GPS: Improvement in the formulation of the TSP for its generalizations type QUBO

Abstract

We propose a new binary formulation of the Travelling Salesman Problem (TSP), with which we overcame the best formulation of the Vehicle Routing Problem (VRP) in terms of the minimum number of necessary variables. Furthermore, we present a detailed study of the constraints used and compare our model (GPS) with other frequent formulations (MTZ and native formulation). Finally, we have carried out a coherence and efficiency check of the proposed formulation by running it on a quantum annealing computer, D-Wave 2000Q6.

Full text

GPS: Improvement in the formulation of the TSP for its generalizations type QUBO
Sa´ul Gonz´alez-Bermejo,1Guillermo Alonso-Linaje,1and Parfait Atchade-Adelomou2, 3
1Universidad de Valladolid C/Plaza de Santa Cruz, 8, 47002 Valladolid (Spain) ∗
2Engineering Department Research Group on Data Science for the Digital Society La Salle -
Universitat Ramon Llull Carrer de Sant Joan de La Salle, 42 08022 Barcelona (Spain) †
3Lighthouse Disruptive Innovation Group, LLC 7 Broadway Terrace,
Apt 1 Cambridge MA 02139 Middlesex County, Massachusetts (USA) ‡
(Dated: Oct 2021)
We propose a new binary formulation of the Travelling Salesman Problem (TSP), with which
we overcame the best formulation of the Vehicle Routing Problem (VRP) in terms of the minimum
number of necessary variables. Furthermore, we present a detailed study of the constraints used and
compare our model (GPS) with other frequent formulations (MTZ and native formulation). Finally,
we have carried out a coherence and efficiency check of the proposed formulation by running it on
a quantum annealing computer, D-Wave 2000Q6.
KeyWords: Quantum Computing, Quantum Annealing, Optimizaci´on combinatoria, QUBO,
TSP, VRP
I. INTRODUCCI ´
ON
It is known that the best current QUBO formulation
of the Traveling Salesman Problem (TSP) requires N2
binary variables (IXA)). However, when this formula-
tion is extended to more general problems such as Vehi-
cle Routing Problem (VRP), its modelling is no longer
efficient in terms of the number of variables because it re-
quires the use of new variables to transform it into QUBO
formulation. The number of the qubits, are crucial in to-
day’s quantum computing, so it is necessary to develop
models focused on reducing the number of required vari-
ables.
The VRP encompasses two different problems: one in
which the distance travelled by vehicles subject to capac-
ity restrictions is minimized [1–4] and another, in which
the time spent it takes vehicles to complete their routes
is minimized. In this article, everything related to the
VRP will optimize the time to complete these routes.
When modelling this problem, it is necessary to estab-
lish relationships between the distances travelled by the
vehicles. For this reason, it is required that the distance
calculation function between journeys has to be a linear
function if we want to reduce the number of auxiliary
variables used.
The best QUBO models of the TSP that can represent
distance linearly use more than N3variables (native TSP
formulation). There is indeed a formulation that uses
N2log2(N) variables (MTZ [5]), but its results are not as
satisfactory when applying the annealing algorithms on
it.
Our purpose in this work is to present a new QUBO
model of the TSP whose the travelled distance’s calcula-
tion is linear, and that uses only 3N2variables, consid-
erably improving the existing TSP models (both of N3
∗saul.gonzalez.bermejo@alumnos.uva.es
†parfait.atchade@salle.url.edu
‡parfait.atchade@lighthouse-dig.com
and N2log2(N) variables). Furthermore, this new formu-
lation of the TSP will be generalized to define the most
efficient formulation in terms of the number of variables
of the VRP.
The document is organized as follows. Section (II)
shows previous work on the TSP algorithm and its deriva-
tives. Section (III) will explore adiabatic computation.
In the section (IV), the general formulations of TSP and
MTZ will be explored. Section (V) presents our TSP
proposal with the improvements in numbers of qubits for
quantum projects of this NISQ era and beyond. A gener-
alization of our contribution is seen in Section (VI) where
we propose an optimal VRP at the qubit level. In Section
(VII), we present the results of the entire study carried
out. In the section (VIII), we discuss the results and con-
trast the contributions. Finally, Section (IX) concludes
the work carried out, and we open ourselves to some lines
of the future of the proposed model.
II. RELATED WORK
In the mid-1920s, these two references, [6, 7] were the
first articles to provide a solution to the minimal span-
ning tree (MST) problem. Based on these works, the
mathematical researcher, Joseph B. Kruskal Jr, applied
these solutions to the TSP [8], giving life to some of the
first solutions to this problem that will arise during the
next decades.
Almost at the end of the sixties, this work [9] offered a
compilation and synthesis of the research on the travel-
ling salesman problem. Its authors began by defining the
problem and presenting several relevant theorems. They
also classified and detailed the solution techniques and
computational results. Before that, in the mid-1960s,
the TSP started to emerge in many different contexts.
This article [10] highlights some applications that began
to gain space in everyday life, such as vehicle routing or
job shop scheduling problems. Other applications such
as planning, logistics and the manufacture of electronic
arXiv:2110.12158v1 [quant-ph] 23 Oct 2021

2
circuits became of particular interest.
By making a few small modifications to the original
TSP, we could apply it in many fields such as SWP [11]
and DNA sequencing [12, 13] among others. In this last
application, the concept of ‘city’ would come to be frag-
ments of DNA and the idea of ‘distance’, a measure of
similarity between the pieces of DNA. In many applica-
tions, additional restrictions such as resource limits or
time windows make the problem considerably difficult.
Computationally, the TSP [14] is an NP-Hard prob-
lem within combinatorial optimization. As an NP-Hard
problem, it is computationally complex, and heuristics
are continually being developed to get as close as possible
to the optimal solution. However, considering the com-
putational complexity nature of these problems, the new
approach that quantum computing takes is very promis-
ing.
Many works are related to the standard/native TSP
or some related variant in a quantum environment within
this new approach. For example, in this work [15], the au-
thors introduced a different quantum annealing scheme
based on a path-integral Monte Carlo processes to ad-
dress the symmetric version of the Travelling Salesman
Problem (sTSP). In these other articles [16, 17], the au-
thors did a comparative study using the D-Wave platform
to evaluate and compare the efficiency of quantum an-
nealing with classical methods for solving standard TSP.
In this reference, [18] several comparisons of heuris-
tic techniques were made for some TSPLIB problems,
both symmetric and asymmetric, and their results have
been compared to other methods such as Self Organizing
Maps and Simulated Annealing. In both cases, the local
search technique was applied to the results found with
Wang’s Recurrent Neural Network with ”Winner Takes
All”. The computational complexity of their heuristics
method got O(n2+n) [19], considered competitive when
compared to Self Organizing Maps, which has a com-
plexity of O(n2) [20]. The authors went on to show that
the CAN technique has a computational complexity of
O(n2log(n)) [21], while the Simulated Annealing tech-
nique has a complexity of O(n4log(n)) [22]. Results that
are quite far from what we propose in this article.
One of the generalizations of the TSP, known as the
VRP, was studied on the D-Wave platform [23, 24]. In
tasks where routing and planning capacity (time) is re-
quired, the TSP with time windows (TSPTW) was gen-
eralized [25, 26], and has high inherent complexity and
presents enormous resolution difficulties. In these refer-
ences [11, 27, 28], the authors modelled combinatorial
optimization problems in which social workers visit their
patients at their respective homes and attend to them at
a specific time, called Social Workers’ Problem (SWP).
SWP is a significant problem because additional time
constraints allow more realistic scenarios to be modelled
than native TSP. The optimal or near-optimal solution
for such issues is important in minimizing distance and
time and environmental problems such as reducing fuel
consumption.
The generalization of the TSP that we will use in work
will be the VRP. However, there are other TSP deriva-
tives, such as the Job Shop Scheduling Problem (JSSP)
[29] that are not included in the study of this work.
During state of the art carried out, we have found sev-
eral articles [23, 24, 30] that solve the TSP and VRP
(focusing on minimizing distance and not time) for an-
nealing computers. However, the number of variables
is still intractable for the current size of quantum com-
puters. For this reason, we propose a new TSP formu-
lation with a representation of the linear distance that
uses only 3N2variables, which we will use to outperform
the current best VRP modelling in terms of the number
of required variables.
III. ADIABATIC QUANTUM COMPUTERS
Adiabatic computation was born from the use of the
adiabatic theorem [31, 32] to perform the calculations.
Using the tunnel effect to go from the global mini-
mum of a simple Hamiltonian to the global minimum
of the problem of interest. One of the market leaders
for this type of computing is D-Wave, which roughly
solves the quadratic unconstrained binary optimization
problem (QUBO). The QUBO formulation (1) is suitable
for running a D-Wave architecture; however, QUBO can
be mapped to the Ising [33] model and thus be used in
computers based on quantum gates, for example, IBMQ,
Rigetti, Xanadu (strawberryfields), etc.
The problems that D-Wave quantum computers are
prepared to solve are those that consist in finding the
minimum of a function of the following form:
n
X
i=1
bixi+
n
X
i=1
n
X
j=1
qi,j xixj,(1)
where the variables xi∈ {0,1}and the coefficients
bi, qi,j ∈R.
We have then that, given a problem, we need to model
it with the above structure where the variables that form
the solution will only take the values 0 or 1. Let us
observe that, by taking the variables xithe values 0 or
1, it is true that x2
i=xi. Therefore, we can group the
linear terms with the quadratic terms and express the
above equation in matrix format:
xtQx, (2)
with x∈ {0,1}nand Q∈Mn×nwhich is compactly
representing the QUBO formulation.
IV. TSP FORMULATION
As discussed in the introduction, we will develop the
QUBO model of the TSP in which we have worked and

3
which improves the number of variables required from
the previous models whose distance calculation function
is linear. To do this, first, we will present the models
that we will improve by analyzing their strengths and
weaknesses.
A. Native Formulation
In this section, we will recall the formulation of the
native TSP [30] to analyze it in detail. This modelling,
which has been defined in [34], despite appearing in a very
natural way which facilitates its understanding, requires
N3variables to be implemented. For today’s quantum
computers, due to qubit number limitations, we could
hardly solve cases with several cities less than 15.
The variables that appear in this model are the
variables xi,j,t such that i, j ∈ {0, ..., N + 1}and
t∈ {0, ..., N }. Let us consider that the variables xi,i,t do
not exist in this model.
The interpretation of the variables xi,j,t is simple, since
xi,j,t = 1 if at instant twe traverse the edge that
connects the cities iand j, and xi,j,t = 0 for all other
cases.
We can define the objective function of the native (Na-
tive in the sense of general, the most used) TSP[30] as:
N+1
X
u=0
N+1
X
v=0
N
X
t=0
xu,v,tdu,v .(3)
where du,v represents the distance between nodes u
and v. This objective function is subject to a series of
restrictions:
•Constraint 1. The salesman must leave each city
once.
For each u∈ {0, ..., N}:
N+1
X
v=1
N
X
t=0
xu,v,t = 1.(4)
•Constraint 2. Each city must be reached once.
For each v∈ {1, .., N + 1}:
N
X
u=0
N
X
t=0
xu,v,t = 1.(5)
•Constraint 3. If the salesman leave a city, he cannot
return to it later. This constraint ensures that no
unconnected cycles are formed as a solution. There
are two ways of posing this constraint.
–Imposing that once he leaves a city he cannot
return to it.
For each u∈ {1, ..., N + 1}:
N+1
X
v=0
N
X
t=0
N+1
X
w=0
N
X
j=t+1
xu,v,txw ,u,j = 0.(6)
–Imposing that once he arrives in a city, at the
next moment, he must leave it.
For each t∈ {0, ..., N −1},u, v ∈ {0, ..., N }:
xu,v,t(1 −
N+1
X
w=1
xv,w,t+1 ) = 0.(7)
This formulation requires N3variables. Next, we will
analyze another model used to define the TSP, which,
although it uses fewer variables, offers worse results when
working with annealing.
B. MTZ formulation.
The idea of this formulation is to consider the vari-
ables xi,j = 1 if the edge that connects the cities iand
jappears in the solution path, where xi,j = 0 for all
other cases. Once we have these variables, we can es-
tablish order on the route employing a set of variables
that will represent the moment the salesman arrives at
that city (the variable uiexpressed in binary format, will
take the integer value tif the city iis reached in the t
-th position.). This model requires N2log2(N), greatly
improving the number of variables in the general formu-
lation. However, when implemented using annealing it
presents surprisingly bad results. This is because the an-
nealing algorithm gets stuck trying to minimize the part
of the objective function generated by the constraint of
the equation (11), since the representation of integers in
their binary format has the disadvantage that close num-
bers such as 2n−1 and 2ndiffer by a large number of
qubits, so from the annealing they are perceived as very
different solutions.
Next, let us define the variables needed for this model.
•xi,j such that i, j ∈ {0, ..., N + 1}. They will repre-
sent the edges that appear in the route.
•Let H:= blog2(N+ 1)c+ 1 and then define ui:=
PH
h=0 ui,h such that i∈ {0, ..., N +1}. As discussed
in the previous paragraph, these variables represent
the order in which the cities are travelled.
•To be able to handle the previous variables, it
is necessary to include the following variables:
slacki,j,h such that i, j ∈ {0, ..., N + 1}and h∈
{0, ..., H1}with H1:= blog2(N+ 1)c+ 2. These
auxiliary variables will work as slack variables to
transform inequalities into equalities.
Once the model variables have been defined, let us ana-
lyze the constraints that must be met.
Similar to the native TSP, we can define the MTZ ob-
jective function as follows:
N+1
X
i=0
N+1
X
j=0
xi,j di,j ,(8)

4
where di,j represents the distance between cities iand j.
In this case, the restrictions to be satisfied are as follows:
•Constraint 1 : The salesman must leave each city
once.
For each i∈ {0, ..., N}:
N+1
X
j=1
xi,j = 1.(9)
•Constraint 2 : Each city must be reached once.
For each j∈ {1, ..., N + 1}:
N
X
i=1
xi,j = 1.(10)
•Constraint 3 : There must be an order between
the cities so that there is a single cycle that
runs through all of them. Then, for each i, j ∈
{0, ..., N + 1}:
ui−uj+ (N+ 2)xi,j +
H1
X
h=0
2hslacki,j,h =N+ 1.(11)
This restriction guarantees that if xi,j = 1, what
means, from city iwe travel to city j, then the
value of ujmust be greater than ui. This ensures
that the uivariables provide a meaningful order.
Once the two most common QUBO models of the TSP
have been presented, let us analyze the formulation with
which we improve the number of variables of the previous
two.
V. GPS FORMULATION
To develop this model, we take the variables xi,j,r with
i, j ∈ {0, ..., N +1}and r∈ {0,1,2}. In all the modelling,
the variables xi,j,r such that i=jare not considered. We
work with directional edges, that is, if in the model the
edge (i, j) appears, we will understand that first we go
through node iand immediately after that we go to j.
Let us analyze what each variable represents:
•xi,j,0= 1 means that the edge (i, j ) does not appear
in the path and the node iis reached earlier than
the j.
•xi,j,1= 1 means that the edge (i, j) appears in the
path, so the node iis reached earlier than the j.
•xi,j,2= 1 means that the edge (i, j ) does not appear
in the path, and the node jis reached earlier than
the i.
From the definition of our variables, we can define the
distance travelled through the following objective func-
tion as:
N+1
X
i=0
N+1
X
j=0
di,j xi,j,1.(12)
The constraints that must be met are:
•Constraint 1 : For each i, j one and only one of the
3 cases of rmust be given, so
For all i, j:
2
X
r=0
xi,j,r = 1.(13)
•Constraint 2 : Each node must be exited once.
For each i∈ {0, ..., N}:
N+1
X
j=0
xi,j,1= 1.(14)
•Constraint 3 : Each node must be reached once.
For each j∈ {1, ..., N + 1}:
N
X
i=0
xi,j,1= 1.(15)
•Constraint 4 : If node iis reached before j,
then node jis reached after i, so, for all i, j ∈
{0, ..., N + 1}such that i6=j:
xi,j,2= 1 −xj,i,2.(16)
It would also have to be specified for r= 0 and
r= 1, however this restriction is sufficient since by
(13) it is implicit.
•Constraint 5 : If node iis reached before node j
and node jis reached before node k, then node i
must be reached before k. This condition will pre-
vent the route from returning to a node from which
it had already exited, thus preventing cycles from
forming. We then arrive at the penalty function
equation (17).
N
X
i=1
N
X
j=1
N
X
k=1
(xj,i,2xk,j,2−xj,i,2xk,i,2−xk,j,2xk,i,2+xk,i,2).
(17)
With only the cases in which i6=j,i6=kand j6=k
will be taken in the summation and in the annex
(X) we will provide the approach followed to arrive
at it.
The following is deduced from the equation (17).
We have xi,j,2= 0 if iis reached before jand
xi,j,2= 1 in the case where jis reached before i.
Thus, with the previous equation we penalize these

5
following cases in which xi,j,2= 0, xj,k,2= 0 and
xi,k,2= 1 and xi,j,2= 1, xj,k,2= 1 and xi,k,2= 0
which lead to cases in which it would be forming
cycles (for these two situations we have that the
value of the parentheses is 1 and for the rest 0). In
(31) we have a similar situation for our VRP formu-
lation, where we offer more details. For this condi-
tion, we must have directly constructed a penalty
function that avoids erroneous cases without first
posing linear conditions through which to generate
its corresponding penalty function.
Formulated in this way we have managed to reduce
the number of variables required from N2log2Nto 3N2,
achieving very noticeable reductions when working with
large problems. Once we have this formulation, let us see
how we can generalize it to the VRP.
VI. VRP FORMULATION
As discussed in the introduction, this model is ori-
ented to be optimal concerning the number of qubits
used. However, this generalization does not appear as
naturally as expected because it requires a delicate step
to get the constraints of the equation (31). To do this,
we will detail each step and explain each constraint step
by step.
A. Original formulation 5N2Q
For this VRP, we will consider that Nis the number
of cities and Qis the number of available vehicles. We
first present the variables that will form the problem. We
then take the following set of variables.
xi,j,r,q with i, j ∈ {0, ..., N + 1},
r∈ {0,1,2,3,4}and q∈ {1, ..., Q}(18)
In all the modelling, the variables xi,j,r such that i=j
are not considered. The variables i,jrefer to the cities
must travel to, and the variable qrefers to the vehicle.
The nodes 0 and N+ 1 correspond to the starting and
ending points. Note that they may be the same node but
we will separate them for convenience in the formulation.
The values di,j with i, j ∈ {0, ..., N + 1}correspond to
the distance between node iand j. Let us dive into the
interpretation of each variable:
•xi,j,0,q = 1 means that the vehicle qtravels to the
cities iand j, does not travel across the edge (i, j)
and arrives at the city ibefore the j.
•xi,j,1,q = 1 means that the vehicle qtravels to the
cities iand jtravels across the edge (i, j) (that is,
once it passes through the city ithe next city it
reaches is the j) and therefore the city iis reached
earlier than the city j.
•xi,j,2,q = 1 means that the vehicle qtravels through
the cities iand jand arrives at the city jearlier
than at the i.
•xi,j,3,q = 1 means that the vehicle qdoes not go
through the cities iand j, and the city iis reached
earlier than the city j. Note that xi,j,3,q can take
the value 1 whether the vehicle qpasses through
one of both cities or neither of them.
•xi,j,4,q = 1 means that the vehicle qdoes not travel
to the cities iand j, and the city jis reached earlier
than the city i.
Even if no vehicle passes through the objects iand j,
the formulation must establish an order between them.
However, this restriction does not make the modelling
meaningless, since we can assume that if the vehicles are
ordered in the order of {1, ..., Q}, then iwill be reached
before jif the vehicle that passes through node ihas a
lower number than the one that passes through node j.
Once the interpretation of each variable is explained, let
us analyse the constraints that must be met.
•Constraint 1: For each i, j, q, one and only one of
the possibilities must be met for r, so:
For all i, j, q:
4
X
r=0
xi,j,r,q = 1,(19)
•Constraint 2: Each vehicle has to fulfill that it
leaves the starting position. For this situation, we
are going to impose that:
For all q:
N+1
X
j=1
x0,j,1,q = 1,(20)
No vehicle can return to the starting position from
a city, so:
For all q:
N+1
X
i=0
xi,0,1,q = 0,(21)
•Constraint 3: Every vehicle must finish in the final
position. For this, it must be fulfilled that:
For all q:
N
X
i=0
xi,N+1,1,q = 1,(22)
No vehicle can leave the final position. We then
have that:
For all q:
N+1
X
j=0
xN+1,j,1,q = 0.(23)
Vehicles that do not travel on any road will
meet all constraints when taking the following con-
dition:
x0,N+1,1,q = 1.

6
•Constraint 4: The vehicle must leave once and only
once from each city, then:
For each i∈ {1, ..., N}:
Q
X
q=1
N+1
X
j=1
xi,j,1,q = 1.(24)
•Constraint 5: The vehicle must arrive once and
only once to each city, then:
For each j∈ {1, ..., N}:
Q
X
q=1
N
X
i=0
xi,j,1,q = 1.(25)
•Constraint 6: The city iis reached before the city j
does not depend on each vehicle. Therefore, for all
the vehicles that either arrive at city iearlier than
j, or arrive at city jearlier than i. Introducing
the auxiliary variables ai,j, we have the following
constraint. For all i, j ∈ {1, ..., N}:
Q
X
q=1
xi,j,0,q +xi,j,1,q +xi,j,3,q =ai,j Q. (26)
It will then be true that for each i, j or ai,j = 1,
which means that the city iis reached earlier than
the city jand therefore for each qwe will have
xi,j,r,q = 1 for any value of the rin which iis
reached before j, or ai,j = 0, and, we will have
xi,j,r,q = 0 for all the vehicles and for values rwhere
iis reached before j.
•Constraint 7: If the vehicle qarrives in the
city j, then the vehicle qmust leave the city
j. For this we impose the constraint that
for i∈ {0, ..., N },j∈ {1, ..., N}and q∈
{1, ..., Q}:
xi,j,1,q(1 −
N+1
X
k=1
xj,k,1,q ) = 0.(27)
Let us now impose the conditions that make vehicles
run on a tour.
•Constraint 8: It must be fulfilled that either the
vehicle pass through the city ibefore the jor ar-
rive before to the city jthan the i. Therefore,
it must be verified that, for i∈ {0, ..., N},j∈
{1, ..., N }and q∈ {1, ..., Q}:
xi,j,0,q +xi,j,1,q +xi,j,3,q = 1 −(xj,i,0,q +xj,i,1,q +xj,i,3,q).
(28)
•Constraint 9: If city iis reached before jand city j
is reached before city k, then city imust be reached
before city k. This condition will prevent the ve-
hicle from returning to a city it has already passed
through and therefore prevents a cycle from form-
ing.
To introduce this constraint, we will directly calcu-
late a penalty function worth 0 in the correct cases
and 1 in those that are not. To facilitate the un-
derstanding of the penalty function, we are going
to take, for i, j, k, q, the following variables:
ai,j =xi,j,0,1+xi,j,1,1+xi,j,3,1
aj,k =xj,k,0,1+xj,k,1,1+xj,k,3,1
ai,k =xi,k,0,1+xi,k,1,1+xi,k,3,1.
(29)
Remember that it is not necessary to introduce
these conditions because the constraint (26)
establishes the correct values of the variables ai,j.
Therefore, ai,j = 1 means that the city iis reached
before the city jand the same with jand k. Also,
it is very important to remember that due to
the same constraint (26), we can take any of the
vehicles as a reference. In this case, we have taken
the first vehicle as a reference.
In this way, fixed i, j, k, we have the 3 variables
ai,j , aj,k, ai,k . Remember that ai,j , aj,k , ai,k only
take the values 0 or 1. Also, let us note that
the cases that lead to values of the variables
for which cycles can be formed and that we
must discard are (ai,j, aj,k , ai,k ) = (0,0,1) and
(ai,j , aj,k, ai,k ) = (1,1,0).
In the case (0,0,1) we would have that the city
jis reached after the i, the kafter the j, and
yet the city kis reached rather than i, which
is absurd. The case (1,1,0) cannot be given ei-
ther, since it reaches ibefore jand jbefore k, so
it cannot be that we also reach kbefore i. We
therefore must construct a penalty function so that
for f(ai,j , aj,k, ai,k ) it holds that f(0,0,1) >0,
f(1,1,0) >0 y f(ai,j , aj,k, ai,k ) = 0 for all other
cases. A function that satisfies these conditions is
from the equation (30).
f(ai,j , aj,k, ai,k ) := ai,j aj,k −ai,jai,k −aj,k ai,k+a2
i,k.(30)
then, adding to the cost function the equation (31)
λ
N
X
i=1
N
X
j=1
N
X
k=1
(ai,j aj,k −ai,j ai,k −aj,kai,k +a2
i,k),(31)
we will have that the best solutions will be those
that comply with this constraint.
•Constraint 10: The objective we seek is to mini-
mize vehicle travel time. What we could do is to see
how long each vehicle takes to complete the route
and try to minimize as much of the time as possi-
ble. However, this function soon becomes complex

7
so we have decided to develop a different idea that
simplifies the process and smoothes the objective
function. If we impose the condition that all vehi-
cles travel less distance than the distance travelled
by vehicle number 1, we will have that minimizing
the maximum of the distances will be equivalent to
minimizing the distance travelled by the first ve-
hicle. We then have the following condition. For
each q∈ {2, ..., Q}:
N+1
X
i=0
N+1
X
j=0
di,j xi,j,1,q ≤
N+1
X
i=0
N+1
X
j=0
di,j xi,j,1,1.(32)
We transform this inequality into equality by tak-
ing once again Dmax := Pn
i=0 maxj{di,j }and the
variables bh,q (the variables bh,q are like in (11)
slack variables and they are in their binary expres-
sion) in:
N+1
X
i=0
N+1
X
j=0
di,j xi,j,1,q +
hmax
X
h=0
2hbh,q −
N+1
X
i=0
N+1
X
j=0
di,j xi,j,1,1= 0.
(33)
Under these conditions the function to be mini-
mized corresponds to:
N+1
X
i=0
N+1
X
j=0
di,j xi,j,1,1.(34)
This condition has the disadvantage that we are
eliminating solutions where it is another vehicle
that travels the longest distance. Let us explore
how to avoid this problem and get more flexibility
in the model to make it easier for the Quantum
Annealing to find the optimum one. We can es-
tablish an auxiliary variable Dand we set that the
distance travelled by each vehicle must be less than
this variable, that is to say:
N+1
X
i=0
N+1
X
j=0
di,j xi,j,1,q ≤D, for all q∈ {1, ..., Q}.(35)
The variable Dis an integer, so we must treat it in
some way in order to include it in the model. As
we explained in the introduction of the section ded-
icated to the formulation of the MTZ model (IV B),
it is convenient to try to avoid the binary represen-
tation of integer variables. To do so, we can ex-
press Das a combination of the distances between
edges by taking D=PN+1
i=0 PN+1
j=0 xi,j bi,j . Thus
after imposing the constraint (35) we have that the
function to minimize is D.
Thanks to this modelling of the VRP we have been able
to reduce the number of variables required to the order
of 5N2Q. However, we have managed to reduce it even
further to 3N2Q, which is detailed in (IX B). However,
we have preferred to present this other model due to its
easy understanding.
VII. RESULTS
Before analyzing the results, we are pleased with our
GPS formulation’s positive results achieved.
Let us observe in the different tables the comparison of
the number of qubits, time during which the D-Wave
Quantum Annealing simulator has been executed, and
the length of the path found. The sign ”-” represents
that the algorithm did not find a possible way during the
elapsed time (in minutes). In this examples, the cities
which form the TSP to solve are the vertex of the regular
polygon with these number of vertex.
GPS General MTZ
Number of qubits 75 100 140
Elapsed Time (min) 0.332 0.08 0.569
Path Length (m) 5.65 5.65 5.65
Table I. In this scenario of 4 cities, we set comparison with
the 3 models, MTZ, native TSP and GPS. The comparison
is based on the number of times to find the solution, the dis-
tance travelled, and the number of qubits. We can appreciate
the good performance of our GPS model, and above all the
savings it offers us in the number of qubits.
GPS General MTZ
Number of qubits 147 294 266
Elapsed Time (min) 0.337 0.39 1.338
Path Length (m) 6.00 6.00 8.46
Table II. In this scenario of 6 cities, we set comparison with
the 3 models, MTZ, native TSP and GPS. The comparison
is based on the number of times to find the solution, the dis-
tance travelled, and the number of qubits. We can appreciate
the good performance of our GPS model, and above all the
savings it offers us in the number of qubits.
These results have been obtained using a simulator be-
cause we would require access to a quantum computer
for a time similar to that needed to perform the simula-
tions (in some cases more than an hour). However, it is
the benefits of modeling with few qubits (such as GPS
modelling) will be much more notable when these prob-
lems are implemented in real quantum computers. Other
studies that did not require many hours of the quantum
computer were carried out on the DW 2000Q 6. In the
discussion section, we detail some interesting cases.

8
GPS General MTZ
Number of qubits 243 648 522
Elapsed Time (min) 1.209 1.177 2.676
Path Length (m) 6.122 9.58 11.46
Table III. In this scenario of 8 cities, we set comparison with
the 3 models, MTZ, native TSP and GPS. The comparison
is based on the number of times to find the solution, the dis-
tance travelled, and the number of qubits. We can appreciate
the good performance of our GPS model, and above all the
savings it offers us in the number of qubits.
GPS General MTZ
Number of qubits 363 1210 770
Elapsed Time (min) 3.316 3.087 4.175
Path Length (m) 12.51 10.978 -
Table IV. In this scenario of 10 cities, we set comparison with
the 3 models, MTZ, native TSP and GPS. The comparison
is based on the number of times to find the solution, the dis-
tance travelled, and the number of qubits. We can appreciate
the good performance of our GPS model, and above all the
savings it offers us in the number of qubits.
GPS General MTZ
Number of qubits 507 2028 1066
Elapsed Time (min) 7.992 9.677 10.578
Path Length (m) 14.286 12.28 -
Table V. In this scenario of 12 cities, we set comparison with
the 3 models, MTZ, native TSP and GPS. The comparison is
based on the number of times to find the solution, the distance
travelled, and the number of qubits.
VIII. DISCUSSIONS
Once the different models had been implemented, we
achieved the following results. Through the results of the
figure (1) up to the figure (4), the good performance of
our formulation compared to the general TSP [30] can
be observed. An almost identical operation is seen with
the generic TSP, except that we are improving at least
the number of qubits for the same cases in our proposal.
Although the time difference is not significant again, the
difference between path lengths is. Let us remember that
the advantage of the formulation in which we have worked
is based on improving the number of qubits used. We
then have that the larger the problems we are working
on, the better this difference will be appreciated in the
number of variables.
The MTZ model does not offer positive results. This
is since Annealing presents many difficulties to find
minimum expressions in which the representation of
integers appears in their binary format. This is because
although the numbers 2k−1 and 2kare close, they
are not close in their binary form since they differ in k
variables, so the annealing tends to present bad results.
Apart from that adjusting, the Lagrange coefficient of
this type of constraint is also a complicated task.
General and GPS modelling show better results. While
it is true that general modelling gives slightly better re-
sults, it requires the use of a higher number of qubits.
This may be since the function to be optimized for this
model has a smaller number of local minima where the
Annealing can get stuck or to a bad fit of the Lagrange
coefficients.
The problem on which the simulations are carried out
consists in finding the optimal path when the points are
placed on the vertices of the regular polygons that have
the same number of vertices as nodes in our problem.
Figure 1. Path length comparison for N = 9. In this graph,
we see how the length of the solution paths for the case of 9
cities is very similar so that both models give good results.
One of the behaviours and results that we believe is
important to mention is the following. We realized that
it is even more important to consider the number of edges
that our model generates. The vertex/connections in
a quantum computer are limited and define our quan-
tum computer’s typology and quality for error mitiga-
tion. Thus, a model that produces many edges (direct
links) may request more from a computer than another
generates few. The figure (5) offers us a comparative
study between our GPS model and the native TSP. This
figure shows the exponential behaviour and the number
of interconnections that each model offers. Our model
improves the number of qubits and gives us a great re-
sult reducing the number of connections a lot. The native
TSP behaves as 0.8(N+ 2)5while the GPS as 2(N+ 2)3.

9
Figure 2. Time comparison for N = 9. This graph shows
the time taken to carry out the executions in the case of 9
cities. Although it seems that there is a lot of difference, it
only represents 10% of the total time, which, as we have seen
in other experiences, is not significant.
Figure 3. Path length comparison for N = 11. For the exam-
ple of 11 cities, we can observe that the outcomes are quite
similar. Although the time difference is not significant again,
the difference between path lengths is. Let us remember that
the advantage of the modelling we have worked is based on
improving the number of qubits used. We then have that
the larger the problems we are working on, the better that
difference will be appreciated in the number of variables.
One aspect of GPS worth commenting on here is
to generalize it also to be used for the Cutting-plane
method. We must change the current constraint (17)
since this methodology only works with linear con-
straints. The way to do this is as follows. For each i, j, k:
•xj,i,2+xk,j,2≤2xk,i,2+w1
i,j,k
•xj,i,2+xk,j,2≥2xk,i,2−w2
i,j,k.
In these equations, the variables wp
i,j,k are auxiliaries.
The purpose of these variables is to satisfy the said con-
strains. These two restrictions are satisfied by all cases of
Figure 4. Time comparison for N = 11. For the example of
11 cities, we can observe that the outcomes are quite similar
because although there is a mean difference of about 20 sec-
onds between the results of both simulations, the experience
with this problem and other similar ones is that this very
small difference does not affect the results on the length of
the solution path.
Figure 5. In this figure we can appreciate the exponential be-
haviour and the number of interconnections that each model
offers. Our model (GPS) improves the number of qubits and
gives us a great result reducing the number of connections a
lot. The Native TSP behaves as 0.8(N+ 2)5while the GPS
as 2(N+ 2)3.
(xj,i,2, xk,j,2, xk,i,2) except for (0,0,1) (because it doesn’t
satisfy the second constraint) and (1,1,0) (because it
doesn’t satisfy the first constraint).
IX. CONCLUSIONS AND FURTHER WORK
The modelling of both the TSP and the VRP de-
veloped in this work has given outstanding results to
be used to solve both problems and the tasks that
derive from them. We have improved the number of
variables over other similar ones that provide good
results in the condition of having a linear distance
calculation function. The improvement in our models
represents a fairly significant order of magnitude be-
cause we went from N3variables to 3N2. The figure (6)
and (7) summarizes the major contribution of this article.

10
Figure 6. Comparison of the different models based on the
number of qubits. This graph shows the behaviour and evo-
lution of the numbers of qubits for each model. We see the
best performance of our GPS model compared to the other
models.
Figure 7. Benchmark between MTZ and GPS model based
on the number of qubits. We can appreciate that for 30 cities,
GPS model needs 2700 qubits while the MTZ 4458.
The results obtained from our VRP formulation and
all the experiments carried out maintain the number of
variables QN2and allow us to offer the community new
formulations that minimize the time it takes for vehicles
to travel.
Future work will apply the ideas developed in the
QUBO model of these problems to similar ones. In
particular, we will look for other variants of the TSP to
use the modelling of this that we have carried out.
Compliance with Ethics Guidelines
Funding: This research received no external funding.
Institutional review: This article does not contain any
studies with human or animal subjects.
Informed consent: Informed consent was obtained from
all individual participants included in the study.
Data availability: Data sharing not applicable. No new
data were created or analyzed in this study. Data sharing
is not applicable to this article.
APPENDIX
A. TSP formulation N2.
There is a TSP modeling that requires N2variables,
where these are the following:
xi,t such as i∈ {0, ..., N +1}and t∈ {0, ..., N +1}.(36)
Under this formulation xi,t = 1 denotes that the city iis
reached at position t. The distance calculation function
with this formulation is as follows
N+1
X
i=0
N+1
X
j=0
N
X
t=0
di,j xi,txj,t+1 ,(37)
where di,j represents the distance between the node iand
the node j. This expression has the problem that the
distance formulation has terms of degree two and when
trying to generalize this idea to other types of problems
such as the VRP it will become a 4 degree constraint
making use of a large number of auxiliary variables to
convert it to QUBO type format.
B. Improved model 3N2Q
In the previous modelling, we can improve the number
of variables used from 5N2Qto 3N2Qsince certain vari-
ables are redundant. Let us see how we can do this. Let
us take the set of variables as follows:
xi,j,r,q with i<j∈ {0, .., N +1}, r ∈ {0,1,2}and q∈ {1, .., Q}
In all the modelling, the variables xi,j,r such that i=j
are not considered. Let us analyze the interpretation of
each variable. For each edge (i, j), different cases de-
pending on whether a vehicle passes through both cities,
which city is visited before the other and whether the
edge is travelled or not.
•xi,j,0,q = 1 means that the city iis reached earlier
than the jand the edge (i, j) is not travelled.
•xi,j,1,q = 1 means that the vehicle qtravels the
cities iand j, it reaches the city ibefore the jand
it travels the edge (i, j).
•xi,j,2,q = 1 means that the city jis reached earlier
than the iand the edge (j, i) is not travelled.
This new simplification keeps constraints (20), (22),
(24), (25), (27) and (32) defined in the same way as the
first proposal of the VRP formulation, so we will only
focus on the changes of the remaining constraints:
•Constraint 1: For each i, j, q, one and only one of
the possibilities must be met for r, so:
For all i, j, q:
2
X
r=0
xi,j,r,q = 1,(38)

11
•Constraint 6: That the city iis reached before the
city jdoes not depend on each vehicle. Therefore,
for all the vehicles that either arrive at city iearlier
than j, or arrive at city jearlier than i. Introducing
the auxiliary variables ai,j, we have the following
constraint. For all i, j ∈ {1, ..., N}:
Q
X
q=1
xi,j,0,q +xi,j,1,q =ai,j Q. (39)
•Constraint 8: It must be fulfilled that either the
vehicle pass through the city ibefore the jor ar-
rive before to the city jthan the i. Therefore,
it must be verified that, for i∈ {0, ..., N},j∈
{1, ..., N }and q∈ {1, ..., Q}:
xi,j,0,q +xi,j,1,q = 1 −(xj,i,0,q +xj,i,1,q ).(40)
•Constraint 9: If the city iis reached before jand
the city jis reached before the city k, then the city
imust be reached before the city k. This condition
will prevent the vehicle from returning to a city it
has already passed through and therefore prevents
a cycle from forming.
λ
N
X
i=1
N
X
j=1
N
X
k=1
(ai,j aj,k −ai,j ai,k −aj,kai,k +a2
i,k),(41)
X. RESTRICTION PENALTY
Let us analyze the system that must be solved
to build the penalty function from the equation
(17). Our penalty function P(ai,j , aj,k, ai,k ) must
satisfy that P(0,0,1) = 1, P(1,1,0) = 1 and
P(ai,j , aj,k, ai,k ) = 0 for the rest of the cases. Let
us call the variables ai,j =x,aj,k =y,ai,k =z
to simplify the notation. Then, we arrive at the
quadratic function P, as is shown as follows:
P(x, y, z) = c1x2+c2xy +c3xz +c4y2+c5yz +c6z2
Imposing the previous restrictions, we have the fol-
lowing system of equations.
–P(0,0,1) = 1 So that c6= 1.
–P(0,1,0) = 0 So that c4= 0.
–P(0,1,1) = 0 So that c5+c6= 1 ⇒c5=−1.
–P(1,0,0) = 0 So that c1= 0.
–P(1,0,1) = 0 So that c1+c3+c6= 0 ⇒c3=
−1
–P(1,1,0) = 1 So that c2= 1.
So far, we have a system of 6 equations with
six certain compatible unknowns. First, how-
ever, an additional restriction must be veri-
fied. Let us verify if it is met.
–P(1,1,1) = 0. P6
i=1 ci= 1 −1−1 + 1 = 0.
So that indeed all the requirements are met.
We then have that the following function which is
a penalty function for the constraint (17).
P(ai,j , aj,k, ai,k ) = ai,j aj,k −ai,j ai,k −aj,k ai,k +a2
i,k

12
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