# Benchmark neutrinoless double-beta decay matrix elements in a light nucleus

R.A.M. Basili  Department of Physics, Iowa State University, Ames, IA 50010,
USA  J.M. Yao  Department of Physics, University of North Carolina, Chapel
Hill, NC 27514, USA  FRIB/NSCL and Department of Physics and Astronomy,
Michigan State University, East Lansing, MI 48824, USA  J. Engel  Department
of Physics, University of North Carolina, Chapel Hill, NC 27514, USA  H.
Hergert  FRIB/NSCL and Department of Physics and Astronomy, Michigan State
University, East Lansing, MI 48824, USA  M. Lockner  Department of Physics,
Iowa State University, Ames, IA 50010, USA  P. Maris  Department of Physics,
Iowa State University, Ames, IA 50010, USA  J.P. Vary  Department of Physics,
Iowa State University, Ames, IA 50010, USA

(August 27, 2024)

###### Abstract

We compute nuclear matrix elements of neutrinoless double-beta decay mediated
by light Majorana-neutrino exchange in the $A=6$ system. The goal is to
benchmark two many-body approaches, the No-Core Shell Model and the Multi-
Reference In-Medium Similarity Renormalization Group. We use the SRG-evolved
chiral N3LO-EM500 potential for the nuclear interaction, and make the
approximation that isospin is conserved. We compare the results of the two
approaches as a function of the cutoff on the many-body basis space. Although
differences are seen in the predicted nuclear radii, the ground-state energies
and neutrinoless double-beta decay matrix elements produced by the two
approaches show significant agreement. We discuss the implications for
calculations in heavier nuclei.

arXiv: xxxx.xxxx

##  I introduction

Since the discovery of the lepton flavor violation in neutrino oscillations
(Ahmad  _et al._ , 2001; Eguchi  _et al._ , 2003; Fukuda  _et al._ , 1998),
identifying whether the neutrino is a Majorana fermion (i.e. its own
antiparticle) has become a priority in nuclear and particle physics. However,
because neutrinos are charge neutral and nearly massless, they are notoriously
difficult to detect, and their properties remain only partly understood. Major
theoretical and experimental collaborative efforts are already underway to
study neutrino properties (Martin-Albo  _et al._ , 2016; Albert  _et al._ ,
2014; Gilliss  _et al._ , 2018; Gando  _et al._ , 2016; Alfonso  _et al._ ,
2015; Agostini  _et al._ , 2017; Aalseth  _et al._ , 2018; Andringa  _et al._
, 2016; Iwata  _et al._ , 2016; Cirigliano  _et al._ , 2017; Contessi  _et
al._ , 2017; Coraggio  _et al._ , 2017; Jiao  _et al._ , 2017; Tiburzi  _et
al._ , 2017; Shanahan  _et al._ , 2017; Horoi and Neacsu, 2017). Determining
whether neutrinos are indeed Majorana particles would not only shed light on
the mechanism behind neutrino mass generation, but would also provide insight
on leptogenesis and the universe’s apparent matter-antimatter asymmetry.

Neutrinoless double-beta ($0\nu\beta\beta$) decay is a hypothetical lepton-
number-violating (LNV) nuclear transition where two neutrons decay to two
protons and two electrons but no anti-neutrinos (or the reverse with leptons
exchanged with their antiparticles). Observing $0\nu\beta\beta$ decay would
confirm the existence of a LNV process, and is commonly viewed as the best
means of learning whether neutrinos are Majorana particles. Experiments
designed to detect $0\nu\beta\beta$ decay in ton-scale amounts of
${}^{76}\text{Ge}$, ${}^{136}\text{Xe}$, and other materials have already put
impressive limits on the $0\nu\beta\beta$-decay half-life (Gilliss  _et al._ ,
2018; Gando  _et al._ , 2016; Agostini  _et al._ , 2017), and these limits
will only become more accurate as additional data is collected. For a more
complete description of current and past efforts as well as some of the
underlying theory, see Refs. Cremonesi and Pavan, 2014; Dell’Oro  _et al._ ,
2016; Engel and Menèndez, 2017; Gomez-Cadenas  _et al._ , 2012; Henning, 2016
and references therein.

While of enormous significance in itself, the experimental detection or non-
detection of $0\nu\beta\beta$ decay will be insufficient to pin down or put
limits on extra-Standard-Model parameters such as the average neutrino mass.
Because the decay rate depends on the $0\nu\beta\beta$-decay nuclear matrix
elements (NMEs), interpreting the experimental results requires the accurate
calculation of those NMEs. However, at present the calculated NMEs in the
heavy nuclei of interest differ by a factor of two to three (Engel and
Menèndez, 2017). In addition, calculated NMEs for $\beta$ decay are usually
smaller than experimental values, and the reasons for these differences are
only now being understood in a quantitative way (Gysbers  _et al._ , 2019). To
shed light on these differences, it is helpful to examine weak processes in
light nuclei, where calculations are better controlled than in the heavy
nuclei we must eventually grapple with. Thus, while not viable for
$0\nu\beta\beta$-decay experiments, light nuclei are a practical option for
benchmarking.

The purpose of this study is to calculate the NMEs (for $0\nu\beta\beta$ decay
mediated by light Majorana-neutrino exchange) in the $A=6$ system.
Benchmarking different many-body methods and identifying important features
that affect the NMEs in these light nuclei will both test the approaches that
we will apply in heavy nuclei and help us anticipate issues that may arise
there. Assessing the convergence behavior of the decay NMEs with increasing
model-space size is of particular importance, as it will help quantify
uncertainties in heavier nuclei where more severe basis truncation is
computationally required. Thus, we consider the ground-state-to-ground-state
$0\nu\beta\beta$ decay of ${}^{6}\text{He}\rightarrow{}^{6}\text{Be}$, which,
while kinematically disallowed, involves the same decay operator that
determines the allowed decay rates in heavy nuclei.

We employ two ab initio many-body approaches: the No-Core Shell Model (NCSM)
and the Multi-Reference In-Medium Similarity Renormalization Group (MR-IMSRG).
The NCSM is a large-scale diagonalization method that yields exact results in
the limit of an infinitely large configuration space. On the other hand, the
MR-IMSRG (a variation of the IMSRG in which the method’s reference state
contains explicitly built-in correlations) yields approximate solutions to the
many-body Schrödinger equation within a systematically improvable truncation
scheme. That is, where the NCSM includes all many-body correlations up to the
given basis cutoff by construction, the MR-IMSRG only includes many-body
correlations up to a cutoff in the many-body expansion. In exchange, the
computational effort of the MR-IMSRG scales much more favorably with particle
number and configuration space size, which makes it capable of modeling both
light and heavy nuclei. While both methods treat all nucleons as active, they
can also be used to generate effective interactions and operators for
traditional Shell-model calculations in heavier nuclei (Bogner  _et al._ ,
2014; Jansen  _et al._ , 2014, 2016; Stroberg  _et al._ , 2016, 2017; Dikmen
_et al._ , 2015; Barrett  _et al._ , 2017).

For both the MR-IMSRG and NCSM calculations performed in this work, we assume
good isospin symmetry to facilitate the comparison of their results, though it
should be noted that we could drop this assumption at the cost of introducing
more complex methods (Song  _et al._ , 2017; Yao  _et al._ , 2018). For both
approaches we adopt the next-to-next-to-next-to-leading chiral order (N3LO)
Entem-Machleidt two-body potential with regulator cutoff $\Lambda=500$ MeV
(referred to as ’N3LO-EM500’) (Entem and Machleidt, 2003; Machleidt and Entem,
2011), to model the nucleon-nucleon (NN) interaction. The potential is
expressed in the harmonic oscillator (HO) basis with energy scale
$\hbar\Omega=20$ MeV, and softened by SRG evolution to the scale of
$\lambda=2.0$ $\text{fm}^{-1}$ (with the relative kinetic energy, $T_{rel}$,
as the generator (Bogner  _et al._ , 2010)) prior to many-body calculations.

Our examination of the $A=6$ system with the NCSM is similar to the studies in
Refs. Cockrell  _et al._ , 2012; Shin  _et al._ , 2017, but differs from both
in: the NN-interaction used, the extrapolations employed, our focus on
$0\nu\beta\beta$ decay, and our comparison with the MR-IMSRG approach. Our
study also offers a point of comparison to the computation of
$0\nu\beta\beta$-decay NMEs arising from an array of LNV mechanisms in light
nuclei by using ab initio Variational Monte-Carlo (VMC) techniques (Pastore
_et al._ , 2018), though our study is distinguished by our use of a different
NN-interaction and our focus solely on $0\nu\beta\beta$ decay mediated by
light Majorana-neutrino exchange.

The rest of this paper is structured as follows: Section II briefly outlines
the derivation of the $0\nu\beta\beta$-decay operator as defined in Refs.
Engel and Menèndez, 2017; Avignone  _et al._ , 2008; Simkovic  _et al._ ,
1999. We provide a brief review of the NCSM in III.1, and the MR-IMSRG in
III.2. Section IV compares the ground-state energy and square radius (in
IV.1), and analyzes the contributions to the total $0\nu\beta\beta$-decay NME
(in IV.2). Finally, Section V reviews our findings and concludes the
discussion. Additional details regarding our extrapolation methods and tables
of calculated values are provided in the appendix.

##  II $0\nu\beta\beta$ Decay with Light Majorana-Neutrinos

Figure 1:  Feynman diagram (modified from Ref. Engel and Menèndez, 2017) for
$0\nu\beta\beta$ decay mediated by light-neutrino exchange. Two neutrons (n)
decay into two protons (p), emitting two electrons ($\text{e}^{-}$). No
neutrinos are emitted, implying that they are Majorana particles
($\nu_{\text{M}}$).

We consider $0\nu\beta\beta$ decay caused by the exchange of the three light
Majorana neutrinos and the Standard-Model weak interaction as depicted in Fig.
(1); all contributions from other LNV processes are neglected.

Drawing on Refs. Engel and Menèndez, 2017; Avignone  _et al._ , 2008 and the
approximations employed there, we write the $0\nu\beta\beta$-decay rate as

$\left[T_{1/2}^{0\nu}\right]^{-1}=G_{0\nu}\left(Q,Z\right)\left|M_{0\nu}\right|^{2}\left|\underset{k}{\sum}m_{k}U_{ek}^{2}\right|^{2}\,,$
(1)

where $Q$ is the difference between initial ($i$) and final ($f$) state
energies, (i.e. $Q\equiv E_{i}-E_{f}$), $Z$ is the proton number of the final
nucleus, $m_{k}$ is the Majorana mass eigenvalue, and $U_{ek}$ is the element
of the neutrino mixing matrix that connects the electron neutrino with mass
eigenstate $k$. $G_{0\nu}\left(Q,Z\right)$ comes from the phase-space
integral, which has been evaluated with improved precision in Refs. Kotila and
Iachello, 2012; Stoica and Mirea, 2013.

In this study, we focus on the ${}^{6}\text{He}\rightarrow{}^{6}\text{Be}$
ground-state-to-ground-state NME, $M_{0\nu}$ (Simkovic  _et al._ , 1999; Horoi
and Stoica, 2010; Rodin  _et al._ , 2006), obtained from the
$0\nu\beta\beta$-decay many-body operator, $O_{0\nu}$, as

$M_{0\nu}=\langle^{6}\text{Be}|O_{0\nu}|^{6}\text{He}\rangle.$ (2)

Our notation follows that of Ref. Suhonen, 2007 unless specified otherwise.

###  II.1 The $0\nu\beta\beta$-decay Matrix Elements

The many-body operator $O_{0\nu}$ is conventionally divided into three
contributions, labeled Fermi, Gamow-Teller (GT), and tensor. We use the symbol
$O$ to generically denote any one of these contributions’ corresponding two-
body operator, which may always be written in second-quantized form as

$\displaystyle O$
$\displaystyle=\frac{1}{4}\underset{\alpha\beta\gamma\delta}{\sum}\langle\alpha\beta|O|\gamma\delta\rangle
a_{\alpha}^{\dagger}a_{\beta}^{\dagger}a_{\delta}a_{\gamma}$ (3)

where $a^{\dagger}$ and $a$ create and annihilate nucleons, respectively, in
single-particle states. A given single-particle state $\alpha$ is defined by
the quantum numbers $n_{\alpha}$, $l_{\alpha}$, $s_{\alpha}$, $j_{\alpha}$,
$t_{\alpha}$, $m_{j\alpha}$, and $m_{t\alpha}$, which correspond to the
radial, angular momentum, spin, total angular momentum, isospin, angular
momentum projection, and isospin projection, respectively. Greek indices
$\alpha,\beta,\gamma,\delta$ are used to denote single-particle states, while
the corresponding Roman indices $a,b,c,d$ refer to the reduced set of quantum
numbers, such that $a_{\alpha}^{\dagger}\equiv
a_{a,m_{j\alpha},m_{t\alpha}}^{\dagger}$. We define spherical tensor/isotensor
versions of the annihilation operators as

$\displaystyle\hat{a}_{\delta}$
$\displaystyle\equiv\left(-1\right)^{j_{\delta}+m_{j\delta}+\frac{1}{2}+m_{t\delta}}a_{d,-m_{j\delta},-m_{t\delta}}\,,$
(4)

such that

$a_{\delta}a_{\gamma}=\left(-1\right)^{j_{\gamma}+j_{\delta}+m_{j\gamma}+m_{j\delta}+1}\hat{a}_{c,-m_{j\gamma},-m_{t\gamma}}\hat{a}_{d,-m_{j\delta},-m_{t\delta}}\,.$
(5)

For the ground-state-to-ground-state portion of the
${}^{6}\text{He}\rightarrow^{6}\text{Be}$ transition, we may narrow our scope
to components of the two-body operators that contribute to $0^{+}\rightarrow
0^{+}$ NMEs. Expanding Eq. (3) into doubly-reduced tensorial components in the
$JT$-coupled two-body isospin representation yields for this transition

$\displaystyle O_{0,-2}^{0,2}$
$\displaystyle=-\frac{1}{4\sqrt{3}}\underset{abcd}{\sum}\underset{J}{\sum}\left(\mathcal{N}_{ab}\left(J,1\right)\mathcal{N}_{cd}\left(J,1\right)\right)^{-1}$
$\displaystyle\cdot(ab;J\,1|||O^{0,2}|||cd;J\,1)$
$\displaystyle\cdot\left[\left[a_{a}^{\dagger}a_{b}^{\dagger}\right]^{J,1}\left[\hat{a}_{c}\hat{a}_{d}\right]^{J,1}\right]_{0,-2}^{0,2}\,,$
(6)

where brackets denote tensor products with tensor, isotensor couplings in
superscripts and their corresponding projections in subscripts,
$\mathcal{N}_{ij}(J,T)\equiv\sqrt{1-\delta_{ij}(-1)^{J+T}}/(1+\delta_{ij})$ is
an antisymmetrization factor, and the triple lines ’|||’ denote doubly-reduced
two-body matrix elements (TBMEs). In Eq. (6) we implicitly include only two-
body states that satisfy the Pauli exclusion principle in the sum over nucleon
states (or, effectively, we only consider values of $i$, $j$, $J$, and $T$
such that $\mathcal{N}_{ij}(J,T)\neq 0$).

We express the total NME ($M_{0\nu}$) as the sum of the Fermi
($M_{0\nu}^{F}$), GT ($M_{0\nu}^{GT}$), and tensor ($M_{0\nu}^{T}$)
contributions

$M_{0\nu}=M_{0\nu}^{F}+M_{0\nu}^{GT}+M_{0\nu}^{T}\,.$ (7)

These three NME contributions are developed for the many-body initial and
final nuclear states from the doubly-reduced TBMEs of the three corresponding
two-body operators. We evaluate the NMEs by summing the two-body contribution
from each unique pair of the system’s nucleons. We calculate the TBMEs with
the two-body operators

$\displaystyle O_{0\nu}^{F}\left(r\right)$ $\displaystyle=\frac{4R}{\pi
g_{A}^{2}}\int_{0}^{\infty}|\mathbf{q}|d|\mathbf{q}|\frac{j_{0}(|\mathbf{q}|r)h_{F}(|\mathbf{q}|)}{|\mathbf{q}|+\bar{E}-(E_{i}+E_{f})/2}\tau_{1}^{+}\tau_{2}^{+}\,,$
$\displaystyle O_{0\nu}^{GT}\left(r\right)$ $\displaystyle=\frac{4R}{\pi
g_{A}^{2}}\int_{0}^{\infty}|\mathbf{q}|d|\mathbf{q}|\frac{j_{0}(|\mathbf{q}|r)h_{GT}(|\mathbf{q}|)\boldsymbol{\sigma}_{1}\cdot\boldsymbol{\sigma}_{2}}{|\mathbf{q}|+\bar{E}-(E_{i}+E_{f})/2}\tau_{1}^{+}\tau_{2}^{+}\,,$
$\displaystyle O_{0\nu}^{T}\left(r\right)$ $\displaystyle=\frac{4R}{\pi
g_{A}^{2}}\int_{0}^{\infty}|\mathbf{q}|d|\mathbf{q}|\frac{j_{2}(|\mathbf{q}|r)h_{T}(|\mathbf{q}|)\boldsymbol{S}_{12}}{|\mathbf{q}|+\bar{E}-(E_{i}+E_{f})/2}\tau_{1}^{+}\tau_{2}^{+}\,,$
(8)

where $\boldsymbol{q}$ is the momentum transfer,
$r=|\boldsymbol{r}_{1}-\boldsymbol{r}_{2}|$ is the magnitude of the inter-
nucleon position vector, and $\hat{\boldsymbol{r}}$ is the corresponding unit
vector. Additionally, $\boldsymbol{r}_{1\backslash 2}$,
$\boldsymbol{\sigma}_{1\backslash 2}$, and $\tau_{1\backslash 2}^{+}$
respectively denote the labeled nucleon’s position operator, spin operator,
and isospin raising operator (transforming neutrons to protons), while
$\boldsymbol{S}_{12}=3\boldsymbol{\sigma}_{1}\cdot\hat{\boldsymbol{r}}\boldsymbol{\sigma}_{2}\cdot\hat{\boldsymbol{r}}-\boldsymbol{\sigma}_{1}\cdot\boldsymbol{\sigma}_{2}$
is the tensor operator. The NMEs contain $r$-dependence through the spherical
Bessel functions $j_{0}$ and $j_{2}$ in Eq. (8), and, for several heavy parent
nuclei, have been shown to vanish at small distances $r$, fall off like
$\nicefrac{{1}}{{r}}$ at large distance, and have a typical range of a few
femtometers (fm) (Simkovic  _et al._ , 2008). Hence, we expect good
convergence with the basis space for these operators in our calculations.

The neutrino potentials, $h$, are defined in momentum space as

$\displaystyle h_{F}\left(|\boldsymbol{q}|\right)$ $\displaystyle\equiv-
g_{V}^{2}(\boldsymbol{q}^{2})\,,$ $\displaystyle
h_{GT}\left(|\boldsymbol{q}|\right)$ $\displaystyle\equiv
g_{A}^{2}(\boldsymbol{q}^{2})-\frac{g_{A}(\boldsymbol{q}^{2})g_{P}(\boldsymbol{q}^{2})\boldsymbol{q}^{2}}{3m_{N}}+\frac{g_{P}^{2}(\boldsymbol{q}^{2})\boldsymbol{q}^{4}}{12m_{N}^{2}}+\frac{g_{M}^{2}(\boldsymbol{q}^{2})\boldsymbol{q}^{2}}{6m_{N}^{2}}\,,$
$\displaystyle h_{T}\left(|\boldsymbol{q}|\right)$
$\displaystyle\equiv\frac{g_{A}(\boldsymbol{q}^{2})g_{P}(\boldsymbol{q}^{2})\boldsymbol{q}^{2}}{3m_{N}}-\frac{g_{P}^{2}(\boldsymbol{q}^{2})\boldsymbol{q}^{4}}{12m_{N}^{2}}+\frac{g_{M}^{2}(\boldsymbol{q}^{2})\boldsymbol{q}^{2}}{12m_{N}^{2}}\,,$
(9)

where
$g_{M}\left(q^{2}\right)=\left(1+\kappa_{1}\right)g_{V}\left(q^{2}\right)\simeq
4.706g_{V}\left(q^{2}\right)$ (with the anomalous nucleon isovector magnetic
moment $\kappa_{1}=3.706$), and the Goldberger–Treiman relation
$g_{P}\left(q^{2}\right)=2m_{N}g_{A}\left(q^{2}\right)/\left(\boldsymbol{q}^{2}+m_{\pi}^{2}\right)$
(with nucleon mass $m_{N}$ and pion mass $m_{\pi}$) connects the pseudoscalar
and axial terms (Simkovic  _et al._ , 2009; Engel and Menèndez, 2017). The
conservation of the vector current implies that
$g_{V}\equiv\left.g_{V}\left(q^{2}\right)\right|_{0}=1$, while the value
$g_{A}\equiv\left.g_{A}\left(q^{2}\right)\right|_{0}\simeq 1.27$ may be
extracted from neutron $\beta$-decay measurements. Their momentum transfer
dependence is
$g_{V}\left(q^{2}\right)=g_{V}\left(1+q^{2}/\Lambda_{V}^{2}\right)^{-2}$ and
$g_{A}\left(q^{2}\right)=g_{A}\left(1+q^{2}/\Lambda_{A}^{2}\right)^{-2}$ where
$\Lambda_{V}=850$ MeV and $\Lambda_{A}=1040$ MeV are the vector and axial
masses, respectively. The nuclear radius $R=1.2A^{\frac{1}{3}}\approx 2.2$ fm
is inserted by convention to make the matrix elements dimensionless, with a
compensating factor absorbed into $G_{0\nu}$ in Eq. (1). Finally, $\bar{E}$ is
an estimate of the average intermediate-state energy, the choice of which has
been shown to have only a mild influence on the decay amplitude (Avignone  _et
al._ , 2008). We employ the value $\bar{E}-(E_{i}+E_{f})/2\equiv
5\,\text{MeV}$ throughout this work.

In other prescriptions, the operators defined by Eq. (8) are sometimes
multiplied by an additional radial function, $f(r)$, designed to take into
account short-range correlations that are omitted by Hilbert-space truncations
performed in the many-body calculations (Miller and Spencer, 1976; Roth  _et
al._ , 2005; Brueckner, 1955; Benhar  _et al._ , 2014; Muther and Polls,
2000). In this work, we assume all relevant nucleon-nucleon correlations are
embedded in the many-body wavefunctions generated in our NCSM and MR-IMSRG
model spaces and employ no additional radial function. We numerically
integrate the inner products of the operators in Eq. (8) using relative HO
states to obtain reduced matrix elements in the relative basis. These elements
are then converted to M-scheme TBMEs via a Moshinsky transformation (Tobocman,
1981; Barrett  _et al._ , 2013) before being employed in many-body
calculations.

###  II.2 $0\nu\beta\beta$ decay in ${}^{6}\text{{He}}$ with Isospin Symmetry

When considering isovector operators, a common challenge shared by many
$ab\,initio$ nuclear approaches (particularly those relying on finite matrix
methods) arises when the initial and final nuclei are not the same, as the
many-body spaces for the two will generally differ. In NCSM calculations, this
problem usually requires the many-body eigenstate wavefunctions of the two
systems to be calculated independently. In the MR-IMSRG, two different unitary
transformation operators must be constructed, one for the initial nucleus and
one for the final nucleus.

While solutions to this problem have been developed for the NCSM and have been
implemented for the MR-IMSRG (Yao  _et al._ , 2018), a careful choice of
transition can circumvent the issue when isospin conservation is a good
approximation. Thus, we assume that isospin symmetry is obeyed in the mirror
nuclei 6He and 6Be.

The ground states of ${}^{6}{\rm Be}$ and ${}^{6}{\rm He}$ are characterized
by total angular momentum $\mathcal{J}=0$ and isospin $\mathcal{T}=1$, with
projections $\mathcal{T}_{z}=-1,+1$, respectively. If isospin symmetry is
obeyed, the two-body density of Eq. (6) may be rewritten in terms of the
${}^{6}\text{He}$ two-body density alone as

$\displaystyle\langle^{6}\text{Be}|\left[\left[a_{a}^{\dagger}a_{b}^{\dagger}\right]^{J,1}\left[\hat{a}_{c}\hat{a}_{d}\right]^{J,1}\right]_{0,-2}^{0,2}|^{6}\text{He}\rangle$
$\displaystyle=\sqrt{6}\langle^{6}\text{He}|\left[\left[a_{a}^{\dagger}a_{b}^{\dagger}\right]^{J,1}\left[\hat{a}_{c}\hat{a}_{d}\right]^{J,1}\right]_{0,0}^{0,2}|^{6}\text{He}\rangle.$
(10)

##  III benchmarked methods

Both the NCSM and MR-IMSRG can provide accurate results when applied in light
nuclei. The MR-IMSRG has the advantage that, with suitable approximations, it
can be applied in heavier systems (Bogner  _et al._ , 2014; Stroberg  _et al._
, 2016, 2017). For the NCSM one may envision applications in heavier systems
by merging it with renormalization approaches or by introducing an inert core
and deriving effective interactions for valence-space Shell model calculations
(see e.g., Refs. Jansen  _et al._ , 2014; Dikmen  _et al._ , 2015; Jansen  _et
al._ , 2016; Barrett  _et al._ , 2017). The approximations involved in these
envisioned approaches to heavier nuclei will also require benchmarking.

Both methods consider the $A$-body nuclear Hamiltonian, $H$, consisting of a
relative kinetic-energy term and interaction terms, i.e.

$H=\frac{1}{2Am_{N}}\underset{i<j}{\overset{A}{\sum}}\left(p_{i}-p_{j}\right)^{2}+V_{NN}+V_{NNN}+...$
(11)

where $m_{N}$ is the average nucleon mass, $V_{NN}$ is the NN-interaction, and
$p_{i}$ denotes the momentum of nucleon $i$. We follow the convention for two-
body operators where summations over nucleon pairs are performed under the
ordering given by $i<j$ to avoid counting the same pair twice. The term
$V_{NNN}$ denotes three-body interactions, also called 3-nucleon forces
(3NFs), which may be supplemented by higher-body interactions. Although
studies have demonstrated that 3NFs can have a significant impact on
calculated nuclear observables (Barrett  _et al._ , 2013), their inclusion
would greatly increase computational cost and is thus deferred to future
efforts. We therefore consider here only the NN-interactions from N3LO-EM500
(Entem and Machleidt, 2003; Machleidt and Entem, 2011), which is charge-
dependent.

###  III.1 No-Core Shell Model

The NCSM (Barrett  _et al._ , 2013) is a configuration-interaction (CI)
approach in which the many-body basis states, $|\Phi\rangle$, are expressed as
Slater determinants of single-particle states occupied by the system’s
nucleons, or

$|\Phi\rangle=\mathcal{\mathcal{A}}\left[\underset{i}{\prod}|\phi_{\alpha_{i}}\rangle\right]\,,$
(12)

where $|\phi_{\alpha_{i}}\rangle$ denotes a single-particle state with quantum
numbers $\alpha_{i}$ occupied by nucleon $i$, and $\mathcal{A}$ is an
antisymmetrization operator that carries both the sign permutations of the
determinant and an overall normalization factor. Our NCSM approach features
separate Slater determinants for the neutrons and protons, and the resulting
many-body basis is specific to the nucleus under consideration. For a given
application, we form total Slater determinants of fixed parity and fixed total
angular momentum projection $M_{J}$.

The infinite HO basis (with energy scale fixed by the usual parameter
$\hbar\Omega$) is the conventional choice of single-particle basis and is used
in this work. Additional details on the HO basis functions may be found in
Ref. Barrett  _et al._ , 2013.

The nuclear many-body wavefunctions, $\Psi\left(r_{1},...,r_{A}\right)$,
satisfy the A-body Schrödinger equation and are obtained by solving the
Hamiltonian matrix eigenvalue problem

$H|\Psi\rangle=E|\Psi\rangle$ (13)

where $E$ is the eigenenergy of nuclear state $|\Psi\rangle$. Beginning with
the kinetic-energy and interaction TBMEs in the HO basis, one constructs the
$A$-body Hamiltonian matrix elements in the many-body basis as
$\langle\Phi_{\mu}|H_{A}|\Phi_{\nu}\rangle$, where the indices $\mu$ and $\nu$
label the many-body basis states. The many-body eigenstates are then linear
combinations of many-body basis states:

$|\Psi\rangle=\underset{\mu}{\overset{\infty}{\sum}}c_{\mu}|\Phi_{\mu}\rangle$
(14)

where $c_{\mu}$ are the normalized coefficients of the many-body basis states
$|\Phi_{\mu}\rangle$. For practical calculations, the infinite many-body basis
requires truncation, which one controls by using a basis cutoff parameter. For
NCSM calculations performed in this study, we employ the cutoff parameter
$N_{\text{max}}$, which denotes the maximum number of HO excitation quanta
allowed in the many-body basis above the minimum number required by the Pauli
principle (Barrett  _et al._ , 2013).

Solving Eq. (13) with the resulting finite many-body Hamiltonian then becomes
a large (but generally sparse) matrix eigenvalue problem. We obtain the
solution with the hybrid OpenMP/MPI CI code Many Fermion Dynamics for nucleons
(MFDn). The code is optimized for solving the large sparse matrix eigenvalue
problem by using a Lanczos-like algorithm to determine the desired lowest-
lying energy eigenvalues and corresponding eigenvectors. The eigenvectors are
then used with other operator matrix elements to calculate that operator’s
expectation values during post-processing. For more details on MFDn, see Refs.
Maris  _et al._ , 2010; Aktulga  _et al._ , 2014; Shao  _et al._ , 2017.

By solving the system in a sequence of increasingly large bases, one can
extrapolate to the result when using the complete basis (i.e. when the matrix
dimension of $H$ goes to infinity and the calculation becomes exact). Any
other observable can also, in principle, be extrapolated to this limit, and
such extrapolations are a distinguishing feature of No-Core Full-Configuration
(NCFC) studies (Maris  _et al._ , 2009).

###  III.2 Multi-Reference In-Medium Similarity Renormalization Group

Here we provide a brief overview of the MR-IMSRG; a more complete description
may be found in Refs. Hergert  _et al._ , 2016; Hergert, 2017; Hergert  _et
al._ , 2018. For an initial Hamiltonian $H$, the flow equation

$\dfrac{dH(s)}{ds}=[\eta(s),H(s)]\,,$ (15)

determines a unitary transformation of the Hamiltonian. Here $\eta$ is called
the generator of scale transformations and $s$ is the flow parameter, defined
such that $H(s)\mid_{s=0}$ is just $H$. The ground-state energy is simply
given by the expectation value of the evolved Hamiltonian $H(s)$ in the
reference state. Instead of solving the set of differential equations for
$H(s)$ in Eq. (7), one can solve a similar flow equation for the unitary
transformation operator $U(s)$,

$\frac{dU\left(s\right)}{ds}=\eta\left(s\right)U\left(s\right)\,,$

whose solution can formally be written in terms of the ${\cal S}$-ordered
exponential

$U\left(s\right)=\mathcal{S}\,\text{exp}\int_{0}^{s}ds^{\prime}\eta\left(s^{\prime}\right)\,,$
(16)

which is short-hand for the Dyson series expansion of $U(s)$. As shown first
by Magnus, it is possible to rewrite the unitary transformation operator as
$U(s)\equiv e^{\Omega(s)}$, a step that transforms the equation for $U(s)$
into one for $\Omega$ (Morris  _et al._ , 2015):

$\frac{d\Omega\left(s\right)}{ds}=\underset{n=0}{\overset{\infty}{\sum}}\frac{B_{n}}{n!}\left[\Omega\left(s\right),\eta\left(s\right)\right]^{\left(n\right)}\,.$
(17)

The nested commutators in this equation are given by

$\displaystyle\left[\Omega\left(s\right),\eta\left(s\right)\right]^{\left(0\right)}$
$\displaystyle=\eta\left(s\right)\,,$ (18a)
$\displaystyle\left[\Omega\left(s\right),\eta\left(s\right)\right]^{\left(n\right)}$
$\displaystyle=\left[\Omega\left(s\right),\left[\Omega\left(s\right),\eta\left(s\right)\right]^{\left(n-1\right)}\right]\,,$
(18b)

and $B_{n=0,1,2,\cdots}$ are the Bernoulli numbers $\\{1,-1/2,1/6,\cdots\\}$.

The expectation value of any operator $O$ is then given by
$\langle\Phi|O\left(s\right)|\Phi\rangle=\langle\Phi|e^{\Omega(s)}O{}^{-\Omega(s)}|\Phi\rangle$,
and can be evaluated with the Baker-Campbell-Hausdorff formula:

$e^{\Omega(s)}Oe^{-\Omega(s)}=\underset{n=0}{\overset{\infty}{\sum}}\frac{1}{n!}\left[\Omega\left(s\right),O\right]^{\left(n\right)}.$
(19)

In the MR-IMSRG calculations performed here, we express all operators in
normal-ordered form with respect to a reference state $|\Phi\rangle$ in order
to control the proliferation of induced terms. We keep up to normal-ordered
two-body operators throughout the calculation, in accordance with the MR-
IMSRG(2) truncation described in Ref. Hergert  _et al._ , 2018. We use
particle-number-projected HFB quasiparticle vacua as reference states, and
adopt the Brillouin generator (Hergert, 2017). We numerically solve the flow
equation for values of $s$ large enough so that the solutions are very close
to their asymptotic limits. The underlying Hamiltonian that defines both the
projected HFB reference state and the starting point for the flow equation is
determined by using the same TBMEs in the single-particle HO basis that are
used in our NCSM calculations. However, unlike the NCSM, the MR-IMSRG is
formulated in the natural orbital basis of the reference state. Since the
reference state results from a projected HFB calculation in a HO basis, the
MR-IMSRG effectively explores a configuration space controlled by the cutoff
parameter $e_{\text{max}}$, which denotes the maximum number of energy quanta
that the HO components of any natural orbital can have. In effect, for a given
cutoff $e_{\text{max}}$, the MR-IMSRG many-body basis will include single-
particle excitations up to $e_{\text{max}}$ (i.e. one-particle-one-hole, or
1p1h), two-particle excitations (i.e. 1p1h+1p1h or 2p2h) up to
$2e_{\text{max}}$, uncorrelated three-body excitations (i.e. 1p1h + 1p1h +
1p1h or 1p1h + 2p2h) up to $3e_{\text{max}}$, and so on.

##  IV results and discussion

Here we discuss the results of the NCSM and MR-IMSRG calculations. We provide
graphical representations of the results, as functions of the basis cutoff
parameters, to analyze the convergence of the operators at the chosen basis
scale of $\hbar\Omega=20$ MeV. Throughout, we use solid dots to represent NCSM
results and open boxes to represent MR-IMSRG results. Similarly, we use solid
lines to denote extrapolations of the NCSM results and dashed lines to denote
extrapolations of the MR-IMSRG results.

In order to compare the convergence behavior of results from the NCSM and MR-
IMSRG, we must consider the differences in their truncation schemes. We recall
that the NCSM’s cutoff parameter $N_{\text{max}}$ denotes the total number of
allowed $excitation$ quanta in the system, and $e_{\text{max}}$ denotes the
maximum number of allowed energy quanta possessed by any single nucleon. Since
in ${}^{6}\text{He }$ at a given $N_{\text{max}}$ the highest number of quanta
possessed by any single-particle state will be $N_{\text{max}}+1$, we equate
the two cutoffs with the assignment $e_{\text{max}}\equiv N_{\text{max}}+1$
for our comparison. While this assignment is not exact, it ensures that for a
given pair of matched cutoffs, we use identical single-particle bases under
both truncation schemes. Moreover, our use of this assignment to compare the
results does not preclude their examination from other perspectives. Instead,
we merely offer this assignment as a reasonable vehicle to present our results
graphically.

We extrapolate our results to obtain predictions of observables at the
continuum limit and to better examine their convergence behavior; the
functional forms and other details of these extrapolations are provided in the
appendix. We extrapolate our NCSM and MR-IMSRG results for energy and square
radii with formulae (Eq. (21) and Eq. (22), respectively) inspired by those
provided in Refs. Furnstahl  _et al._ , 2012; Maris  _et al._ , 2009. Although
these extrapolations were originally designed with the $N_{\text{max}}$
truncation scheme in mind, there is good reason from a theoretical perspective
to expect that the same extrapolation forms effective for the NCSM will be
effective for the results of IMSRG calculations (Hergert  _et al._ , 2016).
Meanwhile, as is the case for many nonscalar operator observables (with the
exception of those in significant investigations on extrapolating E2
observables (Odell  _et al._ , 2016)), precision extrapolation approaches for
$0\nu\beta\beta$-decay observables remain largely unexplored. Guided by the
similarities of the observable’s $r$-dependence seen in Ref. (Simkovic  _et
al._ , 2008) to that of nuclear interactions, we employ the same simple
exponential form applied for the energy to extrapolate the
$0\nu\beta\beta$-decay contributions. While we acknowledge that a thorough
investigation of extrapolating $0\nu\beta\beta$-decay NMEs is warranted for
refined predictions and accurate uncertainty estimates, we find that this form
provides an adequate fit and proves sufficient for this comparative study.

To facilitate our discussion of convergence, we refer to the speed (with
respect to the cutoff parameter) at which an eigenvalue result approaches its
asymptotic value as the result’s “convergence rate”. We gauge the convergence
rate with the value of $N_{\text{max}}$ ($e_{\text{max}}$) at which the
extrapolation is within $5\%$ of its value at the continuum limit, and denote
this generally non-integer value $\widetilde{N}_{5\%}$
($\widetilde{e}_{5\%}$). Though this metric relies heavily on the validity of
the extrapolation, it provides a functional estimate for both the relative
convergence speeds between results and the cutoffs required for reaching well-
converged values.

Finally, in the interest of understanding what the differences between the
extrapolated results of the two $ab\,initio$ calculations signify, we briefly
consider the general $A$-body system. For such a system, the untruncated MR-
IMSRG calculation would include all many-body correlations, and would
therefore provide identical results (within numerical noise) as the NCSM at
the continuum limit. By performing only the MR-IMSRG(2) calculation, we expect
the two approaches’ results to converge to different values that depend on how
significant the neglected three-body (up to $A$-body) correlations are to the
observable in question. Thus, beyond the mild uncertainty introduced by the
extrapolation, differences between the extrapolated results estimate the
significance of many-body correlations neglected by the MR-IMSRG(2)
calculation.

###  IV.1 Ground-State Energy and Nuclear Square Radius

Figure 2:  Ground-State energy (a) and neutron (light blue), proton (light
red), and matter (gray) square radii ($r_{n}^{2}$, $r_{p}^{2}$, and
$r_{m}^{2}$ respectively) (b) of ${}^{6}\text{He}$ with varying basis cutoff
parameter from NCSM (solid circle) and MR-IMSRG(2) (open square) ab initio
calculations. Solid and dashed lines denote the NCSM and MR-IMSRG
extrapolations, respectively. The realistic N3LO-EM500 potential with energy
scale $\hbar\Omega=20$ MeV and SRG evolution scale $\lambda=2.0$
$\text{fm}^{-1}$ is used in all cases. The asymptotic fit parameter, $\langle
r^{2}\rangle_{\infty}$, of the NCSM square radius extrapolations are listed in
the legend of (b) (see Eq. (22) for extrapolation definition). Fit parameters
and plotted values are listed in Table (1) for energy, and in Table (2) for
the square radii.

The initial system is the ${}^{6}\text{He}$ nucleus in its ground state. The
calculated ground-state energy and neutron, proton, and matter square radii
($r_{n}^{2}$, $r_{p}^{2}$, and $r_{m}^{2}$ respectively) varying with basis
truncation are shown in panels (a) and (b) of Fig. (2) respectively.

The NCSM ground-state energy extrapolation has converged to within $5\%$ of
its asymptotic value of -29.132 MeV by $\widetilde{N}_{5\%}\sim 7.9$. The MR-
IMSRG(2) extrapolated energy converges somewhat faster by comparison, with
$\widetilde{e}_{5\%}\sim 4.9$ and the asymptotic value of -28.472 MeV (around
2.3% higher than the NCSM result). Both extrapolated ground-state energies are
underbound compared to the experimental result of -29.272 MeV (Riisager  _et
al._ , 1990), as well as the extrapolated results of -30.0(1) MeV and -29.87
MeV from two similar (but independent) NCFC calculations of the
${}^{6}\text{He}$ ground-state (Bogner  _et al._ , 2008; Furnstahl  _et al._ ,
2012) that used only the charge-independent parts of our strong-interaction
Hamiltonian.

Performing the same calculations for ${}^{4}\text{He}$ yields extrapolated
binding energies of 28.305 and 28.316 MeV for the NCSM and MR-IMSRG(2),
respectively. Subtracted from our ${}^{6}\text{He}$ binding energy results,
this corresponds to 2$n$-separation energies of 0.827 MeV and 0.156 MeV. The
experimental 2n-separation energy of ${}^{6}\text{He}$, by comparison, is
approximately 0.975 MeV (Wang  _et al._ , 2017). The difference between these
results is most likely a consequence of the truncations inherent to the MR-
IMSRG(2). While the method probes a larger space of 2p2h excitations than the
NCSM for a given pair of matched $N_{\text{max}}$ and $e_{\text{max}}$ single-
particle bases, the MR-IMSRG(2) misses correlation energy from 3p3h and higher
excitations that are included in the NCSM. We expect such correlations to play
a more important role in a nucleus with a complex structure, like
${}^{6}\text{He}$, than in a compact nucleus like ${}^{4}\text{He}$. We will
analyze this issue in more detail using improved MR-IMSRG truncations (Hergert
_et al._ , 2018) in the future.

Comparing the convergence rates of the square radii results between
approaches, we see the MR-IMSRG(2) (NCSM) square radii consistently converge
faster (slower), with $\widetilde{e}_{5\%}\sim 14,\,3.5,\,12$
($\widetilde{N}_{5\%}\sim 72,\,30,\,63$) for neutron, proton, and matter
square radii, respectively. This large difference in convergence speed
primarily results from the use of natural orbitals in the MR-IMSRG. However,
we observe the MR-IMSRG(2) results converge to a roughly $34\%,$ $21\%,$ and
$31\%$ smaller value than the NCSM results for the corresponding square radii.
All radii share the slower convergence rate relative to the ground-state
energy that is commonly associated with the $r^{2}$ operator; a consequence of
coming from an effective operator with sensitivity to correlations outside the
characteristic length scale of the chosen HO basis (Barrett  _et al._ , 2013;
Cockrell  _et al._ , 2012; Furnstahl  _et al._ , 2014, 2015; Coon  _et al._ ,
2012). In the NCSM (and to a lesser degree MR-IMSRG), the slow convergence
reflects the basis regularization of the infrared (IR) momentum region from
the HO basis truncation. In effect, because the basis’s length scale is chosen
to favor convergence in energy, the basis requires higher cutoffs to fully
capture the longer-range correlations of the $r^{2}$ operator. The
significantly faster convergence speed of the proton square radius (compared
to those of the neutron and matter square radii) is a consequence of this
effect, as the protons predominantly remain in the core of the 6He ground-
state halo structure (Riisager, 2013). That is, since the protons are only
found in the four-nucleon core, the proton square radius operator correlations
primarily exist at the shorter distances pertinent to the core, and are thus
better encompassed by the scales of the chosen basis.

The benefit of the MR-IMSRG’s renormalization can be seen in the improved
convergence observed in its results. In essence, the renormalization decouples
the NN-correlations existing outside the scales encompassed by the basis, and
distributes those correlations inside those scales. The drawback is that some
induced many-body forces must be neglected in the process, an approximation
that would explain the notable differences seen in the extrapolated square
radii. In the case of a light nucleus such as ${}^{6}\text{He}$ where a
convergence trend can be established, we expect the NCSM extrapolated radii to
be more accurate than those of the MR-IMSRG(2) for the given potential.
Consequently, we conjecture that the smaller MR-IMSRG(2) square radii reflect
meaningful induced many-body correlations that are being lost through the MR-
IMSRG(2) many-body truncation. Specifically, the fast convergence of the MR-
IMSRG(2) results suggests that the 1p1h and 2p2h correlations relevant to
$r^{2}$ are well-accounted for by $e_{\text{max}}=12$ and 24, respectively,
and that the remaining differences with the NCSM results are from higher many-
body correlations omitted by the MR-IMSRG(2) approach.

###  IV.2 $0\nu\beta\beta$-decay Matrix Element

Figure 3:  Ground-state-to-ground-state $0\nu\beta\beta$-decay NME (a) for
${}^{6}\text{He}\rightarrow{}^{6}\text{Be}$ , decomposed into its Fermi (b),
GT (c), and tensor (d) contributions, as a function of the basis cutoff for
NCSM (solid circle) and MR-IMSRG(2) (open square) ab initio calculations.
Solid lines denote NCSM extrapolations. Each vertical axis is expanded for
visibility. The three contributions add to the total as specified by Eq. (7)).
Insets provide the percent difference in magnitude (open triangles) between
the MR-IMSRG results and the NCSM extrapolation as described in the text.
Plotted values and fit parameters are listed in Table (1).

We turn finally to the ground-state-to-ground-state
${}^{6}\text{He}\rightarrow{}^{6}\text{Be}$ $0\nu\beta\beta$-decay NME. As
already mentioned, we assume isospin symmetry so that the initial and final
state are described by the same wavefunction (except for an interchange of
protons and neutrons). We present our results in Fig. (3), where we recall
that discrete points represent results of many-body calculations while lines
represent fits specified by Eq. (21). We decompose the total NME in (a) into
its Fermi, GT, and tensor contributions from Eq. (7) in panels (b), (c) and
(d), respectively. Insets provide estimates for the percent difference,
$\Delta_{\%}$, between results of the two methods within our mapping of their
basis truncation schemes (we omit such estimates for the numerically less
significant tensor contribution). For a given $e_{\text{max}}$, we calculate
these values as

$\left(\Delta_{\%}\right)_{e_{\text{max}}}=200\left|\frac{\left(M_{0\nu}^{\text{IMSRG}}\right)_{e_{\text{max}}}-f\left(\left[N_{\text{max}}\right]_{e_{\text{max}}}\right)}{\left(M_{0\nu}^{\text{IMSRG}}\right)_{e_{\text{max}}}+f\left(\left[N_{\text{max}}\right]_{e_{\text{max}}}\right)}\right|\,,$
(20)

where $\left(M_{0\nu}^{\text{IMSRG}}\right)_{e_{\text{max}}}$ is the NME
result of the MR-IMSRG(2) calculation with cutoff $e_{\text{max}}$, and
$f\left(N_{\text{max}}\right)$ is the NCSM fit described by Eq. (21) evaluated
at the mapped cutoff value
$\left[N_{\text{max}}\right]_{e_{\text{max}}}=e_{\text{max}}-1$ (visualized in
the figure by the intersection of a vertical line between each open square and
the NCSM extrapolation). While, much like the mapping between cutoffs, these
estimates require some level of arbitration, we nevertheless find them a
reasonable and useful tool for gauging the differences between methods.

The $0\nu\beta\beta$-decay NMEs from the NCSM and MR-IMSRG approaches agree
remarkably well. Although the results of the MR-IMSRG(2) calculations reflect
significantly larger fluctuations with each step in the basis cutoff, those
fluctuations consistently remain less than a few percent of the converged
value, and the overall trends remain quite similar to those of the NCSM. More
importantly, the contributions, especially the larger Fermi and GT
contributions, show excellent agreement between approaches.

The relative magnitudes of the contributions agree between approaches. The GT
contribution is around four times greater than the Fermi contribution, while
the tensor contribution is roughly two orders of magnitude smaller and of
opposite sign. The Fermi, GT, tensor, and total NME results have
$\widetilde{N}_{5\%}\sim 12.6,\,10.8,\,10.3,\,11.7$, respectively, which
suggests only slightly slower convergence than that of the energy but still
significantly faster convergence than that of the NCSM square radii.

The MR-IMSRG $0\nu\beta\beta$-decay results resemble a saw-tooth pattern for
results beyond $e_{\text{max}}=4$ that gradually decreases in magnitude as
$e_{\text{max}}$ increases. The maximum deviation of this pattern occurs in
the GT contribution and reaches the order of a few percent. The deviations of
the tensor contribution appear less systematic, though this may be a
consequence of the contribution’s relatively small magnitude. The deviations
in the Fermi and GT results of the MR-IMSRG share a sign and are most visible
at $e_{\text{max}}=8$, where they consistently deviate in the negative
direction.

A mildly similar (though less pronounced) saw-tooth pattern is observed in the
tensor contribution of the NCSM results at the lowest $N_{\text{max}}$
cutoffs. Within NCSM calculations, such patterns (sometimes called “odd-even
effects”) are generally the consequence of alternating signs in the asymptotic
tails of the HO basis wavefunctions that are introduced with each increment in
$N_{\text{max}}$ (Vary  _et al._ , 2018). In such cases, as the tail region of
the calculated wavefunction shifts with each increment, the tail begins to
overlap a region of phase space in which the effective operator is
particularly active (i.e., has dominant correlations). If the span of that
active region is long enough to require multiple steps in $N_{\text{max}}$ for
the tail to pass through, the result is a visible contribution to the
observable that alternates in sign. Naturally, the pattern disappears as
$N_{\text{max}}$ increases enough so that the effective operator’s range is
completely encompassed by that of the basis.

One might wonder if the pattern observed in the MR-IMSRG
$0\nu\beta\beta$-decay results reflect a similar effect. However, considering
our MR-IMSRG(2) calculations employ natural orbitals and not HO wavefunctions,
the pattern’s similarity may be entirely circumstantial. Determining the
origin of these deviations in the MR-IMSRG results will require further study.

Despite these fluctuations making it somewhat challenging to make more than
qualitative observations, the trends of the NCSM and MR-IMSRG results are
remarkably similar. Indeed, the differences in the asymptotic limits of the
square radii in Fig. (2) do not appear indicative of similar differences in
the $0\nu\beta\beta$-decay NME results. Similarly however, the more rapid
convergence observed in the MR-IMSRG(2) ground-state energy and square radii
compared to that of the NCSM does not appear to translate into a more rapid
convergence of the $0\nu\beta\beta$-decay NMEs in Fig. (3). The differences
between the two approaches’ $0\nu\beta\beta$-decay results appear to be of
similar magnitudes as the saw-tooth deviations present in the MR-IMSRG(2)
results, and remain less than $5\%$ of the total $0\nu\beta\beta$-decay NME at
the maximum basis cutoff employed for each method.

Comparing our extrapolated $0\nu\beta\beta$-decay NMEs to those calculated in
the VMC approach with 3N correlations included (Pastore  _et al._ , 2018), we
see that the magnitudes of both the GT and Fermi contributions agree to within
about $15\%$, while those of the tensor contribution agree to within about
$10\%$. For all three contributions, the VMC results are larger. These
differences may suggest a modest correction from 3N correlations, though other
differences between our study and that of Ref. Pastore  _et al._ , 2018 may
play a significant role as well.

##  V conclusion

We find significant agreement between the NCSM and MR-IMSRG results in our
investigation of $0\nu\beta\beta$ decay in the $A=6$ system. The difference in
the calculated ground-state energy is only about $\sim 2.3\%$. We see
measurable differences in the square-radius results that offer an estimate for
the effects of correlations that are omitted by the MR-IMSRG(2) truncation at
the normal-ordered two-body level. It is interesting that these differences do
not extend to the $0\nu\beta\beta$-decay NMEs, which are remarkably similar in
the two approaches, differing by only $\sim 4.3\%$ in the total NME at the
largest basis cutoffs considered. The convergence rate of the
$0\nu\beta\beta$-decay NMEs appears to be comparable to that of the energies.

The GT contribution dominates the $0\nu\beta\beta$-decay NME, comprising $\sim
80\%$ of its total. The Fermi contribution makes up most of the remainder, and
the tensor contribution is roughly two orders of magnitude smaller and of
opposite sign.

Our estimates of the differences in the total $0\nu\beta\beta$-decay NME
between the two approaches do not exceed $9\%$ for any of the basis cutoffs
considered. Fluctuations in the MR-IMSRG results could pose a minor obstacle
for extrapolation, though their consistent saw-tooth appearance may suggest
these fluctuations are systematically correctable. Beyond these fluctuations,
the two approaches result in qualitatively similar convergence for the
$0\nu\beta\beta$-decay NME.

The agreement between the two approaches for $0\nu\beta\beta$-decay NMEs is
encouraging, and warrants additional benchmarking. Unlike the transition
studied in this work, the physically realistic $0\nu\beta\beta$-decay
transition contributions do not possess a uniform sign as a function of the
pair separation (Wang  _et al._ , 2019). That is, the
$(0+,T)\rightarrow(0+,T-2)$ transitions of experimental interest (Gilliss  _et
al._ , 2018; Gando  _et al._ , 2016; Agostini  _et al._ , 2017) have a node in
the transition density, making them much more sensitive to both short- and
long-range correlations in the wavefunctions. This sensitivity has been
recently explored in another benchmark study comparing VMC and shell-model
calculations of $0\nu\beta\beta$ decay in $A=10$ and $A=12$ systems, and has
been seen to generate differences ranging anywhere from $30\%$ to $400\%$
between approaches (Wang  _et al._ , 2019). A similar comparison between the
MR-IMSRG and NCSM approaches, including full isospin dependence, would provide
a more stringent test of the many-body methods, and greater insight into
problems that may appear when modeling the decay in heavier nuclei. The
results observed here warrant such an investigation, and lend support to the
application of MR-IMSRG to $0\nu\beta\beta$ decay in heavier nuclei (Yao  _et
al._ , 2019), where it is computationally more feasible than the NCSM. The
good agreement between the two approaches for $0\nu\beta\beta$-decay NMEs is a
promising development.

###### Acknowledgements.

We offer special thanks to Roland Wirth for his help in validating the results
of this work. We acknowledge fruitful discussions with Sofia Quaglioni, Peter
Gysbers, Soham Pal, Shiplu Sarker, and Weijie Du. This work was supported in
part by the US Department of Energy (DOE), Office of Science, under Grant Nos.
DE-FG02-87ER40371, DE-SC0018223 (SciDAC-4/NUCLEI), DE-SC0015376 (DOE Topical
Collaboration in Nuclear Theory for Double-Beta Decay and Fundamental
Symmetries), DE-SC0017887, and DE-FG02-97ER41019. Computational resources were
provided by the National Energy Research Scientific Computing Center (NERSC),
which is supported by the US DOE Office of Science under Contract No. DE-
AC02-05CH11231.

## appendix: extrapolation methods

In this work, we perform all extrapolations by using a non-linear least-
squares fit to a form that is specific to each observable and varies with
cutoff parameter. The fitting process is iterated until all fit parameters
have converged to at least 10 digits of precision. We apply forms identically
for both NCSM and MR-IMSRG extrapolations, treating the former as functions of
$N_{\text{max}}$ and the latter as functions of $e_{\text{max}}$. We use
$X_{\text{max}}$ to denote either cutoff parameter when defining the
extrapolations provided below. Following a common NCFC practice, we do not
include the $N_{\text{max}}=0$ result when performing fits to any of the NCSM
data sets. The extrapolation for each data set is performed without regard to
any other data sets or their extrapolations. The formulae for ground-state
energy and square radius are applied identically to both NCSM and MR-IMSRG
results. Extrapolations for $0\nu\beta\beta$-decay NMEs are only performed for
the NCSM results because of fluctuations in the MR-IMSRG results.

It should be noted that the extrapolations described here were originally
designed with the $N_{\text{max}}$ truncation scheme in mind, and their
effectiveness for extrapolating results in the $e_{\text{max}}$ truncation
scheme has not yet been fully explored. Nevertheless, the significant
similarities of the two schemes and their quantification of the same
underlying variable (i.e. the content of the many-body basis) suggest the same
extrapolation forms may be effective; an expectation that is supported by the
results of this work.

Motivated by the extrapolations proposed in Ref. Maris  _et al._ , 2009, we
extrapolate the ground-state energy to the form

$\displaystyle f\left(X_{\text{max}}\right)$ $\displaystyle=a+b\cdot
e^{-cX_{\text{max}}}$ (21)

where $a$, $b$, and $c$ are fit parameters. We employ the same form for our
extrapolations of the NCSM $0\nu\beta\beta$-decay results. Values of fit
parameters calculated in this study for energy and $0\nu\beta\beta$-decay NMEs
may be found in the right-most columns of Table (1) alongside their
corresponding data set.

The simple exponential form depicted in Eq. (21) generally provides a poor
prediction for the convergence behavior of square-radius operator observables.
Thus, inspired by the methods discussed in Ref. Furnstahl  _et al._ , 2012, we
extrapolate square radii by fitting to the form

$\displaystyle\langle r^{2}\rangle$ $\displaystyle=\langle
r^{2}\rangle_{\infty}-\left(c_{0}\beta+c_{1}\beta^{3}\right)e^{-\beta}\,,$
(22)

where

$\displaystyle\beta\equiv$ $\displaystyle
2k_{\infty}\frac{\hbar}{m\Omega}\left[\sqrt{2X_{\text{max}}+5}+0.54437\left(2X_{\text{max}}+5\right)^{1/6}\right]\,.$

Here $m=938.92$ MeV is the average mass of a neutron and a proton, and
$\langle r^{2}\rangle_{\infty},$ $c_{0}$, and $c_{1}$ are fit parameters.
Unlike the authors of Ref. Furnstahl  _et al._ , 2012 who determine
$k_{\infty}$ while extrapolating the ground-state energy with their
theoretically-founded “IR formula”, we treat $k_{\infty}$ as an additional fit
parameter when extrapolating each square radius. We provide our calculated
values of the fit parameters for each square radius extrapolation alongside
its corresponding data set in Table (2).

Table 1: MR-IMSRG(2) and NCSM calculated observables and extrapolation parameters (see Eq. (21)). The ground-state energy ($E$) results correspond to the ${}^{6}\text{He}$ ground state, and are plotted in Fig. (2). The $0\nu\beta\beta$-decay NME ($M_{0\nu}$) results and their decomposition into Fermi (F), Gamow-Teller (GT), and tensor (T) contributions correspond to the${}^{6}\text{He}\rightarrow{}^{6}\text{Be}$ ground-state-to-ground-state transition, and are plotted in Fig. (3). In all calculations of $M_{0\nu}$ and its contributions, isospin symmetry has been assumed. Extrapolations for the $0\nu\beta\beta$-decay NME were only performed using the NCSM results. |  |  $N_{\text{max}}$($e_{\text{max}}$) | Fit Parameters
---|---|---|---
Observable | Method | 0(2) | 2(4) | 4(6) | 6(8) | 8(10) | 10(12) | 12 | $a$ | $b$ | $c$
$E$ (MeV) | NCSM | -12.546 | -19.406 | -23.961 | -26.438 | -27.699 | -28.374 | -28.720 | -29.132 | 18.414 | 0.3188
MR-IMSRG | -20.810 | -26.037 | -27.752 | -28.240 | -28.385 | -28.435 |  | -28.472 | 24.375 | 0.5784
$M_{0\nu}^{F}$ | NCSM | -1.0165 | -0.9669 | -0.9287 | -0.8984 | -0.8773 | -0.8604 | -0.8458 | -0.8032 | -0.2135 | 0.1331
MR-IMSRG | -1.0430 | -0.9811 | -0.9335 | -0.9452 | -0.8880 | -0.9110 |  |  |  |
$M_{0\nu}^{GT}$ | NCSM | -4.0553 | -3.7751 | -3.6398 | -3.5256 | -3.4546 | -3.3960 | -3.3471 | -3.2144 | -0.7472 | 0.1429
MR-IMSRG | -3.9576 | -3.7688 | -3.5812 | -3.7742 | -3.4503 | -3.5326 |  |  |  |
$M_{0\nu}^{T}$ | NCSM | 0.0435 | 0.0479 | 0.0325 | 0.0313 | 0.0279 | 0.0265 | 0.0252 | 0.0247 | 0.0441 | 0.3472
MR-IMSRG | 0.0661 | 0.0484 | 0.0441 | 0.0268 | 0.0294 | 0.0301 |  |  |  |
$M_{0\nu}$ | NCSM | -5.0283 | -4.6941 | -4.5361 | -4.3926 | -4.3041 | -4.2299 | -4.1677 | -3.9777 | -0.9355 | 0.1322
MR-IMSRG | -4.9346 | -4.7016 | -4.4706 | -4.6927 | -4.3089 | -4.4134 |  |  |  |
Table 2: MR-IMSRG(2) and NCSM neutron, proton, and matter square radii ($r_{n}^{2}$, $r_{p}^{2}$, and $r_{m}^{2}$, respectively) and corresponding square radius extrapolation parameters (see Eq. (22)) for the ${}^{6}\text{He}$ ground state. The extrapolated fits are plotted alongside their respective results in Fig. (2). |  |  $N_{\text{max}}$($e_{\text{max}}$) | Fit Parameters
---|---|---|---
Observable | Method | 0(2) | 2(4) | 4(6) | 6(8) | 8(10) | 10(12) | 12 | $\langle r^{2}\rangle_{\infty}$ | $c_{0}$ | $c_{1}$ | $k_{\infty}$
$r_{n}^{2}$ ($\text{fm}^{2}$) | NCSM | 3.6286 | 4.0096 | 4.3347 | 4.6604 | 4.9169 | 5.1656 | 5.4014 | 8.0927 | 10.9067 | 0.9965 | 0.1238
MR-IMSRG | 3.8389 | 4.3023 | 4.5616 | 4.7524 | 4.8929 | 4.9903 |  | 5.3226 | 10.601 | 0.2827 | 0.2235
$r_{p}^{2}$ ($\text{fm}^{2}$) | NCSM | 2.5918 | 2.5622 | 2.6955 | 2.8056 | 2.8791 | 2.9532 | 3.0217 | 3.4732 | 3.0146 | 0.1924 | 0.1505
MR-IMSRG | 2.5125 | 2.6540 | 2.6870 | 2.7149 | 2.7394 | 2.7543 |  | 2.7598 | 8.9888 | 0.0807 | 0.3546
$r_{m}^{2}$ ($\text{fm}^{2}$) | NCSM | 3.2830 | 3.5273 | 3.7885 | 4.0421 | 4.2378 | 4.4285 | 4.6083 | 6.5005 | 8.1034 | 0.7285 | 0.1282
MR-IMSRG | 3.3970 | 3.7527 | 3.9367 | 4.0733 | 4.1751 | 4.2448 |  | 4.4563 | 8.4272 | 0.2028 | 0.2326

## References

  * Ahmad  _et al._ (2001) Q. R. Ahmad _et al._ (SNO), Phys. Rev. Lett. 87, 071301 (2001), arXiv:nucl-ex/0106015 [nucl-ex] .
  * Eguchi  _et al._ (2003) K. Eguchi _et al._ (KamLAND), Phys. Rev. Lett. 90, 021802 (2003), arXiv:hep-ex/0212021 [hep-ex] .
  * Fukuda  _et al._ (1998) Y. Fukuda _et al._ (Super-Kamiokande), Phys. Rev. Lett. 81, 1562 (1998), arXiv:hep-ex/9807003 [hep-ex] .
  * Martin-Albo  _et al._ (2016) J. Martin-Albo _et al._ (NEXT), JHEP 05, 159 (2016), arXiv:1511.09246 [physics.ins-det] .
  * Albert  _et al._ (2014) J. B. Albert _et al._ (EXO-200), Nature 510, 229 (2014), arXiv:1402.6956 [nucl-ex] .
  * Gilliss  _et al._ (2018) T. Gilliss _et al._ (MAJORANA),  _Proceedings, 21st International Conference on Particles and Nuclei (PANIC 17): Beijing, China, September 1-5, 2017_ , Int. J. Mod. Phys. Conf. Ser. 46, 1860049 (2018), arXiv:1804.01582 [physics.ins-det] .
  * Gando  _et al._ (2016) A. Gando, Y. Gando, T. Hachiya, A. Hayashi, S. Hayashida, H. Ikeda, K. Inoue, K. Ishidoshiro, Y. Karino, _et al._ (KamLAND-Zen), Phys. Rev. Lett. 117, 082503 (2016), [Addendum: Phys. Rev. Lett.117,no.10,109903(2016)], arXiv:1605.02889 [hep-ex] .
  * Alfonso  _et al._ (2015) K. Alfonso _et al._ (CUORE), Phys. Rev. Lett. 115, 102502 (2015), arXiv:1504.02454 [nucl-ex] .
  * Agostini  _et al._ (2017) M. Agostini _et al._ ,  (2017), 10.1038/nature21717, [Nature544,47(2017)], arXiv:1703.00570 [nucl-ex] .
  * Aalseth  _et al._ (2018) C. E. Aalseth _et al._ (Majorana), Phys. Rev. Lett. 120, 132502 (2018), arXiv:1710.11608 [nucl-ex] .
  * Andringa  _et al._ (2016) S. Andringa _et al._ (SNO+), Adv. High Energy Phys. 2016, 6194250 (2016), arXiv:1508.05759 [physics.ins-det] .
  * Iwata  _et al._ (2016) Y. Iwata, N. Shimizu, T. Otsuka, Y. Utsuno, J. Menèndez, M. Honma, and T. Abe, Phys. Rev. Lett. 116, 112502 (2016), [Erratum: Phys. Rev. Lett.117,no.17,179902(2016)], arXiv:1602.07822 [nucl-th] .
  * Cirigliano  _et al._ (2017) V. Cirigliano, W. Dekens, J. de Vries, M. L. Graesser, and E. Mereghetti, JHEP 12, 082 (2017), arXiv:1708.09390 [hep-ph] .
  * Contessi  _et al._ (2017) L. Contessi, A. Lovato, F. Pederiva, A. Roggero, J. Kirscher, and U. van Kolck, Phys. Lett. B772, 839 (2017), arXiv:1701.06516 [nucl-th] .
  * Coraggio  _et al._ (2017) L. Coraggio, L. De Angelis, T. Fukui, A. Gargano, and N. Itaco, Phys. Rev. C95, 064324 (2017), arXiv:1703.05087 [nucl-th] .
  * Jiao  _et al._ (2017) C. F. Jiao, J. Engel, and J. D. Holt, Phys. Rev. C96, 054310 (2017), arXiv:1707.03940 [nucl-th] .
  * Tiburzi  _et al._ (2017) B. C. Tiburzi, M. L. Wagman, F. Winter, E. Chang, Z. Davoudi, W. Detmold, K. Orginos, M. J. Savage, and P. E. Shanahan, Phys. Rev. D96, 054505 (2017), arXiv:1702.02929 [hep-lat] .
  * Shanahan  _et al._ (2017) P. E. Shanahan, B. C. Tiburzi, M. L. Wagman, F. Winter, E. Chang, Z. Davoudi, W. Detmold, K. Orginos, and M. J. Savage, Phys. Rev. Lett. 119, 062003 (2017), arXiv:1701.03456 [hep-lat] .
  * Horoi and Neacsu (2017) M. Horoi and A. Neacsu, (2017), arXiv:1706.05391 [hep-ph] .
  * Cremonesi and Pavan (2014) O. Cremonesi and M. Pavan, Adv. High Energy Phys. 2014, 951432 (2014), arXiv:1310.4692 [physics.ins-det] .
  * Dell’Oro  _et al._ (2016) S. Dell’Oro, S. Marcocci, M. Viel, and F. Vissani, Adv. High Energy Phys. 2016, 2162659 (2016), arXiv:1601.07512 [hep-ph] .
  * Engel and Menèndez (2017) J. Engel and J. Menèndez, Rept. Prog. Phys. 80, 046301 (2017), arXiv:1610.06548 [nucl-th] .
  * Gomez-Cadenas  _et al._ (2012) J. J. Gomez-Cadenas, J. Martin-Albo, M. Mezzetto, F. Monrabal, and M. Sorel, Riv. Nuovo Cim. 35, 29 (2012), arXiv:1109.5515 [hep-ex] .
  * Henning (2016) R. Henning, Rev. Phys. 1, 29 (2016).
  * Gysbers  _et al._ (2019) P. Gysbers _et al._ , Nature Phys. 15, 428 (2019), arXiv:1903.00047 [nucl-th] .
  * Bogner  _et al._ (2014) S. K. Bogner, H. Hergert, J. D. Holt, A. Schwenk, S. Binder, A. Calci, J. Langhammer, and R. Roth, Phys. Rev. Lett. 113, 142501 (2014), arXiv:1402.1407 [nucl-th] .
  * Jansen  _et al._ (2014) G. R. Jansen, J. Engel, G. Hagen, P. Navràtil, and A. Signoracci, Phys. Rev. Lett. 113, 142502 (2014), arXiv:1402.2563 [nucl-th] .
  * Jansen  _et al._ (2016) G. R. Jansen, M. D. Schuster, A. Signoracci, G. Hagen, and P. Navràtil, Phys. Rev. C94, 011301(R) (2016), arXiv:1511.00757 [nucl-th] .
  * Stroberg  _et al._ (2016) S. R. Stroberg, H. Hergert, J. D. Holt, S. K. Bogner, and A. Schwenk, Phys. Rev. C93, 051301(R) (2016), arXiv:1511.02802 [nucl-th] .
  * Stroberg  _et al._ (2017) S. R. Stroberg, A. Calci, H. Hergert, J. D. Holt, S. K. Bogner, R. Roth, and A. Schwenk, Phys. Rev. Lett. 118, 032502 (2017), arXiv:1607.03229 [nucl-th] .
  * Dikmen  _et al._ (2015) E. Dikmen, A. F. Lisetskiy, B. R. Barrett, P. Maris, A. M. Shirokov, and J. P. Vary, Phys. Rev. C91, 064301 (2015), arXiv:1502.00700 [nucl-th] .
  * Barrett  _et al._ (2017) B. R. Barrett, E. Dikmen, P. Maris, A. M. Shirokov, N. A. Smirnova, and J. P. Vary,  _Proceedings, 14th International Symposium on Nuclei in the Cosmos (NIC-XIV): Niigata, Japan, June 19-24, 2016_ , JPS Conf. Proc. 14, 021006 (2017).
  * Song  _et al._ (2017) L. S. Song, J. M. Yao, P. Ring, and J. Meng, Phys. Rev. C95, 024305 (2017), arXiv:1702.02448 [nucl-th] .
  * Yao  _et al._ (2018) J. M. Yao, J. Engel, L. J. Wang, C. F. Jiao, and H. Hergert, Phys. Rev. C98, 054311 (2018), arXiv:1807.11053 [nucl-th] .
  * Entem and Machleidt (2003) D. R. Entem and R. Machleidt, Phys. Rev. C68, 041001(R) (2003), arXiv:nucl-th/0304018 [nucl-th] .
  * Machleidt and Entem (2011) R. Machleidt and D. R. Entem, Phys. Rept. 503, 1 (2011), arXiv:1105.2919 [nucl-th] .
  * Bogner  _et al._ (2010) S. K. Bogner, R. J. Furnstahl, and A. Schwenk, Prog. Part. Nucl. Phys. 65, 94 (2010), arXiv:0912.3688 [nucl-th] .
  * Cockrell  _et al._ (2012) C. Cockrell, J. P. Vary, and P. Maris, Phys. Rev. C86, 034325 (2012), arXiv:1201.0724 [nucl-th] .
  * Shin  _et al._ (2017) I. J. Shin, Y. Kim, P. Maris, J. P. Vary, C. Forssèn, J. Rotureau, and N. Michel, J. Phys. G44, 075103 (2017), arXiv:1605.02819 [nucl-th] .
  * Pastore  _et al._ (2018) S. Pastore, J. Carlson, V. Cirigliano, W. Dekens, E. Mereghetti, and R. B. Wiringa, Phys. Rev. C97, 014606 (2018), arXiv:1710.05026 [nucl-th] .
  * Avignone  _et al._ (2008) F. T. Avignone, III, S. R. Elliott, and J. Engel, Rev. Mod. Phys. 80, 481 (2008), arXiv:0708.1033 [nucl-ex] .
  * Simkovic  _et al._ (1999) F. Simkovic, G. Pantis, J. D. Vergados, and A. Faessler, Phys. Rev. C60, 055502 (1999), arXiv:hep-ph/9905509 [hep-ph] .
  * Kotila and Iachello (2012) J. Kotila and F. Iachello, Phys. Rev. C85, 034316 (2012), arXiv:1209.5722 [nucl-th] .
  * Stoica and Mirea (2013) S. Stoica and M. Mirea, Phys. Rev. C88, 037303 (2013), arXiv:1307.0290 [nucl-th] .
  * Horoi and Stoica (2010) M. Horoi and S. Stoica, Phys. Rev. C81, 024321 (2010), arXiv:0911.3807 [nucl-th] .
  * Rodin  _et al._ (2006) V. A. Rodin, A. Faessler, F. Simkovic, and P. Vogel, Nucl. Phys. A766, 107 (2006), [Erratum: Nucl. Phys.A793,213(2007)], arXiv:0706.4304 [nucl-th] .
  * Suhonen (2007) J. Suhonen,  _From Nucleons to Nucleus_, Theoretical and Mathematical Physics (Springer, Berlin, Germany, 2007).
  * Simkovic  _et al._ (2008) F. Simkovic, A. Faessler, V. Rodin, P. Vogel, and J. Engel, Phys. Rev. C77, 045503 (2008), arXiv:0710.2055 [nucl-th] .
  * Simkovic  _et al._ (2009) F. Simkovic, A. Faessler, H. Muther, V. Rodin, and M. Stauf, Phys. Rev. C79, 055501 (2009), arXiv:0902.0331 [nucl-th] .
  * Miller and Spencer (1976) G. A. Miller and J. E. Spencer, Annals Phys. 100, 562 (1976).
  * Roth  _et al._ (2005) R. Roth, H. Hergert, P. Papakonstantinou, T. Neff, and H. Feldmeier, Phys. Rev. C72, 034002 (2005), arXiv:nucl-th/0505080 [nucl-th] .
  * Brueckner (1955) K. A. Brueckner, Phys. Rev. 100, 36 (1955).
  * Benhar  _et al._ (2014) O. Benhar, R. Biondi, and E. Speranza, Phys. Rev. C90, 065504 (2014), arXiv:1401.2030 [nucl-th] .
  * Muther and Polls (2000) H. Muther and A. Polls, Prog. Part. Nucl. Phys. 45, 243 (2000), arXiv:nucl-th/0001007 [nucl-th] .
  * Tobocman (1981) W. Tobocman, Nucl. Phys. A357, 293 (1981).
  * Barrett  _et al._ (2013) B. R. Barrett, P. Navràtil, and J. P. Vary, Prog. Part. Nucl. Phys. 69, 131 (2013).
  * Maris  _et al._ (2010) P. Maris, M. Sosonkina, J. P. Vary, E. Ng, and C. Yang, Procedia Computer Science 1, 97 (2010), iCCS 2010\.
  * Aktulga  _et al._ (2014) H. M. Aktulga, C. Yang, E. G. Ng, P. Maris, and J. P. Vary,  _Concurrency and Computation: Practice and Experience_ , 26, 2631 (2014).
  * Shao  _et al._ (2017) M. Shao, H. M. Aktulga, C. Yang, E. G. Ng, P. Maris, and J. P. Vary, Computer Physics Communications 222 (2017), 10.1016/j.cpc.2017.09.004, arXiv:1609.01689 .
  * Maris  _et al._ (2009) P. Maris, J. P. Vary, and A. M. Shirokov, Phys. Rev. C79, 014308 (2009), arXiv:0808.3420 [nucl-th] .
  * Hergert  _et al._ (2016) H. Hergert, S. K. Bogner, T. D. Morris, A. Schwenk, and K. Tsukiyama, Phys. Rept. 621, 165 (2016), arXiv:1512.06956 [nucl-th] .
  * Hergert (2017) H. Hergert, Phys. Scripta 92, 023002 (2017), arXiv:1607.06882 [nucl-th] .
  * Hergert  _et al._ (2018) H. Hergert, J. M. Yao, T. D. Morris, N. M. Parzuchowski, S. K. Bogner, and J. Engel, Journal of Physics: Conference Series 1041, 012007 (2018).
  * Morris  _et al._ (2015) T. D. Morris, N. M. Parzuchowski, and S. K. Bogner, Phys. Rev. C92, 034331 (2015), arXiv:1507.06725 [nucl-th] .
  * Furnstahl  _et al._ (2012) R. J. Furnstahl, G. Hagen, and T. Papenbrock, Phys. Rev. C86, 031301(R) (2012), arXiv:1207.6100 [nucl-th] .
  * Odell  _et al._ (2016) D. Odell, T. Papenbrock, and L. Platter, Phys. Rev. C93, 044331 (2016), arXiv:1512.04851 [nucl-th] .
  * Riisager  _et al._ (1990) K. Riisager _et al._ (ISOLDE), Phys. Lett. B235, 30 (1990).
  * Bogner  _et al._ (2008) S. K. Bogner, R. J. Furnstahl, P. Maris, R. J. Perry, A. Schwenk, and J. P. Vary, Nucl. Phys. A801, 21 (2008), arXiv:0708.3754 [nucl-th] .
  * Wang  _et al._ (2017) M. Wang, G. Audi, F. G. Kondev, W. Huang, S. Naimi, and X. Xu, Chinese Physics C 41, 030003 (2017).
  * Furnstahl  _et al._ (2014) R. J. Furnstahl, S. N. More, and T. Papenbrock, Phys. Rev. C89, 044301 (2014), arXiv:1312.6876 [nucl-th] .
  * Furnstahl  _et al._ (2015) R. J. Furnstahl, G. Hagen, T. Papenbrock, and K. A. Wendt, J. Phys. G42, 034032 (2015), arXiv:1408.0252 [nucl-th] .
  * Coon  _et al._ (2012) S. A. Coon, M. I. Avetian, M. K. G. Kruse, U. van Kolck, P. Maris, and J. P. Vary, Phys. Rev. C86, 054002 (2012), arXiv:1205.3230 [nucl-th] .
  * Riisager (2013) K. Riisager, Phys. Scripta T152, 014001 (2013), arXiv:1208.6415 [nucl-ex] .
  * Vary  _et al._ (2018) J. P. Vary, R. Basili, W. Du, M. Lockner, P. Maris, S. Pal, and S. Sarker, Phys. Rev. C98, 065502 (2018), arXiv:1809.00276 [nucl-th] .
  * Wang  _et al._ (2019) X. Wang, A. Hayes, J. Carlson, G. Dong, E. Mereghetti, S. Pastore, and R. Wiringa, Physics Letters B 798, 134974 (2019).
  * Yao  _et al._ (2019) J. Yao, B. Bally, J. Engel, R. Wirth, T. RodrÃguez, and H. Hergert, (2019), arXiv:1908.05424 [nucl-th] .


[[30]]\label{lemma:ckos-inverter} For every efficient linear extractor $\Ext$, there exists an efficient randomized function $\iExt : \{0, 1\}^l \times \{0, 1\}^d \rightarrow \{0, 1\}^\eta \cup {\bot}$ (termed inverter) such that
\begin{enumerate}
\item $U_\eta, U_d, \Ext(U_\eta;U_d) \equiv \iExt(\Ext(U_\eta;U_d), U_d), U_d, \Ext(U_\eta;U_d)$
\item For each $(s, y) \in \{0, 1\}^d \times \{0, 1\}^l$:
\begin{enumerate}
\item $\mathbb{P}[\iExt(y, s) = \bot] = 1$, if and only if there exists no $w \in \{0, 1\}^\eta$ such that $\Ext(w; s) = y$.
\item $\mathbb{P}[\Ext(\iExt(y, s), s) = y] = 1$, if there exists some $w \in \{0, 1\}^\eta$ such that $\Ext(w; s) = y$
\end{enumerate}
\end{enumerate}
\end{lemma}

\subsection{Code Construction}
\label{subsection:lrss-construction}

Our code construction uses essentially the same ingredients as [30], with small modifications to the locality of their privacy parameters and to their compiler.

\begin{enumerate}
\item $(\mshare, \mrec)$, an $(p, t,\epspriv, 0)$ threshold secret sharing scheme which is $(k, \epsilon_u)$-locally uniform over the message space $\{0, 1\}^l$ and with share size $l'$.
\item $(\sdshare, \sdrec)$, an $(p, k+1, \epspriv', 0)$ threshold secret sharing scheme over the message space $\{0, 1\}^d$ and share size $d'$.
\item $\Ext$, a quantum-proof $(\eta, \tau\leq \eta-\mu, d, l', \epsilon_{ext})$-strong linear extractor. Let $\iExt$ be the inverter function corresponding to $\Ext$ given by \Cref{lemma:ckos-inverter} .
\end{enumerate}

\noindent \textbf{Share} To share a message $m$, we begin by encoding it into $(M_1,\cdots, M_p)\leftarrow \mshare(m)$. Then, for each party $i\in [p]$, we sample a random seed $R_i\in \{0, 1\}^d$, and then use $\iExt$ to get the source $W_i \leftarrow \iExt(M_i, R_i)$. If any of the $W_i=\bot$ rejects, output each of the share to be $(\bot, M_i)$. Else, concatenate the randomness $R = (R_1, \cdots, R_p)$ and share it using $\sdshare(R)$ to get $(S_1, \cdots, S_p)$, and finally set the $i$-th share to be $(W_i, S_i)$. \\

\noindent \textbf{Rec} Assuming the encoding doesn't reject, an authorized party $T\subset [p]$ of size $\geq t> k$ begins by recovering the randomness $R = (R_1, \cdots, R_p)$ using $\sdrec$ on any $k+1$ (honest) shares of $T$. Then, they recover the message shares $M_T =\{M_i:i\in T\}$ by running the extractor on $(W_i, R_i)$ for each $i\in T$. Finally, using $\mrec$ on $M_T$ it decodes the message $m$.

\begin{theorem}\label{theorem:lrss-klocal}
$(\share, \rec)$ defines a $(p, t, \epspriv, 0)$ secret sharing scheme, which is $\leq 2(\epspriv +\epspriv') +2 p\cdot (\epsilon_\Ext+\epsilon_u)$ leakage resilient against $\mathcal{F}_{ k, \mu}^{p, t}$.
\end{theorem}

\subsection{Analysis}
\label{subsection:lrss-analysis}

The correctness and privacy of the scheme are inherited from that of $\mshare, \mrec$. We analyze its rate in the next subsection \Cref{subsubsection:lrss-params}, and dedicate the rest of this subsection to a proof of security.

Fix a leakage channel $(T, K, \Lambda)$ and a message $m$. We assume $|T|\leq t-1$ and $|K|\leq k$ are both unauthorized subsets. We proceed in a sequence of hybrids, where within the encoding map $\Share$ we replace the shares of $K$, $W_i\leftarrow \iExt(M_i, R_i)$, by a uniformly random source $W_i\leftarrow U$:\\

\noindent $\share^0(m)$:  To share a message $m$, we simply encode it into $\share(m)$.\\

\noindent $\share^1(m)$: To share a message $m$, we begin by encoding it into $(M_1,\cdots, M_p)\leftarrow \mshare(m)$. For each $i\in [n]$, sample a random seed $R_i\in \{0, 1\}^d$ and use $\iExt$ to get the source $W_i \leftarrow \iExt(M_i, R_i)$. Then, concatenate the randomness $R = (R_1, \cdots, R_p)$ and share it using $\sdshare(R)$ to get $(S_1, \cdots, S_p)$, and finally set the $i$-th share to be $(W_i, S_i)$.\\

\noindent $\share^2(m)$: To share a message $m$, we begin by encoding it into $(M_1,\cdots, M_p)\leftarrow \mshare(m)$. For each $i\in K$, sample a random seed $R_i\in \{0, 1\}^d$ and source $W_i$. For each $i\in [p]\setminus K$, sample a random seed $R_i\in \{0, 1\}^d$ and use $\iExt$ to get the source $W_i \leftarrow \iExt(M_i, R_i)$. Then, concatenate the randomness $R = (R_1, \cdots, R_p)$ and share it using $\sdshare(R)$ to get $(S_1, \cdots, S_p)$, and finally set the $i$-th share to be $(W_i, S_i)$.\\

Note that $\share^0$ differs from $\share^1$ in that it conditions on the extractor pre-image sampling succeeding. [30] begin by proving that it succeeds with high probability, and thus $\share(m)\approx \share^1(m)$.

\begin{claim}
[[30]]\label{claim:lrss-enc-bot} For any message $m$, $\share(m) = ((\bot, M_1), \cdots,(\bot, M_p))$ with probability $\leq p(\epsilon_\Ext+\epsilon_u)$.
\end{claim}

Moreover, note that $\share^2(m)$ is completely independent of the shares $M_i:i\in K$, and thus the reduced density matrix $\share^{2}_{T\cup K}$ only depends on the shares of $M_i:i\in T$ - where $|T|\leq t-1$. By the privacy of $\mshare$,

\begin{claim}
[[30]]\label{claim:lrss-privacy} For any pair of messages $m, m'$, $\share^2_{T\cup K}(m)\approx_{\epspriv}\share^2_{T\cup K}(m')$.
\end{claim}

It remains to argue that $\share^1(m)$ and $\share^2(m)$ are indistinguishable, given the shares in the unauthorized subset $T$ and the leakage $L$.

\begin{claim}
\label{claim:lrss-hybrid} For any message $m$,
\begin{equation}
(\mathbb{I}_T\otimes \Lambda_K)(\share^1_{T\cup K}(m))\approx_{\delta}(\mathbb{I}_T\otimes \Lambda_K)(\share_{T\cup K}^{2}(m))
\end{equation}

\noindent where $\delta \leq 2(\epspriv'+\epsilon_u+k\cdot \epsilon_\Ext)$.
\end{claim}

By the triangle inequality and the claims above, we conclude that for all $m, m'$:
\begin{equation}
(\mathbb{I}_T\otimes \Lambda_K)(\share_{T\cup K}(m))\approx_{\delta'} (\mathbb{I}_T\otimes \Lambda_K)(\share_{T\cup K}(m'))
\end{equation}

\noindent For some $\delta' \leq \delta + 2\epspriv + 2\cdot p (\epsilon_\Ext+\epsilon_u)\leq 2(\epspriv +\epspriv') +2 p\cdot (\epsilon_\Ext+\epsilon_u)$, which is exactly \Cref{theorem:lrss-klocal}. Now we prove \Cref{claim:lrss-hybrid}:

\begin{proof}

[of \Cref{claim:lrss-hybrid}] Fix a message $m$. Consider the quantum-classical mixed state comprised of the classical shares $M_K = \{M_i:i\in K\}$ of $\mshare$, the seed $R$ and its shares $S_K = \{S_i:i\in K\}$, and the quantum leakage register $L$. Our goal will be to show that this cq density matrix is nearly independent of the ``source" register in the shares of $K$. That is, if we denote as $W_i, i\in K$ as uniformly random sources on $\eta$ bits, then it suffices to show that for some $\delta$,

\begin{equation}\label{equation:main-ckos-claim}
\Lambda\big(\{S_i, \iExt(M_i, R)\}_{i\in K}\big), R, S_K, M_K \approx_{\delta} \Lambda\big(\{S_i, W_i\}_{i\in K}\big), R, S_K, M_K.
\end{equation}

This is since there is a CPTP map $\mathcal{N}^m$ (dependent on the message) which given $R, S_K, M_K$ and the leakage $L$ produces
\begin{gather}
\mathcal{N}^m(\Lambda\big(\{S_i, \iExt(M_i, R)\}_{i\in K}\big), R, S_K, M_K) = (\mathbb{I}_T\otimes \Lambda_K)(\share^1_{T\cup K}(m)) \text{ and } \\ \mathcal{N}^m(\Lambda\big(\{S_i, W_i\}_{i\in K}\big), R, S_K, M_K) = (\mathbb{I}_T\otimes \Lambda_K)(\share^2_{T\cup K}(m)).
\end{gather}

\noindent Note that $\mathcal{N}^m$ simply samples shares $M_T, S_T$ consistent with $m, R$ and $M_K, S_K$.\footnote{This is also known as ``consistent resampling" [30]} Thereby by monotonicity of trace distance we obtain the desired $(\mathbb{I}_T\otimes \Lambda_K)(\share^1_{T\cup K}(m))\approx_{\delta}(\mathbb{I}_T\otimes \Lambda_K)(\share_{T\cup K}^{2}(m))$.

It remains to show \cref{equation:main-ckos-claim}. We begin by replacing the shares in $K$ by fixed shares independent of $R$, $\hat{S}_K\leftarrow \sdshare(0^d)_K$ using the privacy of $\sdshare$, and leveraging the $\epsilon_u$ approximate $k$-wise independence of the shares $M_K$ to replace them by the uniform distribution:
\begin{gather}
\Lambda\big(\{S_i, \iExt(M_i, R)\}_{i\in K}\big), R, S_K, M_K \approx_{\epspriv'} \Lambda\big(\{\hat{S}_i, \iExt(M_i, R)\}_{i\in K}\big), R, \hat{S}_K, M_K, \text{ and }\\
\Lambda\big(\{\hat{S}_i, \iExt(M_i, R)\}_{i\in K}\big), R, \hat{S}_K, M_K \approx_{\epsilon_u} \Lambda\big(\{\hat{S}_i, \iExt(U_{l'}^i, R_i)\}_{i\in K}\big), R, \hat{S}_K, (U_{l'})^{\otimes k}
\end{gather}

Recall $W_1, \cdots, W_K$ are uniformly random $\eta$ bit sources. By \Cref{definition:quantum-proof-extractor}, since $\Ext$ is a strong linear seeded extractor, $\Ext(W_i, R_i)\approx_{\epsilon_\Ext} U_{l'}$, and moreover by \Cref{lemma:ckos-inverter}(b) from [30] we have $W_i = \iExt(\Ext(W_i, R_i), R_i)$. Thus,

\begin{equation}
\Lambda\big(\{\hat{S}_i, \iExt(U_{l'}^i, R_i)\}_{i\in K}\big), R, \hat{S}_K, (U_{l'})^{\otimes k}  \approx_{k\cdot \epsilon_{\Ext}} \Lambda\big(\{\hat{S}_i, W_i\}_{i\in K}), R, \hat{S}_K, \{\Ext(W_i, R_i)\}_{i\in K}
\end{equation}

We now replace each $\Ext(W_j, R_j)$ by $U_l$ in a sequence of hybrids, evoking \textit{quantum-proof} extractor security. This is the main modification to the proof of [30]: Fix $0\leq j\leq k$, and define the collection of classical registers $Z_j$:
\begin{equation}
Z_j = R_{[n]\setminus \{j\}}, \hat{S}_{K}, (U_l)^{\otimes (j-1)}, \{\Ext(W_i, R_i)\}_{j< i\leq k}
\end{equation}

Note that $Z_j$ is independent of $W_j$. If $L$ denotes the $\mu$ qubit leakage register, then from the chain rule for the min entropy of separable states, \Cref{lemma:sep-chain-rule} [34], $H_\infty(W_j|Z_j, L)\geq \eta-\mu\geq \tau$. By \Cref{definition:quantum-proof-extractor},
\begin{equation}
\Lambda(Z_j, W_j), Z_j, R_j, \Ext(W_i, R_i) \approx_{\epsilon_\Ext} \Lambda(Z_j, W_j), Z_j, R_j, U_{l}
\end{equation}

\noindent Which implies through the triangle inequality,
\begin{equation}
\Lambda\big(\{\hat{S}_i, W_i\}_{i\in K}), R, \hat{S}_K, \{\Ext(W_i, R_i)\}_{i\in K} \approx_{k\cdot \epsilon_\Ext} \Lambda\big(\{\hat{S}_i, W_i\}_{i\in K}), R, \hat{S}_K, U^{\otimes k}.
\end{equation}

\noindent By once again appealing to $k$ wise independence of $M_K$ and the privacy of $S_K$, we conclude
\begin{equation}
\Lambda\big(\{S_i, \iExt(M_i, R)\}_{i\in K}\big), R, S_K, M_K \approx_{2(\epspriv'+\epsilon_u+k\cdot \epsilon_\Ext)} \Lambda\big(\{S_i, W_i\}_{i\in K}\big), R, S_K, M_K
\end{equation}

\noindent That is, $\delta \leq 2(\epspriv'+\epsilon_u+k\cdot \epsilon_\Ext)$.
\end{proof}

\subsubsection{Parameters}
\label{subsubsection:lrss-params}

We combine

\begin{enumerate}
\item $(\mshare, \mrec)$ is a $(p, t, \epspriv = 0, \epsilon_c = 0)$-Shamir secret sharing scheme for $l$ bit messages and $l$ bit shares, which is perfectly $t-1$ wise independent.

\item We set $\epsilon_\Ext = 2^{-\lambda^{-1/3}}$, and let $\Ext$ be the $(\eta= l+\mu+O(d), \tau = l+O(d), d, l, \epsilon_\Ext)$ quantum proof strong linear extractor guaranteed by \Cref{lemma:quantum-proof-trevisan}, where $d= O(\log^3 \frac{\eta}{\epsilon_\Ext}) = O(\lambda+\log^3 (l+\mu))$.

\item $(\sdshare, \sdrec)$ is a $(p, k+1, \epspriv'=0, \epsilon'_c= 0)$-Shamir secret sharing scheme for $p\cdot d$ bit messages and $p\cdot d$ bit shares.
\end{enumerate}

The resulting share size of $\share$ to handle $\mu$ qubits of leakage is $p\cdot d+\eta = l+\mu+O(p\cdot \lambda + p\cdot \log^3(l+\mu))$, which is $l+\mu+o(l, \mu)$ whenever $\lambda = o(\frac{l+\mu}{p})$ and $p = O(\frac{l+\mu}{\log^3(l+\mu)})$.

\newpage
\bibliographystyle{alpha}
\newcommand{\etalchar}[1]{$^{#1}$}
\begin{thebibliography}{{\c{C}}GLZR23}

[1]
Scott Aaronson, Yosi Atia, and Leonard Susskind.
\newblock On the hardness of detecting macroscopic superpositions.
\newblock {\em Electron. Colloquium Comput. Complex.}, TR20, 2020.

[2]
Divesh Aggarwal, Naresh~Goud Boddu, and Rahul Jain.
\newblock Quantum secure non-malleable-codes in the split-state model, 2022.
\newblock ArXiv:2202.13354.

[3]
Dorit Aharonov, Michael Ben-Or, and Elad Eban.
\newblock Interactive proofs for quantum computations.
\newblock In {\em International Conference on Supercomputing}, 2008.

[4]
Andris Ambainis, Jan Bouda, and Andreas Winter.
\newblock Nonmalleable encryption of quantum information.
\newblock {\em Journal of Mathematical Physics}, 50(4):042106, 2009.

[5]
Divesh Aggarwal, Stefan Dziembowski, Tomasz Kazana, and Maciej Obremski.
\newblock Leakage-resilient non-malleable codes.
\newblock {\em Electron. Colloquium Comput. Complex.}, TR14, 2015.

[6]
Divesh Aggarwal, Yevgeniy Dodis, and Shachar Lovett.
\newblock Non-malleable codes from additive combinatorics.
\newblock {\em SIAM Journal on Computing}, 47(2):524--546, 2018.
\newblock Preliminary version in STOC 2014.

[7]
Divesh Aggarwal, Ivan Damg{\aa}rd, Jesper~Buus Nielsen, Maciej Obremski, Erick
Purwanto, Jo{\~a}o Ribeiro, and Mark Simkin.
\newblock Stronger leakage-resilient and non-malleable secret sharing schemes
for general access structures.
\newblock In Alexandra Boldyreva and Daniele Micciancio, editors, {\em Advances
in Cryptology -- CRYPTO 2019}, pages 510--539, Cham, 2019. Springer
International Publishing.

[8]
Divesh Aggarwal, Bhavana Kanukurthi, Sai Lakshmi~Bhavana Obbattu, Maciej
Obremski, and Sruthi Sekar.
\newblock Rate one-third non-malleable codes.
\newblock In {\em Proceedings of the 54th Annual ACM SIGACT Symposium on Theory
of Computing}, STOC 2022, page 1364–1377, New York, NY, USA, 2022.
Association for Computing Machinery.

[9]
Gorjan Alagic and Christian Majenz.
\newblock Quantum non-malleability and authentication.
\newblock In Jonathan Katz and Hovav Shacham, editors, {\em Advances in
Cryptology -- CRYPTO 2017}, pages 310--341, Cham, 2017. Springer
International Publishing.

[10]
Anurag Anshu and Chinmay Nirkhe.
\newblock Circuit lower bounds for low-energy states of quantum code
\newblock {\em ArXiv}, abs/2011.02044, 2020.

[11]
Divesh Aggarwal and Maciej Obremski.
\newblock A constant rate non-malleable code in the split-state model.
\newblock In {\em 2020 IEEE 61st Annual Symposium on Foundations of Computer
Science (FOCS)}, pages 1285--1294, Los Alamitos, CA, USA, nov 2020. IEEE
Computer Society.

[12]
Charles~H. Bennett, Gilles Brassard, Claude Cr\'epeau, Richard Jozsa, Asher
Peres, and William~K. Wootters.
\newblock Teleporting an unknown quantum state via dual classical and
einstein-podolsky-rosen channels.
\newblock {\em Phys. Rev. Lett.}, 70:1895--1899, Mar 1993.

[13]
Rishabh Batra, Naresh~Goud Boddu, and Rahul Jain.
\newblock Quantum secure non-malleable randomness encoder and its applications,

[14]
Howard Barnum, Claude Cr{\'e}peau, Daniel Gottesman, Adam~D. Smith, and Alain
\newblock Authentication of quantum messages.
\newblock {\em The 43rd Annual IEEE Symposium on Foundations of Computer
Science, 2002. Proceedings.}, pages 449--458, 2001.

[15]
Thiago Bergamaschi.
\newblock Pauli manipulation detection codes and applications to quantum
communication over adversarial channels, 2023.

[16]
Gianluca Brian, Antonio Faonio, Maciej Obremski, Mark Simkin, and Daniele
\newblock Non-malleable secret sharing against bounded joint-tampering attacks
in the plain model.
\newblock In Daniele Micciancio and Thomas Ristenpart, editors, {\em Advances
in Cryptology -- CRYPTO 2020}, pages 127--155, Cham, 2020. Springer
International Publishing.

[17]
Gianluca Brian, Antonio Faonio, and Daniele Venturi.
\newblock Continuously non-malleable secret sharing: Joint tampering, plain
model and capacity.
\newblock In Kobbi Nissim and Brent Waters, editors, {\em Theory of
Cryptography}, pages 333--364, Cham, 2021. Springer International Publishing.

[18]
Thiago Bergamaschi, Louis Golowich, and Sam Gunn.
\newblock Approaching the quantum singleton bound with approximate error
correction, 2022.

[19]
Naresh~Goud Boddu, Vipul Goyal, Rahul Jain, and João Ribeiro.
\newblock Split-state non-malleable codes and secret sharing schemes for
quantum messages, 2023.

[20]
Saikrishna Badrinarayanan and Akshayaram Srinivasan.
\newblock Revisiting non-malleable secret sharing.
\newblock In Yuval Ishai and Vincent Rijmen, editors, {\em Advances in
Cryptology -- EUROCRYPT 2019}, pages 593--622, Cham, 2019. Springer
International Publishing.

[21]
Anne Broadbent and Evelyn Wainewright.
\newblock Efficient simulation for quantum message authentication.
\newblock In Anderson~C.A. Nascimento and Paulo Barreto, editors, {\em
Information Theoretic Security}, pages 72--91, Cham, 2016. Springer
International Publishing.

[22]
Ronald Cramer, Yevgeniy Dodis, Serge Fehr, Carles Padr{\'o}, and Daniel Wichs.
\newblock Detection of algebraic manipulation with applications to robust
secret sharing and fuzzy extractors.
\newblock In {\em International Conference on the Theory and Application of
Cryptographic Techniques}, 2008.

[23]
Mahdi Cheraghchi and Venkatesan Guruswami.
\newblock Capacity of non-malleable codes.
\newblock {\em IEEE Transactions on Information Theory}, 62:1097--1118, 2013.

[24]
Mahdi Cheraghchi and Venkatesan Guruswami.
\newblock Non-malleable coding against bit-wise and split-state tampering.
\newblock {\em Journal of Cryptology}, 30:191 -- 241, 2013.

[25]
Mahdi Cheraghchi and Venkatesan Guruswami.
\newblock Non-malleable coding against bit-wise and split-state tampering.
\newblock In {\em Proceedings of Theory of Cryptography Conference {(TCC)}},
pages 440--464, 2014.
\newblock Extended version in {Journal of Cryptology.}

[26]
Mahdi Cheraghchi and Venkatesan Guruswami.
\newblock Capacity of non-malleable codes.
\newblock {\em IEEE Transactions on Information Theory}, 62(3):1097--1118,

[27]
Eshan Chattopadhyay, Vipul Goyal, and Xin Li.
\newblock Nonmalleable extractors and codes, with their many tampered
\newblock {\em SIAM Journal on Computing}, 49(5):999--1040, 2020.
\newblock Preliminary version in STOC 2016.

[28]
Alper {\c{C}}akan, Vipul Goyal, Chen-Da Liu-Zhang, and Jo{\~a}o~L. Ribeiro.
\newblock Unbounded leakage-resilience and leakage-detection in a quantum
\newblock {\em IACR Cryptol. ePrint Arch.}, 2023:410, 2023.

[29]
Claude Cr{\'e}peau, Daniel Gottesman, and Adam~D. Smith.
\newblock Approximate quantum error-correcting codes and secret sharing
\newblock In {\em International Conference on the Theory and Application of
Cryptographic Techniques}, 2005.

[30]
Nishanth Chandran, Bhavana Kanukurthi, Sai Lakshmi~Bhavana Obbattu, and Sruthi
\newblock Short leakage resilient and non-malleable secret sharing schemes.
\newblock In Yevgeniy Dodis and Thomas Shrimpton, editors, {\em Advances in
Cryptology - {CRYPTO} 2022}, pages 178--207. Springer, 2022.

[31]
Richard Cleve, Debbie Leung, Li~Liu, and Chunhao Wang.
\newblock Near-linear constructions of exact unitary 2-designs.
\newblock {\em Quantum Info. Comput.}, 16(9–10):721–756, jul 2016.

[32]
Eric Chitambar, Debbie~W. Leung, Laura Mancinska, Maris~A. Ozols, and
Andreas~J. Winter.
\newblock Everything you always wanted to know about locc (but were afraid to
\newblock {\em Communications in Mathematical Physics}, 328:303--326, 2012.

[33]
Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine.
\newblock Exact and approximate unitary 2-designs and their application to
fidelity estimation.
\newblock {\em Phys. Rev. A}, 80:012304, Jul 2009.

[34]
Simon~Pierre Desrosiers and Fr{\'e}d{\'e}ric Dupuis.
\newblock Quantum entropic security and approximate quantum encryption.
\newblock {\em IEEE Transactions on Information Theory}, 56:3455--3464, 2007.

[35]
David~P. DiVincenzo, Patrick~M. Hayden, and Barbara~Maria Terhal.
\newblock Hiding quantum data.
\newblock {\em Foundations of Physics}, 33:1629--1647, 2002.

[36]
Stefan Dziembowski, Tomasz Kazana, and Maciej Obremski.
\newblock Non-malleable codes from two-source extractors.
\newblock In Ran Canetti and Juan~A. Garay, editors, {\em Advances in
Cryptology -- CRYPTO 2013}, pages 239--257, Berlin, Heidelberg, 2013.
Springer Berlin Heidelberg.

[37]
David~P. DiVincenzo, Debbie~W. Leung, and Barbara~Maria Terhal.
\newblock Quantum data hiding.
\newblock {\em IEEE Trans. Inf. Theory}, 48:580--598, 2001.

[38]
D.~P. DiVincenzo, D.~W. Leung, and B.~M. Terhal.
\newblock Quantum data hiding.
\newblock {\em IEEE Transactions on Information Theory}, 48(3):580--598, March

[39]
Anindya De, Christopher Portmann, Thomas Vidick, and Renato Renner.
\newblock Trevisan's extractor in the presence of quantum side information.
\newblock {\em SIAM Journal on Computing}, 41(4):915--940, 2012.

[40]
Stefan Dziembowski, Krzysztof Pietrzak, and Daniel Wichs.
\newblock Non-malleable codes.
\newblock {\em J. ACM}, 65(4), April 2018.
\newblock Preliminary version in ICS 2010.

[41]
Tilo Eggeling and Reinhard~F. Werner.
\newblock Hiding classical data in multipartite quantum states.
\newblock {\em Physical review letters}, 89 9:097905, 2002.

[42]
Antonio Faonio and Daniele Venturi.
\newblock Non-malleable secret sharing in the computational setting: Adaptive
tampering, noisy-leakage resilience, and improved rate.
\newblock In Alexandra Boldyreva and Daniele Micciancio, editors, {\em Advances
in Cryptology -- CRYPTO 2019}, pages 448--479, Cham, 2019. Springer
International Publishing.

[43]
Christopher~A. Fuchs and Jeroen van~de Graaf.
\newblock Cryptographic distinguishability measures for quantum-mechanical
\newblock {\em IEEE Trans. Inf. Theor.}, 45(4):1216–1227, September 2006.

[44]
Sam Gunn, Nathan Ju, Fermi Ma, and Mark Zhandry.
\newblock Commitments to quantum states.
\newblock {\em Proceedings of the 55th Annual ACM Symposium on Theory of
Computing}, 2022.

[45]
Vipul Goyal and Ashutosh Kumar.
\newblock Non-malleable secret sharing.
\newblock In {\em Proceedings of the 50th Annual ACM SIGACT Symposium on Theory
of Computing}, STOC 2018, page 685–698, New York, NY, USA, 2018.
Association for Computing Machinery.

[46]
Vipul Goyal and Ashutosh Kumar.
\newblock Non-malleable secret sharing for general access structures.
\newblock In Hovav Shacham and Alexandra Boldyreva, editors, {\em Advances in
Cryptology -- CRYPTO 2018}, pages 501--530, Cham, 2018. Springer
International Publishing.

[47]
Vipul Goyal, Omkant Pandey, and Silas Richelson.
\newblock Textbook non-malleable commitments.
\newblock In {\em Proceedings of the Forty-Eighth Annual ACM Symposium on
Theory of Computing}, STOC '16, page 1128–1141, New York, NY, USA, 2016.
Association for Computing Machinery.

[48]
Vipul Goyal, Akshayaram Srinivasan, and Chenzhi Zhu.
\newblock Multi-source non-malleable extractors and applications.
\newblock In Anne Canteaut and Fran{\c{c}}ois-Xavier Standaert, editors, {\em
Advances in Cryptology -- EUROCRYPT 2021}, pages 468--497, Cham, 2021.
Springer International Publishing.

[49]
Sumegha Garg, Henry Yuen, and Mark Zhandry.
\newblock New security notions and feasibility results for authentication of
quantum data, 2016.

[50]
Patrick~M. Hayden, Debbie~W. Leung, and Dominic Mayers.
\newblock The universal composable security of quantum message authentication
with key recyling.
\newblock {\em arXiv: Quantum Physics}, 2016.

[51]
Patrick~M. Hayden, Debbie~W. Leung, and Graeme Smith.
\newblock Multiparty data hiding of quantum information.
\newblock {\em Physical Review A}, 71:062339, 2004.

[52]
Patrick~M. Hayden, Debbie~W. Leung, Peter~W. Shor, and Andreas~J. Winter.
\newblock Randomizing quantum states: Constructions and applications.
\newblock {\em Communications in Mathematical Physics}, 250:371--391, 2003.

[53]
Patrick Hayden and John Preskill.
\newblock Black holes as mirrors: quantum information in random subsystems.
\newblock {\em Journal of High Energy Physics}, 2007(09):120, 2007.

[54]
Patrick~M. Hayden and Geoffrey Penington.
\newblock Approximate quantum error correction revisited: Introducing the
\newblock {\em Communications in Mathematical Physics}, 374:369--432, 2017.

[55]
Zahra Jafargholi and Daniel Wichs.
\newblock Tamper detection and continuous non-malleable codes.
\newblock In {\em Theory of Cryptography Conference}, 2015.

[56]
Bhavana Kanukurthi, Sai Lakshmi~Bhavana Obbattu, and Sruthi Sekar.
\newblock Non-malleable randomness encoders and their applications.
\newblock In Jesper~Buus Nielsen and Vincent Rijmen, editors, {\em Advances in
Cryptology -- EUROCRYPT 2018}, pages 589--617, Cham, 2018. Springer
International Publishing.

[57]
Xin {Li}.
\newblock Three-source extractors for polylogarithmic min-entropy.
\newblock In {\em 2015 IEEE 56th Annual Symposium on Foundations of Computer
Science}, pages 863--882, 2015.

[58]
Xin Li.
\newblock Improved non-malleable extractors, non-malleable codes and
independent source extractors.
\newblock In {\em Proceedings of the 49th Annual ACM SIGACT Symposium on Theory
of Computing}, STOC 2017, page 1144–1156, New York, NY, USA, 2017.
Association for Computing Machinery.

[59]
Xin Li.
\newblock Non-malleable extractors and non-malleable codes: Partially optimal
\newblock In {\em Proceedings of the 34th Computational Complexity Conference},
CCC '19, Dagstuhl, DEU, 2019. Schloss Dagstuhl--Leibniz-Zentrum fuer

[60]
Xin {Li}.
\newblock Two source extractors for asymptotically optimal entropy, and (many)
more, 2023.
\newblock Available at \url{https://arxiv.org/abs/2303.06802}. To appear at
FOCS 2023.

[61]
Xin Li.
\newblock Two source extractors for asymptotically optimal entropy, and (many)
\newblock {\em ArXiv}, abs/2303.06802, 2023.

[62]
Feng-Hao Liu and Anna Lysyanskaya.
\newblock Tamper and leakage resilience in the split-state model.
\newblock In Reihaneh Safavi-Naini and Ran Canetti, editors, {\em Advances in
Cryptology -- CRYPTO 2012}, pages 517--532, Berlin, Heidelberg, 2012.
Springer Berlin Heidelberg.

[63]
Ludovico Lami, Carlos Palazuelos, and Andreas~J. Winter.
\newblock Ultimate data hiding in quantum mechanics and beyond.
\newblock {\em Communications in Mathematical Physics}, 361:661--708, 2017.

[64]
S.~Massar and S.~Popescu.
\newblock Amount of information obtained by a quantum measurement.
\newblock {\em Phys. Rev. A}, 61:062303, May 2000.

[65]
W.~Matthews and S.~Wehner.
\newblock Finite blocklength converse bounds for quantum channels.
\newblock {\em IEEE Transactions on Information Theory}, 60(11):7317--7329, Nov

[66]
William Matthews, Stephanie Wehner, and Andreas~J. Winter.
\newblock Distinguishability of quantum states under restricted families of
measurements with an application to quantum data hiding.
\newblock {\em Communications in Mathematical Physics}, 291:813--843, 2008.

[67]
Michael~A. Nielsen and Isaac~L. Chuang.
\newblock {\em Quantum Computation and Quantum Information}.
\newblock Cambridge University Press, Cambridge, UK, 2000.

[68]
Maris Ozols.
\newblock Lecture 5: Quantum information processing protocols.
\newblock Quantum computing: Exercise sheet 2. Available at
\url{https://www.cl.cam.ac.uk/teaching/1617/QuantComp/exercise2.pdf}, 2016.

[69]
Asher Peres and William~K. Wootters.
\newblock Optimal detection of quantum information.
\newblock {\em Physical review letters}, 66 9:1119--1122, 1991.

[70]
Adi Shamir.
\newblock How to share a secret.
\newblock {\em Commun. ACM}, 22(11):612–613, nov 1979.

[71]
Akshayaram Srinivasan and Prashant~Nalini Vasudevan.
\newblock Leakage resilient secret sharing and applications.
\newblock In Alexandra Boldyreva and Daniele Micciancio, editors, {\em Advances
in Cryptology -- CRYPTO 2019}, pages 480--509, Cham, 2019. Springer
International Publishing.

[72]
Barbara~Maria Terhal, David~P. DiVincenzo, and Debbie~W. Leung.
\newblock Hiding bits in bell states.
\newblock {\em Physical review letters}, 86 25:5807--10, 2000.

[73]
Barbara~Maria Terhal and Paweł Horodecki.
\newblock Schmidt number for density matrices.
\newblock {\em Physical Review A}, 61:040301, 1999.

[74]
Marco Tomamichel.
\newblock Quantum information processing with finite resources - mathematical
\newblock {\em ArXiv}, abs/1504.00233, 2015.

[75]
Luca Trevisan.
\newblock Extractors and pseudorandom generators.
\newblock {\em J. ACM}, 48(4):860--879, July 2001.

[76]
John Watrous.
\newblock Chapter 5, {Naimark}’s theorem; characterizations of channels.
\newblock Lecture notes on Theory of Quantum Information, 2011.

[77]
John Watrous.
\newblock {\em The Theory of Quantum Information}.
\newblock Cambridge University Press, Cambridge, UK, 2018.

[78]
Mark~M. Wilde.
\newblock From classical to quantum shannon theory.
\newblock {\em ArXiv}, abs/1106.1445, 2011.

[79]
Andreas {Winter}.
\newblock {Reflections on Quantum Data Hiding}.
\newblock In {\em APS March Meeting Abstracts}, volume 2017 of {\em APS Meeting
Abstracts}, page S52.001, March 2017.

[80]
Jing-Tao Wang, Gang Xu, Xiubo Chen, Xingming Sun, and Hengyue Jia.
\newblock Local distinguishability of dicke states in quantum secret sharing.
\newblock {\em Physics Letters A}, 381:998--1002, 2017.

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