††thanks: These authors contributed equally††thanks: These authors contributed
equally††thanks: These authors contributed equally

# Implementation of a scalable universal two-qubit quantum processor
with electron and nuclear spins in a trapped ion

Ji Bian  School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai
519082, China  Teng Liu  School of Physics and Astronomy, Sun Yat-Sen
University, Zhuhai 519082, China  Qifeng Lao  School of Physics and Astronomy,
Sun Yat-Sen University, Zhuhai 519082, China  Min Ding  School of Physics and
Astronomy, Sun Yat-Sen University, Zhuhai 519082, China  Huiyi Zhang  School
of Physics and Astronomy, Sun Yat-Sen University, Zhuhai 519082, China  Xinxin
Rao  School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai 519082,
China
Pengfei Lu  School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai
519082, China  Shenzhen Research Institute of Sun Yat-Sen University, Shenzhen
518057, China  State Key Laboratory of Optoelectronic Materials and
Technologies, Sun Yat-Sen University, Guangzhou 510275, China  Le Luo
<EMAIL_ADDRESS>School of Physics and Astronomy, Sun Yat-Sen
University, Zhuhai 519082, China  Shenzhen Research Institute of Sun Yat-Sen
University, Shenzhen 518057, China  State Key Laboratory of Optoelectronic
Materials and Technologies, Sun Yat-Sen University, Guangzhou 510275, China
Quantum Science Center of Guangdong-HongKong-Macao Greater Bay Area, Shenzhen
518048, China

###### Abstract

Increasing the quantum information processing power with limited number of
hosts is vital for achieving quantum advantage. Here we propose a novel scheme
that achieves a scalable $n$-ion-$2n$-qubit quantum processor utilizing four
internal levels of each ion, and experimentally implement a $1$-ion-$2$-qubit
universal processor using the valence electron spin and nuclear spin of a
single 171Yb+ ion. Fidelities of single-qubit and two-qubit gates are around
$98\%$ obtained by quantum process tomography. Additionally, the Grover’s
algorithm is implemented with a successful rate exceeding $99\%$. We provide
explicit scaling-up protocols based on standard laser-less and laser-based
frameworks, and further demonstrate that the electron/nuclear-spin scheme
allows less demanding two-qubit entangling gates between different ions. The
replacement of some inter-atomic gates by intra-atomic gates could increase
the fidelity of some quantum circuits. Our work paves the way towards
achieving $2^{n}$-times increase in the size of quantum computational Hilbert
space with $n$ ions.

##  I Introduction

Quantum information processing (QIP) based on two-level qubits naturally
inherits the binary information processing used in classical computers. The
physical hosts of quantum information, for example, atoms [1, 2, 3, 4] and
superconducting circuits [5, 6, 7], however almost always consist of higher-
dimensional Hilbert spaces. Meanwhile, increasing the QIP power with limited
number of hosts and demonstrate quantum advantage remains a major goal in the
noisy intermediate-scale quantum (NISQ) era [8, 9, 10]. These has stimulated
studies on qudit-based QIP, i.e., using $d>2$ levels of the underlying system
as a QIP unit [11], with recent experimental progress in constructing single-
and two-qudit processors [12, 13, 14]. While the d-level structure makes it
more suitable for applications that are naturally formulated in high-
dimensional Hilbert spaces [15, 16], it comes at a cost. First, it increases
complexity in algorithms: While qudits offer the potential for increased
computational power, developing algorithms that effectively utilize this
additional dimensionality can be challenging. Secondly, quantum algorithms and
protocols designed for qubits may not directly translate to qudits. This can
slow down progress in utilizing qudits for quantum information processing, as
it requires additional research and development effort to adapt existing
algorithms or develop new ones, as an example, standard qubit-based quantum
error correction could not be directly implemented on qudit processors.
Instead, people are looking forward to the scheme using $d>2$ levels to
realize a multi-qubit quantum processor in a single host [17, 18].

Here, we propose a novel scheme that utilizes the $4$-levels of a single
trapped ion as _two qubits_ , and construct a scalable $2n$-qubit processor
with $n$ ions. The four levels $|1^{\prime}\sim 4^{\prime}\rangle$ could be
chosen arbitrarily (termed “logical encoding”), e.g.,
$|1^{\prime}\rangle=\left|\uparrow\uparrow\right\rangle$,
$|2^{\prime}\rangle=\left|\uparrow\downarrow\right\rangle$,
$|3^{\prime}\rangle=\left|\downarrow\uparrow\right\rangle$,
$|4^{\prime}\rangle=\left|\downarrow\downarrow\right\rangle$, where
$\left|\uparrow,\downarrow\right\rangle$ represents encoded qubits (omitting
the tensor product symbol). This is also proposed in recent works [17, 19].
Or, as explicitly studied in this work, the two qubits are specifically
represented by two physical $2$-level quantum systems (termed “physical
encoding”), of the electron spin and nuclear spin (subspace). This way of
encoding is implicitly used in color center-based QIP [20, 21, 22], but has
not been applied to atom-based QIP experiments.

We experimentally realize a $1$-ion-$2$-qubit processor using the valence
electron spin, nuclear spin (spin-$1/2$) and their mutal hyperfine interaction
within a single 171Yb+ ion under standard quantization magnetic fields [23],
and implement high-fidelity universal two-qubit gates. We then perform
Grover’s algorithm as an application. We also give explicit scaling-up
protocols based on both laser-less [24, 25, 26, 27, 28] and laser-based
frameworks [29, 30, 31, 32]. We then discuss the readout process, which could
be achieved using well-known techniques [33, 34]. We discover that the large
discrepancy between electron (nuclear) gyromagnetic ratio $\gamma_{e}$
($\gamma_{n}$) allows less demanding two-qubit entangling gates between
different ions under large quantization magnetic fields [35, 36].

The main distinction between the $n$-ion-$2n$-qubit processor and a ($d=4$)
qudit processor is as follows. Although both methods offers $2^{n}$-times
increase in the size of QIP Hilbert space, in the $n$-ion-$2n$-qubit
processor, individual qubit state preparation and readout in a single ion
could be accomplished, without destroying the other qubit in the same ion.
This allows standard quantum algorithms including quantum error correction to
be implemented without modification, whereas this is not the case for qudits.
Besides, the replacement of some inter-atomic gates (fidelity limited by
motional state decoherence [37]) by intra-atomic gates in our scheme could
increase fidelities of certain quantum circuits.

The logical encoding and physical encoding schemes each has their special
features: the logical encoding could use four clock states, and thus offers
relatively long coherence times. But it does not have a clear entanglement
structure, for example, if the four levels are chosen from the spin angular
momentum eigenstates of the same spin, then there is no entanglement at all.
Nevertheless, as is also pointed out in [17], most quantum computing
algorithms do not require the nonlocality of quantum entanglement, and could
therefore obtain the increased processing power within this scheme. The
physical encoding scheme has well defined entanglement structure, and the
identification of nuclear/electron spins enables certain tasks such as the
quantum simulation of non-Abelian lattice gauge theories [38], where the
internal symmetry of nuclear spins could be used to maintain gauge invariance.
The naturally existing entanglement structure in this scheme, e.g., strong
coupling between two kinds of qubits in individual ions plus adjustable
coupling between different ions, also makes it advantageous in certain
applications, such as the preparation of thermofield double states [39, 40],
and studying the dynamics of many-body scars [41, 42]. The number of qubits
and the dimension of QIP Hilbert space are not limited to $2$ and $4$ in a
single ion: $2^{p}$ levels could be used to encode $p>2$ qubits. The higher
angular momentum ($>1/2$) structure (and its subspace) could be explored to
construct hybrid qubit-qudit coupled systems, such as the 25Mg+ (43Ca+) ion
with nuclear spin $5/2$ ($7/2$), to simulate interacting spin ($>1/2$)
systems. Through viewing the information hosts from a different angle and
exploring their underlying physical structures, the $n$-ion-multi-qubit scheme
would become a complement to the earlier developed qudit scheme on the way
toward increasing the QIP power.

##  II Electron and nuclear spin as two qubits in a single trapped ion

Here we take 171Yb+ ion as an example, and demonstrate the physical encoding
scheme (see fig.1).

Figure 1: The $n$-ion-$2n$-qubit processor. The hyperfine interaction between
nuclear spin (qubit-1) and electron spin (qubit-2) results in the 2S1/2
4-level subspace. The 2F7/2 subspace could also be utilized as qubit levels
and to perform readout.

Consider the s-electron (spin-$1/2$) in the outermost shell and the 171Yb
nucleus (spin-$1/2$). Denote the angular momentum operators
$I_{x},I_{y},I_{z}$ as the standard Pauli matrices times $1/2$, and
eigenstates of $I_{z}$ as $\left|\uparrow\right\rangle$,
$\left|\downarrow\right\rangle$, with corresponding eigenvalues $\hbar/2$,
$-\hbar/2$ ( $\hbar=1$ in the following). The magnetic dipole-dipole
interaction between these two spins results in the hyperfine interaction [43]
$H_{A}=A(I_{1,x}I_{2,x}+I_{1,y}I_{2,y}+I_{1,z}I_{2,z}),$ where $A\approx
2\pi\times 12.6$ GHz is the hyperfine interaction constant, $1$ represents
nucleus and $2$ electron.

The ion is typically in an external magnetic field $B_{0}$ providing the
quantization axis (denote it $z$ axis), so the free evolution Hamiltonian is

$\displaystyle H_{0}$
$\displaystyle=A(I_{1,x}I_{2,x}+I_{1,y}I_{2,y}+I_{1,z}I_{2,z})-B_{0}(\gamma_{1}I_{1,z}+\gamma_{2}I_{2,z})$
(1) $\displaystyle=E_{1}|1\rangle\langle 1|+E_{2}|2\rangle\langle
2|+E_{3}|3\rangle\langle 3|+E_{4}|4\rangle\langle 4|,$

where $\gamma_{1}/(2\pi)\approx 7.5\times 10^{6}$ Hz$\cdot$T-1$\cdot$s-1,
$\gamma_{2}/(2\pi)\approx-2.8\times 10^{10}$ Hz$\cdot$T-1$\cdot$s-1 are the
gyromagnetic ratios. $|1\rangle$, $|2\rangle$, $|3\rangle$, $|4\rangle$ are
eigenstates of $H_{0}$ with eigenvalues $E_{1\sim 4}$, they are related to
$\left|\uparrow\uparrow\right\rangle$,
$\left|\uparrow\downarrow\right\rangle$,
$\left|\downarrow\uparrow\right\rangle$,
$\left|\downarrow\downarrow\right\rangle$ by

$\displaystyle|1\rangle=\left|\uparrow\uparrow\right\rangle,\quad|2\rangle=\cos\frac{\theta_{0}}{2}\left|\uparrow\downarrow\right\rangle-\sin\frac{\theta_{0}}{2}\left|\downarrow\uparrow\right\rangle,$
(2)
$\displaystyle|3\rangle=-\sin\frac{\theta_{0}}{2}\left|\uparrow\downarrow\right\rangle-\cos\frac{\theta_{0}}{2}\left|\downarrow\uparrow\right\rangle,\quad|4\rangle=\left|\downarrow\downarrow\right\rangle,$

where $\theta_{0}$ depends on $B_{0}$ ($\theta_{0}\approx-\frac{\pi}{2}$
here). Define the “spin basis” where
$\left|\uparrow\uparrow\right\rangle_{\textrm{s}}=[1,0,0,0]^{\textrm{T}}$,
$\left|\uparrow\downarrow\right\rangle_{\textrm{s}}=[0,1,0,0]^{\textrm{T}}$,
$\left|\downarrow\uparrow\right\rangle_{\textrm{s}}=[0,0,1,0]^{\textrm{T}}$,
$\left|\downarrow\downarrow\right\rangle_{\textrm{s}}=[0,0,0,1]^{\textrm{T}}$,
the subscript indicates spin basis, and it is also the computational basis of
the two-qubit processor. Also define the “number basis” where
$|1\rangle_{\textrm{n}}=[1,0,0,0]^{\textrm{T}}$,
$|2\rangle_{\textrm{n}}=[0,1,0,0]^{\textrm{T}}$,
$|3\rangle_{\textrm{n}}=[0,0,1,0]^{\textrm{T}}$,
$|4\rangle_{\textrm{n}}=[0,0,0,1]^{\textrm{T}}$, the subscript indicates
number basis. Define a mapping operator $R$ (depends on $B_{0}$) satisfying

$\displaystyle
R|k\rangle_{\textrm{n}}=|k\rangle_{\textrm{s}}\quad(k=1,2,3,4),$ (3)

where $|k\rangle_{\textrm{s}}$ represents the matrix form of $|k\rangle$ in
spin basis, i.e., it maps the matrix form of a vector (or an operator) in
number basis to that in spin basis, and vice versa. The explicit form of $R$
reads

$R=\begin{pmatrix}1&0&0&0\cr 0&\cos(\theta_{0}/2)&-\sin(\theta_{0}/2)&0\cr
0&-\sin(\theta_{0}/2)&-\cos(\theta_{0}/2)&0\cr 0&0&0&1\end{pmatrix},$ (4)

where $\theta_{0}=2\textrm{tan}^{-1}(\lambda)$ and

$\lambda=\frac{-B_{0}\gamma_{1}+B_{0}\gamma_{2}-\sqrt{A^{2}+B^{2}_{0}\gamma^{2}_{1}+B^{2}_{0}\gamma^{2}_{2}-2B^{2}_{0}\gamma_{1}\gamma_{2}}}{A}.$
(5)

Eq. (1) corresponds to the ${}^{2}S_{1/2}$ four-level subspace in 171Yb+, as
shown in fig.1. It is clear that we could treat spin $1$, $2$ as two qubits
with coupling, and $\left|\uparrow,\downarrow\right\rangle$ as the two levels
of a qubit.

As Zeeman sublevels are used, the coherence time is degraded compared to
conventional clock-states based qubits. This could be circumvented by
multilevel dynamical decoupling [44] or magnetic shielding [45]. The logical
encoding scheme also works in these four levels, and could potentially achieve
better coherence time if the 2S1/2 $|F=0,1,m_{F}=0\rangle$ and 2F7/2
$|F=3,4,m_{F}=0\rangle$ (lifetime $\approx 1.58$ yrs) clock states are
utilized, at the expense of more complicated optical set-ups [17, 18].

##  III Universal quantum control in the $1$-ion-$2$-qubit system

To demonstrate the universality of the $1$-ion-$2$-qubit processor, we choose
to implement the following operations that form a universal quantum gate set
[46]: Hadamard, Phase, $\pi/8$, and CNOT gates.

There is a number of ways to implement these operations. For example, one
could use analytical method to solve the Schrödinger equation and obtain the
control pulses that achieve the gate set. These pulses require simultaneous
driving of e.g., $|3\rangle\leftrightarrow|1\rangle$ and
$|2\rangle\leftrightarrow|4\rangle$, without driving
$|1\rangle\leftrightarrow|2\rangle$. This would require a large $B_{0}$ (to
magnify the unequal splitting between $|2\rangle\leftrightarrow|1,4\rangle$),
and therefore requires modification of current set-ups ($B_{0}\approx 6$ Gs).
Alternatively, one could even go to regimes where $|B_{0}\gamma_{1,2}|\gg A$.
Single-qubit gates could now be realized by selective excitation based on
different Larmor frequencies ($\gamma_{1,2}B_{0}$), and the secular-
approximated coupling term $I_{1,z}I_{2,z}$ could be used to realize two-qubit
gates [47]. This is similar to those applied in magnetic resonance
experiments, but requires even larger magnetic fields and is therefore not
applicable on current set-ups either [35, 36]. There is yet another
possibility, inspired by universal quantum control of spin systems in ultra-
low magnetic field [48, 49, 50, 51], that using shaped dc-magnetic fields
(without the $\approx 12.6$ GHz carrier frequencies) and standard quantization
fields only, the universal gate set could still be realized. This is directly
applicable on current trapped-ion setups and is worth further experimental
studies.

Numerical optimal control is powerful in obtaining the desired control pulses
[52], it is also applicable in the current situation and is utilized in our
experiment. Consider a control scenario applicable to common trapped-ion
setups: three near-resonant control fields applied simultaneously to each ion.
The control fields could be realized by microwaves, such as in our experiments
among others [53, 54, 55, 56, 57, 28, 58, 59]. Or, they could be realized by
laser fields [23, 60]. Transforming to a rotating frame with frequencies
determined by the control fields and apply the rotating wave approximation
(RWA) (see Appendix A), we get the total Hamiltonian

$\displaystyle H_{c}=$ $\displaystyle c_{31}(t)|3\rangle\langle
1|+c_{32}(t)|3\rangle\langle 2|+c_{34}(t)|3\rangle\langle 4|$ (6)
$\displaystyle+\frac{\delta_{1}}{2}(t)|1\rangle\langle
1|+\frac{\delta_{2}(t)}{2}|2\rangle\langle
2|+\frac{\delta_{4}(t)}{2}|4\rangle\langle 4|+\textrm{H.c.},$

where $c_{31,32,34}$ ($\delta_{1,2,4}$) are complex (real) numbers, and are
determined by the strength and phase (detunings) of the near-resonant control
fields.

Denote $F$ the performance function that characterizes the fidelity of the
implemented operation $U$ relative to the ideal $U_{\textrm{id}}$, the
optimization task is therefore

find {$c_{31,32,34}(t)$, $\delta_{1,2,4}(t)$},

max $F(U)$,

s.t. $\frac{\partial U(t)}{\partial t}=-iH_{c}(t)U(t)$.

While there are many quantum optimal control algorithms developed in these
years [61, 62, 63], we use the Gradient Ascent Pulse Engineering (GRAPE)
algorithm [64], as it behaves well enough in accomplishing our tasks. We
obtain control pulses achieving all the operations in the gate set with
theoretical fidelities $>0.999$. As an example, a pulse sequence achieving
C1NOT2 is illustrated in fig. 2. Here only $c_{31,32,34}$, with corresponding
amplitudes $A_{1,2,3}$ and phases $\theta_{1,2,3}$ are modulated,
$\delta_{1,2,4}(t)$ are kept zero.

Figure 2: A theoretical pulse sequence achieving C1NOT2. $A_{1,2,3}$ and
$\theta_{1,2,3}$ are corresponding amplitudes and phases of $c_{31,32,34}$

Quantum state tomography (QST) is required to obtain the full information of
the state after an operation, and could be combined to implement the quantum
process tomography (QPT) that enables characterization of the performance of
the implemented operation. In our experiment, the direct observable is
$|3\rangle\langle 3|$. Populations on $|1\sim 4\rangle$ and coherence among
them could be obtained by first transferring them to the $|3,1\rangle$,
$|3,2\rangle$ or $|3,4\rangle$ subspace by $\pi$ and $\pi/2$ pulses, and then
apply the standard qubit QST. The obtained $\tilde{\rho}$ is in the number
basis, applying $R$ we obtain $\rho=R\tilde{\rho}R^{\dagger}$ in the spin
basis, which is also the natural computational basis for the two-qubit
processor.

The experiment is performed on the trapped 171Yb+ ion quantum information
processor (details listed in Appendix B). Microwave fields near resonant with
the three transitions $|3\rangle\leftrightarrow|1,2,4\rangle$, with tunable
frequencies, amplitudes and phases are used to control the spin states. The
population $P_{3}$ on $|3\rangle$ could be readout by first measuring the
total population $P_{1}+P_{2}+P_{4}$ on $|1\rangle$, $|2\rangle$, and
$|4\rangle$ through applying the $369.5$ nm detection beam and measuring the
fluorescence, then $P_{3}=1-(P_{1}+P_{2}+P_{4})$. The initial state of each
experiment is $|3\rangle$ prepared by optical pumping. Starting from
$|3\rangle$, the quantum states after application of operations in the gate
set are obtained by QST. For example, the measured density matrix after C1NOT2
is shown in fig. 3. It is demonstrated in the number basis, and could be
easily transferred to the spin basis by $R$. The state fidelities are listed
in table 1. We also implement the SWAP gate which is useful to transfer
quantum information among different types of qubits.

Table 1: QST fidelity. Gate | C1NOT2 | C2NOT1 | Phase1 | Phase2 | $\pi/8_{1}$ | $\pi/8_{2}$ | Hadamard1 | Hadamard2 | SWAP
---|---|---|---|---|---|---|---|---|---
Fidelity (%) | 99.0 $\pm$0.19 | 99.0$\pm$0.45 | 99.0$\pm$0.51 | 99.0$\pm$0.14 | $98.9\pm 0.33$ | $98.3\pm 0.69$ | $98.4\pm 0.43$ | $98.2\pm 0.65$ | $99.7\pm 0.14$
Figure 3: Real parts of the final state density matrix in the number basis
after C1NOT2 Gate. Theoretical imaginary parts equal zero, absolute values of
the measured imaginary parts are all smaller than $0.03$ and are not shown.

The QPT is further applied to characterize the performance of the implemented
gate operations. For example, the $\chi$ matrices of Hadmard1 and C-phase $=$
diag$(1,-1,1,1)$ (spin basis) obtained by QPT are illustrated in fig. 4 and 5.
Here the mapping operator is already applied to obtain the $\chi$ matrices in
spin basis. The fidelities are $97.5$% and $98.5$%, respectively.

Figure 4: Real parts of the $\chi$ matrix of the Hadamard1 Gate in spin basis.
Theoretical imaginary parts equal zero, absolute values of the measured
imaginary parts are all smaller than $0.02$ and are not shown. Figure 5: Real
parts of the $\chi$ matrix of the C-phase Gate in spin basis. Theoretical
imaginary parts equal zero, absolute values of the measured imaginary parts
are all smaller than $0.04$ and are not shown.

Although the control pulses obtained by GRAPE are already made robust against
pulse amplitude errors [64], the fidelities are still below ideal values. We
believe the main limitation is due to magnetic noises, which randomly shift
the energy levels (especially the Zeeman sublevels $|1,4\rangle$). This causes
pulse errors both during gate operations and tomographic pulses, and also
degrades the coherence among the four levels. Reducing magnetic noises via
triggering to the ac-line [45] increases the control performance, as indicated
by our experiment: the state fidelities of Hadamard1,2 with $300$ $\mu$s gate
times are increased from around $85\%$ without line triggering, to around
$99\%$ with line triggering. Higher fidelities could be envisioned with
magnetic shielding.

##  IV Application: Demonstration of the Grover’s algorithm

Based on the universal $1$-ion-$2$-qubit processor, we could realize simple
quantum algorithms, for example, the Grover’s algorithm [46]. As demonstrated
in fig. 6(a), the quantum circuit that searches for
$\left|\uparrow\downarrow\right\rangle$ is implemented. The initial state
$\left|\uparrow\uparrow\right\rangle$ is prepared by optical pumping to
$|3\rangle$ then apply a $\pi$ pulse,
$\textrm{C-00}=\textrm{diag}(1,-1,-1,-1)$ in spin basis. The theoretical and
measured density matrix of the final state (in number basis) are shown in fig.
6(b) and fig. 6(c). The average successful rate is around $99.0\%$.

Figure 6: Implementation of the Grover’s algorithm. (a) The quantum circuit.
(b) Real parts of the theoretical density matrix of the final state in number
basis. Imaginary parts equal zero. (c) Real part of the experimental density
matrix of the final state. Absolute values of measured imaginary parts are all
smaller than $0.06$ and are not shown.

##  V Scale up to $2n$-qubit universal quantum processors with $n$ ions

Here we discuss the scalability of our scheme. That is, in an
$n$-ion-$2n$-qubit processor, demonstrate how to realize arbitrary single-
qubit gates and two-qubit entangling gates, not only within an ion, but also
between two ions. Moreover, demonstrate how to readout the quantum information
(state preparation is done by pumping all the ions to $|3\rangle$ and apply
global $\pi$ pulses on $|3\rangle\leftrightarrow|1\rangle$). The fact that the
two qubits are naturally existing electron and nuclear spins with hyperfine
coupling allows novel and efficient scaling protocols. To simplify discussion,
we define $I^{i}_{j,k}$ as the spin momentum operator in $k$ direction for
qubit $j$ in ion $i$ [i.e., qubit-1(2) means the nuclear (electron) spin qubit
in this ion], define $I^{i}_{j\leftrightarrow l,k}$ as the spin momentum
operator in $k$ direction for transition $|j\rangle\leftrightarrow|l\rangle$
in ion $i$, define $R^{i}_{j\leftrightarrow l,k}(\theta)$
($R^{i}_{j,k}(\theta)$) as the rotation operator in ion $i$ for
$|j\rangle\leftrightarrow|l\rangle$ transition (qubit $j$) with an angle
$\theta$ along $k$.

###  V.1 Laser-less approach

Universal trapped-ion quantum processor based on static or time-varying
magnetic field gradient has been proposed and realized [24, 25, 26, 27, 28].
Our scheme based on electron/nuclear spin-qubits posses unique advantages when
combining with this approach. Here we use one time-varying gradient field near
a normal mode frequency as an example [25].

As demonstrated in [65], single-ion control
($|2\rangle\leftrightarrow|3\rangle$ subspace) could be accomplished by
selective positioning of the ions in a spatially varying microwave magnetic
field. Similarly, as transitions $|3\rangle\leftrightarrow|1,4\rangle$ are
driven by $B_{\perp}$ (this is clear form (19) in Appendix A), the component
of corresponding near-resonant microwave fields perpendicular to $B_{0}$,
single-ion control in $|3\rangle\leftrightarrow|1,4\rangle$ subspace could be
realized by placing the ions to be controlled in configurations $B_{\perp}\neq
0$, and the remaining ions in $B_{\perp}=0$.

Without loss of generality, to realize entangling operations between qubit $2$
in ion $p$ and qubit $2$ in ion $w$, assume the ions are aligned along the $y$
axis, the trap axis orthogonal to $y$ are along $x$ and $z$ directions, and
the quantization field is along $z$. Also denote $q^{n}_{z}$ the displacement
along $z$ of ion $n$ relative to its equilibrium position ($q^{n}_{z}=0$).
$q^{n}_{z}$ could be expanded as a summation over normal modes
$q^{n}_{z}=\sum_{j}b^{n}_{j,z}\epsilon_{j}(a_{j}+a^{\dagger}_{j})$, where
$a^{\dagger}_{j}$, $a_{j}$ are normal-mode creation and annihilation
operators, $b^{n}_{j,z}$ are dimensionless coefficients,
$\epsilon_{j}=\sqrt{\hbar/(2m\omega_{j})}\sim 1$ nm and $\omega_{j}$ are mode
frequencies [66]. At positions with $z$ coordinates $q_{z}$, there is an
oscillating magnetic field gradient $\vec{B}=q_{z}B^{\prime}_{z}\cos(\omega
t+\phi)\vec{e}_{z}$ that serves to couple the internal states and motion,
$\omega=\omega_{j}-\delta$ close to a normal mode frequency. Only the ions
$p,w$ are affected, this is done by selective positioning of the ions [65].
The total Hamiltonian in the lab frame reads
$H_{l}=H^{p}_{0}+H^{w}_{0}+\sum_{j}\omega_{j}a^{\dagger}_{j}a_{j}+H_{g}$,
where $H^{p}_{0}$ and $H^{w}_{0}$ are single-ion-two-qubit Hamiltonian
discussed in (1), and the interaction with the gradient field is

$\displaystyle H_{g}$
$\displaystyle=-\sum_{n=p,w}q^{n}_{z}B^{\prime}_{z}\cos(\omega
t+\phi)(\gamma_{1}I^{n}_{1,z}+\gamma_{2}I^{n}_{2,z}).$ (7)

Transforming to the rotating frame with respect to
$H_{r}=H^{p}_{0}+H^{w}_{0}+\sum_{j}\omega_{j}a^{\dagger}_{j}a_{j}$, neglect
fast oscillating terms, as well as terms $\sim
B_{0}\gamma_{1,2}/A,B_{0}\gamma_{1,2}/A^{2}$ ($|B_{0}\gamma_{1,2}|\ll A$), the
resulting Hamiltonian is

$\displaystyle H_{a}$
$\displaystyle=-\sum_{n,l}\Omega^{n}_{j,z}I^{n}_{l,z}[e^{-i(\delta
t-\phi)}a_{j}+e^{i(\delta t-\phi)}a^{\dagger}_{j}],$ (8)

where

$\Omega^{n}_{j,z}=b^{n}_{j,z}B^{\prime}_{z}\frac{\gamma_{1}+\gamma_{2}}{4}\epsilon_{j}.$
(9)

Set the evolution time $\tau=2k_{1}\pi/\delta$ ($k_{1}$ is an integer) to
disentangle the internal states and motion [67], the resulting evolution is a
global entangling operation

$U_{zz}(\tau)=\textrm{exp}[\frac{2k_{1}\pi
i}{\delta^{2}}(\sum_{n,l}\Omega^{n}_{j,z}I^{n}_{l,z})^{2}].$ (10)

With single-ion addressing available, the desired operation (up to a global
phase factor $\theta_{1}=2k_{1}\pi/\delta^{2}\sum_{n}(\Omega^{n}_{j,z})^{2}$)
could be realized by the following sequence

$\displaystyle U_{2,2,zz}(\tau)=$ $\displaystyle\textrm{exp}[\frac{2k_{1}\pi
i}{\delta^{2}}(8\Omega^{p}_{j,z}\Omega^{w}_{j,z}I^{p}_{2,z}I^{w}_{2,z})]\exp(i\theta_{1})$
(11) $\displaystyle=$ $\displaystyle
U_{zz}(\tau)R^{p}_{1,y}(\pi)R^{w}_{1,y}(\pi)U_{zz}(\tau)R^{w}_{1,y}(-\pi)$
$\displaystyle
U_{zz}(\tau)R^{p}_{1,y}(\pi)R^{w}_{1,y}(\pi)U_{zz}(\tau)R^{w}_{1,y}(-\pi).$

Entangling operation between other combinations of qubits could be achieved by
choosing proper single ion operations in (11). Entangling operations with
arbitrary spin operators, e.g., $U_{2,2,xx}$, could be achieved by the
sequence:
$U_{2,2,xx}=R^{p}_{2,y}(\pi/2)R^{w}_{2,y}(\pi/2)U_{2,2,zz}R^{p}_{2,y}(-\pi/2)R^{w}_{2,y}(-\pi/2)$.
As the 2-ion-4-qubit system, with the available control fields discussed
above, is complete controllable, one could also achieve arbitrary operations
via numerical optimal control.

$U_{2,2,zz}$ could also be achieved with only global control fields on both
ions, provided that the quantization magnetic field is large
($|B_{0}\gamma_{1,2}|\gg A$). This is because given $H_{l}$, one could now
transform to a rotating frame with respect to
$\tilde{H}_{r}=H^{p}_{z}+H^{w}_{z}+A(I^{p}_{1,z}I^{p}_{2,z}+I^{w}_{1,z}I^{w}_{2,z})+\sum_{j}\omega_{j}a^{\dagger}_{j}a_{j}$,
where $H^{n}_{z}=B_{0}(\gamma_{1}I^{n}_{1,z}+\gamma_{2}I^{n}_{2,z})$. After
RWA

$\displaystyle H_{a}$
$\displaystyle=-\sum_{n,l}\tilde{\Omega}^{n}_{l,j,z}I^{n}_{l,z}[e^{-i(\delta
t-\phi)}a_{j}+e^{i(\delta t-\phi)}a^{\dagger}_{j}]$ (12)
$\displaystyle\approx-\sum_{n}\tilde{\Omega}^{n}_{2,j,z}I^{n}_{2,z}[e^{-i(\delta
t-\phi)}a_{j}+e^{i(\delta t-\phi)}a^{\dagger}_{j}],$

where

$\tilde{\Omega}^{n}_{l,j,z}=b^{n}_{j,z}B^{\prime}_{z}\gamma_{l}\epsilon_{j},$
(13)

and the approximation holds because $|\gamma_{2}|\gg|\gamma_{1}|$. This is
advantageous as it allows direct selective-entangling operation on qubit-2,
with only global controls on both ions. Setting the evolution time
$\tau=2k_{1}\pi/(\delta)$ one obtains the desired entangling operation.

###  V.2 Laser-based approach

Using single-ion addressing lasers with adjustable polarization, arbitrary
single-ion-two-qubit operations could be achieved in an $n$-ion processor.
Arbitrary two-qubit entangling operation between qubits in different ions
could be realized by combination of conventional Mølmer–Sørensen (MS)
interaction with single-ion-two-qubit control pulses. This is equivalent to
the two-ion qudit entangling operation [12, 13] plus similar transformation.
For example, the entangling operation
$U_{1,1,xx}(\tau)=\textrm{exp}[-4i(I^{p}_{1,x}I^{w}_{1,x})\tau]$ could be
realized by the following sequence:

$\displaystyle U_{1,1,xx}(\tau)=$ $\displaystyle
R^{\dagger}_{5}R_{4}\tilde{U}_{xx}(\tau)R^{\dagger}_{4}R_{3}\tilde{U}_{xx}(\tau)R^{\dagger}_{3}$
(14) $\displaystyle
R_{2}\tilde{U}_{xx}(\tau)R^{\dagger}_{2}R_{1}\tilde{U}_{xx}(\tau)R^{\dagger}_{1}R_{5},$

where

$\tilde{U}_{xx}(\tau)=\textrm{exp}[-i(I^{p}_{2\leftrightarrow
3,x}I^{w}_{2\leftrightarrow 3,x})\tau]$ (15)

is the conventional MS interaction between two clock qubits
($|2\rangle\leftrightarrow|3\rangle$ subspace of each ion), and

$\displaystyle R_{1}=R^{p}_{1\leftrightarrow
4,y}(\pi/2)R^{p}_{1\leftrightarrow 2,y}(\pi)R^{w}_{1\leftrightarrow
4,y}(\pi/2)R^{w}_{3\leftrightarrow 4,y}(\pi),$ (16) $\displaystyle
R_{2}=R^{p}_{1\leftrightarrow 4,y}(\pi/2)R^{p}_{3\leftrightarrow
4,y}(\pi)R^{w}_{1\leftrightarrow 4,y}(\pi/2)R^{w}_{1\leftrightarrow
2,y}(\pi),$ $\displaystyle R_{3}=R^{p}_{1\leftrightarrow
4,y}(\pi/2)R^{p}_{1\leftrightarrow 2,y}(\pi)R^{w}_{1\leftrightarrow
4,y}(\pi/2)R^{w}_{1\leftrightarrow 2,y}(\pi),$ $\displaystyle
R_{4}=R^{p}_{1\leftrightarrow 4,y}(\pi/2)R^{p}_{3\leftrightarrow
4,y}(\pi)R^{w}_{1\leftrightarrow 4,y}(\pi/2)R^{w}_{3\leftrightarrow
4,y}(\pi),$ $\displaystyle R_{5}=R^{p}_{2\leftrightarrow
3,y}(\theta^{\prime})R^{w}_{2\leftrightarrow 3,y}(\theta^{\prime}),$

are single-ion operations, where

$\theta^{\prime}\approx
0=2\textrm{arctan}[\frac{\sqrt{2}\cos(\frac{\theta_{0}}{2})+\sqrt{2}\sin(\frac{\theta_{0}}{2})}{\sqrt{2}\cos(\frac{\theta_{0}}{2})-\sqrt{2}\sin(\frac{\theta_{0}}{2})}],$
(17)

and could be realized by our method with single-ion-addressing laser beams.
Entangling gates with arbitrary spin operators between arbitrary qubits could
be similarly obtained. If the goal is to realize global entangling gates
among, e.g., the four qubits in two ions, then only global control fields
implemented simultaneously on both ions are required. It is thus promising to
increase the number of spins when simulating Ising models, e.g., the
$300$-spin system simulated in the recent work [68] could be readily upgraded
to a $600$-spin system.

One could also apply hybrid scheme that uses both microwaves and lasers. For
example, to apply entangling gates between qubits in ion $p,w$, one could
first coherently transfer the states of these two ions to metastable levels
(e.g., 2F7/2-subspace [18, 69, 17]), and apply the two-qubit gate via (14).
While $\tilde{U}_{xx}(\tau)$ is still achieved by lasers, $R_{1-5}$ could now
be achieved by microwaves: as the resonance frequencies in 2F7/2 and 2S1/2
subspaces are different, these operations will not affect other ions. At last,
one transfer the states back to 2S1/2 subspace. These transfer operations only
require $\pi$ rotations and are relatively easy to implement.

###  V.3 Readout

If only a small number of ions is of interest, one could apply the method in
sec. III and obtain the full density matrix through quantum state tomography,
as long as the efficiency is not a concern. In general, in an
$n$-ion-$2n$-qubit processor, the diagonal elements of the density matrix
could be obtained by the same method in qudit processors [33]: coherently
transfer the states to a metastable manifold (e.g., 2D5/2 or 2F7/2-subspace
[18, 69, 17]), and sequentially transfer them back to the ground state
manifold and readout. These operations are applied simultaneously to all ions.
In some cases, individual qubit measurement within an ion ($p$) is required,
e.g., to implement mid-circuit error correction through measuring spectator
qubits [70]. One must measure the corresponding qubit without destroying the
other one. This could be achieved by introducing an ancilla ion, applying
motion-sensitive laser fields to ion $p$, and readout the qubit through
measuring the phonon state of the ancilla ion [17]. Alternatively, one could
apply a two-qubit SWAP gate that transfers the qubit state to the ancilla ion,
and then measure the ancilla ion. Note interestingly that different spin
states generate different motional states in a gradient magnetic field,
moreover, this effect is significantly different for electron-qubit states and
nuclear-qubit states ($|\gamma_{e}|\gg|\gamma_{n}|$). This provides the
possibility to design more efficient readout protocols that utilize not only
the “brightness” of an ion, but also its spatial position to distinguish
different qubit states within an ion.

##  VI Conclusion

In conclusion, we construct a scalable $n$-ion-$2n$-qubit quantum processor,
where the four energy levels of a single ion is utilized to encode two qubits.
We experimentally realize a $1$-ion-$2$-qubit universal processor, where the
valence electron spin and nuclear spin of a single 171Yb+ ion constitute the
two-qubit system. Process fidelities of single-qubit and two-qubit gates are
around $98\%$. The Grover’s algorithm is further implemented with a successful
rate exceeding $99\%$. Higher fidelities could be achieved by more
sophisticated control sequences and magnetic shielding (which also brings
benefits to coherence time). The scaling-up scheme is straightforward, and we
give explicit protocols based on both laser-less and laser-based frameworks.
We also discuss the readout process, which could be achieved using well-known
techniques. We further discover that under suitable relative-strength between
the hyperfine interaction and interaction with quantization magnetic-field,
the large discrepancy between $\gamma_{e}$ and $\gamma_{n}$ allows more
efficient two-qubit entangling gates between different ions, and may be
utilized to design novel readout schemes.

Our work paves the way to achieving more information processing capability
($2^{2n}$ Hilbert-space) using $n$ ions, comparable to the qudit $(d=4)$-based
scheme. Compared to qudits, the two-qubit scheme could use standard, binary
quantum algorithms without modification, including qubit quantum error
correction [17]. Besides, the replacement of some inter-atomic gates by intra-
atomic gates in the two-qubit scheme could increase the fidelity of some
quantum circuits. On the other hand, the similarities between these two
schemes allow easy adaptation of control protocols. The two qubits could also
be constructed by any convenient four levels, e.g., the four clock states in
2S1/2 and 2F7/2. The scheme with electron/nuclear spin has clear two-qubit
entanglement structure, and provides potential efficiency and novelty in
certain tasks such as simulating lattice gauge theories [38] and preparing
thermofield double states [39, 40].

## Acknowledgments

This work is supported by the Key-Area Research and Development Program of
Guangdong Province under Grant No.2019B030330001, the National Natural Science
Foundation of China under Grant No.11774436, No.11974434 and No.12074439,
Natural Science Foundation of Guangdong Province under Grant 2020A1515011159,
Science and Technology Program of Guangzhou, China 202102080380, the Shenzhen
Science and Technology Program under Grant No.2021Szvup172 and
JCYJ20220818102003006. Le Luo acknowledges the support from Guangdong Province
Youth Talent Program under Grant No.2017GC010656. Ji Bian acknowledges the
support from the China Postdoctoral Science Foundation under Grant
No.2021M703768.

## Appendix A: Universal quantum control in the 1-ion-2-qubit processor

Here we derive $H_{c}$ in (6) under microwave control fields. Applying a
general near-resonant microwave magnetic field
$\sum_{k}\mathbf{B}^{k}(t)\cos[\omega_{k}(t)t+\phi_{k}(t)]$ to the ion, we get
the Hamiltonian in the lab frame

$H^{\prime}_{l}=H_{0}-\sum_{k}\sum_{j=x,y,z}B^{k}_{j}\cos(\omega_{k}t+\phi_{k})(\gamma_{1}I_{1,j}+\gamma_{2}I_{2,j}),$
(18)

where $B^{k}_{x,y,z}(t)\cos(\omega_{k}t+\phi_{k})$ are components along
$x,y,z$ axis. Assume the following parameter regime which describes the actual
experiment: $|\omega_{1,2,3}-(E_{1,2,4}-E_{3})|\ll E_{1}-E_{2}$ (and
$E_{2}-E_{4}$), $|\gamma_{1,2}B^{k}_{x,y,z}|\ll E_{1}-E_{2}$ (and
$E_{2}-E_{4}$), $\omega_{1,2,3}\gg$ any combination of
$|\gamma_{1,2}B^{k}_{x,y,z}|$, and $B^{k}_{x,y,z}(t)$ varies slowly compared
to $\omega_{1,2,3}$. Set $\omega_{1}=E_{1}-E_{3}-\tilde{\delta}_{1}(t)$,
$\omega_{2}=E_{2}-E_{3}-\tilde{\delta}_{2}(t)$,
$\omega_{3}=E_{4}-E_{3}-\tilde{\delta}_{4}(t)$, where
$\tilde{\delta}_{1,2,4}(t)=1/t\int_{0}^{t}\delta_{1,2,4}(s)ds$. Then
transforming to the rotating frame $r^{\prime}$ with respect to
$H^{\prime}_{r^{\prime}}(t)=(\delta_{1}(t)-E_{1})|1\rangle\langle
1|+(\delta_{2}(t)-E_{2})|2\rangle\langle 2|-E_{3}|3\rangle\langle
3|+(\delta_{4}(t)-E_{4})|4\rangle\langle 4|$ and apply the RWA, we get

$\displaystyle H_{c}$
$\displaystyle=R_{r^{\prime}}H^{\prime}_{l}R^{\dagger}_{r^{\prime}}+i\dot{R}_{r^{\prime}}R^{\dagger}_{r^{\prime}}$
(19) $\displaystyle\approx\frac{\delta_{1}(t)}{2}|1\rangle\langle
1|+\frac{\delta_{2}(t)}{2}|2\rangle\langle
2|+\frac{\delta_{4}(t)}{2}|4\rangle\langle 4|$
$\displaystyle+\frac{1}{2}(B^{1}_{x}+iB^{1}_{y})(\gamma_{1}\cos\frac{\theta_{0}}{2}+\gamma_{2}\sin\frac{\theta_{0}}{2})e^{i\phi_{1}}|3\rangle\langle
1|$
$\displaystyle-\frac{1}{2}B^{2}_{z}\sin\frac{\theta_{0}}{2}\cos\frac{\theta_{0}}{2}(\gamma_{2}-\gamma_{1})e^{i\phi_{2}}|3\rangle\langle
2|$
$\displaystyle+\frac{1}{2}(B^{3}_{x}-iB^{3}_{y})(\gamma_{1}\sin\frac{\theta_{0}}{2}+\gamma_{2}\cos\frac{\theta_{0}}{2})e^{i\phi_{3}}|3\rangle\langle
4|$ $\displaystyle+\textrm{H.c.},$

where
$R_{r^{\prime}}(t)=\textrm{exp}(-i\int_{0}^{t}H^{\prime}_{r^{\prime}}(s)ds)$.
Compared to (6), we could identify $c_{31,32,34}(t)$, $\delta_{1,2,4}(t)$, and
implement them through modulating the amplitudes, frequencies and phases of
the microwave field. It is also clear that the component of the microwave
field parallel (perpendicular) to the quantization axis drives the
$|3\rangle\leftrightarrow|2\rangle$ ($|3\rangle\leftrightarrow|1,4\rangle$)
transition.

$R_{r^{\prime}}$ is itself an entangling operation on electron spin and
nuclear spin, so the degree of entanglement (e.g., the entanglement of
formation [71]) between these two spins in $r^{\prime}$ is different from that
in the lab frame $l$. Nevertheless, the entanglement in $r^{\prime}$ is
physically well defined, as one could straightforwardly transfer from
$r^{\prime}$ to $l$ and obtain the “true degree of entanglement”. If
necessary, it is also straightforward to implement an entangling operation
$U_{l}$ in $l$: one could just search for pulses that implement
$R_{r^{\prime}}U_{l}$ in $r^{\prime}$ by GRAPE.

## Appendix B: Characterization of the two-qubit quantum processor

The ion is confined near the center of the trap, which consists of four blade
electrodes. The ion is trapped through the application of radio frequency (RF)
and direct current (DC) voltages. The quantization magnetic field
($B_{0}\approx 6$ Gs) is provided by the current in Helmholtz coils and is
oriented along the Z-axis. It is further stabilized by a PI controller,
ensuring stable Zeeman levels. The microwave signals ($\approx 2\pi\times
12.6$ GHz) utilized to drive the three transitions
$|3\rangle\leftrightarrow|1,2,4\rangle$ are generated by mixing a standard RF
source with an arbitrary waveform generator (AWG) (see fig. 7(a)). Notably,
the signal from the RF source is divided into three paths by a power divider
and subsequently mixed with the three paths of the AWG. Each of the three
resulting microwaves is then amplified individually before being combined into
a single path. This generates large Rabi frequencies while avoiding the
creation of high-order harmonics. Finally, the combined signal is directed
into the trap via the microwave horn. The calibration of the microwave
amplitudes versus Rabi frequencies for the transitions
$|3\rangle\leftrightarrow|1,2,4\rangle$ is illustrated in fig. 7(b). The
coherence time $T^{*}_{2}$ of the relevant transitions measured by Ramsey
sequences are listed in table 2. Using triggering to AC-line techniques [45],
the coherence time could be significantly increased, as listed in table 2.
Fig.7 (c) depicts our experimental sequence. Initially, the ion is cooled by
Doppler-cooling beams. Subsequently, the state is initialized to $|3\rangle$
by optical pumping. Next, the system is manipulated by the three-path
microwave. Finally, QST is performed to obtain the full density matrix of the
final state.

Figure 7: Experimental setup of the two-qubit quantum processor, with purple representing $|3\rangle\leftrightarrow|2\rangle$, blue representing $|3\rangle\leftrightarrow|1\rangle$, and red representing $|3\rangle\leftrightarrow|4\rangle$. (a) Microwave circuits utilized in the experiment. (b) Calibration curves of the AWG amplitudes versus Rabi frequencies. (c) The timing sequence of the experiment. Table 2: Coherence time without (first row) and with (second row) line triggering. $|2\rangle\leftrightarrow|3\rangle$ | $|1\rangle\leftrightarrow|3\rangle$ | $|4\rangle\leftrightarrow|3\rangle$ | $|1\rangle\leftrightarrow|4\rangle$
---|---|---|---
$20$ ms |  $500$ $\mu$s |  $500$ $\mu$s |  $200$ $\mu$s
$20$ ms |  $7$ ms |  $7$ ms |  $3$ ms

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