# Wilson loops at large $N$ and the quantum M2 brane

###### Abstract

The Wilson loop operator in the $U(N)_{k}\times U(N)_{-k}$ ABJM theory at
large $N$ and fixed level $k$ has a dual description in terms of a wrapped M2
brane in the M-theory background AdS${}_{4}\times S^{7}/\mathbb{Z}_{k}$. We
consider the localization result for the $1\over 2$-BPS circular Wilson loop
expectation value $W$ in this regime, and compare it to the prediction of the
M2 brane theory. The leading large $N$ exponential factor is matched as
expected by the classical action of the M2 brane solution with
AdS${}_{2}\times S^{1}$ geometry. We show that the subleading $k$-dependent
prefactor in $W$ is also exactly reproduced by the one-loop term in the
partition function of the wrapped M2 brane (with all Kaluza-Klein modes
included). This appears to be the first case of an exact matching of the
overall numerical prefactor in the Wilson loop expectation value against the
dual holographic result. It provides an example of a consistent quantum M2
brane computation, suggesting various generalizations.

###### Contents

  1. 1 Introduction
  2. 2 The AdS${}_{2}\times S^{1}$ M2 brane in AdS${}_{4}\times S^{7}/{\mathbb{Z}}_{k}$
    1. 2.1 Classical M2 brane solution
    2. 2.2 One-loop correction
  3. 3 Concluding remarks

##  1 Introduction

The existence of a consistent quantum supermembrane (or M2 brane) theory
remains an enigma (see e.g. [1, 2]). The corresponding 3d world-volume theory
is formally non-renormalizable, apparently requiring a specific definition
like a built-in cutoff. Nevertheless, some simple semiclassical computations
to one-loop order can still be done in a straightforward way, as one-loop
corrections in 3d field theory are free of logarithmic UV divergences, see
e.g. [3, 4, 5, 6] or more recent work in [7].

In this paper, we present another example of one-loop calculation in the M2
brane theory, which provides further evidence that the quantization of the
supermembrane might be under good control, at least within the semiclassical
expansion.

The AdS4/CFT3 duality between the $U(N)_{k}\times U(N)_{-k}$ ABJM theory [8]
and M-theory on AdS${}_{4}\times S^{7}/{\mathbb{Z}}_{k}$ provides a remarkable
opportunity to shed light on the properties of the quantum M2 brane theory by
testing its predictions against exact results in 3d superconformal gauge
theory. In the large $N$ limit with $k$ fixed, the holographic dual of a
Wilson loop in the fundamental representation is expected to be an M2 brane
wrapping the M-theory circle direction. Note that this limit is different from
the standard large $N$ ’t Hooft limit, where $N$ and $k$ are taken to be large
with $\lambda=N/k$ fixed, and in which Wilson loops are described by
fundamental strings in type IIA string theory in AdS${}_{4}\times{\rm
CP}^{3}$.

For fixed $k$, the large $N$ expansion of the Wilson loop operator in the ABJM
theory corresponds to the expansion in the large effective M2 brane tension
$R^{3}T_{2}\sim\sqrt{Nk}$, where $R$ is the curvature radius of
AdS${}_{4}\times S^{7}/{\mathbb{Z}}_{k}$ and $T_{2}={1\over(2\pi)^{2}\ell_{\rm
pl}^{3}}$.

Our starting point will be an analytic expression for the expectation value of
the $1\over 2$-BPS circular Wilson loop in the ABJM theory derived using
supersymmetric localization in [9] (see also [10, 11, 12, 13, 14, 15,
16])111We use the same normalization of $\langle W_{\frac{1}{2}}\rangle$ as in
[17].

$\langle W_{\frac{1}{2}}\rangle=\frac{1}{2\,\sin(\frac{2\pi}{k})}\frac{{\rm
Ai}\left[C^{-\frac{1}{3}}\left(N-\frac{k}{24}-\frac{7}{3k}\right)\right]}{{\rm
Ai}\left[C^{-\frac{1}{3}}\left(N-\frac{k}{24}-\frac{1}{3k}\right)\right]}\ .$
(1.1)

where ${\rm Ai}(z)$ is the Airy function, and $C=2/(\pi^{2}k)$. This
expression resums all of the perturbative $1/N$ corrections at fixed
$k$.222The leading behavior of the Wilson loop in this large $N$, fixed $k$
limit was first discussed in [14], but the $k$-dependent sine prefactor, which
comes from the fluctuations about the large $N$ saddle point, was not derived
there.

In order to compare to the semiclassical expansion in the M2 brane world-
volume theory, one is to expand (1.1) at large $N$ with fixed $k$, which gives

$\langle
W_{\frac{1}{2}}\rangle=\frac{1}{2\sin(\frac{2\pi}{k})}e^{\pi\sqrt{\frac{2N}{k}}}\left[1-\frac{\pi\left(k^{2}+32\right)}{24\sqrt{2}\,k^{3/2}}\frac{1}{\sqrt{N}}+O(\frac{1}{N})\right]\
.$ (1.2)

As we will show in Section 2 below, the exponential factor in (1.2) is
reproduced by the classical action of the M2 brane with AdS${}_{2}\times
S^{1}$ world-volume, while the $k$-dependent prefactor
$(2\sin\frac{2\pi}{k})^{-1}$ is matched precisely by the one-loop correction
coming from the functional determinants of the quantum fluctuations around
this M2 brane solution.

Higher-order $1/(\sqrt{N})^{n}$ terms in (1.2) are expected to represent
higher-loop corrections in the semiclassical expansion of the partition
function of the quantum M2 brane theory, and checking this is a very
interesting but challenging future problem. We comment on this and other
generalizations in Section 3.

##  2 The AdS${}_{2}\times S^{1}$ M2 brane in AdS${}_{4}\times
S^{7}/{\mathbb{Z}}_{k}$

The AdS${}_{4}\times S^{7}/{\mathbb{Z}}_{k}$ metric is given by333Explicitly,
in (2.3) one has $ds^{2}_{{\rm CP}^{3}}={dw^{s}d\bar{w}^{s}\over
1+|w|^{2}}-{w_{r}\bar{w}_{s}\over(1+|w|^{2})^{2}}dw^{s}d\bar{w}^{r}$ and
$dA=i[{\delta_{sr}\over
1+|w|^{2}}-{w_{s}\bar{w}_{r}\over(1+|w|^{2})^{2}}]dw^{r}\wedge d\bar{w}^{s}$
($s,r=1,2,3)$.

$\displaystyle ds^{2}=$ $\displaystyle\frac{R^{2}}{4}ds^{2}_{{\rm
AdS}_{4}}+R^{2}ds^{2}_{S^{7}/{\mathbb{Z}}_{k}}\ ,$ (2.1) $\displaystyle
ds^{2}_{{\rm AdS}_{4}}=$
$\displaystyle\frac{1}{z^{2}}\left(-dt^{2}+dz^{2}+dx_{1}^{2}+dx_{2}^{2}\right)\,,$
(2.2) $\displaystyle ds^{2}_{S^{7}/{\mathbb{Z}}_{k}}=$ $\displaystyle
ds^{2}_{{\rm CP}^{3}}+\frac{1}{k^{2}}\left(d\varphi+kA\right)^{2}\ ,\ \ \ \ \
\ \ \qquad\varphi\equiv\varphi+2\pi\ .$ (2.3)

The 11d supergravity background also includes the 4-form field strength

$F_{4}=dC_{3}=-\frac{3}{8}\frac{R^{3}}{z^{4}}dt\wedge dx_{1}\wedge
dx_{2}\wedge dz\,.$ (2.4)

The radius $R$ in units of the 11d Planck length ${\ell_{\rm pl}}$ is related
to the parameters $N$ and $k$ of the dual ABJM gauge theory by444We are
ignoring higher order corrections to this relation (due to $N\to N-{1\over
24}(k-{k^{-1}})$ [18]) as they will not be relevant for the leading large $N$
M2 brane contribution considered below.

$\left(\frac{R}{\ell_{\rm pl}}\right)^{6}=2^{5}\pi^{2}Nk\,.$ (2.5)

The world-volume action for a probe M2 brane in this background is given by
[19, 20]555The AdS${}_{4}\times S^{7}$ background is a limit of the 11d
supergravity solution describing $N$ M2 branes [21]. Here we consider a probe
M2 brane intersecting the boundary (or coincident M2 branes) over a line,
which is a BPS configuration like a fundamental string intersecting D3 branes
or D2 branes in the analogous $\text{AdS}_{5}\times S^{5}$ and
AdS${}_{4}\times{\rm CP}^{3}$ cases.

$S_{\rm M2}=T_{2}\int d^{3}\sigma\sqrt{-{\rm det}\,g}+T_{2}\int C_{3}\ +\ {\rm
fermionic\ terms}\ ,$ (2.6)

where the M2 brane tension is666The universal expression for the M2 brane
tension in terms of the coefficient $\kappa_{11}$ in the 11d theory action
${1\over 2\kappa_{11}^{2}}\int d^{11}x\sqrt{-g}\,R+...$ is [22]:
$T_{2}=(2\pi)^{2/3}(2\kappa_{11}^{2})^{-1/3}$. Here $\ell_{\rm pl}$ is defined
so that $2\kappa_{11}^{2}=(2\pi)^{8}\ell_{\rm pl}^{9}$ as, e.g., in [23].

$T_{2}=\frac{1}{(2\pi)^{2}}\frac{1}{\ell_{\rm pl}^{3}}\,.$ (2.7)

###  2.1 Classical M2 brane solution

The action (2.6) admits a simple classical solution given by the M2 brane
wrapping the M-theory circle direction (the $\varphi$ angle in (2.1)), and
occupying the AdS2 subspace of AdS4 spanned by the coordinates $t$, $z$ in
(2.2). The resulting membrane has the AdS${}_{2}\times S^{1}$ world-volume
geometry and is dual to the $1\over 2$-BPS Wilson loop along the $t$ direction
at the boundary of AdS4. By an appropriate Wick rotation and coordinate
transformation, one may obtain in the same way the solution dual to the
circular Wilson loop, for which the AdS2 factor is just the Euclidean
hyperbolic disk with circular boundary.

The value of the classical action (2.6) for this AdS${}_{2}\times S^{1}$
solution is simply given by777Note that there is no contribution from the
Wess-Zumino term involving the 3-form field in (2.6).

$S^{\rm cl.}_{\rm M2}=T_{2}R^{3}\frac{1}{4}{\rm vol}({\rm
AdS}_{2})\frac{2\pi}{k}\ ,$ (2.8)

where $1\over 4$ comes from the AdS4 radius in (2.1), and the $2\pi/k$ is the
length of the M-theory $\varphi$-circle in (2.3). Using (2.5) and (2.7), and
the well-known value of the regularized volume of the unit-radius hyperbolic
disk, ${\rm vol}({\rm AdS}_{2})=-2\pi$, this gives (we always assume $k>0$)

$S_{\rm M2}^{\rm cl.}=-\pi\sqrt{\frac{2N}{k}}\,.$ (2.9)

Thus $e^{-S_{\rm M2}^{\rm cl.}}$ precisely matches the exponential in the
localization prediction (1.2).888This AdS${}_{2}\times S^{1}$ solution was
considered in [24] but the calculation of the regularized classical action and
its connection to the Wilson loop expectation value was not discussed there.
Various wrapped BPS M2 brane solutions in AdS${}_{4}\times X^{7}$ and matching
of their classical actions to exponents in the localization prediction for the
corresponding 3d BPS Wilson loops, including the present $1\over 2$-BPS one in
AdS${}_{4}\times S^{7}/{\mathbb{Z}}_{k}$, was demonstrated in [25].
Supersymmetries of similar M2-brane solutions were considered also in [26]).

In the next section we will also compute the one-loop correction to the M2
brane partition function due to the quantum fluctuations about this classical
solution, and will reproduce precisely the prefactor in (1.2).

Let us note that in the case of the $1\over 2$-BPS Wilson loop along the
infinite straight line, one should use in (2.8) the regularized volume of AdS2
in Poincare coordinates which is zero, thus getting $S_{\rm M2}^{\rm cl.}=0$,
consistently with $\langle W_{1\over 2}\rangle=1$ in this case. All quantum
corrections also vanish here since the AdS2 space is homogeneous and hence the
quantum M2 brane free energy is proportional to ${\rm vol}({\rm AdS}_{2})$ to
all orders.999This is the same argument that applies also in the similar
$\text{AdS}_{5}\times S^{5}$ string context [27, 28].

###  2.2 One-loop correction

Starting with the action (2.6) one may expand it near a classical solution to
quadratic order fixing a 3d reparametrization and $\kappa$-symmetry gauge to
get an action for 8+8 physical 3d fluctuation fields. The resulting spectrum
of the quantum fluctuations around the above AdS${}_{2}\times S^{1}$ solution
was obtained in ref. [24], that we follow below.

It is natural to chose a static gauge identifying two membrane coordinates
$\sigma_{1},\sigma_{2}$ in (2.6) with the AdS2 directions and the third
$\sigma_{3}$ with the $S^{1}$ angle $\varphi$. After a Kaluza-Klein (Fourier)
expansion of the 3d fields in the periodic coordinate $\sigma_{3}$, one
obtains a tower of bosonic and fermionic fluctuations that can be viewed as 2d
fields propagating on the (unit-radius) AdS2 background. Thus one gets an
equivalent 2d theory with an infinite number of fields.

The bosonic fluctuations in the two transverse directions within AdS4 give a
tower of complex scalar fields $\eta_{n}$ (two real scalars for each $n$) with
masses

$m_{\eta_{n}}^{2}=\frac{1}{4}(kn-2)(kn-4)\,,\qquad n\in\mathbb{Z}\,,$ (2.10)

while from the fluctuations in the six ${\rm CP}^{3}$ directions one finds a
tower of 3 complex fields $\zeta^{s}_{n}$ ($s=1,2,3$) with masses

$m_{\zeta^{s}_{n}}^{2}=\frac{1}{4}kn(kn+2)\,,\qquad n\in\mathbb{Z}\,.$ (2.11)

For the fermionic fluctuations, the KK reduction leads to a tower of eight
two-component spinors $\vartheta^{A}_{n}$ ($A=1,\ldots,8$) for each value of
the KK mode number $n$, with masses given by101010In more detail, in the
notations of [24] the eight fermions are labelled by
$\vartheta^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}$ where $\alpha_{i}=\pm
1$ and $\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}=-1$. The masses of the KK
fermion tower are then given by
$m_{\vartheta_{n}}=\frac{1}{4}(3\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}-2kn)$,
$n\in\mathbb{Z}$, which leads to the spectrum summarized in (2.12). Also, the
resulting action for the supermembrane fermions that are space-time spinors
may be reinterpreted as an action for 2d Majorana spinors.

$m_{\vartheta^{a}_{n}}=\frac{kn}{2}\pm 1~{}~{}~{}(\mbox{3+3
modes})\,,\qquad\qquad m_{\vartheta^{i}_{n}}=\frac{kn}{2}~{}~{}~{}(\mbox{2
modes})\,,\qquad n\in\mathbb{Z}\,.$ (2.12)

The above masses explicitly depend on the integer $k$ which is the inverse
radius of the $\varphi$-circle in (2.3). Thus, in the type IIA string limit
$k\to\infty$, all KK modes with $n\not=0$ become infinitely heavy.

For $n=0$, this spectrum coincides (as expected upon double dimensional
reduction [29]) with the spectrum of bosonic and fermionic fluctuations around
the corresponding AdS2 string solution in the type IIA superstring theory on
AdS${}_{4}\times{\rm CP}^{3}$ [30, 17]: we get 2 scalars of $m^{2}=2$, 6
scalars of $m^{2}=0$ scalars, 3+3 fermions of $m=\pm 1$ and 2 fermions of
$m=0$.

One can also check that the full spectrum is consistent with 2d supersymmetry.
The bosonic and fermionic masses in a ${\cal N}=1$ supermultiplet in AdS2
containing one real scalar and a Majorana fermion are related as (see e.g.
[31])

$m_{{{}_{B}}}^{2}=m_{{{}_{F}}}(m_{{{}_{F}}}-1)\ .$ (2.13)

Indeed, the bosonic and fermionic modes listed above can be grouped so that
their masses satisfy this relation.

A stronger consistency test of the spectrum is obtained by checking the
vanishing of the vacuum energy in Lorentzian AdS2 in global coordinates (as
that happens also in the simple case of the flat toroidal M2 brane [3, 32]).
The vacuum energies for massive bosons and fermions in AdS2 are given by (see
e.g. [27])

$E_{{}_{B}}(m_{{}_{B}})=-\frac{1}{4}\big{(}m^{2}_{{}_{B}}+\frac{1}{6}\big{)}\,,\qquad\
\ \
E_{{}_{F}}(m_{{}_{F}})=\frac{1}{4}\big{(}m^{2}_{{}_{F}}-\frac{1}{12}\big{)}\,.$
(2.14)

We find that the total vacuum energy in the present case is zero separately
for each KK level $n$

$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad E^{\rm
tot}=\sum_{n=-\infty}^{\infty}E^{\rm tot}_{n}\ ,$ (2.15) $\displaystyle E^{\rm
tot}_{n}=-\frac{1}{4}\left[\frac{2}{4}(kn-2)(kn-4)+\frac{6}{4}kn(kn+2)-3(\frac{kn}{2}+1)^{2}-3(\frac{kn}{2}-1)^{2}-2(\frac{kn}{2})^{2}+2\right]=0\
.$ (2.16)

Using the above spectrum, we can derive the one-loop correction to the
partition function of the M2 brane theory expanded around the Euclidean
AdS${}_{2}\times S^{1}$ solution with circular boundary. The semiclassical
partition function is given by

$Z_{\rm M2}=Z_{1}e^{-S_{\rm M2}^{\rm
cl.}}\left[1+O\Big{(}\frac{1}{R^{3}T_{2}}\Big{)}\right]\ ,$ (2.17)

where the one-loop term $Z_{1}$ is the ratio of the determinants of the
corresponding fluctuation operators

$Z_{1}=\prod_{n\in\mathbb{Z}}\frac{\big{[}{\rm
det}(-\nabla^{2}+\frac{R^{(2)}}{4}+(\frac{kn}{2}+1)^{2})\big{]}^{\frac{3}{2}}\,\big{[}{\rm
det}(-\nabla^{2}+\frac{R^{(2)}}{4}+(\frac{kn}{2}-1)^{2})\big{]}^{\frac{3}{2}}\,{\rm
det}(-\nabla^{2}+\frac{R^{(2)}}{4}+(\frac{kn}{2})^{2})}{{\rm
det}\big{(}-\nabla^{2}+\frac{1}{4}(kn-2)(kn-4)\big{)}\,\big{[}{\rm
det}\big{(}-\nabla^{2}+\frac{1}{4}kn(kn+2)\big{)}\big{]}^{3}}\,.$ (2.18)

Here $R^{(2)}=-2$ is the curvature of AdS2.111111As usual, the form
$-\nabla^{2}+\frac{R^{(2)}}{4}+m^{2}$ of the second order differential
operator assumed to be defined on 2d Majorana spinor fields arises from
squaring the Dirac operator (in our conventions its determinant is counted as
if it acts on a single-component real field).  The $n=0$ factor in (2.18) is
of course the same as the one-loop partition function [30, 17] for the
fluctuations near the corresponding type IIA AdS2 string worldsheet ending on
a circle at the boundary of AdS${}_{4}\times\rm CP^{3}$.

The functional determinants in (2.18) may be computed by the standard AdSd
spectral zeta-function techniques (as was done in the similar AdS2 string case
in e.g. [27, 28, 17]). For a massive boson, one has

$\Gamma_{1_{B}}={1\over 2}\log{\rm det}(-\nabla^{2}+m_{{}_{B}}^{2})=-{1\over
2}\zeta(0;m_{{}_{B}}^{2})\log(\Lambda^{2})-{1\over
2}\zeta^{\prime}(0;m_{{}_{B}}^{2})\ ,$ (2.19)

where $\Lambda$ is a 2d UV cutoff, AdS2 is assumed to have unit radius and

$\displaystyle\zeta_{{}_{B}}(0;m_{{}_{B}}^{2})=\frac{m_{{}_{B}}^{2}}{2}+\frac{1}{6}\
,$ (2.20)
$\displaystyle\zeta_{{}_{B}}^{\prime}(0;m_{{}_{B}}^{2})=-\frac{1}{12}(1+\log
2)+\log{\rm A}-\int_{0}^{m_{{}_{B}}^{2}+\frac{1}{4}}dx\
\psi(\sqrt{x}+\frac{1}{2})\ .$ (2.21)

Here ${\rm A}$ is the Glaisher’s constant and
$\psi(x)=\Gamma^{\prime}(x)/\Gamma(x)$. Similarly, for a massive fermion

$\displaystyle\Gamma_{1_{F}}=-{1\over 2}\log{\rm
det}(-\nabla^{2}+\frac{R^{(2)}}{4}+m_{{}_{F}}^{2})$ $\displaystyle=-{1\over
2}\zeta_{{}_{F}}(0;m_{{}_{F}})\log(\Lambda^{2})-{1\over
2}\zeta^{\prime}_{{}_{F}}(0;m_{{}_{F}})\ ,$ (2.22)
$\displaystyle\zeta_{{}_{F}}(0;m_{{}_{F}})$
$\displaystyle=-\frac{m_{{}_{F}}^{2}}{2}+\frac{1}{12}\ ,$ (2.23)
$\displaystyle\zeta^{\prime}_{{}_{F}}(0;m_{{}_{F}})$
$\displaystyle=-\frac{1}{6}+2\log{\rm
A}+|m_{{}_{F}}|+\int_{0}^{m_{{}_{F}}^{2}}dx\ \psi(\sqrt{x})\,.$ (2.24)

Using these expressions we can first verify the cancellation of the
logarithmically divergent part of the one-loop free energy $\Gamma_{1}=-\log
Z_{1}$ in (2.18). Indeed, from the above calculation of the vacuum energy, one
can see that the sum over the bosonic and fermionic masses at each KK level
$n$ satisfies $\sum(m_{{}_{B}}^{2}-m_{{}_{F}}^{2})=-2$. Then the total
coefficient of the logarithmic divergence in the sum of the corresponding
terms in (2.19),(2.20) and (2.22),(2.23) over the spectrum is

$\zeta_{\rm tot}(0)={1\over
2}\sum_{n\in\mathbb{Z}}\big{(}-2+4\big{)}=\sum_{n\in\mathbb{Z}}1=1+2\zeta_{R}(0)=0\,,$
(2.25)

where we have used the Riemann zeta-function regularization to evaluate the
(linearly divergent) sum. Note that the contribution of all massive KK modes
at non-zero $n$ levels cancels 1 coming from the $n=0$ modes, i.e. cancels the
logarithmic UV divergence that was present in the similar computation in the
AdS${}_{4}\times\rm CP^{3}$ superstring regime [17].

The vanishing of the logarithmic divergence in the free energy was actually
expected, as the M2 brane theory we started with is three-dimensional, and
there are no logarithmic divergences in the corresponding functional
determinants in 3d. The reduction to 2d with all KK modes included cannot
produce logarithmic divergences that were not present in the 3d
formulation.121212In general, to match the divergences in 3d and 2d
formulations one is to regularize the 2d theory in a way consistent with
symmetries of the 3d theory. An analytic regularization like $\zeta$-function
one that discards power divergences is sufficient for this purpose in the
present context.

The one-loop free energy is thus finite and is given by

$\displaystyle\Gamma_{1}=-\log Z_{1}=-\frac{1}{2}\zeta^{\prime}_{\rm tot}(0)\
,$ (2.26)

where according to (2.18)

$\displaystyle\zeta^{\prime}_{\rm
tot}(0)=\sum_{n\in\mathbb{Z}}\zeta^{\prime}_{\rm tot}(0;n)\ ,$ (2.27)
$\displaystyle\zeta^{\prime}_{\rm
tot}(0;n)=2\zeta^{\prime}_{{}_{B}}(0;\frac{1}{4}(kn-2)(kn-4))+6\zeta^{\prime}_{{}_{B}}(0;\frac{1}{4}kn(kn+2))$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+3\zeta^{\prime}_{{}_{F}}(0;\frac{kn}{2}+1)+3\zeta^{\prime}_{{}_{F}}(0;\frac{kn}{2}-1)+2\zeta^{\prime}_{{}_{F}}(0;\frac{kn}{2})\,.$
(2.28)

Summing up the bosonic and fermionic contributions, some remarkable
simplifications occur. Combining the contributions of the positive and
negative modes (so that below $n\geq 0$) we find the following result131313The
vanishing of the $n=0$ contribution, i.e. that of the type IIA string modes,
was already observed in [17].

$\zeta^{\prime}_{\rm tot}(0;n)+\zeta^{\prime}_{\rm
tot}(0;-n)=\begin{cases}&-2\log\left(\frac{k^{2}n^{2}}{4}-1\right)\,,\qquad
kn>2\\\ &\ \ \log(\pi^{2})\,,\qquad\qquad\qquad\ kn=2\\\
&-\log(\frac{9}{4})\,,\qquad\qquad\qquad\ \ kn=1\\\ &\ \
0\,,\qquad\qquad\qquad\qquad\ \ \ \ n=0\end{cases}$ (2.29)

If we assume that $k>2$, only the $n=0$ and $kn>2$ cases in (2.29) occur, and
the complete one-loop free energy is given by the following simple result

$\Gamma_{1}=\sum_{n=1}^{\infty}\log\left(\frac{k^{2}n^{2}}{4}-1\right)=2\sum_{n=1}^{\infty}\log(\frac{kn}{2})+\sum_{n=1}^{\infty}\log\left(1-\frac{4}{k^{2}n^{2}}\right)\,.$
(2.30)

Using again the Riemann zeta-function regularization ($\zeta_{R}(0)=-{1\over
2},\ \zeta^{\prime}_{R}(0)=-{1\over 2}\log(2\pi)$) we get

$2\sum_{n=1}^{\infty}\log(\frac{kn}{2})=2\zeta_{R}(0)\log(\frac{k}{2})-2\zeta^{\prime}_{R}(0)=-\log(\frac{k}{4\pi})\,.$
(2.31)

The second sum in (2.30) is finite and given by

$\sum_{n=1}^{\infty}\log\Big{(}1-\frac{4}{k^{2}n^{2}}\Big{)}=\log\Big{[}\prod_{n=1}^{\infty}\big{(}1-\frac{4}{k^{2}n^{2}}\big{)}\Big{]}=\log\Big{[}\frac{k}{2\pi}\sin\big{(}\frac{2\pi}{k}\big{)}\Big{]}\,.$
(2.32)

Here we used the Euler’s expression for the sine as a product of its zeros,
$\sin(\pi x)=\pi x\prod_{n=1}^{\infty}(1-\frac{x^{2}}{n^{2}})$.

Combining (2.31) and (2.32) we get the final result for the one-loop partition
function for $k>2$

$Z_{1}=e^{-\Gamma_{1}}=\frac{1}{2\sin(\frac{2\pi}{k})}\,,$ (2.33)

which is thus in precise agreement with the localization result in (1.2).

Let us now discuss the special cases of $k=1,2$ which require a separate
treatment.141414Recall that for these values of $k$, the supersymmetry of the
M-theory background (2.1) and of the dual gauge theory is enhanced from ${\cal
N}=6$ to ${\cal N}=8$ [33, 8, 23, 34]. For $k=1$, all of the cases listed in
(2.29) occur in the sum over the KK modes, i.e.

$\Gamma_{1}^{k=1}=\frac{1}{2}\log(\frac{9}{4})-\frac{1}{2}\log(\pi^{2})+\sum_{n=3}^{\infty}\log\Big{(}\frac{n^{2}}{4}-1\Big{)}\,.$
(2.34)

The infinite sum here can be evaluated in a similar way (i.e. using
$\zeta$-function regularization) as explained above for (2.30)

$\sum_{n=3}^{\infty}\log\Big{(}\frac{n^{2}}{4}-1\Big{)}=2\sum_{n=3}^{\infty}\log(\frac{n}{2})+\sum_{n=3}^{\infty}\log(1-\frac{4}{n^{2}})=\log(16\pi)-\log
6\,.$ (2.35)

Then from (2.34) we get $\Gamma_{1}^{k=1}=\log 4$ and thus

$Z_{1}^{k=1}=\frac{1}{4}\,.$ (2.36)

Similarly, for $k=2$ we find

$\displaystyle\Gamma_{1}^{k=2}=-\frac{1}{2}\log(\pi^{2})+$
$\displaystyle\sum_{n=2}^{\infty}\log\left(n^{2}-1\right)=0\,,$ (2.37)
$\displaystyle\ \ \ \ \ Z_{1}^{k=2}$ $\displaystyle=1\,.$ (2.38)

These results can not be directly compared to localization, as the result
(1.1) of [9] is singular for $k=1,2$.151515Remarks that localization result is
divergent for $k=1,2$ appeared, e.g., in [9, 35, 36, 37]. It might be that the
derivation of (1.1) in [9] is to be reconsidered specifically for $k=1,2$. The
matching in these special cases thus remains an open problem.

##  3 Concluding remarks

Extending the above computation to higher loops in the semiclassical expansion
of the partition function (2.17) would allow one to compare the quantum M2
brane prediction with the subleading terms in the expansion of the
localization expression at large $N$. For instance, the term of order
$1/\sqrt{N}$ in (1.2) should come from a 2-loop calculation in the M2 brane
theory (recall that ${1\over R^{3}T_{2}}\sim{1\over\sqrt{N}}$). Apart from
technical difficulties, one issue with this computation is whether the 2-loop
correction will be UV finite.

The cancellation of logarithmic divergences (despite apparent non-
renormalizability) may happen due to the large amount of supersymmetry of the
supermembrane theory (cf. [38]). An example of a cancellation of 2-loop UV
divergences in a formally non-renormalizable theory is provided by the
successful computation of the subleading $1\over\sqrt{\lambda}$ correction to
the cusp anomalous dimension
$f(\lambda)=a_{0}\sqrt{\lambda}+a_{1}+{a_{2}\over\sqrt{\lambda}}+...$ in the
$\text{AdS}_{5}\times S^{5}$ superstring theory [39, 40, 41], that matched the
corresponding term in the strong-coupling expansion of $f(\lambda)$ derived on
the ${\cal N}=4$ SYM side using integrability [42] (the analogous 2-loop
computation in the case of the AdS${}_{4}\times{\rm CP}^{3}$ string was done
in [43]).

An alternative possibility could be that the M2 brane theory has a built-in UV
cutoff $\Lambda\sim{\ell_{\rm pl}}^{-1}\sim T_{2}^{-1/3}$. However, then a
logarithmically divergent term would scale as $\log(R\Lambda)={1\over
6}\log(Nk)+...$ (see (2.5)), but there is no such $\log N$ term in the
localization expansion of the Wilson loop in (1.2).161616The $\log N$ term is
known to be present in the free energy of the ABJM theory on $S^{3}$ and it
was matched to the one-loop 11d supergravity result in [44], where it comes
only from the zero-mode contribution. This suggests that the logarithmic
divergences may cancel at higher loops in the M2 brane theory, at least for
such a $1\over 2$-BPS observable.

Let us now comment on the 10d type IIA string theory limit, which correspond
to $k$ and $N$ both taken to be large, with the ’t Hooft coupling
$\lambda=N/k$ kept fixed. In this regime, the 11d background (2.1) reduces to
AdS${}_{4}\times{\rm CP}^{3}$, and a Wilson loop operator is dual to an open
string ending on a loop at the boundary of AdS4. The corresponding type IIA
string coupling constant $g_{\rm s}$ and the effective string tension
$T={1\over 2\pi}{R^{2}_{\rm s}\over\ell_{\rm s}^{2}}$ are then given by [8]

$g_{\rm s}={\sqrt{\pi}\,(2\lambda)^{5/4}\over N}\ ,\qquad\ \ \ \ \
T={\sqrt{2\lambda}\over 2}\ ,\qquad\ \ \ \lambda={N\over k}\ .$ (3.1)

Note that to be in the $g_{\rm s}\ll 1$ and $\lambda\gg 1$ regime, we need to
assume that $k\ll N\ll k^{5}$.

As was pointed out in [17], the string partition function computed near the
AdS2 minimal surface representing the ${1\over 2}$-BPS circular Wilson loop in
both type IIB $\text{AdS}_{5}\times S^{5}$ and type IIA AdS${}_{4}\times{\rm
CP}^{3}$ theories has an expansion in small $g_{\rm s}$ and then in large
tension $T$ of the following universal form171717The fact that string loop
corrections go as powers of ${g^{2}_{\rm s}\over T}$ not just of $g^{2}_{\rm
s}$ (and thus are suppressed by factors of $1/T$) should be a consequence of
the underlying supersymmetry of the theory.

$\displaystyle\langle W_{1\over 2}\rangle=$ $\displaystyle\
e^{2\pi\,T}\,{\sqrt{T}\over g_{\rm s}}\Big{\\{}{\rm
c}_{0}\big{[}1+O(T^{-1})\big{]}+{\rm c}_{1}{g^{2}_{\rm s}\over
T}\big{[}1+O(T^{-1})\big{]}+{\rm c}_{2}\Big{(}{g^{2}_{\rm s}\over
T}\Big{)}^{2}\big{[}1+O(T^{-1})\big{]}+...\Big{\\}}\ .$ (3.2)

In our present case, we can see that this is consistent with the structure of
the corresponding large $N$, large $k$ expansion of (1.1), according to which
one should get ${\rm c}_{0}={1\over\sqrt{2\pi}}$, ${\rm c}_{1}={\pi\over
12}{\rm c}_{0},$ etc. The presence of the overall $\sqrt{T}$ factor was shown
in [17] to originate from the the leading one-loop string sigma model
correction on the disk (it is related to the $n=0$ contribution in (2.25)).
The precise value of the one-loop coefficient ${\rm
c}_{0}={1\over\sqrt{2\pi}}$ was not so far derived directly on the string side
(that appears to require a careful normalization of the measure in the
superstring path integral). Remarkably, the M2 brane one-loop computation
described above effectively determines this coefficient and, moreover, the
coefficients of all of the leading large tension terms at higher genus (disk
with handles). Indeed, comparing (3.2) to the corresponding large $N$, large
$k$ expansion of (1.1), the leading large tension terms in (3.2) can be seen
[45] to arise from a resummed expression

$\langle W_{1\over 2}\rangle={1\over
2\sin\big{(}\sqrt{\frac{\pi}{2}}\,\frac{g_{\rm
s}}{\sqrt{T}}\big{)}}\,e^{2\pi\,T}\Big{[}1+O(T^{-1})\Big{]}\ ,$ (3.3)

where $\sqrt{\frac{\pi}{2}}\,\frac{g_{\rm
s}}{\sqrt{T}}=2\pi\,\frac{\lambda}{N}={2\pi\over k}$. Here the sine factor is
just the same as the one found in (1.2),(2.33) (the exponential factor is also
the same as in (2.17),(2.9)). Thus, the one-loop M2 brane correction happens
to describe the leading large tension terms at all orders in the genus
expansion in the type IIA string theory!181818This is somewhat analogous to
how a semiclassical AdS${}_{2}\times S^{2}$ D3-brane solution happens to
capture the leading resummed contribution [46] in the $1\over 2$-BPS circular
Wilson loop expectation value $\langle W_{1\over 2}\rangle={1\over
2\pi}\frac{\sqrt{T}}{g_{\rm s}}\,e^{2\pi\,T\ +\ \frac{\pi}{12}\frac{g_{\rm
s}^{2}}{T}}\,\Big{[}1+O(T^{-1})\Big{]}$ in the ${\cal N}=4$ SYM or
$\text{AdS}_{5}\times S^{5}$ string case [47].

One natural generalization of our calculation is to consider the $1\over
6$-BPS Wilson loop [10, 48]. In this case the localization result derived in
[9], expanded in the large $N$ fixed $k$ limit, gives

$\langle W_{\frac{1}{6}}\rangle={i\over 2\sin{({2\pi\over k})}}\sqrt{{2N\over
k}}\ e^{\pi\sqrt{2N\over k}}\big{(}1+\ldots\big{)}\ .$ (3.4)

Note that there is an extra factor $i\sqrt{2N/k}$ compared to ${1\over 2}$-BPS
case. The origin of this factor should be similar to what was discussed in the
corresponding string case, where it was argued [10] that the string solution
should be smeared over a ${\rm CP}^{1}$ in ${\rm CP}^{3}$, leading to two zero
modes and hence an overall factor $(\sqrt{T})^{2}\sim\sqrt{\lambda}$ in the
partition function [13, 17]. For the M2 brane, we similarly expect that the
solution relevant to the $1\over 6$-BPS case should be smeared over a ${\rm
CP}^{1}$, leading again to an extra tension dependent prefactor
$\sim\sqrt{N}$. It would be interesting to study the fluctuation spectrum of
the corresponding M2 brane in detail and reproduce from a one-loop calculation
the remaining normalization factor in (3.4).

Another interesting extension would be to explore the defect CFT defined by
the ${1\over 2}$-BPS Wilson loop in the large $N$, fixed $k$ limit. The
corresponding problem in the type IIA string regime was studied in [49]. In
particular, in that case one finds that the 8+8 fluctuation modes about the
AdS2 string solution form a short supermultiplet containing the displacement
operator. The same multiplet appears for the M2 brane as the $n=0$ mode in the
KK reduction. It would be interesting to understand the interpretation of the
higher KK modes from the defect CFT point of view, and compute their boundary
correlation functions.191919Similar calculations in the context of the D3 and
D5 brane dual to the ${1\over 2}$-BPS Wilson loop in ${\cal N}=4$ SYM were
done in [50], extending the earlier string theory calculation in [51].

## Acknowledgments

We thank Matteo Beccaria and Nadav Drukker for useful comments on the draft,
and also Marcos Marino and Kazumi Okuyama for correspondence. The work of SG
is supported in part by the US NSF under Grant No. PHY-2209997. AAT is
supported by the STFC grant ST/T000791/1.

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# Hybrid Gate-Pulse Model for Variational Quantum Algorithms

Zhiding Liang1 Zhixin Song2 Jinglei Cheng3 Zichang He4 Ji Liu5 Hanrui Wang6
Ruiyang Qin1 Yiru Wang3 Song Han6 Xuehai Qian3 Yiyu Shi1  1University of Notre
Dame 2Georgia Institute of Technology 3Purdue University  4University of
California, Santa Barbara 5Argonne National Laboratory  6Massachusetts
Institute of Technology

###### Abstract

Current quantum programs are mostly synthesized and compiled on the gate-
level, where quantum circuits are composed of quantum gates. The gate-level
workflow, however, introduces significant redundancy when quantum gates are
eventually transformed into control signals and applied on quantum devices.
For superconducting quantum computers, the control signals are microwave
pulses. Therefore, pulse-level optimization has gained more attention from
researchers due to their advantages in terms of circuit duration. Recent
works, however, are limited by their poor scalability brought by the large
parameter space of control signals. In addition, the lack of gate-level
“knowledge” also affects the performance of pure pulse-level frameworks. We
present a hybrid gate-pulse model that can mitigate these problems. We propose
to use gate-level compilation and optimization for “fixed” part of the quantum
circuits and to use pulse-level methods for problem-agnostic parts.
Experimental results demonstrate the efficiency of the proposed framework in
discrete optimization tasks. We achieve a performance boost at most 8% with
60% shorter pulse duration in the problem-agnostic layer.

##  I Introduction

Quantum Computing (QC) is an emerging technique with the potential to achieve
exponential acceleration over classical computation. QC has promised
theoretical speedups for problems including integer factoring [39] and
unstructured database search [11]. However, these algorithms are designed for
fault-tolerant quantum computers. In the current Noisy Intermediate Scale
Quantum (NISQ) [34] era, quantum computers with hundreds of qubits are
available, but hardware noise still limits the depth of quantum circuits that
can be executed reliably. Variational Quantum Algorithms (VQAs) [4] are among
the most promising tasks to demonstrate practical quantum advantages on
current noisy quantum computers. These hybrid algorithms utilize Parametrized
Quantum Circuits (PQC) with external parameters (e.g., angle $\theta$ in
rotation gates such as $R_{y}(\theta)$) as a degree of freedom to explore the
Hilbert space. Then, one can use a classical computer to optimize these
parameters with respect to problem-tailored cost functions $C(\vec{\theta})$.
VQAs have been adopted to explore the opportunities of using quantum resources
to speed up computations in quantum chemistry [33, 31, 28], combinatorial
optimization [9, 12], and machine learning [36, 45].

Figure 1: Overview of our hybrid optimization framework for VQAs. Here we keep
the circuit for initial state preparation in gate-level, since it generally
requires high accuracy and gate-level operations are well calibrated. However,
whether one can efficiently encode information directly on the pulse-level is
an interesting open question.

Typically, researchers use a quantum circuit, filled with gate-level
operations, as a high-level abstraction layer to design and implement quantum
algorithms. For long-term quantum algorithms such as Grover’s algorithm
designed for unstructured search, each part of the circuit is constructed with
specific “knowledge” about the algorithm, which results in less flexibility in
the circuit architecture. On the contrary, VQAs are heuristic algorithms and
usually use a problem-agnostic black-box circuit architecture. We notice one
special case for VQA is the unitary coupled-cluster (UCC) ansatz [41] inspired
by quantum chemistry theory. Since it is problem-aware, it also maintains a
fixed circuit architecture. We illustrate such a difference of algorithm
design in Fig. 2a-b. However, both of them involve quantum operations that are
not native to the hardware. To run these gate-level algorithms on real-world
NISQ machines, we need to compile them into a lower-level abstraction layer of
control signals such as microwave pulses for IBM’s superconducting quantum
processors or laser operations for Quantinuum’s ion-trap quantum computers.

Figure 2: (a) Fixed v.s. (b) Flexible algorithm design. Notice the entangling
layer consists of CX gates can be customized. (c) Visualization of the Max-Cut
for a 4-node graph; (d) A fixed design in QAOA due to problem encoding in the
Hamiltonian layer; (e) The gate-level visualization of QAOA ansatz; (f) Pulse-
level realization of the RZZ gate in (d). The pulse parameters include
amplitude, width, phase, frequency, and duration. The Cross-resonance (CR)
gate is the most native 2-qubit operation to superconducting qubits. All the
CX gates in entanglement layer of (b) has to be compiled into CR plus other
pulse operations, which introduces redundancy as part of the compilation. “D”
means the drive channels and “U” represents the control channels.

In the past few years, we have seen multiple works on designing VQAs with
quantum pulses [25, 21, 23], considering a lower-level abstraction layer for
quantum algorithms. The major advantage of this approach is the reduction of
pulse duration and less compilation overhead since each pulse is directly
implementable on the hardware. However, this abstraction layer makes the
optimization problem in VQAs more challenging. It also leads to poor
scalability due to an increasing search space dimension (each physical pulse
involves more parameters than a quantum gate). Consequently, pulse-level VQAs
introduces higher training costs than their gate-based counterparts. Many
methods have already been proposed to reduce training costs for gate-level
VQAs [13, 42, 43, 44, 19, 15, 35]. One recent relevant work for pulse-level
VQAs is enabling gradient descent training [16].

In this paper, we propose a hybrid gate-pulse framework to combine the
advantages of the two aforementioned abstraction layers. Our approach keeps
part of the VQA that is based on algorithmic design knowledge at the gate
level but directly works on the pulse design when the sub-circuit is problem-
agnostic (see Fig. 1). The pulse-level workflow brings us less latency and
reduces decoherence noise. On the other hand, the gate-level “knowledge” helps
reduce the parameter space and cut down the training cost. We experimentally
evaluate the hybrid framework with the Quantum Approximate Optimization
Algorithm (QAOA) [9] for the Max-Cut problem on four different IBM’s quantum
computers. Specifically, our work has the following contributions:

  * •

We propose the concept of hybrid abstraction layers to the community. The
hybrid of gate and pulse models take the advantage of both abstraction layers:
algorithm design knowledge and hardware friendly implementation.

  * •

We propose a gate-pulse co-optimization workflow for VQAs. We are the first to
experimentally realize hybrid gate-pulse optimization for VQAs on NISQ
machines.

  * •

Experimental results on four real quantum computers demonstrate that the
proposed method outperforms existing methods in terms of algorithmic
performance and duration for solving the Max-Cut problem using QAOA.

##  II Background

QAOA for the Max-Cut problem: In VQAs, a series of parametrized quantum gates
$U(\vec{\theta})=U_{p}(\theta_{p})\cdots U_{2}(\theta_{2})U_{1}(\theta_{1})$
are used to perform unitary transformation on the statevector
$|\psi\rangle(\theta)=U(\vec{\theta})\ket{0}^{\otimes n}$, where $n$ is the
number of qubits involved in the computation. Each layer of
$U_{p}(\theta_{p})$ is usually composed of parametrized single qubit gate
$U_{3}(\theta,\lambda,\varphi)$ and unparametrized entanglement layer
$\\{CX_{i,j}\\}$ among qubit $i$ and $j$ (see Fig. 2b). The computation
results are obtained by repeated measurements of the quantum state as a
probability distribution
$\Pr(|\psi\rangle)=\\{|\langle\psi|z\rangle|^{2}\\}_{z=0\cdots 00}^{1\cdots
11}$.

QAOA is originally proposed to approxiamtely solve combinatorial optimization
problems. One of the most well-known problem is Max-Cut, which aims to find a
partition of the graph’s $G=(V,E)$ vertices into two complementary sets such
that the number of edges between these two sets is the largest (Fig. 2c).
Unlike the black-box ansatz that are used in some other VQAs, QAOA ansatz
(Fig. 2e) involves an alternating structure inspired by the Quantum Adiabatic
Theorem [3]:

$U(\vec{\beta},\vec{\gamma})=U_{M}(\beta_{p})U_{P}(\gamma_{p})\cdots
U_{M}(\beta_{1})U_{P}(\gamma_{1}),$ (1)

where we evolve the quantum state
$|\vec{\beta},\vec{\gamma}\rangle=U(\vec{\beta},\vec{\gamma})|+\rangle^{\otimes
n}$ according to a Hamiltonian layer
$U_{P}(\gamma_{\ell})=e^{-i\gamma_{\ell}H_{P}}$ and a mixer layer
$U_{M}(\beta_{\ell})=e^{-i\beta_{\ell}H_{M}}$. The Hamiltonian $H_{P}$ encodes
the Max-Cut problem such that the ground state is the optimal solution. Then,
we can write the optimization as

$\max
C(\vec{\beta},\vec{\gamma})=\max\langle\vec{\beta},\vec{\gamma}|H_{P}|\vec{\beta},\vec{\gamma}\rangle,$
(2)

where $C(\vec{\beta},\vec{\gamma})$ is the cost function. In this paper, we
follow the original protocol [9] to choose $H_{M}=X^{\otimes n}$ as the mixer
layer and set the initial state as $|+\rangle^{\otimes n}$, a uniform
superposition of all the solutions created by a series of Hadamard gate $H$.
We choose the approximation ratio (AR) $\alpha=C^{*}/C_{max}$ to measure how
close the approximate solution is to the actual optimum, where $C^{*}$ is the
final value of cost function.

Quantum Pulse Control: For superconducting quantum computers, the pulse-level
is the lowest layer of the workflow in quantum computing [7, 5]. A pulse is
defined as a sequence of analog control signals that are applied to quantum
channels of different qubits. In IBM’s quantum devices, multiple channels are
defined. DriveChannel is the primary quantum channel that is associated with
qubits, while ControlChannel only exists for multi-qubit operations and is
generated by the feedback Hamiltonian from the backend. The AcquireChannel
stores the measured data and information. The MeasureChannel generates readout
pulses from another resonator, and readout signals are collected and processed
after the measurement pulses are applied.

##  III Motivation

The quantum programs are usually firstly coded in high-level programming
languages, and then compiled into assembly languages such as .qasm files.
Since the qubits in current quantum hardware are not fully-connected, some
gates in .qasm files can not be performed on a quantum computer. The conflicts
need to be solved through the insertion of SWAP gates. The process is referred
to as “qubit mapping and routing” [18, 32, 40]. The mapped .qasm files are
then translated into control-pulses and executed on a quantum computer. The
transpiling process introduces significant redundancy in the aspect of circuit
duration. Moreover, limited decoherence time in NISQ machines makes it hard to
implement large-scale quantum algorithms. Recently, more researchers have
turned their focus from the gate-level abstraction layer to beyond the gate-
level abstraction layer.

Most of the "beyond gate level" works are on the pulse-level abstraction
layer. Existing pulse-level works include applying insights from pulse
properties to generate pulse-efficient gate circuit transpilation [8, 10]. And
[30, 14, 26] take the advantages of pulse-efficient transpilation to boost the
performance of VQAs. On the other hand, [20, 25] directly generate ansatz and
implement quantum algorithms on pulse level. However, these pulse-level
methods lack knowledge from algorithm design, which results in a large
parameter space that causes troubles for optimizers. To address the challenge,
we provide our solution on a new abstraction layer. We aim to find a good
trade-off between the latency and training cost via gate-pulse co-design. Our
hybrid model is compatible with available gate-level optimization techniques
as well as those on pulse-level.

##  IV Hybrid Gate-Pulse Model

A gate-level VQA with problem-inspired knowledge has the highest latency but
the lowest training cost, while a pulse-level model has the lowest latency but
the highest training cost. As a new abstraction layer for "beyond gate level"
endeavors, we propose a hybrid gate-pulse model. We reduce the redundancy and
shorten the duration compared with gate-level framework. In addition, it
considers gate-level algorithmic design knowledge. Therefore, it needs less
parameters than a pulse-level framework. Our hybrid gate-pulse model takes a
good trade-off between the latency and training cost. We illustrate the
workflow to build a hybrid gate-pulse model in Fig. 3:

  * •

Step I. We adopt pulse-level optimization to reduce the duration.

  * •

Step II. We use gate-level optimization methods like gate cancellation, Sabre
qubits mapping [18], etc.

  * •

Step III. We design an error suppression protocol that includes measurement
error mitigation algorithms which are also compatible with our model. In this
step, general optimization techniques including Conditional Value-at-Risk
(CVaR) [2] can also be applied to further boost the performance of the hybrid
model.

Since the entire optimization process is conducted in a machine-in-loop way,
the framework is resilient to device noises. The proposed hybrid model is
orthogonal to available optimization techniques like dynamic decoupling, noise
injection, etc. These techniques can be applied to our framework.

###  IV-A Basic Model Design and Implementation

In the proposed model, we keep the deterministic part based on quantum
information theory, quantum physics, physical constraints, and domain
knowledge associated with the task at the gate level. For example, in QAOA, we
design the Hamiltonian layer at the gate level to maintain the RZZ structure
which is based on conventional problem encoding. Then, we replace the gates of
the problem-agnostic parts with a constructed native-pulse ansatz. For
example, when dealing with QAOA tasks, we build the mixer layer with the
native pulses since the mixer layer is problem-agnostic and hence relatively
flexible by design. According to the adiabatic theorem, we only need to
prepare an easily accessible ground state $|\psi\rangle_{initial}$ of the
mixer Hamiltonian $H_{M}$ as initialization. Our hybrid model can, in
principle, help find novel mixer designs given a fixed initial state.

####  IV-A1 Pulse Parameters Setting

Prior works at the pulse level demonstrate that we can directly tune
parameters that cannot be accessed at the gate level. Pulse amplitude is one
of the most commonly used parameters in the existing literature. Both the
pulse amplitudes of the DriveChannel and the ControlChannel are directly
related to the signal’s intensity, which determines the drive Hamiltonian.
Concurrently, it has been demonstrated in existing work that the adjustment of
frequency has a significant impact on quantum operations. In addition, we can
access the phase of classical electronics at the pulse level, which
corresponds to the angles of rotation gates at the gate level. Therefore, the
parametric pulses in our gate-pulse model contain more parameters including
frequency, amplitude, and phase. Among these parameters, amplitude and
frequency are invisible to gate-level users.

####  IV-A2 Frequency Tuning for Pulses

In the gate-pulse model, the boundaries of phase and amplitude are clearly
defined. The absolute value of amplitude cannot exceed 1, and the range of
phase is $(0,2\pi)$. In comparison, it is worthwhile to consider how to
determine the range of frequency since the change of the power of drive
frequency could trigger the frequency collision related to Stark shift [38].
Thus, we consider frequency tuning as the operation against noise from
hardware or environment. These noises can lead to variances in the accuracy of
the actual frequency applied to the machine. Therefore, we introduce a
flexible method of frequency modulation, which modifies the frequency of each
pulse in a more flexible manner than the crude methods that applied to an
entire qubit channel and set the range of frequency modulation between -100MHz
to 100MHz. Our method applies frequency shifts as parameters to parametric
pulses, enabling more flexible frequency control for each pulse operation.

###  IV-B Pulse-level Knowledge

The pulse-level optimization is described as Step I in Fig. 3, where we deploy
the principles of quantum optimal control (QOC) [5, 23]. We use a binary
search algorithm for the pulse duration of the pulse layer in our hybrid gate-
pulse model. In the model, we define the parametric pulse and set its duration
to an initial value that is a multiple of 32dt, which is a restriction of the
qiskit pulse for Gaussian waveforms. Based on the initial value, we use a
binary search to search for the minimum duration that is needed for pulses
while we maintain the good performance of the model.

Figure 3: The workflow of gate-pulse co-optimization on proposed hybrid gate-
pulse model. Step I is pulse-level optimization, followed by gate-level
optimization in Step II. Step III aims to design an error suppression protocol
for the proposed model, including measurement error mitigation and so on. The
methods that mark as selected are used in this project.

###  IV-C Gate-level Knowledge

As discussed in the preceding section, pulse-level models with heavy
parameters always exceed the capabilities of the optimizer and can result in a
terrible optimization landscape for specific benchmarks. In addition, for
certain problems, there exist layers with a fixed structure derived from the
knowledge of algorithmic design, such as QAOA, the quantum algorithm for
topological data analysis (TDA) [1], the quantum algorithm for partial
differentiable equations [22], etc. There are two implementation challenges
for these layers at the pulse level: 1) If the circuit is optimized directly
at the pulse level, the final state is unknown unless state tomography is used
to determine the final state. For example, the Hamiltonian layer of QAOA fixes
the RZZ structure, making it difficult to maintain the structure at the pulse
level. This issue could be addressed if the problem is designed with pulse-
level encoding, but it will be long-term work with lots of challenges. 2) Even
if we directly transform the gate-level circuit with algorithmic design
knowledge to pulse, the calibration could be a problem. Typically, both
single-qubit and 2-qubit gates are created by irradiating the qubits with
pulses. These corresponding gate and pulse pairs are carefully calibrated and
can be carefully controlled in hardware. However, if we control the pulse
parameters to form the corresponding circuit, we lose the advantage of careful
calibration. Because the pulse parameters given to the hardware will still
produce a certain variance due to noise interference and thus cannot achieve
precise control.

Therefore, we inherit and retain the knowledge at the gate level in the
problem-specific layer of the corresponding problem. In addition, we employ
some gate-level optimization techniques to confirm the compatibility of the
hybrid gate-pulse model with the gate-level optimization techniques and
further optimize the performance of the model for problem-solving as the Step
II in Figure 3. We used Sabre mapping for better-selecting qubits routing
relation, and commutative cancellation to cancel redundant such as self-joint
gates via commutation relations [40] for the benchmarks chosen for this paper.
Notably, many gate-level optimizations can be further applied here, but we
have selected only a few for demonstration.

###  IV-D General Optimization Technique

In addition to pulse-level and gate-level co-optimization, we can design
effective error suppression protocols for the hybrid gate-pulse model, thereby
enhancing this model’s performance in implementing quantum algorithms on
quantum hardware. For example, dynamic decoupling (DD) [6] sequences are
incorporated at the algorithmic level to mitigate various idling errors,
including dephasing and ZZ crosstalk [27]. As for measurement error
mitigation, all shots returned from the backend undergo a final postprocessing
step. A designed initial calibration program identifies the measurement error
process with the help of an algorithm and stores the corresponding calibration
data. Finally, calibration data and measurement results are considered in
combination at runtime to estimate probability distributions and confidence
intervals for the hardware’s bitstrings. These two are methods commonly used
to design error suppression protocols. And in our work, we applied measurement
error mitigation to demonstrate the usefulness of such protocol on the hybrid
gate-pulse model. In more detail, we use a matrix-free measurement mitigation
(M3) routine [29]. Instead of forming a full assignment matrix or its inverse,
This method operates in a subspace defined by the to-be-corrected noisy input
bit strings. Since the number of distinct bit strings can be smaller than the
multi-bit Hilbert space’s dimensionality, the linearity problem can be solved.

Next, we add methods to the protocol that perform a general performance
optimization using the conditional risk value as an aggregation function,
allowing quantum algorithms to converge on a better solution quickly. Simply
put, in this method, we must define a proportion of the risk value that
determines our payout. Get the counts in the final states corresponding to
that proportion, then divide this number by the product of the total number of
shots and the proportion.

##  V Experiments

###  V-A Experiment Setups

Our experiments are conducted on real NISQ machines, including four IBM
quantum systems: $ibm\\_auckland$, $ibmq\\_toronto$, $ibmq\\_montreal$, and
$ibmq\\_guadalupe$. The calibration data is captured in Table. I. Notice,
$ibmq\\_toronto$ has the lowest CNOT error rate and we expect the best
performance on this backend. $ibm\\_auckland$ has the lowest readout error and
we expect measurement error mitigation has the least effect on it. We choose
level 1 ($p=1$) QAOA for the Max-Cut problem on three-regular six nodes,
three-regular eight nodes, and randomized six nodes graphs as our evaluation
benchmark as shown in Fig. 4. During the optimization procedure, we use the
‘COBYLA’ optimizer and set the maximum iteration to 50. In each iteration, in
order to evaluate the cost function $C(\vec{\beta},\vec{\gamma})$, the hybrid
quantum program is executed 1024 times repeatedly for measurement purpose.
Furthermore, we fixed the logical to physical qubit mapping and set the CVaR
coefficient to 0.3 for fair comparison among experiments.

Figure 4: Graphs used in the QAOA Max-Cut benchmark. (1)(2)(3) corresponding to task 1, task 2, task 3 below. Backends | $auckland$ | $toronto$ | $guadalupe$ | $montreal$
---|---|---|---|---
# qubit | 27 | 27 | 16 | 27
Pauli-X error | 2.229e-4 | 2.774e-4 | 3.023e-4 | 2.780e-4
CNOT error | 1.164e-2 | 9.677e-3 | 1.108e-2 | 1.049e-2
Readout error | 0.011 | 0.031 | 0.025 | 0.015
$T_{1}$ time (ms) | 166.220 | 104.200 | 102.320 | 123.99
$T_{2}$ time (ms) | 145.620 | 120.760 | 102.530 | 95.01
Readout length (ns) | 757.333 | 5962.667 | 7111.111 | 5201.778
TABLE I: Calibration data of quantum computers used in this study obtained at
the time the experiments were performed.

###  V-B Main Results

Pulse-level Optimization: The results collected from $ibmq\\_toronto$ in Fig.
5 verify the pulse-level optimization techniques are compatible with our
proposed model. The binary search optimization method can successfully reduce
the 60% pulse duration of the mixer layer in QAOA with no significant effect
on the final approximation ratio. On the other hand, our hybrid model
outperforms the pulse-level model with a 2.1% higher approximation ratio and
4x faster training time to reach convergence. The pulse-level model is
initialized from the gate-level circuit (see Fig. 2) and we gradually lose the
fixed structure $Z_{i}Z_{j}(\gamma_{\ell})$ in the Hamiltonian layer since
many pulse parameters (amplitude, frequency, and phase) are changing during
optimization. This setup is similar to the VQP approach [21]. Such a loss of
knowledge about algorithm design leads to a larger parameter space and hence
requires longer convergence time (maximum iteration up to 200).

Figure 5: Pulse-level model and hybrid gate-pulse model results for QAOA Max-Cut problem with a three-regular six nodes graph on $ibmq\\_toronto$, with the pulse duration reduction. Backends |  $auckland$ (gate) |  $auckland$ (hybrid) |  $toronto$ (gate) |  $toronto$ (hybrid) |  $guadalupe$ (gate) |  $guadalupe$ (hybrid)
---|---|---|---|---|---|---
Raw AR | 49.1% | 54.2% | 48.8% | 54.1% | 50.5% | 54.5%
GO AR | 53.3% | 55.7% | 49.9% | 57.3% | 52.4% | 55.9%
M3 AR | 50.8% | 55.5% | 51.3% | 60.1% | 53.8% | 56.8%
CVaR AR | 63.8% | 73.5% | 72.3% | 84.3% | 75.0% | 76.1%
Raw Mixer Layer Duration | 320dt | 320dt | 320dt | 320dt | 320dt | 320dt
PO Mixer Layer Duration | - | 128dt | - | 128dt | - | 128dt
TABLE II: Benchmark the hybrid gate-pulse model for QAOA Max-Cut problem on a
three-regular six nodes graph. AR refers to the Approximate Ratio. GO is the
gate-level optimization. M3 is a measurement error mitigation technique. And
PO refers to pulse-level optimization.

Gate-level Optimization and Error Mitigation: In Table. II, we compare the
performance between the hybrid model and standard gate-level QAOA with
experimental results collected from real NISQ machines. We apply gate-level
optimization methods, including Sabre mapping and commutative gate
cancellation, on both models. Then, we implement the M3. From the results on
$ibmq\\_toronto$, we validate that both of the gate-level optimization methods
and measurement error mitigation are compatible with our proposed model,
achieving 3.2% and 2.8% improvement towards the approximation ratio,
respectively. Moreover, our proposed hybrid model consistently outperforms the
gate-level model. For the raw hybrid gate-pulse model, we have the advantage
of 5.3% in the approximation ratio compared to the raw gate-level model. With
the gate-level optimization methods, our model is 7.4% more accurate in
approximation ratio than the gate-level model. With M3 implemented on both
models, our proposed model has a 60.1% approximation ratio, which is 8.8%
higher than the gate-level model obtained.

Figure 6: Optimized Gate-level model and optimized hybrid gate-pulse model for
QAOA Max-Cut problem with three-regular six nodes, randomized six nodes, and
three-regular eight nodes graphson $ibmq\\_toronto$ and $ibmq\\_montreal$.

More NISQ Machine Results: We evaluate the hybrid model on three different
NISQ machines provided by IBM with a three-regular six nodes graph. Table. II
depicts the ability of the proposed model over the gate-level model. For raw
models, with gate-level optimization, with M3, and with CVaR, the hybrid gate-
pulse model demonstrates improvements in approximation ratio by 4.7%, 4.4%,
and 5.5% compared to on gate-level model on the average of results obtained
from NISQ machines. Pulse-level optimization generally reduces duration by 60%
on the mixer layer of QAOA. We further evaluate both the optimized gate-level
model and the optimized hybrid gate-pulse model; for both models, gate-level
optimization and M3 have been applied, and the hybrid gate-pulse model adds
the pulse-level optimization. The results indicate proposed model achieves on
average 7.3%, 4.2%, 3.0% higher than the gate-level model on task 1, task 2,
task 3, respectively.

##  VI Conclusion and Open Questions

In this work, we propose the first hybrid gate-pulse model as a new
abstraction layer to design and implement variational quantum algorithms. With
experimental results from IBM’s near-term quantum processors, we demonstrate
performance improvements on the Max-Cut problem using a hybrid QAOA to reduce
pulse duration and enhance the approximation ratio. Additionally, our
framework integrates multiple general optimization techniques such as
measurement reduction, Conditional Value-at-Risk, and measurement error
mitigation. Finally, we reach a 8% performance boost and 60% duration
reduction on the mixer layer compared to existing methods. We also list many
other optimization methods that can be adopted into our hybrid model for
future work. Our hybrid model can be naturally adopted into many other VQAs
for Quantum Machine Learning (QML) purposes such as quantum generative models
[45] and quantum classifiers [37, 17]. One counterexample is the quantum
autoencoder for data compression [36] since pulse-level contains more
parameters and hence reduces the compression ratio.

Compared to current gate-level VQA designs, we see the potential of using
pulse-level abstraction in the hybrid model to construct more efficient
problem-specific Hamiltonian layers in terms of parameterized multi-quit
gates. Relevant works have the potential to reduce redundancy introduced when
encoding the problem. Notably, in our work, we define the problem-agnostic
part as our pulse layer in the hybrid gate-pulse abstraction layer. However,
we believe the choice of how to use the pulse layer and gate layer in the
hybrid gate-pulse abstraction layer for specific problems can still be an open
question. For example, if we adopt the idea of the hybrid gate-pulse model to
quantum neural network tasks, the pulse-level encoding layer may benefit more
since the pulse has more parameters that may have the potential to allow more
classical data encode to the quantum state, and the gate-level trainable layer
has lots of existing techniques that could be used to gain the advantage. On
the other hand, the general trainability and further optimization of the
pulse-level abstraction layer is still an open question since it involves a
larger parameter space and may lead to problems such as Barren Plateaus [24].

##  VII Acknowledgement

The authors would like to thank Naoki Kanazawa for insightful discussions on
pulse-level algorithm implementation and design through Qiskit-OpenPulse,
Jiaqi Leng and Yuxiang Peng for discussion on pulse-level gradient descent
methods. We acknowledge the use of IBM Quantum services for this work.

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