^{\dagger}+a_{4}^{\dagger}))|0\rangle.$

Combining this with the above formulas, we see

$\displaystyle\xrightarrow{(\ref{eq:1})}\bigg{[}A-\frac{B}{2}(a_{1}^{\dagger}+a_{2}^{\dagger})+\frac{C}{2}(a_{3}^{\dagger}-a_{4}^{\dagger})\bigg{]}|0\rangle$
$\displaystyle\xrightarrow{(\ref{eq:3})}\bigg{[}A-\frac{B}{2}(a_{1}^{\dagger}+a_{2}^{\dagger})+\frac{C}{2}\bigg{(}\frac{C}{B-1}\bigg{)}(a_{1}^{\dagger}+a_{2}^{\dagger})\bigg{]}|0\rangle.$

Grouping by operator yields the output state (IV)

$\displaystyle|\psi_{\text{out}}\rangle=\bigg{[}\bigg{(}\frac{C^{2}}{2B-2}-\frac{B}{2}-\frac{1}{2}\bigg{)}a_{1}^{\dagger}$
$\displaystyle+\bigg{(}\frac{C^{2}}{2B-2}-\frac{B}{2}+\frac{1}{2}\bigg{)}a_{2}^{\dagger}$
$\displaystyle\bigg{]}|0\rangle.$ (12)

Up to a $\pi$ phase shift, the $B$ and $C$ are respectively $r$ and $t$ for
the traditional Michelson. Hence the above calculation illustrates that the
Grover coin nonlinearly maps the scattering parameters of the standard
Michelson interferometer. Because the Grover coin is permutation symmetric
there is no need to consider the initial state $a_{2}^{\dagger}|0\rangle$.
Relabeling ports $1\longleftrightarrow 2$ results in the same permutation of
the output amplitudes so that $r$ and $t$ are the same for input on either
port.

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# Easy Guided Decoding in Providing Suggestions
for Interactive Machine Translation

Ke Wang, Xin Ge∗, Jiayi Wang, Yu Zhao, Yuqi Zhang
Alibaba Group Inc.
{moyu.wk,shiyi.gx,joanne.wjy,kongyu<EMAIL_ADDRESS>
indicates equal contribution. indicates the corresponding author.

###### Abstract

Machine translation technology has made great progress in recent years, but it
cannot guarantee error-free results. Human translators perform post-editing on
machine translations to correct errors in the scene of computer-aided
translation. In favor of expediting the post-editing process, many works have
investigated machine translation in interactive modes, in which machines can
automatically refine the rest of translations constrained by human’s edits.
Translation Suggestion (TS), as an interactive mode to assist human
translators, requires machines to generate alternatives for specific incorrect
words or phrases selected by human translators. In this paper, we utilize the
parameterized objective function of neural machine translation (NMT) and
propose a novel constrained decoding algorithm, namely Prefix-Suffix Guided
Decoding (PSGD), to deal with the TS problem without additional training.
Compared to the state-of-the-art lexically constrained decoding method, PSGD
improves translation quality by an average of $10.87$ BLEU and $8.62$ BLEU on
the WeTS111https://github.com/ZhenYangIACAS/WeTS and the WMT 2022 Translation
Suggestion datasets222https://www.statmt.org/wmt22/translation-suggestion-
task.html, respectively, and reduces decoding time overhead by an average of
$63.4\%$ tested on the WMT translation datasets. Furthermore, on both of the
TS benchmark datasets, it is superior to other supervised learning systems
trained with TS annotated data.

##  1 Introduction and Related Work

The emergence of machine translation technology (Lopez, 2008; Koehn, 2009)
assists human translation to improve translation efficiency (Green et al.,
2014, 2015; Herbig et al., 2020). Even though there is a quality gap between
the outputs of machine translation (MT) systems and manual translations by
professional translators, MT can still practically reduce time in comparison
with translating from scratch (Läubli et al., 2013). Later, the advances at
the sequence-to-sequence model (Sutskever et al., 2014; Bahdanau et al., 2014;
Vaswani et al., 2017) further made a breakthrough in translation technology,
inspiring the industry to transform human translation into computer-aided
translation (CAT) (Knowles and Koehn, 2016; Santy et al., 2019) to a great
extent. CAT usually relies on an MT engine and a platform with a user-friendly
interface (Bowker, 2002; Lengyel et al., 2004; Bowker and Fisher, 2010;
Bowker, 2014; Pal et al., 2016; Chatterjee, 2019), with which humans perform
post-editing (PE) on machine translations to achieve final results with
quality standards.

Figure 1: Examples of Translation Suggestions in computer-aided translation.
Highlighted are spans with incorrect words selected by humans. The TS system
automatically generates alternatives for the spans to obtain final translation
results.

In the past, the post-editing process was typically static and machines would
no longer respond to humans’ modifications once humans started post-editing.
Recent works (Domingo et al., 2016; González-Rubio et al., 2016; Peris et al.,
2017) investigate interactive protocols and algorithms so that humans and
machines can collaborate and machines automatically refine the translations
according to the human’s edits. One promising mode is Translation Suggestion
(TS) proposed by Yang et al. (2021a) as a pioneer, which requires the machine
to provide alternatives for specific spans of incorrect words or phrases
selected by humans, namely making suggestions given prefix and suffix
constraints. In practical applications as shown in Figure 1, it usually
happens when human translators would like to edit part of the MT output. It
can be easily implemented with a user interface if machines correctly provide
suggestions for the selected incorrect spans. Yang et al. (2021a) has proven
the significance of TS in post-editing in terms of resolving two pitfalls of
earlier works. The importance has also been recognized by the Conference of
Machine Translation (WMT), and they released the Naive Translation Suggestion
shared task Yang et al. (2022) in WMT
2022333https://statmt.org/wmt22/translation-suggestion-task.html.

One of the solutions to TS can be training a supervised model with TS
annotated data. Yang et al. (2021a) trained such an end-to-end Transformer-
like model as a benchmark system. Ge et al. (2022) applied model fine-tuning
with TS data augmentation on pre-trained NMT models. However, supervised
learning, which relies on a large amount of labeled data, is too heavy to be
easily adjusted to other domains. In addition, due to the complicated post-
editing process, it is expensive to obtain such limited annotated data.

Our idea is to investigate inference algorithms given prefix and suffix
constraints. We tested the state-of-the-art lexically constrained decoding
algorithm (Post and Vilar, 2018; Hu et al., 2019) on the WeTS dataset, and
found that omissions frequently occur in suggestion generations. There are two
reasons behind: (1) the division of beams in the dynamic beam allocation
narrows the search space so that there would not be enough candidates that
match constraints to be picked. This is more likely to happen when constraints
are much longer than the average length of the span selected, such as in TS
applications; (2) the beam search stops when the probability of eos, the
special token for the end of a sentence, appears largest in the softmax
distribution, but the probability of the entire sentence generation has not
been considered. In terms of efficiency, this decoding algorithm contains
unnecessary calculations since it always generates suggestions step by step
from the beginning of the translation to the end of the sentence, including
prefix and suffix constraints.

In this paper, we propose a neat prefix-suffix guided decoding (PSGD)
algorithm for the TS task. There are three main contributions: (1) PSGD
emphasizes the probability of the entire generation including prefix and
suffix constraints, but only decodes for the incorrect span rather than the
whole translation sentence, which improves both suggestion quality and time
efficiency. (2) PSGD theoretically avoids dividing beams as in Hu et al.
(2019) so that the original beam search space can be used to improve the
quality of generated translation suggestions. (3) PSGD does not require any
additional training/fine-tuning on the original NMT model, which means it can
be applied to any auto-regressive machine translation system with flexibility.

Our experimental observations show that PSGD significantly outperforms the
state-of-the-art lexically constrained decoding method (Post and Vilar, 2018;
Hu et al., 2019) by an average increase of $10.87$ BLEU and $8.62$ BLEU on the
benchmark datasets WeTS and WMT 2022 Naive Translation Suggestion datasets
(WMT22-TS), respectively. Experimental results also demonstrate PSGD’s
superiority in overall time efficiency by a $63.4\%$ time reduction. In
addition, on both the WeTS and WMT22-TS datasets, PSGD is superior over other
supervised learning systems trained with TS annotated data.

##  2 Preliminary

Symbol |  Definition
---|---
$\mathbf{x}$ |  Sentence in source language
$\mathbf{y}$ |  Translated sentence in target language
$\Theta$ |  Model parameters
$\mathbf{p}$ |  A given prefix of $\mathbf{y}$
$\mathbf{s}$ |  A given suffix of $\mathbf{y}$
$\mathbf{r}$ |  The remained part of $\mathbf{y}$ after $\mathbf{p}$ and $\mathbf{s}$ are removed
$t_{p}$ |  The number of tokens in $\mathbf{p}$
$t_{s}$ |  The number of tokens in $\mathbf{s}$
$t_{r}$ |  The number of tokens in $\mathbf{r}$
bos |  The special begin of sentence token used in machine translation
eos |  The special end of sentence token used in machine translation
$y_{i}$ |  The $i$-th token of $\mathbf{y}$, similar for $x_{i},p_{i},s_{i}$,$r_{i}$, etc.
$\hat{\mathbf{y}}$ |  The estimation/prediction of $\mathbf{y}$, similar for $\hat{\Theta}$, $\hat{\mathbf{r}}$, $\hat{r}_{j}$, etc.
$\mathbf{v}_{<t}$ |  The first $t$ tokens of sequence $\mathbf{v}$
$\langle\mathbf{v}_{1},...,\mathbf{v}_{n}\rangle$ |  Concatenation of sequences $\mathbf{v}_{1},...,\mathbf{v}_{n}$
$P_{\text{MT}}$ |  The probabilistic model of machine translation
$P_{r}^{(n)}$ |  The probability of the entire sequence after the $n$-th decoding step for the remain part $r$
$pt$ |  Patience for decoding early stopping
$f(y_{t})$ |  The softmax probability distribution at the $t$-th decoding step
Table 1: Notations

Before introducing the PSGD, we first elaborate on the problem in the scene of
TS with the mathematical notations in Table 1. The sequence-to-sequence
machine translation model is generally a conditional auto-regressive language
model that follows the factorized probability distribution.

$P_{\text{MT}}(\mathbf{y}|\mathbf{x};\Theta)=\prod_{i}P_{\text{MT}}(y_{i}|\mathbf{y}_{<i},\mathbf{x};\Theta)$
(1)

Given training pairs of $(\mathbf{x},\mathbf{y})$, maximizing the probability
in Formula 1 will return an estimate of the model parameter, $\hat{\Theta}$.

During inference, we are supposed to maximize the probability of the generated
sequence:

$\hat{\mathbf{y}}=\operatorname*{argmax}_{\mathbf{y}}P_{\text{MT}}(\mathbf{y}|\mathbf{x};\hat{\Theta})$
(2)

However, the solution space $\\{\mathbf{y}\\}$ is infinite and impossible to
search. Therefore, an auto-regressive decoding process is applied as an
approximation. The decoder of the model decodes the translation step by step
greedily444We do not involve beam-search here to simplify the equations, but
in practice, all our experiments are conducted with beam-search. as following
until eos is met:

$\hat{y_{i}}=\operatorname*{argmax}_{y}P_{\text{MT}}(y|\mathbf{\hat{y}}_{<i},\mathbf{x};\hat{\Theta})$
(3)

In the scene of TS, a span $\mathbf{r}$ is masked for suggestion generation,
and thus we need to guarantee that $\mathbf{y}$ starts with the given prefix
$\mathbf{p}$ and ends with the given suffix $\mathbf{s}$. Then, the problem
becomes:

$\hat{\mathbf{r}}=\operatorname*{argmax}_{\mathbf{r}}P_{\text{MT}}(\langle\mathbf{p},\mathbf{r},\mathbf{s}\rangle|\mathbf{x};\hat{\Theta})$
(4)

Figure 2: An example of applying the PSGD algorithm to generate translation
suggestions

.

##  3 Methodology

As mentioned previously, earlier works of constrained decoding require step-
by-step generations for the whole sentence even with prefix and suffix
constraints. It causes efficiency issues especially when the constraints are
long. What’s worse, the division of beams might bring about low-quality
suggestions. Accordingly, PSGD generates translation suggestions given prefix
and suffix constraints through maximizing the probability of the whole
generated sequence. Meanwhile, PSGD performs a step-by-step beam search
generation only for the span without dividing beams. A brief example of the
PSGD algorithm for TS can be found in Figure 2. With this picture in mind, we
will give details of the PSGD algorithm in this Section. The pseudo codes of
this algorithm are presented in Algorithm 1. And the fairseq based
implementation is available on
github555https://github.com/wangke1996/translation-suggestion-psgd.

Algorithm 1 Algorithm of PSGD

Input: $\mathbf{x}$, $\mathbf{p}$, $\mathbf{s}$
Parameter: $pt$
Output: $\hat{\mathbf{r}}$

1: Let $\hat{\mathbf{r}}=[]$, $n=0$, $maxP_{r}=-\infty$, $bestStep=0$.

2: while $n-bestStep<pt$ do

3: $\hat{\mathbf{y}}=\langle\mathbf{p},\hat{\mathbf{r}},\mathbf{s}\rangle$

4: $\\{f(p_{i})\\},\\{f(\hat{r}_{j})\\},\\{f(s_{k})\\}=$ Output probability
distribution of every tokens in
$\hat{\mathbf{y}}=\langle\mathbf{p},\hat{\mathbf{r}},\mathbf{s}\rangle$

5: calculate $P_{r}^{(n)}$ with Formula 9

6: if $P_{r}^{(n)}>maxP_{r}$ then

7: $maxP_{r}=P_{r}^{(n)}$

8: $bestStep=n$

9: end if

10: calculate $\hat{r}_{n+1}$ with Formula 10

11: append $\hat{r}_{n+1}$ to $\hat{\mathbf{r}}$

12: $n=n+1$

13: end while

14: return the first $bestStep$ elements of $\hat{\mathbf{r}}$

###  3.1 Decoding Process for TS

Similar to the normal translation process, we apply the auto-regressive
decoding process to overcome the issue of infinite solution space
$\\{\mathbf{r}\\}$. We can factorize the entire probability in Formula 4 into
three parts:

$\displaystyle\begin{split}&P_{\text{MT}}(\langle\mathbf{p},\mathbf{r},\mathbf{s}\rangle|\mathbf{x};\Theta)\\\
=&\prod_{i}P_{\text{MT}}(p_{i}|\mathbf{p}_{<i},\mathbf{x};\Theta)\\\
\cdot&\prod_{j}P_{\text{MT}}(r_{j}|\langle\mathbf{p},\mathbf{r}_{<j}\rangle,\mathbf{x};\Theta)\\\
\cdot&\prod_{k}P_{\text{MT}}(s_{k}|\langle\mathbf{p},\mathbf{r},\mathbf{s}_{<k}\rangle,\mathbf{x};\Theta)\end{split}$
(5)

Notice that in Formula 5, as all $p_{i}$ and $s_{k}$ are given, only the
second product requires a step-by-step auto-regressive generation. The first
product, referring to the probability of the prefix sequence $\mathbf{p}$, can
be calculated at the first step when $r_{1}$ is decoded. And when all of the
decoding steps of $\mathbf{r}$ are completed, we can easily obtain the third
product, which refers to the probability of the suffix sequence $\mathbf{s}$.
Hence, the decoding process at each step is actually:

$\hat{r}_{j}=\operatorname*{argmax}_{r}P_{\text{MT}}(r|\langle\mathbf{p},\mathbf{\hat{r}}_{<j}\rangle,\mathbf{x};\hat{\Theta})$
(6)

Theoretically, we only need to go forward through the model for $t_{r}$ times,
which is smaller than that of a normal constraint decoding Hu et al. (2019),
i.e. $t_{p}+t_{r}+t_{s}$. In particular, for the TS task, human translators
might select a short span with incorrect words, and then $t_{p}+t_{r}+t_{s}$
could be much larger than $t_{r}$.

###  3.2 Condition for Stopping Decoding

The decoding process aforementioned is straightforward. Now, it comes to a
problem: when should the inference stop? In Post and Vilar (2018) and Hu et
al. (2019), the model generates the full sequence and the generation stops
when the eos is met. However, in PSGD, only the span $\mathbf{r}$ will be
generated and $\mathbf{r}$ does not end with eos. We cannot utilize the
generation of eos as a signal to stop decoding.

The key to solving this problem can be aligned with the objective function of
NMT training as described in Formula 1. We assume that we have finished
decoding in Formula 6 for $N$ steps, where $N$ is large enough to obtain a
hypothesis sequence ${\mathbf{r}}^{*}$. The solution space $\\{\mathbf{r}\\}$
becomes $\\{\mathbf{r}^{*}_{<n}\\}_{n=1}^{N}$. The most optimal sequence
$\hat{\mathbf{r}}=\mathbf{r}^{*}_{<\hat{n}}$ is obtained by calculating
$\hat{n}$ as:

$\hat{n}=\operatorname*{argmax}_{n}P_{MT}(\langle\mathbf{p},\mathbf{r}^{*}_{<n},\mathbf{s}\rangle|\mathbf{x};\hat{\Theta})$
(7)

The normal decoding process stops when eos is reached, and it does not
necessarily mean that the probability of the entire sequence generation is
maximized. In other words, there might be a better hypothesis if the model
generates fewer or more steps in the normal decoding process. Our condition
for stop decoding in PSGD approaches the training objective function in
Formula 1 as much as possible. This is the reason why PSGD achieves better
performance in suggestions.

####  3.2.1 Early Stopping

Although the above workaround provides a proper condition for stopping
decoding, it might cause efficiency issues since the number of decoding steps
would become $N$ rather than $t_{r}$. Obviously, $N$ should be equal to or
greater than $t_{r}$ to achieve the best performance. However, we have no idea
how large $t_{r}$ is. We have to set $N$ to a large enough value to cover as
many cases as possible, which is unacceptable in practice.

Instead of setting a large $N$, we try to apply an early stopping mechanism to
save unnecessary decoding steps. In Formula 6, after every decoding step, PSGD
calculates and records the averaged probability of entire sequence generation
with both prefix and suffix constraints concatenated, denoted as $P_{r}^{(n)}$
as follows:

$P_{r}^{(n)}=\frac{P_{MT}(\langle\mathbf{p},\hat{\mathbf{r}}_{<n},\mathbf{s}\rangle|\mathbf{x},\hat{\Theta})}{t_{p}+n+t_{s}}$
(8)

PSGD stops decoding when $P_{r}^{(n)}$ does not increase any more within $pt$
consecutive steps, where $pt$ represents patience for early stopping.

Figure 3: Forward path for calculating $P_{r}^{(n)}$

.

####  3.2.2 Parallel Acceleration

Because the early stopping mechanism requires extra calculations for
$P_{r}^{(n)}$ at each step, it seems that we would need to go forward through
the model for $2*(t_{r}+pt)$ times. In fact, by utilizing the future mask in
Transformer’s decoder, we only run $t_{r}+pt$ times since $P_{r}^{(n)}$ and
$\hat{r}_{n+1}$ can be obtained at the same time. Details are explained as
follows.

Figure 3 shows the forward path for calculating $P_{r}^{(n)}$, where
$f(\hat{y}_{t})$ is the softmax probability distribution at the $t$-th
decoding step. $f(y|y_{t})$ indicates the probability of token $y$ at
$(t+1)$-th step, which is short for $P_{MT}(y|\hat{\mathbf{y}}_{\leq
t},\mathbf{x};\hat{\Theta})$. We have:

$\displaystyle\begin{split}P_{r}^{(n)}=&\frac{1}{t_{p}+n+t_{s}}\cdot
f(p_{1}|{\it bos})\cdot f(p_{2}|p_{1})\cdot\dots\\\
\cdot&f(p_{t_{p}}|p_{t_{p}-1})\cdot f(\hat{r}_{1}|p_{t_{p}})\cdot
f(\hat{r}_{2}|\hat{r}_{1})\cdot\dots\\\
\cdot&f(\hat{r}_{n}|\hat{r}_{n-1})\cdot f(s_{1}|\hat{r}_{n})\cdot
f(s_{2}|s_{1})\cdot\dots\\\ \cdot&f(s_{t_{s}}|s_{t_{s}-1})\cdot f({\it
eos}|s_{t_{s}})\end{split}$ (9)

and

$\displaystyle\begin{split}\hat{r}_{n+1}=&\operatorname*{argmax}_{r}P_{\text{MT}}(r|\langle\mathbf{p},\mathbf{\hat{r}}_{\leq
n}\rangle,\mathbf{x})\\\ =&\operatorname*{argmax}_{r}f(p_{1}|{\it bos})\cdot
f(p_{2}|p_{1})\cdot\dots\\\ \cdot&f(p_{t_{p}}|p_{t_{p}-1})\cdot
f(\hat{r}_{1}|p_{t_{p}})\cdot f(\hat{r}_{2}|\hat{r}_{1})\cdot\\\ &\dots\cdot
f(\hat{r}_{n}|\hat{r}_{n-1})\cdot f(r|\hat{r}_{n})\\\
=&\operatorname*{argmax}_{r}f(r|\hat{r}_{n})\end{split}$ (10)

At the $n$-th step, we can simultaneously get the softmax distributions
$\\{f(p_{i})\\}$, $\\{f(\hat{r}_{j})\\}$, and $\\{f(s_{k})\\}$. Then,
according to Formula 9 and 10, $P_{r}^{(n)}$ (the probability of the entire
generation for early stopping) and $\hat{r}_{n+1}$ (the next token generation)
can be obtained together. Therefore, the total step of the decoding process in
PSGD is $t_{r}+pt$.

###  3.3 Summary of Methodology

In this section, we have provided full details of PSGD, and the pseudo code is
presented in Algorithm 1. By utilizing the parameterized objective function,
PSGD maximizes the probability of the entire generation for better suggestion
quality, but only generates the suggestion for the incorrect span instead of
the whole sequence. Besides, it designs a proper early-stopping mechanism for
efficiency improvement. Only $t_{r}+pt$ decoding steps are required, which is
smaller than $t_{p}+t_{r}+t_{s}$ in Hu et al. (2019), since $t_{r}$ is much
smaller than $t_{p}+t_{r}+t_{s}$ on average in the scene of TS, and the value
of the hyper-parameter $pt$ is proven to be quite small in the subsequent
experimental observations.

##  4 Experiments

###  4.1 Generation Performance on Translation Suggestion

Method | Note | De-En | En-De | Zh-En | En-Zh | Avg
---|---|---|---|---|---|---
SA-Transformer Yang et al. (2021a) | Supervised | 31.20 | 29.48 | 25.51 | 36.28 | 30.62
VDBA Hu et al. (2019) | Unsupervised | 23.71 | 31.29 | 12.11 | 12.96 | 20.02
PSGD (Ours) | 34.74 | 38.92 | 20.25 | 29.64 | 30.89
Table 2: BLEU results on the WeTS test sets. PSGD outperforms VDBA by $+10.87$ BLEU on average, and it also outperforms the TS benchmark systems, SA-Transformer, trained with annotated data in German-English and English-German. Method | Note | De-En | En-De | En-Zh | Zh-En | Avg
---|---|---|---|---|---|---
TSMind Ge et al. (2022) | Supervised | 33.23 | 37.14 | 21.20 | 16.44 | 27.00
VDBA Hu et al. (2019) | Unsupervised | 23.16 | 29.86 | 13.37 | 8.97 | 18.84
PSGD (Ours) | 33.87 | 35.99 | 24.64 | 15.32 | 27.46
Table 3: BLEU results on the WMT22-TS test sets. PSGD outperforms VDBA by
$+8.62$ BLEU on average, and it also slightly outperforms the winning system,
TSMind, trained without pseudo corpus on average of 4 language pairs.

Translation suggestion is a significant CAT task where human translators
select spans with incorrect words or phrases in the MT results. It requires
models to automatically provide alternatives for the spans for efficiency
improvement in translation. WeTS (Yang et al., 2021a) and WMT22-TS (Yang et
al., 2022) are the benchmark datasets for the TS task. Each sample consists of
a source sentence, a masked MT sentence, and a golden reference corresponding
to the masked span. The masked spans with potential translation errors are
selected and annotated by humans. The annotations mainly focus on three types
of translation errors: under (over)-translations, grammatical or syntactic
errors, and semantic errors.

We compare the PSGD with three state-of-the-art systems on WeTS and WMT22-TS:
SA-Transformer Yang et al. (2021a), TSMind Ge et al. (2022) (without TS data
augmentation) and VDBA Hu et al. (2019). SA-Transformer is an end-to-end
transformer-like model, which initializes the encoder with weights from the
pre-trained XLM-Roberta Conneau et al. (2020) and fine-tunes it with TS
labeled data. TSMind utilizes the pre-trained NMT model and fine-tunes it with
TS triplets as well, and it wins in three out of four language pairs of the
Naive Translation Suggestion task in WMT 2022 Yang et al. (2022). VDBA is
basically an improved version of the Dynamic Beam Allocation algorithm
proposed in Post and Vilar (2018), that is remarkably capable in lexically
constrained decoding.

We should notice that both of SA-Transformer and TSMind perform supervised
learning with human-annotated data for translation suggestion, while VDBA and
our PSGD generate suggestions solely based on the original NMT model (released
by Yang et al. (2021a)) without any additional training/fine-tuning on any TS-
labeled data. Comparisons between SA-Transformer and TSMind have been reported
in Yang et al. (2021a) and Ge et al. (2022), respectively. VDBA and PSGD are
both implemented with the Fairseq toolkit (Ott et al., 2019) to generate
suggestions. In evaluation, we calculate SacreBLEU metric (Post, 2018) between
the golden references for the masked spans and the alternatives generated by
each system to measure their performances.

Figure 4: BLEU and decoding time cost with different early stopping patience
${pt}$ on the WeTS German-English development set.

(a) German to English

(b) English to German

(c) English to Chinese

(d) Chinese to English

Figure 5:  Histograms of the length ratios of the masked spans on the WeTS
test sets (1000 samples each).

(a) German to English

(b) English to German

(c) English to French

Figure 6: BLEU of the masked span with different mask ratios on WMT news test
sets. PSGD outperforms VDBA in all language pairs with each mask ratios.

(a) German to English

(b) English to German

(c) English to French

Figure 7: Decoding time costs with different mask ratios on WMT news test
sets. PSGD saves $63.4\%$ of time costs compared with VDBA on average.

(a) German to English

(b) English to German

(c) English to French

Figure 8: BLEU of whole sentences with different mask ratios when prefix and
suffix constraints come from MT outputs.

Results on the test set of WeTS are presented in Table 2. Notice that results
of TSMind are not listed in Table 2. This is because TSMind is a system
submitted to the WMT22-TS shared task. Its results on the WeTS dataset are
those of systems trained with a large amount of augmented data. To demonstrate
the performance of our designed unsupervised method, comparisons with models
without data augmentation are more reasonable and much fairer. Therefore, we
compared our model performances with the results of TSMind without data
augmentation on the development set of WMT22-TS in Table 3. It can be observed
that our PSGD outperforms VDBA significantly by an average of $10.87$ BLEU and
an average of $8.62$ BLEU separately on the two datasets for German-English,
English-German, Chinese-English, and English-Chinese. Moreover, without any
extra training, PSGD is even better than the supervised SA-Transformer and
TSMind systems that are trained with TS-labeled data.

###  4.2 Patience for Early Stopping in Decoding

In PSGD, we involve a hyper-parameter $pt$, the number of extra decoding steps
we take for early stopping. Larger $pt$ can usually avoid omissions since when
a local optimum is reached, the model should be encouraged to generate in more
steps to ensure that it is globally optimal. However, a larger $pt$ would slow
down the decoding process.

To determine a proper value for $pt$, we investigate the model performances in
SacreBLEU and the decoding time cost on the German-English development set of
$WeTS$. The results are shown in Figure 4, which indicates that when $pt$
increases from a relatively small value, the performance of PSGD will
significantly increase as well. However, there is no more much gain when $pt$
is over $5$. Taking the trade-off between time efficiency and the suggestion
quality into consideration, we empirically set $pt=5$. For other language
pairs, we found $pt=5$ is also relatively optimal.

###  4.3 Experiments with Different Mask Ratios

Samples in the WeTS dataset have a short length of masked spans on average,
whose masked spans contain less than $10\%$ words of the whole translation
sentence as displayed in the histograms in Figure 5.

To fairly verify the performance of PSGD on longer masked spans, we consider
applying various mask ratios on the translation datasets. We mask a span in
the golden references with a random length ratio from $20\%$ to $90\%$ and
obtain predictions for the spans by PSGD and VDBA separately. Then, we
calculate SacreBLEU scores for comparison.

In practice, we use the WMT 2014 English-French test set Bojar et al. (2014)
and WMT 2021 English-German and German-English test sets Akhbardeh et al.
(2021) in the news translation for evaluation. PSGD and VDBA decode based on
the publicly released NMT models, Ng et al. (2019) for English-German and
German-English and Ott et al. (2018) for English-French.

Experimental results are shown in Figure 6. It shows that the proposed PSGD
outperforms VDBA in all language pairs for each mask ratio. The performance of
VBDA reaches the lowest point at the mask ratio of $40\%$, and then
dramatically increases as the mask ratio increases. By contrast, the
performance of PSGD appears more stable.

###  4.4 Computation Efficiency

In translation suggestion, compared with the decoding steps
$t_{p}+t_{r}+t_{s}$ of Hu et al. (2019), PSGD is more efficient with
$t_{r}+pt$ decoding steps when $pt$ is smaller than $t_{p}+t_{s}$. To
thoroughly evaluate the speed of PSGD’s decoding process, we conduct a
running-time cost analysis of translation suggestions with different mask
ratios, relying on the WMT news translation test sets mentioned in Section
4.3. All experiments are run on a single Tesla P-100 GPU with CUDA V10.2. The
target sequences are generated with a batch size of $10$ and a beam size of
$5$. We obtain the time cost by averaging the running time of each sample in
the test sets. Notice that times for pre-processings, such as tokenization and
subwords segmentation, are not counted because these parts are identical for
both methods. Hyper-parameter $pt$ is empirically set to 5, the same value in
the experiments shown in Figure 6.

The time costs of PSGD and VDBA are displayed in Figure 7. It is clear that
PSGD runs faster than VDBA in all cases. As discussed in Section 4.3, in the
scene of TS, the incorrect spans are usually short, which contain less than
$10\%$ of words in the whole translation sentences on average. In such a case,
referred in Figure 7, PSGD can speed up at least 2x times compared with VDBA.

###  4.5 Robustness Test

In the above experiments, the given prefix and suffix constraints come from
the golden translation references. While in the applications of CAT, prefix
and suffix constraints could include erroneous words even if the incorrect
span has been selected. Therefore, it is necessary to carry out experiments
where the prefix and suffix constraints contain potential errors. We assume
that the performances of a robust decoding algorithm will not decrease too
much in such a circumstance.

Again, we use the machine translation datasets mentioned in Section 4.3 and
4.4. The difference is that rather than masking golden references, we try to
mask a span with random length in the MT result to get the prefix and suffix
constraints. This setup is much closer to that of WeTS’s. In evaluation, the
Sacre-BLEU score can only be calculated between the entire golden reference
and the suggestions for the masked span alone with the prefix and suffix
constraints, since we don’t have the parallel references for the masked spans.

Results of PSGD and VDBA are plotted in Figure 8. It illustrates that PSGD
significantly outperforms VDBA in all the cases of mask ratios. When the
masked span is extremely short, both algorithms will naturally not impact much
on the whole sequence evaluation. On the other hand, when the masked span is
extremely long, the decoding process of PSGD or VDBA is almost equivalent to
that of the original NMT model. Therefore, in both cases, it is unsurprising
to see that the gap between the performances of PSGD and VDBA is small. Except
for these two situations, the performance of PSGD remains at a higher level
than that of VDBA. And VDBA reaches the lowest point when nearly half of MT
words are masked.

###  4.6 Summary of Experiments

In this section, we conduct exhaustive experiments to evaluate the suggestion
quality, time efficiency, and robustness of PSGD. PSGD significantly
outperforms VDBA by an average increase of $10.87$ BLEU and $8.62$ BLEU on the
benchmark datasets, WeTS and WMT22-TS, respectively. Experiments in time
efficiency show its superiority by an overall $63.4\%$ deduction on the WMT
translation datasets. In robustness tests, PSGD remains at a higher level than
VDBA all the time. Finally, on the TS benchmark datasets, PSGD is superior
over two supervised TS systems, SA-Transformer and TSMind, which are trained
with human-annotated data.

##  5 Conclusion

In this paper, we propose a neat prefix-suffix guided decoding (PSGD)
algorithm for the translation suggestion task in computer-aided translation.
It emphasizes the probability of the entire generation including prefix and
suffix constraints and decodes only for incorrect spans selected by human
translators with an early stopping mechanism. Given a pre-trained auto-
regressive NMT model, PSGD can be easily applied during inference without any
additional training/fine-tuning. Comprehensive experimental results
demonstrate that PSGD significantly outperforms the state-of-the-art
constrained decoding algorithm, VDBA, in all of the translation quality, time
efficiency, and robustness. Meanwhile, it is also superior to supervised TS
systems trained with human-annotated data.

## Limitations

PSGD is a straightforward constrained-decoding algorithm for the translation
suggestion task. However, the early-stopping mechanism involves extra time
costs. Though PSGD is more efficient than VDBA in the scene of TS, where only
two constraints (prefix and suffix) appear, it could be slower than VDBA if
there were more short constraints. Besides, even if we take both prefix and
suffix constraints into consideration for emphasis on the whole translation
generation, the decoding process is still auto-regressive from left to right.
The algorithm could be improved if we made better use of the information of
suffix constraints. For example, how to apply PSGD on non-autoregressive
models, such as Gu et al. (2019) and Yang et al. (2021b) will be our future
work.

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