# Mass sum rule of hadrons in the QCD instanton vacuum

Ismail Zahed<EMAIL_ADDRESS>Center for Nuclear Theory,
Department of Physics and Astronomy, Stony Brook University, Stony Brook, New
York 11794–3800, USA

###### Abstract

We briefly review the key aspect of the QCD instanton vacuum in relation to
the quantum breaking of conformal symmetry and the trace anomaly. We use
Ji${}^{\prime}s$ invariant mass decomposition of the energy momentum tensor
together with the trace anomaly, to discuss the mass budget of the nucleon and
pion in the QCD instanton vacuum. A measure of the gluon condensate in the
nucleon, is a measure of the compressibility of the QCD instanton vacuum as a
dilute topological liquid.

##  I Introduction

A remarkable feature of QCD is that in the chiral limit it is a scale free
theory. Yet, all hadrons are massive, composing most of the visible mass in
the universe. The typical hadronic scale is 1 fm, but where does it come from?
The answer appears to be from a subtle quantum effect referred to as
dimensional transmutation, and related to the quantum breaking of the
conformal symmetry of QCD. This mechanism is non-perturbative. On the lattice,
the lattice cutoff along with the running coupling combine to generate this
scale. In the continuum, to achieve this mechanism requires a non-perturbative
description of the vacuum state and its excitations.

The QCD vacuum as a topological liquid of instantons and anti-instantons,
offers by far the most compelling non-perturbative description that is
analytically tractable in the continuum, thanks to its QCD semi-classical
origin and diluteness Diakonov (1996); Schäfer and Shuryak (1998); Nowak  _et
al._ (1996). It is not the only description. Other candidates based on center
vortices and monopoles to cite a few Greensite (2017), are also suggested and
may as well be present in addition to the instantons. However, the latters
appear to trigger the dual breaking of conformal and chiral symmetry breaking,
and dominate the vacuum state and its low-lying hadronic excitations. Center
vortices maybe important for the disordering of the large Wilson loops and
confinement, a mechanism likely at work in the orbitally excited hadrons as
they Reggeize.

The spontaneous breaking of chiral symmetry rather than confinement drives the
formation of the low-lying and stable hadrons such as the nucleon and pion. In
the QCD instanton vacuum conformal symmetry is broken by the density of
instantons: their continuous quantum rate of tunneling in the vacuum. The
breaking of chiral symmetry follows simultaneously from the delocalization of
the light quark zero modes, by leapfrogging the instantons and anti-instantons
much like electrons leapfrogging atoms in a metal. Detailed numerical
simulations of light hadronic correlators in the QCD instanton vacuum Shuryak
(1999) show remarkable agreement with direct lattice measurements Chu  _et
al._ (1993), and a wealth of correlators extracted from data Shuryak (1999).
The universal conductance fluctuations in the zero mode region of the Dirac
spectrum, predicted by random matrix theory Verbaarschot and Zahed (1993) and
confirmed by lattice simulations Wittig (2020), show unequivocally the
topological character of the origin of mass.

In this note, we briefly review the salient aspects of the QCD instanton
vacuum in relation to the quantum breaking of conformal symmetry in section
II. We then discuss the role of the trace anomaly in the nucleon and pion mass
in section III. The quark and gluon composition of the hadronic mass using
Ji${}^{\prime}s$ decomposition Ji (1995a) is discussed in section IV. In
section V, we show that the gluon condensate in the nucleon is tied to the QCD
vacuum compressibility, a measure of the diluteness of the QCD instanton
vacuum as a topological liquid. Our conclusions are in section VI.

##  II QCD instanton vacuum

As we noted above, the chief aspect of the QCD vacuum (meaning quenched
throughout) is its quantum breaking of conformal symmetry with the emergence
of all light hadronic scales. The nature of the gauge fields at the origin of
this breaking were mysterious and the subject of considerable debates and
speculations for many decades, till stunning pictures were developed by
Leinweber and his collaborators using cooling and/or projection techniques
Leinweber (1999); Biddle  _et al._ (2020). Out of the fog of millions of gauge
fluctuations, cooling has revealed a stunning vacuum landscape composed of
inhomogeneous and topologically active gauge fields as shown in Fig. 1.
Remarkably, the key features of this vacuum were predicted long ago by Shuryak
Shuryak (1982)

$n_{I+\bar{I}}\equiv\frac{1}{R^{4}}\approx\frac{1}{{\rm
fm}^{4}}\qquad\qquad\frac{\bar{\rho}}{R}\approx\frac{1}{3}$ (1)

for the instanton plus anti-instanton density and size, respectively. In other
words, the hadronic scale $R=1\,{\rm fm}$ emerges as the mean quantum
tunneling rate of the topological charge in the QCD vacuum. The dimensionfull
parameters (1) combine in the dimensionless packing parameter
$\kappa\equiv\pi^{2}\bar{\rho}^{4}n_{I+\bar{I}}\approx 0.1$, a measure of the
diluteness of the instanton-anti-instanton ensemble in the QCD vacuum.
Fortunatly, the smallness of $\kappa$ is what will allow us to do reliable
analytical calculations. In the cooled landscape shown in Fig. 1, most
hadronic correlations are left unchanged with those before the cooling takes
place Schäfer and Shuryak (1998) (and references therein).

The size distribution of the instantons and anti-instantons density (their
tunneling rate per size) in the QCD vacuum is well captured semi-empirically
by

$\frac{dn(\rho)}{d\rho}\sim{1\over\rho^{5}}\big{(}\rho\Lambda_{QCD}\big{)}^{b}\,e^{-\\#\rho^{2}/R^{2}}$
(2)

with $b=11N_{c}/3-2N_{f}/3$ (one loop). The small size distribution follows
from the conformal nature of the instanton moduli and perturbation theory. The
large size distribution is non-perturbative. A variational analysis of the QCD
instanton vacuum including binary interactions Diakonov (1996), shows that the
large size instantons are cutoff self-consistently by their density as in (2).
Lattice parametrization of the same distribution suggests that the cutoff is
due to the onset of confinement with $R\approx 1\,{\rm fm}\rightarrow
l_{s}\approx 0.2\,{\rm fm}$ (the string length) Hasenfratz (2000); Shuryak
(1999). Here, we favor the former cutoff as it preserves the strictures of the
renormalization group invariance following the quantum breaking of conformal
symmetry, with a single scale overall in the chiral limit ($R$ sets the scale
for both the gluon and chiral condensates).

Figure 1: Instantons (yellow) and anti-instantons (blue) configurations in the
cooled Yang-Mills vacuum Leinweber (1999). They constitute the primordial
gluon epoxy at the origin of the hadronic mass. See text.

###  II.1 Quantum conformal symmetry breaking and the trace anomaly

The quantum breaking of conformal symmetry is best captured through the trace
of the energy momentum tensor. Indeed, consider its symmetric form

$\displaystyle T^{\mu\nu}=$ $\displaystyle\frac{2}{\sqrt{-g}}\frac{\delta
S_{1+3}}{\delta{g_{\mu\nu}}}=F^{a\mu\lambda}F^{a\nu}_{\lambda}-\frac{1}{4}g^{\mu\nu}F^{2}+\frac{1}{4}\overline{\psi}\gamma^{[\mu}i\overleftrightarrow{D}^{\nu]_{+}}\psi$
(3)

with $\overleftrightarrow{D}=\overrightarrow{D}-\overleftarrow{D}$ and
$[]_{+}$ denotes symmetrization. It is conserved $\partial_{\mu}T^{\mu\nu}=0$,
with an anomalous trace

$T^{\mu}_{\mu}=\frac{\beta(g^{2})}{4g^{4}}F^{a}_{\mu\nu}F^{a\mu\nu}+m\overline{\psi}\psi$
(4)

with the Gell-Mann-Low beta-function (2 loops)

$\beta(g^{2})=-\frac{bg^{4}}{8\pi^{2}}-\frac{\bar{b}g^{6}}{2(8\pi^{2})^{2}}+{\cal
O}(g^{8})$ (5)

Throughout, we use the rescaling $gF\rightarrow F$ for all operators in the
instanton and anti-instanton gauge fields. In the QCD instanton vacuum, the
gluon operator $F^{2}/(32\pi^{2})\rightarrow(N_{+}+N_{-})/V$ counts the number
of instantons plus anti-instantons per 4-volume $V$. In the canonical ensemble
with zero theta-angle, it is fixed by the instanton density with
$N_{\pm}/V=\bar{N}/2V$. Therefore we have

$\left<T^{\mu}_{\mu}\right>\approx-b\bigg{(}\frac{\bar{N}}{V}\bigg{)}+m\left<\overline{\psi}\psi\right>\approx-b\bigg{(}\frac{\bar{N}}{V}\bigg{)}\bigg{(}1+{\cal
O}(mR)\bigg{)}\approx-10\,{\rm fm}^{-4}$ (6)

setting the scale of all hadrons. The current mass $m\approx 8$ MeV in (6) is
fixed at the soft renormalization point $\bar{\rho}\approx 0.3$ fm, about
twice the commonly used value at the hard renormalization scale. The scale of
the spontaneous breaking of chiral symmetry is also fixed by the finite
instanton density, but its contribution to the vacuum scalar density is small
since $mR\approx(8\,{\rm MeV})(1\,{\rm fm})\approx 1/25$.

Since the vacuum is Lorentz symmetric, (6) amounts to a negative vacuum energy
density $B=\left<T^{\mu}_{\mu}\right>/4\approx-(250\,{\rm MeV})^{4}$, with no
strict confinement at work. Note that $B\approx-b\left<F^{2}\right>$, where
the important overall negative sign inherited from the scale anomaly is
ultimatly a quantum magnetic effect (sign of the beta function). The gluon
condensate $\left<F^{2}\right>$ is always positive in Euclidean signature.
This is usually referred to as the gluon epoxy (a term coined by the late
Gerry Brown).

Some of the quantum scale fluctuations in QCD are captured in the QCD
instanton vacuum using the grand-canonical description instead of the canical
one. In the former, the instanton number $N=N_{+}+N_{-}$ is allowed to
fluctuate with the measure Diakonov  _et al._ (1996); Kacir  _et al._ (1999);
Nowak  _et al._ (1996)

$\mathbb{P}(N)=e^{\frac{bN}{4}}\bigg{(}\frac{\overline{N}}{N}\bigg{)}^{\frac{bN}{4}}$
(7)

which is stronger than Poisson ($b/4\rightarrow 1$), to reproduce the vacuum
compressibility

$\frac{\langle(N-\bar{N})^{2}\rangle_{\mathbb{P}}}{\bar{N}}=\frac{4}{b}$ (8)

expected from QCD low-energy theorems Novikov  _et al._ (1981).

###  II.2 Spontaneous breaking of chiral symmetry and conductance fluctuations

The quantum breaking of conformal symmetry and the generation of the $R=1$ fm
and a gluon condensate, is a direct measure of the instanton tunneling rate or
topological density in the vacuum. In the quenched approximation, it is solely
a property of the gluon fields. This is a necessary but not a sufficient
mechanism for hadronic mass generation. The sufficient mechanism, which relies
on these topological fields, is the delocalization of the quark zero modes and
the ensuing spontaneous breaking of chiral symmetry. The result is a vacuum
chiral condensate and the emergence of a quark mass, both of which are fixed
by the same $R=1$ fm scale at the origin of the gluon condensate. This
topological mechanism for mass generation leaves behind a fingerprint:
universal conductance-like fluctuations in the quark spectrum Verbaarschot and
Zahed (1993). This is a lesser known fundamental phenomenon, that we now
briefly discuss.

Decades ago, Banks and Casher observed that the spontaneous breaking of chiral
symmetry with a finite chiral condensate $\langle\overline{\psi}\psi\rangle$,
is associated to a huge accumulation of the quark zero modes near the zero
point of the virtual quark spectrum as illustrated in Fig. 2 (left), and
captured by the relation Banks and Casher (1980)

$\displaystyle\langle\overline{\psi}\psi\rangle=-\pi\nu(0)\equiv-\sigma_{C}$
(9)

The quark density of states is

$\displaystyle\nu(\lambda)=\lim_{m\to
0}\lim_{V\to\infty}\frac{1}{V}\left<\sum_{n}\delta(\lambda-\lambda_{n}[A])\right>_{A}\equiv\frac{1}{V}\frac{1}{\Delta\lambda}$
(10)

with the virtual quark eigenstates solution to

$(iD[A]+im)q_{n}[A]=(\lambda_{n}[A]+im)q_{n}[A]$

for a given gauge configuration. A finite $\nu(0)$ means that
$\Delta\lambda\approx R^{3}/V$, as opposed to $\Delta\lambda\approx
1/{}^{4}\sqrt{V}$ in the continuum. The quark spectrum is extremely dense near
$\lambda=0$, as the disordering turn the quark zero modes to quasi-zero modes.

Figure 2: Left: sketch of the virtual Dirac spectrum for the disordered quark
zero modes. Right: the universal spectral or conductance fluctuations (dotted
line) predicted by random matrix theory (11) Verbaarschot and Zahed (1993),
and measured in lattice generated gauge configurations (solid line) Wittig
(2020). See text.

Remarkably, the Banks-Casher relation (9) resembles the Kubo formula for the
DC conductivity in metals with
$\sigma_{C}\leftrightarrow-\langle\overline{\psi}\psi\rangle$ and the zero-
virtuality point $F=\lambda=0$ playing the role of the Fermi-surface. The QCD
vacuum turns metallic when chiral symmetry is spontaneously broken. The same
holds for the QCD instanton vacuum. In the unquenched case, this statement is
extremely non-trivial. A too dilute or too dense topologically active vacuum
would result into either a neutral gas of instanton-anti-instanton molecules,
or a crystal arrangement of instantons-anti-instantons, instead of a liquid,
with no spontaneous breaking of chiral symmetry Diakonov (1996); Schäfer and
Shuryak (1998); Nowak  _et al._ (1996). We are thankfull that nature has
served us a dilute liquid!

In fact there are infinitly many Banks-Casher-like formula which capture the
fluctuations of the conductance $\sigma_{C}$ in the mesoscopic limit, with the
connected moments
$\sigma_{C}^{n}=\langle(\overline{\psi}\psi)^{n}\rangle_{C}$. These
conductance fluctuations are universal, and follow from chiral random matrix
theory, with the result for the mesoscopic spectral density at zero virtuality
given by the master formula Verbaarschot and Zahed (1993)

$\displaystyle\nu_{s}(z=N\lambda)=\frac{z}{2}\bigg{(}J^{2}_{N_{f}}(z)-J_{N_{f}+1}(z)J_{N_{f}-1}(z)\bigg{)}$
(11)

with $N,V\rightarrow\infty$ but fixed $z=N\lambda$ and $N/V$. These predicted
conductance fluctuations were later measured in numerically generated lattice
gauge configurations as shown in Fig. 2 Wittig (2020) (right). The dotted line
is the result (11) and the solid line is the lattice measurement. This is one
of the signature for the topological origin of the hadronic mass scale, as it
centers on the quark zero mode zone. The zero modes only occur in the presence
of topological gauge fields such as the instantons and anti-instantons ’t
Hooft (1976) (or their close cousins the instanton-dyons Chernodub  _et al._
(2000); Liu  _et al._ (2015)), a consequence of the Atiyah-Singer index
theorem (or the Atiyah-Patodi-Singer for their cousins).

##  III Mass identity

We now focus on the anomalous trace of the QCD energy momentum tensor, and its
relation to the hadron mass. It is worth stressing that the ensuing relation
to the mass is just a bulk identity and not a mass decomposition. Having said
that, the trace couples to a scalar dilaton which sources a sigma (2-pion)
meson and/or a $0^{++}$ scalar glueball field. QCD perturbative arguments
suggest that this trace may be accessible in the photo-production of
charmonium at treshold Kharzeev (2021); Joosten and Meziani (2018) (and
references therein), although the coupling to the $2^{++}$ tensor glueball may
still be very active in the treshold region Mamo and Zahed (2020). Recall that
the Reggeized form of the $2^{++}$ exchange transmutes to the Pomeron, and is
dominant asymptotically.

###  III.1 Nucleon

For a nucleon state $|P\rangle$ with the standard normalization $\langle
P|P^{\prime}\rangle=2E_{P}(2\pi)^{3}\delta^{3}(P-P^{\prime})$, one has

$\displaystyle\langle P|T^{\mu\nu}|P\rangle=2P^{\mu}P^{\nu}\ .$ (12)

with the trace in any frame (one-loop)

$\displaystyle\langle P|T^{\mu}_{\mu}|P\rangle=\langle
P|\bigg{(}-\frac{b}{32\pi^{2}}F^{2}+m\bar{\psi}\psi\bigg{)}|P\rangle=2M_{N}^{2}\
.$ (13)

with $g^{2}F^{2}\rightarrow F^{2}$ for the strong instanton and anti-instanton
gauge fields. It is renormalization group invariant. The identity (13) shows
that the nucleon mass is the change of the conformal anomaly or gluon field in
a nucleon state. However, the formation of the state occurs only if chiral
symmetry is spontaneously broken as we noted earlier. (13) is a QCD identity
that is satisfied in the QCD instanton vacuum as we now show.

In the rest frame, the gluon contribution in (13) follows from the normalized
and connected 3-point function asymptotically

$\frac{\langle P|F^{2}|P\rangle}{\langle
P|P\rangle}={\lim_{T\to\infty}}\frac{\left<J_{P}^{\dagger}(T)F^{2}J_{P}(-T)\right>_{C}}{\left<J_{P}^{\dagger}(T)J_{P}(-T)\right>}$
(14)

with $J_{P}$ a pertrinent nucleon source. In the canonical description of the
QCD instanton vacuum $F^{2}/(32\pi^{2})\rightarrow\bar{N}/V$ is a number. It
factors out in the 3-point correlator in (14) (numerator), and the connected
correlator vanishes.

A non-vanishing contribution to the connected 3-point correlator follows from
the grand-canonical description, where $N$ is allowed to fluctuate as we noted
in (7). With this in mind, it is straightforward to see that (14) is dominated
by the variance

$\displaystyle\frac{V}{32\pi^{2}}\frac{\langle P|F^{2}|P\rangle}{\langle
P|P\rangle}\approx\left<(N-\bar{N})^{2}\right>_{\mathbb{P}}\frac{\partial}{\partial\bar{N}}{\rm
Log}\bigg{(}{\lim_{T\to\infty}}{\left<J_{P}^{\dagger}(T)J_{P}(-T)\right>}\bigg{)}$
(15)

with the higher moments suppressed by $1/b^{2}\sim 1/N_{c}^{2}$. The result
(15) was noted in Diakonov  _et al._ (1996) (see Eq. 5.8) using a
fermionization method, and in Kacir  _et al._ (1999) (see Eqs. 91,93) using a
bosonization method, each of the QCD instanton vacuum in the $1/N_{c}$
approximation. The expectation value in the first bracket is carried using the
distribution (7). (15) illustrates how the nucleon scoops the epoxy from the
QCD instanton vacuum.

All dimensions in the QCD instanton vacuum are fixed by the density
$\bar{N}/V=1/R^{4}$ and the current quark masses. The nucleon mass is the sum
of the chirally symmetric (invariant mass) plus the symmetry breaking
contribution (pion-nucleon sigma term),

$M_{N}=M_{\rm inv}+\sigma_{\pi
N}=C\bigg{(}\frac{\bar{N}}{V}\bigg{)}^{\frac{1}{4}}+\bar{C}m\bigg{(}1+{\cal
O}(mR)\bigg{)}$ (16)

with Steele  _et al._ (1995); Alexandrou  _et al._ (2020); Hoferichter  _et
al._ (2016)

$\sigma_{\pi
N}=\frac{\left<P|m\overline{\psi}\psi|P\right>}{\left<P|P\right>}\approx
50\,{\rm MeV}$ (17)

evaluated at the soft renormalization scale $\bar{\rho}=0.6\,{\rm GeV}$, which
is the appropriate scale for hadronic spectroscopy. The right-most relation in
(16) follows from the QCD instanton vacuum. As a result, the anomalous
contribution in the QCD instanton vacuum is

$\displaystyle\frac{V}{2T}\frac{-b}{32\pi^{2}}\frac{\langle
P|F^{2}|P\rangle}{\langle P|P\rangle}=4\frac{\partial M_{N}}{\partial{\rm
Log}\bar{N}}=M_{\rm inv}$ (18)

which is seen to satisfy the sum rule

$\displaystyle\frac{\left<P|T^{\mu}_{\mu}|P\right>}{2M_{N}}=M_{\rm
inv}+\frac{\left<P|m\overline{\psi}\psi|P\right>}{2M_{N}}=M_{N}$ (19)

###  III.2 Pion

The preceding arguments apply also to the pion, with one major difference,

$\displaystyle
m_{\pi}=C\sqrt{m}\bigg{(}\frac{\bar{N}}{V}\bigg{)}^{\frac{1}{8}}(1+{\cal
O}(mR))$ (20)

since it is a Goldstone mode. The ${\cal O}(mR)$ corrections are small in the
QCD instanton vacuum. It follows that

$\displaystyle\frac{V}{2T}\frac{-b}{32\pi^{2}}\frac{\langle\pi|F^{2}|\pi\rangle}{\langle\pi|\pi\rangle}=4\frac{\partial
m_{\pi}}{\partial{\rm Log}\bar{N}}=\frac{1}{2}m_{\pi}$ (21)

which was first observed in Ji (1995b), with the sum rule

$\displaystyle\frac{\left<\pi|T^{\mu}_{\mu}|\pi\right>}{2m_{\pi}}=\frac{1}{2}m_{\pi}+\frac{\left<\pi|m\overline{\psi}\psi|\pi\right>}{2m_{\pi}}=m_{\pi}$
(22)

satisfied, as expected. The pion sigma-term follows from chiral reduction or
the Feynman-Hellmann theorem

$\frac{\left<\pi|m\overline{\psi}\psi|\pi\right>}{2m_{\pi}}=\frac{\partial{\cal
E_{\pi}}}{\partial{\rm log}m}=\frac{1}{2}m_{\pi}$ (23)

##  IV Ji mass sum rule

The trace identity (12) reflects on the general fact that all hadron masses in
QCD are tied to the quantum breaking of conformal symmetry as we noted
earlier, and should be enforced by any non-perturbative quantum description,
wether numerical such as the lattice or analytical such as the QCD instanton
vacuum. However, it does not specifically budget this mass breaking in terms
of the hadron constituents. In a strongly interacting theory, this issue may
be elusive, especially with a soft renormalization scale, as the gluons are
strongly untertwined with the light quarks. This is more so in the unquenched
and screened formulation.

This notwithstanding, a specific and physically motivated proposal to budget
the mass, was put forth by Ji in Ji (1995a, b), and since revisited by many
Lorcé (2018); Roberts (2021); Metz  _et al._ (2021) (and references therein).
The ensuing mass composition involves the sum of partonic contributions, some
of which may be measurable using DIS experiments. The proposal relies on an
invariant decomposition of the energy momentum tensor which we now detail.

The energy-momentum tensor (3) can be decomposed as the sum of a traceless and
tracefull part Ji (1995b, 2021)

$\displaystyle
T^{\mu\nu}\equiv\bar{T}^{\mu\nu}+\hat{T}^{\mu\nu}\equiv\bar{T}^{\mu\nu}+{g^{\mu\nu}}\,\frac{1}{4}{T}^{\alpha}_{\alpha}\
,$ (24)

where the traceless part reads

$\displaystyle\bar{T}^{\mu\nu}=\bigg{(}-F^{a\mu\tau}F^{a\nu}_{\tau}+\frac{1}{4}g^{\mu\nu}F^{2}\bigg{)}+\frac{1}{4}\overline{\psi}\gamma^{[\mu}i\overleftrightarrow{D}^{\nu]_{+}}\psi-{g^{\mu\nu}}\frac{1}{4}m\bar{\psi}\psi,$
(25)

and the tracefull part is

${\hat{T}}^{\mu\nu}=g^{\mu\nu}\frac{1}{4}\bigg{(}\frac{\beta(g^{2})}{4g^{4}}F^{2}+m\overline{\psi}\psi\approx-\frac{b}{32\pi^{2}}F^{2}+m\overline{\psi}\psi\bigg{)}$
(26)

We note that this decomposition is commensurate with the analysis of the
nucleon energy momentum tensor in holographic QCD, through dual gravitons in
bulk Mamo and Zahed (2020). (Holography is a good example of a strong coupling
description of a gauge theory via its gravity dual, where the partonic
structure is elusive).

The tracefull and traceless part of the energy momentum tensor (24-26)
correspond to the spin-2 and spin-0 representations of the Lorentz group, and
do not mix under renormalization by symmetry. Their renormalization at the
instanton size scale $\bar{\rho}\approx 0.3\,{\rm fm}$ is subsumed throughout.
On the lattice, this soft renormalization scale is best achieved using a
cooling procedure where only the UV quantum and non-singular fluctuations are
subtracted (our instantons are classical fields in singular gauge!). Note that
our renormalization scale is softer than the one used in currently fine
lattices with $1/\mu\approx 0.1\,{\rm fm}$ ($\overline{\rm MS}$ scheme) Yang
_et al._ (2018). This difference will be further discussed below.

With this in mind, the matrix elements of the split energy-momentum tensor are
constrained by Lorentz symmetry

$\displaystyle\left<P|\bar{T}^{\mu\nu}|P\right>=$ $\displaystyle
2\bigg{(}P^{\mu}P^{\nu}-\frac{1}{4}g^{\mu\nu}M_{N}^{2}\bigg{)}$
$\displaystyle\left<P|\hat{T}^{\mu\nu}|P\right>=$
$\displaystyle\frac{1}{2}g^{\mu\nu}M_{N}^{2}$ (27)

The corresponding Hamiltonian in Minkowski signature, follows from the
00-component of (24-26) modulo BRST exact and gauge dependent contributions,

$\displaystyle H_{G}=\int d^{3}x\,\bar{T}^{00}_{G}$ $\displaystyle=$
$\displaystyle\int d^{3}x\,\frac{1}{2}(E^{2}+B^{2})$ $\displaystyle
H^{\prime}_{Q}=\int d^{3}x\,\bar{T}^{00}_{Q}$ $\displaystyle=$
$\displaystyle\int d^{3}x\,\bigg{(}\frac{1}{2}\overline{\psi}\gamma\cdot
i\overleftrightarrow{D}\psi+\frac{3}{4}m\bar{\psi}\psi\bigg{)}$ $\displaystyle
H^{\prime}_{A}=\int d^{3}x\,\hat{T}^{00}_{A}$ $\displaystyle=$
$\displaystyle\int
d^{3}x\,\frac{1}{4}\bigg{(}\frac{\beta(g^{2})}{4g^{4}}F^{2}+m\overline{\psi}\psi\approx-\frac{b}{32\pi^{2}}F^{2}+m\overline{\psi}\psi\bigg{)}$
(28)

where the time $t=0$ is subsumed. The mass term can be rearranged so that (IV)
reads

$\displaystyle H_{G}=\int d^{3}x\,\bar{T}^{00}_{G}$ $\displaystyle=$
$\displaystyle\int d^{3}x\,\frac{1}{2}(E^{2}+B^{2})$ $\displaystyle H_{Q}=\int
d^{3}x\,\bar{T}^{00}_{Q}$ $\displaystyle=$ $\displaystyle\int
d^{3}x\,\bigg{(}\frac{1}{2}\overline{\psi}\gamma\cdot
i\overleftrightarrow{D}\psi\bigg{)}$ $\displaystyle H_{A}=\int
d^{3}x\,\hat{T}^{00}_{A}$ $\displaystyle=$ $\displaystyle\int
d^{3}x\,\frac{1}{4}\bigg{(}\frac{\beta(g^{2})}{4g^{4}}F^{2}\approx-\frac{b}{32\pi^{2}}F^{2}\bigg{)}$
$\displaystyle H_{m}=\int d^{3}x\,\bar{T}^{00}_{G}$ $\displaystyle=$
$\displaystyle\int d^{3}x\,m\overline{\psi}\psi$ (29)

The nucleon mass budget is then

$M_{N}=\frac{\left<P|H_{G}+H_{Q}+H_{A}+H_{m}|P\right>}{\left<P|P\right>}\equiv
M^{N}_{G}+M^{N}_{Q}+M^{N}_{A}+M^{N}_{m}$ (30)

which shows that the combination

$M_{\rm inv}=M^{N}_{G}+M^{N}_{Q}+M^{N}_{A}$ (31)

is chirally symmetric and equal to the invariant mass in (16).

In Euclidean signature, whether on the lattice or using the QCD instanton
vacuum, (30) can be evaluated by trading $T^{00}\rightarrow T^{44}$ and
$t=0\rightarrow i0$. In the dilute QCD instanton vacuum, the gluonic operator
in (IV-30) is the sum of multi-instanton contributions of the form

$\displaystyle\bar{T}^{44}_{G}[A]=\sum_{I=1}^{N_{\pm}}\bar{T}^{44}_{G}[A_{I}(\xi_{I})]+\sum_{I\neq
J=1}^{N_{\pm}}\bar{T}^{44}_{G}[A_{I}(\xi_{I}),A_{J}(\xi_{J})]+...=\sum_{I\neq
J=1}^{N_{\pm}}\bar{T}^{44}_{G}[A_{I}(\xi_{I}),A_{J}(\xi_{J})]+...$

Since the first one-instanton contribution in (IV) is composed of self-dual
fields it vanishes. So we are left with only the two and higher multi-
instanton contributions. When averaged over a measure of independent
instantons, the remaining terms in (IV) are suppressed by the diluteness
factor $\kappa\approx 0.1$. As a result, the contribution of $M^{N}_{G}$ is
parametrically small in comparison to $M^{N}_{Q}$ or $4M^{N}_{A}$, i.e.
$M^{N}_{G}/M^{N}_{Q}\approx\kappa\approx 0.1$. The contributions $M^{N}_{Q,m}$
are solely given in terms of the fermionic zero modes (modulo the instanton
gauge fields in the long derivative).

With this in mind, the breakdown in the mass budget (30) for the nucleon
yields the estimates

$\displaystyle\frac{M^{N}_{Q}}{M_{N}}$ $\displaystyle\approx$
$\displaystyle\frac{3}{4}\frac{1}{1+\kappa}\bigg{(}1-\frac{\sigma_{\pi
N}}{M_{N}}\bigg{)}\approx 64\%$ $\displaystyle\frac{M^{N}_{G}}{M_{N}}$
$\displaystyle\approx$
$\displaystyle\frac{3}{4}\frac{\kappa}{1+\kappa}\bigg{(}1-\frac{\sigma_{\pi
N}}{M_{N}}\bigg{)}\approx 7\%$ $\displaystyle\frac{M^{N}_{A}}{M_{N}}$
$\displaystyle=$ $\displaystyle\frac{1}{4}\bigg{(}1-\frac{\sigma_{\pi
N}}{M_{N}}\bigg{)}\approx 24\%$ $\displaystyle\frac{M^{N}_{m}}{M_{M}}$
$\displaystyle=$ $\displaystyle\frac{\sigma_{\pi N}}{M_{N}}\approx 5\%$ (33)

with the empirical pion-nucleon sigma term (17). The anomalous contribution is
scale and scheme independent at one-loop order. The mass contribution is also
renormalization group invariant. (IV) shows that in the QCD instanton vacuum,
about 70% of the nucleon mass stems from the valence quarks (hopping zero
modes), 25% from the gluon condensate or epoxy (displaced vacuum instanton
field), and 7% from emerging valence gluons. The nucleon is composed mostly of
quark constituents hopping and dragging the gluon epoxy. The gluon epoxy in
the nucleon is the quantum anomalous energy in the nucleon discussed recently
in Ji and Liu (2021).

We note that the budgeting of the nucleon mass in (IV) differs from the one
reported on the lattice in Yang  _et al._ (2018), with a noticeably larger
valence gluon fraction in the lattice nucleon. In our analysis, this can only
be accomodated by a stronger instanton packing fraction of $\kappa\approx 0.5$
instead of 0.1, which is unlikely. (Note that larger values of $\kappa$ that
include close instanton-anti-instanton pairs, not responsible for the breaking
of chiral symmetry, were reported when analyzing certain correlations at zero
cooling time Athenodorou  _et al._ (2018)). The harder renormalization scale
$\mu=2\,{\rm GeV}$ used in the reported lattice results, is the likely source
of the valence and perturbative gluon enhancement reported in the lattice
nucleon. Quantum evolution will enhance $M_{G}^{N}$ at the expense of
$M_{Q}^{N}$, which in (IV) would amount to effectively dressing $\kappa\approx
0.1\rightarrow 0.5$ at $\mu=2\,{\rm GeV}$.

Finally, a similar mass decomposition holds for the pion at the same soft
renormalization scale of $\bar{\rho}=0.6\,{\rm GeV}$, with the estimates

$\displaystyle\frac{M^{\pi}_{Q}}{m_{\pi}}$ $\displaystyle\approx$
$\displaystyle\frac{3}{8}\frac{1}{1+\kappa}\approx 34\%$
$\displaystyle\frac{M^{\pi}_{G}}{m_{\pi}}$ $\displaystyle\approx$
$\displaystyle\frac{3}{8}\frac{\kappa}{1+\kappa}\approx 3\%$
$\displaystyle\frac{M^{\pi}_{A}}{m_{\pi}}$ $\displaystyle=$
$\displaystyle\frac{1}{8}\approx 13\%$
$\displaystyle\frac{M^{\pi}_{m}}{m_{\pi}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\approx 50\%$ (34)

About 85% of the pion mass stems from the valence quarks (hopping zero modes),
13% from the gluon condensate or epoxy (displaced vacuum instanton field), and
3% from emerging valence gluons. Needless to say that all mass contributions
in the pion vanish smootly in the chiral limit. Again, quantum evolution will
enhance $M_{G}^{\pi}$ at the expense of $M_{Q}^{\pi}$, with effectively
dressing $\kappa\approx 0.1\rightarrow 0.5$ at $\mu=2\,{\rm GeV}$.

##  V Measuring the QCD vacuum compressibility

While the present discussion has focused on some key aspects of the QCD vacuum
and the hadronic mass sum rule, it is worth noting that the results (15-21)
can be recast in the following form

$\frac{\langle P\left|{F^{2}(0)}\right|P\rangle}{(4\pi(m_{N}-\sigma_{\pi
N}/2))^{2}}\approx-\sigma_{F^{2}}\,$ (35)

with the QCD vacuum compressibility

$\sigma_{F^{2}}=\frac{1}{32\pi^{2}}\int d^{4}x\frac{\langle
F^{2}(x)\,F^{2}(0)\rangle_{C}}{\langle F^{2}(0)\rangle}$ (36)

A measure of the gluon condensate or epoxy inside the proton (left hand-side)
is a measure of the QCD vacuum compressibility $\sigma_{F^{2}}$ (right hand-
side), modulo the pion-nucleon sigma term which is small. Since (35) is a
nucleon connected matrix element, it is natural that it probes the
fluctuations of $F^{2}$. While in the vacuum state the gluon condensate is
positive, (36) shows that it is negative in the nucleon state. The nucleon
state carries less epoxy.

The cooled Yang-Mills vacuum in Fig. 1 is composed of interacting topological
charges. The vacuum compressibility $\sigma_{F^{2}}$ captures the squared
variance of their interactions: $\sigma_{F^{2}}=1$ for a non-interacting gas
phase, $\sigma_{F^{2}}<1$ for an interacting liquid phase, and
$\sigma_{F^{2}}\ll 1$ for a strongly interacting crystal phase. QCD low-energy
theorems suggest $\sigma_{F^{2}}\approx 4/b\approx 4/11$ (one-loop and
quenched) Novikov  _et al._ (1981), so the QCD instanton vacuum appears to be
a dilute quantum topological liquid. A measure of $\sigma_{F^{2}}$ is a
measure of a fundamental and universal parameter of the QCD vacuum.

##  VI Conclusions

The QCD instanton vacuum is populated with topological tunneling
configurations, with each costing zero energy. The way a light quark can
propagate coherently through this maze of tunneling configurations is through
its zero mode, scattering and hopping from an instanton to an anti-instanton
and so on. The scattering through the instanton flips chirality, an amazing
effect caused by a non-perturbative vector interaction (a perturbative gluon
interaction preserves chirality). The hopping generates a very dense band in
the virtual quark spectrum, reminiscent of the conduction band in conductors.
As a result, chiral symmetry is spontaneously broken, a chiral condensate is
formed and a running constituent quark mass emerges.

The QCD instanton vacuum breaks simultaneously conformal symmetry, with a
large and negative vacuum energy density, or equivalently a large and positive
gluon condensate (gluon epoxy). A hadronic excitation in this vacuum, whether
a quark, a meson or a baryon costs energy or mass. A useful and physical way
to budget this mass is Ji′s mass decomposition of the energy momentum tensor
Ji (1995a, b). In the QCD instanton vacuum, we find that the hadronic masses
are largely due to the contribution of the valence quarks as they hop and drag
the gluon epoxy.

Finally, a measure of the gluon condensate or epoxy in the nucleon, is a
measure of the compressibility of the QCD instanton vacuum as a topological
liquid. The diluteness of this liquid is central in our non-perturbative
understanding of the emergence of mass in QCD using analytical methods. This
gluonic content of the proton may be accessible through treshold
electromagnetic production of heavy quarkonia Joosten and Meziani (2018), and
perhaps diffractive cluster production in hadron-hadron collisions Shuryak and
Zahed (2021) at current and future facilities.

Acknowledgements

I thank Xiang-dong Ji, Zein-Eddine Meziani and Edward Shuryak for discussion.
This work is supported by the Office of Science, U.S. Department of Energy
under Contract No. DE-FG-88ER40388.

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2021

[1,2]Stefano Martina

These authors contributed equally to this work.

These authors contributed equally to this work.

[1]Department of Physics and Astronomy, University of Florence, Via Sansone,
1, Sesto Fiorentino, I-50019, FI, Italy

2]European Laboratory for Non-Linear Spectroscopy (LENS), University of
Florence, Via Nello Carrara 1, Sesto Fiorentino, I-50019, FI, Italy

3]Physics of Information and Quantum Technologies Group, Instituto de
Telecomunicações, University of Lisbon, Av. Rovisco Pais, Lisbon, P-1049-001,
Portugal

4]Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea, 265,
Trieste, I-34136, TS, Italy

# Learning the noise fingerprint of quantum devices

<EMAIL_ADDRESS>Lorenzo Buffoni<EMAIL_ADDRESS>Stefano Gherardini
<EMAIL_ADDRESS>Filippo Caruso<EMAIL_ADDRESS>*  [  [  [

###### Abstract

Noise sources unavoidably affect any quantum technological device. Noise’s
main features are expected to strictly depend on the physical platform on
which the quantum device is realized, in the form of a distinguishable
fingerprint. Noise sources are also expected to evolve and change over time.
Here, we first identify and then characterize experimentally the noise
fingerprint of IBM cloud-available quantum computers, by resorting to machine
learning techniques designed to classify noise distributions using time-
ordered sequences of measured outcome probabilities.

###### keywords:

Noise Fingerprint, Noise Quantum Sensing, Support Vector Machines, NISQ,
Machine Learning

##  1 Introduction

In the quantum technologies context, no quantum device can be considered an
isolated (ideal) quantum system. For this reason, the acronym _Noisy
Intermediate-Scale Quantum_ (NISQ) technology has been recently introduced
preskill2018quantum  to identify the class of early devices in which noise in
quantum gates dramatically limits the size of circuits and algorithms that can
be reliably performed deutsch2020harnessing ; BhartiArxiv2021 . As early
quantum devices become more widespread, a question that naturally arises is to
understand, at the experimental level, whether in a generic quantum device the
signature left by inner noise processes exhibits universal features or is
characteristic of the specific quantum platform. Moreover, one may wonder to
determine if such a noise signature has a time-dependent profile or can be
effectively considered stable, in the sense of constant over time, while the
device is operating.

The answers to these questions are expected to be crucial in defining a proper
strategy to mitigate the influence of noise and systematic errors DegenRMP2017
; SzankowskiJPCM2017 ; DoNJP2019 ; MuellerPLA2020 ; wise2021using , possibly
going beyond standard quantum sensing techniques ColeNanotech2009 ;
BylanderNatPhys2011 ; AlvarezPRL2011 ; YugePRL2011 ; Paz-SilvaPRL2014 ;
NorrisPRL2016  and overcoming current limitations on probes dimension and
resolution ColeNanotech2009 ; BylanderNatPhys2011 ; FreyNatComm2017 ;
MuellerSciRep2018 ; HernandezPRB2018 ; GomezFrontiers2021 . On top of that, it
gains even more importance in case one proves that noise signatures are
peculiar to the single device, with the consequence that the issue of
attenuating noise effects may be harder than expected. Indeed, each quantum
technologies platform, ranging from superconducting circuits
devoret2004superconducting ; clarke2008superconducting  to trapped ions
quantum computers wineland2003quantum , photonic chips spring2013boson ;
metcalf2014quantum  and topological qubits freedman2003topological , could
need ad hoc solutions that usually are expensive and incompatible from a
device to another. In addition, if the noise properties of a quantum device
happen to be time-dependent, the system necessarily requires continuous
calibrations, thus hindering not only the available runtimes, but also the
accessibility from the external user and the replicability of the experiments
performed on it. Furthermore, in case the noise fingerprint of the considered
device can be easily discerned and remains unchanged over time, one could be
able to identify from which specific quantum device certain data were
generated just by looking at the noise fingerprint. However, this aspect might
create problems, in principle, for possible future usages of the device in
privacy-sensitive applications.

In this paper, we aim to shed light on the previously discussed aspects by
providing a powerful tool, based on Machine Learning (ML) techniques, for the
classification of noise fingerprints in quantum devices with same technical
specifications but physically placed in different environmental conditions. ML
BishopPRML2006 ; HastieESL2009  – originally introduced in the classical
domain to learn from data, identify distinctive patterns, and then make
decisions with minimal human intervention – has been already proven useful to
characterize open quantum dynamics YoussryArXiv ; luchnikov2020machine ;
fanchini2020estimating  and to carry out quantum sensing tasks niu2019learning
; HarperNatPhys2020 ; MartinaArXiv2021 ; wise2021using , as for example the
learning and classification of non-Markovian noise niu2019learning ;
MartinaArXiv2021  or the detection of qubits correlations HarperNatPhys2020 .

Here, we first design a quantum circuit that mimics the transport dynamics of
a quantum particle on a network of $16$ nodes that are identified by the
states of the computational basis $\\{\lvert 0000\rangle,\lvert
0001\rangle,\dots,\lvert 1110\rangle,\lvert 1111\rangle\\}$. The designed
quantum circuit is measured (by locally applying the Z Pauli operators on some
qubits of the circuit) in $9$ distinct parts that, from now on, we denote as
_measurement steps_. The routine allowing to record all the outcome in each
measurement step is instead denoted as _execution_. Moreover, the repetition
of a given number of executions is called _run_. Employing the open-access
quantum computers offered by the IBM Quantum Experience ibmq , we
experimentally classify a set of quantum devices by executing in all of them
the same quantum transport circuit (testbed circuit). The classification is
enabled by the presence of a peculiar noise fingerprint associated to each
quantum machine. In more details, the ML models are trained by taking as input
the distributions of the outcomes recorded at the $9$ measurement steps of the
testbed circuit. As shown in the next sections, the classification is
successfully achieved with a test accuracy greater than $99\%$, both on
diverse IBM machines and on single devices but at different times from one
execution to another. Indeed, from our experiments we can observe that the
noise fingerprint of each tested quantum devices has also a clear time-
dependence, meaning that executions of a quantum circuit, implemented at
different times, can be associated to distinctive main traits.

These experimental evidences lead us to the conclusion that different IBM
quantum devices exhibit distinctive, and thus distinguishable, noise
fingerprints that, however, can be characterized and predicted by ML methods.
Therefore, the proposed solution might be pivotal to certify the time-
scheduling and the specific machine on which a given quantum computation is
executed. Moreover, learning the noise fingerprint of the quantum device under
analysis could play a key role both for diagnostics purposes – especially in
all those contexts where logic quantum operations cannot be error-corrected
deutsch2020harnessing  and thus need to be “noise-free” as much as possible –
and to accomplish benchmarking and certification eisert2020quantum  of quantum
noise sources within a pre-established error threshold.

##  2 Experimental platform

For our experiments we employ the IBM Quantum cloud services to run remotely
quantum circuits on several machines. To interact with the remote services, we
use the Qiskit Software Development Kit (SDK) qiskit , which is an open-source
Python SDK, useful both to simulate quantum dynamics (with or without noise)
and to program a given set of operations on a real quantum device. Overall, we
have at our disposal up to $11$ superconducting quantum computers ranging from
a single qubit up to $15$ qubits, with different topology and calibration
routines. For all the available devices and their characteristics, we direct
the reader to the IBM documentation ibmq .

The accessibility and availability of the IBM devices allow to carry real
experiments having the flexibility of taking either a lot of samples in a
short amount of time, or collecting samples from the same circuit but at
longer time intervals. As it will be shown below, both these aspects will be
properly exploited in carrying out our experiments. Moreover, one can also run
the same exact circuit not only on a single device but on multiple machines,
thus enabling the creation of complete datasets of quantum experiments to be
fed in ML algorithms. Regarding the generation of our datasets, we refer the
reader to the source codes at the address provided at the end of the
manuscript.

Overall, several experiments (explained in detail later) have been conducted
on different IBM chips (specifically, ‘Yorktown’, ‘Athens’, ‘Bogota’,
‘Casablanca’, ‘Lima’, ‘Quito’, ‘Santiago’, ‘Belem’, and ‘Rome’). The chips
differ by two main aspects. The first is the architecture (or connectivity) of
the qubits, which ranges from a simple line topology to a ladder or a star
topology. The second important difference is the so-called quantum volume
cross2019validating  ($8$, $16$, $32$ for the machines used in our
experiments) that quantifies the maximum dimension of a circuit that can be
effectively executed, and is correlated also with the noise affecting each
device. Indeed, some quantum machines are inherently noisier than other, and
even single qubits inside a machine can have a distinctive noise profile. All
these peculiar differences in noise and topology represent the fingerprint
that we aim to exploit using our method.

Before proceeding, it is worth stressing that, albeit the proposed experiments
are carried out on superconducting devices, the gate-model approach adopted
here is valid in principle for a large class of NISQ devices.

##  3 Testbed quantum circuit

To learn the noise fingerprint of IBM quantum devices, we employ a quantum
walker on a network of $16$ nodes realized by the quantum states $\lvert
0000\rangle,\lvert 0001\rangle,\dots,\lvert 1110\rangle,\lvert 1111\rangle$
through the circuit in Fig. 1. Notice that, for our purposes, the number of
qubits of the testbed circuit can be just a few; however, this does not imply
that the proposed solutions cannot be applied to circuits with generic
dimension.

Figure 1: On the left, circuit implementation of the quantum transport
dynamics employed as a testbed. The quantum circuit, which involves 4 qubits,
is repeated more than once (in our experiments, $3$ times), and 2 of the 4
qubits are measured at regular steps. The outcome probabilities obtained by
our measurements, which together form the datasets to train, validate and test
the used ML models, are fed into a Support Vector Machine (SVM) –
schematically represented on the right – in order to be classified.

The idea of our testbed circuit is to simulate a quantum transport dynamics,
whereby, by initialising a quantum particle in one node of the network
(specifically, in the state $\lvert 0000\rangle$), the particle “flows” across
the allowed pathways thanks to the action of local operations and of
_controlled NOT_ (CNOTs) and _Toffoli_ gates (denoted in Fig. 1 by a light
blue and light purple rounds, respectively, with the symbol ‘plus’ inside). We
recall that the CNOT is a two-qubit quantum operation, commonly used to
entangle/disentangle Bell states, that flips the second qubit when the first
qubit is in $\lvert 1\rangle$. Instead, the Toffoli gate is a universal
“controlled-controlled-not” (3-qubit) operation where a third qubit is flipped
when two control qubits are both in $\lvert 1\rangle$. In our circuit in Fig.
1, two qubits (i.e., $q_{3}$ and $q_{2}$ in the figure) are used to get
information on the particle, providing at each measurement the pair of bits
$(0,0),(0,1),(1,0),(1,1)$, where the first and second bits correspond,
respectively, to the outcomes measured on $q_{3}$ and $q_{2}$. Conversely,
qubits $q_{0}$ and $q_{1}$ are employed as ancilla qubits to move, and thus
control, the particle. Then, this quantum circuit is repeated $3$ times, with
the aim to collect data on the quantum dynamics in each IBM device. As already
mentioned in the Introduction, the resulting quantum circuit (given by
repeating $3$ times the circuit in Fig. 1) is locally measured in $9$ distinct
parts (corresponding to the measurement steps) thanks to the simultaneous
application of Z Pauli operators $\sigma_{z}$ on the qubits $q_{3}$ and
$q_{2}$, from which the measurement outcomes are collected. It is worth noting
that the procedure we are proposing is not based on repeated measurements as
in a quantum monitoring protocol or in Zeno quantum dynamics FischerPRL2001 ;
SchaferNatComm2014 ; GherardiniQST2017 ; VirziArXiv2021 , since, each time a
measurement is performed at a given measurement step (say the $k$-th, with
$k=1,\ldots,9$), the whole testbed quantum circuit is regenerated and then
(locally) measured at the subsequent step, i.e., the $(k+1)$-th.

In a single repetition, the quantum circuit is initialized in $\lvert
0000\rangle$ that corresponds to the measurement outcomes $(0,0)$, and then
two Hadamard gates (blue squares ‘H’ in Fig. 1) are applied to both $q_{0}$
and $q_{1}$. Thus, since the two CNOT gates are conditioned to $q_{0}$ and
$q_{1}$ respectively, the probability to get $1$ or $0$ in $q_{2}$ and $q_{3}$
after the CNOTs is $0.5$. In this way, after the Pauli-X rotation (green
squares ‘X’ in Fig. 1) and the Toffoli gate, the system is in the state
$\frac{1}{2}(\lvert 0110\rangle+\lvert 0111\rangle+\lvert 1001\rangle+\lvert
1100\rangle)$ before that the qubits $q_{2}$, $q_{3}$ are measured along the
$z$-axis (black squares in Fig. 1). This entails that, at the end of the
circuit, measuring $q_{3}$ and $q_{2}$ provides the results
$\\{(0,0),(0,1),(1,0),(1,1)\\}$ with probabilities respectively
$\\{0,0.5,0.25,0.25\\}$. Of course, such a dynamic only occurs under _ideal_
unitary evolution, which is not the case of the implementation on real
experimental devices. In our case, the noisy environment, in which the
machines are immersed, alters each realization of the simulated quantum
(transport) dynamics, thus making stochastic the evolution of the particle
within the circuit. As we will prove below, this randomness is a specific
feature of each machine and changes from one device to another, thus allowing
us to perform classification tasks. Specifically, are the discrepancies
between the measured outcome probabilities (from qubits $q_{2}$ and $q_{3}$)
on one or more IBM machines that enable to learn the corresponding noise
fingerprint, and then classify from which device the input data have been
generated. Here, it is worth noting that, despite from one implementation to
another a slight different physical Hamiltonian may be implemented in the
chips of each quantum device, the variations observed in the measurement
outcome distributions – having a prominent random nature – are not ascribable
to such a deterministic aspect, but to a stochastic cause thus pertaining to
an external noise source. However, a same stochastic process can affect
differently two equivalent quantum dynamics but originated by two distinct
physical Hamiltonian operators. Therefore, the fingerprint that we leverage
for the classification can be due not only to differences in the noise
profiles affecting the quantum devices, but also on their dependence on the
way the testbed circuit is physically implemented.

While our picture of considering the implemented quantum dynamics as the ones
of a quantum walker may be pretty useful for illustrative purposes, we want to
stress that this kind of dynamics has been chosen just for its simplicity and
generalizability with other types of NISQ devices. Indeed, our results shown
in the following are quite general, since they do not depend on specific
dynamics and do not require initial assumptions. Accordingly, we expect that
such results may be re-obtained in other quantum devices, even ones not
necessarily designed to carry out computing tasks.

##  4 Machine learning model

Let us provide some details on the adopted ML model, i.e., the popular Support
Vector Machine (SVM) HastieESL2009 .

The dataset yielded as input to the SVM is a set of $n$ points
$\mathbf{x}_{q}\in\mathbb{R}^{p}$, with $q=1,\ldots,n$, each of them living in
the $p$-dimension space of the data features, where a feature is a distinctive
attribute of the data set elements.

In binary classification problems, to each $\mathbf{x}_{q}$ with $q=1,\dots,n$
is associated a class $y_{q}\in\\{-1,1\\}$ that represents the desired output
of the SVM. By contextualizing it to our problem, the binary classes $y_{q}$
denote if a given set of points $\mathbf{x}_{q}$ have been generated $(+1)$ or
not $(-1)$ by a specific machine or in a time window/interval. A SVM for
binary classification is trained such that the two classes of points (provided
as input to the ML model) are separated by the hyperplane that maximises the
distance between the hyperplane itself and the nearest points of the classes
(commonly denoted as _margin_). If the points $\mathbf{x}_{q}$ of the data set
are not linearly separable (which is most often the case), then the value of
the margin is negative and the points cannot be classified. To circumvent this
problem, SVMs employ a clever mapping in an higher-dimensional space (called
feature-space) with polynomial or Radial Basis Function (RBF) kernels that
allows for an easy classification as in Fig.1. The extension to multiclass
classification problems is then obtained by associating a class with multiple
values to each $\mathbf{x}_{q}$. In our experiments, part of the generated
dataset is used as a validation set to choose the best mapping among the
kernels: linear (meaning that the data is already linearly separable),
polynomial with degree 2, 3 and 4, and RBF. In many cases, just the simple
linear kernel is enough to successfully perform the classification, but in
other cases (e.g., in multiclass classification) the more complex kernels may
be beneficial.

Finally, in our experiments, the classification _accuracy_ is computed by
comparing the predictions $\hat{\mathbf{y}}$ returned by the ML models with
the desired classes $\mathbf{y}$ of the test set:

${\rm
accuracy}(\hat{\mathbf{y}},\mathbf{y})\equiv\frac{1}{n}\sum_{i=1}^{n}\mathds{1}\left\\{\hat{\mathbf{y}}_{i}=\mathbf{y}_{i}\right\\}$
(1)

where $\mathds{1}\\{\cdot\\}$ is the _indicator function_ such that
$\mathds{1}\\{c\\}=1$ if $c$ is true, and $\mathds{1}\\{c\\}=0$ otherwise. In
this regard, to clarify the naming convention for the reader, we refer to:
“training”, “validation” and “test” sets, to identify three non-overlapping
partitions of the data. These partitions are used respectively to: train the
model, validate the best parameters, and test the performance on unseen data.
In the experiments we randomly select $60\%$ of the data to train the SVM
model, $20\%$ to validate different configurations (i.e., SVM kernel type),
and $20\%$ to report the results on unseen data.

##  5 Experiments description

The results, which we are going to show, concern three series of ML
experiments that use two different datasets, obtained from the IBM quantum
chips mentioned above.

In the first two experiment series, the ML models are trained both to
discriminate the noise fingerprint of different quantum devices and to
identify a time-dependence in each of them. The training of some of the models
is performed on the dataset here denoted as FAST that collects the outcome
distributions measured in temporally-close executions of the testbed quantum
circuit on $7$ different IBM quantum machines (i.e., ‘Athens’, ‘Bogota’,
‘Casablanca’, ‘Lima’, ‘Quito’, ‘Santiago’, ‘Yorktown’). In these experiments,
$20$ parallel tasks (corresponding to the maximum allowed number) are appended
to the IBM fair-share queue, and, once a task is concluded, another task is
immediately added. For each task the testbed circuit is run $8\,000$ times for
each one of the $9$ different steps, and the probabilities to get the
measurement outcomes are computed over $1\,000$ shots among the total $8\,000$
to obtain $8$ different outcome probabilities times $9$ steps per task.

Conversely, in the third ML experiment series, we perform a robustness
analysis by making stricter the time constraints on the employed datasets.
Specifically, in those experiments, and in part of the previous ones, we
employ a second dataset, called SLOW, which is composed of measurement
distributions extracted from executions in two different quantum machines
(‘Belem’ and ‘Quito’) more “slowly” than the data in the first dataset. As
represented in Fig. 2, more “slowly” means that only one task per time is
appended to the queue and then run, waiting at least $2$ minutes from the
conclusion of the previous task. Moreover, for each task the testbed circuit
is executed, for each one of the $9$ steps, $1\,000$ times that corresponds to
the number of shots set to compute the outcome distributions.

Figure 2:  Elapsed hours to collect all the measurement outcomes on the IBM
machines ‘Belem’ and ‘Quito’ (solid blue and dashed red lines, respectively)
for the dataset SLOW. Each point of the curves, obtained over $1\,000$
executions of the testbed quantum circuit for each measurement step
$k=1,\ldots,9$, is associated to the relative physical/real time in which the
measurement probabilities are computed in a single run. Notice that, if
compared with the time scale of the vertical axis (y-axis), which is expressed
in hours, the computation of the $9\,000$ executions of each run can be
considered practically instantaneous, i.e., in the order of some seconds.
Moreover, the anomalous behaviour of the curves after $1\,500$ runs has to be
attributed to the policy of the IBM fair-share queue.

We recall that in each execution, for both the FAST and SLOW datasets, the
qubits $q_{3}$, $q_{2}$ of the testbed quantum circuit (the full circuit is
obtained by repeating 3 times the circuit in Fig. 1) are measured iteratively
after each CNOT and Toffoli gate, for a total of $9$ outcome probabilities at
the consecutive measurement steps $k=1,\dots,9$. Overall, for each machine, we
have collected $2\,000$ sequences of $9$ probability distributions built with
the measurement outcomes from the qubits $q_{3}$ and $q_{2}$ of the testbed
circuit. This means that a total of $2\,000\,000\times 9$ single executions
have been run on each quantum machine that we employed to generate the FAST
dataset, and similarly for the SLOW one.

As final remark, let us note that the FAST dataset is employed for the
experiments illustrated in section 6 and part of section 7, while the SLOW
dataset to complete the experiments in section 7 and perform in section 8 a
robustness analysis at different time scales.

##  6 Quantum devices classification

As first, we present binary classification experiments. For each pair of IBM
machines, a SVM model is trained using the dataset FAST (introduced in section
5) with the aim to identify on which device the executions of the testbed
quantum circuit are run. The inputs of the SVM model are the distributions of
the measurement outcomes from qubits $q_{3}$ and $q_{2}$ recorded at the
discrete measurement steps $k=1,\dots,9$. Specifically, two different kinds of
inputs are set: In the first we consider only the outcome distributions
measured at the single step $k$ with $k\in[1,9]$, while in the second we
concatenate all the measurement probabilities in ordered sequences
$1,\dots,k$. Then, our ML experiments are performed by alternatively taking
the two types of inputs; we will report below the resulting accuracy values
for both of them.

Table 1: Classification accuracy, denoted as $\alpha({\cdot})$, of all the
possible binary SVMs trained with the measurement probabilities collected in
the dataset FAST. For each experiment, a large number of executions are run on
two different IBM machines (whose names are in the $1$st column and in the
$1$st row of the table) in correspondence of the measurement steps $k$ ($1$st
column of each sub-table). Then, two different inputs are tested: Outcome
distributions at single steps (whose accuracy values are in the $2$nd column
of the sub-tables) and sequences of measurement probabilities obtained at each
$k$ (accuracy values in the $3$rd column of each sub-table).

From the results of our experiments – reported in table 1 – we observe that it
is sufficient to use only the outcome probabilities corresponding to the first
three measurements at $k=1,2,3$ to reach more than $99\%$ of accuracy in
discriminating all the pairs of tested machines. This implies that, in a
realistic deployment scenario, one needs less data than the amount acquired
here to reach good classification performances. An additional observation we
can make is that the accuracy is _not monotonic_ in $k$ when considering the
classifier using single measurement data. This can be due to the fact that, at
various measurement steps, to distinguish the noise fingerprint from a single
measurement probability might be easier or harder. On the other hand, we can
also observe that the accuracy is steadily increasing when as input is set the
sequence of all outcome distributions up to any measurement step $k$. Hence,
from this we can deduce that, to identify the noise fingerprint of IBM quantum
devices, sequences of outcome distributions recorded at more than one
measurement steps need to be taken into account. This is also the reason why
we deem important to frame the issues addressed in this paper as belonging to
a noise fingerprint _in time_ instead of single shot measures.

Table 2: Classification accuracy, denoted as $\alpha({\cdot})$, of multiclass
SVMs trained with the measurement probabilities collected in the dataset FAST.
A large number of executions are run on 7 different IBM machines ($1$st column
of the table) in correspondence of the measurement steps $k\in[1,9]$ ($2$nd
column). Different inputs are tested: Outcome distributions at single steps
($3$rd column), sequences of measurement probabilities computed on windows of
width from $2$ to $5$ steps before each $k$ (from $4$th to $7$th columns), and
the sequences of all measurement probabilities obtained from the $1$st to the
$k$th step ($8$th column). Finally, the last row of the table reports, for all
the $k$s, the averages of the accuracy values in the rows above; the average
of the last column is omitted since the accuracy values therein are calculated
on models with different numbers of input measurement steps.

Let us now extend the binary SVM algorithms to multiclass classification
problems, in which more quantum devices are simultaneously discriminated. In
our experiments, the so-called _one-vs-rest_ strategy is adopted
BishopPRML2006 , where for $n$ distinct classes we train $n$ different binary
classifiers that discriminate the elements of a class from the others. In
particular, our multiclass SVM is trained with the aim to identify to which
IBM quantum machines, among the $7$ that have been used, belongs a given set
of measured outcome probabilities (from the testbed quantum circuit) of the
FAST dataset. The results in table 2 reports the test accuracy values returned
by the models that are trained with different input data. As in binary
classification, for one kind of input data, the model is trained with the
outcome distributions obtained at single step $k$ with $k\in[1,9]$ ($3$rd
column of table 2), while another set of input data is provided by
concatenated measurement probabilities $1,\dots,k$ ($8$th column). Moreover,
for the purpose of multiclass classification, further input are also adopted:
At each step $k$ the model is trained not only with the outcome distributions
at the $k$th step, but also with a window of preceding measurement
probabilities belonging to $[k-s,k]$ with $s$ integer number. Regarding $s$,
the range from $1$ ($4$th column of table 2) to $5$ ($7$th column of table 2)
is considered. As for the binary case, the SVM is able to successfully
discriminate between the tested machines just by using the measurement
outcomes taken in few measurement steps. While the accuracy using the outcomes
at single-time measurement steps oscillates, the time-ordered sequence
monotonically increases. That confirms our previous observations about the
need of a time sequence to have a reliable fingerprint. In addition, the
models trained with the input data on sliding windows allow us to understand
the effective need of outcome distributions taken from more than a single
measurement step for the classification of the noise fingerprint. In such
case, we observe that the accuracy at each step $k$ steadily increases with
the size of the set of considered steps, and this holds also by looking at the
average of the accuracy values computed over all the measurement steps. It is
worth noting that the last column on table 2 expresses a similar strategy,
where the single accuracy values are provided as output of the models trained
on a window (with increasing dimension) that always starts from the $1$th to
the $k$th step. In other words, the first accuracy values on top of columns
from $3$ to $7$ correspond to the elements of the last columns for $k$ from
$1$ to $5$.

The high-level of accuracy (even more than $99$%) in carrying out binary and
multiclass classification of the IBM quantum machines is an evidence for the
presence of a strong underlying noise fingerprint in the dynamics of NISQ
devices. Indeed, this is the key feature that can allow one to identify,
basically in a deterministic way, from which quantum machine a given set of
measurement has been obtained.

##  7 Noise fingerprint at different time scales

Since the environment of the IBM quantum devices changes quite often (e.g.,
the machines are calibrated up to multiple times in an hour), we have slightly
modified our experiments to prove also the existence of a noise fingerprint
that pertains to the temporal evolution of the chip on which a given quantum
circuit is executed.

Table 3: Classification accuracy, denoted as $\alpha(\cdot)$, of SVMs –
trained with two sets of outcome distributions from the dataset FAST,
temporally separated by 24 hours – to predict in the IBM machine ‘Casablanca’
which executions were implemented the first day and which the second day. Also
in this case, the inputs to the SVMs at the measurement steps $k$ ($2$nd
column of the table) are the outcome distributions at single steps ($3$rd
column) or the sequences of measurement probabilities computed at each $k$
($4$th column).

To confirm this hypothesis, we have designed a temporal classification setting
that we employ with data from both the FAST and SLOW datasets.

Regarding the experiments using the FAST dataset, two sets of measurement
outcome distributions are collected for the machine ‘Casablanca’, one
temporally separated from the other by 24 hours. After that, similarly to what
done in the previous experiments, a SVM model is used to discriminate the
executions implemented the first day on the IBM device from the ones performed
on the second day. From these experiments, whose results are shown in table 3,
we observe that the designed ML algorithms are able to detect a characteristic
fingerprint, still induced by the presence of noise sources, in a single
quantum device but in measurement steps separated by a quite long ($24$ hours)
time interval. In such classification tasks, an accuracy of $95\%$ is achieved
by the ML models, just by taking as input the sequence of outcome
distributions at the first measurement steps $k=1,2,3$.

Table 4: Binary classification accuracy, denoted as $\alpha(\cdot)$, of SVMs
trained to classify the outcome distributions belonging to distinct two sets
of data. One set is composed by the runs of ‘Belem’ in the SLOW dataset by
numbering them from $1$ to $200$ in temporal ordering. Also the other set is
composed by runs of ‘Belem’ in the SLOW dataset, but collected within temporal
windows specified on the columns title (from run $201$ to run $400$, from run
$401$ to run $600$, etc…). In the top sub-table the models are trained with
the outcome distributions taken at the $k$th measurement step, while in the
bottom sub-table the inputs are the sequences of measurement probabilities
from step $1$ to step $k$.

In order to better quantify the evolution in time of the noise fingerprint, we
use data from the runs of ‘Belem’ in the SLOW dataset. Respect to the previous
dataset, the data from the runs in SLOW are more evenly distributed in time so
that we have decided to split the data in $10$ adjacent windows, each of them
containing $200$ consecutive runs. Subsequently, the SVMs models are trained
to classify if a run has been computed on the first window (from run $1$ to
run $200$) or in another window of the remaining $9$. From the results in
table 4, we can observe that is difficult to distinguish the runs pertaining
to the first window from the runs in the adjacent window (i.e., runs from
$201$ to $400$ in the third column), either considering as input the single
outcome distributions at the $k$th measurement step (the top part of table 4)
or the sequences of measurement probabilities from step $1$ to step $k$
(bottom part). As a matter of fact, we do not reach $90\%$ in neither case.
Conversely, when we consider the subsequent windows (runs after $400$ on the
next columns), thus at a greater distance from the first window, the
classification task becomes easier.

Analogously to the previous experiments, the single measurement outcomes do
not seem to carry enough information on the noise fingerprint and the
classification accuracy depends on the choice of $k$. Instead, when we
consider the sequences of outcomes for all the steps, we can observe that the
noise fingerprint in the first window of runs can be much better distinguished
from the corresponding fingerprint in all the subsequent windows, except the
neighbouring one. The window from run $1401$ to run $1600$ seems more
challenging to classify with respect to the others. One possible reason for
this can be that, as one can see from fig. 2, around the run $1500$ the
policies of the IBM fairshare queue caused a discontinuity in time. This means
that the data distribution inside the aforementioned window has more variance
with respect to the data in the other windows and for the ML models can be
find more difficult to classify the data. However, even in that case the
classification accuracy reaches $100\%$ when using the sequence of measurement
probabilities for all the steps $k=1,\dots,9$.

Figure 3:  Maximum reached accuracy for SVM models trained on sequences of
measurement outcomes for all the steps $k=1,\dots,9$ taken from the ‘Belem’
quantum machine and collected in the dataset SLOW. The model is trained to
classify the executions in the window of runs from $1$ to $200$ from the ones
in a subsequent window of $200$ runs. Initially, the latter is adjacent to the
first window, then it is moved by increasing the gap between the two windows.
The plotted curve is then obtained by drawing the accuracy values for the
corresponding gaps, expressed in hours. Note that a gap of $6$ hours
correspond to approximately $90$ runs.

In these experiment, the execution times for all the runs in each window is
approximately $12$ hours (except for the previously-discussed window from run
$1400$ to $1600$). Thus, we can deduce that $12$ hours of time distance
between the windows are sufficient to distinguish the noise fingerprint at
different times with $100\%$ of accuracy. To find the minimum necessary hours
gap, in fig. 3 we report the reached accuracy of a SVM model trained to
distinguish the runs in the first window (from run $0$ to $200$) of ‘Belem’
within the SLOW dataset from the runs in another window with an increasing
time gap among them. We can observe that, in this case, already after $6$
hours the noise fingerprint is distinguishable with an accuracy of $100\%$. In
general, we can observe that even starting from different windows in time, and
using different window sizes, more than $95\%$ of accuracy is reached after a
few hours (in the order of one day).

Overall, we can thus conclude that a clear temporal dependence of noise
fingerprint is present in our experiments, even when the same quantum machine
is taken into account.

##  8 Robustness analysis

Finally, we investigate the robustness of the learned fingerprint at different
time scales. For this purpose, taking the IBM machines ‘Belem’ and ‘Quito’, we
temporally order all the executions of the testbed quantum circuit, by
dividing them in $10$ distinct windows of $400$ consecutive runs, i.e., $200$
runs per machine. The elapsed time between runs has been already reported in
Fig. 2. In this way, after have generated the SLOW dataset (introduced in
section 5) with $2000$ runs per machine, the SVMs are trained to classify on
which device, among ‘Belem’ or ‘Quito’, the testbed quantum circuit has been
executed. Specifically, in any experiment designed for the robustness
analysis, the ML model is trained over the data collected in a time window of
$200$ consecutive runs (overall, we consider $10$ distinct time windows), and
then tested in all the considered time windows including the one used for the
training.

Table 5: Classification accuracy of SVMs trained to classify on which quantum
device, among ‘Belem’ or ‘Quito’, a given set of data has been generated. The
training of the models is performed with the outcome distributions collected
in the dataset SLOW, and then divided in $10$ distinct time windows of $200$
runs (the first window includes the runs from $1$ to $200$, the second from
$201$ to $400$, etc). We recall that each run contains the outcomes from all
the $9$ measurement steps in each execution. The row and column indexes
denote, respectively, the number of time windows whose data are used to train
and test the ML model. Finally, the reported accuracy values are calculated by
using the outcome distributions computed at all the measurement steps
$k=1,\ldots,9$.

All the obtained results – summarized in table 5 – point out the following
peculiar feature. Unsurprisingly, the SVM reaches $100\%$ of accuracy in the
time window used for the training of the ML model (corresponding to the
diagonal of the table), and then, in proximity of the time windows on the
diagonal, the accuracy decreases monotonically. This corresponds to the
intuition that the machine-related noise fingerprint “fade” with time, due to
the evidence – discussed in the previous section – that the noise fingerprint
of the IBM quantum devices exhibits a quite prominent time-dependence.
However, surprisingly, we observe that the accuracy returns to $100\%$ for
time windows of runs far from the training one. We conjecture that this
counter-intuitive phenomenon may be due either to the periodic calibration of
the machines or to the slowdown induced by the fair-share queue. The latter,
indeed, may be also observed in the last part of the SLOW dataset in Fig. 2,
and is supported by the evidence that, if we restrict the experiment to the
runs from $1$ to $1\,000$ (i.e., the range where the execution times of the
tested machines are more homogeneous as shown in Fig. 2), the resulting
accuracy values decrease with time.

The general result that can be deduced from the experiments of the robustness
analysis is that, by training our ML model on just $200$ runs (corresponding
to the diagonal time windows of the table), we are able to identify the
device-related noise fingerprint with high accuracy for all the 1 800
remaining ones. In this regard, it is worth noting that, between the training
samples and the last test ones, there is up to a week in real-time execution
(as one can see in Fig. 2). This means that we can consider our classifier to
be fairly robust in time, despite the changes in the environment and
calibration of the machines that might occur even at time-scales of weeks.

##  9 Conclusions

In this work we prove the existence of a noise fingerprint – also admitting a
clear time-dependent profile – in the tested IBM quantum machines, which are
just a particular class of NISQ devices. We have also demonstrate that such
noise fingerprints can be exploited to reliably distinguish the machines by
means of SVM models. As general results, our experiments confirm that (i) all
the analysed quantum devices exhibit a clear machine-related noise fingerprint
that is _robust_ , in the sense that the fingerprint is highly predictable
over time in windows of consecutive runs; (ii) the noise fingerprint has also
a _time dependence_ , namely it changes over time and after few hours becomes
different enough to be distinguished from fingerprint in the past; (iii) in
each quantum device, sequences of measurement outcome distributions are
required for the accurate learning of the corresponding noise fingerprint. One
may conjecture that possible reason behind the latter aspect may be that the
noisy dynamics in the IBM machines can be non-Markovian due to the presence of
time-correlations among consecutive samples of the noise field. However, it is
worth observing that the SVMs we successfully used in this work are memory-
less ML models, which thus ignores possible temporal relations across the
measurement steps. Therefore, the gathered data and the adopted ML models are
not indicated to validate any hypothesis on non-Markovianity. These aspects,
deserving further investigations, will be addressed in another contribution in
which memory-less ML models will be compared with other ML architectures
processing time series data with variable memory length. In conclusion,
despite the microscopic reasons for the existence of a machine-related noise
fingerprint are still unknown (indeed, the IBM machines are partly
inaccessible), we can now affirm that one can reliably leverage such noise
profiles to distinguish, and possibly in the future characterize, different
NISQ quantum devices.

As an outlook, learning the noise fingerprint of quantum devices from time-
ordered measurements of testbed quantum circuits is expected to open the way,
in the next future, to many other experiments and ideas. The proposed
methodology, indeed, may be applied not only to IBM quantum machines, but even
to a larger class of quantum devices, both in commercial or laboratory
scenarios. In all of them, classification ML model, exploiting the presence of
intrinsic noise sources that give rise to an identifiable noise fingerprint in
the devices, may be employed to predict on which machine, and at which time, a
given quantum circuit or algorithm was executed. Moreover, our procedures
could be adopted to predict if and when the noise fingerprint of a specific
quantum device changes over time, e.g., due to calibration actions. Such a
knowledge will help in mitigating (time-varying) errors occurring in the
computation and, possibly, performing ad hoc error corrections.

### Data and code availability

The datasets and the source codes for the ML experiments are available on
_GitHub_ at the following address:
https://github.com/trianam/learningQuantumNoiseFingerprint .

### Acknowledgments

This work was financially supported from Fondazione CR Firenze through the
project QUANTUM-AI, from University of Florence through the project Q-CODYCES,
and from the European Union’s Horizon 2020 research and innovation programme
under FET-OPEN Grant Agreement No. 828946 (PATHOS). We acknowledge the access
to advanced services provided by the IBM Quantum Researchers Program.

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  *[ML]: Machine Learning
  *[SDK]: Software Development Kit
  *[SVM]: Support Vector Machine
  *[SVMs]: Support Vector Machine
  *[RBF]: Radial Basis Function