# Robust Macroscopic Schrödinger’s Cat on a Nucleus

Pragati Gupta  Institute for Quantum Science and Technology, University of
Calgary, Alberta T2N 1N4, Canada  Arjen Vaartjes  School of Electrical
Engineering and Telecommunications, UNSW Sydney, Sydney, New South Wales 2052,
Australia  Xi Yu  School of Electrical Engineering and Telecommunications,
UNSW Sydney, Sydney, New South Wales 2052, Australia  Andrea Morello  School
of Electrical Engineering and Telecommunications, UNSW Sydney, Sydney, New
South Wales 2052, Australia  Barry C. Sanders  Institute for Quantum Science
and Technology, University of Calgary, Alberta T2N 1N4, Canada

###### Abstract

We propose a scheme to generate spin cat states, i.e., superpositions of
maximally separated quasiclassical states on a single high-dimensional nuclear
spin in a solid-state device. We exploit a strong quadrupolar nonlinearity to
drive the nucleus significantly faster than usual gate sequences, achieving
collapses and revivals two orders of magnitude faster than the dephasing
timescale. Furthermore, these states are engineered without entanglement with
an ancilla, hence, are robust against error propagation. With our multitone
control, we can realize arbitrary high-spin rotations within an experimentally
feasible regime, as well as transform a spin coherent state to a spin cat
state using only phase modulation, opening the possibility of storing and
manipulating high-fidelity cat states.

##  I Introduction

A Schrödinger cat state [1, 2, 3] is an equal superposition of two distinct
“classical” states of a system [4, 5, 6], that is a probe for fundamental
aspects of quantum physics, including the measurement problem [7], macro-
realism [8, 9], quantum-classical transition [10], and non-locality [11]. Cat
states also serve as important resources for quantum error correction [12, 13,
14, 15], quantum sensing [16, 17, 18], and quantum communication [19, 20].
Thus, practical realization of cat states is of interest for both foundational
and technological applications.

In continuous-variable systems, cat states are generated using nonlinearity
induced by light-matter interactions in the Jaynes-Cummings model [21, 22],
e.g., a harmonic oscillator coupled to a superconducting qubit [23, 24, 25] or
an atom in an optical cavity [26]. However, errors occurring in the qubit can
propagate to the cavity, corrupting such nonclassical states [25].
Alternatively, Greenberger-Horne-Zeilinger (GHZ) states, which are spin cat
states (SCSs) for permutationally symmetric systems, can be realized by
entangling multiple qubits [27, 28, 29, 30], but such SCSs are fragile—with
their decoherence rate proportional to the square of the number of qubits
[31].

We propose a new platform to generate cat states deterministically, on-demand
and without post selection using a nucleus, which is controlled without
entanglement with an ancilla, hence, is robust against error propagation,
unlike continuous-variable systems [12, 23, 24, 25, 26]. Here, we consider
SCSs (implying macroscopic to differentiate them from “kitten” states [32]) on
a nuclear spin, e.g., shown in Fig. 1, but, more generally, a superposition of
any two maximally separated quasi-classical states, which are minimum
uncertainty states that follow a classical evolution under a linear drive [5,
6]. We are motivated by recent experiments on high-spin nuclei (spin $I>1$) in
a solid-state device [33, 34] accompanied by an enhanced electric field-driven
quadrupole interaction, two orders of magnitude faster than the dephasing
timescale, that could achieve ultrafast non-linear dynamics.

A highly coherent nuclear spin could open the possibility of robust generation
and long-lived storage of SCSs. So far, quadrupole interaction has only been
utilized for realizing spin squeezed states (SQSs), that too, on an ensemble
of nuclear spins [35, 36]. However, individual addressing of high-spin systems
is important for applications like quantum information processing and error
correction [15]. We present two key results: i) a scheme to go beyond squeezed
states towards generating SCSs and ii) an experimentally feasible control
protocol for storing and manipulating SCSs on an individual nucleus.

Figure 1: Superposition of spin-up and spin-down instances of a high-spin
nucleus ($I>1$), comprising protons (red) and neutrons (blue).

##  II Setup

We consider a high-spin donor (Group-V element with a high-spin nucleus)
implanted in a nanoelectronic device based on enriched 28Si (or any other
spinless Group-IV atoms), as described in [33]. The chip is affixed to a
dilution refrigerator maintained at a low temperature ($T\approx 0.01$ K) and
placed inside a superconducting coil that produces a static magnetic field
$\bm{B}_{0}\approx 1$ T. As shown in Fig. 2(a), the chip includes
electrostatic gates to control the electrochemical potential of the donor, a
microwave antenna to supply electromagnetic pulses, and a single electron
transistor (SET) for reading out the donor spin state [37].

An implanted Group-V donor contains five electrons in its outermost shell,
four of which form bonds with neighboring Si atoms and the remaining electron
can be removed by ionizing the donor using electrostatic gates. The quantum
state of the nuclear spin is initialized using flip-flop drive [38], where an
electron and nuclear spin pair is flipped together in opposite directions. The
donor atom is subject to a static potential, and a target nuclear-spin
eigenstate can be achieved by applying a combination of flip-flop pulses and
electron spin-resonance pulses that move population from higher and lower
nuclear spin states towards a target state [34].  Then, the electrostatic
potential is increased to ionize the donor atom by removing the electron and
only the nuclear spin is manipulated using RF pulses. [39]. The readout
process using electron spin resonance can be made dependent on the nuclear
spin state, thus achieving nuclear readout via spin dependent tunneling of the
electron [40, 33].

The differential thermal expansion between the silicon substrate and the
electrostatic gates induce lattice strain in the chip [33], thereby distorting
the bonds between the donor atom and neighboring Si atoms, as shown in Fig.
2(b). The electric-field gradient (EFG) generated from distortion of the bonds
results in a quadrupole interaction of the high-spin nucleus, which has a non-
zero quadrupole moment due to its asymmetric nuclear charge distribution. For
a spatially varying potential $V(x,y,z)$, the EFG tensor comprises elements
$V_{\alpha\beta}:=\frac{\partial^{2}V(x,y,z)}{\partial\alpha\partial\beta},\,\alpha,\beta\in\\{x,y,z\\},$
which is real, traceless and symmetric. Thus, $V_{\alpha\beta}$ can be
diagonalized in a principal axis system $\\{x^{\prime}$, $y^{\prime}$,
$z^{\prime}\\}$, such that
$|\mathcal{V}_{z^{\prime}z^{\prime}}|\geq|\mathcal{V}_{y^{\prime}y^{\prime}}|\geq|\mathcal{V}_{x^{\prime}x^{\prime}}|$,
and the EFG is characterized by the asymmetry parameter
$\eta:=\frac{\mathcal{V}_{x^{\prime}x^{\prime}}-\mathcal{V}_{y^{\prime}y^{\prime}}}{\mathcal{V}_{z^{\prime}z^{\prime}}}$
for $0\leq\eta\leq 1$. For a nucleus with electric quadrupole moment
$q_{\text{n}}$, the quadrupole interaction is

$\hat{H}_{\text{q}}=\omega_{\text{q}}\left[\hat{I}_{z^{\prime}}^{2}+\frac{\eta}{3}(\hat{I}_{x^{\prime}}^{2}-\hat{I}_{y^{\prime}}^{2}-\mathbb{I}^{2})\right],\omega_{\text{q}}=\frac{3\text{e}q_{\text{n}}\mathcal{V}_{z^{\prime}z^{\prime}}}{4I(2I-1)\hbar},$
(1)

where $\omega_{\text{q}}$ is the quadrupole coupling strength, e the
elementary charge, $\hbar$ the reduced Planck’s constant, the nuclear spin
operator vector
$\hat{\bm{I}}:=[\hat{I}_{x^{\prime}},\hat{I}_{y^{\prime}},\hat{I}_{z^{\prime}}]$
and invariant $\mathbb{I}^{2}=\bm{I}\cdot\bm{I}$.

Figure 2: a) High-spin donor implanted in a fabricated silicon-based chip. b)
Bond distortion due to lattice strain leads to an electric-field gradient at
the donor site. c) Energy-level diagram of a high-spin nucleus.

In addition to the quadrupole interaction, we manipulate the nuclear spin
state using a static magnetic dipole interaction due to $\bm{B}_{0}$ and time
dependent perturbation driven by magnetic field $\bm{B}_{1}$ with time-
dependence $\epsilon(t)$ ($|\epsilon(t)|\leq 1$). For a nucleus with
gyromagnetic ratio $\gamma$, the nuclear spin dynamics are described by

$\hat{H}=\gamma\bm{\hat{I}}\cdot\bm{B}_{0}+\hat{H}_{\text{q}}+\gamma\bm{\hat{I}}\cdot\bm{B}_{1}\epsilon(t).$
(2)

The $2I+1$ degenerate levels of a nuclear spin undergo Zeeman splitting due to
$\gamma\bm{\hat{I}}\cdot\bm{B}_{0}$ which creates an equal spacing between the
energy levels. The electric quadrupole interaction $\hat{H}_{\text{q}}$ causes
an unequal shift of the levels such that the transition frequency
$\omega_{i,i-1}$ between two consecutive levels $i$ and $i-1$, for
$i\in[I,-I+1]$, is unique, as shown in Fig. 2(c). We represent spin states as
$\ket{I,m_{I}}$, where $m_{I}\in[-I,I]$, and RF pulses drive transitions which
follow the selection rule $\Delta m_{I}=\pm 1$. For numerical simulations
[41], we use the example of a spin-$\nicefrac{{7}}{{2}}$ 123Sb donor [33, 34],
with Zeeman splitting $\gamma B_{0}=2\pi\times 8.25$ MHz,
$\omega_{\text{q}}=2\pi\times 40$kHz and perturbation strength $\gamma
B_{1}=2\pi\times 800$ Hz.

##  III One-Axis Twisting

Quadrupole interaction is equivalent to one-axis twisting (OAT) due to
$\hat{I}^{2}_{z^{\prime}}$ if $\eta=0$, two-axis counter twisting (TACT) due
to $\hat{I}^{2}_{z^{\prime}}$ and $\hat{I}^{2}_{y^{\prime}}$ if $\eta=1$ [36,
42], and partial TACT otherwise. Generally, TACT is preferred over OAT as TACT
generates faster squeezing in comparison to OAT and can help produce SQSs for
metrology [43]. However, TACT cannot produce SCSs, and, in general,
quadrupolar non-linear evolution with $\eta>0$ cannot go beyond SQSs on a
nuclear spin [35].

We devise a way to convert TACT to OAT by adding a large linear term to the
quadrupole interaction, induced here by the Zeeman coupling. The resulting
dynamics depend on the orientation of the EFG and strength of magnetic field
$B_{0}$. When the Zeeman splitting is much larger than quadrupole coupling,
i.e. $\gamma\bm{B}_{0}\gg\omega_{\text{q}}$, the component perpendicular to
$\bm{B}_{0}$ does not affect the nuclear spin because quadrupolar transitions
($\Delta m_{I}=\pm 2$) are prevented by the large energy difference. In this
regime, only the component of the quadrupole interaction parallel to
$\bm{B}_{0}$ generates non-linear rotation, which effectively results in OAT,
regardless of the value of $\eta$. In contrast to TACT, our scheme allows
generation of SCSs for any orientation of the quadrupolar axes of symmetry
$\\{x^{\prime},y^{\prime},z^{\prime}\\}$ with respect to linear interaction
along the $z$ axis. For numerical simulations in this work, we set
$\bm{B}_{0}=B_{0}\hat{z}$, $\bm{B}_{1}=B_{1}\hat{y}$, $\eta=0$, and EFG to be
symmetric about $z$ axis, i.e. $\hat{z}^{\prime}=\hat{z}$, but, our results
can be generalized to other configurations.

Figure 3: Degree of superposition $N_{\text{eff}}^{\text{r}}$ under free
evolution ($\epsilon(t)=0$) of a coherent spin state to a SCS at
$t=\nicefrac{{\pi}}{{2\omega_{\text{q}}}}$. For differing nuclear spins, the
maximum value $N_{\text{eff}}^{\text{r}}\propto 2I$, showing the
macroscopicity of these SCSs, but the $I=\nicefrac{{3}}{{2}}$ case lacks a
“hill-on-mesa” and is actually a “spin ‘kitten’ state”.

Despite the femtometer scale of a nucleus, nuclear-SCSs are macroscopic with
their size, quantified by the relative quantum Fisher information [44, 45],
scaling linearly with $I$. The size or degree of superposition
$N_{\text{eff}}^{\text{r}}$ of a nuclear-spin state $\psi$ for measurement
operator $\hat{O}$ is

$N_{\text{eff}}^{\text{r}}=\frac{2}{I}\left(\bra{\psi}\hat{O}^{2}\ket{\psi}-\bra{\psi}\hat{O}\ket{\psi}^{2}\right),$
(3)

i.e., proportional to the variance of the observable. For an appropriate
$\hat{O}$, the variance is $I^{2}$ for bimodal SCSs and we calculate the size
of a nuclear SCS to be $2I$, which denotes the maximum quantum Fisher
information of the system and is equivalent to a GHZ state [46] made of $N=2I$
entangled qubits [47]. A nuclear SCS with $2I+1$ dimensions, e.g., a
spin-$\nicefrac{{7}}{{2}}$ SCS with 8 dimensions is equivalent to a $2I=7$
qubit GHZ state with permutation symmetry, and not a $\log_{2}(2I+1)=3$ qubit
state, and can have macroscopicity up to $2I=9$ for a 209Bi donor.

The degree of superposition $N_{\text{eff}}^{\text{r}}$ (Eq. (3)) increases
and decreases periodically during OAT, as shown in Fig. 3.
$N_{\text{eff}}^{\text{r}}=1$ for a spin coherent state on the $\hat{I}_{x}$
axis of the Bloch sphere at $t=0$, increases as the spin state squeezes, then
reaches the flat-top of the “mesa” where it interferes with itself and finally
increases again (“hill”) as two distinct components form resulting in a SCS at
$t=\nicefrac{{\pi}}{{2\omega_{\text{q}}}}$. Under further evolution, the SCS
returns to a coherent state at $t=\nicefrac{{\pi}}{{\omega_{\text{q}}}}$,
which again squeezes to a SCS at $t=\nicefrac{{3\pi}}{{2\omega_{\text{q}}}}$,
and so on. This periodic process of collapse-and-revival of the SCS has a time
period of $\nicefrac{{\pi}}{{\omega_{\text{q}}}}$ and is a signature of
coherence [47]. From the plots for different values of nuclear spin in Fig. 3,
we note the macroscopicity of the SCSs through the linear scaling of their
size $(N^{\text{r}}_{\text{eff}})_{\text{max}}=2I$.

##  IV Control Scheme

External control of a high-spin nucleus is required for initialization,
manipulation and detection of nuclear-SCSs.  For initialization, the
eigenstate $\ket{I,I}$, which is also a coherent state, should first be
rotated to reside on the $\hat{I}_{x}$ axis of the Bloch sphere. Then, OAT
results in a SCS on the $\hat{O}=\hat{I}_{y}$ axis for a half-integer nuclear
spin, but, for donors in solid-state devices, nuclear-spin initialization and
measurement is limited to the eigenstates of $\hat{I}_{z}$. Observing nuclear
SCSs could be possible with angular momentum operations, which are trivial
when $\gamma B_{1}\gg\omega_{\text{q}}$ [42, 48] or with a tunable non-
linearity [49]. However, quantum control is needed as, experimentally, $\gamma
B_{1}=2\pi\times 800$ Hz $\ll\omega_{\text{q}}=2\pi\times 40$ kHz such that a
RF pulse can only selectively drive a particular nuclear spin transition
leading to Givens rotations, and the quadrupole coupling cannot be turned on-
and-off.

For applying global SU(2) rotations with an always-on non-linearity and low
drive strength, we use a multi-tone (MT) pulse that is a superposition of $2I$
tones of equal amplitudes, corresponding to the $2I$ nearest neighbor sub-
spaces with transition frequencies represented by frequency vector
$\bm{\omega}:=\\{\omega_{i,i-1}\\}_{i=I}^{-I-1}$. Each tone has a global phase
$\phi$ and a pulse envelope switched on at a time $t_{0}$ is described by the
amplitude
$\epsilon^{\text{MT}}(t;\bm{\omega},\phi,t_{0})=\frac{1}{2I}\sum_{\omega_{j}\in\bm{\omega}}\cos(\omega_{j}(t-t_{0})+\phi)$.
Here, $\phi$ tunes the axis of rotation on the Bloch sphere while preserving
the relative phase between different energy levels. The frequency $\Omega$ of
SU(2) rotations is generalized from the Rabi frequency $\nicefrac{{\gamma
B_{1}}}{{2}}$ for a two-level nuclear spin, to a high-spin nucleus by dividing
the Rabi frequency by the number of tones $2I$, i.e. $\Omega=\nicefrac{{\gamma
B_{1}}}{{4I}}$. Applying this pulse for time $\Delta t$ rotates the nuclear
spin by angle $\Theta$ given by $\Theta=\Omega\Delta t$. In contrast to other
schemes that employ numerical techniques with multi-tone control [50, 51], we
derive exact expressions for phase-locked pulses, which can enable novel
experimental applications such as full-state tomography and interferometry
with a high-spin nucleus.

Figure 4: a) Generalized Ramsey-like protocol for creating a SCS (Eq. (IV))
using multi-tone pulses. b) Observing SCS collapses-and-revivals by varying
$T$, where $\Delta\phi=\nicefrac{{\pi}}{{2}}+\omega T$ (black) is used for
rotating-frame transformation. c) Collapses-and-revivals under decoherence
with $\Gamma_{\text{m}}=10$ Hz and $\Gamma_{\text{e}}=0.1$ Hz. d) Dynamics
illustrated by Husimi Q-function plots.

Figure 4(a) shows our Ramsey-like control sequence to realize SCSs, where a
square-shaped MT RF pulse is applied to the initial state $\ket{I,I}$ between
time $t_{0}=0$ and $t_{1}=t_{\nicefrac{{\pi}}{{2}}}$. Then the pulse is
switched off for time $T$, and again applied between
$t_{2}=T+t_{\nicefrac{{\pi}}{{2}}}$ and $t_{3}=T+2t_{\nicefrac{{\pi}}{{2}}}$,
with the amplitude

$\displaystyle\epsilon^{\text{cat}}(t;\bm{\omega},\Delta\phi,T)=$
$\displaystyle\epsilon^{\text{MT}}(t;\bm{\omega},0,t_{0})\sqcap\left({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t};t_{0},t_{1}\right)$
$\displaystyle+\epsilon^{\text{MT}}(t;\bm{\omega},\Delta\phi,t_{2})\sqcap\left({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}t};t_{2},t_{3}\right)\big{]}$
(4)

for $\sqcap$ the top-hat function. Here, the global rotation of
$\Theta=\nicefrac{{\pi}}{{2}}$ would take time $\Delta
t_{\nicefrac{{\pi}}{{2}}}=\frac{\pi}{2}\cdot\frac{4I}{\gamma B_{1}}$ and
setting the phase difference $\Delta\phi=\nicefrac{{\pi}}{{2}}$ between the
first and the second global rotations accounts for the angle between a
coherent state and the resulting SCS on the equator. The resulting dynamics
are shown in Fig. 4(d), where we note that SCSs along any axis can be realized
using global rotations (Fig. 4(d)(vi)). The quadrupole coupling cannot be
turned off and these states squeeze during rotations. A SCS can be stored for
long times by aligning it to the fixed points on the $z$ axis (Fig.
4(d)(vii)).

For a spin-$\nicefrac{{7}}{{2}}$ 123Sb nucleus, Fig. 4(b) shows the change in
$N_{\text{eff}}^{\text{r}}$, for $\hat{O}=\hat{I}_{z}$, as $T$ is varied,
similar to Fig. 3. When $\Delta\phi=\nicefrac{{\pi}}{{2}}$, we observe a rapid
variation of $N_{\text{eff}}^{\text{r}}$ with a frequency of
$\nicefrac{{\gamma B_{0}}}{{\pi}}$, because nuclear-spin states generated by
OAT undergo Larmor precession and when the axis of rotation corresponds with
the axis of the SCS, the second global rotation is unable to move the state.
To remedy this immobility, we assign a $T$-dependent shift to the phase
difference: $\Delta\phi=\nicefrac{{\pi}}{{2}}+\omega T$ in (IV), which is
equivalent to a rotating-frame transformation, and results in the periodic
collapses-and-revivals of a SCS solely due to OAT, which would enable
efficient experimental detection.

##  V Virtual Phase Control

In this section, we show that cat-state generation is possible with just phase
modulation of multi-tone pulses in the generalized rotating frame of
reference. First, we note that, in the limit of large Zeeman interaction
$\gamma B_{0}\gg\omega_{\text{q}}$, we can neglect non-commuting terms, such
as $\hat{I}_{x^{\prime}}^{2}$ of the quadrupole interaction (1) so that the
non-linear evolution is only described in terms of the strength of quadrupole
tensor, $\omega_{\text{q}}^{\text{eff}}$, parallel to the magnetic field
$\bm{B}_{0}$. Under this approximation and using the notation
$[\ell,\ell^{\prime}]:=\\{\ell,\ell+1,\dots,\ell^{\prime}\\}$, the effective
Hamiltonian is

$\hat{H}_{\text{eff}}=\gamma
B_{0}\hat{I}_{z}+\omega_{\text{q}}^{\text{eff}}\hat{I}_{z}^{2}+\gamma
B_{1}\hat{I}_{x}\sum_{j\in[1,2I]}\epsilon_{j}\cos(\omega_{j}t+\phi_{j}),$ (5)

where $\omega_{j}$, $\epsilon_{j}$, and $\phi_{j}$ are respectively the
frequency, amplitude and phase of the $j^{\text{th}}$ component of the multi-
tone control. The eigenstates $\ket{I,k}$ of the nuclear spin have energy

$e_{k}=\gamma B_{0}k+\omega_{\text{q}}^{\text{eff}}k^{2}$ (6)

for $k\in[-I,I]$.

Now we transform the Hamiltonian to the generalized rotating frame, defined by
the diagonal matrix [50]

$U_{\text{rot}}=\text{diag}\left(e^{-ie_{I}t},e^{-ie_{I-1}t},\ldots\right)$
(7)

and obtain

$\hat{H}_{\text{rot}}=\gamma
B_{1}\begin{bmatrix}0&h_{1}\epsilon_{1}e^{-i\phi_{1}}&0&\ldots\\\
h_{1}\epsilon_{1}e^{i\phi_{1}}&0&h_{2}\epsilon_{2}e^{-i\phi_{2}}&\ldots\\\
0&h_{2}\epsilon_{2}e^{i\phi_{2}}&0&\ldots\\\
\vdots&\vdots&\vdots&\ddots\end{bmatrix},$ (8)

where

$h_{j}=\bra{I,I-(j-1)}\hat{I}_{x}\ket{I,I-j}.$ (9)

We note that global rotations are applied by setting
$\epsilon_{j}=\nicefrac{{1}}{{2I}}$ and $\phi_{j}=\phi$; for example, using
$\phi=0$ is equivalent to

$\hat{H}_{\text{rot}}=\frac{\gamma B_{1}\hat{I}_{x}}{2I}$ (10)

and $\phi=\nicefrac{{\pi}}{{2}}$ is equivalent to
$\hat{H}_{\text{rot}}=\nicefrac{{\gamma B_{1}\hat{I}_{y}}}{{2I}}$.

One-axis twisting can be realized by virtually updating the phases of a multi-
tone pulse. To see this, we note that that the effective quadrupolar operator
applied for duration $T$ generates the operation
$e^{-iT\omega_{\text{q}}^{\text{eff}}\hat{I}_{z}^{2}}$, which results in a
non-linear phase shift between the different eigenstates of a nuclear-spin
Hamiltonian. Using the dummy variable $k$ for integers in the interval
$[-I,I]$, a state

$\ket{\psi}=\sum_{k\in[-I,I]}c_{k}\ket{I,k}$ (11)

is transformed as

$e^{-iT\omega_{\text{q}}^{\text{eff}}\hat{I}_{z}^{2}}\ket{\psi}=\sum_{k\in[-I,I]}c_{k}e^{-iT\omega_{\text{q}}^{\text{eff}}k^{2}}\ket{I,k},$
(12)

where $-T\omega_{\text{q}}^{\text{eff}}k^{2}$ is the resulting phase shift on
the energy eigenstate $\ket{I,k}$.

Now, we can factor out the global phase
$-T\omega_{\text{q}}^{\text{eff}}I^{2}$ and instead of freely evolving a state
under one-axis twisting, apply equivalent phase updates,

$\phi_{j}\mapsto\phi_{j}+T\omega_{\text{q}}^{\text{eff}}\left((I-j)^{2}-I^{2}\right),$
(13)

to the subsequent multi-tone pulse. Such virtual phase updates would have same
effect as free non-linear evolution and would result in cat state formation
when $T=\nicefrac{{\pi}}{{2\omega_{\text{q}^{\text{eff}}}}}$. Thus, in the
generalized rotating frame, we can start with the initial state $\ket{I,I}$,
apply a global $\nicefrac{{\pi}}{{2}}$ rotation with $\phi_{j}=\phi$, followed
by a second by a global $\nicefrac{{\pi}}{{2}}$ rotation with

$\phi_{j}=\phi+\nicefrac{{\pi\left((I-j)^{2}-I^{2}\right)}}{{2}},$ (14)

realizing cat states by only phase modulation of multi-tone pulses.

##  VI Decoherence

Cat states with a lifetime of several milliseconds can be achieved using
silicon-based chips with an implanted spin-$\nicefrac{{7}}{{2}}$ 123Sb donor
atom, that have a nuclear spin coherence time $T_{2}^{*}\approx 100$ ms for
two-level transitions inferred from experiments on nuclear electric resonance
[33]. SCSs along the $z$-axis would have a reduced coherence time, lower by a
factor of $2I$, due to the large separation between the two components,
resulting in a coherence time of $\nicefrac{{100}}{{7}}\approx 14$ ms for
123Sb. Even for a modest driving strength $\gamma B_{1}=2\pi\times 800$ Hz, a
$\nicefrac{{\pi}}{{2}}$-global rotation would take $\Delta
t_{\nicefrac{{\pi}}{{2}}}=\frac{\pi}{2}\cdot\frac{4I}{\gamma B_{1}}=4.375$ ms.
Thus, our approach, using two MT pulses and OAT, can yield highly coherent
SCSs in less than 9 ms—within the dephasing time on current hardware. Faster
Rabi frequencies are easily achieved for nuclear magnetic resonance (NMR)
methods [40, 34], that could reduce the time needed for MT pulses and lead to
faster generation and detection of nuclear SCSs.

We compare the performance of OAT to an intuitively appealing gate sequence
for creating SCSs, achieved by sequential Givens rotations [52]. Starting with
$\ket{I,I}$, we apply a $\nicefrac{{\pi}}{{2}}$-pulse of frequency
$\omega_{I,I-1}$ to form the superposition $\ket{I,I}+\ket{I,I-1}$, with unit
norm implied. Then, we apply a sequence of $2I-1$ $\pi$-pulses to create the
SCS $\ket{I,I}+\ket{I,-I}$. For observing a collapse of the SCS, repeating the
same sequence evolves the SCS into the coherent state $\ket{I,-I}$, opposite
to the initial state $\ket{I,I}$. In total, $4I$ pulses would be needed for
each collapse-and-revival using Givens rotations, that would take $\sim 9$ ms
for a $I=\nicefrac{{7}}{{2}}$ with Rabi frequency $2\pi\times 800$ Hz, in
contrast to OAT which takes just
$\nicefrac{{\pi}}{{\omega_{\text{q}}}}=12.5\mu$s when
$\omega_{\text{q}}=2\pi\times 40$ kHz.

We simulate our scheme under decoherence from a magnetic field fluctuating at
rate $\Gamma_{\text{m}}$, modeled using Lindblad operator
$\hat{L}_{\text{m}}=\hat{I}_{z}$, and an electric field fluctuating at rate
$\Gamma_{\text{e}}$, modeled using $\hat{L}_{\text{e}}=\hat{I}_{z}^{2}$.
Figure 4(c) shows $N_{\text{eff}}^{r}$ vs evolution time beyond 9 ms
(approximate time to form the first SCS). We note that OAT is two orders of
magnitude faster than both Givens rotations and the dephasing timescale,
opening the possibility of realizing hundreds of SCS collapses-and-revivals by
using a strong non-linearity relative to decoherence, which makes our SCS
robust, in contrast to alternative platforms [53]. Additionally, SCSs
generated by Givens rotations lie on the $\hat{I}_{z}$ axis, where the
dephasing rate of the cat state increases with the nuclear spin increases; in
comparison, cat states produced by one-axis twisting lie on the equator,
perpendicular to the dephasing axis, and are highly protected against
decoherence.

##  VII Conclusion

In summary, Schrödinger cat states are important for investigating foundations
of quantum mechanics as well as for applications such as quantum error
correction but are fragile and hard to realize. We propose a new platform for
realizing macroscopic spin cat states by converting quadrupolar two-axis
counter twisting to one-axis twisting on a high-spin donor in a solid-state
device. Our scheme can deterministically generate spin cat states on-demand
without post-selection and is two orders of magnitude faster than the
dephasing timescale, resulting in highly coherent states. We show that, in the
generalized rotating frame, a spin coherent state can be converted to a cat
states by only phase modulation of multi-tone pulses. Unlike continuous-
variable systems, we generate cat states without entanglement with an ancilla,
which renders robustness against error propagation [25] and opens the
possibility of storing long-lived nuclear-spin cat states. Furthermore, our
cat states can be experimentally realized on a single nucleus, which could
lead to novel applications in quantum information processing.

###### Acknowledgements.

We thank Dipankar Home for useful discussion on macroscopicity of nuclear cat
states. P.G. and B.C.S. acknowledge support from NSERC. Work at UNSW was
funded by an Australian Research Council (DP210103769). A.V. and X.Y.
acknowledge support from the Sydney Quantum Academy.

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##  Appendix A Dynamics

###  A.1 Quadrupole interaction

Our system is described by the Hamiltonian

$\hat{H}=\gamma
B_{0}\hat{I}_{z}+\sum_{\alpha,\beta}Q_{\alpha\beta}\hat{I}_{\alpha}\hat{I}_{\beta}+\gamma
B_{1}\hat{I}_{y}\epsilon(t),$ (15)

where $\gamma$ is the nuclear gyromagnetic ratio, $B_{0}$ is the strength of
the static magnetic field oriented along $z$ axis, $B_{1}$ is the strength of
perturbative magnetic field and $\epsilon(t)$ the time-dependent pulse
amplitude. The quadrupole moment tensor components are

$Q_{\alpha\beta}=\frac{3\text{e}q_{\text{n}}\mathcal{V}_{\alpha\beta}}{4I(2I-1)\hbar},$
(16)

with $q_{\text{n}}$ the electric quadrupole moment, which characterizes the
spherical asymmetry of the nuclear charge distribution, e the elementary
charge, $\hbar$ the reduced Planck’s constant, and
$V_{\alpha\beta}:=\frac{\partial^{2}V(x,y,z)}{\partial\alpha\partial\beta}$
the electric field gradient. The quadrupole moment tensor ((16)) is real,
traceless and symmetric; thus, a set of principal axes of symmetry (PAS)
$\\{x^{\prime},y^{\prime},z^{\prime}\\}$ exist for which the quadrupole moment
tensor can be diagonalized.

The quadrupole term in the Hamiltonian can be expressed in the PAS as

$\hat{H}_{\text{q}}=\omega_{\text{q}}\left[\hat{I}_{z^{\prime}}^{2}+\frac{\eta}{3}(\hat{I}_{x^{\prime}}^{2}-\hat{I}_{y^{\prime}}^{2}-\mathbb{I}^{2})\right],$
(17)

where
$\eta:=\frac{\mathcal{V}_{x^{\prime}x^{\prime}}-\mathcal{V}_{y^{\prime}y^{\prime}}}{\mathcal{V}_{z^{\prime}z^{\prime}}}$
is the asymmetry parameter for $0\leq\eta\leq 1$, and
$\mathbb{I}^{2}=\bm{I}\cdot\bm{I}$. The orientation of the PAS with respect to
the lab frame of reference can be expressed using Euler angles
$\\{\delta,\mu,\nu\\}$ as

$\begin{split}\hat{I}_{x^{\prime}}=&\left(\cos\nu\cos\delta-\sin\nu\cos\mu\sin\delta\right)\hat{I}_{x}+\\\
&\left(\cos\nu\sin\delta-\sin\nu\cos\mu\cos\delta\right)\hat{I}_{y}+\\\
&\sin\nu\sin\mu\hat{I}_{z},\end{split}$ (18)
$\begin{split}\hat{I}_{y^{\prime}}=&\left(-\sin\nu\cos\delta+\cos\nu\cos\mu\sin\delta\right)\hat{I}_{x}+\\\
&\left(-\sin\nu\sin\delta+\cos\nu\cos\mu\cos\delta\right)\hat{I}_{y}+\\\
&\cos\nu\sin\mu\hat{I}_{z},\end{split}$ (19)
$\hat{I}_{z^{\prime}}=\sin\mu\sin\delta\hat{I}_{x}-\sin\mu\cos\delta
I_{y}+\cos\mu I_{z}.$ (20)

Together with $\eta$, the three Euler angles completely specify the symmetric
and traceless quadrupole tensor for a given nucleus.

The EFG depends on several factors including microscopic interactions within
the crystal lattice, the position of the donor, the magnitude of the applied
electric field and the fabrication process of the device. In general, it is
hard to pre-determine the EFG orientation or quadrupole splitting; however,
finite-elements modelling can be used to estimate the strength of quadrupole
interaction. In recent experiments [33, 34], a quadrupole splitting of up to
$2\pi\times 60$ kHz was observed.

###  A.2 Macroscopicity

Figure 5 shows the Husimi Q-function plots of superpositions of maximally
separated coherent states along the $\hat{I}_{z}$ axis of the Bloch sphere. We
note that spin-$1$ case lacks bi-modality and cannot form cat states.
Spin-$\nicefrac{{3}}{{2}}$ case forms spin kitten states and higher spins
($\geq\nicefrac{{5}}{{2}}$) are required for forming cat states (implying
macroscopic).

Figure 5: Husimi Q-function plots of SCSs for different values of the spin.
The spin-$1$ case lacks bi-modality as the spin coherent states overlap with
each other. The spin-$\nicefrac{{3}}{{2}}$ case corresponds to a “spin
‘kitten’ state” with small overlap between its two components. Higher spin
($\geq\nicefrac{{5}}{{2}}$) form macroscopic cat states.

##  Appendix B Experimental Control

###  B.1 Initialization and readout

The nuclear spin of the donor atom can be coherently controlled through
electromagnetic interaction using the electrostatic gates and microwave
antenna fabricated on the chip. The donor atom is subjected to a static
potential, with the nuclear spin coupled to an electron spin, and
initialization of the nuclear spin is done via electron-nuclear hyperfine
coupling. A typical experiment uses flip-flop driving [38], in which an
electron spin is flipped and a nuclear spin is flopped in opposite directions
and high-power chirp pulses, corresponding to different transitions, are
applied sequentially. As shown in Fig. 6 (a), to reach a target state in the
energy ladder, first, the nuclear spin is pumped down from all higher-lying
energy levels using flip-flop transitions (orange). Then, the nuclear spin
state is pumped up from all the lower-lying energy levels using a combination
electron-spin resonance pulses (green) and flip-flop pulses (orange).

The initialization and readout schemes benefit from the coupling of a nuclear
spin to an electron spin, which enables fast initialization of the nuclear
spin to any energy eigenstate $\ket{I,m_{I}}$, of which $\ket{I,\pm I}$ are
spin coherent states. We use the ground state $\ket{I,I}$ as the initial state
in our scheme. After initialization, the electrostatic potential is increased
to ionize the donor atom by removing the electron and only the nuclear spin is
manipulated using RF pulses [39]. When the nuclear spin reaches the desired
final state, the donor atom is de-ionized such that the nuclear spin again
couples to an electron. As shown in Fig. 6 (b), the nuclear-spin state is read
out using single-shot detection with pulsed electron spin resonance and spin-
dependent tunneling of the electron to the SET, which acts as a cold-electron
reservoir [54].

Figure 6: Initialization (a) and readout (b) of a nuclear spin by coupling it
to an electron spin and using flip-flop pulses (orange arrows) and electron-
spin resonance pulses (green arrows)

.

###  B.2 Control

We manipulate the nucleus using RF magnetic pulses that cause NMR transitions
with selection rule $\Delta m_{I}=\pm 1$. The quadrupole interaction shifts
the energy levels of a nuclear spin qudit, as shown in Fig. 2 of the main
text, which allows selectively addressed transitions by tuning the frequency
of Rabi pulses, applied through the time-dependent perturbation $\epsilon(t)$
in Eq. (15). At off-resonant drives, the probability of a Rabi flip is given
by,

$\text{P}_{\text{flip}}=\frac{(\gamma B_{1})^{2}}{\Omega^{2}}\sin^{2}\Omega
t,$ (21)

where $\gamma B_{1}$ is the drive strength (15), $\Omega^{2}=\sqrt{(\gamma
B_{1})^{2}+\Delta\omega^{2}/4}$ with $\Delta\omega$ the difference between the
detuning of the drive from the transition frequency. With quadrupole
interaction $\omega_{\text{q}}\approx 2\pi\times 40$ kHz achieved in
experiments [33, 34] and drive strength $\gamma B_{1}=2\pi\times 0.8$ kHz,
frequency difference between two transitions $\Delta\omega\geq
2\omega_{\text{q}}$, thus, $\text{P}_{\text{flip}}<0.0004$. Thus, cross talks
are highly suppressed. Additionally, the quadrupole interaction generates fast
collapse and revivals due to non-linear evolution having time-period of a few
microseconds, which can be resolved by pulses applied using an Arbitrary
Waveform Generator (AWG) with a resolution of 1 ns. While lower-order non-
linearity would slow down SCS generation, higher-order non-linearity, for
example in the GHz range, would generate OAT at a rate faster than the
resolution of an AWG.

##  Appendix C Decoherence

###  C.1 Master equation for a high-spin nucleus

We model decoherence using a Lindblad master equation of the form

$\begin{split}\dot{\rho}=-i[\hat{H},\rho]&+\Gamma_{\text{m}}\left(\hat{L}_{\text{m}}\rho\hat{L}_{\text{m}}^{\dagger}-\frac{1}{2}\\{\hat{L}_{\text{m}}^{\dagger}\hat{L}_{\text{m}},\rho\\}\right)\\\
&+\Gamma_{\text{e}}\left(\hat{L}_{\text{e}}\rho\hat{L}_{\text{e}}^{\dagger}-\frac{1}{2}\\{\hat{L}_{\text{e}}^{\dagger}\hat{L}_{\text{e}},\rho\\}\right),\end{split}$
(22)

where $\rho$ is the density matrix of the nuclear spin, $\Gamma_{\text{m}}$ is
the decay rate due to magnetic field fluctuations, modeled using Lindblad
operator $\hat{L}_{\text{m}}=\hat{I}_{z}$ and $\Gamma_{\text{e}}$ is the decay
rate due to electric field fluctuations, modeled using Lindblad operator
$\hat{L}_{\text{e}}=\hat{I}_{z}^{2}$. Here, we take the electric field
fluctuation only in the $z$ direction, i.e. parallel to the Zeeman splitting,
because other components of the electric quadrupole interaction are washed
out, as described above.

We simulate the cat-state collapse-and-revival control scheme for various
values of $\Gamma_{m}$ and $\Gamma_{e}$, shown in Fig. 7. Here, magnetic
fluctuations slowly dephase the nuclear spin and the coherence of the SCS
decreases during collapse-and-revivals. On the other hand, electric field
fluctuations have a more drastic effect on SCSs generated by quadrupole
interaction, resulting in an overall decrease of the variance, i.e.
macroscopicity of the SCSs.

Figure 7: Collapse and revivals under decoherence (22). The line joining the
revival points (where $N^{\text{r}}_{\text{eff}}$ is maximum) can be
extrapolated to estimate the number of revivals possible within the coherence
time of a nuclear spin. The decoherence rates are specified in units of kHz
and time in ms.

###  C.2 Coherence time of a qudit SCS

Figure 8: Decrease in coherence, given by the off diagonal component of the
density matrix, of a SCS with increase in qudit dimension $d=2I+1$.

We study the scaling of the coherence of a SCS with respect to the Hilbert
space dimension of a qudit [55] and consider dephasing purely due to magnetic
field fluctuations, of the state $\ket{I,I}+\ket{I,-I}$ for different values
of $I$. The density matrix of such a state consists only of four elements: two
diagonal elements corresponding to the population in $\ket{I,\pm I}$ and two
off-diagonal components that quantify the coherence of the superposition
state. In Fig. 8, we plot the magnitude of the off-diagonal component after a
fixed time of 1 ms under an amplified dephasing at rate of
$\Gamma_{\text{m}}=1$ kHz ($\Gamma_{\text{m}}\propto 1/T_{2}^{*}$). We note
that coherence decreases with the dimension $d=2I+1$ of the nuclear spin and
results in a linear decrease in coherence time as the dimension of a qudit
increases. Such a trade-off between lifetime and the macroscopicity of a SCS
is similarly found in other systems such as harmonic oscillators and maximally
entangled GHZ states [56]. However, compared to other systems, a nuclear spin,
with naturally long coherence times, can help increase the lifetime of
practically achievable SCSs.

##  Appendix D Converting two-axis counter twisting to one-axis twisting

###  D.1 Two-axis counter twisting

Substituting
$\mathbb{I}^{2}=\hat{I}_{x^{\prime}}^{2}+\hat{I}_{y^{\prime}}^{2}+\hat{I}_{z^{\prime}}^{2}$
in Eq. (17), the quadrupole interaction can be described as

$\hat{H}_{\text{q}}=\omega_{\text{q}}\left(1-\frac{\eta}{3}\right)\left[\hat{I}_{z^{\prime}}^{2}-a\hat{I}_{y^{\prime}}^{2}\right];a=\left(\frac{2\eta}{3-\eta}\right),$
(23)

which is equivalent to one-axis twisting (OAT) when $\eta=0$, two-axis counter
twisting (TACT) when $\eta=1$, and partial TACT otherwise. Under TACT, the
nuclear spin is simultaneously twisted about two axes, which increases the
squeezing speed and the nuclear spin returns to a coherent state faster than
in the case of OAT (as can be noted in Fig. 9 for $\eta=1$). During this, the
spin does not spread over the equator of the Bloch sphere due to counter
twisting, which prevents interference and cat-state formation.

The quadrupole interaction can only generate squeezed states for $\eta\neq 0$,
as can be seen in Fig. 9. For small $\eta$, partial TACT can create cat-like
states, as shown for $\eta=0.2$ in Fig. 9; however, its two components are not
spin coherent states and the separation between them is smaller than that for
a macroscopic SCS. In practice, there is little to no control over the in-situ
EFG generated by the lattice distortions, and $\eta$ depends on several
experimental factors. Thus, quadrupole interaction alone cannot be used for
realizing SCSs.

###  D.2 Adding a linear interaction

We convert TACT to OAT by adding a large linear term, introduced here by the
Zeeman interaction, that “washes out” TACT. To understand this, we note that
spin squeezing is caused by non-linear transitions induced by
$\hat{I}_{z^{\prime}}^{2}$ and $\hat{I}_{y^{\prime}}^{2}$, and, for the simple
case of $\\{x^{\prime},y^{\prime},z^{\prime}\\}\equiv\\{x,y,z\\}$,
$\hat{I}_{z}^{2}$ causes non-linear phase shifts and $\hat{I}_{y}^{2}$ causes
quadrupolar transitions between different $\hat{I}_{z}$ energy eigenstates
with $\Delta m_{I}=\pm 2$. TACT can be converted to OAT if
$\hat{I}_{y}^{2}$-induced quadrupolar transitions can be prevented. This is
achieved when the Zeeman splitting is much larger than quadrupole coupling,
i.e. $\gamma\bm{B}_{0}\gg\omega_{\text{q}}$, and non-linear transitions
induced by the relatively small $\hat{I}_{y}^{2}$ term are prevented by the
large energy level splitting. In such a case, only the component of the
quadrupole interaction parallel to Zeeman interaction generates squeezing,
which results in an effective one-axis twisting, regardless of the value of
$\eta$. Figure 10 shows nuclear spin squeezing under different strengths of
quadrupole interaction, where we note that SCSs are formed only for a very
large value of $\gamma B_{0}$.

###  D.3 Effect of quadrupolar orientation

Above, we analyzed the special case of
$\\{x^{\prime},y^{\prime},z^{\prime}\\}\equiv\\{x,y,z\\}$; however, the
conversion of TACT to OAT holds for any set of Euler angles
$\\{\delta,\mu,\nu\\}$ and our scheme can create SCSs in-situ for any
configuration of the nuclear quadrupolar interaction. In the presence of a
large Zeeman interaction, effective spin squeezing is only caused by $Q_{zz}$
(15) and any other non-linear effect is washed-out. $Q_{zz}$ is always non-
zero except when the principal axis $z^{\prime}$ is perpendicular to the
Zeeman interaction, i.e. $\mu=\pi/2$ (20) and the asymmetry parameter
$\eta=0$, resulting in a quadrupole interaction similar to $\hat{I}_{x}^{2}$.
In this corner case of OAT, an initial nuclear spin coherent state can form a
spin SCS without any additional control.

Figure 9: Nuclear spin squeezing under non-linear quadrupole interaction for
different values of asymmetry parameter $\eta$. The quadrupole interaction can
produce spin cat states (SCSs) for $\eta=0$, however, for higher values of
$\eta$, only spin squeezed states are formed. One can note that the period of
non-linear evolution decreases as $\eta$ increases, for e.g. the coherent
state reappears faster for $\eta=1$ (third figure from right), in comparison
to $\eta=0.5$ (second figure from right). Figure 10: Dynamics when $\eta=1$
for different Zeeman interaction strength $\gamma B_{0}$ and quadrupole
strength $\omega_{\text{q}}=2\pi\times 40\approx 250$ kHz. SCSs are formed by
converting two-axis counter twisting to one-axis twisting by adding a large
linear interaction induced by the Zeeman interaction ($\gamma
B_{0}\gg\omega_{\text{q}}$).


# Summation formulas of $q$-hyperharmonic numbers

Takao Komatsu
Department of Mathematical Sciences, School of Science
Zhejiang Sci-Tech University
Hangzhou 310018 China
<EMAIL_ADDRESS>

Rusen Li
School of Mathematics
Shandong University
Jinan 250100 China
<EMAIL_ADDRESS>
###### Abstract

In this paper, several weighted summation formulas of $q$-hyperharmonic
numbers are derived. As special cases, several formulas of hyperharmonic
numbers of type $\sum_{\ell=1}^{n}{\ell}^{p}H_{\ell}^{(r)}$ and
$\sum_{\ell=0}^{n}{\ell}^{p}H_{n-\ell}^{(r)}$ are obtained.
Keywords: Hyperharmonic numbers, Stirling numbers, $q$-generalizations

##  1 Introduction

Spieß[15] gives some identities including the types of
$\sum_{\ell=1}^{n}\ell^{k}H_{\ell}$, $\sum_{\ell=1}^{n}\ell^{k}H_{n-\ell}$ and
$\sum_{\ell=1}^{n}\ell^{k}H_{\ell}H_{n-\ell}$. In particular, explicit forms
for $r=0,1,2,3$ are given. In this paper, several identities including
$\sum_{\ell=1}^{n}\ell^{k}H_{\ell}^{(r)}$ and
$\sum_{\ell=1}^{n}\ell^{k}H_{n-\ell}^{(r)}$ are shown as special cases of more
general results, where $H_{\ell}^{(r)}$ denotes hyperharmonic numbers defined
in (4). When $r=1$, $H_{n}=H_{n}^{(1)}$ is the original harmonic number
defined by $H_{n}=\sum_{j=1}^{n}1/j$. This paper is also motivated from the
summation $\sum_{\ell=1}^{n}\ell^{k}$, which is related to Bernoulli numbers.
In [1], Stirling numbers are represented via harmonic numbers and
hypergeometric functions related to Euler sums. In this paper, the sums
involving harmonic numbers and their $q$-generalizations are expressed by
using Stirling numbers and their $q$-generalizations.

There are many generalizations of harmonic numbers. Furthermore, some
$q$-generalizations of hyperharmonic numbers have been proposed. In this
paper, based upon a certain type of $q$-harmonic numbers $H_{n}^{(r)}(q)$
defined in (3), several formulas of $q$-hyperharmonic numbers are also derived
as $q$-generalizations. These results are also motivated from the
$q$-analogues of the sums of consecutive integers ([9, 14, 16]).

In order to consider the weighted summations, we are motivated by the fact
that the sum of powers of consecutive integers $1^{k}+2^{k}+\cdots+n^{k}$ can
be explicitly expressed in terms of Bernoulli numbers or Bernoulli
polynomials. After seeing the sums of powers for small $k$:

$\sum_{\ell=1}^{n}\ell=\frac{n(n+1)}{2},~{}\sum_{\ell=1}^{n}\ell^{2}=\frac{n(n+1)(2n+1)}{6},~{}\sum_{\ell=1}^{n}\ell^{3}=\left(\frac{n(n+1)}{2}\right)^{2},~{}\dots\,,$

the formula can be written as

$\displaystyle\sum_{\ell=1}^{n}\ell^{k}$
$\displaystyle=\frac{1}{k+1}\sum_{j=0}^{k}\binom{k+1}{j}B_{j}n^{k+1-j}$ (1)
$\displaystyle=\frac{1}{k+1}(B_{k+1}(n+1)-B_{k+1}(1))\quad\hbox{\cite[cite]{[\@@bibref{}{CFZ}{}{}]}}\,,$
(2)

where Bernoulli numbers $B_{n}$ are determined by the recurrence formula

$\sum_{j=0}^{k}\binom{k+1}{j}B_{j}=k+1\quad(k\geq 0)$

or by the generating function

$\frac{t}{1-e^{-t}}=\sum_{n=0}^{\infty}B_{n}\frac{t^{n}}{n!}\,,$

and Bernoulli polynomials $B_{n}(x)$ are defined by the following generating
function

$\frac{te^{xt}}{e^{t}-1}=\sum_{n=0}^{\infty}B_{n}(x)\frac{t^{n}}{n!}\,.$

If Bernoulli numbers $\mathfrak{B}_{n}$ are defined by

$\frac{t}{e^{t}-1}=\sum_{n=0}^{\infty}\mathfrak{B}_{n}\frac{t^{n}}{n!}\,,$

we can see that $B_{n}=(-1)^{n}\mathfrak{B}_{n}$. Then

$\sum_{\ell=1}^{n}\ell^{k}=\frac{1}{k+1}\sum_{j=0}^{k}\binom{k+1}{j}(-1)^{j}\mathfrak{B}_{j}n^{k+1-j}\,.$

We recall the well-known Abel’s identity, which is frequently used in the
present paper.

######  Lemma 1.

(Abel’s identity) For any positive integer $n$,

$\sum_{\ell=1}^{n}a_{\ell}b_{\ell}=s_{n}b_{n}+\sum_{\ell=1}^{n-1}s_{\ell}(b_{\ell}-b_{\ell+1})\,.$

where

$s_{n}=\sum_{\ell=1}^{n}a_{\ell}.$

In the weight of harmonic numbers $H_{n}$, we have the following formulas.

######  Proposition 1.

For $n,k\geq 1$,

$\sum_{\ell=1}^{n}\ell^{k}H_{\ell}=\frac{H_{n}}{k+1}\sum_{j=0}^{k}\binom{k+1}{j}B_{j}n^{k+1-j}-\sum_{\ell=1}^{n-1}(H_{n}-H_{\ell})\ell^{k}\,.$

###### Proof.

Set $a_{\ell}={\ell^{k}}$ and $b_{\ell}={H_{\ell}}$ in Lemma 1. With

$\displaystyle\sum_{\ell=1}^{n-1}s_{\ell}(H_{\ell}-H_{\ell+1})$
$\displaystyle=s_{1}(H_{1}-H_{2})+\cdots+s_{n-1}(H_{n-1}-H_{n})$
$\displaystyle=1^{k}H_{1}+\cdots+{(n-1)}^{k}H_{n-1}-s_{n-1}H_{n}$
$\displaystyle=-\sum_{\ell=1}^{n-1}(H_{n}-H_{\ell})\ell^{k}\,,$

formula (1) gives the result. ∎

######  Proposition 2.

For $n,k\geq 1$,

$\sum_{\ell=1}^{n}\ell^{k}H_{\ell}=\frac{H_{n}}{k+1}(B_{k+1}(n+1)-B_{k+1}(1))-\sum_{\ell=1}^{n-1}\frac{B_{k+1}(\ell+1)-B_{k+1}(1)}{(k+1)(\ell+1)}\,.$

###### Proof.

Set $a_{\ell}={\ell^{k}}$ and $b_{\ell}={H_{\ell}}$ in Lemma 1. Formula (2)
gives the result. ∎

##  2 Weighted summations of $q$-hyperharmonic numbers

Many types of $q$-generalizations have been studied for harmonic numbers
(e.g.,[11, 17]). In this paper, a $q$-hyperharmonic number $H_{n}^{(r)}(q)$
(see [12]) is defined by

$H_{n}^{(r)}(q)=\sum_{j=1}^{n}q^{j}H_{j}^{(r-1)}(q)\quad(r,n\geq 1)$ (3)

with

$H_{n}^{(0)}(q)=\frac{1}{q[n]_{q}}$

and

$[n]_{q}=\frac{1-q^{n}}{1-q}\,.$

Note that

$\lim_{q\to 1}[n]_{q}=n\,.$

In this $q$-generalization,

$H_{n}(q)=H_{n}^{(1)}(q)=\sum_{j=1}^{n}\frac{q^{j-1}}{[j]_{q}}$

is a $q$-harmonic number. When $q\to 1$, $H_{n}=\lim_{q\to 1}H_{n}(q)$ is the
original harmonic number and $H_{n}^{(r)}=\lim_{q\to 1}H_{n}^{(r)}(q)$ is the
$r$-th order hyperharmonic number, defined by

$H_{n}^{(r)}=\sum_{\ell=1}^{n}H_{\ell}^{(r-1)}\quad\hbox{with}\quad
H_{n}^{(1)}=H_{n}\,.$ (4)

Mansour and Shattuck [12, Identity 4.1, Proposition 3.1] give the following
identities

$\displaystyle H_{n}^{(r)}(q)$
$\displaystyle=\binom{n+r-1}{r-1}_{q}\bigl{(}H_{n+r-1}(q)-H_{r-1}(q)\bigr{)}$
(5)
$\displaystyle=\sum_{j=1}^{n}\binom{n+r-j-1}{r-1}_{q}\frac{q^{rj-1}}{[j]_{q}},$
(6)

where

$\binom{n}{k}_{q}=\frac{[n]_{q}!}{[k]_{q}![n-k]_{q}!}$

is a $q$-binomial coefficient with $q$-factorials
$[n]_{q}!=[n]_{q}[n-1]_{q}\cdots[1]_{q}$. Note that the identities (5) and (6)
are $q$-generalization of the identities (7) and (8), respectively.

$\displaystyle H_{n}^{(r)}$
$\displaystyle=\binom{n+r-1}{r-1}(H_{n+r-1}-H_{r-1})\quad\hbox{\cite[cite]{[\@@bibref{}{Conway}{}{}]}}$
(7)
$\displaystyle=\sum_{j=1}^{n}\binom{n+r-j-1}{r-1}\frac{1}{j}\quad\hbox{\cite[cite]{[\@@bibref{}{BGG}{}{}]}}\,.$
(8)

So, we can see the recurrence relation for $r\geq 1$

$H_{n}^{(r+1)}=\frac{n+r}{r}H_{n}^{(r)}-\frac{1}{r}\binom{n+r-1}{r}\,.$

The generating function of this type of $q$-hyperharmonic numbers is given by

$\sum_{n=1}^{\infty}H_{n}^{(r)}(q)z^{n}=\frac{-\log_{q}(1-q^{r}z)}{q(z;q)_{r}}\quad(r\geq
0)$ (9)

([12, Theorem 3.2]), where

$-\log_{q}(1-t)=\sum_{m=1}^{\infty}\frac{t^{m}}{[m]_{q}}$

is the $q$-logarithm function and

$(z;q)_{k}:=\prod_{j=0}^{k-1}(1-zq^{j})$

is the $q$-Pochhammer symbol. When $q\to 1$, (9) is reduced to the generating
function of hyperharmonic numbers:

$\sum_{n=1}^{\infty}H_{n}^{(r)}z^{n}=\frac{-\log(1-z)}{(1-z)^{r}}\quad(r\geq
0)\,.$

In fact, the same form is given by Knuth [10] as

$\sum_{n=r-1}^{\infty}\binom{n}{r-1}(H_{n}-H_{r-1})z^{n-r+1}=\frac{-\log(1-z)}{(1-z)^{r}}\quad(r\geq
0)\,.$

By (5), we have

$H_{n}^{(r+1)}(q)-\frac{[n+r]_{q}}{[r]_{q}}H_{n}^{(r)}(q)=-\frac{q^{r-1}}{[r]_{q}}\binom{n+r-1}{r}_{q}\,.$

Hence,

$H_{n}^{(r+1)}(q)=\frac{[n+r]_{q}}{[r]_{q}}H_{n}^{(r)}(q)-\frac{q^{r-1}}{[r]_{q}}\binom{n+r-1}{r}_{q}\,.$
(10)

By replacing $n$ by $n+1$ and $r$ by $r-1$ in (10), together with the
definition in (3), we have

$\displaystyle[n+r]_{q}H_{n}^{(r)}(q)$
$\displaystyle=[n+1]_{q}H_{n+1}^{(r)}(q)-q^{n+r-1}\binom{n+r-1}{r-1}_{q}\,.$
(11)

Mansour and Shattuck [12, Theorem 3.3] also give the following formula,

$H_{n}^{(r)}(q)=\sum_{j=1}^{n}q^{j(r-m)}\binom{n+r-m-j-1}{r-m-1}H_{j}^{(m)}(q).$
(12)

When $q\to 1$, (12) is reduced to

$H_{n}^{(r)}=\sum_{j=1}^{n}\binom{n+r-m-j-1}{r-m-1}H_{j}^{(m)}$

(see also [2],[3, 2.4.Theorem]). When $m=0$, (12) is reduced to (6).

We prove a more general result of (5).

######  Theorem 1.

For nonnegative integers $n$ and $k$ and a positive integer $r$, we have

$\binom{k+r-1}{k}_{q}H_{n}^{(k+r)}(q)=\binom{n+k}{n}_{q}H_{n+k}^{(r)}(q)-\binom{n+k+r-1}{n}_{q}H_{k}^{(r)}(q)\,.$

Remark. If $r=1$ and $k$ is replaced by $r-1$ in Theorem 1, we have the
identity (5). If $q\to 1$ in Theorem 1, we have the version of the original
hyperharmonic numbers in [13, Theorem 1].

###### Proof of Theorem 1..

The proof is done by induction on $k$. When $k=0$, the identity is clear since
both sides are equal to $H_{n}^{(r)}(q)$. Assume, then, that the identity has
been proved for $0,1,\cdots,k$. We give some explanations for the following
calculation. Firstly, by replacing $r$ by $k+r$ in (10), we get the first
identity. Secondly, by using the inductive assumption, we get the second
identity. Thirdly, by replacing $n$ by $n+k$ and $n$ by $k$ respectively in
(11), we get the third identity. Then, we have

$\displaystyle\binom{k+r}{k+1}_{q}H_{n}^{(k+r+1)}(q)$
$\displaystyle=\binom{k+r}{k+1}_{q}\frac{[n+k+r]_{q}}{[k+r]_{q}}H_{n}^{(k+r)}(q)-\binom{k+r}{k+1}_{q}\frac{q^{k+r-1}}{[k+r]_{q}}\binom{n+k+r-1}{k+r}_{q}$
$\displaystyle=\frac{[n+k+r]_{q}}{[k+1]_{q}}\binom{n+k}{n}_{q}H_{n+k}^{(r)}(q)-\frac{[n+k+r]_{q}}{[k+1]_{q}}\binom{n+k+r-1}{n}_{q}H_{k}^{(r)}(q)$
$\displaystyle\quad-\frac{q^{k+r-1}}{[k+r]_{q}}\binom{k+r}{k+1}_{q}\binom{n+k+r-1}{k+r}_{q}$
$\displaystyle=\frac{[n+k+1]_{q}}{[k+1]_{q}}\binom{n+k}{n}_{q}H_{n+k+1}^{(r)}(q)-\frac{q^{n+k+r-1}}{[k+1]_{q}}\binom{n+k}{n}_{q}\binom{n+k+r-1}{r-1}_{q}$
$\displaystyle\quad-\frac{[n+k+r]_{q}}{[k+1]_{q}}\binom{n+k+r-1}{n}_{q}\frac{[k+1]_{q}}{[k+r]_{q}}H_{k+1}^{(r)}(q)$
$\displaystyle\quad+\frac{[n+k+r]_{q}}{[k+1]_{q}}\binom{n+k+r-1}{n}_{q}\frac{q^{k+r-1}}{[k+r]_{q}}\binom{k+r-1}{r-1}_{q}$
$\displaystyle\quad-\frac{q^{k+r-1}}{[k+r]_{q}}\binom{k+r}{k+1}_{q}\binom{n+k+r-1}{k+r}_{q}$
$\displaystyle=\binom{n+k+1}{n}_{q}H_{n+k+1}^{(r)}(q)-\binom{n+k+r}{n}_{q}H_{k+1}^{(r)}(q)\,.$

We used the relation $[n+k+r]_{q}-q^{n}[k+r]_{q}=[n]_{q}$ in the final part. ∎

Cereceda [5] gives the following formula,

$\lim_{n\to\infty}\frac{H_{n+1}^{(n+1)}}{H_{n}^{(n)}}=4\,.$

However, the ratio of $q$-hyperharmonic numbers of type $H_{n}^{(n)}(q)$ has a
different phenomenon.

######  Proposition 3.

For $|q|<1$, we have

$\lim_{n\to\infty}\frac{H_{n+1}^{(n+1)}(q)}{H_{n}^{(n)}(q)}=q\,.$

###### Proof.

Since

$\frac{(1-q^{2n+1})(1-q^{2n})}{(1-q^{n+1})(1-q^{n})}\to
1\quad(|q|<1,~{}n\to\infty)$

and

$\displaystyle\frac{H_{2n+1}(q)-H_{n}(q)}{H_{2n-1}(q)-H_{n-1}(q)}$
$\displaystyle=\frac{q\left(\dfrac{1}{[n+1]_{q}}+\dfrac{q}{[n+2]_{q}}+\cdots+\dfrac{q^{n}}{[2n+1]_{q}}\right)}{\dfrac{1}{[n]_{q}}+\dfrac{q}{[n+1]_{q}}+\cdots+\dfrac{q^{n-1}}{[2n-1]_{q}}}$
$\displaystyle\to q\quad(n\to\infty)\,,$

from (5),

$\displaystyle\frac{H_{n+1}^{(n+1)}(q)}{H_{n}^{(n)}(q)}$
$\displaystyle=\frac{\binom{2n+1}{n}_{q}\bigl{(}H_{2n+1}(q)-H_{n}(q)\bigr{)}}{\binom{2n-1}{n-1}_{q}\bigl{(}H_{2n-1}(q)-H_{n-1}(q)\bigr{)}}$
$\displaystyle\to 1\cdot q=q\,.$

∎

######  Theorem 2.

For positive integers $n$ and $r$,

$\displaystyle\sum_{\ell=1}^{n}q^{\ell-1}[\ell]_{q}H_{\ell}^{(r)}(q)$
$\displaystyle=\frac{[n]_{q}[n+r]_{q}}{[r+1]_{q}}H_{n}^{(r)}(q)-\frac{q^{r}[n-1]_{q}[n]_{q}}{([r+1]_{q})^{2}}\binom{n+r-1}{r-1}_{q}$
$\displaystyle=\frac{[n]_{q}[r]_{q}}{[r+1]_{q}}H_{n}^{(r+1)}(q)+\frac{q^{r-1}}{[r+1]_{q}}\binom{n+r}{r+1}_{q}\,.$
(13)

###### Proof.

Set $a_{\ell}=q^{\ell-1}\binom{\ell+r-1}{r}_{q}$ and
$b_{\ell}=H_{\ell+r-1}(q)$. By using Lemma 1, we have

$\displaystyle\sum_{\ell=1}^{n}q^{\ell-1}\binom{\ell+r-1}{r}_{q}H_{\ell+r-1}(q)$
$\displaystyle=\sum_{\ell=1}^{n}q^{\ell-1}\binom{\ell+r-1}{r}_{q}H_{n+r-1}(q)-\sum_{\ell=1}^{n-1}\frac{q^{\ell+r-1}}{[\ell+r]_{q}}\binom{\ell+r}{r+1}_{q}$
$\displaystyle=\binom{n+r}{r+1}_{q}H_{n+r-1}(q)-\frac{q^{r}}{[r+1]_{q}}\binom{n+r-1}{r+1}_{q}\,.$
(14)

Hence,

$\displaystyle\sum_{\ell=1}^{n}q^{\ell-1}{[\ell]_{q}}H_{\ell}^{(r)}(q)$
$\displaystyle=\sum_{\ell=1}^{n}q^{\ell-1}{[\ell]_{q}}\binom{\ell+r-1}{r-1}_{q}(H_{\ell+r-1}(q)-H_{r-1}(q))$
$\displaystyle=[r]_{q}\sum_{\ell=1}^{n}q^{\ell-1}\binom{\ell+r-1}{r}_{q}(H_{\ell+r-1}(q)-H_{r-1}(q))$
$\displaystyle=[r]_{q}\sum_{\ell=1}^{n}\binom{\ell+r-1}{r}_{q}H_{\ell+r-1}(q)-[r]_{q}H_{r-1}(q)\binom{n+r}{r+1}_{q}\,.$
(15)

With the help of (5), (14) and (15), we get the desired result. ∎

When $q\to 1$, Theorem 2 is reduced to the following.

######  Corollary 1.

For $n,r\geq 1$,

$\displaystyle\sum_{\ell=1}^{n}\ell H_{\ell}^{(r)}$
$\displaystyle=\frac{n(n+r)}{r+1}H_{n}^{(r)}-\frac{(n-1)^{(r+1)}}{(r-1)!(r+1)^{2}}$
$\displaystyle=\frac{nr}{r+1}H_{n}^{(r+1)}+\frac{1}{r+1}\binom{n+r}{r+1}\,,$

where $(x)^{(n)}=x(x+1)\cdots(x+n-1)$ ($n\geq 1$) denotes the rising factorial
with $(x)^{(0)}=1$.

In order to establish similarly structured theorems of $q$-hyperharmonic
numbers, we recall the $q$-Stirling numbers of the second kind, denoted by
$S_{q}(n,m)$, defined by Carlitz (see e.g. [4]) as

$\displaystyle([x]_{q})^{n}=\sum_{m=0}^{n}q^{\binom{m}{2}}S_{q}(n,m)([x]_{q})_{(m)},\quad(n\in\mathbb{N}),$
(16)

where $([x]_{q})_{(m)}=[x]_{q}[x-1]_{q}\cdots[x-m+1]_{q}$ denotes the
$q$-falling factorial with $([x]_{q})_{0}=1$. The $q$-Stirling numbers of the
second kind $S_{q}(n,m)$ satisfy the recurrence relation

$S_{q}(n+1,m)=S_{q}(n,m-1)+[m]_{q}\cdot S_{q}(n,m)$

with boundary values

$S_{q}(n,0)=S_{q}(0,n)=\delta_{n0},\quad(n\geq 0)$

([8]).

We need a $q$-version of the relation by Spieß[15], which is essential in the
proof of the following structured theorem of $q$-hyperharmonic numbers of type
$\sum_{\ell=0}^{n}q^{\ell-1}{([\ell]_{q})}^{p}H_{\ell}^{(r)}(q)$.

######  Lemma 2.

Given summation formulas
$\sum_{\ell=0}^{n}q^{\ell-1}\binom{\ell}{j}_{q}[c_{\ell}]_{q}=F_{q}(n,j)$ for
$n,j\in\mathbb{N}$, one has

$\sum_{\ell=0}^{n}q^{\ell-1}{([\ell]_{q})}^{p}[c_{\ell}]_{q}=\sum_{\ell=0}^{p}q^{\binom{\ell}{2}}S_{q}(p,\ell)\cdot{[\ell]_{q}}!\cdot
F_{q}(n,\ell)\,.$

where $S_{q}(p,\ell)$ denote the $q$-Stirling numbers of the second kind.

###### Proof.

Using (16), we have

$\displaystyle\sum_{\ell=0}^{n}q^{\ell-1}{([\ell]_{q})}^{p}[c_{\ell}]_{q}$
$\displaystyle=\sum_{\ell=0}^{n}q^{\ell-1}[c_{\ell}]_{q}\sum_{j=0}^{p}q^{\binom{j}{2}}S_{q}(p,j)\cdot([\ell]_{q})_{(j)}$
$\displaystyle=\sum_{j=0}^{p}q^{\binom{j}{2}}S_{q}(p,j){[j]_{q}}!\sum_{\ell=0}^{n}q^{\ell-1}\binom{\ell}{j}_{q}[c_{\ell}]_{q}$
$\displaystyle=\sum_{j=0}^{p}q^{\binom{j}{2}}S_{q}(p,j)\cdot{[j]_{q}}!\cdot
F_{q}(n,j)\,.$

∎

We introduce some notations. For $n,r,p\in\mathbb{N}$, set

$\displaystyle\sum_{\ell=0}^{n}q^{\ell-1}{[\ell]_{q}}^{p}H_{\ell}^{(r)}(q)=A_{q}(p,r,n)H_{n}^{(r)}(q)-B_{q}(p,r,n)\,.$

From (10), for $p=0$,
$A_{q}(0,r,n)=\frac{[n+r]_{q}}{[r]_{q}},B_{q}(0,r,n)=\frac{q^{r-1}}{[r]_{q}}\binom{n+r-1}{r}_{q}$.
From Theorem 2, for $p=1$, we know that

$\displaystyle A_{q}(1,r,n)=\frac{[n]_{q}[n+r]_{q}}{[r+1]_{q}},$
$\displaystyle
B_{q}(1,r,n)=\frac{q^{r}[n-1]_{q}[n]_{q}}{([r+1]_{q})^{2}}\binom{n+r-1}{r-1}_{q}\,.$

######  Theorem 3.

For $n,r,p\geq 1$,

$\sum_{\ell=0}^{n}q^{\ell-1}{[\ell]_{q}}^{p}H_{\ell}^{(r)}(q)=A_{q}(p,r,n)H_{n}^{(r)}(q)-B_{q}(p,r,n)\,,$

where

$\displaystyle A_{q}(p,r,n)$
$\displaystyle=\sum_{\ell=0}^{p}q^{\binom{\ell}{2}+p-1}S_{q}(p,\ell){[\ell]_{q}!}\binom{n+r-1}{r-1}_{q}^{-1}\binom{r+\ell-1}{\ell}_{q}\binom{r+n}{r+\ell}_{q},$
$\displaystyle
B_{q}(p,r,n)=\sum_{\ell=0}^{p}\frac{q^{\binom{\ell}{2}+r+2p-2}}{[r+\ell]_{q}}S_{q}(p,\ell){[\ell]_{q}!}\binom{r+\ell-1}{\ell}_{q}\binom{r+n-1}{r+\ell}_{q}\,.$

###### Proof.

Set $[c_{\ell}]_{q}=H_{\ell}^{(r)}(q)$ in Lemma 2. Then by using Lemma 1, we
have

$\displaystyle F_{q}(n,p)$
$\displaystyle=\sum_{\ell=0}^{n}q^{\ell-1}\binom{\ell}{p}_{q}H_{\ell}^{(r)}(q)$
$\displaystyle=\sum_{\ell=1}^{n}q^{\ell-1}\binom{\ell}{p}_{q}\binom{\ell+r-1}{r-1}_{q}(H_{\ell+r-1}(q)-H_{r-1}(q))$
$\displaystyle=\sum_{\ell=1}^{n}q^{\ell-1}\binom{r+p-1}{p}_{q}\binom{\ell+r-1}{r+p-1}_{q}(H_{\ell+r-1}(q)-H_{r-1}(q))$
$\displaystyle=q^{p-1}\binom{r+p-1}{p}_{q}\binom{r+n}{r+p}_{q}H_{n+r-1}(q)$
$\displaystyle\quad-q^{p+r-1}\binom{r+p-1}{p}_{q}\sum_{\ell=1}^{n-1}\frac{q^{\ell-1}}{[\ell+r]_{q}}\binom{r+\ell}{r+p}_{q}$
$\displaystyle\quad\quad-\binom{r+p-1}{p}\binom{r+n}{r+p}H_{r-1}$
$\displaystyle=q^{p-1}\binom{r+p-1}{p}_{q}\binom{r+n}{r+p}_{q}(H_{n+r-1}(q)-H_{r-1}(q))$
$\displaystyle\quad\quad-\frac{q^{r+2p-2}}{[r+p]_{q}}\binom{r+p-1}{p}_{q}\binom{r+n-1}{r+p}_{q}.$
(17)

With the help of (5) and (17), Lemma 2 gives the result. ∎

When $q\to 1$, Theorem 3 is reduced to the following.

######  Corollary 2.

For $n,r,p\geq 1$,

$\sum_{\ell=0}^{n}{\ell}^{p}H_{\ell}^{(r)}=A(p,r,n)H_{n}^{(r)}-B(p,r,n)\,,$

where

$\displaystyle
A(p,r,n)=\sum_{\ell=0}^{p}S(p,\ell){\ell!}{\binom{n+r-1}{r-1}^{-1}\binom{r+\ell-1}{\ell}\binom{r+n}{r+\ell}},$
$\displaystyle
B(p,r,n)=\sum_{\ell=0}^{p}\frac{1}{r+\ell}S(p,\ell){\ell!}\binom{r+\ell-1}{\ell}\binom{r+n-1}{r+\ell}\,.$

######  Example 1.

$p=2$ gives

$\displaystyle\sum_{\ell=1}^{n}q^{\ell-1}([\ell]_{q})^{2}H_{\ell}^{(r)}(q)$
$\displaystyle=\frac{[n]_{q}[n+r]_{q}(1+q[r+1]_{q}[n]_{q})}{[r+1]_{q}[r+2]_{q}}H_{n}^{(r)}(q)$
$\displaystyle\quad-q^{r}[n-1]_{q}[n]_{q}\binom{n+r-1}{r-1}_{q}\frac{q[r+1]_{q}^{2}[n]_{q}-q^{3}[r]_{q}^{2}+[2]_{q}}{[r+1]_{q}^{2}[r+2]_{q}^{2}}\,.$
(18)

Note that $[\ell+1]_{q}=1+q\cdot[\ell]_{q}$ and
$[\ell+2]_{q}=[2]_{q}+q^{2}\cdot[\ell]_{q}$. With the help of Theorem 3 and
identities (13) and (18), we have the following identities. For positive
integers $n$ and $r$,

$\displaystyle\sum_{\ell=1}^{n}q^{\ell-1}[\ell]_{q}[\ell+1]_{q}H_{\ell}^{(r)}(q)$
$\displaystyle=\frac{[n]_{q}[n+r]_{q}([2]_{q}[n+2]_{q}+q^{3}[r-1]_{q}[n+1]_{q})}{[r+1]_{q}[r+2]_{q}}H_{n}^{(r)}(q)$
$\displaystyle\quad-q^{r}[n-1]_{q}[n]_{q}\binom{n+r-1}{r-1}_{q}\frac{[2]_{q}[r+2]_{q}^{2}+q^{4}[r+1]_{q}^{2}[n-2]_{q}}{[r+1]_{q}^{2}[r+2]_{q}^{2}}\,.$
$\displaystyle\sum_{\ell=1}^{n}q^{\ell-1}[\ell]_{q}[\ell+1]_{q}[\ell+2]_{q}H_{\ell}^{(r)}(q)$
$\displaystyle=\frac{[n]_{q}[n+r]_{q}\bigl{(}(r+1)(r+2)n^{2}+3(r+1)(r+4)n+2(r^{2}+6r+11)\bigr{)}}{[r+1]_{q}[r+2]_{q}[r+3]_{q}}H_{n}^{(r)}(q)$
$\displaystyle\quad-q^{r}[n-1]_{q}[n]_{q}\binom{n+r-1}{r-1}_{q}\frac{(r+1)^{2}(r+2)^{2}n^{2}+(r+1)^{2}(r^{2}+16r+34)n+12(3r^{2}+12r+11)}{([r+1]_{q})^{2}([r+2]_{q})^{2}([r+3]_{q})^{2}}\,.$

To give a more general result, we need the $q$-unsigned Stirling numbers of
the first kind $s_{uq}(n,k)$ defined by

$[\ell]_{q}^{(n)}=[\ell]_{q}[\ell+1]_{q}\cdots[\ell+n-1]_{q}=\sum_{k=0}^{n}s_{uq}(n,k)([\ell]_{q})^{k},\quad(n\in\mathbb{N}).$

The $q$-unsigned Stirling numbers of the first kind $s_{uq}(n,k)$ are well
defined since $[\ell+m]_{q}=[m]_{q}+q^{m}\cdot[\ell]_{q}$.

######  Theorem 4.

For positive integers $n,p$ and $r$,

$\displaystyle\sum_{\ell=1}^{n}q^{\ell-1}[\ell]_{q}^{(p)}H_{\ell}^{(r)}(q)=A_{1q}(p,r,n)H_{n}^{(r)}-B_{1q}(p,r,n)\,,$

where

$\displaystyle A_{1q}(p,r,n)=\sum_{m=0}^{p}s_{uq}(p.m)A_{q}(m,r,n),$
$\displaystyle B_{1q}(p,r,n)=\sum_{m=0}^{p}s_{uq}(p.m)B_{q}(m,r,n).$

###### Proof.

$\displaystyle\sum_{\ell=1}^{n}q^{\ell-1}[\ell]_{q}^{(p)}H_{\ell}^{(r)}(q)$
$\displaystyle=\sum_{\ell=1}^{n}q^{\ell-1}\sum_{m=0}^{p}s_{uq}(p,m){[\ell]_{q}}^{m}H_{\ell}^{(r)}(q)$
$\displaystyle=\sum_{m=0}^{p}s_{uq}(p,m)\sum_{\ell=1}^{n}q^{\ell-1}{[\ell]_{q}}^{m}H_{\ell}^{(r)}(q)$
$\displaystyle=\sum_{m=0}^{p}s_{uq}(p,m)(A_{q}(m,r,n)H_{n}^{(r)}(q)-B_{q}(m,r,n))$
$\displaystyle=\left(\sum_{m=0}^{p}s_{uq}(p,m)A_{q}(m,r,n)\right)H_{n}^{(r)}(q)-\left(\sum_{m=0}^{p}s_{uq}(p,m)B_{q}(m,r,n)\right).$

∎

When $q\to 1$, Theorem 4 is reduced to the following.

######  Corollary 3.

For positive integers $n,p$ and $r$,

$\displaystyle\sum_{\ell=1}^{n}(\ell)^{(p)}H_{\ell}^{(r)}=A_{1}(p,r,n)H_{n}^{(r)}-B_{1}(p,r,n)\,,$

where

$\displaystyle A_{1}(p,r,n)=\sum_{m=0}^{p}(-1)^{p+m}s(p.m)A(m,r,n),$
$\displaystyle B_{1}(p,r,n)=\sum_{m=0}^{p}(-1)^{p+m}s(p.m)B(m,r,n).$

###  2.1 Backward summations

Now we consider backward summations of $q$-hyperharmonic numbers.

######  Theorem 5.

For positive integers $n$ and $r$,

$\sum_{\ell=1}^{n}q^{2n-2\ell}[\ell]_{q}H_{n-\ell}^{(r)}(q)\\\
=\frac{[n]_{q}[n+r]_{q}}{[r]_{q}[r+1]_{q}}H_{n}^{(r)}(q)-\binom{n+r}{r+1}_{q}\left(\frac{q^{r-1}}{[r]_{q}}+\frac{q^{r}}{[r+1]_{q}}-\frac{q^{n+r-1}}{[n+r]_{q}}\right)\,.$

###### Proof.

Set $a_{\ell}=q^{n-\ell}{H_{n-\ell}^{(r)}(q)}$, and
$b_{\ell}=q^{n-\ell}{[\ell]_{q}}$. By using Lemma 1 and
$[\ell+1]_{q}-q[\ell]_{q}=1$, we have

$\displaystyle\sum_{\ell=1}^{n}q^{2n-2\ell}[\ell]_{q}H_{n-\ell}^{(r)}(q)$
$\displaystyle=[n]_{q}\cdot
H_{n-1}^{(r+1)}(q)+\sum_{\ell=1}^{n-1}(H_{n-1}^{(r+1)}(q)-H_{n-\ell-1}^{(r+1)}(q))(q^{n-\ell}{[\ell]_{q}}-q^{n-\ell-1}{[\ell+1]_{q}})$
$\displaystyle=[n]_{q}\cdot
H_{n-1}^{(r+1)}(q)+\sum_{\ell=1}^{n-1}H_{n-1}^{(r+1)}(q)(q^{n-\ell}{[\ell]_{q}}-q^{n-\ell-1}{[\ell+1]_{q}})$
$\displaystyle\quad+\sum_{\ell=1}^{n-1}H_{n-\ell-1}^{(r+1)}(q)(-q^{n-\ell}{[\ell]_{q}}+q^{n-\ell-1}{[\ell+1]_{q}})$
$\displaystyle=q^{n-1}H_{n-1}^{(r+1)}(q)+\sum_{\ell=1}^{n-1}q^{n-\ell-1}H_{n-\ell-1}^{(r+1)}(q)$
$\displaystyle=H_{n-1}^{(r+2)}(q)\,.$

With the help of (5), we get the desired result. ∎

When $q\to 1$, Theorem 5 is reduced to the following.

######  Corollary 4.

For positive integers $n$ and $r$,

$\sum_{\ell=1}^{n}\ell
H_{n-\ell}^{(r)}=\frac{n(n+r)}{r(r+1)}H_{n}^{(r)}-\frac{(n)^{(r)}\bigl{(}(2r+1)n+r^{2}\bigr{)}}{(r-1)!r^{2}(r+1)^{2}}\,.$

It is more complicated to get a summation formula for the backward summations
of higher power. In the case where $q\to 1$, we have more relations, including
the following.

######  Theorem 6.

For positive integers $n,p$ and $r$,

$\displaystyle\sum_{\ell=0}^{n}{\ell}^{p}H_{n-\ell}^{(r)}=A_{2}(p,r,n)H_{n}^{(r)}-B_{2}(p,r,n)\,.$

where $A_{2}(p,r,n)$ and $B_{2}(p,r,n)$ satisfy the following relations:

$\displaystyle
A_{2}(p,r,n)=A_{2}(0,r,n)\left(1+\sum_{j=0}^{p-1}\binom{p}{j}A_{2}(j,r+1,n-1)\right),$
$\displaystyle B_{2}(p,r,n)$
$\displaystyle=B_{2}(0,r,n)\left(1+\sum_{j=0}^{p-1}\binom{p}{j}A_{2}(j,r+1,n-1)\right)+\sum_{j=0}^{p-1}\binom{p}{j}B_{2}(j,r+1,n-1)\,,$

with the initial values $A_{2}(0,r,n)=\frac{n}{r}$ and
$B_{2}(0,r,n)=\frac{1}{r}\binom{n+r-1}{r}$.

Nevertheless, we can have a different backward summation formula without
weights.

######  Theorem 7.

For positive integers $n,p$ and $r$,

$\sum_{\ell=1}^{n}q^{p(n-\ell)}H_{n-\ell}^{(r)}(q)=C_{q}(p,r,n)H_{n}^{(r)}(q)-D_{q}(p,r,n)\,,$

where $C_{q}(p,r,n)$ and $D_{q}(p,r,n)$ satisfy the following recurrence
relation.

$\displaystyle
C_{q}(p,r,n)=\frac{[n]_{q}}{[r]_{q}}\left(q^{(p-1)(n-1)}+(1-q^{p-1})C_{q}(p-1,r+1,n-1)\right)$
$\displaystyle D_{q}(p,r,n)$
$\displaystyle=\frac{q^{r-1}[n]_{q}}{([r]_{q})^{2}}\binom{n+r-1}{r}_{q}\left(q^{(p-1)(n-1)}+(1-q^{p-1})C_{q}(p-1,r+1,n-1)\right)$
$\displaystyle\quad+(1-q^{p-1})D_{q}(p-1,r+1,n-1).$

###### Proof.

Set $a_{\ell}=q^{n-\ell}{H_{n-\ell}^{(r)}(q)}$ and
$b_{\ell}=q^{(p-1)(n-\ell)}$. By using Lemma 1 and
$[\ell+1]_{q}-q[\ell]_{q}=1$, we have

$\displaystyle\sum_{\ell=1}^{n}q^{p(n-\ell)}H_{n-\ell}^{(r)}(q)$
$\displaystyle=H_{n-1}^{(r+1)}(q)+\sum_{\ell=1}^{n-1}(H_{n-1}^{(r+1)}(q)-H_{n-\ell-1}^{(r+1)}(q))(q^{(p-1)(n-\ell)}-q^{(p-1)(n-\ell-1)})$
$\displaystyle=H_{n-1}^{(r+1)}(q)+\sum_{\ell=1}^{n-1}H_{n-1}^{(r+1)}(q)(q^{(p-1)(n-\ell)}-q^{(p-1)(n-\ell-1)})$
$\displaystyle\quad+\sum_{\ell=1}^{n-1}H_{n-\ell-1}^{(r+1)}(q)(-q^{(p-1)(n-\ell)}+q^{(p-1)(n-\ell-1)})$
$\displaystyle=q^{(p-1)(n-1)}H_{n-1}^{(r+1)}(q)+(1-q^{p-1})\sum_{\ell=1}^{n-1}q^{(p-1)(n-\ell-1)}H_{n-\ell-1}^{(r+1)}(q)$
$\displaystyle=q^{(p-1)(n-1)}H_{n-1}^{(r+1)}(q)$
$\displaystyle\quad+(1-q^{p-1})\left(C_{q}(p-1,r+1,n-1)H_{n-1}^{(r+1)}(q)-D_{q}(p-1,r+1,n-1)\right)\,.$

With the help of (5), we get the desired result. ∎

## Acknowledgement

Authors are grateful to the anonymous referee for helpful comments.

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