# Transition slow-down by Rydberg interaction of neutral atoms and a fast
controlled-NOT quantum gate

Xiao-Feng Shi  School of Physics and Optoelectronic Engineering, Xidian
University, Xi’an 710071, China

###### Abstract

Exploring controllable interactions lies at the heart of quantum science.
Neutral Rydberg atoms provide a versatile route toward flexible interactions
between single quanta. Previous efforts mainly focused on the excitation
annihilation (EA) effect of the Rydberg blockade due to its robustness against
interaction fluctuation. We study another effect of the Rydberg blockade,
namely, the transition slow-down (TSD). In TSD, a ground-Rydberg cycling in
one atom slows down a Rydberg-involved state transition of a nearby atom,
which is in contrast to EA that annihilates a presumed state transition. TSD
can lead to an accurate controlled-NOT (CNOT) gate with a sub-$\mu$s duration
about $2\pi/\Omega+\epsilon$ by two pulses, where $\epsilon$ is a negligible
transient time to implement a phase change in the pulse and $\Omega$ is the
Rydberg Rabi frequency. The speedy and accurate TSD-based CNOT makes neutral
atoms comparable (superior) to superconducting (ion-trap) systems.

##  I introduction

There are exciting advances in Rydberg atom quantum science recently Jaksch
_et al._ (2000); Lukin  _et al._ (2001); Saffman  _et al._ (2010); Saffman
(2016); Weiss and Saffman (2017); Firstenberg  _et al._ (2016); Adams  _et
al._ (2020); Browaeys and Lahaye (2020) because of the feasibility to
coherently and rapidly switch on and off the strong dipole-dipole interaction.
Such interaction enables simulation of quantum many-body physics Gurian  _et
al._ (2012); Tretyakov  _et al._ (2017); De Léséleuc  _et al._ (2019); Schauß
_et al._ (2015); Labuhn  _et al._ (2016); Zeiher  _et al._ (2016, 2017);
Bernien  _et al._ (2017); de Léséleuc  _et al._ (2018); Guardado-Sanchez  _et
al._ (2018); Kim  _et al._ (2018); Keesling  _et al._ (2019); Ding  _et al._
(2020); Borish  _et al._ (2020), probing and manipulation of single photons
Dudin and Kuzmich (2012); Peyronel  _et al._ (2012); Firstenberg  _et al._
(2013); Li  _et al._ (2013); Gorniaczyk  _et al._ (2014); Baur  _et al._
(2014); Tiarks  _et al._ (2014); Li  _et al._ (2016); Busche  _et al._ (2017);
Ripka and Pfau (2018); Liang  _et al._ (2018); Li  _et al._ (2019); Thompson
_et al._ (2017), large-scale entanglement generation Ebert  _et al._ (2015);
Omran  _et al._ (2019), and quantum computation Ebert  _et al._ (2015); Omran
_et al._ (2019); Wilk  _et al._ (2010); Isenhower  _et al._ (2010); Zhang  _et
al._ (2010); Maller  _et al._ (2015); Zeng  _et al._ (2017); Levine  _et al._
(2019); Graham  _et al._ (2019); Madjarov  _et al._ (2020); Jau  _et al._
(2016); Levine  _et al._ (2018); Picken  _et al._ (2019); Tiarks  _et al._
(2019); Jo  _et al._ (2020). To date, however, most effort focused on the
effect of excitation annihilation (EA) proposed in Jaksch  _et al._ (2000);
Lukin  _et al._ (2001) and reviewed in Saffman  _et al._ (2010); Saffman
(2016); Weiss and Saffman (2017); Firstenberg  _et al._ (2016); Adams  _et
al._ (2020); Browaeys and Lahaye (2020), although there are other categories
as summarized in Table 1. Belonging to the Rydberg blockade regime, EA
involves single-atom Rydberg excitation and hence is not sensitive to the
fluctuation of interaction. Besides EA, one can also excite both qubits to
Rydberg states Jaksch  _et al._ (2000), or explore the antiblockade regime
Ates  _et al._ (2007); Amthor  _et al._ (2010), or use the resonant dipole-
dipole flip Thompson  _et al._ (2017); De Léséleuc  _et al._ (2019). These
latter processes, however, involve two-atom Rydberg excitation and are
sensitive to the fluctuation of qubit separation that can substantially
reduces the fidelity of a quantum control by using them Jo  _et al._ (2020).
It is an open question whether there is another means other than EA to explore
the Rydberg blockade regime for efficient and high-fidelity quantum control.

Here, we study an unexplored transition slow-down (TSD) effect of the dipole-
dipole interaction between Rydberg atoms. When the state of one atom
oscillates back and forth between ground and Rydberg states, its Rydberg
interaction does not block a Rydberg-involved state swap in a nearby atom, but
slows it down, and the fold of slow-down is adjustable. The resulted TSD
denotes the slow-down of the state transfer of one of the two atoms although
the collective Rabi frequency is enhanced due to the many-body effect. Albeit
appeared as a slow-down, a controlled TSD on the contrary can drastically
speed up certain crucial element in a quantum computing circuit, such as the
controlled-NOT (CNOT) which is the very important two-qubit entangling gate in
the circuit model of quantum computing in both theory Nielsen and Chuang
(2000); Williams (2011); Ladd  _et al._ (2010); Shor (1997); Bremner  _et al._
(2002); Shende  _et al._ (2004) and experiment Peruzzo  _et al._ (2014);
Debnath  _et al._ (2016). TSD is a new route toward efficient, flexible, and
high-fidelity quantum control over neutral atoms because it is implemented in
the strong blockade regime which is robust to the fluctuation of interactions.

Category | Blockade ($V\gg\Omega$) | Frozen interaction
---|---|---
Feature | Robust to fluctuation of $V$ | Sensitive to fluctuation of $V$
Type of $V$ |  | Dipole-dipole; Van der Waals
---
Van der Waals | Dipole-dipole
Application | TSD | Excitation annihilation | Phase shift | Antiblockade |  | State flip
---
Theoretical proposal | Here | Jaksch  _et al._ (2000); Lukin  _et al._ (2001) | Jaksch  _et al._ (2000) | Ates  _et al._ (2007) | (Intrinsic)
Realized |  CNOT; Toffoli |  |  |  Isenhower  _et al._ (2010); Zhang  _et al._ (2010); Maller  _et al._ (2015); Zeng  _et al._ (2017); Levine  _et al._ (2019); Graham  _et al._ (2019); Tiarks  _et al._ (2019); Levine  _et al._ (2019)
---
|  |
Entanglement |  |  Ebert  _et al._ (2015); Omran  _et al._ (2019); Wilk  _et al._ (2010); Isenhower  _et al._ (2010); Zhang  _et al._ (2010); Maller  _et al._ (2015); Jau  _et al._ (2016); Zeng  _et al._ (2017); Levine  _et al._ (2018); Picken  _et al._ (2019); Levine  _et al._ (2019); Graham  _et al._ (2019); Madjarov  _et al._ (2020), for photons:Tiarks  _et al._ (2019)
---
Jo  _et al._ (2020) |  |
| Many-body; Optics
---
|  Schauß  _et al._ (2015); Labuhn  _et al._ (2016); Zeiher  _et al._ (2016,
2017); Bernien  _et al._ (2017); de Léséleuc  _et al._ (2018); Guardado-
Sanchez  _et al._ (2018); Kim  _et al._ (2018); Keesling  _et al._ (2019);
Ding  _et al._ (2020); Borish  _et al._ (2020); Dudin and Kuzmich (2012);
Peyronel  _et al._ (2012); Firstenberg  _et al._ (2013); Li  _et al._ (2013);
Gorniaczyk  _et al._ (2014); Baur  _et al._ (2014); Tiarks  _et al._ (2014);
Li  _et al._ (2016); Busche  _et al._ (2017); Ripka and Pfau (2018); Liang
_et al._ (2018); Li  _et al._ (2019)
---
|  | Ates  _et al._ (2007); Amthor  _et al._ (2010)
---
|  Gurian  _et al._ (2012); Tretyakov  _et al._ (2017); De Léséleuc  _et al._
(2019); Thompson  _et al._ (2017)
---
Table 1:  Summary of quantum science and technology based on two-body Rydberg
interaction of neutral atoms (references are not complete but representative).
There are in general two categories, the strong blockade regime and the frozen
interaction regime, where the former was widely studied because of its
robustness to fluctuation of interactions.

Though Rydberg atoms have kindled the flame for the ambition to large-scale
quantum computing Jaksch  _et al._ (2000); Lukin  _et al._ (2001); Saffman
_et al._ (2010); Saffman (2016); Weiss and Saffman (2017), and recent
experiments demonstrated remarkable advances Wilk  _et al._ (2010); Isenhower
_et al._ (2010); Zhang  _et al._ (2010); Maller  _et al._ (2015); Jau  _et
al._ (2016); Zeng  _et al._ (2017); Levine  _et al._ (2018); Picken  _et al._
(2019); Levine  _et al._ (2019); Graham  _et al._ (2019); Madjarov  _et al._
(2020); Tiarks  _et al._ (2019); Jo  _et al._ (2020), further progress toward
neutral-atom quantum computing is hindered by the difficulty to prepare a fast
and accurate CNOT. This is partly because each of those CNOT gates was
realized via combining an EA-based controlled-Z ($C_{Z}$) and a series of
single-qubit gates Isenhower  _et al._ (2010); Zhang  _et al._ (2010); Maller
_et al._ (2015); Zeng  _et al._ (2017); Graham  _et al._ (2019); Levine  _et
al._ (2019), leading to CNOT durations dominated by single-qubit operations
(e.g., over $4~{}\mu$s in Levine  _et al._ (2019); Graham  _et al._ (2019)).
In contrast, an TSD-based CNOT has a duration about $2\pi/\Omega+\epsilon$ by
only two Rydberg pulses, i,e., needs no single-qubit rotations, where
$\epsilon$ is a transient moment to implement a phase change in the pulses and
$\Omega$ the Rydberg Rabi frequency. Using values of $\Omega$ and $\epsilon$
from Graham  _et al._ (2019); Levine  _et al._ (2019), the TSD-based CNOT
(duration $\sim 0.3~{}\mu$s) would be orders of magnitude faster than those in
Graham  _et al._ (2019); Levine  _et al._ (2019), which means that a neutral-
atom CNOT can be much faster than the CNOT (or ground Bell-state gate) by
trapped ions Cirac and Zoller (1995); Sørensen and Mølmer (1999); Ballance
_et al._ (2015, 2016); Gaebler  _et al._ (2016) (notice that the fast ion-trap
gates in Schäfer  _et al._ (2018); Zhang  _et al._ (2020) are phase gates).
Although still inferior to superconducting circuits You and Nori (2005);
Devoret and Schoelkopf (2013) where CNOT gate times can be around $50$ ns
Barends  _et al._ (2014), the TSD-based CNOT is applicable in scalable
neutral-atom platforms that are ideal for long-lived storage of quantum
information in room temperatures.

##  II A two-state case

The simplest model of TSD consists of two nearby atoms, each pumped by a laser
pulse that induces a transition between a ground state $|1\rangle$ and a
Rydberg state $|r\rangle$. The Rabi frequency is $\Omega_{\text{c(t)}}$ for
the control (target) qubit, and the dipole-dipole interaction $V$ in
$|rr\rangle$ is assumed large compared to $\Omega_{\text{c(t)}}$ so that
$|rr\rangle$ is not populated. For Rydberg interaction of the van der Waals
type, $V$ is limited because in this regime the native dipole-dipole
interaction should be much smaller than the energy gaps between nearby two-
atom Rydberg states Walker and Saffman (2008), and thus the qubit spacing
should be large enough. On the other hand, a direct dipole-dipole interaction
can be huge for high-lying Rydberg states although it is no longer a pure
energy shift. In this sense, the residual blockade error of the order
$\sim\Omega_{\text{c(t)}}^{2}/V^{2}$ Saffman and Walker (2005) can be
negligible in the strong dipole-dipole interaction regime as long as the two-
atom spacing is beyond the LeRoy radius.

Figure 1 shows a contrast between EA and a two-state TSD, where the pulse sent
to the control atom is applied during $t\in[t_{0},~{}t_{0}+t_{1})$, and that
to the target atom is during $t\in[0,~{}2t_{1})$ with
$t_{0}<t_{1}\equiv\pi/\Omega_{\text{t}}$. Starting from an initial two-atom
state $|11\rangle$, the wavefunction at $t=t_{0}$ becomes

$\displaystyle|\psi(t_{0})\rangle$ $\displaystyle=$
$\displaystyle\cos(\Omega_{\text{t}}t_{0}/2)|11\rangle-i\sin(\Omega_{\text{t}}t_{0}/2)|1r\rangle.$
(1)

During $t\in[t_{0},~{}t_{0}+t_{1})$, the system Hamiltonian is

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle[(\Omega_{\text{c}}|r1\rangle+\Omega_{\text{t}}|1r\rangle)\langle
11|/2+\text{H.c.}]+\hat{H^{\prime}},$ (2)

where $\hat{H^{\prime}}$ includes excitation between
$\\{|1r\rangle,~{}|r1\rangle\\}$ and $|rr\rangle$ and the dipole-dipole flip
from $|rr\rangle$. Focusing on the strong interaction regime,
$\hat{H^{\prime}}$ can be discarded because $|rr\rangle$ is not coupled Jaksch
_et al._ (2000) and hence $\hat{H}=\bar{\Omega}|\mathbb{R}\rangle\langle
11|/2+\text{H.c.}$, where
$\bar{\Omega}\equiv\sqrt{\Omega_{\text{c}}^{2}+\Omega_{\text{t}}^{2}}$ and
$|\mathbb{R}\rangle\equiv(\Omega_{\text{c}}|r1\rangle+\Omega_{\text{t}}|1r\rangle)/\bar{\Omega}$.
When $\alpha\equiv\Omega_{\text{c}}/\Omega_{\text{t}}=\sqrt{15}$, we have
$\bar{\Omega}t_{1}=4\pi$, so that the wavefunction at $t=t_{0}+t_{1}$, given
by $e^{-it_{1}\hat{H}}|\psi(t_{0})\rangle$, is equal to Eq. (1), and it is
like that nothing happens to the target qubit upon the completion of the drive
in the control qubit. Then, because the pulse for the control qubit ends at
$t=t_{0}+t_{1}$, the continuous pumping on the target qubit drives the state
to $|1r\rangle$ at $t=2t_{1}$, i.e., a $2\pi$ pulse, instead of a $\pi$ pulse,
completes the transition $|11\rangle\rightarrow|1r\rangle$, which corresponds
to a 2-fold slow-down.

Figure 1: Comparison between excitation annihilation (EA) and transition slow-
down (TSD) in two states. Black and blue (gray) circles represent ground and
Rydberg states, respectively. Hollow circles denote Rydberg states not
populated. (a) The first and second pulses in EA are sent to the control and
target qubits, respectively, both for the transition between a ground state
$|1\rangle$ and a Rydberg state $|r\rangle$. (b) State evolution starting from
the two-qubit state $|11\rangle$ in EA. The blockade effect is in the second
pulse, where the transition $|r1\rangle\leftrightarrow|rr\rangle$ is
annihilated when $\Omega_{\text{t}}\ll V$. (c) In a two-state TSD, one pulse
is applied to the target qubit with duration $2t_{\pi}$, within which another
pulse is applied to the control qubit with duration $t_{\pi}$, where
$t_{\pi}=\pi/\Omega_{\text{t}}$. The delay between the pulses for the two
qubits shall be smaller than $t_{\pi}$. (d) The parameter
$\alpha\equiv\Omega_{\text{c}}/\Omega_{\text{t}}$ depends on the desired
extent of TSD. With $\alpha=\sqrt{15}$, the Rydberg pumping in the target
qubit, $|11\rangle\rightarrow|1r\rangle$, requires a $2\pi$ pulse, i.e., twice
of that when the control qubit is not pumped.  Figure 2: A three-state TSD and
a fast CNOT by two pulses. (a) A three-state TSD is realized by sending to
each qubit two pulses, each with equal duration $\pi/\Omega_{\text{c}}$. The
Rabi frequencies are $\Omega_{\text{c}}~{}(\Omega_{\text{c}})$ and
$\Omega_{\text{t}}~{}(-\Omega_{\text{t}})$ for the control and target qubits
in the first (second) pulse, where
$\Omega_{\text{t}}=\sqrt{1.5}\Omega_{\text{c}}$. (b) Relevant transitions for
$|10\rangle\rightarrow|11\rangle$. Transitions involving $|00\rangle$ and
$|01\rangle$ are not shown because they do not involve dipole-dipole
interaction. That the arrows point left or right does not mean change of
angular momentum but only for clarifying the protocol. (c) The input state
$|00\rangle$ transitions back to itself because of the spin echo by
$\Omega_{\text{t}}\rightarrow-\Omega_{\text{t}}$. (d) With the same pumping,
the input state $|10\rangle$ transitions to $|11\rangle$ at the end of the
pulse sequence. The time for implementing the phase change between the two
pulses is not shown for brevity (see text). The duration of the two pulses is
$2\pi/\Omega_{\text{c}}=\sqrt{6}\pi/\Omega_{\text{t}}$, corresponding to a
$\sqrt{3}$-fold slow-down because the state swap
$|0\rangle\rightarrow|r\rangle\rightarrow|1\rangle$ in the target qubit can
complete by a pulse duration $\sqrt{2}\pi/\Omega_{\text{t}}$ if no TSD is
used.

##  III A fast CNOT

To show the strength of TSD in quantum information, we would like to consider
TSD in three states. This is because the two-state TSD ends up in a Rydberg
state which is not stable, while quantum information shall be encoded with two
qubit states $|0\rangle$ and $|1\rangle$ in the more stable ground manifold.
For frequently used rubidium and cesium atoms, $|0\rangle$ and $|1\rangle$ can
be chosen from the two hyperfine-split ground levels with a frequency
difference of several gigahertz.

Consider a ground-Rydberg-ground transition chain
$|0\rangle\leftrightarrow|r\rangle\leftrightarrow|1\rangle$, i.e., a state
swap between the two long-lived qubit states via a metastable Rydberg state
$|r\rangle$. The three-state TSD is implemented by pumping the control and
target qubits with respective Rabi frequencies $\Omega_{\text{c}}$ and
$\Omega_{\text{t}}$ for the same duration, with transition
$|1\rangle\xleftrightarrow{\Omega_{\text{c}}}|r\rangle$ for the control qubit
and
$|0\rangle\xleftrightarrow{\Omega_{\text{t}}}|r\rangle\xleftrightarrow{\Omega_{\text{t}}}|1\rangle$
for the target qubit. This requires a setup capable of pumping both qubit
states to a common Rydberg state in the target qubit, which is feasible as
experimentally demonstrated for two-atom entanglement about a decade ago Wilk
_et al._ (2010) and for $W$-state preparation in atom ensembles later Ebert
_et al._ (2015). Because $|0\rangle$ is not pumped in the control qubit, the
Hamiltonian is
$\hat{\mathcal{H}}_{c0}(\Omega_{\text{t}})=\Omega_{\text{t}}(|00\rangle+|01\rangle)\langle
0r|/2+$H.c. for the input states $|00\rangle$ and $|01\rangle$, where the
subscript $c0$ ($c1$) denotes that the Hamiltonian applies when the input
state for the control qubit is $|0(1)\rangle$. For the remaining input states,
we consider the ordered basis
$\\{|1r\rangle,|r1\rangle,|r0\rangle,|11\rangle,|10\rangle\\}$ for the
following Hamiltonian

$\displaystyle\hat{\mathcal{H}}_{c1}(\Omega_{\text{c}},\Omega_{\text{t}})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{ccccc}0&0&0&\Omega_{\text{t}}&\Omega_{\text{t}}\\\
0&0&0&\Omega_{\text{c}}&0\\\ 0&0&0&0&\Omega_{\text{c}}\\\
\Omega_{\text{t}}&\Omega_{\text{c}}&0&0&0\\\
\Omega_{\text{t}}&0&\Omega_{\text{c}}&0&0\end{array}\right),$ (8)

where we ignore the two-atom Rydberg state $|rr\rangle$ with reasons shown
below Eq. (2). By diagonalizing Eq. (8) one has

$\displaystyle|10\rangle$ $\displaystyle=$
$\displaystyle(|\mathscr{R}_{4}\rangle-|\mathscr{R}_{3}\rangle+|\mathscr{R}_{2}\rangle-|\mathscr{R}_{1}\rangle)/2,$
$\displaystyle|11\rangle$ $\displaystyle=$
$\displaystyle(|\mathscr{R}_{4}\rangle-|\mathscr{R}_{3}\rangle-|\mathscr{R}_{2}\rangle+|\mathscr{R}_{1}\rangle)/2,$
(9)

where $|\mathscr{R}_{k}\rangle$ with $k=1-4$ are four eigenvectors of Eq. (8),
with eigenvalues
$(\Omega_{\text{c}},~{}-\Omega_{\text{c}},~{}\overline{\Omega},-\overline{\Omega})/2$,
respectively [the fifth eigenvector does not enter Eq. (9)], where
$\overline{\Omega}\equiv\sqrt{\Omega_{\text{c}}^{2}+2\Omega_{\text{t}}^{2}}$
and

$\displaystyle|\mathscr{R}_{1,2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}[|r1\rangle-|r0\rangle\pm(|11\rangle-|10\rangle)],$
(10) $\displaystyle|\mathscr{R}_{3,4}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2\overline{\Omega}}[\Omega_{\text{c}}(|r1\rangle+|r0\rangle)\pm\overline{\Omega}(|11\rangle+|10\rangle)+2\Omega_{\text{t}}|1r\rangle].$

We proceed to describe an accurate and exceedingly fast CNOT by a spin-echo
assisted TSD. The sequence consists of two pulses, each with duration
$\pi/\Omega_{\text{c}}$ and condition
$|\Omega_{\text{t}}|/\Omega_{\text{c}}=\sqrt{6}/2$, shown in Fig. 2(a). A
$\pi$ phase change (requiring a time $\epsilon$) is inserted between the
pulses sent to the target qubit so as to induce spin echo, where $\epsilon$
can be around $10$ ns Levine  _et al._ (2019). The spin echo suppresses the
state swap of the target qubit if the control qubit is initialized in
$|0\rangle$ Shi (2018a, 2020a); but when the control qubit is initialized in
$|1\rangle$, it is pumped to $|r\rangle$ which results in TSD in the target
qubit, i.e., the transition
$|0\rangle\rightleftharpoons|r\rangle\rightleftharpoons|1\rangle$ in the
target qubit, which can occur with a pulse duration
$\sqrt{2}\pi/\Omega_{\text{t}}$ if no TSD is used Shi (2018b), will be slowed
down by a fold of $\sqrt{3}$. The mechanism is understood in two steps. First,
during $t\in[0,~{}t_{\pi})$ with $t_{\pi}=\pi/\Omega_{\text{c}}$, the input
state $|\psi_{c0}(0)\rangle=|00\rangle$ or $|01\rangle$ evolves according to
$e^{-it\mathcal{H}_{c0}(\Omega_{\text{t}})}|\psi_{c0}\rangle$, and after a
$\pi$ phase change to the Rydberg Rabi frequency which may require a finite
transient time $\epsilon$, the state evolution becomes
$e^{-i(t-t_{\pi}-\epsilon)\mathcal{H}_{c0}(-\Omega_{\text{t}})}|\psi_{c0}(t_{\pi})\rangle$
during the second pulse, leading to
$e^{-it\mathcal{H}_{c0}(-\Omega_{\text{t}})t_{\pi}}|\psi_{c0}(t_{\pi})\rangle=e^{-it\mathcal{H}_{c0}(-\Omega_{\text{t}})t_{\pi}}e^{-it\mathcal{H}_{c0}(\Omega_{\text{t}})t_{\pi}}|\psi_{c0}(0)\rangle=|\psi_{c0}(0)\rangle$
at the end of the sequence. Second, according to Eq. (10), the first pulse
during $t\in[0,~{}t_{\pi})$ evolves the input state
$|\psi_{c1}(0)\rangle=|10\rangle$ according to

$\displaystyle\frac{1}{2}(e^{\frac{it\overline{\Omega}}{2}}|\mathscr{R}_{4}\rangle-e^{\frac{-it\overline{\Omega}}{2}}|\mathscr{R}_{3}\rangle+e^{\frac{it\Omega_{\text{c}}}{2}}|\mathscr{R}_{2}\rangle-e^{-\frac{it\Omega_{\text{c}}}{2}}|\mathscr{R}_{1}\rangle),$
(11)

which becomes

$\displaystyle|\psi_{c1}(t_{\pi})\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}(i|\mathscr{R}_{1}\rangle+i|\mathscr{R}_{2}\rangle+|\mathscr{R}_{3}\rangle-|\mathscr{R}_{4}\rangle)$
(12)

at $t=t_{\pi}$ because
$(t_{\pi}\Omega_{\text{c}},~{}t_{\pi}\overline{\Omega})=(\pi,~{}2\pi)$. In Eq.
(12), the eigenvectors $|\mathscr{R}_{j}\rangle$, $j=1-4$, are defined in Eq.
(10) with Rabi frequencies $(\Omega_{\text{c}},~{}\Omega_{\text{t}})$ during
$t\in[0,~{}t_{\pi})$. During $t\in t_{\pi}+\epsilon+[0,~{}t_{\pi})$, the
Hamiltonian has Rabi frequencies $(\Omega_{\text{c}},~{}-\Omega_{\text{t}})$
with the eigenvectors $|\mathscr{R}_{1,2}\rangle$ the same as in Eq. (10), but
the expressions for $|\mathscr{R}_{3,4}\rangle$ become

$\displaystyle|\mathscr{R}_{3,4}^{\prime}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2\overline{\Omega}}[\Omega_{\text{c}}(|r1\rangle+|r0\rangle)\pm\overline{\Omega}(|11\rangle+|10\rangle)-2\Omega_{\text{t}}|1r\rangle].$

Using Eqs. (10) and (LABEL:eq08) we cast Eq. (12) into

$\displaystyle|\psi_{c1}(t_{\pi})\rangle=e^{-it_{\pi}\mathcal{H}_{c1}(\Omega_{\text{c}},-\Omega_{\text{t}})}|\psi_{c1}(0)\rangle,$
(14)

i.e., the state at $t=t_{\pi}$ is like if Rabi frequencies
$(\Omega_{\text{c}},~{}-\Omega_{\text{t}})$ were used during the first pulse.
Equation (14) is the key feature enabling a fast CNOT. After the second pulse,
the state at $t=2t_{\pi}+\epsilon$ becomes
$e^{-it_{\pi}\mathcal{H}_{c1}(\Omega_{\text{c}},-\Omega_{\text{t}})}|\psi_{c1}(t_{\pi})\rangle$
that can be rewritten as
$e^{-2it_{\pi}\mathcal{H}_{c1}(\Omega_{\text{c}},-\Omega_{\text{t}})}|\psi_{c1}(0)\rangle$
thanks to Eq. (14), which further reduces to $|11\rangle$ according to Eqs.
(9) and (11). A similar analysis shows that if the initial state is
$|11\rangle$, it maps to $|10\rangle$ upon the completion of the pulse. So the
following map is realized

$\displaystyle\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}\rightarrow\\{|00\rangle,|01\rangle,|11\rangle,|10\rangle\\}$
(15)

which is the standard CNOT. A numerical simulation of the population evolution
by $\mathcal{H}_{c0}$ and $\mathcal{H}_{c1}$ in Eq. (8) is shown in Figs. 2(c)
and 2(d) for the input states $|00\rangle$ and $|10\rangle$, respectively,
where the transient time $\epsilon$ is ignored for brevity.

High fidelity is possible with the TSD-based CNOT. To estimate the intrinsic
fidelity limited by Rydberg-state decay and Doppler dephasing, we consider
atomic levels and Rydberg laser Rabi frequencies in recent experiments Graham
_et al._ (2019); Levine  _et al._ (2019), and numerically found that the
fidelity of the TSD-based CNOT or Bell state would reach $99.9\%$ ($99.8\%$)
with experimentally affordable effective temperatures $T_{a}=5~{}(15)~{}\mu$K
of qubit motion Picken  _et al._ (2019); Zeng  _et al._ (2017); Graham  _et
al._ (2019). As detailed in Appendix A.2, this estimate assumes that the field
for $|0\rangle\rightarrow|r\rangle$ and that for
$|1\rangle\rightarrow|r\rangle$ in the target qubit copropagate, leading to
opposite phases in the two transitions of the chain
$|0\rangle\xrightarrow{\Omega_{\text{t}}e^{itkv_{t}}}|r\rangle\xrightarrow{\Omega_{\text{t}}e^{-itkv_{t}}}|1\rangle$,
where $k$ is the wavevector and $v_{t}$ the speed of the target qubit along
the propagation direction of light. Then, the state transfer from $|0\rangle$
to $|1\rangle$ (or the reverse) picks up two opposite phases $\pm tkv_{t}$
which partly suppresses the dephasing. Fortunately, our method is not strictly
dependent on the above dephasing-resilient configuration. For example, for the
worst case of counterpropagating fields so that the sequential state transfer
from $|0\rangle$ to $|1\rangle$ picks up a total phase $2tkv_{t}$, simulation
in Appendix A shows that fidelity over $99.6\%$ is achievable for $T_{a}\sim
15~{}\mu$K in room temperatures. The robustness against the Doppler dephasing
benefits from the avoidance of shelving the control atom on the Rydberg level
in free flight during pumping the target atom as required in an EA-based CNOT
Müller  _et al._ (2009); Shi (2018b).

The above analysis assumes $V/2\pi=500$ MHz corresponding to an interatomic
distance $l=3~{}\mu$m. When $V/2\pi=100$ MHz with larger $l=4.6~{}\mu$m (where
crosstalk can be negligible Graham  _et al._ (2019)), the fidelity would be
$99.8\%$ ($99.7\%$) with $T_{a}=5~{}(15)~{}\mu$K in room temperatures (see
Appendix B for details). This benefits from that our CNOT is not sensitive to
the change of interaction $V$ in the strong blockade regime. Finally, with
fluctuation (of relative Gaussian width $\sigma$) of the two Rabi frequencies
for the two transitions $|0(1)\rangle\leftrightarrow|r\rangle$ in the target
qubit, very small extra error $\in(0.02,~{}0.4)\%$ arises for
$\sigma\in(1,~{}5)\%$ as detailed in Appendix C. So, high fidelity is
achievable since the power fluctuation of Rydberg lasers can be well
suppressed Levine  _et al._ (2018, 2019).

##  IV Discussion and conclusions

The CNOT in Eq. (15) is implemented within a short time
$2\pi/\Omega_{\text{c}}=\sqrt{6}\pi/\Omega_{\text{t}}$ plus a transient moment
$\epsilon$ for a phase change between the two pulses, where $\epsilon$ can be
negligible as in Levine  _et al._ (2019). Although a phase twist is used in
both Levine  _et al._ (2019) and here, our TSD method is drastically different
in physics from the method of Levine  _et al._ (2019) as discussed in Appendix
D. Note that Eq. (15) does not depend on any single-qubit gates if one starts
from Eq. (8). If $s$ or $d$-orbital Rydberg states are used, two-photon
transitions can be applied which leads to ac Stark shifts for the ground and
Rydberg states. As shown in Appendix E, these shifts are annulled in Eq. (8)
which is achievable by choosing appropriate ratio between the detuning at the
intermediate state and the magnitudes of the fields for the lower and upper
transitions Maller  _et al._ (2015).

Our CNOT gate duration can be $\lesssim 0.3~{}\mu$s with Rydberg Rabi
frequencies ($2\pi\times(3.5\sim 4.6)$ MHz) like those realized in Graham  _et
al._ (2019); Levine  _et al._ (2019) that showed CNOT gate durations orders of
magnitude longer than here. Previous CNOT gates Zhang  _et al._ (2010); Maller
_et al._ (2015); Zeng  _et al._ (2017); Graham  _et al._ (2019); Levine  _et
al._ (2019); Isenhower  _et al._ (2010) required two or more single-qubit
rotations to convert $C_{Z}$ to CNOT, which was usually achieved with two-
frequency Raman light Isenhower  _et al._ (2010); Zhang  _et al._ (2010); Zeng
_et al._ (2017); Levine  _et al._ (2019), or with microwave driving assisted
by Stark shift of laser fields Graham  _et al._ (2019); Maller  _et al._
(2015). High-fidelity realizations of such gates came with long gate durations
Xia  _et al._ (2015); Wang  _et al._ (2016); Graham  _et al._ (2019); Levine
_et al._ (2019), although they can also be carried out rapidly in proof-of-
principle experiments Yavuz  _et al._ (2006); Jones  _et al._ (2007) where
high fidelity is not concerned. The atom traps have finite lifetimes Saffman
(2016) and a faster protocol can lead to more CNOT cycles before the qubit
arrays should be reloaded. Because a practical quantum processor tackling real
problems uses a series of quantum gates including CNOT Nielsen and Chuang
(2000); Williams (2011); Ladd  _et al._ (2010); Shor (1997); Bremner  _et al._
(2002); Shende  _et al._ (2004); Peruzzo  _et al._ (2014); Debnath  _et al._
(2016), certain computation tasks can be executed only with fast enough CNOT
in each cycle of laser cooling and loading of atom array. The studied speedup
of the minimal functional quantum circuit shows a possibility to confront the
problems of finite lifetime of atom traps which limits the capability of a
neutral-atom quantum processor. Moreover, concerning the ratio between the
coherence time and the CNOT gate time, the TSD method makes neutral-atom
systems competitive with superconducting and ion-trap systems as compared in
detail in Appendix F.

In conclusion, we have explored another effect of Rydberg blockade by dipole-
dipole interaction in neutral atoms, namely, the effect of transition slow-
down (TSD). We show that TSD can speed up a neutral-atom quantum computer by
proposing an exceedingly fast TSD-based CNOT realized by two pulses separated
by a short transient moment for changing the phase of the pulse. The exotic
TSD is an excellent character of dipole-dipole interactions other than the
well-known excitation annihilation effect that can push Rydberg-atom quantum
science to another level.

## ACKNOWLEDGMENTS

The author thanks Yan Lu for useful discussions. This work is supported by the
National Natural Science Foundation of China under Grant No. 11805146 and
Natural Science Basic Research plan in Shaanxi Province of China under Grant
No. 2020JM-189.

##  Appendix A Gate fidelity

Before analyzing the gate error, we would like to emphasize that TSD is
particularly useful for speeding up the minimal functional quantum circuit,
namely, the CNOT. The CNOT protocols by EA in previous methods involve
multiple switchings of the external control fields which brings extra
complexity to the experimental implementation, elongate the gate duration, and
introduces extra errors from the fluctuation of the control fields and from
the intrinsic atomic Doppler broadening. Previous effort for suppressing these
errors includes a $C_{Z}$ gate based on quantum interference Shi (2019);
Levine  _et al._ (2019), but going from $C_{Z}$ to the more useful CNOT (or to
create Bell states in the ground-state manifold) still needs several single-
qubit operations Graham  _et al._ (2019); Levine  _et al._ (2019); Isenhower
_et al._ (2010); Zhang  _et al._ (2010); Maller  _et al._ (2015); Zeng  _et
al._ (2017). For example, the first experimental neutral-atom CNOT needed five
or seven pulses Isenhower  _et al._ (2010), and the recent CNOT in Levine  _et
al._ (2019) used two (four) pulses in the control (target) qubit besides
several short pulses for phase compensation. The central procedure of the CNOT
in Levine  _et al._ (2019) is via combining (i) a two-qubit $C_{Z}$-like gate
by two pulses of duration about $2.7\pi/\Omega$ with a phase change inserted
between the pulses that requires an extra transient time $\epsilon$, (ii) a
short pulse to compensate an intrinsic phase to recover a $C_{Z}$ from the
$C_{Z}$-like gate, and (iii) two single-qubit rotations of duration
$\pi/\Omega_{\text{hf}}$ ($\approx 2~{}\mu$s therein) in the target qubit,
where $\Omega$ is the Rydberg laser Rabi frequency and $\Omega_{\text{hf}}$ is
the hyperfine laser Rabi frequency between the two qubit states. Page 4 of
Ref. Levine  _et al._ (2019) indicates that the $C_{Z}$-like gate needs
$0.4~{}\mu$s, and it had
$(\Omega_{\text{hf}},~{}\Omega)/(2\pi)=(0.25,~{}3.5)~{}$MHz (see page 1 and 2
of Levine  _et al._ (2019)), thus
$\epsilon=0.4~{}\mu$s$-2.732\pi/\Omega\approx 9.7$ ns; we assume such a fast
phase change time here. Figure 3(d) of Ref. Levine  _et al._ (2019) presents a
CNOT sequence with durations equal to those of two $X(\pi/2)$, one $X(\pi)$,
and the $C_{Z}$-like, i.e.,
$2\times\pi/(2\Omega_{\text{hf}})+\pi/\Omega_{\text{hf}}+2.7\pi/\Omega$, and
thus the CNOT sequence needs about $4.4~{}\mu$s therein (the actual gate
durations should be larger when accounting for gaps between pulses therein).
In contrast, the TSD-based CNOT needs a duration about
$\sqrt{6}\pi/\Omega_{\text{t}}$ which is only $0.27~{}(0.35)~{}\mu$s with
$\Omega_{\text{t}}/(2\pi)=4.6~{}(3.5)$ MHz from Ref. Graham  _et al._ (2019)
(Ref. Levine  _et al._ (2019)).

One may image that if the single-qubit rotations in Refs. Graham  _et al._
(2019); Levine  _et al._ (2019) are implemented by the transition chain
$|0\rangle\leftrightarrow|r\rangle\leftrightarrow|1\rangle$, then their gate
durations can also be small. But there will be problems in this assumption.
This is because the single-qubit rotations necessary to transform $C_{Z}$ to
CNOT are two $X(\pi/2)$ gates that transfer $\\{|0\rangle,~{}|1\rangle\\}$ to
$\\{|0\rangle-i|1\rangle,~{}-i|0\rangle+|1\rangle\\}/\sqrt{2}$ (see Fig. 5(a)
of Graham  _et al._ (2019) or Fig. 3(d) of Levine  _et al._ (2019)). But the
pumping by
$\hat{\mathbb{H}}(\Omega_{\text{t}})=\Omega_{\text{t}}(|0\rangle+|1\rangle)\langle
r|/2+$H.c can not achieve this since one can easily prove that starting from
$|\varphi\rangle=|0\rangle$, the populations in
$\\{|0\rangle,~{}|r\rangle,~{}|1\rangle\\}$ will be
$\frac{1}{2}\\{2\cos^{4}\frac{\theta}{2},\sin^{2}\theta,~{}2\sin^{4}\frac{\theta}{2}\\}$,
where $\theta=\frac{t\Omega_{\text{t}}}{\sqrt{2}}$. This means that there is
no way to use the resonant transition chain
$|0\rangle\leftrightarrow|r\rangle\leftrightarrow|1\rangle$ for the $X(\pi/2)$
rotation. On the other hand, one may also imagine a succession of a $\pi$
pulse on $|1\rangle\rightarrow|r\rangle$, a $\pi/2$ pulse on
$|r\rangle\rightarrow|0\rangle$, and a $\pi$ pulse on
$|r\rangle\rightarrow|1\rangle$ can, e.g., realize a $X(\pi/2)$ rotation.
However, the extra time to shelve an atom in Rydberg state leads to extra
Doppler dephasing, and the frequent turning on and off of Rydberg lasers can
lead to extra atom loss Maller  _et al._ (2015).

Below, we analyze gate imperfections due to the prevailing intrinsic Rydberg-
state decay and Doppler broadening. These are the dominant intrinsic errors in
gate operations Graham  _et al._ (2019) while technical issues such as laser
noise is in principle not fundamental. From here to Appendix C, we take, as an
example, 87Rb qubits and consider the intermediate level $6P_{3/2}$ for
Rydberg pumping, where the two detunings at $6P_{3/2}$ should be different for
the two transitions $|0\rangle\xleftrightarrow{\Omega_{\text{t}}}|r\rangle$
and $|1\rangle\xleftrightarrow{\Omega_{\text{t}}}|r\rangle$. For an
$s$-orbital rubidium Rydberg state with principal quantum number around $70$,
the lifetime of $|r\rangle$ is about $\tau=400~{}(150)~{}\mu$s at a
temperature of $4~{}(300)$ K by the estimate in Beterov  _et al._ (2009). When
the lower and upper fields counterpropagate along, e.g., $\mathbf{z}$, the
wavevector is $k=2\pi(1/420.3-1/1012.7)$nm-1 Shi (2020b). We assume that the
two qubits are initially located at the centers of the traps at $(0,0,x_{0})$
and $(0,0,0)$ respectively. The traps are usually turned off during Rydberg
pumping, and the free flight of the qubits leads to time-dependent Rabi
frequencies $\Omega_{\text{c}}e^{it(kv_{c}+x_{0})}$ for the control qubit, and
$(\Omega_{\text{t}}e^{itkv_{t}},\Omega_{\text{t}}e^{itkv_{t}})$ for the two
transitions $|0(1)\rangle\leftrightarrow|r\rangle$ in the target qubit, where
$(v_{c},~{}v_{t})$ are the projection of velocity along $\mathbf{z}$ for the
control and target qubits, respectively. With a finite atomic temperature
$T_{\text{a}}$, there is a finite distribution
$\mathscr{D}(v_{c})\mathscr{D}(v_{t})$ for the speeds $(v_{c},v_{t})$, where
$\mathscr{D}(v)$ is a Gaussian Graham  _et al._ (2019); Shi (2020b).

###  A.1 Rydberg-state decay

Although the gate duration, when neglecting the transient time ($<10$ ns as
shown above) for phase change in the pulses, is
$t_{\text{g}}=\frac{2\pi}{\Omega_{\text{c}}}$, the main decay error arises
when the atoms are in Rydberg state Saffman and Walker (2005) supposing the
intermediate state is largely detuned. By using the estimate in Zhang  _et
al._ (2012), the decay error of the TSD-based CNOT gate can be approximated as

$\displaystyle E_{\text{decay}}$ $\displaystyle=$
$\displaystyle\frac{1}{4\tau}\int
dt\Big{[}\sum_{|\psi(0)\rangle=|00\rangle,|01\rangle}|\langle
0r|\psi(t)\rangle|^{2}$ (16)
$\displaystyle+\sum_{|\psi(0)\rangle=|10\rangle,|11\rangle}\big{(}|\langle
r0|\psi(t)\rangle|^{2}+|\langle r1|\psi(t)\rangle|^{2}$
$\displaystyle+|\langle 1r|\psi(t)\rangle|^{2}\big{)}\Big{]},$

which is $0.39t_{\text{g}}/\tau$ by numerical simulation. Consider a set of
experimentally feasible Graham  _et al._ (2019) values of Rydberg Rabi
frequencies $\Omega_{\text{c}}=\Omega_{\text{t}}/\sqrt{1.5}=2\pi\times 3.6$
MHz, we have $E_{\text{decay}}=2.7~{}(7.2)\times 10^{-4}$ for qubits in an
environment temperature of $4~{}(300)$ K. Alternatively, a more detailed
numerical simulation by using the optical Bloch equation in the Lindblad form
with correct branching ratios Shi (2019) can predict a slightly lower
$E_{\text{decay}}$ because some population can decay to qubit states that will
again contribute to the gate operation.

###  A.2 Doppler dephasing

For the TSD-based CNOT to be resilient to Doppler dephasing, the two fields
for the lower transitions $|0\rangle\rightarrow|p\rangle$ and
$|1\rangle\rightarrow|p\rangle$ shall copropagate along $\mathbf{z}$, and
those for their upper transitions $|p\rangle\rightarrow|r\rangle$ shall
copropagate along $-\mathbf{z}$, so that the wavevectors for
$|0(1)\rangle\rightarrow|r\rangle$ have (approximately) the same value $k$.
Here $|p\rangle$ is symbolic for the intermediate $6P_{3/2}$ state and one
shall bare in mind that the detunings at $|p\rangle$ for the two transition
chains must have a large difference; alternatively one can use different fine
states in the $6p$ manifold for the two transition chains (the numerical
results in Tables 2 and 3 stay similar). Then, the transition
$|0\rangle\xrightarrow{\Omega_{\text{t}}}|r\rangle\xrightarrow{\Omega_{\text{t}}}|1\rangle$
becomes

$\displaystyle|0\rangle\xrightarrow{\Omega_{\text{t}}e^{itkv_{t}}}|r\rangle\xrightarrow{\Omega_{\text{t}}e^{-itkv_{t}}}|1\rangle$
(17)

when accounting for the atom drift. The above transition mainly transfers the
population between the two hyperfine states, which means that if negligible
population stays at $|r\rangle$, the two phases $tkv_{t}$ and $-tkv_{t}$ add
up for any moment, leading to negligible phase noise because
$tkv_{t}-tkv_{t}=0$ for the population transfer from $|0\rangle$ to
$|1\rangle$. However, this is not ideal since there is always some population
at $|r\rangle$ during the process. Nonetheless, that there is partial phase
cancellation can suppress the Doppler dephasing compared to usual cases.

For the control qubit, the transition
$|1\rangle\xrightarrow{\Omega_{\text{c}}e^{itkv_{c}}}|r\rangle$ still has the
usual Doppler dephasing. However, the pumping
$|1\rangle\xrightarrow{\Omega_{\text{c}}e^{itkv_{c}}}|r\rangle$ in the control
is immersed in the TSD and does not put much population in the Rydberg state.
Numerical simulation shows $\int dt[|\langle r0|\psi(t)\rangle|^{2}+|\langle
r1|\psi(t)\rangle|^{2}]=0.31t_{\text{g}}$ for either
$|\psi(0)\rangle=|00\rangle$ or $|01\rangle$ in each gate sequence.

To show the robustness of the TSD-based CNOT against the Doppler dephasing, we
numerically simulate the state evolution by using
$\hat{\mathcal{H}}_{c0}(\Omega_{\text{t}}e^{itkv_{t}})=\Omega_{\text{t}}(e^{-itkv_{t}}|00\rangle+e^{-itkv_{t}}|01\rangle)\langle
0r|/2+$H.c. for the input states $|00\rangle$ and $|01\rangle$, and

$\displaystyle\hat{\mathcal{H}}_{c1}(\Omega_{\text{c}}e^{itkv_{c}},\Omega_{\text{t}}e^{itkv_{t}})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{cccccc}2V&\Omega_{\text{c}}e^{itkv_{c}}&\Omega_{\text{t}}e^{itkv_{t}}&\Omega_{\text{t}}e^{itkv_{t}}&0&0\\\
\Omega_{\text{c}}e^{-itkv_{c}}&0&0&0&\Omega_{\text{t}}e^{itkv_{t}}&\Omega_{\text{t}}e^{itkv_{t}}\\\
\Omega_{\text{t}}e^{-itkv_{t}}&0&0&0&\Omega_{\text{c}}e^{itkv_{c}}&0\\\
\Omega_{\text{t}}e^{-itkv_{t}}&0&0&0&0&\Omega_{\text{c}}e^{itkv_{c}}\\\
0&\Omega_{\text{t}}e^{-itkv_{t}}&\Omega_{\text{c}}e^{-itkv_{c}}&0&0&0\\\
0&\Omega_{\text{t}}e^{-itkv_{t}}&0&\Omega_{\text{c}}e^{-itkv_{c}}&0&0\end{array}\right),$
(24)

for the input states $|10\rangle$ and $|11\rangle$, where Eq. (24) is written
with the basis
$\\{|rr\rangle,|1r\rangle,|r1\rangle,|r0\rangle,|11\rangle,|10\rangle\\}$,
where $V$ represents the interaction of the state $|rr\rangle$. Because of the
Doppler dephasing, the gate map in the basis
$\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$ changes from the ideal
form

$\displaystyle U$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\
0&0&1&0\\\ \end{array}\right),$ (29)

to

$\displaystyle\mathscr{U}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}a&b&0&0\\\ c&d&0&0\\\ 0&0&e&f\\\
0&0&g&h\\\ \end{array}\right),$ (34)

where $a,b,c$, and $d$ can be calculated by sequentially using
$\hat{\mathcal{H}}_{c0}(\Omega_{\text{t}})$ and
$\hat{\mathcal{H}}_{c0}(-\Omega_{\text{t}})$ for the input states $|00\rangle$
and $|01\rangle$, while $e,f,g$, and $h$ can be calculated using,
sequentially, $\hat{\mathcal{H}}_{c1}(\Omega_{\text{c}},\Omega_{\text{t}})$
and $\hat{\mathcal{H}}_{c1}(\Omega_{\text{c}},-\Omega_{\text{t}})$, for the
input states $|10\rangle$ and $|11\rangle$. To study the robustness to Doppler
dephasing, we would like to see errors mainly from the Doppler dephasing if
the blockade error is negligible. So we adopt a large blockade interaction
$V/(2\pi)=500$ MHz as in Ref. Saffman  _et al._ (2020). We choose
$\Omega_{\text{c}}=\Omega_{\text{t}}/\sqrt{1.5}=2\pi\times 3.6$ MHz, define
the rotation error by Pedersen  _et al._ (2007)

$\displaystyle E_{\text{ro}}$ $\displaystyle=$ $\displaystyle
1-\frac{1}{20}\left[|\text{Tr}(U^{\dagger}\mathscr{U})|^{2}+\text{Tr}(U^{\dagger}\mathscr{U}\mathscr{U}^{\dagger}U)\right],$
(35)

and evaluate the ensemble average with

$\displaystyle\overline{E_{\text{ro}}}\approx\frac{\sum_{v_{c}}\sum_{v_{t}}E_{\text{ro}}(v_{c},~{}v_{t})\mathscr{D}(v_{c})\mathscr{D}(v_{t})}{\sum_{v_{c}}\sum_{v_{t}}\mathscr{D}(v_{c})\mathscr{D}(v_{t})},$
(36)

where the sum is over $10^{4}$ sets of speeds $(v_{c},~{}v_{t})$, where
$v_{c(t)}$ applies $101$ values equally distributed from $-0.5$ to $0.5$ m/s
because the atomic speed has little chance to be over $0.5$ m/s for the
temperatures $T_{a}\leq 50~{}\mu$K considered in this work. The approximation
by Eq. (36) has little difference from a rigorous integration. More details
can be found in Shi (2020b) for the method of numerical simulation.

Case 1.–By using $\hat{\mathcal{H}}_{c0}(\Omega_{\text{t}}e^{itkv_{t}})$ (for
$\\{|00\rangle,|01\rangle\\}$) and
$\hat{\mathcal{H}}_{c1}(\Omega_{\text{c}}e^{itkv_{c}},\Omega_{\text{t}}e^{itkv_{t}})$
(for $\\{|10\rangle,|11\rangle\\}$) for the first pulse, and
$\hat{\mathcal{H}}_{c0}(\Omega_{\text{t}}e^{itkv_{t}})$ and
$\hat{\mathcal{H}}_{c1}(\Omega_{\text{c}}e^{itkv_{c}},-\Omega_{\text{t}}e^{itkv_{t}})$
for the second pulse, the ensemble-averaged rotation errors are given in the
second row of Table 2, with effective atomic temperatures above
$T_{a}=5~{}\mu$K which was achievable in experiments Picken  _et al._ (2019).
From these results, one can see that the CNOT fidelity
$1-E_{\text{decay}}-\overline{E_{\text{ro}}}$ can reach $99.9\%$ with qubits
cooled to around $T_{a}=10~{}(5)~{}\mu$K in a $4~{}(300)~{}$K environment.

Case 2.–To further suppress the Doppler dephasing, we consider switching
propagation directions of fields for the control qubit between the two pulses.
In other words, the Hamiltonians (for $\\{|00\rangle,|01\rangle\\}$) are
$\hat{\mathcal{H}}_{c0}(\Omega_{\text{t}}e^{itkv_{t}})$ and
$\hat{\mathcal{H}}_{c0}(-\Omega_{\text{t}}e^{-itkv_{t}})$ for the first and
second pulses, respectively. The ensemble-averaged rotation errors are shown
in the third row of Table 2. From the results in Table 2, one can see that the
CNOT fidelity $1-E_{\text{decay}}-\overline{E_{\text{ro}}}$ has a slight
improvement if this latter case of configuration is employed. The mechanism
for the extra suppression of Doppler dephasing in this latter case lies in
that the population going, e.g., from $|0\rangle$ to $|1\rangle$, obtains some
phase error due to the action
$\Omega_{\text{t}}e^{-it^{\prime}kv_{t}}|1\rangle\langle r|$ and
$\Omega_{\text{t}}e^{itkv_{t}}|r\rangle\langle 0|$; subsequently, the second
pulse pumps $|1\rangle$ to $|0\rangle$ and induces some phase error due to the
action $-\Omega_{\text{t}}e^{-it^{\prime}kv_{t}}|r\rangle\langle 1|$ and
$-\Omega_{\text{t}}e^{itkv_{t}}|0\rangle\langle r|$, which means that the
phase terms between the two pulses are continuous which is better for the
desired transition to occur. For case 1, the pumping from $|0\rangle$ to
$|r\rangle$ has a phase term $tkv_{t}$ at the end of pulse 1, which becomes
$-tkv_{t}$ when the population begins to go back at the start of pulse 2,
i.e., there is a phase jump. More details about the influence on gate fidelity
from atom-drift-induced phase change in Rabi frequencies can be found in Shi
(2020b). Since this latter case is a little involved and the increase of
fidelity is marginal, it is only of interest in the future when technical
issues are resolved so that a fidelity level beyond $99\%$ is technically
achievable.

|  $T~{}(\mu$K) | 5 | 10 | 15 | 20 | 50
---|---|---|---|---|---|---
Case 1 | $10^{4}\times\overline{E_{\text{ro}}}$ | $4.31$ | $8.09$ | $11.9$ | $15.6$ | $38.2$
Case 2 | $10^{4}\times\overline{E_{\text{ro}}}$ | $3.11$ | $5.69$ | $8.26$ | $10.8$ | $26.2$
Table 2:  Rotation error (scaled up by $10^{4}$; excluding Rydberg-state decay) of the TSD-based CNOT by $\Omega_{c}/2\pi=3.5$ MHz with two cases, where case 2 corresponds to that the propagation directions for the fields on the target qubit are switched between the first and second pulses, while no such switching occurs in case 1. Here Eq. (35) was used that accounts for the phase errors and the truth table errors.  |  $T~{}(\mu$K) | 5 | 10 | 15 | 20 | 50
---|---|---|---|---|---|---
Case 1 | $10^{4}\times\overline{E_{\text{Bell}}}$ | $2.86$ | $5.27$ | $7.67$ | $10.1$ | $24.4$
Case 2 | $10^{4}\times\overline{E_{\text{Bell}}}$ | $2.57$ | $4.67$ | $6.78$ | $8.88$ | $21.4$
Table 3:  Fidelity error (scaled up by $10^{4}$; excluding Rydberg-state
decay) of the Bell state by TSD-based CNOT, with the same pulse sequence as
used for Table 2. $\overline{(\cdots)}$ denotes ensemble average as Eq. (36).

The result in Table 2 accounts for both the population errors and the phase
errors. In experiments, usually the phase errors are not important especially
in the characterization of the Bell state Levine  _et al._ (2019); Graham  _et
al._ (2019). Thus we continue to study the strength of TSD-based CNOT for
achieving high-fidelity Bell states.

###  A.3 High-fidelity Bell states

A TSD-based CNOT can map the initial product state
$|\psi(0)\rangle=(|00\rangle+|10\rangle)/\sqrt{2}$ to the Bell state
$|\Phi\rangle=(|00\rangle+|11\rangle)/\sqrt{2}$ without resorting to extra
single-qubit gates (except the state initialization). The Rydberg
superposition time is equal to that in Sec. A.1. We follow Ref. Levine  _et
al._ (2019) by evaluating the infidelity as
$E_{\text{Bell}}=1-\langle\Phi|\rho(t)|\Phi\rangle$, where
$\rho(t)=|\psi\rangle\langle\psi|$ with $|\psi\rangle$ evolved by using the
pulse sequence as in Sec. A.2 for the input state $|\psi(0)\rangle$. The
numerical results are given in Table 3, which shows that in room temperatures,
the Bell state can be created with a fidelity $99.88\%$ if qubits are cooled
to the level of $T_{a}=10~{}\mu$K, which is affordable according to the atom
cooling achieved in previous experiments Picken  _et al._ (2019); Zeng  _et
al._ (2017).

Finally, we would like to emphasize that it is not so crucial to have the
Doppler-resilient configuration (with copropagating fields) for the two
transition chains in the target qubit, as discussed around Eq. (17). Consider
the worst case with largest dephasing, i.e., if the two sets of fields
counterpropagate so that Eq. (17) becomes
$|0\rangle\xrightarrow{\Omega_{\text{t}}e^{itkv_{t}}}|r\rangle\xrightarrow{\Omega_{\text{t}}e^{itkv_{t}}}|1\rangle$,
then the motional dephasing error should be larger. For example, we
numerically found that the Bell-state errors in the second row (case 1) of
Table 3 become $(9.72\times 10^{-4},~{}1.90\times 10^{-3},~{}2.82\times
10^{-3},~{}3.74\times 10^{-3},~{}9.22\times 10^{-3})$ for
$T_{a}=(5,~{}10,~{}15,~{}20,~{}50)~{}\mu$K; similar results can be found for
the TSD-based CNOT. With $E_{\text{decay}}=7.2\times 10^{-4}$ for qubits in an
environment of $300$ K, the Bell-state fidelity would be $99.65\%$ at
$T_{a}=15~{}\mu$K. This still large fidelity means that for experimental
convenience, high fidelity is possible with any configuration for the
propagation directions for the two sets of fields if errors other than the
intrinsic Doppler dephasing and Rydberg-state decay can be avoided.

Figure 3: Energy spectrum for two 87Rb atoms lying along the quantization axis
when initialized in the two-atom pair state
$|ss\rangle\equiv|70S_{1/2},m_{J}=1/2;70S_{1/2},m_{J}=1/2\rangle$ Šibalić  _et
al._ (2017). Here 1 and 2 label the two states that have largest overlap with
$|ss\rangle$ and nearest to the unperturbed energy (labeled as zero).

##  Appendix B Interatomic distances

The dipole-dipole interaction $V$ is a function of interatomic distance
$\mathcal{L}$. For $V$ to be large enough so that the blockade error is
negligible, $\mathcal{L}$ should be small enough. In order to avoid
wavefunction overlap, the distance between the nuclei of the two Rydberg atoms
shall be larger than the Le Roy distance which can be calculated by using the
open source library of Ref. Šibalić  _et al._ (2017). For the parameters
chosen as an example in this work, the Le Roy distance is $1.50~{}\mu$m for
two rubidium atoms in the state $70S_{1/2}$, and we can consider longer
interatomic distance in the range of $\mathcal{L}>2.6\mu$m. The dipole-dipole
interaction will couple the two-atom state
$|70S_{1/2},m_{J}=1/2;70S_{1/2},m_{J}=1/2\rangle$ to many other states. To
find the interaction, we consider states $|nL;n^{\prime}L^{\prime}\rangle$
coupled from the initial states ($n$ and $L$ denote the principal and angular
momentum quantum numbers, respectively), with $|n-70|,|n^{\prime}-70|\leq 5$,
and $|L|,|L^{\prime}|\leq 4$ (because $ss$ states couple to $pp$ states, which
couple to $d$-orbital and to $f$-orbital states, and so on), and consider
energy gaps between pair states within $2\pi\times 25$ GHz. After
diagonalization, the energy map is shown in Fig. 3. In Fig. 3, the color
denotes the population of the state
$|70S_{1/2},m_{J}=1/2;70S_{1/2},m_{J}=1/2\rangle$ in the diagonalized state;
the color can also be understood as how possible if one atom is already in the
state $|70S_{1/2},m_{J}=1/2\rangle$, what will the chance be to populate the
state if the other atom is Rydberg pumped via, e.g., the $5P_{3/2}$
intermediate state.

For the TSD to hold, we focus on the strong blockade regime where the two-atom
Rydberg state shown in Fig. 3 is barely populated. In this case, the smallest
energy of the diagonalized eigenstate matters, or more accurately, the state
with the smallest eigenenergy that can be coupled (marked by red) plays the
role. In Fig. 3, one can find that for $\mathcal{L}\in[2.6,~{}4.6]\mu$m, there
are mainly three eigenstates, one with a positive energy (labeled as state 1),
and the other two with negative energy. For those two that have negative
energy, the one we focus on is the state with more overlap with
$|ss\rangle\equiv|70S_{1/2},m_{J}=1/2;70S_{1/2},m_{J}=1/2\rangle$, whose
energy is sometimes lower and sometimes higher than the energy of the other
with less population in $|ss\rangle$. The two states we focus on are labeled
by 1 and 2 in Fig. 3. State 1 has the largest component in $|ss\rangle$: at
$\mathcal{L}=\\{2.6,~{}3.0,~{}3.6,~{}4.0,~{}4.6\\}\mu$m, the amplitude overlap
between state 1 and $|ss\rangle$ is
$\\{0.64,~{}0.74,~{}0.84,~{}0.9,~{}0.95\\}$. For state 2, its overlap with
$|ss\rangle$ is $\\{0.49,~{}0.43,~{}0.32\\}$ when
$\mathcal{L}=\\{2.6,~{}3.0,~{}3.6\\}\mu$m and is negligible when $\mathcal{L}$
is beyond $4.0~{}\mu$m. The eigenenergy of these two states are listed in
Table 4, from which one can see that around $\mathcal{L}=3.0~{}\mu$m, the
interaction $V=2\pi\times 510$ can be represented by the eigenenergy of state
2, which is largest for the cases shown in Table 4.

|  $\mathcal{L}~{}(\mu$m) | 2.6 | 3.0 | 3.6 | 4.0 | 4.6
---|---|---|---|---|---|---
State 1 |  $V/(2\pi)$ (MHz) | 1600 | 780 | 340 | 180 | 94
State 2 |  $V/(2\pi)$ (MHz) | -280 | -510 | -590 | -650 | -670
Table 4:  Eigenenergy of the two eigenstates of the dipole-coupled two-atom states that have the largest overlap with the state $|70S_{1/2},m_{J}=1/2;70S_{1/2},m_{J}=1/2\rangle$ and that are nearest to zero, labeled as state 1 and state 2 in Fig. 3.  $V/(2\pi)$ (MHz) |  $T~{}(\mu$K) | 5 | 10 | 15 | 20 | 50
---|---|---|---|---|---|---
50 | $10^{4}\times\overline{E_{\text{ro}}}$ | $56.5$ | $60.3$ | $64.1$ | $67.8$ | $90.3$
100 | $10^{4}\times\overline{E_{\text{ro}}}$ | $17.0$ | $20.8$ | $24.5$ | $28.3$ | $58.6$
200 | $10^{4}\times\overline{E_{\text{ro}}}$ | $7.09$ | $10.9$ | $14.6$ | $18.4$ | $41.0$
300 | $10^{4}\times\overline{E_{\text{ro}}}$ | $5.25$ | $9.03$ | $12.8$ | $16.6$ | $39.1$
400 | $10^{4}\times\overline{E_{\text{ro}}}$ | $4.61$ | $8.39$ | $12.2$ | $15.9$ | $38.5$
Table 5:  Rotation error (scaled up by $10^{4}$; excluding Rydberg-state
decay) of the TSD-based CNOT by $\Omega_{c}/2\pi=3.5$ MHz with different
values of $V$.

To have $V=2\pi\times 500$ MHz, the above study shows that placing the two
qubits with a distance around $\mathcal{L}=3.0~{}\mu$m seems necessary. With
such a distance, the analysis in Ref. Graham  _et al._ (2019) shows that the
crosstalk error is about $0.5\%$ if the waist ($1/e^{2}$ intensity radii) of
the laser beams is $w=3.0~{}\mu$m. To reduce the crosstalk, it is possible to
use super-Gaussian Rydberg beams, as detailed in Ref. Gillen-Christandl  _et
al._ (2016). Another possibility is to use higher Rydberg states so that the
dipole-dipole interaction can be as large as $V=2\pi\times 500$ MHz at a
larger interatomic distance, and hence the laser-beam crosstalk can be
avoided.

If only commonly used Gaussian Rydberg beams are employed for a Rydberg state
of principal quantum number around $n=70$, then smaller values of $V$ can be
used for larger interatomic distances so as to avoid crosstalk. For the values
$V/(2\pi)\in[50,~{}400]$ MHz, the fidelity of our gate is shown in Table 5,
which shows that with $V=2\pi\times 100$ MHz, the rotation error is about
$1.7\times 10^{-3}$ with $T_{a}=5~{}\mu$K, and the interatomic distance should
be around $\mathcal{L}=4.6\mu$m where crosstalk can be ignored safely
($\Omega^{\prime}/\Omega\sim e^{-2\mathcal{L}^{2}/w^{2}}=0.009$) if the waists
for the lower and upper lasers are both $w=3.0~{}\mu$m. Taking into account
the decay error, this means that our gate would have a fidelity $99.76\%$ with
$T_{a}=5~{}\mu$K in room temperatures.

Figure 4: Fidelity error from the fluctuation in the Rabi frequencies for the
two transitions $|0\rangle\leftrightarrow|r\rangle$ and
$|1\rangle\leftrightarrow|r\rangle$; the fluctuation obeys a Gaussian
distribution of width $\sigma_{\Omega}$. The error is $0.00376$ for
$\sigma_{\Omega}=5\%$.

##  Appendix C Amplitude fluctuation of laser fields

Our TSD-based CNOT requires the Rabi frequencies for the two transitions
$|0\rangle\leftrightarrow|r\rangle\leftrightarrow|1\rangle$ to be equal in the
target qubit. Here we investigate the impact on the gate fidelity if this
condition is not satisfied.

We assume that the Rabi frequency $\Omega$ obeys a Gaussian distribution

$\displaystyle\mathscr{G}(\Omega)$ $\displaystyle=$
$\displaystyle\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(\Omega-\Omega_{k})^{2}}{2\sigma^{2}}}$
(37)

around the desired value $\Omega_{k}$, where $k\in\\{$t1, t2$\\}$ denotes the
two channels $|0\rangle\leftrightarrow|r\rangle$ and
$|1\rangle\leftrightarrow|r\rangle$. By using the Gaussian distributed Rabi
frequencies in the target qubit, one can evaluate the averaged fidelity by

$\displaystyle\overline{E_{\text{ro}}}\approx\frac{\sum_{\Omega_{\text{t1}}}\sum_{\Omega_{\text{t2}}}E_{\text{ro}}(\Omega_{\text{t1}},\Omega_{\text{t2}})\mathscr{G}(\Omega_{\text{t1}})\mathscr{G}(\Omega_{\text{t2}})}{\sum_{\Omega_{\text{t1}}}\sum_{\Omega_{\text{t2}}}\mathscr{G}(\Omega_{\text{t1}})\mathscr{G}(\Omega_{\text{t2}})},$
(38)

where $E_{\text{ro}}(\Omega_{\text{t1}},\Omega_{\text{t2}})$ is evaluated by
using the gate fidelity with $\Omega_{\text{t1}}$ and $\Omega_{\text{t2}}$ for
the two transitions $|0\rangle\leftrightarrow|r\rangle$ and
$|1\rangle\leftrightarrow|r\rangle$. In Eq. (38), the integration is
approximated by the sum over 121 sets of
$(\Omega_{\text{t1}},\Omega_{\text{t2}})$, each of which applies values
$(\Omega-\Omega_{k})/\Omega_{k}\in\\{\pm 5,~{}\pm 4,~{}\pm 3,~{}\pm 2,~{}\pm
1,~{}0\\}\sigma$. With $\sigma\in\\{0.01,~{}0.02,~{}0.03,~{}0.04,~{}0.05\\}$,
the result is shown in Fig. 4. One can see that with quite large relative
fluctuation of Gaussian width $5\%$, the fidelity is still larger than
$99.6\%$. There is little relation between the Doppler dephasing and
fluctuation of Rabi frequency, and one can expect that the gate fidelity can
be evaluated by combining Table 2 and Fig. 4 if the fluctuation of the Rabi
frequencies is not quenched.

In Ref. Levine  _et al._ (2018), the power fluctuation of the Rydberg lasers
was suppressed below $1\%$ for preparing ground-Rydberg entanglement, which
means that the Rydberg Rabi frequencies were suppressed to have relative noise
below $10\%$ in Ref. Levine  _et al._ (2018). More than one year later, in
Ref. Levine  _et al._ (2019), preparation of high-fidelity ground-state
entanglement was reported by the same group, which was a significant progress.
We suppose that the laser noise was even more suppressed in Ref. Levine  _et
al._ (2019) compared to the earlier experiment in Ref. Levine  _et al._
(2018), and thus we consider the relative Gaussian width of the fluctuation up
to $5\%$.

##  Appendix D Comparison with other fast Rydberg gates

It is useful to compare our CNOT with other fast entangling methods with
Rydberg atoms. The comparison focuses on physical mechanism and their
application in quantum computing.

A popular method to generate entanglement by Rydberg interactions is to use
the EA mechanism Jaksch  _et al._ (2000). The standard way to use it is a
three-pulse sequence for a $C_{Z}$ gate in the form of

$\displaystyle C_{Z}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&-1&0&0\\\ 0&0&-1&0\\\
0&0&0&-1\\\ \end{array}\right)$ (43)

in the basis $\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$, where the
blockade takes effect in the state $|11\rangle$. To use it for quantum
computing in the circuit model, the $C_{Z}$ gate in Eq. (43) needs single-
qubit gates to become a CNOT. By this method, a most recent experiment in Ref.
Graham  _et al._ (2019) realized a CNOT of duration more than 100 $\mu$s with
Rydberg Rabi frequencies $2\pi\times 4.6$ MHz because of the slow single-qubit
gates. Physically, this method depends on a missed $\pi$ phase accumulation in
an annihilated Rabi cycle for the input state $|11\rangle$ by the Rydberg
blockade.

The other recent experiment in Ref. Levine  _et al._ (2019) used detuned Rabi
cycles for entanglement combined with detunings found by optimal control. The
method is essentially identical to the interference method proposed in Ref.
Shi (2019), as can be easily verified by looking at the similar structure of
the gate matrix [with the same basis as in Eq. (43)]

$\displaystyle C_{\text{phase}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&e^{i\alpha}&0&0\\\
0&0&e^{i\alpha}&0\\\ 0&0&0&e^{i\beta}\\\ \end{array}\right)$ (48)

in Refs. Levine  _et al._ (2019); Shi (2019) (Ref. Shi (2019) proposed two
gates among which the first is quoted here). Compared to the initial
interference method, Ref. Levine  _et al._ (2019) used optimal-control-found
parameters and phase twist in the Rydberg pumping within the blockade regime
so as to realize $2\alpha-\beta=\pi$ (and thus the gate in Ref. Levine  _et
al._ (2019) can be called a $C_{Z}$-like gate). Because of the detuned Rabi
cycles used, there is no way to realize a gate with
$\\{\alpha,~{}\beta\\}/\pi=\\{N_{1},N_{2}\\}$, where $N_{j}$ is an integer
with $j=1,2$. To use it in quantum computing, several single-qubit gates must
be used to transform the gate in Ref. Levine  _et al._ (2019) to the CNOT so
their final CNOT duration was more than $4~{}\mu$s with Rydberg Rabi
frequencies $2\pi\times 3.5$ MHz. Without using the detuned Rabi cycles, the
method in Ref. Levine  _et al._ (2019) can not work. The physical mechanism of
Ref. Levine  _et al._ (2019) is that dynamical phases in detuned Rabi cycles
are accumulated.

In order to compare with the gates in Eqs. (43) and (48) which have a diagonal
form, the TSD-based CNOT in the basis of
$\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$ can be rewritten as

$\displaystyle C_{\text{TSD}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\ 0&0&-1&0\\\
0&0&0&1\\\ \end{array}\right),$ (53)

with the basis
$\\{|0\overline{0}\rangle,|0\overline{1}\rangle,|1\overline{0}\rangle,|1\overline{1}\rangle\\}$,
where $|\overline{1}(\overline{0})\rangle=(|0\rangle\pm|1\rangle)/\sqrt{2}$.
The CNOT duration would be less than $0.4~{}\mu$s with Rydberg Rabi
frequencies $\leq 2\pi\times 3.5$ MHz by the TSD method. The blockade
interaction is in $|1\overline{1}\rangle$ and the pumping on the target qubit
induces a transition from $|1\overline{1}\rangle$ to
$|\mathscr{S}_{+}\rangle\equiv(\Omega_{\text{c}}|r\overline{1}\rangle+\sqrt{2}\Omega_{\text{t}}|1r\rangle)/\overline{\Omega}$
with a Rabi frequency
$\overline{\Omega}=\sqrt{\Omega_{\text{c}}^{2}+2\Omega_{\text{t}}^{2}}$ while
further excitation to $|rr\rangle$ is blocked. Each of the two pulses in the
TSD-based CNOT sequence induces a complete $2\pi$ rotation
$|1\overline{1}\rangle\rightarrow-i|\mathscr{S}_{+}\rangle\rightarrow-|1\overline{1}\rangle$
which leads to a $\pi$ phase change to $|1\overline{1}\rangle$. The change of
the sign of $\Omega_{\text{t}}$ between the two pulses does not change this
picture, and thus a total phase $2\pi$ is accumulated in
$|1\overline{1}\rangle$. Since $e^{2i\pi}=1$, $|1\overline{1}\rangle$ acquires
no phase term in practice. On the other hand, the input state
$|1\overline{0}\rangle$ only experiences two $\pi$ pulses (i.e., a $2\pi$
pulse) in the control qubit which results in a $\pi$ phase change to it. The
pumping of the target qubit experiences spin echo for the input state
$|0\overline{1}\rangle$, thus no phase appears for it. So, the physical
mechanism of the gate in Eq. (53) is fundamentally different from those in
Eqs. (43) and (48); in fact, the drastically different forms of the three gate
maps reveal this. To sum up in one word, Eq. (43) relies on a missed $\pi$
phase change in an annihilated Rabi cycle, Eq. (48) relies on three phase
changes in three detuned Rabi cycles, and (53) relies on a $\pi$ phase change
in a resonant Rabi cycle. For quantum computing, the gate in Eq. (53) is
exactly a CNOT in the basis of
$\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$, and thus is more useful
compared to Eqs. (43) and (48).

Because a phase twist is used in realizing both Eq. (48) and Eq. (53), one may
guess that they are similar. But the following facts show their distinct
physics: (i) detuned Rydberg pumping is used in Eq. (48), but resonant Rydberg
pumping is used in Eq. (53); (ii) three input states acquire phase terms in
Eq. (48), but only one input state acquires a phase term in Eq. (53); (iii)
the population in Rydberg state can reach 1 for the fourth input state
$|1\overline{1}\rangle$ in Eq. (53), while there is no way to realize such an
effect in Eq. (48). This last effect basically means that the mechanism for
realizing Eq. (53) can be used to realize a high-fidelity multi-qubit gate:
one can use TSD to excite the input state $|1\overline{1}\rangle$ to
$|\mathscr{S}_{+}\rangle$ which can block the Rydberg pumping in a nearby atom
(of course care shall be taken for the design of such a gate); notice that the
three-qubit gate in Ref. Levine  _et al._ (2019) requires exciting the two
edge atoms to block the Rydberg pumping in the middle atom which results in
error due to the residual blockade between the two edge atoms. For the method
in Eq. (48), detuned Rabi cycles are used and it can not excite one Rydberg
excitation in the blocked excitation [for the input state $|11\rangle$ in Eq.
(48)] , and thus it is not possible to extend the method in Eq. (48) to a
high-fidelity multi-qubit gate. This means that the underlying physics in Eq.
(53) is different from that of Eq. (48).

Note that the basis transform used to write the TSD-based CNOT in Eq. (48) is
used only for clarifying the physics, but not for practical use since in a
large-scale quantum computer the way to encode quantum information shall be
based on a commonly used qubit basis in all registers (otherwise the
information loading and retrieving will face trouble). For example, to
initialize the atomic arrays, the quantization axis is fixed by applying,
e.g., an external magnetic field, and the qubit states $|0\rangle$ and
$|1\rangle$ are chosen from two hyperfine levels with different energy. So,
the states
$|\overline{1}(\overline{0})\rangle=(|0\rangle\pm|1\rangle)/\sqrt{2}$ are no
longer eigenstates because of the applied external fields, and new state
detection schemes shall be designed. In a large-scale quantum computer, to
play the role of control or target is not fixed for any qubit: a computation
task is divided into a series of unitary operations, in different unitary
operations, the same qubit sometimes serves as a control, and sometimes as a
target. This means that if one uses a hybrid coding with $|0(1)\rangle$ for
the control and $|\overline{1}(\overline{0})\rangle$ for the target, an
exceedingly large amount of transforming operations between $|0(1)\rangle$ and
$|\overline{1}(\overline{0})\rangle$ should be used in the quantum circuit.
So, it is impractical to use $|0(1)\rangle$ for information coding in the
control, and to use $|\overline{1}(\overline{0})\rangle$ for information
coding in the target; for a proof-of-principle study either in theory or in
experiment, it may be of interest, but not for a realistic large-scale quantum
computer.

##  Appendix E AC Stark shifts

There is a detailed study about the ac Stark shifts in a two-photon Rydberg
excitation in Ref. Maller  _et al._ (2015). Appendix B of Maller  _et al._
(2015) presents a detailed calculation of the ac Stark shifts accounting for
the hyperfine splitting of the intermediate state, where one can find that
compensation of the ac Stark shifts is feasible. Below, we ignore the
hyperfine splitting of the intermediate state to give a brief introduction.
More details can be found in Ref. Maller  _et al._ (2015).

Here, we choose the qubit states by $|1\rangle=|6S_{1/2},F=3,m_{F}=3\rangle$
and $|0\rangle=|6S_{1/2},F=4,m_{F}=4\rangle$ of cesium. The state $|1\rangle$
is driven to $|p\rangle$ (in the $7P_{1/2}$ manifold) with laser fields that
are left-hand circularly polarized (note that this is a little different from
Ref. Maller  _et al._ (2015)), which is further excited to
$|r\rangle=|82S_{1/2},m_{J},m_{I}\rangle$. The Hamiltonian for a two-photon
Rydberg excitation is (in this section we explicitly put $\hbar$ in
Hamiltonians)

$\displaystyle\hat{H}_{\text{2-pho;0}}$ $\displaystyle=$
$\displaystyle\hbar[\Omega_{1}|p\rangle\langle 1|/2+\Omega_{2}|r\rangle\langle
p|/2+\text{H.c.}]$ (54) $\displaystyle+\hbar\Delta|p\rangle\langle p|,$

where $\Delta$ is defined as the frequency of the laser field deducted by the
frequency of the atomic transition. Note that we have not included the
nonresonant shift in the equation above. When $\Delta$ is very large compared
to the decay rate of $|p\rangle$, the intermediate state can be adiabatically
eliminated, leading to

$\displaystyle\hat{H}_{\text{2-pho}}$ $\displaystyle=$
$\displaystyle\hbar\\{[\Omega_{\text{eff}}|r\rangle\langle
1|/2+\text{H.c.}]+\Delta_{r}|r\rangle\langle r|$ (55)
$\displaystyle+\Delta_{q1}|1\rangle\langle 1|\\},$

where $\Omega_{\text{eff}}=\Omega_{1}\Omega_{2}/(2\Delta)$, and according to
Ref. Maller  _et al._ (2015) the effective detuning at the level $|r\rangle$
is

$\displaystyle\Delta_{r}(\omega_{1},\omega_{2},\Delta,\mathcal{E}_{1},\mathcal{E}_{2})$
$\displaystyle=$
$\displaystyle\frac{\Omega_{2}^{2}}{4\Delta}-\frac{e^{2}}{4m_{e}\hbar}\left(\frac{\mathcal{E}_{1}^{2}}{\omega_{1}^{2}}+\frac{\mathcal{E}_{2}^{2}}{\omega_{2}^{2}}\right),$
(56)

where $e$ is the elementary charge, $m_{e}$ the mass of the electron, and
$\omega_{j}$ and $\mathcal{E}_{j}$ are the frequency and electric field
amplitude of the laser field, where $j=1$ and $2$ for the lower and upper
transitions, respectively. Similarly, for the ground state, the effective
detuning is

$\displaystyle\Delta_{q1}(\omega_{1},\omega_{2},\Delta,\mathcal{E}_{1},\mathcal{E}_{2})$
$\displaystyle=$
$\displaystyle\frac{\Omega_{1}^{2}}{4\Delta}-\frac{1}{4\hbar}\left(\alpha_{1}\mathcal{E}_{1}^{2}+\alpha_{2}\mathcal{E}_{2}^{2}\right),$
(57)

where $\alpha_{1}$ and $\alpha_{2}$ are the nonresonant polarizabilities from
the lower and upper fields that can be calculated via the sum over transitions
to other states except the intermediate state $|p\rangle$. Meanwhile, there is
a shift on the ground state $|0\rangle$

$\displaystyle\Delta_{q0}(\omega_{1},\omega_{2},\Delta,\mathcal{E}_{1},\mathcal{E}_{2})$
$\displaystyle=$
$\displaystyle\frac{\mathscr{C}^{2}\Omega_{1}^{2}}{4(\Delta+\omega_{q})}-\frac{1}{4\hbar}(\alpha_{1}\mathcal{E}_{1}^{2}$
(58) $\displaystyle+\alpha_{2}\mathcal{E}_{2}^{2}),$

where $\omega_{q}$ is the frequency separation between $|0\rangle$ and
$|1\rangle$, and $\mathscr{C}$ is a factor determined by the selection rules.
Because $\alpha_{2}/\alpha_{1}=-16.3$ and $\alpha_{1}$ is negative, the off-
resonant shift will be negative, and thus it is necessary for the resonant
shift to be positive. So, both $\Delta$ in Eq. (57) and $\Delta+\omega_{q}$ in
(58) shall be positive, which further means that $\mathscr{C}^{2}$ shall be
larger than 1.

For the case when $\Delta$ is much larger than the hyperfine splitting of the
intermediate $7P_{1/2}$ state, we ignore its hyperfine splitting and then it
can be written as $|7P_{1/2},m_{J},m_{I}\rangle$. When right-hand polarized
fields are used for coupling qubit states and the $7P_{1/2}$ states, the
square of the ratio of the coupling between $|0\rangle$ and $|p\rangle$ and
that between $|1\rangle$ and $|p\rangle$ is

$\displaystyle\mathscr{C}^{2}$ $\displaystyle=$
$\displaystyle\frac{\sum_{m_{J},m_{I}}\left(\sum_{m_{e}}C_{m_{J},1,m_{e}}^{1/2,1,1/2}C_{m_{e},m_{I},0}^{1/2,7/2,4}\right)^{2}}{\sum_{m_{J},m_{I}}\left(\sum_{m_{e}}C_{m_{J},1,m_{e}}^{1/2,1,1/2}C_{m_{e},m_{I},0}^{1/2,7/2,3}\right)^{2}},$
(59)

which is 8. To shift
$\Delta_{r}(\omega_{1},\omega_{2},\Delta,\mathcal{E}_{1},\mathcal{E}_{2})$ to
zero, one can simply adjust the frequency of the laser fields for addressing
the upper transition. To compensate the Stark shifts at $|0\rangle$ and
$|1\rangle$, Eqs. (57) and (58) indicates that
$\mathscr{C}^{2}=1+\omega_{q}/\Delta$, which means that we shall choose
$\Delta=\omega_{q}/(\mathscr{C}^{2}-1)\approx 2\pi\times 1.3$ GHz (for
rubidium it would be $2\pi\times 0.97$ GHz) which is near to values used in
experiments Isenhower  _et al._ (2010); Maller  _et al._ (2015). With
$\Delta=\omega_{q}/(\mathscr{C}^{2}-1)$ satisfied, an appropriate set of
$(\Delta,\mathcal{E}_{1},\mathcal{E}_{2})$ satisfying Eq. (57) would satisfy
Eq. (58), too. With the given data for the radial coupling between ground and
$|p\rangle$ states and the values of $\alpha_{1,2}$ (page 6 of Maller  _et
al._ (2015)), Eq. (57) reduces to
$\overline{\Delta}[16.3\left(\frac{\mathcal{E}_{2}}{\mathcal{E}_{1}}\right)^{2}-1]=2.98$,
where $\overline{\Delta}=\Delta\times 10^{-9}s$, which can be used to set up
the ratio $|\mathcal{E}_{2}/\mathcal{E}_{1}|$.

We have shown the method to compensate the Stark shifts if pumping applies
only to the transition $|1\rangle\leftrightarrow|r\rangle$. For the target
qubit, there will be another transition required,
$|0\rangle\leftrightarrow|r\rangle$. Then, four fields lead to Stark shifts in
the two qubit states (two equations), and there is a condition that the Rabi
frequencies for $|0(1)\rangle\leftrightarrow|r\rangle$ shall be equal (a third
equation). Moreover, there will be two detunings at the intermediate states
for these two transitions. Altogether there are three equations for six
variables and hence the problem is solvable. The Stark shift at $|r\rangle$
can be compensated by adjusting the central frequencies of the laser fields.
So, Eq. (24), or Eq. (3) in the main text, can be realized.

For one-photon excitation of $p$-orbital Rydberg states, the off-resonant ac
Stark shift on $|0\rangle$ or $|1\rangle$ from the resonant field is positive,
but one can use another field of larger wavelength to induce a negative shift.
Then, a similar set of equations like Eqs. (57) and (58) can be established to
compensate the Stark shifts. For the shift on the Rydberg state, one can shift
the wavelength of the laser field to recover the resonance condition for
$|r\rangle$.

##  Appendix F CNOT in other systems

It is useful to compare the speed of CNOT in different physical systems by
using the ratio $\Xi=T_{\text{coh}}/T_{\text{g}}$, where $T_{\text{coh}}$ is
the coherence time (pure spin dephasing $T_{2}$ or the inhomogeneously
broadened $T_{2}^{\ast}$), and $T_{\text{g}}$ is the duration of CNOT. First,
for neutral atom qubit systems, in Ref. Wang  _et al._ (2016) a coherence time
of 7 seconds has been measured (see the left column of page 2 of Wang  _et
al._ (2016)) in a large neutral atom array. Ref. Wang  _et al._ (2016)
realized such a coherence time so as to have high fidelity in the single-qubit
gates that had gate durations 80 $\mu$s; in principle, much longer coherence
times can be achieved. To have a conserved estimation, we assume that a
coherence time of 7s for neutral atoms. With the protocol in this manuscript,
the CNOT duration would be about 0.3 $\mu$s with Rydberg Rabi frequencies like
those realized in Refs. [43, 44]. Then, the figure-of-merit would be
$\Xi=2.3\times 10^{7}$.

Second, for trapped ions, several most recent and most advanced results can be
inferred from literatures. However, one should bare in mind that some fast
entangling gates are not CNOT gates. Any two-qubit entangling gate can be
repeated several times to form a CNOT when assisted by single-qubit gates, as
demonstrated in Bremner  _et al._ (2002), but the CNOT is the gate directly
helpful to quantum computation in the circuit model Shor (1997); Bremner  _et
al._ (2002); Shende  _et al._ (2004); Peruzzo  _et al._ (2014); Debnath  _et
al._ (2016). So, we focus on data for trapped ions with CNOT gates
demonstrated. (i) In Ref. Ballance  _et al._ (2015), Bell states of trapped
ions were created with gate times 27.4 $\mu$s. The coherence times were not
mentioned in Ref. Ballance  _et al._ (2015). However, near the end of Ref.
Ballance  _et al._ (2015) it stated coherence times of 60s which indicates
that the coherence times of their qubits were around 60s. So the figure-of-
merit would be $60s/(27.4~{}\mu$s) which is about $\Xi=2.2\times 10^{6}$. (ii)
In Ref. Ballance  _et al._ (2016), Bell states of trapped ions were prepared
by pulses with durations from 50$\mu$s to 100$\mu$s; it also showed very short
gate times of about 3.8$\mu$s. In the Supplemental Material of Ref. Ballance
_et al._ (2016), it showed that the coherence time is about 6s. So the figure-
of-merit is 6s/(3.8$\mu$s) which is about $\Xi=1.6\times 10^{6}$. (iii) In
Ref. Gaebler  _et al._ (2016), Bell states of trapped ions were prepared by
pulses of duration about 30$\mu$s (Fig. 6 of Ref. Gaebler  _et al._ (2016)
showed gates with durations even longer), and the coherence time is about
1.5s. So the figure-of-merit would be 1.5s/(30$\mu$s) which is about
$\Xi=5\times 10^{4}$.

Third, for superconducting qubits, very fast CNOT gates were reported in Ref.
Barends  _et al._ (2014). The energy relaxation time $T_{1}$ is usually
shorter than the phase coherence time $T_{2}^{\ast}$ in superconducting
circuits. In Barends  _et al._ (2014) the measured value of $T_{1}$ was
$20-40~{}\mu$s and values up to $57~{}\mu$s were recorded. The Supplementary
Information of Ref. Barends  _et al._ (2014) showed that the $C_{Z}$ gate
durations can be as short as $38$ ns, and the single-qubit gate times can be
as small as $10$ ns. So the gate duration of a CNOT can be $T_{\text{g}}=58$
ns, leading to a figure-of-merit $\Xi=10^{3}$. Suppose the fast gate in Ref.
Barends  _et al._ (2014) was realized in the superconducting system with a
longer coherence time $T_{1}=70~{}\mu$s Rigetti  _et al._ (2012), the figure-
of-merit $\Xi=1.2\times 10^{3}$ is still much smaller than those in neutral
atoms.

Concerning the ratio between the coherence time and the CNOT duration, the
above comparison shows that TSD makes neutral atoms advantageous compared to
trapped ions and superconducting systems.

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