# Next-to-Next-to-Leading-Order QCD Prediction for the Photon-Pion Form Factor

Jing Gao<EMAIL_ADDRESS>School of Physics, Nankai University,
Weijin Road 94, 300071 Tianjin, China  Naturwissenschaftlich-Technische
Fakultät, Universität Siegen,
Walter-Flex-Str. 3, 57068 Siegen, Germany  Institute of High Energy Physics,
CAS, P.O. Box 918(4) Beijing 100049, China  School of Physics, University of
Chinese Academy of Sciences, Beijing 100049, China  Tobias Huber
corresponding author<EMAIL_ADDRESS>Naturwissenschaftlich-
Technische Fakultät, Universität Siegen,
Walter-Flex-Str. 3, 57068 Siegen, Germany  Yao Ji  corresponding author:
<EMAIL_ADDRESS>Naturwissenschaftlich-Technische Fakultät, Universität
Siegen,
Walter-Flex-Str. 3, 57068 Siegen, Germany  Yu-Ming Wang  corresponding author:
<EMAIL_ADDRESS>School of Physics, Nankai University,
Weijin Road 94, 300071 Tianjin, China

###### Abstract

We accomplish the complete two-loop computation of the leading-twist
contribution to the photon-pion transition form factor
$\gamma\,\gamma^{\ast}\to\pi^{0}$ by applying the hard-collinear factorization
theorem together with modern multi-loop techniques. The resulting predictions
for the form factor indicate that the two-loop perturbative correction is
numerically important. We also demonstrate that our results will play a key
role in disentangling various models of the twist-two pion distribution
amplitude thanks to the envisaged precision at Belle II.

Keywords: Hard exclusive processes, multi-loop computations, pion distribution
amplitudes

††preprint: SI-HEP-2021-16, P3H-21-036

##  I introduction

It is widely accepted that the exclusive two-photon production of a light
pseudoscalar meson $\pi^{0}$ is of utmost importance for probing the partonic
landscape of composite hadrons with unprecedented precision and for deepening
our understanding of the factorization properties of hard exclusive QCD
reactions in general. Historically, the photon-pion transition form factor
(TFF) had been explored intensively (see, for instance Cornwall (1966); Gross
and Treiman (1971); Brodsky  _et al._ (1971)) even before the advent of QCD
due to its apparent connection to the axial anomaly Adler (1969); Bell and
Jackiw (1969); Bardeen (1969) when taking the vanishing virtuality limits for
the two photons. Likewise, for the energetic photo-production at the light-
like distance, the dynamical behavior of the TFF
$\gamma^{\ast}\,\gamma^{(\ast)}\to\pi^{0}$ can be predicted by applying the
operator-product-expansion (OPE) technique for the time-ordered product of two
electromagnetic currents Lepage and Brodsky (1979, 1980). The appearing
correlation function with the distinct kinematic set-up concurrently provides
the theory description of a variety of two-photon processes including deeply
inelastic lepton-hadron scattering (DIS) Feynman (1969); Bjorken (1969);
Bjorken and Paschos (1969) and deeply virtual Compton scattering (DVCS) Müller
_et al._ (1994); Ji (1997); Radyushkin (1996), whose importance in
establishing QCD as the theory of strong interactions and in accessing the
transverse-distance structure of the target hadron cannot be exaggerated (see,
for instance Gross (1975); Collins (2013); Dudek  _et al._ (2012); Accardi
_et al._ (2016)). In particular, the double-virtual photon-to-pion form factor
has been demonstrated to be an indispensable ingredient for determining the
hadronic light-by-light scattering (HLbL) contribution to the anomalous
magnetic moment of the muon in the dispersive framework Colangelo  _et al._
(2014). The comprehensive data-driven strategy of determining the pion-pole
contribution to the HLBL scattering further illustrates that the yielding
hadronic uncertainties of $(g-2)_{\mu}$ can be reduced substantially with the
improved determination of the single-virtual photon-pion form factor
Hoferichter  _et al._ (2018a, b). Moreover, the collinear QCD dynamics
dictating the TFF at large momentum transfer is encapsulated in the universal
light-cone distribution amplitude (DA), which is undoubtedly of decisive
importance for the model-independent description of semileptonic and
nonleptonic $B$-meson decays, such as $B\to\pi\,\ell^{-}\,\bar{\nu}_{\ell}$,
$B\to\pi\,\ell^{+}\,\ell^{-}$ and $B\to\pi\,\pi$, on the basis of QCD
factorization demanding the detailed information on the very pion DA as the
fundamental non-perturbative input Beneke and Feldmann (2001, 2004); Beneke
_et al._ (2005, 2001). Additionally, the lattice QCD technique has been
applied to address the photon-pion form factor at low photon virtualities with
very encouraging predictions Gérardin  _et al._ (2016, 2019).

Experimentally, the pion TFF $\gamma\,\gamma^{\ast}\to\pi^{0}$ with one on-
shell and one off-shell photon turns out to be more accessible from the
“single-tagged” measurements of the differential $e^{+}\,e^{-}\to
e^{+}\,e^{-}\,\pi^{0}$ cross section Gronberg  _et al._ (1998); Aubert  _et
al._ (2009); Uehara  _et al._ (2012). The unexpected scaling violation of the
TFF reported by the BaBar measurements Aubert  _et al._ (2009) triggered a
storm of enthusiasm in the theory community with diverse speculations from
both perturbative and non-perturbative QCD perspectives. The subsequent Belle
data Uehara  _et al._ (2012) covering the same kinematical region
$Q^{2}\in[4,40]\,{\rm GeV^{2}}$ (where $Q^{2}$ stands for the virtuality of
the off-shell photon), however, did not reveal the pronounced increase of the
pion TFF in the high-$Q^{2}$ region thus creating a moderate tension at the
level of $2\sigma$ in comparison with the BaBar results. The substantial
improvement on the integrated luminosity and the trigger efficiency of the
Belle II experiment at SuperKEKB Altmannshofer  _et al._ (2019) will evidently
enable a clarification of the observed discrepancy between the BaBar and the
Belle data and urgently demand the high-precision theory computation of the
$\gamma\,\gamma^{\ast}\to\pi^{0}$ TFF in QCD.

At leading power (LP) in $\Lambda_{\rm QCD}^{2}/Q^{2}$ the photon-pion TFF can
be expressed in terms of the perturbatively calculable coefficient function
(CF) and the twist-two pion DA Lepage and Brodsky (1980) (see also Li and
Sterman (1992); Musatov and Radyushkin (1997); Li  _et al._ (2014)). The
short-distance CF at the next-to-leading order (NLO) in $\alpha_{s}$ had been
determined more than three decades ago del Aguila and Chase (1981); Braaten
(1983); Kadantseva  _et al._ (1986). The next-to-next-to-leading-order (NNLO)
computation of the perturbative matching coefficient was further carried out
in the large-$\beta_{0}$ approximation Melic  _et al._ (2002) and in the so-
called conformal scheme Melic  _et al._ (2003). Consequently, the
factorization-scale invariance of the resulting expression for the pion TFF
was not achieved in the modified minimal subtraction ($\overline{\rm MS}$)
scheme at two loops. Accomplishing the full NNLO QCD prediction to the TFF
$\gamma\,\gamma^{\ast}\to\pi^{0}$ is therefore of the top priority, on the one
hand, for developing the perturbative factorization formalism of hard
exclusive reactions; and on the other hand, for elevating our capability to
confront the yielding theory predictions with the forthcoming precision of
Belle II measurements. In this Letter, we will take advantage of the QCD
factorization program to obtain analytically the two-loop hard function by
evaluating an appropriate bare QCD matrix element at ${\cal
O}(\alpha_{s}^{2})$ by means of dedicated multi-loop computational strategies
and by implementing the ultraviolet (UV) renormalization and the infrared (IR)
subtractions, with an exploratory phenomenological analysis regarding the
numerical significance of the predicted TFF.

##  II The photon-pion form factor

We first set up the theory framework for establishing the hard-collinear
factorization formula of the TFF $F_{\gamma\pi}(Q^{2})$. The latter is defined
in terms of the matrix element of the electromagnetic current

$j_{\mu}^{\rm em}=\sum_{q}\,g_{\rm em}\,e_{q}\,\bar{q}\,\gamma_{\mu}\,q$ (1)

between an on-shell photon with momentum $p^{\prime}$ and a pion with momentum
$p$

$\displaystyle\langle\pi(p)|j_{\mu}^{\rm em}|\gamma(p^{\prime})\rangle=g_{\rm
em}^{2}\epsilon_{\mu\nu\alpha\beta}q^{\alpha}p^{\beta}\epsilon^{\nu}(p^{\prime})F_{\gamma\pi}(Q^{2})\,,$
(2)

where $q=p-p^{\prime}$ refers to the transfer momentum,
$\epsilon^{\nu}(p^{\prime})$ is the polarization vector of the on-shell
photon, and $e_{q}$ denotes the electric charge of the quark field in units of
$g_{\rm em}$ (the positron charge). Moreover, we have employed the convention
$\epsilon_{0123}=-1$ and the notation $Q^{2}=-q^{2}$. Introducing further two
light-cone vectors $n_{\mu}$ and $\bar{n}_{\mu}$ with $n^{2}=\bar{n}^{2}=0$
and $n\cdot\bar{n}=2$ allows for the decomposition $p_{\mu}=(\bar{n}\cdot
p)/2\,n_{\mu}$ and $p^{\prime}_{\mu}=(n\cdot p)/2\,\bar{n}_{\mu}$.

Applying the hard-collinear factorization theorem results in the LP
contribution to the $\gamma\,\gamma^{\ast}\to\pi^{0}$ form factor

$\displaystyle F^{\rm LP}_{\gamma\pi}(Q^{2})$
$\displaystyle=\frac{(e_{u}^{2}-e_{d}^{2})f_{\pi}}{\sqrt{2}\,Q^{2}}\int^{1}_{0}dx\,T_{2}(x,Q^{2},\mu_{F})\,\phi_{\pi}(x,\mu_{F})$
$\displaystyle\equiv\,\frac{(e_{u}^{2}-e_{d}^{2})f_{\pi}}{\sqrt{2}\,Q^{2}}T_{2}(x,Q^{2},\mu_{F})\otimes\phi_{\pi}(x,\mu_{F})\,,$
(3)

where the pion decay constant $f_{\pi}=(130.50\pm 0.02\pm 0.03\pm 0.13)~{}{\rm
MeV}$ is determined from the experimental measurement Zyla  _et al._ (2020)
and $\mu_{F}$ represents the factorization scale for which we tacitly adopt
$\mu_{F}=\mu_{\rm UV}\equiv\mu$ in (3). Here the $\pi^{0}$ wavefunction
$|{\pi^{0}}\rangle=\frac{1}{\sqrt{2}}(|{u\bar{u}}\rangle-|{d\bar{d}}\rangle)$
with exact isospin symmetry is employed. The CF $T_{2}$ can be expanded
perturbatively in the form (similarly for any other partonic quantity)

$\displaystyle
T_{2}=\sum_{\ell=0}^{\infty}a_{s}^{\ell}\,T_{2}^{(\ell)}\,,\qquad
a_{s}\equiv\frac{\alpha_{s}}{4\pi}\,.$ (4)

The twist-two pion DA $\phi_{\pi}(x,\mu_{F})$ is defined by the renormalized
matrix element on the light-cone

$\displaystyle\langle{\pi(p)}|\,[\bar{q}(z\bar{n})[z\bar{n},0]\gamma_{\mu}\gamma_{5}q(0)]_{R}\,|{0}\rangle$
$\displaystyle=-if_{\pi}p_{\mu}\int^{1}_{0}dx\,\mathop{\rm
e}\nolimits^{ixz\bar{n}\cdot p}\,\phi_{\pi}(x,\mu_{F})\,,$ (5)

where $[z\bar{n},0]$ is the Wilson line to ensure gauge invariance. We also
note that $\phi_{\pi}(x,\mu_{F})=\phi_{\pi}(\bar{x},\mu_{F})$ due to charge
symmetry, where $\bar{x}\equiv 1-x$. The corresponding renormalization group
(RG) equation at one loop Efremov and Radyushkin (1980); Lepage and Brodsky
(1980) implies the series expansion of $\phi_{\pi}(x,\mu_{F})$ in terms of
Gegenbauer polynomials,

$\displaystyle\phi_{\pi}(x,\mu_{F})=6\,x\,\bar{x}\,\sum_{n=0}^{\infty}a_{n}(\mu_{F})\,C_{n}^{3/2}(2\,x-1)\,,$
(6)

with the vanishing odd-moments $a_{1,3,...}(\mu_{F})$.

It is convenient to evaluate the CF by inspecting the following correlation
function

$\displaystyle\frac{g_{\rm em}^{2}e_{q}^{2}}{2\,\bar{n}\cdot
p}\,\Pi_{\mu\nu}=i\int\\!\\!d^{4}z\mathop{\rm e}\nolimits^{-iq\cdot z}$
$\displaystyle\qquad\qquad\times\langle{\bar{q}(\bar{x}p)q(xp)}|{\rm
T}\\{j^{\rm em}_{\mu}(z),\,j^{\rm em}_{\nu}(0)\\}|{0}\rangle\,,$ (7)

which can be parameterized by the two form factors for the bilinear quark
currents with the spin structures Wang and Shen (2017)

$\displaystyle\Gamma_{A}^{\mu\nu}$
$\displaystyle=\gamma_{\perp}^{\mu}\,\not{\bar{n}}\,\gamma_{\perp}^{\nu}\;,$
$\displaystyle\Gamma_{B}^{\mu\nu}$
$\displaystyle=\gamma_{\perp}^{\nu}\,\not{\bar{n}}\,\gamma_{\perp}^{\mu}\;,$
(8)

thanks to the QED Ward identity and charge symmetry. The perpendicular
component of any four-vector $p^{\mu}$ is defined by
$p^{\mu}=(np)\bar{n}^{\mu}/2+(\bar{n}p)n^{\mu}/2+p^{\mu}_{\perp}$. We will
dedicate the next section to the two-loop calculation of the bare quantity
$\Pi_{\mu\nu}^{(2)}$ and subsequently derive the master formula for the CF
$T_{2}$ in (3).

##  III Two-loop calculation

The techniques entering the calculation of the bare two-loop amplitude have
become standard in contemporary multi-loop computations. We first generate the
Feynman diagrams with both Feynarts Hahn (2001) and by means of an in-house
routine. Sample diagrams are shown in Fig. 1. After taking into account that
certain diagrams have color factor zero, vanish due to the Furry theorem
and/or represent the flavor-singlet contributions, a set of $42$ diagrams
(plus the ones with the two photons exchanged) has to be computed. The entire
calculation is carried out in dimensional regularization with $d=4-2\epsilon$.

Figure 1: Sample Feynman diagrams.

The Dirac and tensor reductions are performed with in-house routines, yielding
two-loop scalar integrals, which get further processed with FIRE Smirnov
(2008), an implementation of Laporta’s algorithm Laporta and Remiddi (1996);
Laporta (2000) relying on integration-by-parts identities Chetyrkin and
Tkachov (1981); Tkachov (1981). In addition, we exploit the fact that for the
quark momenta $p_{1}\equiv xp\propto\bar{x}p\equiv p_{2}$, which yields
relations between integrals based on momentum conservation and enables us to
arrive at the minimal set of master integrals. At this stage, additional Dirac
structures besides $\Gamma_{A,B}^{\mu\nu}$ disappear from the sum of all
diagrams, making the QED Ward identity and charge symmetry manifest and
providing a valuable check of our calculation.

In total, we get $12$ independent master integrals, the analytic
$\epsilon$-expansion of which can be written in terms of harmonic
polylogarithms (HPLs) Remiddi and Vermaseren (2000); Maitre (2006, 2012). We
relegate the explicit analytic results of all master integrals to a
forthcoming longer write-up.

##  IV Ultraviolet Renormalization and Infrared Subtractions

We are now prepared to extract the two-loop contribution to $T_{2}$ by first
introducing the operator basis
$\\{O_{1}^{\mu\nu},\,O_{2}^{\mu\nu},\,O_{E}^{\mu\nu}\\}$ with

$\displaystyle O_{j}^{\mu\nu}(x)=\frac{\bar{n}\cdot p}{2\pi}\int
d\tau\,\mathop{\rm e}\nolimits^{{i\bar{x}\tau\bar{n}\cdot
p}}\bar{q}(\tau\bar{n})\,[\tau\bar{n},0]\,\Gamma_{j}^{\mu\nu}\,q(0)\,,$ (9)

where

$\displaystyle\Gamma_{1}^{\mu\nu}$
$\displaystyle=g_{\perp}^{\mu\nu}\not{\bar{n}}\,,\quad\Gamma_{2}^{\mu\nu}=i\epsilon_{\perp}^{\mu\nu}\not{\bar{n}}\gamma_{5}\,,\quad\Gamma_{E}^{\mu\nu}=\not{\bar{n}}\sigma_{\perp}^{\mu\nu}-\Gamma_{2}^{\mu\nu}\,,$
(10)

with
$\sigma_{\perp}^{\mu\nu}=\frac{1}{2}[\gamma_{\perp}^{\mu},\gamma_{\perp}^{\nu}]$
and then by exploiting the matching equation

$\displaystyle\Pi^{\mu\nu}=\sum\limits_{a=1,2,E}T_{a}\otimes\langle
O_{a}^{\mu\nu}\rangle\,.$ (11)

It is evident from the definitions
$g_{\perp}^{\mu\nu}=g^{\mu\nu}-n^{\mu}\bar{n}^{\nu}/2-n^{\nu}\bar{n}^{\mu}$/2
and
$\epsilon^{\mu\nu}_{\perp}\equiv\epsilon^{\mu\nu\alpha\beta}\bar{n}_{\alpha}\,n_{\beta}/2$
that $O_{E}^{\mu\nu}$ is an evanescent operator vanishing at $d=4$, and the
CP-even operator $O_{1}^{\mu\nu}$ cannot couple to the pseudoscalar $\pi^{0}$
state. We therefore encounter a unique physical operator $O_{2}^{\mu\nu}$ Wang
and Shen (2017).

The correlator $\Pi^{\mu\nu}$ assumes the following form to all orders in
$\alpha_{s}$ in terms of the tree-level matrix elements of the effective
operators $O_{1,2,E}^{\mu\nu}$

$\displaystyle\Pi^{\mu\nu}=\\!\\!\sum_{k=1,2,E}\,\sum_{\ell=0}^{\infty}(Z_{\alpha}a_{s})^{\ell}\,A_{k}^{(\ell)}(x)\,\langle{\bar{q}(\bar{x}p)\Gamma_{k}^{\mu\nu}q(xp)}\rangle^{(0)}\,,$
(12)

where $Z_{\alpha}=1-a_{s}\beta_{0}/\epsilon+{\cal O}(a_{s}^{2})$. Note that,
hereafter, we disregard $O_{1}^{\mu\nu}$ completely due to parity. The
expansion of $T_{2}$ was already given in Eq. (4), and due to scaleless
integrals in dimensional regularization the matrix elements of the light-cone
operators are expanded as

$\displaystyle\langle{O_{a}^{\mu\nu}}\rangle=$
$\displaystyle\sum_{\ell=0}^{\infty}a_{s}^{\ell}\,Z_{ab}^{(\ell)}\otimes\langle{O_{b}^{\mu\nu}}\rangle^{(0)}$
(13) $\displaystyle=$
$\displaystyle\left\\{\delta_{ab}+a_{s}\,Z_{ab}^{(1)}+a_{s}^{2}\,Z_{ab}^{(2)}+{\cal
O}(a_{s}^{3})\right\\}\otimes\langle O_{b}^{\mu\nu}\rangle^{(0)}\,.$

Sums over repeated indices are understood to run over $\\{2,E\\}$, $\langle
O_{a}^{\mu\nu}\rangle^{(0)}$ is the tree-level matrix element of
$O_{a}^{\mu\nu}$, and the renormalization constants $Z_{22}^{(\ell)}$ can be
extracted from the Efremov-Radyushkin-Brodsky-Lepage (ERBL) kernel with the
one- and two-loop result given in Efremov and Radyushkin (1980); Lepage and
Brodsky (1980) and Sarmadi (1984); Dittes and Radyushkin (1984); Katz (1985);
Mikhailov and Radyushkin (1985); Belitsky  _et al._ (1999), respectively.
Inserting Eqs. (4), (12), and (13) into Eq. (11) and comparing coefficients of
$\langle O_{2,E}^{\mu\nu}\rangle^{(0)}$ at ${\cal O}(a_{s}^{2})$ leads to the
following “master formula” for the CF at NNLO

$\displaystyle T_{2}^{(2)}$
$\displaystyle=A_{2}^{(2)}+Z_{\alpha}^{(1)}A_{2}^{(1)}-\sum_{a=2,E}\sum_{k=0}^{1}Z_{a2}^{(2-k)}\otimes
T_{a}^{(k)}\,.$ (14)

All quantities multiplied by a divergent term in (14) must be expanded beyond
${\cal O}(\epsilon^{0})$ to correctly capture all finite terms. It remains
important to emphasize the role of the evanescent operator in the master
formula (14). Despite the vanishing of evanescent operators in four
dimensions, their non-vanishing bare matrix element and evanescent-to-physical
mixing (captured by $Z_{E2}$ Dugan and Grinstein (1991)) are crucial for
correctly implementing the IR subtraction and establishing the hard-collinear
factorization in our $d$-dimensional framework. These steps are necessary in
view of the fact that taking the limit $d\to 4$ does not commute with the
hard-collinear factorization for the correlation function $\Pi^{\mu\nu}$ (see
also Braun  _et al._ (2004); Beneke  _et al._ (2010)).

Substituting the newly derived two-loop correction to the hard function, whose
explicit expression is given in the supplemental material, into the
factorization formula (3) and adopting further the asymptotic pion DA enable
us to write down

$\displaystyle F^{\rm LP,\,Asy}_{\gamma\pi}(Q^{2})$
$\displaystyle=\frac{(e_{u}^{2}-e_{d}^{2})f_{\pi}}{\sqrt{2}\,Q^{2}}\,\bigg{\\{}6-30\,a_{s}\,C_{F}-a_{s}^{2}\,\bigg{[}C_{F}\beta_{0}\,\left(\left(31+12\log\frac{\mu^{2}}{Q^{2}}\right)\zeta_{2}+6\zeta_{3}+7\right)$
(15)
$\displaystyle\quad\left.\left.+\,C_{F}^{2}\,\left(24(\zeta_{2}+\zeta_{3})\log\frac{\mu^{2}}{Q^{2}}+42\zeta_{4}+54\zeta_{3}+37\zeta_{2}-\frac{85}{2}\right)-\frac{C_{F}}{N_{c}}\,\left(6\zeta_{4}-12\zeta_{3}-2\zeta_{2}+13\right)\right]+{\cal
O}(\alpha_{s}^{3})\right\\}\,.$

Generalizing (15) to account for the non-asymptotic correction will give rise
to lengthy expressions due to the intricacy of the NNLO CF. Consequently, we
will only evaluate such higher moment effect numerically in the
phenomenological exploration. In particular, it has been verified explicitly
that the photon-pion form factor computed from the factorization formula (3)
is independent of the factorization scale at ${\cal O}(\alpha_{s}^{2})$, with
the aid of the two-loop evolution equation of the twist-two pion DA Sarmadi
(1984); Dittes and Radyushkin (1984); Katz (1985); Mikhailov and Radyushkin
(1985); Belitsky  _et al._ (1999). Subsequently, the complete next-to-next-to-
leading-logarithmic (NNLL) resummation of $\log(Q^{2}/\Lambda_{\rm QCD}^{2})$
appearing in (3) is achieved by virtue of the three-loop evolution equation of
$\phi_{\pi}(x,\mu_{F})$ in the naive dimensional regularization (NDR) scheme
Braun  _et al._ (2017), which is implemented throughout our computation.

##  V Numerical analysis

We now turn to explore the numerical implications of the achieved NNLL
computation of $F_{\gamma\pi}(Q^{2})$ with three phenomenological models of
$\phi_{\pi}(x,\mu_{0})$ at $\mu_{0}=1\,{\rm GeV}$,

$\displaystyle\allowdisplaybreaks{\rm Model\,\,I:}\,\,\,$
$\displaystyle\phi_{\pi}(x,\mu_{0})=\frac{\Gamma(2+2\,\alpha_{\pi})}{\Gamma^{2}(1+\alpha_{\pi})}\,(x\,\bar{x})^{\alpha_{\pi}}\,,$
$\displaystyle{\rm
with}\,\,\alpha_{\pi}(\mu_{0})=0.422^{+0.076}_{-0.067}\,;\qquad$
$\displaystyle{\rm Model\,\,II:}\,\,\,$
$\displaystyle\\{a_{2},a_{4}\\}(\mu_{0})=\\{0.269(47),0.185(62)\\}\,,$
$\displaystyle\\{a_{6},a_{8}\\}(\mu_{0})=\\{0.141(96),0.049(116)\\}\,;$
$\displaystyle{\rm Model\,\,III:}\,\,\,$
$\displaystyle\\{a_{2},a_{4}\\}(\mu_{0})=\\{0.203^{+0.069}_{-0.057}\,,-0.143^{+0.094}_{-0.087}\\}\,.$

Model I is inspired from the AdS/QCD correspondence Brodsky and de Teramond
(2008) with modifications from the most recent lattice result $a_{2}(2\,{\rm
GeV})=0.116^{+0.019}_{-0.020}$ Bali  _et al._ (2019), which corresponds to
$a_{2}(\mu_{0})=0.176^{+0.027}_{-0.029}$. The construction of Model II is
achieved by matching the light-cone QCD sum rules of the pion electromagnetic
form factor in the space-like region onto the modified dispersion integral of
the modulus of the time-like form factor Cheng  _et al._ (2020). The intervals
of $a_{2}$ and $a_{4}$ in Model III Bakulev  _et al._ (2001); Mikhailov  _et
al._ (2016); Stefanis (2020) are determined from the method of QCD sum rules
Mikhailov and Radyushkin (1992). To facilitate the implementation of the
three-loop evolution of the leading-twist pion DA Braun  _et al._ (2017);
Strohmaier (2018), we will apply the Gegenbauer expansion for the
“holographic”-type Model I by discarding $a_{2n}(\mu_{0})$ with $n\geq 7$ (see
also Agaev  _et al._ (2011)).

Figure 2: The twist-2 predictions of $F_{\gamma\pi}$ at the leading-
logarithmic (LL) order [black dotted], at the next-to-leading-logarithmic
(NLL) order [green dashed], and at the NNLL order [blue dot-dashed] in QCD for
Model I given in (V), where the red solid curve is obtained by adding further
the subleading power contributions evaluated in Khodjamirian (1999); Wang and
Shen (2017).

Adopting Model I as our default choice, we present in Fig. 2 the resulting
impacts of QCD radiative corrections to the TFF at LP as well as the numerical
features of the power suppressed terms from the twist-four collinear dynamics
of the pion system Khodjamirian (1999) and from the long-distance photon
correction Wang and Shen (2017). Inspecting the numerical patterns of the
various dynamical mechanisms displayed in Fig. 2 implies that the NNLL twist-
two correction can reduce the corresponding NLL prediction by an amount of
$(4\sim 7)\,\%$ at $Q^{2}\in[3,40]~{}{\rm GeV^{2}}$ (in agreement with the
previous estimate Mikhailov  _et al._ (2016)), while the genuine one-loop
perturbative correction is responsible for a constant $\sim 10\%$ reduction of
the LL contribution in the same kinematic domain. By contrast, the above-
mentioned subleading contributions are observed to bring about the negligible
corrections Wang and Shen (2017). The full two-loop computation of the LP
effect in the TFF is therefore indeed of extraordinary phenomenological
significance. Additionally, the three-loop QCD correction estimated with the
“naïve non-Abelianization” prescription Beneke and Braun (1995) is expected to
be approximately ${\cal O}(10-20)\%$ of the one-loop radiative effect without
adding further the systematic uncertainty.

Figure 3: Theory predictions of the TFF $\gamma\gamma^{\ast}\to\pi^{0}$ for
the three models presented in (V). The color bands are due to variation of the
factorization/renormalization scale $\mu$. For a comparison, we also display
the experimental measurements from the CLEO Gronberg  _et al._ (1998) (purple
squares), BaBar Aubert  _et al._ (2009) (orange circles) and Belle Uehara  _et
al._ (2012) (brown spades) Collaborations.

We proceed to present in Fig. 3 the theory predictions of the TFF for the
three phenomenological models of the leading-twist pion DA. As the dominating
non-perturbative uncertainties of our numerical analysis from
$\phi_{\pi}(x,\mu_{0})$ can be inferred from the discrepancies of the obtained
theoretical results for the distinct sample models, the quoted uncertainties
in (V) are not included in the three shaded (red, green, gray) bands in Fig. 3
to highlight the perturbative uncertainty from the variation of
$\mu^{2}=\langle x\rangle\,Q^{2}$ with $1/4\leq\langle x\rangle\leq 3/4$ Agaev
_et al._ (2011). The distinctive snapshot of the well-separated uncertainty
bands for the three models is particularly encouraging to differentiate
numerous speculations on the intricate behaviors of the twist-two pion DA.
Furthermore, the theory predictions from the incomplete two-loop computation
of $F_{\gamma\pi}(Q^{2})$ in the large $\beta_{0}$ approximation differ from
our full ${\cal O}(\alpha_{s}^{2})$ results by almost a factor of two at
high-$Q^{2}$. Additionally, the NNLL improved hard-collinear factorization
tends to validate the scaling behaviour of the photon-pion form factor at
$Q^{2}\geq 20\,{\rm GeV^{2}}$ with an impressive precision of ${\cal
O}(1\,\%)$, which will be confronted with the future Belle II data. As
expected, such scaling behaviour is broken down at low momentum transfer,
numerically $Q^{2}\leq 5\,{\rm GeV^{2}}$, due to the pronounced subleading
power contributions. A more detailed investigation of the asymptotic behaviour
of $F_{\gamma\pi}(Q^{2})$ in the hard-collinear factorization framework can be
pushed forward by introducing the scaling-rate quantity $\Omega(Q^{2})$ as
proposed in Stefanis (2019, 2020).

We finally address the actual impact of the NNLL correction to the photon-pion
form factor on the model-independent determination of the twist-two pion DA in
anticipation of the upcoming precision Belle II measurements. To this end, we
investigate the intrinsic sensitivity of the $\gamma\,\gamma^{\ast}\to\pi^{0}$
form factor with regard to the poorly constrained Gegenbauer moment
$a_{4}(\mu_{0})$, by employing the improved lattice result for
$a_{2}(\mu_{0})$ Bali  _et al._ (2019) and by further leaving out the higher
moments $a_{n\geq 6}(\mu_{0})$ for illustration purpose. The numerical
implication displayed in Fig. 4 reveals that the two-loop computation of the
photon-pion TFF is indeed highly beneficial for pinning down the theory
uncertainty of the extracted value of $a_{4}(\mu_{0})$ (up to a factor of two
improvement approximately when compared with the corresponding one-loop
result).

Figure 4: Illustration of the $a_{4}(\mu_{0})$ dependence of the photon-pion
form factor at $Q^{2}=27.30\,{\rm GeV^{2}}$ with the second Gegenbauer moment
determined from Bali  _et al._ (2019) and with the vanishing higher Gegenbauer
moments (i.e., $a_{n\geq 6}(\mu_{0})=0$).

##  VI Conclusions

In summary, we have endeavored to carry out, for the first time, the complete
two-loop computation of the leading-twist contribution to the photon-pion form
factor in the hard-collinear factorization framework. The phenomenological
significance of such higher-order perturbative effect was further addressed by
employing the three non-perturbative models of the twist-two pion DA. The
genuine ${\cal O}(\alpha_{s}^{2})$ correction to the hard matching coefficient
was demonstrated to be numerically important at $Q^{2}\in[3,40]\,{\rm
GeV^{2}}$, and in particular, the previously evaluated radiative correction at
${\cal O}(\alpha_{s}^{2}\,\beta_{0})$ only accounts for a moderate portion
(approximately ${\cal O}(50\,\%)$) of the full NNLL QCD effect. Moreover, our
improved theory predictions of the form factor
$\gamma\,\gamma^{\ast}\to\pi^{0}$ appear to be promising for better
constraining the shape parameters of the pion DA. The established two-loop QCD
factorization formula of the photon-pion form factor is then naturally
expected to be of notable importance for exploiting the many facets of the
strong interaction dynamics embedded in hard exclusive reactions.

###### Acknowledgements.

## Acknowledgements

We would like to thank Ulrich Haisch for useful correspondence, and Vladimir
Braun and Alexander Manashov for sharing their results Braun  _et al._ (2021)
with us prior to publication. We have found full agreement with their
expression for the two-loop coefficient function. Y.J. is also grateful to
Alexander Manashov and Goutam Das for illuminating discussions. J.G. is
partially supported by the Deutscher Akademischer Austauschdienst (DAAD). The
research of T.H. and Y.J. was supported in part by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under grant 396021762
- TRR 257. Y.M.W. acknowledges support from the National Youth Thousand
Talents Program, the Youth Hundred Academic Leaders Program of Nankai
University, the National Natural Science Foundation of China with Grant No.
11675082, 11735010 and 12075125, and the Natural Science Foundation of Tianjin
with Grant No. 19JCJQJC61100.

## References

  * Cornwall (1966) J. M. Cornwall, Phys. Rev. Lett. 16, 1174 (1966).
  * Gross and Treiman (1971) D. J. Gross and S. B. Treiman, Phys. Rev. D 4, 2105 (1971).
  * Brodsky  _et al._ (1971) S. J. Brodsky, T. Kinoshita, and H. Terazawa, Phys. Rev. D 4, 1532 (1971).
  * Adler (1969) S. L. Adler, Phys. Rev. 177, 2426 (1969).
  * Bell and Jackiw (1969) J. S. Bell and R. Jackiw, Nuovo Cim. A 60, 47 (1969).
  * Bardeen (1969) W. A. Bardeen, Phys. Rev. 184, 1848 (1969).
  * Lepage and Brodsky (1979) G. P. Lepage and S. J. Brodsky, Phys. Lett. B 87, 359 (1979).
  * Lepage and Brodsky (1980) G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22, 2157 (1980).
  * Feynman (1969) R. P. Feynman, Conf. Proc. C 690905, 237 (1969).
  * Bjorken (1969) J. D. Bjorken, Phys. Rev. 179, 1547 (1969).
  * Bjorken and Paschos (1969) J. D. Bjorken and E. A. Paschos, Phys. Rev. 185, 1975 (1969).
  * Müller  _et al._ (1994) D. Müller, D. Robaschik, B. Geyer, F. M. Dittes, and J. Hořejši, Fortsch. Phys. 42, 101 (1994), arXiv:hep-ph/9812448 .
  * Ji (1997) X.-D. Ji, Phys. Rev. D 55, 7114 (1997), arXiv:hep-ph/9609381 .
  * Radyushkin (1996) A. V. Radyushkin, Phys. Lett. B 380, 417 (1996), arXiv:hep-ph/9604317 .
  * Gross (1975) D. J. Gross, Conf. Proc. C 7507281, 141 (1975).
  * Collins (2013) J. Collins,  _Foundations of perturbative QCD_ , Vol. 32 (Cambridge University Press, 2013).
  * Dudek  _et al._ (2012) J. Dudek _et al._ , Eur. Phys. J. A 48, 187 (2012), arXiv:1208.1244 [hep-ex] .
  * Accardi  _et al._ (2016) A. Accardi _et al._ , Eur. Phys. J. A 52, 268 (2016), arXiv:1212.1701 [nucl-ex] .
  * Colangelo  _et al._ (2014) G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, JHEP 09, 091 (2014), arXiv:1402.7081 [hep-ph] .
  * Hoferichter  _et al._ (2018a) M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold, and S. P. Schneider, Phys. Rev. Lett. 121, 112002 (2018a), arXiv:1805.01471 [hep-ph] .
  * Hoferichter  _et al._ (2018b) M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold, and S. P. Schneider, JHEP 10, 141 (2018b), arXiv:1808.04823 [hep-ph] .
  * Beneke and Feldmann (2001) M. Beneke and T. Feldmann, Nucl. Phys. B 592, 3 (2001), arXiv:hep-ph/0008255 .
  * Beneke and Feldmann (2004) M. Beneke and T. Feldmann, Nucl. Phys. B 685, 249 (2004), arXiv:hep-ph/0311335 .
  * Beneke  _et al._ (2005) M. Beneke, T. Feldmann, and D. Seidel, Eur. Phys. J. C 41, 173 (2005), arXiv:hep-ph/0412400 .
  * Beneke  _et al._ (2001) M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, Nucl. Phys. B 606, 245 (2001), arXiv:hep-ph/0104110 .
  * Gérardin  _et al._ (2016) A. Gérardin, H. B. Meyer, and A. Nyffeler, Phys. Rev. D 94, 074507 (2016), arXiv:1607.08174 [hep-lat] .
  * Gérardin  _et al._ (2019) A. Gérardin, H. B. Meyer, and A. Nyffeler, Phys. Rev. D 100, 034520 (2019), arXiv:1903.09471 [hep-lat] .
  * Gronberg  _et al._ (1998) J. Gronberg _et al._ (CLEO), Phys. Rev. D 57, 33 (1998), arXiv:hep-ex/9707031 .
  * Aubert  _et al._ (2009) B. Aubert _et al._ (BaBar), Phys. Rev. D 80, 052002 (2009), arXiv:0905.4778 [hep-ex] .
  * Uehara  _et al._ (2012) S. Uehara _et al._ (Belle), Phys. Rev. D 86, 092007 (2012), arXiv:1205.3249 [hep-ex] .
  * Altmannshofer  _et al._ (2019) W. Altmannshofer _et al._ (Belle-II), PTEP 2019, 123C01 (2019), [Erratum: PTEP 2020, 029201 (2020)], arXiv:1808.10567 [hep-ex] .
  * Li and Sterman (1992) H.-n. Li and G. F. Sterman, Nucl. Phys. B381, 129 (1992).
  * Musatov and Radyushkin (1997) I. V. Musatov and A. V. Radyushkin, Phys. Rev. D56, 2713 (1997), arXiv:hep-ph/9702443 [hep-ph] .
  * Li  _et al._ (2014) H.-N. Li, Y.-L. Shen, and Y.-M. Wang, JHEP 01, 004 (2014), arXiv:1310.3672 [hep-ph] .
  * del Aguila and Chase (1981) F. del Aguila and M. K. Chase, Nucl. Phys. B 193, 517 (1981).
  * Braaten (1983) E. Braaten, Phys. Rev. D 28, 524 (1983).
  * Kadantseva  _et al._ (1986) E. P. Kadantseva, S. V. Mikhailov, and A. V. Radyushkin, Yad. Fiz. 44, 507 (1986).
  * Melic  _et al._ (2002) B. Melic, B. Nizic, and K. Passek, Phys. Rev. D 65, 053020 (2002), arXiv:hep-ph/0107295 .
  * Melic  _et al._ (2003) B. Melic, D. Mueller, and K. Passek-Kumericki, Phys. Rev. D 68, 014013 (2003), arXiv:hep-ph/0212346 .
  * Zyla  _et al._ (2020) P. A. Zyla _et al._ (Particle Data Group), PTEP 2020, 083C01 (2020).
  * Efremov and Radyushkin (1980) A. V. Efremov and A. V. Radyushkin, Phys. Lett. B 94, 245 (1980).
  * Wang and Shen (2017) Y.-M. Wang and Y.-L. Shen, JHEP 12, 037 (2017), arXiv:1706.05680 [hep-ph] .
  * Hahn (2001) T. Hahn, Comput. Phys. Commun. 140, 418 (2001), arXiv:hep-ph/0012260 .
  * Smirnov (2008) A. V. Smirnov, JHEP 10, 107 (2008), arXiv:0807.3243 [hep-ph] .
  * Laporta and Remiddi (1996) S. Laporta and E. Remiddi, Phys. Lett. B 379, 283 (1996), arXiv:hep-ph/9602417 .
  * Laporta (2000) S. Laporta, Int. J. Mod. Phys. A 15, 5087 (2000), arXiv:hep-ph/0102033 .
  * Chetyrkin and Tkachov (1981) K. G. Chetyrkin and F. V. Tkachov, Nucl. Phys. B 192, 159 (1981).
  * Tkachov (1981) F. V. Tkachov, Phys. Lett. B 100, 65 (1981).
  * Remiddi and Vermaseren (2000) E. Remiddi and J. A. M. Vermaseren, Int. J. Mod. Phys. A 15, 725 (2000), arXiv:hep-ph/9905237 .
  * Maitre (2006) D. Maitre, Comput. Phys. Commun. 174, 222 (2006), arXiv:hep-ph/0507152 .
  * Maitre (2012) D. Maitre, Comput. Phys. Commun. 183, 846 (2012), arXiv:hep-ph/0703052 .
  * Sarmadi (1984) M. H. Sarmadi, Phys. Lett. B 143, 471 (1984).
  * Dittes and Radyushkin (1984) F. M. Dittes and A. V. Radyushkin, Phys. Lett. B 134, 359 (1984).
  * Katz (1985) G. R. Katz, Phys. Rev. D 31, 652 (1985).
  * Mikhailov and Radyushkin (1985) S. V. Mikhailov and A. V. Radyushkin, Nucl. Phys. B 254, 89 (1985).
  * Belitsky  _et al._ (1999) A. V. Belitsky, D. Mueller, and A. Freund, Phys. Lett. B 461, 270 (1999), arXiv:hep-ph/9904477 .
  * Dugan and Grinstein (1991) M. J. Dugan and B. Grinstein, Phys. Lett. B 256, 239 (1991).
  * Braun  _et al._ (2004) V. M. Braun, D. Y. Ivanov, and G. P. Korchemsky, Phys. Rev. D 69, 034014 (2004), arXiv:hep-ph/0309330 .
  * Beneke  _et al._ (2010) M. Beneke, T. Huber, and X.-Q. Li, Nucl. Phys. B832, 109 (2010), arXiv:0911.3655 [hep-ph] .
  * Braun  _et al._ (2017) V. M. Braun, A. N. Manashov, S. Moch, and M. Strohmaier, JHEP 06, 037 (2017), arXiv:1703.09532 [hep-ph] .
  * Brodsky and de Teramond (2008) S. J. Brodsky and G. F. de Teramond, Phys. Rev. D 77, 056007 (2008), arXiv:0707.3859 [hep-ph] .
  * Bali  _et al._ (2019) G. S. Bali, V. M. Braun, S. Bürger, M. Göckeler, M. Gruber, F. Hutzler, P. Korcyl, A. Schäfer, A. Sternbeck, and P. Wein (RQCD), JHEP 08, 065 (2019), [Addendum: JHEP 11, 037 (2020)], arXiv:1903.08038 [hep-lat] .
  * Cheng  _et al._ (2020) S. Cheng, A. Khodjamirian, and A. V. Rusov, Phys. Rev. D 102, 074022 (2020), arXiv:2007.05550 [hep-ph] .
  * Bakulev  _et al._ (2001) A. P. Bakulev, S. V. Mikhailov, and N. G. Stefanis, Phys. Lett. B 508, 279 (2001), [Erratum: Phys.Lett.B 590, 309–310 (2004)], arXiv:hep-ph/0103119 .
  * Mikhailov  _et al._ (2016) S. V. Mikhailov, A. V. Pimikov, and N. G. Stefanis, Phys. Rev. D 93, 114018 (2016), arXiv:1604.06391 [hep-ph] .
  * Stefanis (2020) N. G. Stefanis, Phys. Rev. D 102, 034022 (2020), arXiv:2006.10576 [hep-ph] .
  * Mikhailov and Radyushkin (1992) S. V. Mikhailov and A. V. Radyushkin, Phys. Rev. D 45, 1754 (1992).
  * Strohmaier (2018) M. Strohmaier,  _Conformal symmetry breaking and evolution equations in Quantum Chromodynamics_ , Ph.D. thesis, Regensburg U. (2018).
  * Agaev  _et al._ (2011) S. S. Agaev, V. M. Braun, N. Offen, and F. A. Porkert, Phys. Rev. D 83, 054020 (2011), arXiv:1012.4671 [hep-ph] .
  * Khodjamirian (1999) A. Khodjamirian, Eur. Phys. J. C 6, 477 (1999), arXiv:hep-ph/9712451 .
  * Beneke and Braun (1995) M. Beneke and V. M. Braun, Phys. Lett. B 348, 513 (1995), arXiv:hep-ph/9411229 .
  * Stefanis (2019) N. G. Stefanis, (2019), arXiv:1904.02631 [hep-ph] .
  * Braun  _et al._ (2021) V. M. Braun, A. N. Manashov, S. Moch, and J. Schoenleber, Phys. Rev. D 104, 094007 (2021), arXiv:2106.01437 [hep-ph] .

Supplemental Materials to Next-to-Next-to-Leading-Order QCD Prediction for the
Photon-Pion Form Factor

##  Appendix A Analytic expressions in the $\overline{\rm MS}$-scheme

The master formula (14) was derived under the assumption that dimensional
regularization is used on both sides of Eq. (11) for both UV and IR
divergences. However, to determine the UV-renormalization constants
$Z_{ab}^{(\ell)}$ we apply the following procedure (see e.g. Beneke  _et al._
(2010); Wang and Shen (2017)). The renormalized matrix elements of the
effective operators are expressed as follows

$\displaystyle\langle
O_{a}^{\mu\nu}\rangle=\,\left\\{\delta_{ab}+a_{s}\,\left[M_{ab}^{(1)}+Z_{ab}^{(1)}\right]+a_{s}^{2}\,\left[M_{ab}^{(2)}+Z_{ab}^{(2)}\right.\right.\left.\left.+Z_{a2}^{(1)}\otimes
M_{2b}^{(1)}\right]+{\cal O}(a_{s}^{3})\right\\}\otimes\langle
O_{b}^{\mu\nu}\rangle^{(0)}\,,$ (S1)

and the matrix elements $M_{ab}^{(\ell)}$ are obtained with dimensional
regularization applied only to the UV divergences but with the IR
regularization scheme being different from the dimensional one. Renormalizing
the matrix elements of the evanescent operator to zero Dugan and Grinstein
(1991) yields the relations

$\displaystyle Z_{E2}^{(1)}$ $\displaystyle=-M_{E2}^{(1)}\,,$ $\displaystyle
Z_{E2}^{(2)}$ $\displaystyle=-M_{E2}^{(2)}+M_{E2}^{(1)}\otimes
M_{22}^{(1)}\,,$ (S2)

where $Z_{E2}^{(2)}$ is IR finite as expected albeit both $M_{E2}^{(2)}$ and
$M_{22}^{(1)}$ are IR divergent.

We now present the explicit expressions of the two-loop hard coefficient
function $T_{2}^{(2)}$ with the NDR scheme of $\gamma_{5}$

$\displaystyle T_{2}^{(2)}(x)=$ $\displaystyle\,\beta_{0}\,C_{F}\left({\cal
K}_{\beta}^{(2)}(x)/x+{\cal
K}_{\beta}^{(2)}(\bar{x})/\bar{x}\right)+C_{F}^{2}\left({\cal
K}_{F}^{(2)}(x)/x+{\cal
K}_{F}^{(2)}(\bar{x})/\bar{x}\right)+C_{F}/N_{c}\left({\cal
K}_{N}^{(2)}(x)/x+{\cal K}_{N}^{(2)}(\bar{x})/\bar{x}\right)\,,$ (S3)

where

$\displaystyle{\cal K}_{\beta}^{(2)}(x)=$
$\displaystyle-L^{2}\left(H_{0}(x)+\frac{3}{2}\right)+L\left(-\frac{10}{3}H_{0}(x)-H_{1}(x)+2H_{0,0}(x)-2H_{1,0}(x)-2\zeta_{2}-\frac{19}{2}\right)-\zeta_{2}H_{1}(x)-\frac{19}{9}H_{0}(x)$
$\displaystyle-\frac{1}{2}H_{1}(x)+\frac{10}{3}H_{0,0}(x)-\frac{14}{3}H_{1,0}(x)-H_{1,1}(x)-2H_{0,0,0}(x)+2H_{1,0,0}(x)-H_{1,1,0}(x)-\frac{14}{3}\zeta_{2}-\zeta_{3}-\frac{457}{24}\,,$
(S4) $\displaystyle{\cal K}_{F}^{(2)}(x)=$ $\displaystyle
L^{2}\left(6H_{0}(x)-2H_{1}(x)+4H_{0,0}(x)+2H_{1,0}(x)+\frac{9}{2}\right)+L\left(8\zeta_{2}H_{0}(x)+\frac{38}{3}H_{0}(x)+4\zeta_{2}H_{1}(x)-17H_{1}(x)\right.$
$\displaystyle\left.-6H_{0,0}(x)+8H_{1,0}(x)-2H_{1,1}(x)-12H_{0,0,0}(x)+4H_{1,1,0}(x)-4\zeta_{3}+6\zeta_{2}+\frac{47}{2}\right)+6\zeta_{2}H_{0}(x)+4\zeta_{2}H_{1}(x)$
$\displaystyle-2\zeta_{2}H_{2}(x)-8\zeta_{2}H_{0,0}(x)-2\zeta_{2}H_{1,0}(x)+2\zeta_{2}H_{1,1}(x)+32\zeta_{3}H_{0}(x)-4\zeta_{3}H_{1}(x)-\frac{64}{9}H_{0}(x)-\frac{71}{2}H_{1}(x)$
$\displaystyle-\frac{38}{3}H_{0,0}(x)+\frac{34}{3}H_{1,0}(x)-11H_{1,1}(x)-2H_{1,2}(x)-8H_{1,0,0}(x)+4H_{1,1,0}(x)-2H_{1,1,1}(x)$
$\displaystyle-2H_{1,2,0}(x)-4H_{2,0,0}(x)-2H_{2,1,0}(x)+12H_{0,0,0,0}(x)-2H_{1,0,0,0}(x)-2H_{1,1,0,0}(x)$
$\displaystyle+2H_{1,1,1,0}(x)+3\zeta_{2}^{2}+\frac{34}{3}\,\zeta_{2}+39\zeta_{3}+\frac{701}{24}\,,$
(S5) $\displaystyle{\cal K}_{N}^{(2)}(x)=$ $\displaystyle
L\left(4\zeta_{2}H_{0}(x)-\frac{8}{3}H_{0}(x)-4H_{1}(x)-4H_{3}(x)+4H_{2,0}(x)+12\zeta_{3}-1\right)+12\,x\left(\zeta_{2}H_{0}(x)-H_{3}(x)+H_{2,0}(x)\right)$
$\displaystyle-6\zeta_{2}H_{1}(x)-4\zeta_{2}H_{2}(x)-4\zeta_{2}H_{0,0}(x)+4\zeta_{2}H_{1,0}(x)+2\zeta_{2}H_{1,1}(x)+14\zeta_{3}H_{0}(x)-\frac{32}{9}H_{0}(x)+11H_{1}(x)$
$\displaystyle+4H_{2}(x)+8H_{4}(x)+\frac{8}{3}H_{0,0}(x)+\frac{2}{3}H_{1,0}(x)+2H_{1,1}(x)+6H_{1,2}(x)-2H_{1,3}(x)+6H_{2,2}(x)-4H_{3,0}(x)$
$\displaystyle-4H_{3,1}(x)-6H_{1,1,0}(x)-2H_{1,1,2}(x)+4H_{1,2,0}(x)-6H_{2,0,0}(x)-4H_{2,1,0}(x)-2H_{1,1,0,0}(x)$
$\displaystyle+2H_{1,1,1,0}(x)+\frac{1}{5}\,\zeta_{2}^{2}-\frac{22}{3}\,\zeta_{2}+54\zeta_{3}-\frac{73}{12}\,,$
(S6)

with $L=\log(\mu^{2}/Q^{2})$. The harmonic polylogarithms (HPLs) are defined
recursively by

$\displaystyle H_{\vec{0}_{n}}(x)$ $\displaystyle=\frac{1}{n!}\log^{n}(x)\,,$
$\displaystyle H_{a_{1},a_{2},\ldots,a_{n}}(x)$
$\displaystyle=\int_{0}^{x}dt\;f_{a_{1}}(t)\,H_{a_{2},\ldots,a_{n}}(t)\,.$
(S7)

The subscript of $H$ is called the weight vector. For the weight functions we
have $f_{0}(t)=1/t$, $f_{1}(t)=1/(1-t)$, and $f_{-1}=1/(1+t)$, where the
latter does not occur in the present calculation. A common abbreviation for
the weight vector is to replace occurrences of $k$ zero(s) to the left of $\pm
1$ by $\pm(k+1)$. For example, $H_{0,0,1,0,-1}(x)=H_{3,-2}(x)$. Moreover, all
the HPL functions that we have are representable in the form of simple
$\mathop{\rm Li}\nolimits_{n}$ functions of argument
$\\{x,\bar{x},-x/\bar{x}\\}$ with $n=1,2,3,4$. Our result for the color
structure $C_{F}\beta_{0}$ is in full agreement with Melic  _et al._ (2002).
Besides, we note that the terms in (S6) proportional to $x$ are reproducible
directly from the coefficient function in DIS.


# A unification of the coding theory and OAQEC perspective on hybrid codes

Shayan Majidy<EMAIL_ADDRESS>Department of Mathematics & Statistics,
University of Guelph, Guelph, Ontario, Canada  Institute for Quantum Computing
and Department of Physics and Astronomy, University of Waterloo, Waterloo,
Ontario, Canada

###### Abstract

It has been shown that there is an advantage in transmitting both quantum and
classical information simultaneously over a quantum channel, compared to
independent transmissions. The characterization and construction of such
codes, which we refer to as hybrid codes, has been done from a coding theory
and an operator algebra quantum error correction (OAQEC) perspective. In this
work we unify these two perspectives, and show that the coding theory
formulation is a specific case of the OAQEC perspective. We also generalize
the quantum hamming bound to the hybrid case. To date no such hybrid codes
have been physically implemented. In this work we develop a hybrid code and
provide the encoding and decoding circuit.

##  I Hybrid Codes

The simultaneous transmission of quantum and classical information over a
quantum channel was initially explored in (Devetak and Shor, 2005), where it
was shown that there exist an advantage in transmitting both quantum and
classical information simultaneously compared to independent transmissions.
This work has since been followed up by other groups (Hsieh and Wilde, 2010a,
b; Yard, 2005; Kremsky  _et al._ , 2008).The construction of such hybrid codes
has been developed from a coding theory perspective by (Grassl  _et al._ ,
2017) and from the OAQEC perspective by (Bény  _et al._ , 2007).

An understanding of hybrid codes can be developed by first considering a
trivial example and then a simple one. A two qubit system can be split into
two identical one qubit systems, the first labelled $0$ and the latter $1$.
The sender chooses one of the two qubits to send depending on which classical
bit they want to send, and effectively transmits a single bit of classical
information and a single qubit of quantum information. In this trivial case
$2$ unprotected qubits are converted to $1$ unprotected qubit and $1$
classical bit. A result of superdense coding (Bennett and Wiesner, 1992), it
is known that $1$ qubit of information can be converted to at least $2$
classical bits, which is why such a hybrid construction is trivial.

A simple case would be to consider four states available to a two qubit
system; $|{00}\rangle$, $|{01}\rangle$, $|{10}\rangle$, $|{11}\rangle$. These
four states can provide two orthogonal sets of codewords, labeled with the
classical bits $0$ and $1$ respectively.

$|{\bar{0}}\rangle_{0}=|{00}\rangle\quad|{\bar{1}}\rangle_{0}=|{01}\rangle,\quad|{\bar{0}}\rangle_{1}=|{10}\rangle\quad|{\bar{1}}\rangle_{1}=|{11}\rangle$

This gives two codes which both can correct $Z_{1}$ errors. In this simple
case $2$ unprotected qubits are converted to $1$ partially protected qubit and
$1$ classical bit. From one perspective, quantum error correction can be seen
as a problem of trying to maximize the number of logical qubits that can be
transmitted and the robustness of their protection while minimizing the number
of physical qubits that are required. Hybrid codes add an additional dimension
that one may wish to maximize. This dimension is the amount of classical
information being transmitted. Many other simple examples of hybrid codes can
be found in (Majidy, 2017).

###  I.1 Coding theory perspective

The characterization and construction of hybrid codes was formulated from a
coding theory perspective in (Grassl  _et al._ , 2017). A quantum error-
correcting code that encodes $k$ qubits into $n$ qubits with a distance $d$
can be denoted by $C=[[n,k,d]]$ or equivalently as $C=((n,K,d))$ where $K$ is
the dimensions of the subspace of the $k$ qubits, $K=2^{k}$. A classical code
can be denoted identically, except with $m$ and $M$ representing the number of
classical bits and dimension of the bits subspace respectively, $C=[[n,m,d]]$
or $C=((n,M,d))$ . In this notation a hybrid code can be denoted as
$C=[[n,k:m,d]]$ or $C=((n,K:M,d))$.

Using this notation we can describe three hybrid code constructions which do
not provide an advantage over the independent solutions. First, given a
quantum code, $C=((n,KM,d))$, one can factor the code space into two
subsystems of dimension $K$ and $M$, and use these subspaces to transmit $K$
dimensional quantum and $M$ dimensional classical information seperately. This
effectively sacrifices quantum bits for classical bits. A second trivial
construction comes from assuming one already has a hybrid code $C=[[n,k:m,d]]$
and constructing a new code $C^{\prime}=[[n,k−1:m+1,d]]$. A qubit can always
be used to transmit classical information, making this construction trivial as
well. Finally, given a quantum code $C_{q}=[[n1,k,d]]$ and classical code
$C_{c}=[[n2,m,d]]$ one can form a hybrid code $C=[[n1+n2,k+m,d]]$, which again
would not provide any advantage. Our goal then, in developing hybrid codes is
to find codes with better parameters than those provided.

A hybrid quantum code $C=((n,K:M,d))$ can be described by a collection of $M$
quantum codes $\\{C^{(\nu)}:\nu=1,...,M\\}$, where $\nu$ is the classical
information that determines which of the $C^{(\nu)}$ is used. Each code has an
orthonormal basis $\\{|{c_{i}^{\nu}}\rangle:i=1,...,K\\}$ and for each code to
correct the linear span of errors $E_{A}$ each codes basis must obey the
Knill-Laflamme condition (Knill and Laflamme, 1997):

$\langle{c_{i}^{(\nu)}}|E_{A}^{\dagger}E_{B}|{c_{j}^{(\nu)}}\rangle=\alpha_{kl}^{\nu}\delta_{ij}$
(1)

Equation 1 is different from the original Knill-Laflamme condition in that the
constant $\alpha_{kl}^{(\nu)}$ depends on the classical information being
transmitted as well. An additional condition these hybrid codes must satisfy
is that each quantum code must also be simultaneously distinguishable from all
others in the hybrid code to be able to retrieve the classical information.
This provides a second error correction condition on hybrid codes

$\langle{c_{i}^{(\nu)}}|E_{A}^{\dagger}E_{B}|{c_{j}^{(\mu)}}\rangle=0,\quad\text{for}\quad\mu\neq\nu$
(2)

Equations 1 and 2 can be written succiently as

$\langle{c_{i}^{\nu}}|E_{a}^{\dagger}E_{b}|{c_{j}^{\mu}}\rangle=\alpha_{kl}^{\nu}\delta_{ij}\delta_{\nu\mu}$
(3)

The proof for this condition is outlined in (Grassl  _et al._ , 2017).

###  I.2 OAQEC perspective

The operator quantum error correction (OQEC) is the predecessor to OAQEC
(Kribs  _et al._ , 2005). OQEC is a unification of the standard model and the
noiseless subsystems model of error correction. The standard model (Knill,
1997; Shor, 1995; Steane, 1996; Bennett, 1996) consists of the 3-tuple
$(\mathcal{R},\mathcal{E},\mathcal{C})$. This tuple consist of a quantum code,
$\mathcal{C}$, which can correct errors $\mathcal{E}$ by the action of a
recovery operation $\mathcal{R}$

$(\mathcal{R}\circ\mathcal{E})(\rho)=\rho\quad\forall\rho\in\mathcal{C}$ (4)

If $P_{\mathcal{C}}$ is the projection of a Hilbert space onto $\mathcal{C}$
then such an $\mathcal{R}$ exists for a given $\mathcal{E}=\\{E_{a}\\}$ and
$\mathcal{C}$ if

$P_{\mathcal{C}}E_{a}^{\dagger}E_{b}P_{\mathcal{C}}=\lambda_{ab}P_{\mathcal{C}}\quad\forall
a,b$ (5)

In the noiseless subsystem model (Knill and Laflamme, 1997; Palma  _et al._ ,
1996; Duan and Guo, 1997; Zanardi and Rasetti, 1997; Lidar  _et al._ , 1998)
one considers the error set they wish to correct $\\{E\\}$. This error set
generates an algebra $\mathcal{A}$ which is unitarily equivalent to a direct
sum of full matrix algebras, $\mathcal{A}=\bigoplus_{J}M_{m_{j}}$, which can
be written as $\bigoplus_{J}M_{m_{j}}\otimes I_{n_{j}}$. This decomposes the
Hilbert space into a noisy and noiseless subsystem, elements in the noiseless
subsystem, the noise commutant, will commute with $\mathcal{A}$ and be immune
to errors. The information one wishes to protect is encoded in the noise
commutant $\mathcal{A^{\prime}}=\bigoplus_{J}I_{m_{j}}\otimes M_{n_{j}}$. By
defining the projection operators
$P_{kl}=|{\alpha_{k}}\rangle\langle{\alpha_{l}}|\otimes I_{n}$, one can define
a map into this noiseless subspsace with the following properties:

$\Gamma(\rho)=\sum_{k,l}P_{kl}\rho
P^{\prime}_{kl}\in\mathcal{A}^{\prime}\quad\Gamma(\rho_{A}\otimes\rho_{B})\propto
I_{A}\otimes\rho_{B}$ (6)

This can be further generalized since it is not necessary to protected the
entire space $\rho_{A}\otimes\rho_{B}$, only $\rho_{B}$. Thus instead of being
confined to $\mathcal{A^{\prime}}=I_{A}\otimes\rho_{B}$ we consider the space
$\mathcal{U}=\rho_{A}\otimes\rho_{B}$. Using the earlier defined projector, we
define $P_{k}=P_{kk}$ and can then define $P_{\mathcal{U}}=\sum_{k}P_{k}$ and
$P^{\perp}_{\mathcal{U}}=I-P_{\mathcal{U}}$. With these we can give three
equivalent definitions for a noiseless subsystem:

$\displaystyle\forall\rho^{A},\rho^{B},\exists\sigma^{A}$
$\displaystyle:\mathcal{E}(\rho^{A}\otimes\rho^{B})=\sigma^{A}\otimes\rho^{B}$
(7) $\displaystyle\forall\rho^{B}\exists\sigma^{A}$
$\displaystyle:\mathcal{E}(I^{A}\otimes\rho^{B})=\sigma^{A}\otimes\rho^{B}$
(8) $\displaystyle\forall\rho\in\mathcal{U}$
$\displaystyle:(\text{tr}_{A}\circ\mathcal{P}_{\mathcal{U}}\circ\mathcal{E})(\rho)=\text{tr}_{A}(\rho)$
(9)

The subspace $\mathcal{H}^{B}$ is noiseless if it satisfies any, and thus all,
of the above conditions. There exists such a semigroup $\mathcal{U}$ for a
channel $\mathcal{E}$ if and only if:

$\displaystyle P_{k}E_{a}P_{l}$ $\displaystyle=\lambda_{akl}P_{kl}\forall
a,k,l$ (10) $\displaystyle P_{\mathcal{U}}^{\perp}E_{a}P_{\mathcal{U}}$
$\displaystyle=0\quad\forall a$ (11)

OQEC consists of the 3-tuple $(\mathcal{R},\mathcal{E},\mathcal{C})$. The
noiseless subspace model is a specific case where $\mathcal{R}=I$ and the
standard model is a specific case where $\mathcal{C}=\mathcal{U}$. In OQEC
such a 3-tuple is correctable if

$(\text{tr}_{A}\circ\mathcal{P}_{\mathcal{U}}\circ\mathcal{R}\circ\mathcal{E})(\rho)=\text{tr}_{A}(\rho)$
(12)

A necessary condition for the existence of such a $\mathcal{U}$ is

$P_{k}E_{i}^{\dagger}E_{j}P_{l}=\alpha_{ijkl}P_{k}l\quad\forall i,j,k,l$ (13)

Moving from OQEC to OAQEC will require another generalization. In the
Schrodinger picture our states are changing with time. Thus the expectation
value of a fixed operator, $A$, acting on a system, $\rho$, which is
experiencing an error, $E_{i}$, is given by $\text{tr}({AE_{i}\rho
E_{i}^{\dagger}})$. Due to the cyclic properties of trace this equivalent to
the expectation value of an operator which is experiencing an error acting on
a fixed system $\text{tr}({E_{i}^{\dagger}AE_{i}\rho})$. This form implies
that the error is acting on the observable while the system stays fixed, this
is the Heisenberg picture. For every trace preserving channel in the
Schrodinger picture that acts on state $\rho$, there exists a corresponding
dual map which is unital acting on the observable $A$.

In the Schrodinger picture a subspace was noiseless for a channel
$\mathcal{E}$ if
$\mathcal{E}(\rho^{A}\otimes\rho^{B})=\rho^{A}\otimes\sigma^{B}$, equivalently
a space is noiseless for an error channel $\mathcal{E}^{\dagger}$ in the
Heisenberg picture if and only if $P\mathcal{E}^{\dagger}(X\otimes
I)P=X\otimes I$ for all $X$ which are observables. Where $P$ is the projector
of the Hilbert space onto the subspace $A\otimes B$. This gives two equivalent
definitions for a noiseless subsystem, when one is satisfied the other one is
as well.

We say a set of operators $S$ on $\mathcal{H}$ are conserved by $\mathcal{E}$
for states on some subspace $\mathcal{H}_{S}$ if every element of $S$
satisfies $P\mathcal{E}^{\dagger}(X_{a})P=PX_{a}P$. These observables can
generate an algebra that we wish to protect from errors. This can equivalently
be done via a theorem from (Bény  _et al._ , 2007), which states: Let
$\mathcal{A}$ be a subalgebra of $\mathcal{L}(\mathcal{H}_{s})$ $\mathcal{A}$
is conserved by $\mathcal{E}$ if and only if $E_{a}P$ commutes with every
element of the algebra. This can be expanded to the subalgebra $\mathcal{A}$
being correctable for $\mathcal{E}$ if and only if $PE_{a}^{\dagger}E_{b}P$
commutes with every element of the algebra for every combination of errors.
This generalization , when now considered from the perspective of the
Schrodinger picture, shows that the algebra $\mathcal{A}$ is correctable for
$\mathcal{E}$ for subspaces of the Hilbert space $\mathcal{H}_{S}$ if there
exists a recovery operation $\mathcal{R}$ such that for any density operator
which can be seperated into a sum of tensor products of operators in the
seperate spaces, $\rho=\sum_{k}\alpha_{k}(\rho_{k}\otimes\tau_{k})$ for
$\sum_{k}\alpha_{k}=1$ the following equation holds.

$(\mathcal{R}\circ\mathcal{E})(\rho)=\sum_{k}\alpha_{k}\mathcal{R}(\mathcal{E}(\rho_{k}\otimes\tau_{k}))=\sum_{k}\alpha_{k}(\rho_{k}\otimes\tau_{k}^{\prime})$
(14)

Not that for $\alpha_{1}=1$ this reduces to the OQEC condition.
$(\mathcal{R}\circ\mathcal{E})(\rho_{k}\otimes\tau_{k})=\rho_{k}\otimes\tau_{k}^{\prime}$.
There exists such a correction operation if and only if for all $a,b$ there
are operators $X_{abk}\in\mathcal{L}(B_{k})$ such that:

$PE_{a}^{\dagger}E_{b}P=\sum_{k}I_{A_{k}}\otimes X_{abk}$ (15)

###  I.3 Unified perspective

We now show that the coding theory perspective is a special case of the OAQEC
formulation. The condition that each individual quantum code obeys the Knill-
Laflamme condition and that each codes codewords are distinguishable from one
another are both given with equation 3.

The coding theory construction included three restrictions that do not exist
in the OAQEC model. First, the error set in the coding theory construction is
restricted to containing only unitary errors, particularly the Pauli channel.
Second, in the coding theory model each quantum code is viewed as a subspace
where the OAQEC model deals with subsystems. Finally, the coding theory model
restricts each quantum channel to be of equal dimension. These three
restrictions can be summarized as:

  1. 1.

$\mathcal{E}\subseteq P_{n}$

  2. 2.

$C^{\nu}\subseteq\mathcal{H},\forall\quad 1\leq\nu\leq M$ subspaces

  3. 3.

dim $C^{\nu}=K$ $\forall$ $\nu$,
$C^{\nu}=\text{span}\\{|{c_{i}^{\nu}}\rangle:1\leq i\leq K\\}$

The coding theory error correction condition acts on codewords in Hilbert
space while the OAQEC models acts on operators. In unifying these two models
it is necessary to rewrite the coding theory condition in terms of operators.
This can be done by considering the two equivalent forms of the regular Knill-
Laflamme condition.

$\langle{c_{i}}|E_{a}^{\dagger}E_{b}|{c_{j}}\rangle=\alpha_{kl}\delta_{ij}$
$\iff$ $PE_{a}^{\dagger}E_{b}P=\alpha_{kl}P$

Either form can be used. With this, we can define the projector onto the
hybrid codeword space as:

$P=\sum_{i,\nu}|{c_{i}^{\nu}}\rangle\langle{c_{i}^{\nu}}|$ (16)

This projector can then be used to rewrite equation 3 as

$PE_{a}^{\dagger}E_{b}P=\sum_{i}\sum_{\nu}\alpha_{kl}^{\nu}|{c_{i}^{\nu}}\rangle\langle{c_{i}^{\nu}}|$
(17)

Unlike the general Knill-Laflamme condition, $\alpha$ depends on the codewords
so it must be included “with in” the projector on the right hand side of
equation 16. Consider substituting equation 16 into equation 17.

$\displaystyle PE_{a}^{\dagger}E_{b}P$
$\displaystyle=\sum_{i}\sum_{j}\sum_{\nu}\sum_{\mu}|{c_{i}^{\nu}}\rangle\langle{c_{i}^{\nu}}|E_{a}^{\dagger}E_{b}|{c_{j}^{\mu}}\rangle\langle{c_{j}^{\mu}}|$
(18) $\displaystyle PE_{a}^{\dagger}E_{b}P$
$\displaystyle=\sum_{i}\sum_{j}\sum_{\nu}\sum_{\mu}\alpha_{ab}^{\nu}\delta_{ij}\delta_{\nu\mu}|{c_{i}^{\nu}}\rangle\langle{c_{j}^{\mu}}|$
(19) $\displaystyle PE_{a}^{\dagger}E_{b}P$
$\displaystyle=\sum_{i}\sum_{\nu}\alpha_{ab}^{\nu}|{c_{i}^{\nu}}\rangle\langle{c_{i}^{\nu}}|$
(20)

By starting from the operator form and applying the two requirements outlined
in having a hybrid quantum code we arrive at a condition similar to the
regular Knill-Laflamme condition but with $\alpha$ contained in the
summations. Finally, if we consider the OAQEC model for the case where the
entire space is correctable, namely that the noisy subspace is $1$. Then
equation 13 simplifies to:

$PE_{a}^{\dagger}E_{b}P=\sum_{k}^{M}\alpha_{kl}^{\nu}P_{Ak}$

Which is equivalent to equation 20

###  I.4 The hybrid hamming bound

An important question in the discussion of hybrid codes is when hybrid codes
will provide an advantage over codes which transmit quantum and classical
information separately. Constructing the hamming bound for hybrid codes
provides us with one means with which to compare the parameters of a hybrid
and quantum code. The quantum hamming bound applies to non-degenerate codes
with the error set consisting of the Pauli matrices. The bound is given by:

$\displaystyle\sum_{j=0}^{t}\binom{n}{j}3^{j}2^{k}\leq 2^{n}$ (21)

The bound can be reconstructed for hybrid codes. The quantum hamming bound is
essentially a packing argument. The bound states that the total space
available to the system of qubits must be greater than the total space the
errors can map codewords to along with the amount of space taken by the
codewords themselves.

A code with $n$ physical qubits will have $2^{n}$ orthogonal subspaces
available. Some of this space will be used by the logical codewords
themselves, and the rest can be used by the space that errors map these
codewords to. A code with $k$ qubits will have $2^{k}$ codewords. If the code
can correct up to $j$ errors then there are $\binom{n}{j}$ sets of locations
where an error can occur. At each location any of the three possible Pauli
errors can occur, giving $3^{j}$ possible errors for each set of locations.
These errors can occur on any of the $2^{k}$ codewords. This gives a total of
$\sum_{j=1}^{t}\binom{n}{j}3^{j}2^{k}$ possible errors. This gives the bound:

$\displaystyle\sum_{j=1}^{t}\binom{n}{j}3^{j}2^{k}+2^{k}$ $\displaystyle\leq
2^{n}$ $\displaystyle\sum_{j=0}^{t}\binom{n}{j}3^{j}2^{k}$ $\displaystyle\leq
2^{n}$

Which is the quantum hamming bound. A hybrid code with $n$ physical qubits
will also have a total of $2^{n}$ orthogonal subspaces available. For a hybrid
code with $M$ codes, there will be $M2^{k}$ logical codewords. Since each
quantum code making up the hybrid code must correct the same error set, then
the number of locations an error can occur and the number of possible errors
does not change for the non-degenerate case. The number of codewords this
error can occur on has changed though to $M2^{k}$, thus the total number of
errors which can occur is $\sum_{j=1}^{t}\binom{n}{j}3^{j}M2^{k}$. Therefore
the quantum hamming bound for hybrid codes is given by:

$\displaystyle\displaystyle\sum_{j=1}^{t}\binom{n}{j}3^{j}M2^{k}+M2^{k}\leq
2^{n}$ $\displaystyle M\displaystyle\sum_{j=0}^{t}\binom{n}{j}3^{j}2^{k}\leq
2^{n}$

Thus a non-degenerate hybrid code can not provide an advantage over an
equivalent quantum code. Therefore, degeneracy will be necessary for
constructing non-trivial hybrid codes.

##  II Example

In this work we construct a degenerate hybrid code with parameters
$[[4,1:1,2]]$. This code can detect the error set
$E=\\{X_{i},Y_{i},Z_{i},Z_{1}Z_{2},Z_{3}Z_{4}\\}\quad\forall i=1,2,3,4$, or
equivalent in can correct the given error set, given the location of errors
are known. This hybrid code has codewords

$|{\bar{0}}\rangle_{0}=|{0000}\rangle+|{1111}\rangle\quad|{\bar{1}}\rangle_{0}=|{0011}\rangle-|{1100}\rangle$
$|{\bar{0}}\rangle_{1}=|{0101}\rangle+|{1010}\rangle\quad|{\bar{1}}\rangle_{1}=|{1001}\rangle-|{0110}\rangle$

Using the procedure outlined in… The following circuit encodes the above
hybrid code. $|{\psi}\rangle_{q}$ is the quantum bit and $|{\psi}\rangle_{c}$
the classical bit.

$\Qcircuit<EMAIL_ADDRESS>\lstick{|{\psi}\rangle_{q}}\ctrl{-1}\qwx[1]\gate{H}\ctrl{-1}\targ\ctrl{1}\qw\qw\\\
\lstick{|{0}\rangle}\targ\targ\qw\ctrl{-1}\targ\ctrl{1}\qw\\\
\lstick{|{\psi}\rangle_{c}}\qw\ctrl{-1}\qw\qw\qw\targ\qw\\\ }$

Figure 1: The encoding circuit for the $[[4,1:1:,3]]$ hybrid code. The quantum
information is encoded in the second qubit and the classical information is
encoded in the fourth qubit.

Since the code has $d=2$, the code can detect errors or correct errors if the
location is known. The following four circuits provide the decoding sequence
for this code if an error occured on qubit $1,2,3$ or $4$ respectively.

Qubit 1

$\Qcircuit<EMAIL_ADDRESS>\qw\ctrl{1}\targ\ctrl{-1}\gate{H}\ctrl{-1}\qwx[1]\qw\targ\qw\qw\\\
\ctrl{1}\targ\ctrl{-1}\qw\targ\targ\qw\qw\ctrl{1}\qw\\\
\targ\qw\qw\qw\ctrl{-1}\qw\targ\qw\targ\qw\\\ }$

Qubit 2

$\Qcircuit<EMAIL_ADDRESS>\qw\ctrl{1}\targ\ctrl{-1}\gate{H}\ctrl{-1}\qwx[1]\qw\qw\targ\qw\gate{Z}\qw\\\
\ctrl{1}\targ\ctrl{-1}\qw\targ\targ\qw\ctrl{1}\ctrl{-1}\gate{X}\ctrl{-1}\qw\\\
\targ\qw\qw\qw\ctrl{-1}\qw\targ\targ\qw\qw\qw\qw\\\ }$

Qubit 3

$\Qcircuit<EMAIL_ADDRESS>\qw\ctrl{1}\targ\ctrl{-1}\gate{H}\ctrl{-1}\qwx[1]\targ\gate{Z}\qw\qw\\\
\ctrl{1}\targ\ctrl{-1}\qw\targ\targ\qw\qw\qw\qw\\\
\targ\qw\qw\qw\ctrl{-1}\qw\qw\qw\gate{Z}\qw\\\ }$

Qubit 4

$\Qcircuit<EMAIL_ADDRESS>\qw\ctrl{1}\targ\ctrl{-1}\gate{H}\ctrl{-1}\qwx[1]\targ\gate{Z}\qw\\\
\ctrl{1}\targ\ctrl{-1}\qw\targ\targ\ctrl{-1}\ctrl{-1}\qw\\\
\targ\qw\qw\qw\ctrl{-1}\qw\qw\qw\qw\\\ }$

Figure 2: The decoding circuits for the hybrid code. A distance $2$ code can
detect errors or correct errors if the location of the error is known. With
the location known the decoding circuits given can correct any single qubit
error.

##  III Conclusion

In this work we outlined the characterization and construction of hybrid codes
from a coding theory perspective and from the OAQEC perspective. We then
showed that the coding theory construction is a special case of the broader
OAQEC model. In the coding theory construction there exists three practical
restrictions; namely that the errors are elements of the Pauli group, each
codeword is a subspace of a Hilbert space instead of being a subsystem and
that the dimension of each of the quantum codes in a hybrid code are equal. We
generalized the Hamming bound to the hybrid case, showing that it will be
necessary for hybrid codes to be degenerate to provide an advantage over their
strictly quantum counterparts. We have designed a hybrid code that transmits
one qubit and one classical bit that detects any single Pauli error. We
created a procedure for encoding quantum circuits, and used this procedure to
create an encoding circuit for the codewords.

There has been no physical implementations of such hybrid codes in the
literature. The circuit put forth in this work could provide as a first
implementation. Also, continuing the discussion on hybrid bounds it would be
of interest to develop the hybrid forms of other quantum bounds, particularly
the quantum singleton bound and the quantum Gilbert-Varshamov bound.

##  IV Acknolwedgments

I would like to thank Prof. B. Zeng and D. Kribs for serving as co-advsiors on
my thesis work which this paper is based on.

## References

  * Devetak and Shor (2005) I. Devetak and P. W. Shor, Communications in Mathematical Physics 256, 287 (2005).
  * Hsieh and Wilde (2010a) M.-H. Hsieh and M. M. Wilde, IEEE Transactions on Information Theory 56, 4682 (2010a).
  * Hsieh and Wilde (2010b) M.-H. Hsieh and M. M. Wilde, IEEE Transactions on Information Theory 56, 4705 (2010b).
  * Yard (2005) J. Yard, arXiv preprint quant-ph/0506050 (2005).
  * Kremsky  _et al._ (2008) I. Kremsky, M.-H. Hsieh, and T. A. Brun, Physical Review A 78, 012341 (2008).
  * Grassl  _et al._ (2017) M. Grassl, S. Lu, and B. Zeng, in  _Information Theory (ISIT), 2017 IEEE International Symposium on_ (IEEE, 2017) pp. 1718–1722.
  * Bény  _et al._ (2007) C. Bény, A. Kempf, and D. W. Kribs, Physical Review A 76, 042303 (2007).
  * Bennett and Wiesner (1992) C. H. Bennett and S. J. Wiesner, Physical review letters 69, 2881 (1992).
  * Majidy (2017) S. Majidy, “The implementation of hybrid codes and nuclear magnetic resonance quantum information processing,” (2017).
  * Knill and Laflamme (1997) E. Knill and R. Laflamme, Physical Review A 55, 900 (1997).
  * Kribs  _et al._ (2005) D. Kribs, R. Laflamme, and D. Poulin, Physical review letters 94, 180501 (2005).
  * Knill (1997) E. Knill, Phys. Rev. A 55, 900 (1997).
  * Shor (1995) P. W. Shor, Physical review A 52, R2493 (1995).
  * Steane (1996) A. Steane, Phys. Rev. Lett. 77, 793 (1996).
  * Bennett (1996) C. H. Bennett, Phys. Rev. A 54, 3824 (1996).
  * Palma  _et al._ (1996) G. M. Palma, K.-A. Suominen, and A. K. Ekert, Proc. R. Soc. Lond. A 452, 567 (1996).
  * Duan and Guo (1997) L.-M. Duan and G.-C. Guo, Physical Review Letters 79, 1953 (1997).
  * Zanardi and Rasetti (1997) P. Zanardi and M. Rasetti, Physical Review Letters 79, 3306 (1997).
  * Lidar  _et al._ (1998) D. A. Lidar, I. L. Chuang, and K. B. Whaley, Physical Review Letters 81, 2594 (1998).