# Early Human Capital Accumulation and Decentralization

Guy Tchuente
University of Kent
School of Economics, University of Kent. Comments from Roland Pongou, Pierre
NguimKeu and seminar participant from the Havard University are gratefully
acknowledged.

( April 2021)

###### Abstract

Decentralization is a centerpiece in Cameroonian’s government institutions’
design. This chapter elaborates a simple hierarchy model for the analysis of
the effects of power devolution. The model predicts overall positive effects
of decentralization with larger effects when the local authority processes
useful information on how to better allocate the resources. The estimation of
the effects of the 2010’s power devolution to municipalities in Cameroon
suggests a positive impact of decentralization on early human capital
accumulation. The value added by decentralization is the same for Anglophone
and Francophone municipalities; the effects of decentralization are larger for
advanced levels of primary school.

Keywords: Human Capital, Decentralization, Resources Allocation, Difference-
in-Differences design.

JEL Codes: H41, H52, H72, H75, H77.

##  1 Introduction

Decentralization and human capital are heavily investigated topics in the
economics literature.111 Channa and Faguet (2016) propose a literature review
on the importance of decentralization while, Hanushek and Woessmann (2008)
discusses the role of human capital for development. The geographical and
ethnic diversity of Cameroon makes the invention of uniformly efficient
institutions challenging.222 Cameroon has around 250 groups see
https://www.worldatlas.com/articles/ethnic-groups-of-cameroon.html visited on
06/04/2021. The Cameroonian government has been implementing a regionalization
process organized by Cameroon’s Decentralization Law of 2019. The reliance on
decentralization and democracy is based on studies establishing their crucial
role for economic development (Myerson (2014) and Myerson (2019)). In
Cameroon, causal empirical pieces of evidence on decentralization effects are
rare. This chapter proposed a methodological framework for the evaluation of
decentralization policy and evaluated the effects of the 2010’s power
devolution in Cameroon on early human capital accumulation.

In 2010, the Cameroonian government decentralized a substantial amount of
input allocation decisions at the primary public school level from the central
government to municipal authorities. For example, the creation of schools, the
rehabilitation of the classrooms, the construction of toilets, the water
provision, and equipment maintenance are put under municipal authorities. In
addition to the new competencies in infrastructure, each municipal authority
is in charge of the allocation of the “minimum package" to schools.333The
“minimum package” is an allocation of didactic material introduced in the year
2000 when primary public schools were declared free of charge by the
government.

The assumption supporting this devolution of power by the central government
is that proximity of the local authority will improve the allocation of the
resources. The interest in evaluating this decentralization policy is twofold.
First, the 2010’s decentralization policy is one of the first large-scale
decentralization policies in Cameroon. Second, it affects the supply side of
the national instruments for early human capital production. Indeed, there are
pieces of evidence in microeconomics and macroeconomics on the importance of
the quality and quantity of human capital for the creation of wealth in low-
income countries (Hanushek and Woessmann (2008), Hanushek and Woessmann
(2012), and Dahlum and Knutsen (2017)). Thus, evaluating the effect of the
policy is crucial to understand the impact of future decentralization policy.

The economy of contract theory models the decentralization problems using
hierarchical models. In these models, the principal (the central authority/
government) decides on whether to delegate the decision power to a supervisor
(local authority/ municipality) on the contract to adopt with an agent
(school).444Mookherjee (2006) provides a review of the use of mechanism design
in an hierarchical model. This chapter uses a model with three types of
players: the government (the principal), the supervisor (municipality) and the
agents (schools). A school produces aggregated human capital as a function of
basic effort exercised by pupils and resources allocated by the authority
(government, municipality and schools).555The production function of human
capital is similar to Dal Bó, Finan, Li, and Schechter (2021) but the model
differs with the role of the supervisor. The productivity of the resources
allocated by the authority depends on how compatible they are with the needs
of the school. Resources are assumed limited and a uniform allocation is
supposed when the government is in charge. The municipality is allowed to have
a more precise information on the needs of local schools and could, thus,
realises improvement in the aggregated human capital production. The model
predicts an overall positive effect of decentralization on human capital
production. The effects are heterogeneous and increases with the ability of
the local authority to better allocate resources.

The predictions of the model are tested using data from Cameroon. The
government’s power devolution affected directly pupils in public schools.
Thus, public school pupils are considered as the treatment group in this
quasi-experiment while the private school’s pupils are the control group. The
year 2005 observations are gathered before policy intervention while the 2014
observations are collected after the policy intervention. A difference-in-
differences methodology, based on PASEC666Programme d’Analyse des Systemes
Educatifs de la Confemen. 2005 and 2014 pupil’s mathematics, and literacy
ability test scores data, is used to evaluate the policy intervention. The
ability of the difference-in-differences design to identify a policy-relevant
parameter is explained and the effects of the 2010’s decentralization policy
are estimated using the ordinary least squared (OLS) with different
specifications.

The estimation of the effects of the decentralization policy suggests a
positive effect of decentralization on early human capital accumulation. The
devolution of power to municipalities has increased the tests score of primary
school pupils by 10.20% in mathematics and 15.39% in literacy. The
decomposition of the effect by level shows that the effect is driven by
changes at the end of the cycle level. Indeed, the estimation suggests small
policy effects for Grade 2 pupils. This is a confirmation of the prediction of
the theoretical model. Indeed, it will be difficult to clearly know the
specific need of pupils at such an early level, while the overall school needs
can be better assessed by the municipality. Thus, around the end of the cycle
(Grades 5 or 6), the pupils have enough time to benefit from better-targeted
resources allocation by the municipal authority. The decomposition along the
linguistic line suggests a small difference in the effect of the
decentralization between the Anglophone and the Francophone primary school
system. This may be driven by the fact that there is virtually no difference
between the information that can be acquired by Francophone and Anglophone
municipalities.

Human capital is accounted as a main contributor to wealth creation for
countries (see Manuelli and Seshadri (2014) and Schoellman (2016) ). This
chapter contributes to the literature assessing how government institutions
affect human capital production by analyzing how decentralization in a low-
income country. It, thus, adds to contributions such as Davies, Harber, and
Dzimadzi (2003) who study decentralization in Malawi’s education sector, while
Pomuti and Weber (2012) investigate the role plays by decentralization policy
in Namibia. There are some empirical pieces of evidence suggesting that
decentralized supervision improves the professional development of teachers
(see Esia-Donkoh and Ofosu-Dwamena (2014) for the study of Ghana, Pomuti and
Weber (2012) in Namibia, and Tamukong (2004) presenting a decentralization
model for Cameroon). But, some studies find that decentralization may have
negative effects on funding allocation, with the largest portion of funding
not reaching the target population (see Otieno and Nyangechi (2013) and
Dembélé (2015)). In a review of the literature on decentralization in
education and health proposed by Channa and Faguet (2016) none of the studies
of decentralization in the educational sector is considered to have a “Very
strongly credible identification strategy". Moreover, among the studies deemed
as having a “Strongly credible identification" none of them are from an
African country. This chapter aims to implement a rigorous quasi-experimental
design to evaluate the effect of decentralization in Cameroon. Thus, for
Cameroonian researchers and policymakers, this chapter develops a simple
framework that can guide them in the process of implementation and evaluation
of future decentralization policies.

The remaining chapter is organized as follows. Section 2 presents some
background information on the primary education sector in Cameroon as well as
some useful knowledge regarding 2010’s power devolution. Section 3 presents a
simple model of decentralization in the context of human capital production.
Section 4 discusses the empirical strategy used to evaluate the effect of the
decentralization policy and tests the implications of the model. It also
explores the estimation results and presents relevant policy implications. A
brief conclusion follows in Section 5.

##  2 Background and Data

The development of the education system in Cameroon since 1990 has been
oriented by international commitments and local ambitions. At the
international level, the Education For All (EFA) conference of Jomtien,
Thailand (1990), organized by UNESCO and the Dakar’s (2000) meeting where
several countries expressed their commitment to the Millennium Development
Goals (MDG) are the most prominent. At the local level, the policies are
implemented to fit government growth and poverty reduction strategies as
express in the Poverty Reduction Strategy Paper (PRSP, 2003)and the Growth and
Employment Strategy Paper (GESP, 2009). The government vision for education
consists of providing the youths with quality education, requisite
competencies, and professional attitudes that will facilitate their insertion
and competitiveness in the professional world (Alemnge (2019)). The education
system is divided into four stages: pre-primary (4 to 5 years old), primary (6
to 11 years old), secondary (12 to 18 years old), Tertiary (19 to 23 years
old). This chapter evaluates the effect of decentralization on primary public
schools in Cameroon.

###  2.1 Cameroon Primary Schools: A Brief Overview

The primary school service in Cameroon has three main providers: the
government (public schools), private, and private confessional (Catholic
schools, Islamic schools, protestant schools, etc). Each official language
(French and English) has its independent primary school system. Both systems
are represented in all regions of the country with a predominance of French
schools in the French-speaking region, while English schools form the majority
of English-speaking regions (North-West and South-West regions) schools.
According to the PASEC (2016)’s report on the state of education in Cameroon,
during the academic year 2014-2015, the francophone system has 71.6% of pupils
while the anglophone has 28.4%. Among pupils, the share attending public
institutions is around 75% overall. This share is larger in the francophone
system (79%) compare to the Anglophone system (62%).

Figure 1: Enrolment Rate 1994 to 2019 from the World Bank data.

Figure 1 shows a steady increase in the primary school enrolment rate between
1996 and 2016 for boys and girls. Accelerations of the enrolment rate are
notable in 2000 and 2010. These years correspond respectively to the year of
the announcement of the free public primary school by the government and the
devolution of substantial allocation powers to the municipal level. Since the
independence, the Francophone and the Anglophone systems have studied with
different curricula with some harmonization policy by the government. These
efforts have led to a joint curriculum effective for implementation from the
2018/2019 school year. The intention to harmonize the systems has been
manifested by the government since the days of independence and materialized
by the 1960 law on the organization of education. After these early steps, in
1990 the government launched its first national curriculum the “New
pedagogical approach". After nearly a decade of implementation of the “new
pedagogical approach", the government started, between 1998 and 2000, a new
syllabus based on the National Forum on Education held 1995. Following that
reform, in 2003 the government decided to implement a pedagogic approach
called the Competence-Based Approach (CBA), in all primary schools of the
country Alemnge (2019).

Between 2005 and 2018 there have not been any major change in the primary
school curriculum in Cameroon.777It is important to note that all schooling
the country must follow the official curriculum. The supply of education
services has been marked during the same period by steady recruitment of
teachers in the Anglophone and Francophone systems (the ratio of pupils per
teacher was 47.8 in 2005 and 45.8 in 2014). The most important change in the
period between 2005 and 2014 is a decentralization policy applied by the
government to all public schools.

###  2.2 The 2010’s Power Devolution Policy

The public primary education in Cameroon was inherited from colonial times
with a very high level of centralization. Sine 1990, the Cameroonian
government has adopted decentralization as an instrument to achieve the
Education For All goal. The commitment to apply decentralization in education
was reinforced by several laws. The year 2010 marks a substantial step in the
implementation of the government decentralization ambitions. The
municipalities are granted the right to allocate resources to public primary
schools in their area of authority. For example, the municipality has to right
to create schools (following the government’s school map guidance). They are
in charge of the maintenance of public primary school equipment, they recruit
support staff for the school. They also participate in the acquisition of
didactic material. This includes the provision to the public schools of the
“minimum package" necessary for the functioning of the schools.888See
http://bibliotheque.pssfp.net/index.php/institutions/institutions-
nationales/senat/les-collectivites-territoriales-decentralisees/les-
decrets/951-decret-n-2010-0247-pm-du-26-fevrier-2010-fixant-les-modalites-
dexercice-de-certaines-competences-tranaferees-par-letat-aux-communes-en-
matirer-education-de-base/file, visited on 08/04/2021, for the details of the
decentralization decision.

Alongside the resource allocation power, the municipalities were granted the
right to produce public school infrastructure: the construction and
rehabilitation of classrooms, construction of latrine blocks, water points and
fences, maintenance and equipment of public nursery and primary schools are
the responsibility of the municipality. Figure 1 suggests that the 2010
devolution of power was followed by an acceleration in the enrolment rate in
primary schools.

##  3 A Simple Model of Hierarchy

The decentralization problem can be represented using a hierarchical model
where the government is the principal, the municipality is the supervisor, and
the public schools are the agents (there are $\displaystyle N$ schools in the
municipality).

Each school is a unit mass agent producing human capital $\displaystyle
h_{i}$. The production of human capital is a function of

$\displaystyle h_{i}(l_{i})=l_{i}s_{i}+e_{i}$

where $\displaystyle l_{i}$ is the resource allocated by the municipality,
$\displaystyle s_{i}$ measures the compatibility of the resource allocated to
the school $\displaystyle i$ with its needs and, $\displaystyle e_{i}$ is the
level of effort made by the school. All the variables have bounded support.

In a private school, the human capital production is only a function of the
school effort ($\displaystyle e_{i}$) as they receive do not rely on the
government or municipality to finance their enterprise. For public schools,
the resource granted by the municipality or the government could a positive or
a negative effect on human capital production. This means that $\displaystyle
s_{i}$ can be either negative or positive.

Let $\displaystyle l_{i0}$ be the initial resources endowment of school
$\displaystyle i$ and $\displaystyle l_{i1}$ the resource endowment after a
small increase. Let $\displaystyle\Delta l_{i}=l_{i1}-l_{i0}>0$, the human
capital production is

$h_{i}=e_{i}+s_{i}l_{i0}+s_{i}\Delta l_{i}$ (1)

In general, $\displaystyle\Delta l_{i}$ additional resource will produce
$\displaystyle s_{i}\Delta l_{i}$ additional units of human capital.

As the compatibility of resources with local needs is not expected under
central government allocation. Let assume a uniform increased in allocation to
schools. $\displaystyle s_{i},e_{i}$ are considered independent and
identically distributed across all schools. The expected increase in human
capital

$\displaystyle\delta=E(s_{i}\Delta l_{i}).$

Under central government, $\displaystyle\Delta=\Delta l_{i}$. Thus,
$\displaystyle\delta=E(s_{i})\Delta$. The total additional resource to be
allocated is given $\displaystyle N\Delta$ where $\displaystyle N$ is the
number of schools. The quantity $\displaystyle N\Delta$ is the total amount of
additional resources available that can be granted to the municipality for
allocation.

The local authority (municipality) can choose the distribution of extra
resources such that $\displaystyle s_{i}\Delta l_{i}\geq
s_{i}\sum_{i=1}^{N}\Delta l_{i}/N$ and $\displaystyle\sum_{i=1}^{N}\Delta
l_{i}=N\Delta$.

Let consider the set of allocation $\displaystyle M_{i}=\\{s_{i}\Delta
l_{i}\geq s_{i}\sum_{i=1}^{N}\Delta l_{i}/Nand\sum_{i=1}^{N}\Delta
l_{i}=N\Delta\\}$. These allocations will improve production of individual
with the larger compatibility factor ($\displaystyle s_{i}$). $\displaystyle
M_{i}$ is a no-empty set in presence of decentralization. It can be shown that

$\delta=E(s_{i})\Delta\leq E(s_{i}\Delta l_{i}|M_{i})=\rho_{i}.$ (2)

The quantity $\displaystyle\lambda_{i}=(\rho_{i}-\lambda)$ measures the
expected gain from decentralization. The inequality in Equation (2) implies
the following points:

  1. 1.

An additional unit of resource allocated under decentralization is at least as
productive as under centralization.

  2. 2.

If the central authority is able to meet the local schools’ needs the
decentralization will bring no profit.

  3. 3.

The effect of decentralization is heterogeneous and depends on the ability of
the authority to distinguish different types of schools ($\displaystyle
s_{i}$).

The model predicts an overall positive effect of decentralization on human
capital production. To test the predictions of our model, an ideal solution
would have been to run a randomized control trial (RCT). This will imply the
creation of representative treatment and control groups of municipalities. The
schools in the treatment group are put under decentralization and those in the
control leave in the status quo. This chapter takes an alternative route. It
takes advantage of the quasi-experimental nature of the power devolution to
municipalities in 2010 to measure the causal effect of decentralization on
human capital accumulation using a difference-in-difference design.

##  4 Empirical Investigation of the 2010’s Decentralization

###  4.1 Research Design: Difference-in-differences

In 2010 the Cameroonian government decided to implement a decentralization
policy. This policy affected public primary schools in several aspects.
However, private and private confessional schools were not affected by the
policy. Let consider public primary schools as the treatment group
($\displaystyle P_{i}=1$) and private schools as the control group
($\displaystyle P_{i}=0$).999Given the difference in the production function
of human capital, they may not be suitable control as pupils may select in
different school depending on their unobserved characteristics.

The human capital outcome of pupils from both groups is observed before and
after 2010. Consider $\displaystyle dT=1$ if an observation is collected in
2014 and $\displaystyle dT=0$ otherwise. This allows the estimation of the
average treatment effect on the treated (ATT) of decentralization.

Let $\displaystyle h_{it}$ be the human capital variable of a school/pupil
$\displaystyle i$ and time $\displaystyle t$. The model predicts that

$h_{it}=e_{it}+(\rho_{i}-\delta)dTP_{i}+s_{it}l_{i0t}+E(s_{it})\Delta-E(s_{it}\Delta
l_{it})+s_{it}\Delta l_{it}$ (3)

where $\displaystyle dT$ is a dummy indicating if the observation is in a
period with decentralisation.

Let consider $\displaystyle\lambda_{i}=(\rho_{i}-\delta)$ be the benefit from
decentralization. Based on the model, if $\displaystyle P_{i}=0$ then
$\displaystyle M_{it}$ is empty and $\displaystyle\rho_{i}$ is not defined.
However, it is possible to define
$\displaystyle\lambda_{0}=E(\lambda_{i}|P_{i}=1)$. This is the ATT of
decentralization.

Let $\displaystyle X_{it}=(Z_{it},W_{it})$ be the set of exogenous
characteristics. $\displaystyle Z_{it}$ is a set of individual characteristics
affecting the school efforts $\displaystyle
e_{it}=e(Z_{it},\varepsilon_{0it})$. $\displaystyle W_{it}$ are schools
features related to the compatibility of resources $\displaystyle
s_{it}=s(W_{it},\varepsilon_{1it})$. The assumption of linearity of
$\displaystyle s$ and $\displaystyle e$ implies that

$h_{it}=Z_{it}^{\prime}\theta_{1}+(\rho_{i}-\delta)dTP_{i}+W_{it}^{\prime}\theta_{2}l_{i1t}+\theta_{0}+\varepsilon_{0it}+\varepsilon_{1it}l_{1it}$
(4)

Let assume that $\displaystyle
l_{1it}=\mu_{0}+\mu_{1}dT+\mu_{2}P_{i}+\varepsilon_{2it}$. The structural
Equation (4) is approximated by

$h_{it}=\tilde{\mu}_{1}dT+\tilde{\mu}_{2}P_{i}+Z_{it}^{\prime}\theta_{1}+(\rho_{i}-\delta)dTP_{i}+W_{it}^{\prime}\tilde{\theta}_{2}+\theta_{0}+\varepsilon_{0it}+\varepsilon_{1it}(\mu_{0}+\mu_{1}dT+\mu_{2}P_{i}+\varepsilon_{2it}).$
(5)

The transformed structural equation is

$h_{it}=\beta_{0}+X_{it}^{\prime}\beta_{1}+\alpha_{0}P_{i}+\alpha_{1}dT+\lambda_{0}dT\times
P_{i}+v_{it}.$ (6)

with $\displaystyle v_{it}=\varepsilon_{it}+(\lambda_{i}-\lambda_{0})dT\times
P_{i}$ where
$\displaystyle\varepsilon_{it}=\varepsilon_{0it}+\varepsilon_{1it}(\mu_{0}+\mu_{1}dT+\mu_{2}P_{i}+\varepsilon_{2it})$.
An OLS estimator of $\displaystyle\lambda_{0}$ is unbiased if $\displaystyle
v_{it}$ is uncorrelated with $\displaystyle dT\times P_{i}$ under the
assumption the extra resources allocation is not selective based on unobserved
school characteristic ($\displaystyle
E(\varepsilon_{1it}\varepsilon_{2it}|P_{i},dT,X_{it})=0$).

The set-up described above corresponds to a difference-in-differences design.
The following section presents the data and discusses the parallel trend
assumption in the PASEC sample.

###  4.2 Data and Identification Assumption Verification

The data used for the empirical investigation are 2005 and 2014 school survey
data from the Programme d’Analyse des Systèmes Educatifs de la CONFEMEN
(PASEC) for Cameroon. The is a representative sample of pupils from schools in
French and English- speaking Cameroon. The 2005 sample has pupils from Grade 2
and Grade 5 classes (15 pupils are then randomly drawn, interviewed, selected,
and tested in a class of each grade). In 2014 another random sample is drawn
from Grade 2 and Grade 6. The tests administrated by the PASEC are in language
and maths. They are designed to allow comparison over time and across systems.
Alongside the test score, data contains school characteristics, socioeconomic
characteristics of the pupils, their parents, and some useful household
information.

A crucial assumption of our empirical strategy is that in absence of
decentralization in 2010, the private and public schools’ pupil’s scores in
language and maths would have had the same trend. This assumption is not
directly testable. The curriculum followed by private schools is the same as
public schools. Also, there has not been another major policy intervention in
the sector of public primary schools. This coupled with the heterogeneity of
private school practice, implies that there is a very high chance of
validation of the parallel trend assumption.

###  4.3 Empirical Results and Policy Implications

The estimation of the decentralization effects is carried out using the model
represented in Equation (6). The accumulation of early human capital is
measured by mathematics and literacy test scores. The model’s parameters are
estimated with and without school fixed-effects and on the sub-sample of Grade
2 test scores.

The main parameter of interest is $\displaystyle\lambda_{0}$. It measures the
effects of decentralization of early human capital accumulation in Cameroon.
Our results suggest a positive effect of decentralization. The models that
explain most of the variability in the human capital accumulation in the full
sample are models with school fixed-effects (FE). Thus, these are the
preferred specifications. They are in column Math with FE and Lit. with FE.
These results suggest that decentralization has increased Math scores by 10.2%
and Literacy scores by 15.39%. These effects are significant both
statistically and economically.

The columns Math Gr 2, and Lit. Gr2, are the estimation of the parameters
using the sub-population of Grade 2 pupils. The estimation of the
decentralization effects suggests small but statistically significant effects.
The relatively low-level pupils may make a better allocation of recourse to
this particular group more challenging to the municipality. This
interpretation is in line with the finding of our model of Section 3
suggesting no difference between the central and decentralized regimes when
the municipality cannot target the best match for the resources.

Expect for literacy in Grade 2, decentralization has a positive effect on all
the human capital accumulation measures. Thus, the allocation of the
municipality could be view to be at least as good as the central authority
allocation in our sample.

As Cameroon has the particularity of being divided into two school systems,
namely the anglophone and francophone system, Table 2 presents the assessment
of the heterogeneity of the decentralization effect along that dimension. The
result suggests small but statistically significant difference between
francophone and anglophone. Decentralization seems to have a relatively small
effect effect in Anglophone Municipalities compare to Francophone ones.
However, the difference in score is below 5%. It also worth-nothing that,
everything being equal, Anglophone pupils have roughly 10% higher scores than
a pupil from the Francophone system. Thus, the small relative better
performance of Francophone municipality could come from the fact that they had
more room fro growth. Also, there is no reason to expect the municipality to
better match the needs of pupils in the anglophone system compare to the
francophone one, thus, the model in Section 3 predicts similar
decentralization effects.

The positive effects of decentralization suggested by the regression results
demonstrate the importance of the proximity of the decision-making to the
population. These results should motivate the government to accelerate the
decentralization process prescribed by the Cameroon’s Decentralization Law of
2019. This chapters’ model and empirical findings suggest that
decentralization should prioritized sectors where the local authorities can
have better quality information on the production of the goods and services.

Table 1: Estimation of the Decentralization Effect

| Math | Lit. | Math | Lit. | Math Gr2 | Lit. Gr2
---|---|---|---|---|---|---
After Treatment | -2.759∗∗∗ | -3.424∗∗∗ | -16.44∗∗∗ | -14.80∗∗∗ | -22.23∗∗∗ | -19.30∗∗∗
| (-3.71) | (-4.19) | (-53.68) | (-46.67) | (-77.35) | (-74.60)
Public Schools | -6.617∗∗∗ | -6.160∗∗∗ | -11.93∗∗∗ | -16.35∗∗∗ | 1.146 | -7.590∗∗∗
| (-8.61) | (-7.42) | (-28.41) | (-35.23) | (1.27) | (-9.51)
$\displaystyle\lambda_{0}$ | 1.705∗ | 0.825 | 10.20∗∗∗ | 15.39∗∗∗ | -4.848∗∗∗ | 5.780∗∗∗
| (2.12) | (0.94) | (19.71) | (27.43) | (-5.45) | (7.57)
Anglophone | -1.094∗∗∗ | -3.261∗∗∗ | 9.388∗∗∗ | 10.88∗∗∗ | 7.110∗∗∗ | 10.22∗∗∗
| (-3.42) | (-9.75) | (23.54) | (24.40) | (9.29) | (14.38)
Age | -0.551∗∗∗ | -1.090∗∗∗ | -0.0295 | -0.273 | 0.560∗ | 0.129
| (-5.27) | (-10.03) | (-0.22) | (-1.91) | (2.11) | (0.50)
Dummy for Girl | -0.539 | 0.466 | -0.680∗ | 0.118 | -0.548 | 0.657
| (-1.78) | (1.48) | (-1.99) | (0.37) | (-0.78) | (1.12)
Books at Home | 3.374∗∗∗ | 3.068∗∗∗ | 1.394∗∗ | 1.971∗∗∗ | -0.217 | 0.552
| (9.97) | (8.79) | (2.87) | (4.19) | (-0.25) | (0.77)
Electricity at Home | 2.198∗∗∗ | 2.488∗∗∗ | -0.949∗ | -1.362∗∗ | 1.723∗ | 1.166
| (6.43) | (6.76) | (-2.33) | (-2.88) | (2.00) | (1.48)
Dummy for higher grades | -7.493∗∗∗ | -4.901∗∗∗ | -10.16∗∗∗ | -9.422∗∗∗ |  |
| (-13.16) | (-8.34) | (-9.14) | (-8.44) |  |
School Fixed-effects | No | No | Yes | Yes | Yes | Yes
Observations | 10078 | 10078 | 10078 | 10078 | 4076 | 4076
$\displaystyle R^{2}$ | 0.127 | 0.131 | 0.393 | 0.433 | 0.434 | 0.551
Notes: statistics in parentheses. ∗ $\displaystyle p<0.05$, ∗∗ $\displaystyle p<0.01$, ∗∗∗ $\displaystyle p<0.001$. The outcome variables are pupils’ test scores in mathematics and literacy in 2005 and 2014. The year 2014 is the year after the treatment and public schools pupils are in the treatment group, pupils are either in Grade 2, 5 or 6. Standard errors are clustered at the school level. |  |

Table 2: Estimation of the Decentralization Effect Anglophone vs. Francophone

| Math | Lit. | Math Gr2 | Lit. Gr2
---|---|---|---|---
After Treatment | -16.44∗∗∗ | -14.80∗∗∗ | -22.23∗∗∗ | -19.30∗∗∗
| (-53.68) | (-46.67) | (-77.35) | (-74.60)
Public School | -11.93∗∗∗ | -16.35∗∗∗ | 1.146 | -7.590∗∗∗
| (-28.41) | (-35.23) | (1.27) | (-9.51)
$\displaystyle\lambda_{0}$ | 10.20∗∗∗ | 15.39∗∗∗ | -4.848∗∗∗ | 5.780∗∗∗
| (19.71) | (27.43) | (-5.45) | (7.57)
Decent $\displaystyle\times$ Anglophone | -4.163∗∗∗ | -4.819∗∗∗ | 2.352∗∗∗ | -1.964∗∗∗
| (-9.93) | (-11.01) | (20.13) | (-19.76)
Anglophone | 9.388∗∗∗ | 10.88∗∗∗ | 7.110∗∗∗ | 10.22∗∗∗
| (23.54) | (24.40) | (9.29) | (14.38)
Age | -0.0295 | -0.273 | 0.560∗ | 0.129
| (-0.22) | (-1.91) | (2.11) | (0.50)
Dummy for Girl | -0.680∗ | 0.118 | -0.548 | 0.657
| (-1.99) | (0.37) | (-0.78) | (1.12)
Books at Home | 1.394∗∗ | 1.971∗∗∗ | -0.217 | 0.552
| (2.87) | (4.19) | (-0.25) | (0.77)
Electricity at Home | -0.949∗ | -1.362∗∗ | 1.723∗ | 1.166
| (-2.33) | (-2.88) | (2.00) | (1.48)
Dummy for higher grades | -10.16∗∗∗ | -9.422∗∗∗ |  |
| (-9.14) | (-8.44) |  |
School fixed effects | Yes | Yes | Yes | Yes
Observations | 10078 | 10078 | 4076 | 4076
$\displaystyle R^{2}$ | 0.393 | 0.433 | 0.434 | 0.551
Notes: statistics in parentheses. ∗ $\displaystyle p<0.05$, ∗∗ $\displaystyle
p<0.01$, ∗∗∗ $\displaystyle p<0.001$. The outcome variables are pupils’ test
scores in mathematics and literacy in 2005 and 2014. The year 2014 is the year
after the treatment and public schools pupils are in the treatment group,
pupils are either in Grade 2, 5 or 6. Standard errors are clustered at the
school level.

##  5 Conclusion and Related Future Issues

This chapter has investigated the impact of the 2010 decentralization in the
Cameroonian education sector on early human capital accumulation. This policy
is one of the major government recent actions in the primary education
section.

A simple model of hierarchy is proposed to represent the government
decentralization problem. The theoretical model suggests that when the
municipality is able to achieve a better allocation of resources,
decentralization has a positive impact. It also suggests heterogeneity in the
returns to decentralization.

Data from PASEC containing pupils’ test scores in math and literacy, measures
to ensure comparability, are used to estimate and test the predictions of the
model. The application of a difference-in-differences design enables the
estimation of the causal effect of decentralization. Indeed, decentralization
has a positive effect on early human capital accumulation. The effect of power
devolution in 2010 is larger for literacy (15.39%) than in mathematics
(10.2%). These effects are only noticeable at the end of the primary cycle.
The also no difference between the decentralization effect for Anglophone and
Francophone systems.

## References

  * (1)
  * Alemnge (2019) Alemnge, F. L. (2019): “Curriculum Reform in Cameroon: An Analysis of the New Primary School Curriculum,” _International Journal of Trend in Scientific Research and Development_ , 3(6), 902–913.
  * Channa and Faguet (2016) Channa, A., and J.-P. Faguet (2016): “Decentralization of health and education in developing countries: a quality-adjusted review of the empirical literature,” _The World Bank Research Observer_ , 31(2), 199–241.
  * Dahlum and Knutsen (2017) Dahlum, S., and C. H. Knutsen (2017): “Do democracies provide better education? Revisiting the democracy–human capital link,” _World Development_ , 94, 186–199.
  * Dal Bó, Finan, Li, and Schechter (2021) Dal Bó, E., F. Finan, N. Y. Li, and L. Schechter (2021): “Information Technology and Government Decentralization: Experimental Evidence from Paraguay,” _Econometrica_ , 89(2), 677–701.
  * Davies, Harber, and Dzimadzi (2003) Davies, L., C. Harber, and C. Dzimadzi (2003): “Educational decentralisation in Malawi: a study of process,” _Compare: A Journal of Comparative and International Education_ , 33(2), 139–154.
  * Dembélé (2015) Dembélé, M. E. (2015): “Le financement de l’éducation dans les collectivités locales, quels défis?,” _Africa Development_ , 40(4), 43–60.
  * Esia-Donkoh and Ofosu-Dwamena (2014) Esia-Donkoh, K., and E. Ofosu-Dwamena (2014): “Effects of educational supervision on professional development: Perception of public basic school teachers at Winneba, Ghana,” _British Journal of Education_ , 2(6), 63–82.
  * Hanushek and Woessmann (2008) Hanushek, E. A., and L. Woessmann (2008): “The role of cognitive skills in economic development,” _Journal of economic literature_ , 46(3), 607–68.
  * Hanushek and Woessmann (2012) (2012): “Do better schools lead to more growth? Cognitive skills, economic outcomes, and causation,” _Journal of economic growth_ , 17(4), 267–321.
  * Manuelli and Seshadri (2014) Manuelli, R. E., and A. Seshadri (2014): “Human capital and the wealth of nations,” _American economic review_ , 104(9), 2736–62.
  * Mookherjee (2006) Mookherjee, D. (2006): “Decentralization, hierarchies, and incentives: A mechanism design perspective,” _Journal of Economic Literature_ , 44(2), 367–390.
  * Myerson (2014) Myerson, R. B. (2014): “Democratic Decentralization and Economic Development,” in _The Oxford Handbook of Africa and Economics_.
  * Myerson (2019) (2019): “The Strength of American Federal Democracy,” in _The Oxford Handbook of Structural Transformation_.
  * Otieno and Nyangechi (2013) Otieno, S., and E. O. Nyangechi (2013): “Effectiveness of internal control procedures on management efficiency of Free Primary Education Funds: a case of Public Primary schools in Kisii central District, Kenya,” .
  * PASEC (2016) PASEC (2016): “Performances du système éducatif camerounais : Compétences et facteurs de réussite au primaire.,” _PASEC, CONFEMEN, Dakar_.
  * Pomuti and Weber (2012) Pomuti, H., and E. Weber (2012): “Decentralization and school management in Namibia: The ideologies of education bureaucrats in implementing government policies,” _International Scholarly Research Notices_ , 2012.
  * Schoellman (2016) Schoellman, T. (2016): “Early childhood human capital and development,” _American Economic Journal: Macroeconomics_ , 8(3), 145–74.
  * Tamukong (2004) Tamukong, J. A. (2004): “Towards Better Management of Public Education in Cameroon: the case for decentralisation,” _Africa development_ , 29(2).


# On holomorphic curves into complex projective varieties

Giang Le

###### Abstract.

In this paper, we study holomorphic curves satisfying the Fubini-Study
derivative $\|f^{\prime}(z)\|=O(|z|^{\sigma})$ for some $\sigma>-1$ from the
viewpoint of Nevanlinna theory.

Department of Mathematics, Hanoi National University of Education,

136-Xuan Thuy, Cau Giay, Hanoi, Vietnam.

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
††footnotetext:  2010 Mathematics Subject Classification: 32H30 (14J70 32H04
32H25).
Key words and phrases: Second main theorem over an angular domain, Julia
directions, Brody curves.

##  1\. Introduction

In this paper, we study holomorphic curves
$f:{\mathbf{C}}\rightarrow{\mathbf{P}}^{N}({\mathbf{C}})$ satisfying the
Fubini-Study derivative $\|f^{\prime}(z)\|=O(|z|^{\sigma})$ for some
$\sigma>-1.$ The explicit expression is:

$\|f^{\prime}\|^{2}=\dfrac{\sum_{i\neq
j}|f_{i}^{\prime}f_{j}-f_{i}f_{j}^{\prime}|^{2}}{\|f\|^{4}},$

where ${\bf f}=(f_{0},\ldots,f_{N})$ is a homogeneous representation of $f$
(that is, the $f_{j}$ are entire functions which never simultaneously vanish)
and

$\|f\|^{2}=\sum_{j=0}^{N}|f_{j}|^{2}.$

When $\sigma=0$, $f$ is called a Brody curve. The interest of these curves
comes from Brody’s lemma. It is stated that for every non constant holomorphic
curve $f$, there exists a sequence of affine map
$a_{k}:{\mathbf{C}}\rightarrow{\mathbf{C}}$ such that the limit $\int f\circ
a_{k}$ exists and is a non constant Brody curves. Much attention has been
given to study Brody curves from the viewpoint of Nevanlinna theory [3, 1, 13,
10]. In [3, 1, 12], some versions of Second Main Theorem have been established
for Brody curves intersecting hypersurfaces located in general position. The
first aim of this paper is to generalize such the results to the case of
holomorphic curves satisfying the Fubini-Study derivative
$\|f^{\prime}(z)\|=O(|z|^{\sigma})$ for some $\sigma>-1$ and arbitrary
hypersurfaces.

We recall that the Nevanlinna characteristic is defined as follows

$T_{f}(r)=\int_{0}^{r}\frac{dt}{t}\left(\frac{1}{\pi}\int_{|z|\leq
t}\|f^{\prime}\|^{2}(z)dm_{z}\right),$

where $dm$ is the area element in ${\mathbf{C}}$.

Obviously, the condition $\|f^{\prime}(z)\|=O(|z|^{\sigma})$ for some
$\sigma>-1$ implies that

$T_{f}(r)=O(r^{2\sigma+2}).$

Clunie and Hayman [2] found that if the spherical derivative
$\|f^{\prime}(z)\|$ of an entire function satisfies
$\|f^{\prime}(z)\|=O(|z|^{\sigma})$, then $T_{f}(r)=O(r^{\sigma+1})$. Barret
and Eremenko [3] generalized this to the case of holomorphic curves in
projective space of dimension $n$ omitting $n$ hyperplanes in general
position. The first main result of this paper is to show that this phenomenon
persists in complex projective variety omitting hypersurfaces satisfying the
intersection of them consisting of finite points.

In this paper, we always assume that $V$ be a complex projective algebraic
variety in ${\mathbf{P}}^{N}({\mathbf{C}})$.

######  Theorem 1.

Let $D_{1},\ldots,D_{q}$ be $q$ hypersurfaces such that $\dim(\bigcap_{1\leq
i\leq q}\mathrm{Supp}\,D_{i}\bigcap V)=0$. Let $f:{\mathbf{C}}\rightarrow V$
satisfying $\|f^{\prime}(z)\|=O(|z|^{\sigma})$ for some $\sigma>-1.$ Assume
that $(\bigcup_{1\leq i\leq q}\mathrm{Supp}\,D_{i})\bigcap
f({\mathbf{C}})=\emptyset.$ Then,

$T_{f}(r)=O(r^{\sigma+1}).$

Combined with a result of Tsukamoto [13, Theorem 1.9], Theorem 1 implies that

######  Corollary 2.

Mean dimension in the sense of Gromov [6] of the space of Brody curves in

$V\backslash\text{ $\dim V$ hypersurfaces in general position}$

is zero.

In [1], Da Costa and Duval proved a second main theorem for Brody curves in
projective space intersecting hyperplanes. Later, Thai and Mai [12]
established a second main theorem over an angular domain for Brody curves into
a complex projective algebraic variety intersecting hypersurfaces in general
position. Our second purpose is to generalize Thai-Mai’s results [12] to the
case of arbitrary hypersurfaces and holomorphic curves satisfying
$\|f^{\prime}(z)\|=O(|z|^{\sigma})$ for some $\sigma>-1.$ To state our result,
we recall some notations.

######  Definition 3.

Let $D_{1},\ldots,D_{q}$ be hypersurfaces in ${\mathbf{P}}^{N}({\mathbf{C}})$.

a) They are said to be in $m-$subgeneral position in $V$ ($\dim V\leq
m\in\mathbb{N}$) if for any subset $I\subset\\{1,\ldots,q\\}$ with $\sharp
I\leq m+1$, we have

$\dim\bigcap_{i\in I}\mathrm{Supp}\,D_{i}\bigcap V\leq m-\sharp I.$

b) They are said to be in general position in $V$ if they are in $\dim
V$-subgeneral position.

Let $f:{\mathbf{C}}\rightarrow{\mathbf{P}}^{N}({\mathbf{C}})$ be a holomorphic
curve. We choose a homogeneous representation ${\bf f}=(f_{0},\ldots,f_{N})$
of our curve. Set

$u(z)=\log\sqrt{|f_{0}(z)|^{2}+\ldots+|f_{N}(z)|^{2}}.$

Obviously, $u$ is a positive subharmonic function . Let $\triangle u$ be the
Riesz measure of $u$, that is the measure with the density

$\triangle u=\dfrac{1}{\pi}\|f^{\prime}\|^{2}.$

Let

$\mathbb{D}=\\{z\in{\mathbf{C}}:|z|<1\\},r\mathbb{D}=\\{z:|z|<r\\}$

throughout the paper. Let $\Omega$ be a domain in $\mathbb{C}.$ The Ahlfors-
Shimizu characteristic function of $f$ is defined as follows

$\displaystyle T_{f}(r,\Omega)=\int_{1}^{r}\dfrac{\triangle u(\Omega\bigcap
t\mathbb{D})}{t}dt+O(1).$

It is well-known that

$T_{f}(r,\mathbb{C})=T_{f}(r)=\dfrac{1}{2\pi}\int_{0}^{2\pi}u(re^{i\theta})d\theta+O(1).$

Let $D$ be a hypersurface in ${\mathbf{P}}^{N}({\mathbf{C}})$ defining by a
homogeneous polynomial $Q$. For a domain $\Omega$ denote by
$n_{f}(r,\Omega,D)$ the number of zeros of $Q\circ{\bf f}$ in the domain
$\\{|z|<r\\}\bigcap\Omega,$ counting multiplicity. The counting function is
defined as follows:

$N_{f}(r,\Omega,D)=\int_{0}^{r}\dfrac{n_{f}(r,\Omega,D)-n_{f}(0,\Omega,D)}{t}dt+n_{f}(0,\Omega,D)\log
r.$

In this paper, we consider angular domains
$\Omega(\theta,\epsilon),0\leq\theta\leq 2\pi,0<\epsilon<\pi$ defined as
follows

$\Omega(\theta,\epsilon)=\\{z:\theta-\epsilon<\arg z<\theta+\epsilon\\}.$

Sometimes, we write simply $\Omega$ instead of $\Omega(\theta,\epsilon)$ if
there is no confusion.

######  Theorem 4.

Let $f:{\mathbf{C}}\rightarrow V$ be a holomorphic curve satisfying
$\|f^{\prime}(z)\|=O(|z|^{\sigma})$ for some $\sigma>-1$. Let
$D_{1},\ldots,D_{q}$ be hypersurfaces of degrees $d_{1},\ldots,d_{q}$ in
${\mathbf{P}}^{N}({\mathbf{C}})$. Let $\Omega=\Omega(\theta,\epsilon)$ be any
angular domain in ${\mathbf{C}}.$

(i) Let $\dim V\leq m\in\mathbb{N}.$ If $D_{1},\ldots,D_{q}$ are located in
$m-$subgeneral position in $V$ then we have

$(q-m+1)T_{f}(r,\Omega)\leq\sum_{1\leq i\leq
q}\dfrac{1}{d_{i}}N_{f}(r,\Omega,D_{i})+o(r^{2\sigma+2}).$

(ii) If
$V\bigcap\mathrm{Supp}\,D_{1}\bigcap\ldots\bigcap\mathrm{Supp}\,D_{q}=\emptyset$
then we have

$2T_{f}(r,\Omega)\leq\sum_{1\leq i\leq
q}\dfrac{1}{d_{i}}N_{f}(r,\Omega,D_{i})+o(r^{2\sigma+2}).$

iii) If
$\dim(V\bigcap\mathrm{Supp}\,D_{1}\bigcap\ldots\bigcap\mathrm{Supp}\,D_{q})=0$
then we have

$T_{f}(r,\Omega)\leq\sum_{1\leq i\leq
q}\dfrac{1}{d_{i}}N_{f}(r,\Omega,D_{i})+o(r^{2\sigma+2}).$

Remark that, we obtain the part (ii) from the part (i) by observing that
$D_{1},\ldots,D_{q}$ satisfying the condition
$V\bigcap\mathrm{Supp}\,D_{1}\ldots\bigcap\mathrm{Supp}\,D_{q}=\emptyset$
could be considered to be located in $(q-1)-$subgeneral position. Similarly,
the part (iii) comes from the part (i) by observing that $D_{1},\ldots,D_{q}$
satisfying the condition
$\dim(V\bigcap\mathrm{Supp}\,D_{1}\bigcap\ldots\bigcap\mathrm{Supp}\,D_{q})=0$
could be considered to be located in $q-$subgeneral position.

As a consequence of Theorem 4, for holomorphic curve satisfying
$\limsup\limits_{r\rightarrow\infty}\dfrac{T_{f}(r)}{r^{2\sigma+2}}>0$, we
have the defect relation

$\sum_{1\leq i\leq q}\delta_{D_{i}}(f)\leq m-1$

for a family of hypersurfaces $D_{1},\ldots,D_{q}$ located in $m-$subgeneral
position. For a general holomorphic curve, the best result available at
present is due to Quang[11], where the defect relation is $(m-n+1)(n+1)$ for
the case of hypersurfaces $D_{1},\ldots,D_{q}$ located in $m-$subgeneral
position. For the case of arbitrary family of hypersurfaces, we do not know
any result on that subject.

The last aim of this paper is to study singular directions of holomorphic
curves. The existence of Julia directions was proved by G. Julia [8] in 1920
for all entire function and by Milloux in 1924 and Valiron in 1938 for most of
meromorphic functions. For general holomorphic curves from ${\mathbf{C}}$ to
${\mathbf{P}}^{N}({\mathbf{C}})$, it has been studied by Eremenko [4],
Zheng[16],…In [4], Eremenko showed the existence of Julia directions for some
special classes of holomorphic curves into projective space and gave a
definition of general Julia directions for the “subvariety” case.

######  Definition 5.

[4, p.4] A number $\theta\in[0;2\pi)$ is called a Julia direction for a
holomorphic curve $f:{\mathbf{C}}\rightarrow V$ if for every system of
divisors $D_{1},\ldots,D_{q}$ such that any $n+1$ of them has empty
intersection, and for any $\pi>\epsilon>0$ all but at most $2n$ of these
divisors have infinitely many preimages in the angle
$\\{z:|argz-\theta|<\epsilon\\}$ .

Recently, using the ideas of Eremenko [4], Zheng [16, p.867], Thai and Mai
[12] gave the notion on singular directions for a holomorphic curve into
${\mathbf{P}}^{N}({\mathbf{C}})$ intersecting hypersurfaces in general
position.

######  Definition 6.

[16, Def 3.1, p. 867] A ray $J(\theta)=\\{z\in{\mathbf{C}}:\arg z=\theta\\}$
is a $T-$ direction for a holomorphic curve
$f:{\mathbf{C}}\rightarrow{\mathbf{P}}^{N}({\mathbf{C}})$ if for any
$\epsilon(0<\epsilon<\pi)$, we have

$\limsup\limits_{r\rightarrow\infty}\dfrac{N_{f}(r,\Omega,H)}{T_{f}(r)}=0$

for at most $2N$ hyperplanes $H$ in general position in
${\mathbf{P}}^{N}({\mathbf{C}}).$

######  Definition 7.

[12]Let $V$ be a complex projective variety in
${\mathbf{P}}^{N}({\mathbf{C}})$ of dimension $n\geq 1.$ Let
$f:{\mathbf{C}}\rightarrow V$ be a holomorphic curve. A ray $J(\theta)$ is
called a $\bar{T}-direction$ for $f$ if for any $\epsilon(0<\epsilon<\pi)$, we
have

$\limsup_{r\rightarrow\infty}\dfrac{N_{f}(r,\Omega,D)}{T_{f}(r)}=0$

for at most $n-1$ hypersurfaces $D$ located in general position in $V.$

Obviously, $\bar{T}-$direction is a $T-$ direction and they are Julia
directions in the sense of Eremenko [4]. However, this definition depends on
the dimension of variety which is different from Eremenko’s notion. We now
extend Thai-Mai’s notion on singular directions to not depend on the dimension
of variety. Only the intersection pattern is relevant.

######  Definition 8.

Let $V$ be a complex projective variety in ${\mathbf{P}}^{N}({\mathbf{C}})$.
Let $f:{\mathbf{C}}\rightarrow V$ be a holomorphic curve. A ray $J(\theta)$ is
called a $\bar{T}-direction$ for $f$ if for any $\epsilon(0<\epsilon<\pi)$ and
for any hypersurfaces $D_{1},\ldots,D_{q}$ in ${\mathbf{P}}^{N}({\mathbf{C}})$
satisfying $\bigcap_{1\leq i\leq q}\mathrm{Supp}\,D_{i}\bigcap V$ consisting
of finite points, we have

$\max_{1\leq i\leq
q}\limsup\limits_{r\rightarrow\infty}\dfrac{N_{f}(r,\Omega,D_{i})}{T_{f}(r)}>0$

Based on Theorem 4, we state the third main result of this paper.

######  Theorem 9.

Let $f:{\mathbf{C}}\rightarrow V$ be a holomorphic curve satisfying
$\|f^{\prime}(z)\|=O(|z|^{\sigma})$ for some $\sigma>-1$. If
$\limsup\limits_{r\rightarrow\infty}\dfrac{T_{f}(r)}{r^{2\sigma+2}}>0$ then
$f$ has at least one $\bar{T}-$direction.

We would like to remark that Theorem 4 and Theorem 9 imply Theorem [12,
Theorem 1.7] and Theorem [12, Theorem 1.8], respectively.

##  2\. Some Lemmas

Let $u$ be a subharmonic function in a domain $\Omega\subset{\mathbf{C}}.$ Let
$\triangle u$ be the Riesz measure of $u$. We recall the following lemma due
to Duval and Da Costa [1, p.1599] reformulated by Thai-Mai [12, Lemma 2.3].

######  Lemma 10.

[12, Lemma 2.3](see also [1, p.1599]) Let $q,m$ be positive integers. Let
$\nu_{i},1\leq i\leq q$ and $\nu$ be subharmonic functions in $\Omega.$ Assume
that $\triangle\nu$ is $L^{\infty}$ on $\Omega$ and $\nu=\max_{I}\nu_{i}$ for
any subset $I\subset\\{1,\ldots,q\\},|I|=m.$ Then $\sum_{1\leq i\leq
q}\nu_{i}-(q-m+1)\nu$ is subharmonic in $\Omega.$

Let $f:{\mathbf{C}}\rightarrow V$ be a holomorphic curve. We choose a
homogeneous representation ${\bf f}=(f_{0},\ldots,f_{N})$ of our curve.

Let $D_{1},\ldots,D_{q}$ be hypersurfaces of degrees $d_{1},\ldots,d_{q}$ in
${\mathbf{P}}^{N}({\mathbf{C}})$, defining by homogeneous polynomials
$Q_{1},\ldots,Q_{q}\in{\mathbf{C}}[X_{0},\ldots,X_{N}]$, located in
$m-$subgeneral position in $V.$ Replacing $Q_{i}$ by $Q_{i}^{d/d_{i}}$ if
necessary, where $d$ is the l.c.m of $d_{i},1\leq i\leq q,$ without loss of
generality, we can assume that $Q_{i},1\leq i\leq q$ have the same degree $d$.
Define

$u=\log\|{\bf f}\|=\log\sqrt{|f_{0}|^{2}+\ldots+|f_{N}|^{2}},$
$u_{i}=\dfrac{1}{d}\log|Q_{i}(f_{0},\ldots,f_{N})|,1\leq i\leq q.$

We will prove the following.

######  Lemma 11.

Let $f:{\mathbf{C}}\rightarrow V$ be a holomorphic curve satisfying
$\|f^{\prime}(z)\|=O(|z|^{\sigma})$ for some $\sigma>-1.$ Let $r_{k}$ be a
sequence of positive real numbers which converges to $+\infty.$ Let $h_{k}$ be
the smallest harmonic majorant of $u$ in the disk $2r_{k}\mathbb{D}$ and
$\lambda_{k}:{\mathbf{C}}\rightarrow{\mathbf{C}}$ be the map given by
$\lambda_{k}(z)=r_{k}z.$ Then, there exist subsequences of the sequences
$\frac{1}{r_{k}^{2\sigma+2}}(u_{i}-h_{k})\circ\lambda_{k}$ and
$\frac{1}{r_{k}^{2\sigma+2}}(u-h_{k})\circ\lambda_{k}$ which converge to
subharmonic functions $\nu_{i}$ and $\nu$, respectively, and
$\nu=\max_{I}\nu_{i}$ for any subset $I\subset\\{1,\ldots,q\\}$ such that
$|I|=m.$

###### Proof.

For simplicity, we can assume that $I=\\{1,\ldots,m\\}$. If $q\geq m+1$, from
the fact that since $D_{i},1\leq i\leq q$ are located in $m-$subgeneral
position, we have $Q_{i},1\leq i\leq m+1$ have no common zeros in $V$. In the
case of $q=m,$ we can choose a hypersuface $D_{m+1}$ of
${\mathbf{P}}^{N}({\mathbf{C}})$ defining by a homogeneous polynomial
$Q_{m+1}$ such that $Q_{i},1\leq i\leq m+1$ have no common zeros in $V$.
Therefore, there exist two positive constants $C_{1},C_{2}$ such that

$C_{1}\leq\max_{1\leq i\leq m+1}|Q_{i}(\omega)|^{1/d}\leq
C_{2},\forall\omega\in\pi^{-1}(V)\bigcap\\{\|z\|\leq 1\\},$

where
$\pi:{\mathbf{C}}^{N+1}\backslash\\{0\\}\mapsto{\mathbf{P}}^{N}({\mathbf{C}})$
be the standard projection. Hence,

(2.1) $\displaystyle C_{1}\|{\bf f}(z)\|\leq\max_{1\leq i\leq
m+1}|Q_{i}\circ{\bf f}(z)|^{1/d}\leq C_{2}\|{\bf f}(z)\|,\forall
z\in{\mathbf{C}}.$

It follows that

(2.2) $\displaystyle u=\max_{1\leq i\leq m+1}u_{i}+O(1)$

Consider the sequence of non-positive subharmonic functions on $2\mathbb{D}$
defined by $\dfrac{1}{r_{k}^{2\sigma+2}}(u-h_{k})\circ\lambda_{k}.$ Note that
$h_{k}(0)=\dfrac{1}{2\pi}\int_{0}^{2\pi}u(2r_{k}e^{i\theta})d\theta\leq
O(r_{k}^{2\sigma+2})$ because $T_{f}(r)=O(r_{k}^{2\sigma+2}).$ Hence
$\liminf\limits_{k\rightarrow\infty}\dfrac{1}{r_{k}^{2\sigma+2}}(u(0)-h_{k}(0))>-\infty$.
So the sequence $\dfrac{1}{r_{k}^{2\sigma+2}}(u-h_{k})\circ\lambda_{k}$ does
not converge locally uniformly to $-\infty.$ By [7, Theorem 4.1.9], there
exists a subsequence of the sequence
$\dfrac{1}{r_{k}^{2\sigma+2}}(u-h_{k})\circ\lambda_{k}$ (still denoted by $k$)
such that

$\dfrac{1}{r_{k}^{2\sigma+2}}(u-h_{k})\circ\lambda_{k}\rightarrow\nu,$

where $\nu$ is a subharmonic function in $2\mathbb{D}.$

For each $i,$ from (2.2), we have $u_{i}-h_{k}\leq u-h_{k}+O(1)<O(1).$
Therefore, $\dfrac{1}{r_{k}^{2\sigma+2}}(u_{i}-h_{k})\circ\lambda_{k}$ is
bounded above. Since $Q_{i}\circ{\bf f}\not=0$, there exists $y\in
2\mathbb{D}$ such that $u_{i}\circ\lambda_{k}(y)\not=-\infty.$ For
sufficiently large $k$, $u(2r_{k}e^{i\theta})>0$ for all $\theta$. By
Poisson’s formula, we have

$\displaystyle h_{k}\circ\lambda_{k}(y)$
$\displaystyle=\dfrac{1}{2\pi}\int_{0}^{2\pi}\dfrac{4-|y|^{2}}{|2e^{i\theta}-y|^{2}}u\circ\lambda_{k}(2e^{i\theta})d\theta$
$\displaystyle\leq\dfrac{2+|y|}{2\pi(2-|y|)}\int_{0}^{2\pi}u(2r_{k}e^{i\theta})d\theta\leq
O(r_{k}^{2\sigma+2}).$

Therefore,
$\liminf\limits_{k\rightarrow\infty}\dfrac{1}{r_{k}^{2\sigma+2}}(u_{i}-h_{k})\circ\lambda_{k}(y)>-\infty$.
Hence, there exists a subsequence of the sequence
$\dfrac{1}{r_{k}^{2\sigma+2}}(u_{i}-h_{k})\circ\lambda_{k}$ (still denoted by
$k$) such that

$\dfrac{1}{r_{k}^{2\sigma+2}}(u_{i}-h_{k})\circ\lambda_{k}\rightarrow\nu_{i},$

where $\nu_{i}$ is a subharmonic function in $2\mathbb{D}.$ From (2.2), we
have

(2.3) $\displaystyle\nu=\max_{1\leq i\leq m+1}\nu_{i}.$

We now prove that

$\nu\leq\max_{1\leq i\leq m}\nu_{i}.$

Indeed, let $Q:V\rightarrow{\mathbf{P}}^{m}({\mathbf{C}})$ be a morphism
defined by

$z\mapsto Q(z)=(Q_{m+1}(z),Q_{1}(z)\ldots,Q_{m}(z)).$

Since $D_{i},1\leq i\leq q$ are located in $m-$subgeneral position,
$Q_{i},1\leq i\leq m+1$ have no common zeros in $V$. Hence, $Q$ is a
holomorphic mapping on $V$. Combining with the fact that $V$ is a compact
complex projective variety and $\|f^{\prime}(z)\|=O(|z|^{\sigma}),$ we have

$g=(g_{0},\ldots,g_{m}):=Q\circ{\bf
f}:{\mathbf{C}}\rightarrow{\mathbf{P}}^{m}({\mathbf{C}})$

satifying $\|g^{\prime}(z)\|=O(|z|^{\sigma})$. By composing $g$ with an
automorphism of ${\mathbf{P}}^{m}({\mathbf{C}})$, for example replace $g_{0}$
by $g_{0}+cg_{1},c\in{\mathbf{C}}$ and leave all other $g_{i}$ unchanged, we
can assume that $g_{0}$ has infinitely many zeros. Then, we can apply [3,
Proposition 2] due to Barret and Eremenko [3] to $g$. Then, for every
$\epsilon>0,$ we have for $z$ sufficiently large,

(2.4)
$\displaystyle\log\sqrt{|g_{0}|^{2}+\ldots+|g_{m}|^{2}}(z)\leq\max_{1\leq
i\leq m}\log|g_{i}|(z)+K(2+\epsilon)^{\sigma+1}(m+1)|z|^{\sigma+1}.$

where $K$ is a constant depending only on $f$ and $Q.$ Hence,

(2.5) $\displaystyle\max_{1\leq i\leq m+1}\log|Q_{i}\circ{\bf f}(z)|\leq
d\max_{1\leq i\leq m}u_{i}(z)+K(2+\epsilon)^{\sigma+1}(m+1)|z|^{\sigma+1}.$

From (2.5) and (2.1), we have for $z$ sufficiently large

(2.6) $\displaystyle u(z)\leq\max_{1\leq i\leq
m}u_{i}(z)+K(2+\epsilon)^{\sigma+1}(m+1)|z|^{\sigma+1}.$

From (2.6), we have

(2.7) $\displaystyle\nu\leq\max_{1\leq i\leq m}\nu_{i}.$

From (2.3) and (2.7), we have $\nu=\max_{1\leq i\leq m}\nu_{i}.$ ∎

######  Lemma 12.

$\sum_{1\leq i\leq q}\nu_{i}-(q-m+1)\nu$ is subharmonic in $2\mathbb{D}.$

###### Proof.

In view of Lemma 11 and Lemma 10, it suffices to prove that the Riesz measure
$\triangle\nu$ is $L^{\infty}$ on $2\mathbb{D}.$

Indeed, since
$\dfrac{1}{r_{k}^{2\sigma+2}}(u-h_{k})\circ\lambda_{k}\rightarrow\nu$, we have
$\dfrac{1}{r_{k}^{2\sigma+2}}\int\phi\triangle(u\circ\lambda_{k})\rightarrow\int\phi\triangle\nu$
for any test function $\phi\in C_{0}^{\infty}(2\mathbb{D}).$

We denote by $B(z,r)$ the open disc of radius $r$ centered at $z$. Hence, for
any $B(z,\delta)\subset 2\mathbb{D},$ we have

$\triangle\nu(B(z,\delta))\leq\liminf_{r\rightarrow\infty}\dfrac{\triangle
u(B(rz,r\delta))}{r^{2\sigma+2}}\leq\liminf_{r\rightarrow\infty}\dfrac{\delta^{2\sigma+2}r^{2\sigma+2}O(1)}{r^{2\sigma+2}}=O(1)\delta^{2\sigma+2},$

where $O(1)$ is a constant depending only on $f.$ Hence $\triangle\nu$ is
$L^{\infty}$ on $2\mathbb{D}.$ ∎

##  3\. Proof of main theorems

In this section, we provide the proofs of main theorems. More precisely,
Theorem 1 follows from [3] and inequality 2.2. Theorem 4 is obtained from
Lemma 12 and Theorem 9 is a consequence of Theorem 4. The details are given as
follows.
Proof of Theorem 4.

We first prove that

(3.1)
$\displaystyle\limsup_{r\rightarrow\infty}\frac{1}{r^{2\sigma+2}}\left[(q-m+1)\triangle
u(\mathbb{D}_{r})-\sum_{1\leq i\leq q}\triangle
u_{i}(\mathbb{D}_{r})\right]\leq 0,$

where $\mathbb{D}_{r}=\Omega\bigcap\\{|z|<r\\}.$ Indeed, assume that
$r_{k}\rightarrow+\infty$ such that

$\displaystyle\frac{1}{r_{k}^{2\sigma+2}}\left[(q-m+1)\triangle
u(\mathbb{D}_{r_{k}})-\sum_{1\leq i\leq q}\triangle
u_{i}(\mathbb{D}_{r_{k}})\right]$

converges.

Applying Lemma 12, we have $\sum_{1\leq i\leq q}\nu_{i}-(q-m+1)\nu$ is
subharmonic in $2\mathbb{D}\bigcap\Omega$. Fix $\delta>0$ with
$0<\delta<\min\\{1/2,\epsilon\\}$. We choose a nonnegative smooth function
$\chi$ in $\mathbb{D}\bigcap\Omega$ such that $0\leq\chi\leq 1$ and $\chi=1$
in
$\mathbb{D}^{\delta}=(1-\delta)\mathbb{D}\bigcap\Omega(\theta,\epsilon-\delta)\bigcap\\{|z|>\delta\\}$
and $\mathrm{Supp}\,\chi$ is a compact subset in $\mathbb{D}\bigcap\Omega.$
Since $\sum_{1\leq i\leq q}\nu_{i}-(q-m+1)\nu$ is subharmonic, we have

$(q-m+1)\int\chi\triangle\nu\leq\sum_{1\leq i\leq q}\int\chi\triangle\nu_{i},$

where the integration is taken over $\mathbb{D}\bigcap\Omega.$ It follows that
for $k$ sufficient large, we get

$\displaystyle(q-m+1)\dfrac{1}{r_{k}^{2\sigma+2}}\int\chi\triangle(u\circ\lambda_{k})\leq\sum_{1\leq
i\leq
q}\dfrac{1}{r_{k}^{2\sigma+2}}\int\chi\triangle(u_{i}\circ\lambda_{k})+\delta.$

Therefore,

(3.2) $\displaystyle(q-m+1)\triangle
u(r_{k}\mathbb{D}^{\delta})\leq\dfrac{1}{d}\sum_{1\leq i\leq q}\triangle
u_{i}(r_{k}\mathbb{D}\bigcap\Omega)+\delta r_{k}^{2\sigma+2},$

where
$r_{k}\mathbb{D}^{\delta}=(1-\delta)r_{k}\mathbb{D}\bigcap\Omega(\theta,\epsilon-\delta)\bigcap\\{|z|>\delta
r_{k}\\}$. Furthermore, since $\|f^{\prime}(z)\|=O(|z|^{\sigma})$, we have

(3.3) $\displaystyle\triangle u(r_{k}\mathbb{D}\bigcap\Omega)\leq\triangle
u(r_{k}\mathbb{D}^{\delta})+(1+3\epsilon)O(1)\delta(r_{k})^{2\sigma+2}.$

From (3.2) and (3.3) and letting $\delta\rightarrow 0$, we have

$\displaystyle\lim_{k\rightarrow}\dfrac{1}{r_{k}^{2\sigma+2}}\left[(q-m+1)\triangle
u(\mathbb{D}_{r_{k}})-\sum_{1\leq i\leq q}\triangle
u_{i}(\mathbb{D}_{r_{k}})\right]\leq 0.$

Hence, (3.1) holds. From (3.1), we have

$\displaystyle(q-m+1)\triangle u(r\mathbb{D}\bigcap\Omega)$
$\displaystyle\leq\sum_{1\leq i\leq q}\triangle
u_{i}(r\mathbb{D}\bigcap\Omega)+o(r^{2\sigma+2})$
$\displaystyle=n_{f}(r,r\mathbb{D}\bigcap\Omega,D_{i})+o(r^{2\sigma+2}),$

where $n_{f}(r,r\mathbb{D}\bigcap\Omega,D_{i})$ is the number of zeros of
$Q_{i}\circ{\bf f}$ in the domain $r\mathbb{D}\bigcap\Omega$, counting
multiplicity. Dividing both sides of above inequality to $r$ and integrating
it, we get the conclusion.
Proof of Theorem 9. Suppose that $f$ has no $\bar{T}-$direction. Then, for
each ray $J(\theta)$, we have an angle containing it such that

$\limsup_{r\rightarrow\infty}\dfrac{N_{f}(r,\Omega(\theta,\epsilon),D_{i})}{T_{f}(r)}=0$

for hypersurfaces $D_{1},\ldots,D_{q}$ in ${\mathbf{P}}^{N}({\mathbf{C}})$
satisfying $\bigcap_{1\leq i\leq q}\mathrm{Supp}\,D_{i}\bigcap V$ consisting
of finite points. Hence, we can choose finite family of angles
$\Omega(\theta_{i},\epsilon_{i})(1\leq i\leq p)$ such that

${\mathbf{C}}\subset\cup_{1\leq i\leq p}\Omega(\theta_{i},\epsilon_{i}))$

and

(3.4)
$\displaystyle\limsup_{r\rightarrow\infty}\dfrac{N_{f}(r,\Omega(\theta_{i},\epsilon_{i}),D_{i})}{T_{f}(r)}=0.$

for hypersurfaces $D_{1},\ldots,D_{q}$ in ${\mathbf{P}}^{N}({\mathbf{C}})$
satisfying $\bigcap_{1\leq i\leq q}\mathrm{Supp}\,D_{i}\bigcap V$ consisting
of finite points. Therefore,

(3.5) $\displaystyle T_{f}(r)\leq\sum_{1\leq i\leq
p}T_{f}(r,\Omega(\theta_{i},\epsilon_{i})).$

Applying Theorem 4 to $D_{i},1\leq i\leq q$, we have

(3.6) $\displaystyle\sum_{1\leq k\leq
p}T_{f}(r,\Omega(\theta_{k},\epsilon_{k}))\leq\sum_{1\leq k\leq p}\sum_{1\leq
i\leq
q}\dfrac{1}{d_{i}}N_{f}(r,\Omega(\theta_{k},\epsilon_{k}),D_{i})+o(r^{2\sigma+2}).$

From (3.5) and (3.6), we have

(3.7) $\displaystyle T_{f}(r)\leq\sum_{1\leq k\leq p}\sum_{1\leq i\leq
q}\dfrac{1}{d_{i}}N_{f}(r,\Omega(\theta_{k},\epsilon_{k}),D_{i})+o(r^{2\sigma+2}).$

From (3.4) and (3.7) and the fact that
$\limsup\limits_{r\rightarrow\infty}\dfrac{T_{f}(r)}{r^{2\sigma+2}}>0$, we get
a contradiction.
Proof of Theorem 1. Without loss of generality, we can assume that
$D_{1},\ldots,D_{q}$ have the same degree $d$. Let $Q_{1},\ldots,Q_{q}$ be the
homogeneous polynomials of degree $d$ defining $D_{1},\ldots,D_{q}$. Since
$\bigcap_{1\leq i\leq q}\\{Q_{i}=0\\}\bigcap V$ consists of finite points,
there exists a hypersurface $D_{0}$ of $\mathbb{P}^{N}(\mathbb{C})$ defining
by a homogeneous polynomial $Q_{0}\in\mathbb{C}[X_{0},\ldots,X_{N}]$ such that
$\bigcap_{0\leq i\leq q}\\{Q_{i}=0\\}\bigcap V=\emptyset.$ Consider the map

$\displaystyle
Q=(Q_{0},\ldots,Q_{q}):V\rightarrow{\mathbf{P}}^{q}({\mathbf{C}}),z\mapsto(Q_{0}(z),\ldots,Q_{q}(z)).$

This map is a holomorphic map between two compact complex varieties. Together
with the fact that $\|f^{\prime}(z)\|=O(|z|^{\sigma})$, we have $Q\circ
f:\mathbb{C}\rightarrow{\mathbf{P}}^{q}({\mathbf{C}})$ satisfies
$\|Q^{\prime}\circ f(z)\|=O(|z|^{\sigma})$. Since $\bigcap_{1\leq i\leq
q}\\{Q_{i}=0\\}\bigcap f({\mathbf{C}})=\emptyset,$ $Q\circ f$ omits $q$
hyperplanes in general position. Hence, the main theorem in [3] is applicable
to $Q\circ f$. Then, we have $T_{Q\circ f}(r)=O(r^{\sigma+1}).$ Using the same
arguments as in Lemma 11 to obtain inequality 2.2, we have $T_{Q\circ
f}(r)=dT_{f}(r)$. This completes the proof of Theorem 1.

##  4\. Acknowledgements

This work was supported by a NAFOSTED, grant of Vietnam (Grant No.
101.02-2021.16).

## References

  * [1] B. F. P. Da Costa, J. Duval Sur les courbes de Brody dans $\mathbb{P}^{n}(\mathbb{C})$. Math. Ann. 355(4), 1593-1600 (2013).
  * [2] J. Clunie and W. Hayman, The spherical derivative of integral and meromorphic functions, Comment. math. helv., 40 (1966) 373-381.
  * [3] Barrett. M; Eremenko. A, Generalization of a theorem of Clunie and Hayman. Proc. Amer. Math. Soc. 140(2012), 1397–1402.
  * [4] A. Eremenko, Julia directions for holomorphic curves. Preprint in A. Eremenko’s homepage.
  * [5] M. Gromov, Topological invariants of dynamical systems and spaces of harmonic maps, Math. Phys. Anal. and Geom. 2 (1999) no. 4, 323-415.
  * [6] M. Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps: I, Math. Phys. Anal. Geom., 2(1999), 323–415.
  * [7] L. Hörmander, Analysis of linear partial differential operators I, Distribution theory and Fourier analysis, Springer, Berlin, 1983.
  * [8] G. Julia, Sur quelques proprietes nouvelles des functions entieres ou meromorphes (premier mémoire). Ann. Sci. Ec. Norm. Sup. 36, 93–125 (1916).
  * [9] S. Lang, Introduction to complex hyperbolic spaces, Springer-Verlag, New York, 1987.
  * [10] S. Matsuo, M. Tsukamoto, Brody curves and mean dimension, J. Am. Math. Soc. 28, 159-182 (2015).
  * [11] S.D. Quang, Degeneracy second main theorems for meromorphic mappings into projective varieties with hypersurfaces. Trans. Amer. Math. Soc. 371(2019), no. 4, 2431–2453.
  * [12] D. D. Thai, P. N. Mai, Singular directions of Brody curves, J. Geom. Anal. 31 (2021), 1721–1731.
  * [13] M. Tsukamoto, Moduli space of Brody curves, energy and mean dimension. Nagoya Math. J. 192 (2008), 27–58.
  * [14] M. Tsukamoto, Mean dimension of the dynamical system of Brody curves. Invent. Math. 211, 935-968 (2018).
  * [15] J. Winkelmann, On Brody and entire curves. Bull. Soc. Math. France 135 (2007), no. 1, 25–46.
  * [16] J. H. Zheng, Value distribution of holomorphic curves on an angular domain. Mich. Math. J. 64, 849-879 (2015).


Within this region the bound is saturated as $G(1,2,P)=1$ for all $P$ and $G(n,M,1)=1$ for all $n\geq 1, M\geq 2$. Explicitly, we have for $n=1$

\begin{equation}
G(P,n=1,M)= \frac{1}{2^{2M}} \frac{\left[(1+\sqrt{2P-1})^M-(1-\sqrt{2P-1})^M\right]^2}{2P-1}\,.
\end{equation}

For $n\geq2$ and $M=2$

\begin{align}
G(n\geq2,M=2,P) = \min\left(\frac{P^2}{P-\frac{1}{2^n}}\left(1-\frac{1}{2^n}\right),\
\frac{4}{2^{2n}} + \frac{4}{2^{n/2}}g_2\sqrt{P-\frac{1}{2^n}} + 2^n g^2_2\left(P-\frac{1}{2}\right)\right)\,,
\end{align}

where we denote $g_k=\left(\frac{2^n-1}{2^n}\right)^k+\left( \frac{1}{2^n} \right)^k$. Further, for $n\geq2$ and $M\geq3$ we have

\begin{align}
G(n\geq2,M\geq3,P) = \min\left(\frac{P^M}{P-\frac{1}{2^n}}\left(1-\frac{1}{2^n}\right),\
\frac{2^{n}}{4} \frac{\left[ \left(\sqrt{2\left(P-\frac{1}{2^n}\right)}+\frac{1}{2^n}\right)^M - \left(\frac{1}{2^n}\right)^M \right]^2}{P-\frac{1}{2^n}} \right)\,.
\end{align}

From Definition <ref> we have

\begin{equation}\label{eq:avshotmitigationIIcopy}
\dbloverline{\rchi}_{\boldsymbol{\lambda}} = \frac{1}{\gamma(\boldsymbol{\lambda})} \frac{\big\langle\big(C_m(\rho_{\boldsymbol{\lambda}},U_i) - \Tr[O]/2^n \big)^2\big\rangle_{U_i}}{\big\langle\big(\widetilde{C}(\rho_{\boldsymbol{\lambda}},U_i) - \Tr[O]/2^n \big)^2\big\rangle_{U_i} }\,.
\end{equation}

Let us first evaluate the required averages over unitary 2-designs. The relevant first moments for virtual distillation are given by

\begin{align}
\langle \Tr[U\rho_{\boldsymbol{\lambda}} U^\dag O] \rangle_U &= \Tr[O]/2^n\,, \\
\langle \Tr[U\rho_{\boldsymbol{\lambda}}^M U^\dag O] \rangle_U &= \Tr[\rho_{\boldsymbol{\lambda}}^M]\Tr[O]/2^n\,,
\end{align}

where we have used Lemma <ref>. Thus we can see that the numerator and denominator of (<ref>) correspond to variances which we now evaluate. Again, utilizing Lemma <ref>, the second moments are given by

\begin{align}
\big\langle\big(\widetilde{C}(U_i) - \langle \widetilde{C}(U_j) \rangle_{U_j} \big)^2\big\rangle_{U_i} &= \langle (\Tr[U\rho_{\boldsymbol{\lambda}} U^\dag O])^2 \rangle_U - (\Tr[O]/2^n)^2 \\
&= \frac{\left(\Tr[O^2]- \frac{1}{2^n}\Tr[O]^2\right)  \left(\Tr[\rho_{\boldsymbol{\lambda}}^2]-\frac{1}{2^n}\Tr[\rho_{\boldsymbol{\lambda}}]^2 \right)}{2^{2n}-1} \\[1em]
\big\langle\big(C_m(U_i) - \langle C_m(U_j) \rangle_{U_j} \big)^2\big\rangle_{U_i} &= \bigg\langle \left(\frac{\Tr[U\rho_{\boldsymbol{\lambda}}^M U^\dag O]}{\Tr[\rho_{\boldsymbol{\lambda}}^M]}\right)^2 \bigg\rangle_U - (\Tr[O]/2^n)^2 \\
&=\frac{\left(\Tr[O^2]- \frac{1}{2^n}\Tr[O]^2\right)  \left(\frac{\Tr[\rho_{\boldsymbol{\lambda}}^{2M}]}{\Tr[\rho_{\boldsymbol{\lambda}}^M]^2}-\frac{1}{2^n} \right)}{2^{2n}-1}\,,
\end{align}

where in the final equality we have used the fact that $\Tr\left[\frac{\rho_{\boldsymbol{\lambda}}^{M}}{\Tr[\rho_{\boldsymbol{\lambda}}^M]}\right]=1 $
Using the definition of the averaged relative resolvability II (Definition <ref>), we can arrive at a bound written explicitly in terms of $\rho_{\boldsymbol{\lambda}}$ as

\begin{equation} \label{eq:appdx_VD_avbound1}
\dbloverline{\chi}_{\boldsymbol{\lambda}} = \frac{1}{\gamma} \frac{\big\langle\big(C_m(U_i) - \langle C_m(U_j) \rangle_{U_j} \big)^2\big\rangle_{U_i}}{\big\langle\big(\widetilde{C}(U_i) - \langle \widetilde{C}(U_j) \rangle_{U_j} \big)^2\big\rangle_{U_i}} \leq  \frac{\Tr[\rho_{\boldsymbol{\lambda}}^{2M}]-\frac{1}{2^n}\Tr[\rho_{\boldsymbol{\lambda}}^M]^2}{\Tr[\rho_{\boldsymbol{\lambda}}^2]-\frac{1}{2^n}\Tr[\rho_{\boldsymbol{\lambda}}]^2}\,,
\end{equation}

where we have used the fact that the error mitigation cost $\gamma \geq 1/(\Tr[\rho_{\boldsymbol{\lambda}}^M])^2$.

The goal is to now investigate whether or not $f(M) = \Tr[\rho_{\boldsymbol{\lambda}}^{2M}]-\frac{1}{2^n}\Tr[\rho_{\boldsymbol{\lambda}}^M]^2$ is monotonically decreasing for $M\in \mathbb{N}_+$. This quantity has two interpretations. First, it can be seen to be a Hilbert Schmidt distance between $\rho_{\boldsymbol{\lambda}}^M$ and $\Tr[\rho_{\boldsymbol{\lambda}}^M]\frac{\id}{2^n}$. Second, by considering the eigenvalue decomposition of $\rho$, it can be seen to be proportional to the population variance of the distribution $\{\lambda^M_i \}$, where $\lambda_i$ are the eigenvalues of $\rho_{\boldsymbol{\lambda}}$, that is,

\begin{equation}
f(M) = 2^n \Var[\boldsymbol{\lambda}^{(M)}] = \sum_i \lambda_i^{2M} - \frac{1}{2^n}\left(\sum_i \lambda_i^M\right)^2\,,
\end{equation}

where here $\Var[(.)]$ denotes the population variance of the contained vector. Thus, we can rewrite Eq. (<ref>) as

\begin{equation}\label{eq:VD_chi_av2}
\dbloverline{\chi}_{\boldsymbol{\lambda}}(M)\leq \frac{f(M)}{f(1)} = \frac{\Var[\boldsymbol{\lambda}^{(M)}]}{\Var[\boldsymbol{\lambda}^{(1)}]}\,.
\end{equation}

Let us first treat the qubit setting of $n=1$. Consider eigenvalue decomposition $\rho_{\boldsymbol{\lambda}} = \lambda\ket{\psi}\bra{\psi} + (1-\lambda)\ket{\psi_\perp}\bra{\psi_\perp}$, where we have defined $\lambda_1 = 1-\lambda$, $\lambda_2 = \lambda$ and without loss of generality we fix $1-\lambda \geq \lambda$. We define $G(1,M,P)=f(M)/f(1)$ and will determine $f(M)$ exactly for single-qubit states. For generic $M$ we have

\begin{align}
f(M) &= \lambda^{2M} + (1-\lambda)^{2M} - \frac{1}{2}\left(\lambda^M + (1-\lambda)^M \right)^2\\
&= \frac{1}{2}((1-\lambda)^M-\lambda^{M})^2 \\
&= \frac{1}{2^{2M+1}} \left[(1+\sqrt{2P-1})^M-(1-\sqrt{2P-1})^M\right]^2\,,
\end{align}

where in the final equality we have used the fact that for single-qubit states $\lambda = \frac{1}{2}(1-\sqrt{2P-1})$. Further, using $f(1)=P-\frac{1}{2}$ we have the bound as required.

Now let us consider the setting of $n\geq2$. We will construct two bounds, for the respective high purity and low purity limits. We start with the bound for high purity states.
We can write the right hand side of Eq. (<ref>) explicitly as

\begin{align}
\frac{\Var[\boldsymbol{\lambda}^{(M)}]}{\Var[\boldsymbol{\lambda}^{(1)}]} &= \frac{\frac{1}{2^n}\sum_i\lambda_i^{2M}-(\frac{1}{2^n}\sum_i\lambda_i^M)^2}{\frac{1}{2^n}\sum_i\lambda_i^2-\frac{1}{2^{2n}}} \\
&= \frac{\frac{1}{2^n}\sum_i\lambda_i^{2M}-\frac{1}{2^{2n}}\sum_{i}\lambda_i^{2M}-\frac{1}{2^{2n}}\sum_{i\neq j}\lambda_i^M\lambda_j^M}{\frac{1}{2^n}\sum_i\lambda_i^2-\frac{1}{2^{2n}}} \\
&\leq \frac{(2^n-1)(\sum_i\lambda_i^{2M})}{2^n\sum_i\lambda_i^{2}-1} \\
&\leq \frac{(2^n-1)(\sum_i\lambda_i^{2})^M}{2^n\sum_i\lambda_i^{2}-1} \\
&= \frac{P^M}{P-\frac{1}{2^n}}\left(1-\frac{1}{2^n}\right) \,,
\end{align}

where in order to obtain the first inequality we have dropped the cross terms $\frac{1}{2^{2n}}\sum_{i\neq j}\lambda_i^M\lambda_j^M$, and in the second inequality we have introduced new cross terms. The final equality comes by substituting in the definition of the purity $P$. We note this first bound is upper-bounded by $1$ for all $P\geq\frac{1}{2^n-1}$.

We can now construct our second bound for strongly mixed states. We will consider bounds on $\Var[X^M]$ where a random variable $X$ when it is known that it takes values close to its mean $\mu$. We consider the decomposition

\begin{align}
X^M &= \left((X-\mu)-\mu\right)^M \\
&= \mu^M + \sum^M_{k=1}Y_k
\end{align}

where we have defined the random variables $Y_k=\binom{M}{k} \mu^{M-k}(X-\mu)^k$. Further, we can write

\begin{align}
\Var[X^M] &= \Var\bigg[\sum^M_{k=1}Y_k \bigg] \\
&= \mathbb{E}\left[ \bigg(\sum_k Y_k - \mathbb{E}\Big[ \sum_k Y_k \Big] \bigg) \bigg(\sum_j Y_j - \mathbb{E}\Big[ \sum_j Y_j \Big] \bigg) \right] \\
&= \sum_{k,j}\mathbb{E}\left[ \Big( Y_k - \mathbb{E}[  Y_k ] \Big) \Big( Y_j - \mathbb{E}[ Y_j ] \Big) \right] \\
&=\sum_{k,j} \Cov \big[ Y_k, Y_j \big] \\
&\leq \sum_{k,j} \sqrt{\Var\big[Y_k\big]\Var\big[Y_j\big]}\,,\label{eq:VD_highlvl-varbound}
\end{align}

where the inequality is due to Cauchy-Schwarz. We now take $X$ to be the random variable which takes values $\{\lambda_i\}_i$ with uniform probability and mean $\mu=\frac{1}{2^n}$. We will bound $\Var\big[Y_k\big]$ under the assumption that $\{\lambda_i\}_i$ are close in value to the maximally mixed value $\frac{1}{2^n}$.

First, we note that each $Y_k$ is a function of $(X-\mu)^k$, and so we must investigate the shifted spectrum which we denote $\hat{\boldsymbol{\lambda}}$ where $\hat{\lambda}_i = \lambda_i - \frac{1}{2^n}$ for all $i$. Using Popoviciu's inequality, we have the bound

\begin{equation}\label{eq:VD_Popoviciu}
\Var\left[(X-\mu)^k\right] \leq \frac{1}{4}\left( \hat{\lambda}^k_{max}-\hat{\lambda}^k_{min} \right)^2\,.
\end{equation}

Now suppose that we have the constraint

\begin{equation}\label{eq:VD_lam-constraint}
\lambda_{max}-\lambda_{min}=2b
\end{equation}

for some $b\geq0$. For any $k$, we have

\begin{equation}\label{eq:VD_lambda-hat}
\hat{\lambda}^k_{max}-\hat{\lambda}^k_{min} \leq \big|\hat{\lambda}_{max}\big|^k+\big|\hat{\lambda}_{min}\big|^k\,.
\end{equation}

Let us now bound the quantity on the right by considering its maximum value over all spectra with constraint (<ref>). The quantity on the right is maximized by the choice of vector $\big(\big|\hat{\lambda}_{max}\big|,\big|\hat{\lambda}_{min}\big|\big)$ that majorizes all others, given some fixed value of $\big|\hat{\lambda}_{max}\big|+\big|\hat{\lambda}_{min}\big|$. Indeed, $\big|\hat{\lambda}_{max}\big|+\big|\hat{\lambda}_{min}\big|=b$ is fixed by our constraint (<ref>) ($\hat{\lambda}_{min}$ must be negative in order to preserve trace). Thus the quantity on the right hand side of (<ref>) can be bounded by maximizing $\hat{\lambda}_{max}$ and minimizing $\big|\hat{\lambda}_{min}\big|$. This is achieved by setting all other $\hat{\lambda}_i$ equal to $\hat{\lambda}_{min}$. We then have pair of constraints

\begin{align}
&\hat{\lambda}_{max} + (2^n-1)\hat{\lambda}_{min} = 0 \,,\\
&\hat{\lambda}_{max} - \hat{\lambda}_{min} = 2b\,,
\end{align}

where the first constraint comes from preservation of trace, and the second is our original constraint. This is a linear system of equations with solution

\begin{equation}
\hat{\lambda}^*_{max} = 2b\frac{2^n-1}{2^n}\,,\quad \hat{\lambda}^*_{min} = -2b\frac{1}{2^n}
\end{equation}

substituting these values into (<ref>) we have the bound

\begin{align}
\hat{\lambda}^k_{max}-\hat{\lambda}^k_{min} &\leq (2b)^k\left( \left( \frac{2^n-1}{2^n}\right)^k + \left( \frac{1}{2^n}\right)^k \right)\label{eq:VD_itm-bound1}\\
&\leq (2b)^k\label{eq:VD_itm-bound2}\,.
\end{align}

We will find it necessary to use the tighter bound (<ref>) in the case of $M=2$, but the looser bound (<ref>) will enable us to write a bound with a more compact form for $M\geq3$.

We now relate $b$ to the purity. We can write a general spectrum that satisfies the constraint in (<ref>) as $\boldsymbol{\lambda}_{b,c,\boldsymbol{a}}= (\frac{1}{2^n}+b+c,\,\frac{1}{2^n}-b+c,\,\frac{1}{2^n}-a_1,\,...,\,\frac{1}{2^n}-a_{d-2})$, for some $c$ and set $\{a_i\}_i$ that satisfy $\sum^{d-2}_i a_i = 2c$ (in order to preserve trace). The purity that corresponds to this spectrum is given by

\begin{align}
P(\boldsymbol{\lambda}_{b,c,\boldsymbol{a}}) &= \left(\frac{1}{2^n}+b+c \right)^2 + \left(\frac{1}{2^n}-b+c \right)^2 + \sum_{i=1}^{d-2}\left(\frac{1}{2^n}-a_i \right)^2 \\
&= \frac{1}{2^n} + 2b^2 + c^2 + \sum_i a^2_i + \frac{2}{2^n}\bigg[ 2c - \sum^{d-2}_{i=1}a_i \bigg]\\
&\geq \frac{1}{2^n} + 2b^2\,.
\end{align}

Moreover, this purity bound is achievable by the spectrum $\boldsymbol{\lambda}_{b,0,\boldsymbol{0}}= (\frac{1}{2^n}+b,\frac{1}{2^n}-b,\frac{1}{2^n},...,\frac{1}{2^n})$ if we have $b \leq \frac{1}{2^n}$. We conclude that for any spectrum $\boldsymbol{\lambda}_b$ that satisfies the constraint (<ref>), we have

\begin{equation}\label{eq:VD_b-bound}
b\leq \sqrt{\frac{1}{2}\left(P(\boldsymbol{\lambda}_b)-\frac{1}{2^n}\right)}\,.
\end{equation}

And we now have all the tools to bound $\Var\big[Y_k\big]$ for all $k$ and subsequently $\Var[X^M]$

By combining the bounds (<ref>) and (<ref>) we have

\begin{equation}\label{eq:VD_varXk-bound}
\Var\left[(X-\mu)^k\right] \leq \frac{1}{4} (2b)^{2k} g_k^2
\end{equation}

where we have denoted $g_k= \left( \frac{2^n-1}{2^n}\right)^k + \left( \frac{1}{2^n}\right)^k \leq 1$. This allows us to bound $\Var[Y_k]$ by writing

\begin{align}\label{eq:VD_var-gk}
\sqrt{\Var[Y_k]} &= \binom{M}{k} \mu^{M-k} \sqrt{\Var\left[(X-\mu)^k\right]}\\
&\leq \frac{1}{2}\binom{M}{k} \mu^{M-k}(2b)^{k} g_k\,.
\end{align}

We first pursue a bound for general $M\in\mathbb{N}$ and replace each $g_k$ with $1$. We observe that the quantities $\{\binom{M}{k} \mu^{M-k}(2b)^{k}\}^M_{k=1}$ are simply the terms in the expansion of $(2b-\mu)^M - \mu^M$, that is,

\begin{equation}
\sum_k \sqrt{\Var[Y_k]} \leq \frac{1}{2}\left((2b-\mu)^M - \mu^M \right)\,.
\end{equation}

Returning to (<ref>), we have

\begin{align}
\Var[X^M]&\leq\frac{1}{4}\left((2b-\mu)^M - \mu^M \right)^2\\
&\leq \frac{1}{4}\left[\left(2\sqrt{\frac{1}{2}\left(P-\frac{1}{2^n}\right)}-\mu\right)^M - \mu^M \right]^2
\end{align}

where in order to obtain the second inequality we have used (<ref>) to substitute $b$ with its bound in terms of the purity. We further note that $\Var[X] = \frac{1}{2^n}(P-\frac{1}{2^n})$, and dividing the two quantities we obtain

\begin{equation}
\dbloverline{\chi}\leq\frac{2^{n}}{4} \frac{\left[ \left(\sqrt{2\left(P-\frac{1}{2^n}\right)}+\frac{1}{2^n}\right)^M - \left(\frac{1}{2^n}\right)^M \right]^2}{P-\frac{1}{2^n}}
\end{equation}

as required. To summarize, combining the two bounds for high purity and low purity, so far we have

\begin{align}\label{eq:VD_prelim-bound}
G'(n\geq2,M\geq2,P) = \min\left(\frac{P^M}{P-\frac{1}{2^n}}\left(1-\frac{1}{2^n}\right),\
\frac{2^{n}}{4} \frac{\left[ \left(\sqrt{2\left(P-\frac{1}{2^n}\right)}+\frac{1}{2^n}\right)^M - \left(\frac{1}{2^n}\right)^M \right]^2}{P-\frac{1}{2^n}} \right)\,.
\end{align}

Now we discuss the magnitude of our bound obtained thus far, as well as its monotonicity with respect to $M$. In particular, we will show that its value can exceed 1 for $M=2$, and so we will pursue a tighter bound for $M=2$. We can evaluate $G'(n\geq2,M\geq2,P)$ explicitly at $P=\frac{1}{2^n-1}$ as

\begin{align}\label{eq:VD_prelim-bound2}
G'\Big(n\geq2,M\geq2,P=\frac{1}{2^n-1}\Big) = \min\left(\frac{(2^n-1)^2}{(2^n-1)^M},\
\frac{2^{2n}(2^n-1)}{4} \left[ \left(\sqrt{\frac{2}{2^n(2^n-1)}}+\frac{1}{2^n}\right)^M - \left(\frac{1}{2^n}\right)^M \right]^2 \right)\,.
\end{align}

Firstly, by inspection this is a decreasing function in $n$ for all $M\geq 2$, so in order to bound its magnitude we can consider $n=2$. At $M=2$ we have

\begin{align}
G'\Big(2,2,\frac{1}{2^2-1}\Big) = \min\left(1,\
\frac{5+2\sqrt{6}}{6} \right)= 1\,,
\end{align}

where we note $\frac{5+2\sqrt{6}}{6}\geq 1$. As the first function in the minimization of (<ref>) has negative gradient for $P<\frac{1}{2^n-1}$ for $n\geq2, M=2$, this implies that the exists a set of values $P=\frac{1}{2^n-1}-\delta$, where $\delta>0$ is small, such that the first function has value exceeding 1. The second function also has value exceeding $1$ in such a region as it is continuous. Thus, there exist values of $P$ for which the bound $G'>1$ at $M=2$. Moving on to $M=3$, we can numerically verify that $ G'\Big(2,3,\frac{1}{2^n-1}\Big)\leq1$ with both functions in the minimization having value below $1$. As functions of the form $f(x) = a^x - b^x$ where $b\leq a \leq 1$ only have one stationary point which is a maximum, this implies that $G'\Big(2,M,\frac{1}{2^n-1}\Big)$ is decreasing for all $M\geq 3$ and thus $G'\Big(2,M\geq3,\frac{1}{2^n-1}\Big)\leq 1$.

We will replace $G'\Big(n\geq2,2,P\Big)$ with a tighter bound that is less than 1 for all $n\geq2$. We return to (<ref>) and now explicitly consider the $g_k$ terms. Substituting this into (<ref>) for $M=2$ we have

\begin{align}
\Var[X^2] &\leq \Var[Y_1] + \Var[Y_2] + 2\sqrt{\Var[Y_1]\Var[Y_2]} \\
&= (2\mu)^2\Var[X-\mu] + \Var\left[\left(X-\mu \right)^2\right] + {4\mu}\sqrt{\Var\left[\left(X-\mu \right)^2\right]\Var[X-\mu]}\\
&\leq (2\mu)^2\Var[X] + \frac{1}{4} (2b)^{4} g_2^2 + {4\mu}\sqrt{\frac{1}{4} (2b)^{4} g_2^2 \Var[X]} \\
&\leq \frac{4}{2^{2n}}\Var[X] + \left(P-\frac{1}{2^n} \right)^2g_2^2 + \frac{4}{2^n}g_2\left({P-\frac{1}{2^n}}\right)  \sqrt{\Var[X]}
\end{align}

where in the first equality we use the definition of $Y_k$ for $M=2$, in the first inequality we use (<ref>) along with the fact that $g_1 =1$, and in the final inequality we use (<ref>). Finally, dividing by $\Var[X]$ we have

\begin{align}
\dbloverline{\rchi}(M=2) &\leq \frac{4}{2^{2n}} + \frac{1}{\Var[X]}\left(P-\frac{1}{2^n} \right)^2g_2^2 + \frac{4}{2^n}g_2\sqrt{P-\frac{1}{2^n}}  \sqrt{\frac{1}{\Var[X]}} \\
&= \frac{4}{2^{2n}} + 2^n g^2_2\left(P-\frac{1}{2^n}\right) + \frac{4}{2^{n/2}}g_2\sqrt{P-\frac{1}{2^n}}\,,
\end{align}

where we have used $\Var[X] = \frac{1}{2^n}(P-\frac{1}{2^n})$.

We plot our bounds on the 2-design-averaged resolvability in Fig. <ref>. First, in the left figure we plot the intermediate bound $\frac{\Var[\boldsymbol{\lambda}^{(M)}]}{\Var[\boldsymbol{\lambda}^{(1)}]}$ in (<ref>) for states with 100 randomly generated spectra for increasing number of qubits $n$ and number of state copies $M$. This visualizes the exponential scaling with $M$ and we observe that broadly, the bound is decreasing with increasing number of qubits. Further, as expected, the bound is always less than or equal to 1. Second, in order to demonstrate the behaviour of our final upper bound (<ref>) we plot increasing number of state copies $M$ ranging from $2$ to $4$ for $n=2$. For each $M$, we randomly generate 10000 states and plot $\frac{\Var[\boldsymbol{\lambda}^{(M)}]}{\Var[\boldsymbol{\lambda}^{(1)}]}$ against the purity of the state as separate points. The final upper bound is then plotted as a line for each value of $M$.

Bounds on $\dbloverline{\rchi}$ for VD. (Left): We plot the intermediate upper bound $\frac{\Var[\boldsymbol{\lambda}^{(M)}]}{\Var[\boldsymbol{\lambda}^{(1)}]}$ for randomly generated states with increasing number of qubits $n$ and number of state copies $M$. (Right): We plot as points the intermediate upper bound $\frac{\Var[\boldsymbol{\lambda}^{(M)}]}{\Var[\boldsymbol{\lambda}^{(1)}]}$ against purity for randomly generated states at different values of $M$ at $n=2$. We also plot the final bound  (<ref>), which is a function of purity, as a line.

§.§ Probabilistic Error Cancellation

§.§.§ Error mitigation of multiple gates

We first consider the error mitigation cost of mitigating multiple noise channels. Suppose we have two noisy gates which we represent as the channel

\begin{equation}
\mathcal{N}'\circ\mathcal{U}'\circ\mathcal{N}\circ\mathcal{U}
\end{equation}

where $\{\mathcal{U}',\mathcal{U}\}$ are channels that represent the ideal gates and $\{\mathcal{N}',\mathcal{N}\}$ are noise channels. Note that this framework also includes as a special case the scenario where two gates act in parallel on different subsystems. Given set of basis gates $\{\mathcal{B}_\alpha\}_\alpha$, we can construct a quasiprobability distribution for the ideal channel as

\begin{equation}\label{eq:appdx_doubleQPdistribution}
\mathcal{U}'\circ\mathcal{U} = \sum_{\alpha,\beta} k_\alpha k_\beta\, \mathcal{B}_\alpha\circ\mathcal{N}'\circ\mathcal{U}'\circ\mathcal{B}_\beta\circ\mathcal{N}\circ\mathcal{U}\,.
\end{equation}

where we have used (<ref>).
From (<ref>) we see that the error mitigation cost is

\begin{equation}
\gamma_{tot} = \sum_{\alpha,\beta} k^2_\alpha k^2_\beta = \gamma \gamma'
\end{equation}

where $\gamma, \gamma'$ are the individual error mitigation costs for each gate. We can see the above reasoning can be extended inductively to show that the error mitgation cost of a collection of gates with the probabilistic error cancellation is equal to the product of the individual error mitigation costs.

§.§.§ Global depolarizing noise

Consider a quasi-probability method that corrects global depolarizing noise of the form (<ref>). For any pair of states corresponding to points on the cost function landscape, the optimal quasiprobability scheme gives

\begin{equation}
\rchi_{depol} = \frac{2^{2n}}{2^{2n}-p(2-p)} \geq 1 \,,
\end{equation}

for all $n\,\geq1,\,  p\in[0,1]$, which is achieved with access to noisy Pauli gates.

Ref. <cit.> gives the optimal quasi-probability decomposition for the inverse noise channel as

\begin{equation}
\mathcal{D}^{-1} =  \left(1 + \frac{(2^{2n}-1)p}{2^{2n}(1-p)} \right)\mathcal{I} - \sum_{i=1}^{2^{2n}-1} \frac{p}{2^{2n}(1-p)}\mathcal{P}_i\,,
\end{equation}

where $\mathcal{I}$ is the identity channel and $\mathcal{P}_i$ is the Pauli channel corresponding to the $i$th Pauli tensor product. This has corresponding error mitigation cost

\begin{equation}\label{eq:QP_gamma-global}
\gamma =
%\frac{1+ \left(1-\frac{2}{2^{2n}} \right)}{1-p}\,.
\frac{2^{2n}-2p+p^2}{2^{2n}(1-p)^2}.
\end{equation}

Assuming perfect correction we have $\Delta \widetilde{C} = (1-p)\Delta C$ which implies

\begin{equation}
\rchi_{depol} = \frac{2^{2n}}{2^{2n}-2p+p^2}\,,
\end{equation}

which is greater than or equal to 1 as $-2p + p^2 \leq0$ for all $0\leq p\leq1$.

§.§.§ Local depolarizing noise

Here we consider a model of cost concentration due to a single instance of local depolarizing noise in a circuit. We presume that the concentration follows a similar form of scaling to global depolarizing noise and a tensor product of local depolarizing noise (see Eq. (<ref>)). We show that, under this assumption, the relative resolvability has regimes of being greater than 1 or less than 1, depending on the strength of the cost concentration.

[Relative resolvability of Probabilistic Error Cancellation with one instance of local depolarizing noise]
Consider a single instance of local depolarizing noise occurring with error probability $p$ acting at an arbitrary point in the parameterized circuit. Suppose that due to this noise channel we have

\begin{equation}\label{eq:QP_local-assumption}
\langle\Delta \widetilde{C}(\boldsymbol{\theta}_{i,*})\rangle_i \geq (1-b_\alpha p) \langle\Delta C(\boldsymbol{\theta}_{i,*})\rangle_i
\end{equation}

for all $p\in [0,1]$ where $\langle\cdot\rangle_i$ denotes an average over all avaliable parameters and $b_\alpha$ where is some positive constant. Then the optimal quasiprobability scheme gives:

* for $b_\alpha\leq \frac{3}{4}$,

\begin{equation}
\overline{\rchi}\leq 1\,,\quad \forall p \in [0,1],
\end{equation}

* for $\frac{3}{4}<b_\alpha\leq1$,

\begin{align}
&\overline{\rchi}\leq 1 + \frac{1}{4}p(2-p)+\mathcal{O}(p^2)\,,\quad \forall p \in [0,1]\,, \\
&\overline{\rchi}>1\,,\quad \forall p \in \Big(0,\,1-\sqrt[3]{3(b^{-1}-1)}\Big]\,,
\end{align}

* for $b_\alpha>1$,

\begin{equation}
\overline{\rchi}> 1+\frac{p(2-p)}{4-p(2-p)}\,, \quad \forall p \in \big(0,\,1/b_\alpha\big]\,.
\end{equation}

From Eq. (<ref>), we can write the optimal error mitigation cost for one instance of local depolarizing noise acting on one qubit as

\begin{equation}\label{eq:QP_gamma-local}
\gamma =
\frac{4-2p+p^2}{4(1-p)^2}\,.
\end{equation}

Now, due to our assumption (<ref>) and assuming perfect implementation of the basis of noisy gates (leading to perfect correction of the noise) we have

\begin{equation}
\overline{\rchi} \leq \frac{4(1-p)^2}{(4-2p+p^2)(1-bp)^2}\,,
\end{equation}

and we denote the quantity on the right hand side as $h(p)$. Note that for any value of $b$, $h(p=0)=1$ and $h(p=1)=0$. The partial derivative can be found to satisfy

\begin{equation}\label{eq:QP_derivative}
\frac{\partial h}{\partial p} \propto \left(1-p\right)\left(1-bp\right)\left(-p^3 + 3p^2 - 3p + \frac{1}{b} -4\left(\frac{1}{b}-1\right)\right)\,,
\end{equation}

where the proportionality factor we omit is positive for all $b\geq0$ and $p\in[0,1]$. The third bracket is a cubic form with discriminant

\begin{equation}
\Delta = \frac{108}{b^2}\left(-8b^2 + 11b- 4\right)\,,
\end{equation}

which is strictly negative for all $b$. Thus, the cubic form only has one real root and, inspecting its behaviour for large $p$, we can conclude it has negative gradient for all $p$.

The cubic form has root at $p=0$ when $b=3/4$. More generally, the root can be found to take the form

\begin{equation}
\end{equation}

By evaluating the second derivative of $h(p)$, this root can be seen to correspond to a local maximum of $h(p)$. We now find the maximum value of $h(p)$ over the interval $p\in [0,1]$ for different regimes of cost concentration strength $b$.

First, we inspect the regime where $b\leq3/4$. In this case $p'\leq0$ and thus $\frac{\partial h}{\partial p}\leq 0$ for $p\in[0,1]$. Thus the maximum value of $h(p)$ on the interval $p\in[0,1]$ is $h(0)=1$. We can then conclude that $\overline{\rchi}\leq1$ with bound saturated in the limit of zero error probability.

Now consider the regime $3/4<b\leq1$. In this case $0< p' \leq1$ and $\frac{\partial h}{\partial p}> 0$ for small values of $p$. Specifically, it is clear that $\overline{\rchi} >1$ for $0<p\leq 1+\sqrt[3]{3(1-b^{-1})}$. The upper limit on $p$ can be raised, however, the exact interval is obtained by solving a quartic equation which we omit here as it is not very insightful. Moreover, the upper limit is tight in the limit $b\rightarrow 1$ and we obtain the result that when $b=1$, $\overline{\rchi} >0$ for all $p \in (0,1)$.

Finally, consider the regime $b\geq1$. Now $\frac{\partial h}{\partial p}$ has a different root $p''=1/b$ due to the second bracket in (<ref>). Again, this can be shown to correspond to a maximum of $\overline{\rchi}$ and we can write $\overline{\rchi}>1$ for $0<p\leq1/b$.

Consider tensor-product local depolarizing noise with local depolarizing probability $p$ to act in $L$ instances through a depth $L$ circuit as in Eq. <ref>. Suppose that the effect of this noise is to cause cost concentration

\begin{equation}\label{eq:appdx_QP_localscaling-assumption}
\langle\Delta \widetilde{C}(\boldsymbol{\theta}_{i,*})\rangle_i = Aq^L \langle\Delta C(\boldsymbol{\theta}_{i,*})\rangle_i\,,
\end{equation}

for some constant $A$ and noise parameter $q\in [0,1)$. The optimal quasiprobability method to mitigate the depolarizing noise in the circuit yields

\begin{equation}
\overline{\rchi} = \frac{1}{A^2q^{2L}}\left(Q(p) \right)^{nL} \,,
\end{equation}

where $0\leq Q(p)\leq 1$ for all $p$. Thus, the average relative resolvability has unfavourable scaling with system size.

As shown in Section <ref>, error mitigation cost of multiple gates with probabilistic error cancellation is the product of the individual error mitigation costs. Thus, for the collection of gates considered, we have total error mitigation cost

\begin{equation}
\gamma_{tot} = \left(\frac{4(1-p)^2}{4-2p+p^2} \right)^{nL}\,,
\end{equation}

where we have used Eq. (<ref>). We suppose that mitigation perfectly corrects the error, such that $\Delta C_m(\thv_{i,*}) = \Delta C(\thv_{i,*})$ Combining this with our assumption (<ref>) we obtain the desired result, where we denote

\begin{equation}
Q(p)= \frac{4-2p+p^2}{4(1-p)^2} = 1 - \frac{3p(2-p)}{4-p(2-p)}\,,
\end{equation}

which clearly satisfies $0\leq Q(p) \leq 1$.

§.§ Linear Ansatz Methods

§.§.§ Global depolarizing noise is exactly correctable

Consider the linear ansatz

\begin{equation}
C_m(\boldsymbol{a}) = a_1\widetilde{C} + a_2\,,
\end{equation}

where we denote $\boldsymbol{a} = (a_1,a_2)$. As shown in <cit.>, this ansatz is particularly suited to global depolarizing noise and the ansatz can correct the noise exactly. Namely, the $n$-qubit noise channel

\begin{equation}
\rho \xrightarrow[]{\mathcal{D}} (1-p)^L\rho + (1-(1-p)^L)\frac{\id}{2^n}
\end{equation}

can be exactly corrected by using

\begin{equation}
a_1 = \frac{1}{(1-p)^L}\,,\quad a_2=-\frac{(1-(1-p)^L)}{(1-p)^L}\Tr[O]/2^n\,.
\end{equation}

As correction is exact, $\Delta C_m =\Delta C$. It can also be seen that $\Delta \widetilde{C} = (1-p)^L\Delta C$ and the error mitigation cost is $\gamma = 1/(1-p)^{2L}$. This gives $\rchi =1$ for any pair of cost function points.

§.§.§ Relative resolvability between two points with same ansatz applied

Consider any error mitigation strategy that mitigates noisy cost function value $\widetilde{C}(\boldsymbol{\theta})$ by constructing an estimator $C_m(\boldsymbol{\theta})$ of the form
(<ref>). For any two noisy cost function points to which the same ansatz is applied, we have

\begin{equation}
\rchi = 1\,,
\end{equation}

for any noise process.

By applying the same ansatz of the form (<ref>) to two noisy cost function points corresponding to parameter sets $\thv_1, \thv_2$, one can write

\begin{align}
C_m(\thv_1,\boldsymbol{a}) = a_1\widetilde{C}(\thv_1) + a_2\,, \\
C_m(\thv_2,\boldsymbol{a}) = a_1\widetilde{C}(\thv_2) + a_2\,.
\end{align}

This gives $\gamma = a_1$ and $\Delta C_m = a_1\Delta\widetilde{C} $. Thus, substituting these quantities into Definition <ref> one obtains $\rchi=1$ as required.

§ NUMERICAL SIMULATIONS - IMPLEMENTATION DETAILS

We perform our optimizations using the MATLAB implementation of the Nelder-Mead algorithm <cit.>. For each MaxCut graph we perform optimization independently  for $N_i$ random choices of an initial simplex. We evaluate the cost function by performing perfect sampling of the simulated state with $N_s$ shots. After each iteration of the Nelder-Mead algorithm we compute the total cost of optimization per graph $N_{\rm tot}$ by summing the shot budget spent for all $N_i$ instances of the optimization.  To  analyze convergence of results with $N_{\rm tot}$ as shown in Figs. <ref>, <ref> we take as the result of optimization  after $N_{\rm tot}$ shots the best of $N_i$ instances     determined according to the optimized  cost function. The optimization is terminated when  $N_{\rm tot}$ exceeds $1.5\times10^8$.

§.§ CDR-mitigated optimization

We perform CDR-mitigated optimization with $N_i=30$ and $N_s=1024$. We use  training circuits  constructed with a  non-Clifford gates projection algorithm of <cit.>. To construct the  training circuits we decompose $e^{i \gamma_j H_{\rm MaxCut}}$ to native gates of an IBM quantum computer using a decomposition from <cit.>.  In order to account for linear connectivity of the simulated device we use SWAP gates to implement $ e^{-i \gamma_j Z_k Z_l} $ for non  nearest-neighbors terms.    The training circuits contain $100$ near-Clifford circuits with at most $30$ non-Clifford gates. In the case of circuits with  fewer than $60$ non-Clifford gates we construct training circuits with half of the non-Clifford gates replaced by Clifford gates. We evaluate the cost function for the training circuits using perfect sampling and $N_s=1024$ shots. We perform CDR mitigation for each 2-body  term of $H_{\rm MaxCut}$
independently. In general, in order to maximize the quality of the mitigation one should construct the training circuits for each of the terms of $H_{\rm MaxCut}$, for each new set of QAOA angles. Here, for the sake of shot efficiency, for each new set of parameters we compute the training set from scratch only if the 1-norm distance of its QAOA angles  $(\gamma_1,\beta_1, \gamma_2,\beta_2, \dots \gamma_p,\beta_p)$ from the closest point of a simplex is larger than $0.01$.  Otherwise, we use the CDR linear ansatz  for the closest point of the simplex.

For the noisy (no error mitigation) optimization we benchmark various combinations of
$N_i$ and $N_s$ values for $p=4$. In particular we consider increasing $N_s$ for $N_i=30$ and increasing $N_i$ for $N_s=1024$.   We gather the results in Fig. <ref>. We  find that  for $N_{\rm tot}$ ranging from $10^7$ to $1.5\times10^8$ (as considered in  Fig. <ref>) the best results are obtained for $N_i=3000, N_s=1024$.  We use these values for the noisy optimization presented in Fig. <ref>.

Benchmarking various implementations of the noisy optimization for $\boldsymbol{p=4}$ 5-qubit MaxCut QAOA. We plot the approximation ratio averaged over $36$ Max-Cut graphs    chosen randomly from  the Erdös-Rényi ensemble  as a function of $N_{\rm tot}$. We compare the results for various numbers $N_i$ of optimization instances initialized randomly and various numbers of shots $N_s$ per cost function evaluation. As a reference we show the results of CDR optimization for $p=4$.  We consider $N_{\rm tot}=10^7-1.5\times10^8$ as in Fig. <ref>.  Additionally, as in Fig. <ref> we use the approximation ratio computed with the exact energy to benchmark the  optimization, and in the case of $N_i>1$ we choose as the result of optimization the best instance determined according to the optimized cost function. We consider various values of $N_s = 1024, 16384, 262144$ for $N_i=30$ and various values of $N_i=30,150,3000$ for $N_s=1024$. We find that $N_i=3000, N_s=1024$ yields the best results although differences of quality in between most of the noisy optimization implementations are relatively small in comparison to the CDR mitigated optimization.

§.§ Optimization with Virtual Distillation

In this Appendix we compare 5-qubit MaxCut QAOA optimization of VD-mitigated cost function with optimization of the noisy and CDR mitigated cost function for $p=2,4$. We perform the comparison for $36$ randomly chosen graphs  from the Erdös-Rényi ensemble as in Fig. <ref>.  We perform VD mitigation for each expectation value of a 2-site term of $H_{\rm MaxCut}$ according to (<ref>). Therefore, a key assumption is that we neglect derangement noise, which would affect realistic VD implementation on hardware <cit.>. We use the Nelder-Mead algorithm  as described in Appendix <ref>.
We have $N_i=30$ as for CDR simulations from  Fig. <ref> and assign $65536$ shots in order to estimate  $\Tr[\widetilde{\rho}^M Z_iZ_j]$ for each 2-site term of  $H_{\rm MaxCut}$ and  $\Tr[\widetilde{\rho}^M]$. Consequently, the total shot cost of the mitigated cost function estimation is $(n_e+1)\times 65536$, where $n_e$ is the number of Max-Cut graph edges. We consider $M=2,3$ state copies  as  the shot cost of VD mitigation for given precision grows with increasing $M$ <cit.> and $M=2,3$ was shown to be sufficient for typical applications <cit.>. We find that $M=2$ gives better results than $M=3$ as suggested by our analytical results.  Here we allow for $N_{\rm tot}$  up to $1.5\times10^9$, i.e. up to $10$ times more shots than considered  for CDR mitigated and noisy optimization in Fig. <ref>.

We gather the results in Fig. <ref> comparing them with noisy and CDR mitigated optimization from Fig. <ref>.  We find that even with smaller $N_{\rm tot}$ the  noisy and CDR mitigated optimization outperforms the VD  mitigated optimization. This example shows that even for  circuits outside of the NIBP regime there is no guarantee that mitigating an error mitigated cost function  outperforms noisy cost function optimization. We note that  this result does not prohibit  VD-mitigated optimization advantage   for different choices of $N_{\rm tot},  M$
or the shot number per cost function evaluation.

Virtual Distillation mitigated optimization for $\boldsymbol{p=2,4}$, 5-qubit MaxCut QAOA. We plot the approximation ratio averaged over $36$ Max-Cut graphs    chosen randomly from  Erdös-Rényi ensemble  as a function of total shot number $N_{\rm tot}$. The results were obtained with $N_i=30$ initializations and $65536$ shots per $\Tr[\widetilde{\rho}^M Z_i Z_j]$ and $\Tr[\widetilde{\rho}^M]$ estimation. For reference we also present our results of CDR-mitigated and noisy optimization from Fig. <ref>. We observe that for this setting  the optimization with Virtual Distillation does not outperform the noisy or CDR-mitigated optimization.