# Quantum walk-based portfolio optimisation

N. Slate  Department of Physics, The University of Western Australia, Perth WA
6009, Australia  E. Matwiejew  Department of Physics, The University of
Western Australia, Perth WA 6009, Australia  S. Marsh
<EMAIL_ADDRESS>Department of Physics, The University of
Western Australia, Perth WA 6009, Australia  J. B. Wang
<EMAIL_ADDRESS>Department of Physics, The University of Western
Australia, Perth WA 6009, Australia

###### Abstract

This paper proposes a highly efficient quantum algorithm for portfolio
optimisation targeted at near-term noisy intermediate-scale quantum computers.
Recent work by Hodson et al. (2019) explored potential application of hybrid
quantum-classical algorithms to the problem of financial portfolio
rebalancing. In particular, they deal with the portfolio optimisation problem
using the Quantum Approximate Optimisation Algorithm and the Quantum
Alternating Operator Ansatz. In this paper, we demonstrate substantially
better performance using a newly developed Quantum Walk Optimisation Algorithm
in finding high-quality solutions to the portfolio optimisation problem.

##  1 Introduction

Quantum computers are powerful devices that utilise intrinsic properties of
quantum mechanics, such as superposition and entanglement, to provide
substantial speedups for solving computationally hard problems [1, 2]. A
recent influx of interest and technological advancements in this field have
lead to the discussion of practical applications especially in the Noisy
Intermediate-Scale Quantum (NISQ) era [3]. This includes solving difficult
financial problems [4, 5, 6, 7]; one such problem is portfolio optimisation
and periodic re-balancing [8]. When considering a set of $n$ assets there are
$3^{n}$ different portfolio combinations when we consider three different
discrete asset positions:

  1. 1.

Long position: the buying of an asset such as a stock, commodity or currency
with the expectation that it will rise in value;

  2. 2.

Short position: the selling of an asset with the expectation that it will drop
in value;

  3. 3.

No position: neither a long nor short position is taken.

In this work, we take the mean-variance Markowitz model [9, 10] as the basis
for portfolio optimisation, which is fundamental to modern portfolio theory.
Although this model was developed in the 1950s, its simplicity, relative
accuracy and relevance persists as an area of exploration for the quantum
computing community [8].

Financial portfolio optimisation and the Markowitz model have been shown, when
discrete asset constraints are involved, to fall into the category of NP-hard
combinatorial optimisation problems [11, 12]. Thus, the portfolio optimisation
problem provides a real-world model to investigate quantum speedups through
quantum approximate optimisation algorithms. Of interest are implementations
appropriate for near-term noisy intermediate-scale quantum (NISQ) computers,
which have become increasingly available in the cloud, and are fast
approaching sufficient scale and fidelity [3].

The problem of portfolio optimisation and rebalancing using the Markowitz
model with discrete asset constraints has been previously evaluated using the
Quantum Approximate Optimisation Algorithm and Quantum Alternating Operator
Ansatz [8]. Collectively known as QAOA, the algorithms were developed by [13]
and [14] to solve combinatorial optimisation problems. The latter algorithm
generalises the original QAOA to incorporate optimisation constraints. Since
we aim to compare these algorithms in this paper, we distinguish them as QAOA
and QAOAz respectively. These algorithms are known as hybrid quantum-classical
algorithms, as they utilise both quantum computing advantages and classical
optimisation in order to minimise a given objective function. QAOA can be
thought of as a Trotterised approximation to the Quantum Adiabatic Algotithm
(QAA) [14], where the system is evolved into the ground state of some operator
that encodes the problem solution. As a heuristic or approximation algorithm,
QAOA accepts high-quality solutions when the optimal solution is not found.

Specifically, QAOA-based algorithms utilise an alternating state evolution
consisting of solution-quality-dependant phase shifts and a mixing of
probability amplitude across a state-space of possible solutions. Using a
hybrid quantum-classical variational scheme, the expectation value of an
operator encoding the objective function of an associated scalar optimisation
problem is minimised; such that the probability of measuring the system in a
state corresponding to a high quality solution is amplified.

This paper evaluates a further development in QAOA schema known as the Quantum
Walk Optimisation Algorithm (QWOA) [15]. QWOA utilises an efficient indexing
algorithm in conjunction with a generalisation of the QAOA mixing operator to
a continuous-time quantum walk over a circulant graph that connects all
feasible solutions. Our earlier work indicated that QWOA offers significant
advantages over pre-existing methods through a reduction in the search space
and an unbiased encoding of optimisation constraints [15]. In this paper we
provide numerical evidence for the efficacy of QWOA through its application to
portfolio optimisation.

The paper is organised as follows. In Sec. IIA, we introduce the Markowitz
model for portfolio optimisation, along with its quantum encoding for
approximate optimisation. In Sec. IIB, we compute the size of the search space
for QAOAz and QWOA. This is followed by a detailed description and circuit
comparison of the three quantum approximate optimisations under consideration.
Finally, in Sec. III and Sec. IV, we present numerical results and analysis.

##  2 Portfolio Optimisation Problem Formulation

The formalisation and models used in this paper are based upon the work done
by Hodson et al. [8] as to provide a basis for comparison between QAOA, QAOAz,
and QWOA.

###  2.1 The Markowitz model

The discrete mean-variance Markowitz model can be expressed through
minimisation of the objective function

$c(z)=\lambda\sum_{i,j=1}^{n}\sigma_{ij}z_{i}z_{j}-(1-\lambda)\sum_{i=1}^{n}r_{i}z_{i}\,$
(1)

which we subject to the optimisation constraint

$\sum_{i=1}^{n}z_{i}=A\,.$ (2)

In the above formulation, $z=z_{1}\ldots z_{n}$ encodes a particular choice of
portfolio from $n$ assets, where for each asset $i$ we have
$z_{i}\in\\{1,-1,0\\}$ representing the choice of a long position, a short
position, or no position. Associated with each asset is an expected return
$r_{i}$, and the correlation between two assets $i$ and $j$ is given by the
covariance $\sigma_{ij}$. These values are derived from historical data. The
risk aversion parameter $\lambda$, taking a value between $0$ and $1$,
reflects the balance between returns and risk. As this risk parameter
approaches $0$, the optimal portfolio is based purely on obtaining maximum
returns. As $\lambda$ approaches 1, the only consideration becomes
minimisation of risk (for example, leading to a preference for government
bonds over real estate). The value $A$ defines the net total of discrete lots
to be invested for the portfolio. Note that in the definition of $A$, the
signed quantity $z_{i}$ is summed rather than the absolute value
$\absolutevalue{z_{i}}$, which is a formulation generally used in portfolio
rebalancing to treat the relative net position with respect to an existing
portfolio [8].

The mean-variance formulation is standard and well-studied in finance, both in
the continuous and discrete domain [16, 17]. The model formalises the idea of
portfolio diversification, where an investor can reduce risk by owning a
number of assets having low correlations on returns. The application of
discrete variables commonly applies to situations where the class of assets is
traded in discrete quantities (e.g. real estate), and when the cost of a
single asset is substantial (bonds often meet this criteria). It additionally
is applicable where one wishes to subdivide a fixed budget equally amongst the
assets in the portfolio.

In order to ‘quantise’ the formulation and encode each portfolio using a
quantum register, we require two qubits per asset as shown in Table 1,
representing the three possible position choices for each asset.

‘Short’ qubit | ‘Long’ qubit |  $z_{i}$ value |  $\ket{z}$ encoding
---|---|---|---
$\ket{0}$ | $\ket{0}$ | $0$ | $\ket{00}$
$\ket{0}$ | $\ket{1}$ | $1$ | $\ket{01}$
$\ket{1}$ | $\ket{0}$ | $-1$ | $\ket{10}$
$\ket{1}$ | $\ket{1}$ | $0$ | $\ket{11}$
Table 1: Qubit encoding of the possible asset positions.

###  2.2 Counting valid portfolios

For the QWOA, it is a necessary condition [15] to identify the cardinality of
the solution set for any given parameters $(n,A)$. It is also useful to see
the difference in the size of the constrained search subspace to compare
between QAOA, QAOAz and QWOA.

As there are $n$ assets under consideration, and each asset uses two qubits to
encode its position, the Hilbert space is of size $N=2^{2n}$, which is the
ordinary QAOA search space. However, only a subset of these states correspond
to valid portfolio configurations that satisfy the constraint given in Eq. 2.
Furthermore, portfolio configurations that include a stock with both position
qubits set to $\ket{1}$ are degenerate – the $\ket{11}$ configuration is
interpreted as equivalent to the $\ket{00}$ configuration, i.e. no position.
This means there can be a large amount of degeneracy amongst the ‘valid’
states.

We aim to count the number of valid and non-degenerate states, which we will
later see corresponds to the size of the search subspace for QWOA. First, we
note that the number of valid nondegenerate $(n,A)$-portfolios has an exact
correspondence with the number of lattice paths from $(0,0)$ to $(n,A)$ with
steps taken from the set $\\{U=(1,1),D=(1,-1),H=(1,0)\\}$. An example of this
correspondence is given in Fig. 1. Choosing a long position on a stock is
equivalent to stepping diagonally up, while choosing a short position is
equivalent to stepping diagonally down. Choosing no position on a stock takes
a step horizontally. To satisfy the investment constraint of having $A$ more
long positions than short positions, the path must end at $(n,A)$.

Figure 1: An illustrative lattice representation of three example portfolios
which contain four assets and require one more short position than long
position. The red, blue and green paths represent the portfolios 01101000,
10000110 and 00101001 respectively. There are a total of
$\mathcal{M}(4,-1)=16$ paths that reach this endpoint.

######  Theorem 1.

The number of such lattice paths, plotted in Fig. 2, is given by

$\mathcal{M}(n,A)=\sum\limits_{j=0}^{n}\binom{n}{j}\binom{n-j}{\frac{1}{2}(n+A-j)}$
(3)

where the second binomial coefficient is set to $0$ if its bottom parameter is
not an integer.

Figure 2: Log-number of valid non-degenerate portfolios for values of $n$ and
$A$.

###### Proof.

We follow the techniques used in [18]. In particular, the lattice path is
similar to a so-called Motzkin path, for which the number of paths is given by
Eq. (10.43) in [18]. Suppose that there are $j$ $H$-moves in a given path from
$(0,0)$ to $(n,A)$. If these are removed, we are left with a path from $(0,0)$
to $(n-j,A)$ using only diagonal moves. Performing the coordinate mapping
$(x,y)\mapsto(\frac{1}{2}(x+y),\frac{1}{2}(x-y))$ transforms the problem into
finding a path to $(\frac{1}{2}(n+A-j),\frac{1}{2}(n-A-j))$ using standard
basis steps $(1,0)$ and $(0,1)$. The number of paths to this endpoint with
standard basis steps, using Eq. (10.3) in [18], is

$\binom{n-j}{\frac{1}{2}(n+A-j)}\,,$ (4)

where the binomial coefficient is 0, if the bottom component is fractional.
There are $\binom{n}{j}$ ways to re-insert the originally removed $H$-moves,
and after summing over $j$ we arrive at the desired result. ∎

The search space of QAOAz is larger than $\mathcal{M}(N,A)$, since we will
later see that it incorporates valid but also degenerate solutions. The number
of all valid portfolio encodings, including degeneracies, can be expressed as

$\mathcal{V}(n,A)=\sum\limits_{j=0}^{n}2^{j}\binom{n}{j}\binom{n-j}{\frac{1}{2}(n+A-j)}=\binom{2n}{n+A}\,,$
(5)

since if there are $j$ $H$-moves the corresponding $j$ stocks may each be
encoded as $\ket{00}$ or $\ket{11}$. More simply, the second expression is
derived from choosing $(n+A)$ bits to flip in the $2n$-qubit state encoding.
In the following section it will be shown that this is the size of the search
space for QAOAz.

###  2.3 Quantum Approximate Optimisation Algorithms

The Quantum Approximate Optimisation Algorithms evolve a given initial state
as

$\ket{\psi_{f}}=\prod_{i=1}^{p}U_{\text{mix}}(t_{i})U_{C}(\gamma_{i})\ket{\psi_{0}}\,.$
(6)

Here, $U_{C}(\gamma)\ket{z}=e^{-i\gamma c(z)}\ket{z}$ for any portfolio qubit
encoding $\ket{z}$ as per Table 1, applying a relative phase shift to the
solution proportional to the variational parameter $\gamma$ and to the value
of the portfolio with respect to the objective function. This operator, as
defined based on the objective function $c(z)$ given in Eq. 1, is the same for
all methods discussed in this section. We also define the corresponding
Hamiltonian $C$ such that $U_{C}(\gamma)=e^{-i\gamma C}$, which encodes the
values of the objective function $c(z)$ as its diagonal elements. The operator
$U_{\text{mix}}$ is the mixing operator that differs depending on the choice
of connectivity between solution states. In this section, we contrast three
different algorithms that have differing $U_{\text{mix}}$ operators and
associated initial states $\ket{\psi_{0}}$. The integer parameter $p$ is the
number of QAOA iterations, with increased $p$ providing improved solution
quality at the cost of increased circuit depth and $(\vec{t},\vec{\gamma})$
are circuit parameters which are classically optimised in order to minimise
the expectation value $\bra{\psi_{f}}C\ket{\psi_{f}}$. In this framework, a
lower expectation value corresponds to a greater probability of measuring a
portfolio with an optimal, or near optimal, balancing of predicted future
returns and risk.

####  2.3.1 Quantum Approximate Optimisation Algorithm (QAOA)

For the original QAOA scheme, the mixing operator $U_{\text{mix}}$ is defined
as $U_{\text{QAOA}}(t)=e^{-it\sum_{i}X^{(i)}}$. One method of encoding the
investment constraint is to incorporate it within the objective function. This
so-called ‘soft constraint’ occurs as the addition of a penalty function to
Eq. 1,

$\zeta(z)=\epsilon(A-\sum_{i=1}^{n}{z_{i}})^{2}$ (7)

where $\epsilon>\max(c(z))-\min(c(z))$. Thus for QAOA applied to portfolio
optimisation, $C\ket{z}=(c(z)+\zeta(z))\ket{z}$. Final states with minimal
expectation value correspond to ‘good’ solutions to the given optimisation
problem. The penalty function penalises portfolios with a net long position of
more than or less than $A$ assets. With the use of the $\epsilon$ inequality,
this penalty function ensures that the minimum objective function value
corresponds to a state which satisfies the constraint. The soft constraint
method enables the QAOA algorithm to optimise given the constraint, but there
are states in which the constraint is not satisfied which are still being
considered by the algorithm.

The initial state for the soft constraint method is simply an equal
superposition across all states,

$\ket{\psi_{0}}=\frac{1}{\sqrt{2^{2n}}}\sum_{i=0}^{N-1}\ket{i}\,.$ (8)

####  2.3.2 Quantum Alternating Operator Ansatz (QAOAz)

This extension of the QAOA provides a means of constraining the optimisation
process to the subspace of valid solutions by modification of the mixing
operator. The phase unitary, $U_{C}(\gamma)$, maintains the same form for all
algorithms discussed in this paper. However, the mixing operator is now
replaced by $U_{\text{QAOAz}}$, which is an approximation to the time
evolution of the ‘dual parity ring’ Hamiltonian as given in [8, 19],

${H_{\text{ring}}=\sum\limits_{a=1}^{2n}\left(X^{(a)}X^{(a+2)}+Y^{(a)}Y^{(a+2)}\right)\,,}$
(9)

where all addition is modulo $n$. This Hamiltonian preserves the Hamming
weight of both the short and long qubits independently, and consequently meets
the $A$-constraint. Hadfield et al. [19] provide a method for approximating
the time evolution of the dual parity ring mixer Hamiltonian by decomposing it
into three non-commuting unitaries. The QWOAz mixer for portfolio optimisation
is thus defined as

$U_{\text{QAOAz}}(t)=U_{\text{last}}(t)U_{\text{even}}(t)U_{\text{odd}}(t)\approx
e^{-itH_{\text{ring}}}$ (10)

where

$\displaystyle U_{\text{odd}}(t)$ $\displaystyle=\prod_{\begin{subarray}{c}a\
\text{odd},\\\ a\neq
n\end{subarray}}e^{-it\left(X^{(2a+1)}X^{(2(a+2))}+Y^{(2a+1)}Y^{(2(a+2))}\right)}e^{-it\left(X^{(2(a+1))}X^{(2(a+2))}+Y^{(2(a+1))}Y^{(2(a+2))}\right)}\,,$
(11) $\displaystyle U_{\text{even}}(t)$ $\displaystyle=\prod_{a\
\text{even}}e^{-it\left(X^{(2a+1)}X^{(2(a+2))}+Y^{(2a+1)}Y^{(2(a+2))}\right)}e^{-it\left(X^{(2(a+1))}X^{(2(a+2))}+Y^{(2(a+1))}Y^{(2(a+2))}\right)}\,,$
(12) $\displaystyle U_{\text{last}}(t)$
$\displaystyle=\begin{dcases}e^{-it(X^{(2n)}X^{(1)}+Y^{(2n)}Y^{(1)})}e^{-it(X^{(2n-1)}X^{(2)}+Y^{(2n-1)}Y^{(2)})},&n\
\text{odd},\\\ I,&n\ \text{even}.\\\ \end{dcases}$ (13)

where again all addition is modulo $n$. In the above expressions, the first
exponential acts independently on the short position qubits, whilst the second
acts on the long position qubits. When given an initial state satisfying Eq.
2, the action of the dual parity ring mixer ensures that QAOAz has non-zero
amplitude only in solution states satisfying the net investment constraint.

The mixing operator $U_{\text{QAOAz}}$ creates a structure known as parity
bands, which are a result of the preservation of both the long and short
qubits independently. Given there are more than one way to satisfy the
investment constraint (e.g. for $n=5$ and $A=4$, two valid portfolios are 4
long positions and 1 no position, or 5 long positions and 1 short position),
the parity bands are disconnected. Consequently, there is no ability to
transfer amplitude between parity bands through this mixing operator. We must
thus ensure the initial state is in a superposition across all possible valid
parity bands. Given $n$ assets with a net portfolio position of $A$, a simple
initial state given in [8] that meets this criteria is

$\ket{\psi_{0}}=\frac{1}{2^{(n-A)/2}}\ket{01}^{\otimes
A}\otimes(\ket{00}+\ket{11})^{\otimes(n-A)}\,,$ (14)

which represents an equal superposition over all the (degenerate) portfolios
having exactly $A$ more long positions than short positions and $(n-A)$ no-
positions. This initial state is efficient to prepare, but at the cost of bias
– the probability amplitude is binomially distributed across parity bands [8].

####  2.3.3 Quantum Walk Optimisation Algorithm (QWOA)

QWOA generalises the original QAOA mixer as a continuous-time quantum walk
(CTQW) over the subspace of valid solutions [15]. QWOA again evolves the
initial state as given by Eq. 6, but with a new quantum walk mixer
$U_{\text{QWOA}}$. The quantum analogy to the classical random walk, a CTQW is
the evolution of a quantum system under a Hamiltonian defined by a graph
adjacency matrix [20, 21]. The advantage of QWOA for the portfolio
optimisation problem lies in its flexibility in ‘connecting’ only the
solutions in the valid subspace, the ability to eliminate degenerate portfolio
states (thus significantly reducing the search space), and complete global
symmetry amongst valid solutions (eliminating bias of one valid solution over
another due to mixing asymmetry).

In order to implement a QWOA mixer on a desired subspace of solutions
$\mathcal{S}$, we first design an efficiently computable bijection
$\mathcal{S}\xrightarrow{\text{id}}\\{0,1,\ldots,\absolutevalue{\mathcal{S}}-1\\}$
[15], as illustrated in Table 2. In the case of portfolio optimisation, we
need a classical algorithm $\text{id}_{n,A}(x)$ to index the valid and
canonical (non-degenerate) portfolio encodings $x$. In the following, we
provide such an algorithm to compute $\text{id}_{n,A}(x)$ for any given valid
portfolio $x$ and $\text{id}_{n,A}^{-1}(j)$ for any given index $j$.

$\left[\begin{array}[]{cc}x&\text{id}(x)\\\ \hline\cr 01010000_{2}&0\\\
01000100_{2}&1\\\ 00010100_{2}&2\\\ 01000001_{2}&3\\\ 00010001_{2}&4\\\
00000101_{2}&5\\\ 10010101_{2}&6\\\ 01100101_{2}&7\\\ 01011001_{2}&8\\\
01010110_{2}&9\\\ \end{array}\right]$ Table 2: Indexing function for
$n=4,A=2$, a bijection between the portfolio binary encodings and the
integers.

We use a simple recursion relation to index, based on the counting function in
Eq. 3. Observe that
$\mathcal{M}(n,A)=\mathcal{M}(n-1,A)+\mathcal{M}(n-1,A-1)+\mathcal{M}(n,A+1)$.
Using the lattice analogy, the number of paths reaching $(n,A)$ is the sum of
the number of paths one step from $(n,A)$. This inspires a recursive ranking
algorithm as per Algorithm 1. The un-indexing algorithm works similarly,
mapping an integer index to a binary portfolio representation as per Algorithm
2.

Algorithm 1 $\text{id}(n,A,x)$

if $x=0$ then return 0 end if

$x_{f}\leftarrow x\mathbin{\&}11_{2}$ $\triangleright$ last two bits of $x$

$x^{\prime}\leftarrow x\gg 2$ $\triangleright$ remove last two bits

switch $x_{f}$ do

case $00_{2}$: $r\leftarrow\text{id}(n-1,A,x^{\prime})$ $\triangleright$
$H$-move

case $01_{2}$: $r\leftarrow\mathcal{M}(n-1,A)+\text{id}(n-1,A-1,x^{\prime})$
$\triangleright$ $U$-move

case $10_{2}$:
$r\leftarrow\mathcal{M}(n-1,A)+\mathcal{M}(n-1,A-1)+\text{id}(n-1,A+1,x^{\prime})$
$\triangleright$ $D$-move

return r

Algorithm 2 $\text{id}^{-1}(n,A,j,x=0)$

if $n=0$ then return $x$ end if

if $j<\mathcal{M}(n-1,A)$ then $\triangleright$ $H$-move

$r\leftarrow\text{id}^{-1}(n-1,A,j,x)\ll 2$

else if $j<\mathcal{M}(n-1,A)+\mathcal{M}(n-1,A-1)$ then $\triangleright$
$U$-move

$j^{\prime}\leftarrow j-\mathcal{M}(n-1,A)$

$r\leftarrow 01_{2}\mathbin{|}(\text{id}^{-1}(n-1,A-1,j^{\prime},x)\ll 2)$

else$\triangleright$ $D$-move

$j^{\prime}\leftarrow j-(\mathcal{M}(n-1,A)+\mathcal{M}(n-1,A-1))$

$r\leftarrow 10_{2}\mathbin{|}(\text{id}^{-1}(n-1,A+1,j^{\prime},x)\ll 2)$

end if

return r

The quantum circuit shown in Fig. 3 performs the unitary mapping
$U_{\\#}^{\dagger}\ket{j}=\ket{\text{id}^{-1}(n,A,j)}$, i.e. un-indexes an
integer to the corresponding portfolio. The correctness of the circuit follows
directly from Algorithm 2, where we use the property that $y=0$ and $A=0$ at
the end of the recursive sequence to ensure that there are no registers
entangled with the output. Clearly by reversing the circuit, the indexing
unitary $U_{\\#}$ is obtained. In the sub-circuit shown in Fig. 3(a), we rely
on a unitary implementation of the counting function
$\mathcal{M}^{j}_{k}\ket{A}\ket{0}=\ket{A}\ket{\mathcal{M}(j,A+k)}$. Since $j$
is known classically for each sub-circuit, and we must have $-j\leq A+k\leq
j$, this can be implemented by classically pre-computing the unique possible
values of $\mathcal{M}(j,A+k)$ and applying at most $j+1=\mathcal{O}(n)$
controlled additions of these possible values (where the control is on
equality with each possible value of $A+k$). In addition, the given indexing
sub-circuit uses $O(1)$ quantum comparators and other controlled
subtraction/addition operations. This leads to a sub-circuit complexity of
$\mathcal{O}(n)$ as per Fig. 3(a) and thus an overall indexing gate complexity
of $\mathcal{O}(n^{2})$ as per Fig. 3(b), omitting polylogarithmic factors.

$\Qcircuit@C=1em@R=0.7em{\lstick{\ket{A}}&{/}\qw\multigate{1}{\mathcal{M}^{j}_{0}}\qw\qw\qw\multigate{1}{(\mathcal{M}^{j}_{0})^{\dagger}}\push{\rule{3.00003pt}{0.25pt}}\qw\multigate{1}{\mathcal{M}^{j}_{-1}}\qw\qw\qw\multigate{1}{(\mathcal{M}^{j}_{-1})^{\dagger}}\gate{+1}\gate{-1}\qw\rstick{\ket{A^{\prime}}}\\\
\lstick{\ket{0}}{/}\qw\ghost{\mathcal{M}^{j}_{0}}\ctrl{1}\qw\multigate{1}{-}\ghost{(\mathcal{M}^{j}_{0})^{\dagger}}\qw\ghost{\mathcal{M}^{j}_{-1}}\ctrl{1}\qw\multigate{1}{-}\ghost{{(\mathcal{M}^{j}_{-1})^{\dagger}}}\qw\qw\qw\rstick{\ket{0}}\\\
\lstick{\ket{y}}{/}\qw\push{\rule{0.0pt}{10.00002pt}}\qw\ctrl{3}\qw\ghost{-}\qw\qw\qw\ctrl{2}\qw\ghost{-}\qw\qw\qw\qw\rstick{\ket{y^{\prime}}}\\\
\lstick{\cdots}\\\
\lstick{\ket{0_{j}}}\qw\qw\push{\rule{0.0pt}{10.00002pt}}\qw\qw\qw\qw\qw\qw\gate{<}\qw\ctrl{-2}\ctrl{1}\qw\ctrl{-4}\qw\rstick{\ket{x_{j}}}\\\
\lstick{\ket{0_{j+1}}}\qw\push{{\rule{0.0pt}{10.00002pt}}}\qw\gate{<}\qw\ctrl{-3}\qw\qw\push{{\rule{0.0pt}{10.00002pt}}}\qw\qw\qw\qw\targ\ctrl{-5}\qw\qw\rstick{\ket{x_{j+1}}}\\\
\lstick{\cdots}\gategroup{1}{3}{3}{15}{1em}{.}\gategroup{5}{4}{6}{15}{1em}{.}}$
(a)

$\Qcircuit<EMAIL_ADDRESS>\lstick{\ket{y}}{/}\qw\ghost{\tilde{U}_{n-1}}\qw\ghost{\tilde{U}_{1}}\qw\rstick{\ket{0}}\\\
\lstick{\ket{0_{1}}}\qw\qw\qw\multigate{1}{\tilde{U}_{1}}\qwx[-1]\qw\rstick{\ket{x_{1}}}\\\
\lstick{\ket{0_{2}}}\qw\qw\qw\ghost{\tilde{U}_{1}}\qw\rstick{\ket{x_{2}}}\\\
\lstick{\cdots}\push{\cdots}\rstick{\cdots}\rstick{\ket{\text{id}^{-1}(n,A,y)}}\\\
\lstick{\ket{0_{n-1}}}\qw\multigate{1}{\tilde{U}_{n-1}}\qwx[-4]\qw\qw\qw\rstick{\ket{x_{n-1}}}\\\
\lstick{\ket{0_{n}}}\qw\ghost{\tilde{U}_{n-1}}\qw\qw\qw\rstick{\ket{x_{n}}}\push{\rule{30.00005pt}{0.0pt}}\gategroup{3}{9}{7}{9}{1em}{\\}}}$
(b)

Figure 3: (a) The $j$th sub-circuit for QWOA portfolio un-indexing, which we
label as $\tilde{U}_{j}$. This circuit, given the index $y$, decodes the $j$th
and $(j+1)$th bits of the portfolio representation. (b) Illustration of the
quantum un-indexing circuit $U_{\\#}^{\dagger}$ for QWOA portfolio
optimisation. This performs the unitary mapping
$\ket{y}\mapsto\ket{\text{id}^{-1}(n,A,y)}$.

The quantum circuit for quantum portfolio indexing relies on three main
registers. An ancilla register of size $\mathcal{O}(\log n)$ is used to track
the value of $A$ through the indexing process. The input index is held in a
register of size $\mathcal{O}(\log\mathcal{M}(n,A))=\mathcal{O}(n)$, and the
output portfolio encoding is also $\mathcal{O}(n)$. Note that the second
ancilla register shown in Fig. 3(a) is not required and is shown for graphical
convenience – the un-set bits of the output register can be used instead.

With the indexing unitary in hand, the $\mathcal{M}$ valid portfolios can be
‘connected’ using any $\mathcal{M}$-vertex graph over which an efficient
quantum walk can be implemented. In particular, [15] describes an efficient
approach for using an arbitrary circulant graph as a mixer, since this class
of graphs can be simulated efficiently using the Quantum Fourier Transform
[22, 23, 24, 25, 26, 27, 28, 29]. In this work, we choose the complete graph,
$K_{\mathcal{M}}$, due to its efficiency of implementation and global
symmetry. Details of the circuit for quantum walk over $K_{\mathcal{M}}$ can
be found in [15, 30, 31], having asymptotic gate complexity $\mathcal{O}(n)$
to simulate the walk with exponential precision, again omitting
polylogarithmic factors. Thus, we have

$U_{\text{QWOA}}(t)=U_{\\#}e^{-itK_{\mathcal{M}}}U_{\\#}^{\dagger}\,,$ (15)

where the depth of the mixing unitary is dominated by that of the indexing
procedure, leading to an overall complexity of $\mathcal{O}(n^{2})$. The
associated equal superposition initial state is

$\ket{\psi_{0}}=\frac{1}{\sqrt{\mathcal{M}}}U_{\\#}^{\dagger}\sum\limits_{x=0}^{\mathcal{M}-1}\ket{x}\,.$
(16)

####  2.3.4 Comparison of mixing circuits

In Fig. 4, we represent the Hamiltonian underlying each mixing operator as the
adjacency matrix of a graph for a 3-asset example problem, to illustrate the
connectivity between solutions. The generic QAOA mixing operator can be
considered as a quantum walk on the $2n$-dimensional hypercube shown in Fig.
4(a). QAOAz mixes only valid solutions, but as per Fig. 4(b) there are a
number of disconnected graph components, asymmetry in vertex degree and
connectivity, and degeneracy amongst some of the valid portfolio solutions.
Finally, associated with QWOA is a complete graph connecting the canonical
valid solutions.

(a)

(b)

(c)

Figure 4: An example $(n=3,A=1)$ of the graph representation of each
underlying mixing Hamiltonian. Black vertices are valid non-degenerate
solutions, orange vertices are degenerate solutions, and red vertices
correspond to invalid portfolio configurations. (a) QAOA operates on the full
$2^{6}$-dimensional Hilbert space, connected as a hypercube. (b) QAOAz uses
the dual parity ring mixer on long and short qubits, and thus each graph
component connects solutions of equal Hamming weight. (c) QWOA mixes uniformly
between valid non-denegerate solutions, with full symmetry and transitivity
amongst vertices and edges.

In Fig. 5 we contrast the search space and gate complexity for the three
approaches. Also shown is a plot of the search space for varying $n$, with
fixed $A=0$. QAOA scales like $4^{n}$, while QAOAz scales like
$\frac{4^{n}}{\sqrt{\pi n}}$ for large $n$. QWOA in contrast reduces the
search space by more than half on mean, scaling approximately as
$\frac{3^{n}}{2\sqrt{\pi n}}$ (this result is obtained by observing that for
even $n$, $\mathcal{M}(n,0)$ are the sum of squared trinomial coefficients).
We argue that the significant reduction in the size of the search space is
worth the quadratic increase in mixing circuit complexity as per Fig. 5(a). As
shown in the following section, QWOA converges to high-quality solutions far
more quickly, with a small choice of $p$ producing near-optimal solutions.

| QAOA | QAOAz | QWOA
---|---|---|---
Search space | $2^{2n}$ | $\binom{2n}{n+A}$ | $\mathcal{M}(n,A)$
Gate complexity | $\mathcal{O}(n)$ | $\mathcal{O}(n)$ | $\mathcal{O}(n^{2})$
(a)

(b)

Figure 5: (a) comparison of the mixing operators for the three approaches,
where the QWOA complexity is derived in Section 2.3.3. (b) comparison of the
search space size for $A=0$.

##  3 Numerical Results and Analysis

Numerical simulation of the portfolio optimisation problem was carried out for
QAOA, QAOAz and QWOA, using the software package QuOP_MPI [32, 33]. For all
reported results, the $\vec{\gamma}$ and $\vec{t}$ parameters were optimised
using the Broyden-Fletcher-Goldfarb-Shanno algorithm. The Newton-Conjugate-
Gradient, Nelder-Mead Simplex and Powell’s method algorithms were also
considered. However, these yielded equivalent, or uniformly poorer, results.

The simulations utilised daily share prices from two data sets, Data Set A and
B, which selected $n=8$ stocks from the ASX.20 index. The adjusted close price
was used to ensure that all possible corporate actions were considered in the
share performance, such as dividends. Data Set A contained the daily adjusted
close prices from 01/01/2017 to 31/12/2018 for the shares: AMP, ANZ, AMC, BHP,
BXB, CBA, CSL and IAG. Data Set B ranged over the period of 24/03/2020 to
06/09/2020 and selected shares based on sectors heavily impacted by the
COVID-19 pandemic: FLT, QAN, WEB, REX, AIZ, SYD, SCG and CTD. Both data sets
were obtained from Yahoo Finance using a Python based API [34]. For all
simulations, $A=4$ and $\lambda=0.5$. In each case, the presented results
correspond to the mean and standard deviation for 15 repeats of each
algorithm. These used the same set of randomly generated initial
$\vec{\gamma}$ and $\vec{t}$ values, uniformly distributed between $0$ to
$2\pi$.

With $N=8$ and $A=4$, the search spaces of the three algorithms are $2^{16}$
for QAOA, 1820 for QAOAz and 266 for QWOA. The size of the disconnected QAOAz
parity bands are shown in Fig. 6(a) for $N=8$ along with the distribution of
$\ket{\psi_{0}}$ across all bands satisfying $A=4$. We note that the largest
connected components correspond to states satisfying $A$ equal or close to
$0$, and that the binomial distribution of the initial state centers on these
larger components. As mixing of probability amplitude does not occur across
the parity bands, the probabilities shown in Fig. 6(b) are the maximum
possible for a single state in each of the parity bands, as opposed to QAOA
and QWOA which converge to a single state as $p\rightarrow\infty$. For Data
Set A, this influences the minimum possible objective function value which is
$-0.318$ for QAOA and QWOA, and $-0.305$ for QAOAz. For Data Set B the minimum
is $-1.25$ for QAOA, QWOA and QAOAZ as the optimal portfolio exists in all
four of the populated QAOAz parity bands. The small and fully-connected search
space of QWOA is expected to result in it outperforming QAOA and QAOAz at
sufficiently high $p$.

(a)

‘

(b)

Figure 6: (a) QAOAz parity band size and associated constraint value, $A$, for
$N=8$ assets. Parity bands satisfying $A=4$ are outlined in black. (b)
Probability distribution of $\ket{\psi_{0}}$ over the QAOAz parity bands for
$A=4$ as a function of Hamming weight. This distribution occurs across the $5$
outlined parity bands in Fig. 6(a) where the Hamming weight increases down the
outlined diagonal.

Of critical note is that the classical optimisation component suffers from the
curse of dimensionality as the number of optimisation parameters increase
(i.e. with increasing $p$). Each time $p$ increases by 1, two additional
independent classical optimisation parameters are needed to maximally explore
the available search subspace. In hybrid quantum-classical optimisation an
estimation of $\bra{\psi_{f}}C\ket{\psi_{f}}$ is obtained by repeated sampling
of the $\ket{\psi_{f}}$ state, and the optimisation parameters are tuned to
optimise this quantity. Consequently, with increasing dimensionality the
classical optimisation deteriorates [35], making less progress per
optimisation iteration, and thus increasing the overall number of quantum
circuit shots required. A practical hybrid quantum-classical variational
algorithm must therefore demonstrate rapid convergence at low $p$. In this
sense, the polynomial difference in circuit depth per iteration given in Fig.
5 is negligible compared to the exponentially increasing classical parameter
space with increasing $p$. In the following numerical results we demonstrate
that QWOA is by far the best candidate in this regard out of the three
considered algorithms, needing drastically smaller $p$ to reach a given
expected solution quality $\bra{\psi_{f}}C\ket{\psi_{f}}$.

###  3.1 Data Set A (01/01/2017 to 31/12/2018)

Fig. 7(a) shows the final value of the optimised objective function for QAOA,
QAOAz and QWOA. On average, QAOA performs poorly as compared to the other two
algorithms. Additionally, QAOA consistently exhibits the largest standard
deviation in the optimised objective function value
$\bra{\psi_{f}}C\ket{\psi_{f}}$ with a maximum of $12.96$ as compared to
$0.038$ for QAOAz and $0.011$ for QWOA. This is consistent with the inclusion
of invalid portfolio states and the QAOA mixing operator’s action over the
complete Hilbert space, as per Fig. 5, which increases the likelihood of the
classical optimiser converging to a poor local minima when compared to QAOAz
or QWOA. Note that the minimum objective function value (red line), derived
from the globally optimal portfolio selection, is obtained by classical brute-
force iteration through all feasible solutions and shown for the purpose of
comparison.

(a)

(b)

Figure 7: Expected portfolio quality for Data Set A as a function of $p$. (a)
Comparison of QAOA, QAOAz and QWOA. (b) Excluding QAOA.

(a)

(b)

Figure 8: Comparison of output state probabilities for QWOA and QAOAz using
$p=19$ with Data Set A, with optimal solutions indicated by arrow. (a) QWOA
amplifies the optimal portfolio to over 40% probability, as well as boosting
other high-quality portfolios. (b) QAOAz does not amplify the (degenerate)
optimal solutions in this parity band, instead converging to solutions of
comparatively lower quality.

The optimised expectation value after $p$ iterations for QAOAz and QWOA is
shown in Fig. 7(b). QAOAz shows diminishing improvement past $p\approx 8$,
reaching an expected portfolio value approximately $0.2$ above the optimum
portfolio objective function value after $19$ iterations. To explore the
reason behind this performance, Fig. 8(b) plots the QAOAz output state
probabilities for solutions contained in the parity band of Hamming weight 8,
see also Fig. 6, which corresponds to a parity band containing the optimal
solution. It is clear that degeneracy plays a part, with the two highest-
probability states in the parity band representing the same portfolio. Even
though this is a parity band containing the (degenerate) optimal solution
state, its associated probability has not been amplified, remaining below
$10^{-9}$. Conversely, QWOA exhibits rapid improvement with $p$ over the range
considered as per Fig. 7(b) and, as shown in Fig. 8(a), is able to amplify the
optimal portfolio configuration to above 40% probability by $p=19$.

Overall, the superior performance of QWOA at low $p$ is consistent with the
algorithm’s reduction of the search space to 0.406% of QAOA and 14.61% of
QAOAz; along with a reduction in state and mixing bias. Given $n=8$ with an
investment constraint of $A=4$, there are $1554$ degenerate states, which are
not equally distributed over the valid solutions. As per Eq. 5, the number of
the degenerate states for a given portfolio is directly related to the number
of ‘no positions’ it contains, with a higher number corresponding to more
degenerate states. This clearly effects the performance of QAOA and QAOAz,
producing lower-quality portfolios on average for a given choice of $p$.

Fig. 9(a) and Fig. 9(b) show the expected annual return and expected annual
risk for Data Set A. These values were obtained by multiplying the probability
of a each portfolio configuration by the corresponding annual return and risk
respectively. QWOA performs significantly better than QAOA and QAOAz with
respect to the annual return. We note that, as the mean-variance model is
trying to find the best combination of return vs risk, the optimal portfolio
will not necessarily exhibit the lowest possible risk - as low risk assets are
typically associated with decreased return. Fig. 9(c) displays the historic
mean return and the projected return of the final QWOA state at $p=19$. It is
clear that the obtained data is a good match for the historic data used, with
an accurate representation of the trend and volatility of the data.

(a)

(b)

(c)

Figure 9: Expected annual portfolio return (a) and expected annual risk (b)
for the algorithms with increasing $p$ for Data Set A. The historic mean
return for Data Set A is shown in (c) along with the projected expected return
given by the final QWOA state at $p=19$; with the yellow region being the
1-$\sigma$ risk region and the dotted line the expected value.

###  3.2 Data Set B (24/03/2020 to 06/09/2020)

Data Set B is consistent with the pattern of performance observed for Data Set
A. As shown in Fig. 10(a) and Fig. 10(b), QWOA consistently finds the best
expected solution quality, followed by QAOAz and then QAOA. The same trend in
the standard deviation of $\bra{\psi_{f}}C\ket{\psi_{f}}$ is also observed
with QAOA having a maximum of $12.75$, followed by $0.61$ for QAOAz and
$0.115$ for QWOA.

(a)

(b)

Figure 10: Expected portfolio quality for Data Set B as a function of $p$. (a)
Comparison of QAOA, QAOAz and QWOA. (b) Excluding QAOA.

As shown in Fig. 10(b), QWOA again offers significant advantage over QAOAz. As
previously discussed, the QAOAz parity mixer is expected to lead to
performance variation dependant on the distribution of the optimal solutions
amongst the disconnected parity bands. However, for Data Set B, the optimal
solution is present in each of the 5 parity bands, as opposed to Data Set A,
which contained the optimal solution in only $3$ of the 5 graphs (of sizes
$448$, $784$ and $448$ respectively). A convergence to the optimal solution is
observed in the QWOA numerical simulations, shown in Fig. 7(b) and Fig. 10(b).

Reinforcing the above observations, we note significant differences between
QAOAz and QWOA in the probability distribution of the optimised state
$\ket{\psi_{f}}$ for $p=19$, as shown in Fig. 11. QWOA manages to boost the
probability of the optimal portfolio state to approximately $20\%$. It also
amplifies other high-quality portfolios, as demonstrated in Fig. 11(a).
Examining the parity band shown in Fig. 11(b), the optimised parameters
amplify solutions of comparatively lower quality than the optimum with the
probability for the degenerate optimal solutions remaining below $10^{-6}$.
Also notable is the presence of solution degeneracy, with the 12 significantly
amplified portfolios corresponding to four distinct degenerate solutions.

(a)

(b)

Figure 11: Comparison of output state probabilities for QWOA and QAOAz using
$p=19$ with Data Set B, with optimal solutions indicated by arrow. (a) QWOA
amplifies the optimal portfolio to almost 20% probability, as well as other
high-quality portfolios. (b) QAOAz does not amplify the (degenerate) optimal
solutions in this parity band, instead converging to solutions of
comparatively lower quality.

(a)

(b)

(c)

Figure 12: Expected annual portfolio return (a) and expected annual risk (b)
for the considered algorithms with increasing $p$ for Data Set B. The historic
mean return for Data Set A is shown in (c) along with the projected expected
return given by the final QWOA state at $p=19$; with the yellow region being
the 1-$\sigma$ risk region and the dotted line the expected value.

Fig. 12(a) and Fig. 12(b) display the expectation value for the returns and
expected risk. QWOA yields the highest expected return for $p>2$. Unlike Data
Set A, QAOA is observed to provide the next best expected return for $p>7$.
However, this is associated with a consistently higher expected risk as
compared to QAOAz and QWOA, which indicates convergence to a sub-optimal local
minima. The mean expected return and risk across across the occupied QAOAz
parity bands is $490\%$ and $542\%$, taking into account the binomial
distribution of the initial state. This is significantly different from the
expected risk and return observed for QAOAz in Fig. 12(a) and Fig. 12(b) for
$p\geq 3$ which, in combination with the relatively flat QAOAz response to
increasing $p$, is indicative of convergence to a state highly influenced by
solution degeneracy. As shown in Fig. 12(c), the projected return of the final
QWOA state at $p=19$ is a good match for the historic data used, as it again
accurately represents the trend and volatility of the data.

###  3.3 Analysis

The above results illustrate that the QWOA has an improved rate of convergence
(expected solution quality gained per iteration) over the other two
approaches. Additionally, the stability of the classical parameter
optimisation procedure is higher, as indicated by the smaller error bars. The
reasons behind this are multifaceted. The primary improvement is derived from
the fact that the combinatorial domain consisting of $\mathcal{M}(n,A)$ valid
portfolios is not a binary power in general – if it were, there would be no
‘leftover’ computational basis states that do not represent a valid solution.
In this case, the QWOA would not gain an advantage from considering the valid
subspace of solutions. Note that this same advantage holds in general, even
under various modifications of the objective function. For example, even if
the number of possible asset positions is increased from 3 to 4 such that
there is no degeneracy in the qubit encoding, introducing an investment
constraint will again reduce the number of valid solutions down from $4^{n}$
to (in general) a non-binary power. Thus, QWOA will achieve this advantage
over any QAOA-based algorithm that cannot restrict to only the subspace of
non-degenerate valid solutions.

The secondary advantage is the choice of the complete graph as the mixer
between valid solutions. This leads to an improvement over QAOAz, which has a
non solution-transitive connectivity as shown in Fig. 4(b) and does not have a
connected solution space. Furthermore, the complete graph is likely the
optimal choice of mixer for combinatorial optimisation, based on its proven
optimality with respect to spatial database search [36, 37, 38]. This holds
intuitively, considering the maximal symmetry and connectivity, and the
minimal possible graph diameter of 1. These conclusions are supported by the
smooth convergence of the QWOA optimisation procedure, where the optimisation
of the variational parameters $\vec{t}$ represent optimisation of quantum walk
times on the complete graph.

Note that the above comments do not depend on portfolio optimisation-specific
assumptions. The ‘energy penalty’ constraint encoding approach of the QAOA is
applicable to any optimisation problem, and the parity ring mixer formulation
of QAOAz is applicable to a wide range of nontrivial combinatorial
optimisation domains [19]. Thus, we suggest that these results apply to
general ‘black-box’ optimisation problems where the number of valid solutions
is not a power of 2. An example of another applicable real-world combinatorial
optimisation problem is the well-known Travelling Salesman Problem on $n$
cities [39], where the $n!$ feasible solutions necessarily incur leftover
states for any qubit solution representation. Rather than introducing
degenerate solutions or energy penalties, it is advantageous to restrict
strictly to the non-degenerate subspace of valid solutions using the QWOA. As
demonstrated by the results in this section, the modest polynomial increase in
circuit depth of the QWOA is worth the alleviation of the exponential curse of
dimensionality with respect to the classical parameter optimisation.

##  4 Conclusion

This paper carries out a detailed comparison between the performance of the
well-known Quantum Approximate Optimisation Algorithm (QAOA), the Quantum
Alternating Operator Ansatz (QAOAz), and the newly-developed Quantum Walk
Optimisation Algorithm (QWOA) on the NP-hard problem of portfolio optimisation
with discrete asset constraints. We perform a detailed analysis of the
different mixing operators involved with each technique, and the associated
search subspaces.

Our numerical simulations highlight key advantages of QWOA when compared to
both QAOA and QAOAz. QWOA reduces the search space by a significant factor,
leading to consistently improved performance in obtaining a high quality
portfolio configuration using fewer iterations, and with significantly smaller
standard deviation across numerical simulations. Additionally, the global
symmetry of the QWOA mixing operator leads to clear advantages in convergence
rate and expected solution quality, while QAOA and QAOAz are hindered by bias
in the mixing operator over nontrivial feasible solution spaces. These results
show not only the applicability of quantum combinatorial optimisation
algorithms to an important real-world problem in the financial realm, but also
express the advantages of using the QWOA on arbitrary optimisation problems
with complex discrete constraints and associated solution domains.

##  5 Acknowledgements

This work was supported by resources provided by the Pawsey Supercomputing
Centre with funding from the Australian Government and the Government of
Western Australia. JBW and SM thank Yuying Li for valuable discussions on
computational finance and optimisation. EM and SM acknowledge the support of
the Australian Government Research Training Program Scholarship. SM’s research
was also supported by a Hackett Postgraduate Research Scholarship at the
University of Western Australia.

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# Rigidity results for mean curvature flow graphical translators moving in
non-graphical direction

John Man Shun Ma  Department of Mathematical Sciences, University of
Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark<EMAIL_ADDRESS>,  Yuan Shyong Ooi  Department of Mathematics, Pusan National University,
Busan 46241, Korea<EMAIL_ADDRESS>and  Juncheol Pyo  Department of
Mathematics, Pusan National University, Busan 46241, Korea<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the rigidity results of complete graphical translating
hypersurfaces when the translating direction is not in the graphical
direction. We proved that any entire graphical translating surface in the
translating direction not parallel to the graphical one is flat if either the
translating surface is mean convex or the entropy of the translating surface
is smaller than $2$. For higher dimensional case, we show that the same
conclusion holds if the graphical translating hypersurface satisfies certain
growth condition.

###### 2010 Mathematics Subject Classification:

53C24, 53C44

##  1\. Introduction

A smooth hypersurface $\Sigma^{n}$ immersed in $\mathbb{R}^{n+1}$ is called a
translating soliton (or translator) if its mean curvature $H$ satisfies the
equation

(1.1) $H=\langle\nu,T\rangle$

where $\nu$ is the unit normal of $\Sigma$ and $T$ is any constant unit vector
in $\mathbb{R}^{n+1}$.

In the study of Type II singularity formation of mean curvature flow (MCF),
translating soliton arises as one of the singularity model after parabolic
rescaling the flow. In term of MCF, any translating soliton gives us a
translating solution $\Sigma_{s}=\Sigma+sT$ for any $s\in\mathbb{R}$. This
means under MCF, the shape of a translating soliton remains unchange but only
moves in the $T$ direction. On the other hand, translating soliton can also be
viewed as a minimal hypersurface in $(\mathbb{R}^{n+1},\bar{g})$ where
$\bar{g}$ is a conformally flat metric [Ilm94]. Therefore, we may expect that
translator shares some analog properties as minimal hypersurface in Euclidean
space.

We say that $\Sigma$ is a _graphical translator_ if it can be expressed as a
graph of a smooth function
$u:\Omega\subseteq\mathbb{R}^{n}\rightarrow\mathbb{R}$. In this setting, if we
write $T=(T^{\prime},T_{n+1})$ where $T^{\prime}\in\mathbb{R}^{n}$ and
$\Sigma=\\{(x,u(x)):x\in\Omega\subseteq\mathbb{R}^{n}\\}$ then the function
$u$ satisfies the following quasilinear elliptic PDE in divergence form

(1.2)
$\sum_{i=1}^{n}D_{i}\left(\frac{D_{i}u}{\sqrt{1+|Du|^{2}}}\right)=\frac{-\langle
Du,T^{\prime}\rangle+T_{n+1}}{\sqrt{1+|Du|^{2}}}$

Note that in the graphical case, we always choose the orientation as upward
unit normal $\nu=\frac{(-Du,1)}{\sqrt{1+|Du|^{2}}}$. We say that $\Sigma$ is
an _entire graph_ if its domain $\Omega=\mathbb{R}^{n}$.

Let us denote $\\{e_{i}\\}$ as the standard basis in $\mathbb{R}^{n+1}$. The
case for $T=e_{n+1}$ is well studied in the graphical setting. We shall call
this type of translator a vertical graphical translator. From the translator
equation (1.1), it is clear that graphical condition implies strictly mean-
convexity $H=\langle\nu,e_{n+1}\rangle>0$. So graphical translator is always
(strictly) mean-convex when the translating direction is upward.

Now let us give some examples of vertical graphical translator. When $n=1$,
the only complete graphical translating curve in the plane is the grim reaper
$\Gamma_{gr}$ i.e. the graph of the function

(1.3) $y=-\log\cos x,\quad x\in\left(\frac{-\pi}{2},\frac{\pi}{2}\right)$

We can also produce new translator by taking Cartesian product of a translator
with euclidean factor. For example the Cartesian product of $\Gamma_{gr}$ with
euclidean factor will give us a graphical translator
$\Gamma_{gr}\times\mathbb{R}$ known as the grim reaper surface. By rescaling
and the rotating grim reaper surface, we can produce a family of graphical
translator known as the tilted grim reaper surface
$\\{\Sigma^{\theta}_{tgr}\\}_{\theta\in(0,\pi/2)}$ i.e.

(1.4)
$\Sigma^{\theta}_{tgr}:=\left\\{\left(x,y,-\frac{\log\cos(x\cos\theta)}{\cos^{2}\theta}-y\tan\theta\right):x\in\left(-\frac{\pi}{2\cos\theta},\frac{\pi}{2\cos\theta}\right),y\in\mathbb{R}\right\\}$

Note that the family of tilted grim reaper surface is defined on a slab of
width $\frac{\pi}{\cos\theta}$ which range from $\pi$ to $\infty$ as $\theta$
goes from $0$ to $\frac{\pi}{2}$.

Besides the tilted grim reaper, another example of family of graphical
translator defined on a slab, known as the $\triangle$-wing, is constructed by
Hoffman et al. [HIMW19]. They show that for each $b>\pi/2$, there exist a
unique, complete, strictly convex graphical translator
$u^{b}:(-b,b)\times\mathbb{R}\rightarrow\mathbb{R}$. Note that this family of
translating graph is different from tilted grim reaper surface because tilted
grim reaper surface is not strictly convex.

In [CSS07], Clutterbuck-Schürer-Schulze constructed the unique entire,
rotationally symmetry, vertical graphical translator known as the bowl soliton
(see also [AW94]). The bowl soliton is not only mean-convex but is strictly
convex and is asymptotic to a paraboloid.

For higher dimension, any Cartesian product of complete minimal hypersurface
with a Euclidean factor will give us a translator with translating direction
parallel to the Euclidean factor direction. In particular we can take the non-
planar entire graphical solution of minimal surface equation
$\Sigma^{8}\subseteq\mathbb{R}^{9}$ constructed by Bombieri et al. [BDGG69]
and product it with $\mathbb{R}$. $\Sigma\times\mathbb{R}$ then gives us a
non-planar, entire graphical translator in $\mathbb{R}^{10}$ with horizontal
translating direction. Note that this example of translator is static under
MCF since it is also a minimal hypersurface.

There are also various other example of translator which are not graphical. To
name a few, Clutterbuck et al. [CSS07] construct the winglike translator which
can be viewed as the union of two bowl soliton with a neck; Nguyen [Ngu09]
uses gluing construction to construct the translating tridents; Kim-Pyo [KP18]
study helicoidal type of translating soliton and completely classified all of
them; Hoffman, Martin and White [HMW22] construct and classify semi-graph
translating soliton.

We can now state the known classification result for two dimensional complete
vertical graphical translator. First of all, by the work of Shahriyari
[Sha15], the domain of any complete, two dimensional, vertical translating
graph can only be $\mathbb{R}^{2}$, half-plane or slab
$(-b,b)\times\mathbb{R}$. Building upon the work of Shariyari, Spruck-Xiao and
Wang[SX20, Wan11], Hoffmann et.al [HIMW19] classify all the vertical
translating graph as grim reaper surface, tilted grim reaper surface, bowl
soliton and $\triangle$-wing surface. Among them, only bowl soliton is the
entire example, the others are all defined on a slab. They also show that
there does not exist any complete vertical graphical translator whose domain
is a half plane.

For higher codimensional translator related result, one can refer to the work
of Xin and Kunikawa [Xin15, Kun15, Kun17].

We are interested in the rigidity result of complete graphical translator
which is not translating in the vertical direction. Our first observation is,
if the domain $\Omega\subset\mathbb{R}^{n}$ is bounded and the velocity $T$ is
not vertical then by applying the half space type result of Kim-Pyo [KP21], we
know that such graphical translator cannot exist. Therefore we only need to
study the unbounded domain cases. In particular we shall focus on the entire
domain case. By assuming mean-convexity condition, we are able to show that
entire, mean convex graphical translator not translating vertically must be a
hyperplane:

######  Theorem 1.1.

Let $\Sigma^{2}=\\{(x,y,u(x,y)):\forall
x,y\in\mathbb{R}\\}\subseteq\mathbb{R}^{3}$ for some $u\in
C^{\infty}(\mathbb{R}^{2})$ be an entire graphical surface satisfying the
translator equation

$H=\langle\nu,T\rangle$

where $T\in\mathbb{R}^{3}$ is a unit vector not parallel to $e_{3}$ and $\nu$
is upward unit normal. If $H\geq 0$ then $\Sigma$ is a plane parallel to $T$
direction.

In general, dimension restriction is needed since we could have non-planar
example in higher dimension. (see Section 6 for counter example)

On the other hand, Hershkovits [Her20] classified translator with entropy less
than or equal to the entropy of a cylinder. In particular he shows that if a
translating surface $\Sigma^{2}$ satisfies the entropy bound

$\lambda(\Sigma)\leq\sqrt{\frac{2\pi}{e}}\approx 1.52$

then it can only be a plane with entropy 1 or the bowl soliton with entropy
$\sqrt{\frac{2\pi}{e}}$. If $\Sigma$ lies in a slab and is simply connected,
Chini [Chi20] can classify those translators with entropy less than 3. He
showed that vertical plane, tilted grim reaper surface, grim reaper surface or
$\triangle$-wing surface are the only possible examples. In our case, we can
classify all complete non-vertical graphical translator with entropy less than
2:

######  Theorem 1.2.

Let $\Omega$ be an open subset of $\mathbb{R}^{2}$ and let
$\Sigma=\\{(x,y,u(x,y)):(x,y)\in\Omega\\}$ be a complete graphical translator
in the direction $T$ not parallel to $e_{3}$. If $\lambda(\Sigma)<2$, $\Sigma$
is a hyperplane.

Another way to look at the rigidity problem for translator is to study their
Gauss map image. In this direction, Bao-Shi [BS14] studied translating soliton
whose image of Gauss maps are contained in compact subsets of upper hemisphere
of the standard $\mathbb{S}^{n}$. Their result says that such translator can
only be a hyperplane. For graphical setting, image of Gauss map is closely
related to the asypmtotic behaviour of the gradient of the graph. In
particular, any entire graphical translator with bounded gradient must have
Gauss images contained in compact subsets of upper hemisphere and hence must
be a hyperplane. If we assume that $u$ satisfies certain growth rate, we can
show the following rigidity result:

######  Theorem 1.3.

Assume that $T$ is perpendicular to $e_{n+1}$, and let $\Sigma$ be a graphical
entire translator with velocity $T$ given by
$\Sigma=\\{(x,u(x)):x\in\mathbb{R}^{n}\\}$. If there are positive numbers
$C_{1},C_{2}$ so that

(1.5) $|u(q)|\leq C_{1}+C_{2}\sqrt{|\langle q,T\rangle|},\ \ \ \forall
q\in\mathbb{R}^{n}$

then $\Sigma$ is the stationary horizontal plane.

By assuming a slow growth rate of the gradient, we can also obtain the
following result:

######  Theorem 1.4.

Let $\Sigma$ be an entire graphical translator given by
$\Sigma=\\{(x,u(x)):x\in\mathbb{R}^{n}\\}$ with unit velocity $T$ not parallel
to $e_{n+1}$ direction. If the gradient of $u$ satisfies the growth rate

(1.6) $|Du(x)|=o(|x|^{1/4})$

then $\Sigma$ is a hyperplane.

Recently, Gama, Martin and Møller [GMM22] studied non-vertical graphical
translator in $\mathbb{R}^{3}$ that lies between two parallel planes in the
translating direction. The conditions they impose on are the entropy and the
width of the two parallel planes i.e. the distance between the two planes.
They can show that if the width is finite and if the translating direction is
not horizontal then the graphical translating surface can only be a plane
[GMM22, Proposition 3.13]. We can think of finite width condition as the graph
function has at most linear growth. Besides that, they also classify those
simply connected translator $\Sigma^{2}$ with finite width and with entropy
satisfying $3\leq\lambda(\Sigma)<4$. Although we impose stronger growth rate
condition in Theorem 1.3, we consider the rigidity result of their complement
case which is when the translating direction is horizontal and our method also
works for any dimension.

The organization of this paper is as follows. We provide some background
needed in the proofs of our results in Sec. 2. In Sec. 3, we give a prove for
Theorem 1.1. The proofs for Theorem 1.2 is given in Sec. 4. In Sec. 5 we give
a proof to Theorem 1.3 and Theorem 1.4. The final section is to provide
counter a example for our results in higher dimension showing that the
dimension restriction for the first two results is optimal.

##  2\. Prelimenary

###  2.1. Colding-Minicozzi’s entropy

Let $\Sigma^{n}\subseteq\mathbb{R}^{n+1}$ be any hypersurface. Following the
work of Colding-Minicozzi [CM12], given $x_{0}\in\mathbb{R}^{n+1}$ and
$t_{0}>0$, we define the F-functional $F_{x_{0},t_{0}}$ (see also [Hui90],
[AIC95]) by

(2.1) $F_{x_{0},t_{0}}(\Sigma):=(4\pi
t_{0})^{-\frac{n}{2}}\int_{\Sigma}e^{-\frac{|x-x_{0}|^{2}}{4t_{0}}}d\mu$

and its entropy functional $\lambda=\lambda(\Sigma)$ given by

(2.2) $\lambda=\sup_{x_{0},t_{0}}F_{x_{0},t_{0}}(\Sigma)$

In [CM12], the F-functional is used by Colding-Minicozzi to study the
singularities of MCF. The key property about this functional is that its
critical point is precisely when $\Sigma$ is a $t=-t_{0}$ slice of self-
shrinking solution that becomes extinct at $x_{0}$ and $t=0$.

The reason to work with entropy is as follow. For any properly embedded smooth
hypersurface $\Sigma$ in $\mathbb{R}^{n+1}$, we have $\lambda(\Sigma)\geq 1$
and it is invariant under dilations and rigid motions of $\Sigma$. Moreover
$\lambda(\Sigma)=1$ exactly when $\Sigma$ is a flat hyperplane.

In particular, when $\Sigma_{t}=\Sigma+tT$ is a MCF translating solution with
$\Sigma_{0}=\Sigma$ with finite entropy, and
$\Sigma^{\delta}_{t}:=\delta\Sigma_{\delta^{-2}t}$ is any parabolic re-scaling
of $\Sigma_{t}$, by translation and dilation invariant properties of
$\lambda(\Sigma)$ all their entropy is preserved i.e.

(2.3) $\lambda(\Sigma^{\delta}_{t})=\lambda(\Sigma_{t})=\lambda(\Sigma)$

Finite entropy also gives us some control on the area of hypersurface inside
an extrinsic ball in the following sense,

######  Lemma 2.1.

Let $\Sigma$ be any smooth hypersurface in $\mathbb{R}^{n+1}$. If
$\lambda(\Sigma)$ is finite then $\Sigma$ has extrinsic Euclidean area growth.
This means for any $R>1$, there exist a constant $C=C(n)>0$ such that

$\text{Area}(\Sigma\cap B^{n+1}_{R}(0))\leq C\dot{\lambda}(\Sigma)R^{n}$

###### Proof.

From the definition of entropy, by choosing $t_{0}=R^{2}$ and $x_{0}=0$

$\lambda(\Sigma)\geq(4\pi R^{2})^{-n/2}\int_{\Sigma\cap
B_{R}}e^{-|x|^{2}/4R^{2}}\geq
e^{-1/4}(4\pi)^{-n/2}R^{-n}\text{Area}(\Sigma\cap B_{R})$

Hence

(2.4) $\text{Area}(\Sigma\cap B_{R})\leq
e^{1/4}(4\pi)^{n/2}\lambda(\Sigma)R^{n}$

where our constant $C(n)=e^{1/4}(4\pi)^{n/2}$ ∎

We also recall the definition of a blow-down of a translator with finite
entropy. By Lemma 2.1 and (2.3), we have

(2.5) $\text{Area}(\Sigma^{\delta}_{t}\cap B_{R})\leq C\lambda(\Sigma)R^{2}$

for all $t<0$ and $\delta>0$. Arguing as in [Ilm93], by the compactness
theorem on Brakke flow [Ilm94, 7.1], there is a sequence $\delta_{i}\to 0$ and
a limit Brakke flow $\\{\nu_{t}\\}_{t<0}$ so that $M^{\lambda_{i}}_{t}$
converges weakly to $\nu_{t}$ for a.e. $t$. Using Huisken’s Monotonicity
formula, we have for all $a<b<0$,

$\int_{a}^{b}\int_{M^{\lambda}_{t}}\left|\vec{H}(x)+\frac{x^{\perp}}{-2t}\right|^{2}\Phi(x,t)d\mu_{t}^{\lambda}dt\to
0$

as $\lambda\to 0$, where

$\displaystyle\Phi(x,t)=\frac{1}{(-4\pi
t)^{\frac{n}{2}}}e^{-\frac{|x|^{2}}{-4t}}.$

From there we see that the limit $\\{\nu_{t}\\}$ is self-similar:

$\vec{H}(x)=-\frac{x^{\perp}}{2t},\ \ \ \forall t<0.$

For surfaces in $\mathbb{R}^{3}$, one obtains better regularity on
$\\{\nu_{t}\\}$: Using the local Gauss-Bonnet Estimate [Ilm93, Theorem 3], it
is proved that [Ilm93, Theorem 1,2] the support of $\\{v_{t}\\}$ is embedded:
that is, $|v_{t}|=\sqrt{-t}\Sigma_{\infty}$ and $\Sigma_{\infty}$ is an
embedded self-shrinker. Also, there is $\tau_{i}\to-1$ so that
$\Sigma^{\delta_{i}}_{\tau_{i}}$ converges locally smoothly (possibly with
multiplicity) as $i\to+\infty$ to $\Sigma_{\infty}$ away from a discrete set
in $\mathbb{R}^{3}$. Moreover, $g(\Sigma_{\infty})\leq g(\Sigma)$ (In [Ilm93],
the theorem is stated for blow-ups, i.e. $\delta\to\infty$, but the same
argument also works for blow-down under the finite entropy assumption).

###  2.2. Properties of complete graphical translator

It is known that complete vertical graphical translator cannot have bounded
domain [Sha15]. We show the same result holds true if the translating
direction is not vertical.

######  Proposition 2.2.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain and

$\Sigma:=\\{(x,u(x)):\forall x\in\Omega\\}\subseteq\mathbb{R}^{n+1}$

be a complete graphical translator with translating direction $T$ not parallel
to $e_{n+1}$. Then such $\Sigma$ does not exist.

###### Proof.

By a rotation of $SO(n+1)$ fixing the $e_{n+1}$ direction, we may assume that
$T=\cos\theta e_{1}+\sin\theta e_{n+1}$ for some $\theta\in(-\pi/2,\pi.2)$. We
can also assume that the domain $\Omega\subseteq\\{x\in\mathbb{R}^{n}:a\leq
x_{1}\leq b\\}$ for some finite number $a,b$. If $\Sigma$ exists, it must lie
in the half space $H=\\{x\in\mathbb{R}^{n+1}:\langle x,e_{1}\rangle\leq b\\}$.
Since $\langle e_{1},T\rangle=\cos\theta>0$, by half-space result for
translator of Kim and Pyo [KP21, Theorem 1.1], we know that $\Sigma$ cannot
exist. ∎

##  3\. Mean convex graphical translator

In this section, we will prove the rigidity result for entire mean convex non-
vertical graphical translator. We show that plane parallel to the translating
direction is the only possible case under this assumption. As a corollary, we
obtain the complete classification of mean-convex graphical translator in any
translating direction.

######  Theorem 3.1 (Theorem 1.1).

Let $\Sigma^{2}=\\{(x,y,u(x,y)):\forall
x,y\in\mathbb{R}\\}\subseteq\mathbb{R}^{3}$ for some $u\in
C^{\infty}(\mathbb{R}^{2})$ be an entire graphical surface satisfying the
translator equation $H=\langle\nu,T\rangle$ where $T\in\mathbb{R}^{3}$ is a
unit vector not parallel to $e_{3}$ and $\nu$ is unit upward normal. If $H\geq
0$ then $\Sigma$ is a plane parallel to $T$ direction.

###### Proof.

First of all, recall the structural equation for mean curvature of a
translating soliton (see for example [MSHS15, Lemma 2.1(f)]):

(3.1) $\Delta H+\langle T,\nabla H\rangle+|A|^{2}H=0\ $

Since $H\geq 0$ we have $\Delta H+\langle T,\nabla H\rangle\leq 0$. By strong
maximum principle, either $H\equiv 0$ or $H>0$.

If $H\equiv 0$ then $\langle\nu,T\rangle=0$ on $\Sigma$, hence $T$ is a
parallel constant tangential vector field on $\Sigma$. The integral curve of
$T$ gives a foliation of $\Sigma$ hence $\Sigma$ splits as
$\Gamma\times\mathbb{R}$ where the Euclidean factor is parallel to $T$
direction and $\Gamma$ is minimal geodesic in $\mathbb{R}^{2}$. Therefore
$\Sigma$ has to be a totally geodesic plane parallel to the $T$ direction.

Next, we consider the case when $H>0$ everywhere. We write $T^{\perp}$ as a
plane with unit normal $T$ passing through the origin. From
$H=\langle\nu,T\rangle>0$, $\Sigma$ is graphical over a domain $\Omega$ inside
the plane $T^{\perp}$. By a $SO(n+1)$ rotation, we can assume that
$T^{\perp}=\mathbb{R}^{2}\times\\{0\\}$. From [Sha15, Corollary 4.3], $\Omega$
can only be the entire plane $\mathbb{R}^{2}$, half plane or slab
$\mathbb{R}\times(-b,b)$ with $b\geq\frac{\pi}{2}$. Moreover, we can also
assume that $\Sigma$ is convex since a result of [SX20, Theorem 1.1] says that
complete mean convex translating surface has to be convex (but not necessarily
strictly convex). We now divide into three cases and discuss each of them
separately.

Case(i): $\Omega$ is $\mathbb{R}^{2}$, then $\Sigma$ is an entire convex
graphical translator over $T^{\perp}$. Wang [Wan11] shows that such $\Sigma$
must be a bowl soliton which is a graph over $T^{\perp}$. However bowl soliton
cannot be entire graphical over two transversal planes at the same time. It is
a contradiction.

Case(ii): $\Omega$ is half plane. According to [HIMW19, Theorem 6.7] there
exist no complete graphical translator over half plane. Hence $\Omega$ cannot
be a half plane.

Case(iii): The remaining case is when $\Omega$ is a slab
$\mathbb{R}\times(-b,b)$ for some $b\geq\frac{\pi}{2}$. By convexity of
$\Sigma$, its Gauss curvature satisfies $K\geq 0$. Using [HIMW19, Theorem
2.2], if $K=0$ at some point then it vanishes everywhere and $\Sigma$ can only
be grim reaper surface or tilted grim reaper surface (as a translating graph
over strip in $T^{\perp}$ plane) both of which are not entire graphical over
$xy$-plane (see also [MSHS15, Theorem 2.7]). Hence we can assume $K>0$
everywhere or equivalently $\Sigma$ is strictly convex. The classification
result of Hoffmann et al [HIMW19, Theorem 7.1] on complete strictly convex
translator inside a slab tells us that $\Sigma$ can only be $\triangle$-wing,
grim reaper surface or tilted grim reaper surface (as a translating graph over
slab in $T^{\perp}$ plane) in which none of them are entire graphical over
$xy$-plane.

From the three cases above, we conclude that $H>0$ cannot occur and $\Sigma$
has to be minimal. ∎

Combining our result above and the rigidity result of Spruck-Xiao [SX20], we
obtain the following rigidity result for entire graphical translator with
arbitrary translating direction:

######  Corollary 3.2.

Let $\Sigma^{2}\subseteq\mathbb{R}^{3}$ be an entire graphical translator with
any unit translating velocity $T$. If $H\geq 0$ then $\Sigma$ is either a
plane parallel to the $T$ direction or a bowl soliton.

Using our result and a classification result by Martin et al., we can also
obtain the following immediate consequence

######  Corollary 3.3.

Let $\Sigma^{2}\subseteq\mathbb{R}^{3}$ be an entire graphical translator with
unit translating velocity $T$ satisfying $0\leq\langle T,e_{3}\rangle<1$. If
$H\geq 0$ outside a compact subset of $\Sigma$ then $\Sigma$ is a plane.

###### Proof.

We only need to show that $\\{H=-1\\}=\emptyset$ then [MSHS15, Theorem D]
implies that $\Sigma$ is either a plane or $H>0$. However the later cannot
happen by Theorem 1.1.

Suppose there exist $p\in\Sigma$ such that $H(p)=-1$, by (1.1) we have
$\nu(p)=-T$ where $\nu$ is upward unit normal. Then
$\langle\nu,e_{3}\rangle=-\langle T,e_{3}\rangle\leq 0$, contradict with the
graphical assumption of $\Sigma$. ∎

######  Remark 3.4.

In [MSHS15, Theorem D], Martin et al. show that a translating soliton $\Sigma$
satisfying $H>-1$ everywhere and $H\geq 0$ outside a compact set is either
isometric to a plane or $H>0$ everywhere. Without further assumption, it is
not possible to exclude the case for $\langle\nu,T\rangle=H>0$ since a priori
a rotation of non-planar vertical graphical translator could exists. On the
other hand, with additional graphical condition $\langle\nu,e_{3}\rangle>0$,
we can rule out the case for $H>0$.

##  4\. Graphical translator with small entropy

In this section, we study the rigidity problem of non-vertical graphical
translating soliton when its entropy is less than 2.

######  Theorem 4.1 (Theorem 1.2).

Let $\Omega$ be an open subset of $\mathbb{R}^{2}$ and let

$\Sigma=\\{(x,y,u(x,y)):(x,y)\in\Omega\\}$

be a complete graphical translating soliton in the direction $T$ not parallel
to $e_{3}$. If $\lambda(\Sigma)<2$, $\Sigma$ is a hyperplane.

###### Proof.

For all $t<0$, let $\Sigma_{t}=\Sigma+tT$. For any $\delta>0$, consider the
re-scaling $\Sigma^{\delta}_{t}=\delta\Sigma_{\delta^{-2}t}$. As discussed in
section 2, there is $\delta_{i}\to 0$, $\tau_{i}\to-1$ so that
$\\{\Sigma^{\delta_{i}}_{\tau_{i}}\\}$ converges locally smoothly to an
embedded self-shrinker $\Sigma_{\infty}$, possibly with multiplicity.
Moreover, $\lambda(\Sigma_{\infty})\leq\lambda(\Sigma)<2$.

Since $\Sigma$ is graphical, it is simply connected and thus $\Sigma_{\infty}$
is a genus $0$ embedded self-shrinker. By the classification of Brendle
[Bre16], $\Sigma_{\infty}$ is either the round sphere of radius $2$, the
cylinder of radius $\sqrt{2}$ or the plane. Since $\Sigma$ is non-compact,
$\Sigma_{\infty}$ cannot be the round sphere. Since
$\lambda(\Sigma_{\infty})<2$, we conclude that $\Sigma_{\infty}$ is either the
multiplicity one cylinder or the multiplicity one plane. In particular, the
convergence $\Sigma^{\delta_{i}}_{\tau_{i}}\to\Sigma_{\infty}$ is locally
smooth.

If $\Sigma_{\infty}$ is the multiplicity one plane, than $\lambda(\Sigma)=1$
and thus $\Sigma$ is itself a hyperplane.

Next we assume that $\Sigma_{\infty}$ is the multiplicity one cylinder and
derive a contradiction. Since $\Sigma$ is graphical in the $e_{3}$ direction,
$\Sigma^{\delta}_{t}$ is graphical in the $e_{3}$ direction for all $\delta>0$
and $t<0$. Since $\Sigma^{\delta_{i}}_{\tau_{i}}$ is close to
$\Sigma_{\infty}$ in large balls, the axis of the cylinder $\Sigma_{\infty}$
must be parallel to $e_{3}$, that is

$\Sigma_{\infty}=C=\\{(x,y,z)\in\mathbb{R}^{3}:x^{2}+y^{2}=2\\}.$

Let $q_{0}=cT$, where $c>0$ is fixed so that $q_{0}$ lies outside of $C$. Let
$S$ be the sphere in $\mathbb{R}^{3}$ centered at $q_{0}$ with radius $r_{0}$.
Here $r_{0}$ is chosen small so that $S$ is disjoint from the cylinder $C$.
Let $\epsilon>0$ be the distance between $S$ and $C$. Let $K$ be a fixed large
box in $\mathbb{R}^{3}$ centered at $(0,0,0)$ and contains $q_{0}$ and $S$.
Since $\Sigma^{\delta_{i}}_{\tau_{i}}$ converges smoothly in $K$ to $C$, there
is $N\in\mathbb{N}$ so that if $i\geq N$, $\Sigma^{\delta_{i}}_{\tau_{i}}\cap
K$ can be represented as a graph of a function $u_{i}$ on $C\cap K$ and
$|u_{i}|<\epsilon$. By the choice of $\epsilon$, $S$ is disjoint from
$\Sigma^{\delta_{i}}_{\tau_{i}}$ for all $i\geq N$. Since $\Sigma$ is a
translator moving with velocity $T$, $\Sigma^{\delta_{i}}_{t}$ is a
translating solution moving with velocity $\delta_{i}^{-2}T$. For any $i\geq
N$, there is $q_{i}\in\Sigma^{\delta_{i}}_{\tau_{i}}$ lying in the straight
line joining $(0,0,0)$ to $q_{0}=cT$. In particular, the translating soliton
$\Sigma^{\delta_{i}}_{\tau_{i}+t_{i}}$ contains $q_{0}$, where

$t_{i}=\frac{|q_{0}-q_{i}|}{|\delta_{i}^{-2}T|}=\delta_{i}^{2}\frac{|q_{0}-q_{i}|}{|T|}\leq\delta_{i}^{2}c.$

Note that the MCF starting at the sphere $S$ collapses to $q_{0}$ at time
$t=r_{0}^{2}/4$. Hence if $\delta_{i}^{2}c<r_{0}^{2}/4$, the MCF starting at
$M^{\delta_{i}}_{\tau_{i}}$ and $S$ respectively intersect at some time
$\tilde{t}_{i}\leq t_{i}$. This contradicts to the avoidance principle for the
MCF and thus $\Sigma_{\infty}$ cannot be the cylinder. ∎

##  5\. Graphical translator with certain growth rate

In this section, we consider general dimensional graphical translator
satisfying certain growth condition.

######  Theorem 5.1 (Theorem 1.3).

Assume that $T$ is perpendicular to $e_{n+1}$, and let $\Sigma$ be an entire
graphical translator with velocity $T$ given by
$\Sigma=\\{(x,u(x)):x\in\mathbb{R}^{n}\\}$. If there are positive numbers
$C_{1},C_{2}$ so that

(5.1) $|u(q)|\leq C_{1}+C_{2}\sqrt{|\langle q,T\rangle|},\ \ \ \forall
q\in\mathbb{R}^{n}$

then $\Sigma$ is the stationary horizontal plane.

###### Proof.

By a rotation of $\mathbb{R}^{n+1}$ fixing the $e_{n+1}$ direction, we may
assume that $T=e_{n}$. For any $q\in\mathbb{R}^{n}$, write
$q=(\vec{q},q_{n})$, where $\vec{q}=(q_{1},\cdots,q_{n-1})$. Then (5.1) is the
same as

(5.2) $|u(\vec{q},q_{n})|\leq C_{1}+C_{2}\sqrt{|q_{n}|},\ \ \ \text{for all
}(\vec{q},q_{n})\in\mathbb{R}^{n}.$

For any $t\in\mathbb{R}$, $\Sigma_{t}=\\{x+tT:x\in\Sigma\\}$ is also an entire
graph on $\mathbb{R}^{n}$,

$\Sigma_{t}=\\{(q,u_{t}(q)):q\in\mathbb{R}^{n}\\}$

and $u_{t}:\mathbb{R}^{n}\to\mathbb{R}$ is given by

(5.3) $u_{t}(\vec{q},q_{n})=u(\vec{q},q_{n}-t).$

Since $\Sigma$ is a translator with velocity $T$, $\\{\Sigma_{t}\\}$ is a
solution to the mean curvature flow and $(t,q)\mapsto u_{t}(q)$ satisfies the
graphical mean curvature flow equation,

(5.4) $\frac{\partial u_{t}}{\partial
t}=\sqrt{1+|Du_{t}|^{2}}D_{i}\left(\frac{D_{i}u_{t}}{\sqrt{1+|Du_{t}|^{2}}}\right)$

For any $q=(\vec{q},q_{n})\in\mathbb{R}^{n}$ with $q_{n}\neq 0$, consider the
graphical mean curvature flow $u_{t}$ restricted to the domain
$\Omega=B_{|q_{n}|}(q)\times[-|q_{n}|,|q_{n}|]\subseteq\mathbb{R}^{n}\times\mathbb{R}$.
Then by (5.3) and (5.2), we have

(5.5) $M:=\sup_{\Omega}|u_{t}(q)|\leq C_{1}+C_{3}\sqrt{|q_{n}|},$

where $C_{3}=\sqrt{2}C_{2}$. Using the gradient estimates for graphical mean
curvature flow [ES92, Corollary 5.3], one has

(5.6) $|Du(q)|=|Du_{0}(q)|\leq
C\left(1+\frac{M}{q_{n}}\right)e^{CM^{2}\left(q_{n}^{-2}+q_{n}^{-1}\right)},$

where $C$ is a dimensional constant. By (5.5) this implies when $q_{n}\neq 0$,

(5.7) $|Du(q)|\leq C_{4}$

for some constant $C_{4}$. By continuity of $Du$, (5.7) holds for all
$q\in\mathbb{R}^{n}$. Hence the Gauss map of $\Sigma$ lies in
$B^{S^{n}}_{\lambda}(y_{0})$ for some $\Lambda<\pi/2$ and
$y_{0}=(0,\cdots,0,1)$. By [BS14, Theorem 1.1], $\Sigma$ must be a hyperplane.

∎

######  Remark 5.2.

For the vertical graphical translator case, by using bowl soliton as a
barrier, the growth rate at infinity of $u$ can only be
$|u(x)|=O(|x|^{\alpha})$ where $\alpha\geq 2$ (see Remark 3.2(a) in [MSHS15]).
The situation is quite different for non-vertical graphical translator case
where we don’t have bowl soliton as a barrier.

Using Ecker-Huisken interior gradient estimate for graphical solution of MCF,
we can also obtain rigidity result for graphical translator with slow gradient
growth.

######  Theorem 5.3 (Theorem 1.4).

Let $\Sigma$ be an entire graphical translator given by
$\Sigma=\\{(x,u(x)):x\in\mathbb{R}^{n}\\}$ with unit velocity $T$ not parallel
to $e_{n+1}$ direction. If the gradient of $u$ satisfies the growth rate

(5.8) $|Du(x)|=o(|x|^{1/4})$

then $\Sigma$ is a hyperplane.

###### Proof.

We can assume that $T=\cos\theta e_{n}+\sin\theta e_{n+1}$, for some
$\theta\in(\pi/2,\pi/2)$, after a rotation of $\mathbb{R}^{n+1}$ fixing the
$e_{n+1}$ direction. Similar as before, for any $t\in\mathbb{R}$,
$\Sigma_{t}:=\Sigma+tT=\\{(x,u_{t}(x):x\in\mathbb{R}^{n}\\}$ is an entire
graphical solution of MCF where $u_{t}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is
given by

(5.9)
$u_{t}(x_{1},\cdots,x_{n})=u(x_{1},\cdots,x_{n}-t\cos\theta)+t\sin\theta$

and the gradient of $u_{t}$ is

(5.10) $Du_{t}(x)=Du(x-te_{n})$

Using Ecker-Huisken interior estimate [EH91, Corollary 3.2(ii)] for any $R>0$
and $t=R^{2}$, we obtain an estimate for the curvature of $\Sigma_{t}$ in term
of $Du_{t}$ i.e. for any $y\in\mathbb{R}^{n}$

(5.11) $\sup_{B_{R/2}(y)}|A|^{2}(R^{2})\leq
C\frac{1}{R^{2}}\sup_{B_{R}(y)\times[0,R^{2}]}|Du_{t}|^{4}$

From growth rate assumption and $t\in[0,R^{2}]$, we have

(5.12) $|Du_{t}(x)|=o(|x-te_{n}|^{1/4})=o(R^{1/2})$

Also since $\Sigma_{t}$ is a graphical translating solution, we know that for
any $t$

(5.13) $\sup_{B_{R}(y)}|A|^{2}(t)=\sup_{B_{R}(y-t\cos\theta e_{n})}|A|^{2}(0)$

Letting $R\rightarrow\infty$ will imply that $\Sigma$ is totally geodesic. We
can argue the previous statement by contradiction. Suppose $\Sigma$ is not
totally geodesic then there exist $y_{0}\in\pi(\Sigma)$, the projection of
$\Sigma$ to its domain, such that $|A|(y_{0})\neq 0$. Now we choose $R$ large
enough such that RHS of (5.11) is less than $|A|(y_{0})/2$ and pick
$y=y_{0}+R^{2}\cos\theta e_{n}$. Our choice of $R$, $y$ and also (5.13) give
us a contradiction to (5.11). ∎

##  6\. Counter example in higher dimension

In this section, we give an example to illustrate why the growth condition in
Theorem 1.3 is necessary and that Theorem 1.2 and Theorem 1.1 does not hold
for $n\geq 3$. Let $\mathcal{B}$ be the bowl soliton in $\mathbb{R}^{3}$ given
as a graph $\mathcal{B}=\\{(x,y,b(x,y)):(x,y)\in\mathbb{R}^{2}\\}$. Then for
each $n\geq 3$, consider the translator $\mathcal{B}\times\mathbb{R}^{n-2}$.
That is,

(6.1)
$\mathcal{B}\times\mathbb{R}^{n-2}=\\{(x,y,z_{1},\cdots,z_{n-2},b(x,y)):(x,y,z_{1},\cdots,z_{n-2})\in\mathbb{R}^{n}\\}.$

Let $R\in SO(n+1)$ be the rotation

$R(x,y,z_{1},\cdots,z_{n-2},w)=\left(x,y,z_{1},\cdots,z_{n-3},\frac{1}{\sqrt{2}}(z_{n-2}-w),\frac{1}{\sqrt{2}}(z_{n-2}+w)\right)$

and let $M=R(\mathcal{B}\times\mathbb{R}^{n-2})$.

Note that $M$ is an entire graph over $\mathbb{R}^{n}\times\\{0\\}$. Indeed,

(6.2)
$M=\\{(x,y,z_{1},\cdots,z_{n-3},\tilde{z},\tilde{z}+\sqrt{2}b(x,y)):(x,y,z_{1},\cdots,z_{n-3},\tilde{z})\in\mathbb{R}^{n}\\}.$

$M$ is a translator in the direction
$T=Re_{n+1}=\frac{1}{\sqrt{2}}(-e_{n}+e_{n+1})$. Since

$\lambda(M)=\lambda(\mathcal{B}\times\mathbb{R}^{n-2})=\lambda(\mathcal{B})=\lambda(\mathbb{S}^{1})<2$

this implies that Theorem 1.2 does not hold for $n\geq 3$. Since $M$ is convex
and thus mean convex, this also implies that the Theorem 1.1 cannot be
generalized to $n\geq 3$. Lastly, since

$T_{1}:=-\sqrt{2}e_{n}=T-\frac{1}{\sqrt{2}}(e_{n}+e_{n+1})$

and $e_{n}+e_{n+1}$ is tangential to $M$, so

$T_{1}^{\perp}=T^{\perp}$

If we consider $\bar{M}=\sqrt{2}M$, since $M$ is a graphical translator moving
in $T$ direction then $\bar{M}$ will be a graphical translator moving in
$-e_{n}$ direction. This also implies that Theorem 1.3 does not hold without
extra growth assumption when $n\geq 3$.

## Acknowledgement

The first author is supported in part by DFF Sapere Aude 7027-00110B, by CPH-
GEOTOP-DNRF151 and by CF21-0680 from respectively the Independent Research
Fund Denmark, the Danish National Research Foundation and the Carlsberg
Foundation.The second and third author are supported in part by the National
Research Foundation of Korea (NRF-2021R1A4A1032418).

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