# Fast and Robust Geometric Two-Qubit Gates for Superconducting Qubits and
beyond

F. Setiawan<EMAIL_ADDRESS>Current Affiliation: Riverlane Research
Inc., Cambridge, MA 02142, USA Pritzker School of Molecular Engineering,
University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, USA
Peter Groszkowski  Current Affiliation: National Center for Computational
Sciences, Oak Ridge National Laboratory, TN 37831, USA Pritzker School of
Molecular Engineering, University of Chicago, 5640 South Ellis Avenue,
Chicago, Illinois 60637, USA  Aashish A. Clerk  Pritzker School of Molecular
Engineering, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois
60637, USA

###### Abstract

Quantum protocols based on adiabatic evolution are remarkably robust against
imperfections of control pulses and system uncertainties. While adiabatic
protocols have been successfully implemented for quantum operations such as
quantum state transfer and single-qubit gates, their use for geometric two-
qubit gates remains a challenge. In this paper, we propose a general scheme to
realize robust geometric two-qubit gates in multilevel qubit systems where the
interaction between the qubits is mediated by an auxiliary system (such as a
bus or coupler). While our scheme utilizes Stimulated Raman Adiabatic Passage
(STIRAP), it is substantially simpler than STIRAP-based gates that have been
proposed for atomic platforms, requiring fewer control tones and ancillary
states, as well as utilizing only a generic dispersive interaction. We also
show how our gate can be accelerated using a shortcuts-to-adiabaticity
approach, allowing one to achieve a gate that is both fast and relatively
robust. We present a comprehensive theoretical analysis of the performance of
our two-qubit gate in a parametrically-modulated superconducting circuits
comprising two fluxonium qubits coupled to an auxiliary system.

##  I Introduction

Geometric quantum gates [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] are robust
against a range of parameter uncertainties and pulse imperfections. One
powerful technique that allows one to construct such geometric gates is
adiabatic evolution. While single-qubit gates based on adiabatic evolution
have been implemented in a number of platforms, including quantum dots [14],
trapped ions [15] and nitrogen-vacancy centers in diamond [16], the
implementation of two-qubit gates based on geometric phases and adiabatic
evolution remains a challenge. A powerful adiabatic protocol well suited for
operations on isolated qubit levels is stimulated Raman adiabatic passage
(STIRAP) [17]. While typically used for state transfer, STIRAP can also be
exploited for single-qubit gates [4, 6, 18, 19]. STIRAP has also been proposed
as a way of realizing geometric two-qubit gates in atomic systems, both for
trapped ions [4] and Rydberg atoms [5]. These protocols however use extremely
specific kinds of qubit-qubit interactions (e.g., phonon-sideband processes in
Ref. [4] and the Rydberg blockade in Ref. [5]) as well as large numbers of
control fields and ancillary states, making them ill suited for other kinds of
qubit platforms such as superconducting circuits.

In this paper, we present an alternate, platform-agnostic approach for
designing geometric two-qubit gates using STIRAP that is considerably simpler
than the proposals of Refs. [4, 5], and that does not require a special form
of qubit-qubit interaction. Furthermore, our approach is directly compatible
with shortcuts to adiabaticity methods [20, 21, 22, 23, 24, 18], and hence can
be much faster than a naive adiabatic gate. We explicitly show that this
acceleration can be performed while still retaining some of the robustness
that makes adiabatic protocols so attractive. ur approach is general and hence
realizable in a variety of systems. It is especially well suited to setups
comprising two remote qubits that interact in a tunable manner with a common
auxiliary system (see Fig. 1), something that has been realized in numerous
experiments (see, e.g., Refs. [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
37, 38, 39, 40]).

Figure 1:  (a) $\Lambda$-system formed by the logical qubit state $|11\rangle$
and ancillary states $|a_{1}\rangle$ and $|a_{2}\rangle$. Depicted transitions
are realized via parametrically modulated couplings with envelopes
$\Omega_{A}(t)$ and $\Omega_{B}(t)$. A double-STIRAP protocol in this system
generates a geometric phase, forming the basis of a two-qubit gate. (b)
Schematic setup for implementing our STIRAP-based gate: qubits $A$ and $B$ are
coupled via effectively tunable couplings $g_{A}(t)$ and $g_{B}(t)$ to a
single mode of an auxiliary system. The tunable couplings could be realized
directly, or indirectly by modulating the auxiliary system (or qubit)
frequencies.

To demonstrate the effectiveness of our approach, we explore a superconducting
circuit implementation consisting of two fluxonium qubits [41, 42, 43, 44]
tunably coupled to a common auxiliary system (representing a bus or coupler).
We consider cases where the required tuning is realized either (1) directly by
using tunable couplings [25, 26, 27, 28, 29] between the auxiliary system and
the qubits, or by (2) using static couplings to the qubits, but frequency
modulating the auxiliary mode [30, 31, 32, 33, 34, 35]. Using experimentally
realistic parameters, we demonstrate, via full master-equation simulations
that take into account both non-rotating-wave approximation (non-RWA) effects
and dissipation, that our accelerated two-qubit gates yield a competitive gate
fidelity. For a direct scheme of realizing tunable interactions using time-
dependent couplings, we obtain a gate fidelity of approximately 0.9995 for
gate times in the range of $t_{g}=45$–$60$ ns; this is achieved without any
fine-tuning of $t_{g}$, though we did not model the internal structure of the
couplers. If we instead use static couplings but parametrically modulate the
auxiliary system frequency, we obtain a gate fidelity of approximately 0.999
at a gate time $t_{g}=130$ ns (where here we model all key elements of the
system). Our proposed accelerated adiabatic controlled-$Z$ (CZ) gates together
with its corresponding arbitrary one-qubit gate proposed in Ref. [19] pave the
way towards universal quantum computation that are fast and robust against
imperfections in the control fields.

The paper is organized as follows. We begin in Sec. II by briefly reviewing
the established idea of the STIRAP geometric gate [4, 6] and then outlining
our general approach to implementing two-qubit geometric gates. In Sec. III,
we review the acceleration protocols as well as methods to minimize the
effects of coherent non-RWA errors. We show the robustness and fundamental
performance of our geometric gates in Sec. IV. Readers who are already
familiar with the STIRAP protocol as well as its accelerated version and who
are interested in the details of their application to implementing two-qubit
gates can skip directly to Sec. V. In that section, we explore the
implementation and performance of our STIRAP gates in realistic
superconducting circuits comprising two fluxonium qubits connected via an
auxiliary system. We conclude in Sec. VI.

##  II Two-qubit STIRAP quantum gates

In this section we outline the general concept behind our geometric two-qubit
gate that complements the geometric one-qubit gate proposed in Refs. [4, 6]
and its accelerated version [18, 19]. To set the stage, in what follows we
first discuss the basic physics behind our adiabatic two-qubit gate by
focusing on a simplified RWA Hamiltonian. Following Ref. [19], we show in the
next section how our adiabatic geometric gate can be accelerated and enhanced
for its implementation in realistic multilevel qubit settings where non-RWA
effects cannot be neglected.

###  II.1 Double-STIRAP two-qubit gates in an ideal $\Lambda$ system

The basic building block of our two qubit gate is the canonical state transfer
protocol STIRAP [17] that utilizes a $\Lambda$-level structure [see Fig.
1(a)]. An ideal $\Lambda$ system comprises three energy levels, two of which
($|q\rangle$ and $|a_{2}\rangle$) are resonantly coupled to a common
(typically) excited state $|a_{1}\rangle$ via tunable couplings. Denoting
these (in general complex) tunable couplings as $\Omega_{j}(t)$ ($j=A,B$), the
Hamiltonian of the $\Lambda$ system within the RWA is ($\hbar=1$)

$\displaystyle\hat{H}_{\mathrm{0}}(t)$
$\displaystyle=\frac{1}{2}\left[\Omega_{A}(t)|a_{1}\rangle\langle
q|+\Omega_{B}(t)|a_{1}\rangle\langle a_{2}|+\mathrm{H.c.}\right].$ (1)

In this paper, we always take state $|q\rangle$ to be a target logical qubit
state that we would like to print a geometric phase on, while $|a_{1}\rangle$
and $|a_{2}\rangle$ are the ancillary states that are utilized for the gate
operations.

An adiabatic geometric gate is realized by using a cyclic evolution of the
zero-energy adiabatic eigenstate of $\hat{H}_{\mathrm{0}}(t)$ to generate a
nontrivial geometric phase on the qubit state $|q\rangle$. This can be
understood more clearly by first writing the control pulses as [18]

$\displaystyle\Omega_{A}(t)$ $\displaystyle=\Omega_{0}\sin[\theta(t)],$ (2a)
$\displaystyle\Omega_{B}(t)$
$\displaystyle=\Omega_{0}\cos[\theta(t)]e^{i\gamma(t)}.$ (2b)

The relative magnitudes and phases between the two pulses are controlled by
the time-dependent angles $\theta(t)$ and $\gamma(t)$, respectively. The
overall scale of the pulse amplitude $\Omega_{0}$ (which we set to be a
constant for the duration of the protocol and zero otherwise) determines the
instantaneous adiabatic gap of $\hat{H}_{\mathrm{0}}(t)$:

$\Omega_{\mathrm{ad}}(t)\equiv\frac{1}{2}\sqrt{|\Omega_{A}(t)|^{2}+|\Omega_{B}(t)|^{2}}=\frac{\Omega_{0}}{2}.$
(3)

This gap separates the instantaneous zero-energy dark state (which is
orthogonal to the upper level $|a_{1}\rangle$) of Hamiltonian
$\hat{H}_{\mathrm{0}}(t)$ from its instantaneous bright states with energy
$\pm\Omega_{0}/2$.

By performing a “double-STIRAP protocol” on the $\Lambda$ system, one evolves
the dark state

$|\mathrm{d}(t)\rangle=\cos[\theta(t)]|q\rangle-e^{i\gamma(t)}\sin[\theta(t)]|a_{2}\rangle$
(4)

cyclically as $|q\rangle\rightarrow|a_{2}\rangle\rightarrow|q\rangle$. This
corresponds to a cyclic variation of the pulse parameter $\theta(t)$ (see Ref.
[18] or Appendix A for details) that results in a geometric phase being
written on the qubit state $|q\rangle$. If we pick the phase of our pulse
$\gamma(t)$ to be

$\gamma(t)=\gamma_{0}\Theta\left(t-\frac{t_{g}}{2}\right),$ (5)

with $\Theta(t)$ being the Heaviside step function, the geometric phase is
simply $\gamma_{0}$ [18]. In this paper, we denote the initial and final gate
times as $t=0$ and $t=t_{g}$, respectively. Since the pulse $\Omega_{B}(t)=0$
at $t_{g}/2$, the discontinuity of $\gamma(t)$ at $t_{g}/2$ [Eq. (5)] does not
affect the adiabaticity condition.

Figure 2: A STIRAP-based two-qubit gate proposed in Ref. [4] for trapped-ion
settings. This scheme is more complicated than our scheme (Fig. 1). It
requires a $\Lambda$-level structure for each qubit, which consists of the
logical qubit state $|1\rangle$ and an ancillary state $|a\rangle$
controllably coupled to another ancillary state $|e\rangle$. It also requires
a total of eight control tones (shown by red arrows); four for each qubit.
This scheme also utilizes a phonon-mediated interaction, a resource that is
not available in most quantum computing platforms including superconducting
qubit setups.

If the computational state $|q\rangle$ is a single-qubit state then the
double-STIRAP protocol will implement a single-qubit phase gate [4, 6, 18,
19]. As pointed out in Ref. [4], if, instead, the computational state
$|q\rangle$ is chosen to be a logical two-qubit state $|11\rangle$, the double
STIRAP protocol then implements a two-qubit gate (assuming that no other qubit
states are affected by the dynamics). For the adiabatic limit
[$\dot{\theta}(t)/\Omega_{0}\rightarrow 0$], the resulting gate unitary in the
logical qubit subspace is [18]

$\displaystyle\hat{U}_{\mathrm{G,q}}=|00\rangle\langle 00|+|01\rangle\langle
01|+|10\rangle\langle 10|+e^{i\gamma_{0}}|11\rangle\langle 11|.$ (6)

As an example, the conditional-$Z$ (CZ) gate is obtained by setting the angle
parameter $\gamma_{0}=\pi$.

As shown by Eq. (6), for the double-STIRAP protocol to realize a two-qubit
gate, it is crucial that the geometric phase is written on only one of the
two-qubit states (e.g., the logical state $|11\rangle$) and not on other
logical qubit states ($|00\rangle$, $|01\rangle$, and $|10\rangle$). For most
gate schemes, this is accomplished using a static qubit-qubit interaction;
this is the approach taken in previous proposals for STIRAP gates in atomic
systems [4, 5]. However, such static interactions are often problematic as the
qubits continue to interact even after all control pulses are turned off.
Surprisingly, as we show below, our two-qubit gate setup does not suffer from
similar drawbacks as it does not require a static qubit-qubit interaction.

We stress that there are several crucial differences (both conceptual and
practical) between the two-qubit gate scheme we present, and the STIRAP gates
proposed in Ref. [4] for trapped ions and Ref. [5] for Rydberg atoms. As we
will see, our scheme requires only two ancillary states, and two independent
control tones. In contrast, the previous atomic proposals require four
ancillary states and either four [5] or eight [4] control tones (cf. Fig. 2).
Our approach also ultimately requires only a generic dispersive interaction,
as opposed to the more specific interactions required in the atomic proposals
(phonon-mediated interactions [4] or the Rydberg blockade [5]).

###  II.2 Generic realization of STIRAP two-qubit gates using time-dependent
couplings

In this section, we discuss in detail our geometric two-qubit gate design
based on the STIRAP protocol. Our scheme is conceptually different from the
scheme in Refs. [4, 5] and is compatible with remotely connected qubit setups
[39, 26, 27, 28, 29]. The basic working principle of our two-qubit gate relies
on time-dependent couplings [25, 45, 26, 27, 46, 47, 28, 29] that mediate the
excitation transfer between the qubits and a common auxiliary system that
serves as a bus or coupler [see Fig. 1(b)]. The $\Lambda$-level structure
required for our double-STIRAP protocol is formed by utilizing two levels in
qubit $A$ (levels $g$ and $e$), qubit $B$ (levels $e$ and $f$), and the
auxiliary system (levels 0 and 1). In the order of increasing energy, we label
the energy levels in each qubit by $g,e,f,h,\cdots$ and those in the auxiliary
system by $0,1,2,\cdots$. In particular, we use the state of composite qubit-
auxiliary system $|ge,1\rangle\equiv|a_{1}\rangle$ for the upper level and
$|ee,0\rangle\equiv|11\rangle$ and $|gf,0\rangle\equiv|a_{2}\rangle$ for the
two lower levels of the $\Lambda$ system [see Fig. 3(a)]. (Note that here the
words “upper” and “lower” states are used to refer to the schematic diagram of
the $\Lambda$ system in Fig. 3 and do not necessarily reflect the energies of
the levels.) We use the notation $|ab,c\rangle$ to denote the composite state
of two qubits coupled to a common auxiliary system where $a,b$, and $c$ are
the states of qubit $A$, qubit $B$, and the auxiliary system, respectively.

Our protocol assumes that the $g\rightarrow e$ transition in qubit $A$, the
$0\rightarrow 1$ transition in the auxiliary system, and the $e\rightarrow f$
transition in qubit $B$ have nondegenerate transition frequencies. As a
result, static couplings between the qubits and auxiliary system cause at most
weak hybridization. The gate is activated by modulating the couplings such
that the modulation frequencies
$\omega_{\mathrm{mod},A}\equiv\varepsilon_{ge1}-\varepsilon_{ee0}$ and
$\omega_{\mathrm{mod},B}\equiv\varepsilon_{ge1}-\varepsilon_{gf0}$ resonantly
drive the left and right transitions of the $\Lambda$ system, respectively.
Here, $\varepsilon_{k}$ denotes the energy of state $|k\rangle$. This
modulation can be described by the Hamiltonian of the $\Lambda$ system,

$\displaystyle\hat{H}_{\Lambda}(t)=\hat{H}_{\mathrm{static},\Lambda}+\hat{H}_{\mathrm{mod},\Lambda}(t),$
(7)

where $\hat{H}_{\mathrm{static},\Lambda}$ represents the static (time-
independent) contributions and

$\displaystyle\hat{H}_{\mathrm{mod},\Lambda}(t)=$
$\displaystyle\frac{1}{2}\biggl{[}g^{\mathrm{ac}}_{A}(t)e^{-i\omega_{\mathrm{mod},A}t}n_{A,eg}\hat{\Pi}_{A,eg}\hat{a}+$
$\displaystyle\qquad
g^{\mathrm{ac}}_{B}(t)e^{-i\omega_{\mathrm{mod},B}t}n_{B,fe}\hat{\Pi}_{B,fe}\hat{a}\biggr{]}+\mathrm{H.c.}$
(8)

captures the modulation of the couplings. Here, $g^{\mathrm{ac}}_{j}(t)$ is
the strength of the tunable coupling between qubit $j$ and the auxiliary
system, $\omega_{\mathrm{mod},j}$ is the frequency of the modulation tone $j$,
$n_{j,kl}$ is the matrix element for the
$|l\rangle_{j}\rightarrow|k\rangle_{j}$ transition of qubit $j$,
$\hat{\Pi}_{j,kl}$ is the outer product $|k\rangle_{j}{}_{j}\langle l|$ for
states in qubit $j$, and $\hat{a}$ is the annihilation operator of the
auxiliary mode. Note that many choices are possible for the auxiliary system;
for notational simplicity, we take it to be a (possibly nonlinear) bosonic
mode with annihilation operator $\hat{a}$. Moreover, while we assume modulated
couplings here to illustrate the basic physics, we show below that one can
also realize our gate using static couplings and an auxiliary system whose
transition frequency is modulated.

For a double-STIRAP protocol to implement a two-qubit gate, one needs to
ensure that the protocol imprints a geometric phase onto only one of the qubit
states ($|ee,0\rangle\equiv|11\rangle$) and not onto other qubit states. To
this end, we use the left-resonant modulation tone to move the excitation in
qubit $A$ to the auxiliary system only if qubit $B$ is in state $|e\rangle$,
and not in state $|g\rangle$. Formally, this means that the transition
$|eg,0\rangle\leftrightarrow|gg,1\rangle$ has to be energetically detuned from
the $|ee,0\rangle\leftrightarrow|ge,1\rangle$ transition. Note that the
transitions $|eg,0\rangle\leftrightarrow|gg,1\rangle$ and
$|ge,0\rangle\leftrightarrow|gg,1\rangle$ form another $\Lambda$ configuration
[Fig. 3(b)], where the modulation introduces population leakage and unwanted
phase to be imprinted on the computational states $|eg,0\rangle$ and
$|ge,0\rangle$. Since this $\Lambda$ system is not part of our ideal gate
protocol, we call it a “bad” $\Lambda$ system (labeled $\Lambda_{\rm bad}$
throughout), to differentiate it from the “good” $\Lambda$ system that is used
to implement the gate.

Figure 3:  (a) The “good”’ $\Lambda$ system as in Fig. 1(a) containing the
target computational state $|11\rangle\equiv|ee,0\rangle$, which ideally
acquires a geometric phase. (b) The “bad” $\Lambda$ system that has the
computational state $|10\rangle\equiv|eg,0\rangle$; we do not want any phase
to be written on this state. To realize this selectivity and a two-qubit gate,
the transition $|eg,0\rangle\leftrightarrow|gg,1\rangle$ in the left-arm of
the $\Lambda_{\rm bad}$ system needs to be detuned from the left modulation
tone $\Omega_{A}(t)$. This can be realized by introducing a dispersive
interaction $\chi$ between qubit $B$ and the auxiliary mode.

The degeneracy between the $|ee,0\rangle\leftrightarrow|ge,1\rangle$ and
$|eg,0\rangle\leftrightarrow|gg,1\rangle$ transitions can be lifted by
introducing a dispersive shift $\chi$ between states in the auxiliary system
and qubit $B$, where the required dispersive interaction has the form

$\hat{H}_{\mathrm{disp}}=\chi(\hat{\Pi}_{B,ee}-\hat{\Pi}_{B,gg})\hat{a}^{\dagger}\hat{a}.$
(9)

This dispersive shift can be realized by introducing a nonresonant static
coupling $g^{\mathrm{dc}}_{BC}$ between qubit $B$ and the auxiliary system.
Ultimately, our gate is protected from the coherent errors arising from the
$\Lambda_{\mathrm{bad}}$ system by the energetic detuning ($2\chi$) between
the left-arm transitions of the $\Lambda$ and $\Lambda_{\rm bad}$ systems. As
such, the dispersive interaction strength $\chi$ sets a limit to the gate
speed. We stress that $\chi$ is not a qubit-qubit interaction, but rather an
effective interaction just between the auxiliary system and qubit $B$. Thus,
this speed limit on our gate is very different from more conventional
approaches, where the gate speed is typically limited by the size of a direct
qubit-qubit interaction. As mentioned, the lack of any static qubit-qubit
interaction provides us with another advantage: when control pulses are off,
there is in principle no residual qubit-qubit interaction.

###  II.3 Schemes to implement effective tunable interactions

Figure 4:  Different schemes for realizing tunable interactions for our two-
qubit gate: A. Direct method: (I) time-dependent couplings, static
intermediate auxiliary mode. B. Indirect method: (II) static couplings, time-
dependent intermediate auxiliary mode, and (III) static couplings, time-
dependent modulated qubits.

As discussed above, our protocol uses appropriately engineered tunable
interactions between the auxiliary system and the qubits; these can be
implemented in various ways. The most direct method was already introduced in
Sec. II.2: use explicitly time-modulated coupling elements (tunable couplings)
between each qubit and the auxiliary system [bus or coupling mode; see method
(I) of Fig. 4]. Such tunable couplings have been realized in superconducting-
circuit experiments involving transmon qubits [25, 26, 27, 28, 29] and
fluxonium qubits [48]. This direct method is favorable, since it requires only
one static qubit-bus coupling, and hence there is in principle no residual
qubit-qubit coupling when the gate is off.

In addition to the direct method [method (I) of Fig. 4], one can also
indirectly realize effective tunable interactions by combining static qubit-
auxiliary system interactions with either parametric modulation of the
auxiliary system [method (II) of Fig. 4], and/or the qubits themselves [method
(III) of Fig. 4]. The auxiliary coupler here can be a tunable transmon,
fluxonium, etc. An advantage of this scheme is that the modulation of
frequency-tunable couplers have been demonstrated in a number of experiments
[30, 31, 32, 33, 34, 36, 35]. Note that in contrast to the direct method,
these indirect schemes require two static qubit-auxiliary system couplings. As
a result, there is a mediated static qubit-qubit interaction that remains on
even when the gate is over. This is common to most existing protocols using
couplers. As we show in our detailed simulations, through judicious parameter
choice and circuit design, this spurious coupling can be minimized and even in
some cases eliminated.

##  III Mitigation of Coherent Errors

In this section, we discuss accelerating our gate using the shortcuts-to-
adiabaticity (STA) protocol based on the superadiabatic transitionless driving
(SATD) method [49, 18]. Moreover, we go beyond the RWA settings and use a two-
pronged analytical approach proposed in Ref. [19] to enhance the performance
of our protocol in the presence of the following types of errors.

  1. 1.

Nonadiabatic errors that arise when the gates are accelerated.

  2. 2.

Non-RWA errors due to modulation crosstalk as well as spurious couplings
between computational and noncomputational states.

###  III.1 Nonadiabatic errors

In the adiabatic limit where the protocol time becomes much longer than the
inverse adiabatic gap $1/\Omega_{0}$ [cf. Eq. (3)], the geometric gate
approaches a perfect gate. However, any dissipation of the lower
$\Lambda$-system levels makes long protocol times incompatible with high
fidelity. If one naively accelerates the adiabatic protocol without additional
pulse shaping, the resulting nonadiabatic transitions will result in
substantial errors. To alleviate this problem, one can turn to STA approaches
(see, e.g., Refs. [20, 21, 22, 49, 18, 50]).

Following Ref. [18], we use the SATD shortcuts method [49, 18] that allows
fast operation without nonadiabatic errors, by having the system evolution
follow a dressed adiabatic eigenstate (see Appendix B). The accelerated
protocol is implemented through pulse shaping of the original control fields
[49, 51, 52, 18, 53]. One appealing feature of the SATD method is that this
pulse modification can be described analytically, with a modification of the
form [18]

$\displaystyle\Omega_{A}(t)$
$\displaystyle\rightarrow\tilde{\Omega}_{A}(t)\equiv\Omega_{0}\left[\sin[\theta(t)]+4\frac{\cos[\theta(t)]\ddot{\theta}(t)}{\Omega_{0}^{2}+4\dot{\theta}^{2}(t)}\right],$
(10a) $\displaystyle\Omega_{B}(t)$
$\displaystyle\rightarrow\tilde{\Omega}_{B}(t)\equiv\Omega_{0}e^{i\gamma(t)}\left[\cos[\theta(t)]-4\frac{\sin[\theta(t)]\ddot{\theta}(t)}{\Omega_{0}^{2}+4\dot{\theta}^{2}(t)}\right],$
(10b)

where the angle $\gamma(t)$ [Eq. (5)] remains the same as the adiabatic
version. One can show [18] that the accelerated protocol derived using Eqs.
(10) gives rise to the same unitary $\hat{U}_{\mathrm{G},\mathrm{q}}$ in the
qubit subspace as in the adiabatic limit [cf. Eq. (6)]. For an ideal (RWA)
Hamiltonian of the $\Lambda$ system [Eq. (1)] and a fixed gate time, our SATD
method gives an infinite number of perfect gate protocols (each with a
different pulse shape, characterized by a different value of $\Omega_{0}$). We
can use this degeneracy as a resource to mitigate non-RWA errors in realistic
systems as discussed in the following subsection and Ref. [19].

###  III.2 Non-RWA errors

Coherent errors also arise from nonresonant couplings that would be neglected
within the RWA. We can partially mitigate their effects by using the strategy
introduced in Ref. [19]. This strategy consists of two steps (see Appendix C
for details): (1) using the power optimal $\Omega_{0}=\Omega_{\mathrm{opt}}$
where $\Omega_{\mathrm{opt}}/2\pi\simeq 1.135/t_{g}$, and (2) frequency
chirping the control fields.

##  IV Gate performance

We start by considering the fundamental advantages and limitations of our two-
qubit gate. The system evolution generated by our control pulses, including
coherent and possibly dissipative errors, corresponds to a quantum map
$\mathcal{M}$. We wish to quantify how close this is to the ideal unitary two-
qubit gate we are interested in. We do this via the state-averaged gate
fidelity given by [54, 55]

$\displaystyle\bar{F}$
$\displaystyle=\frac{1}{4}+\frac{1}{80}\sum_{\mu,\nu=\\{0,x,y,z\\}|\\{\hat{\sigma}^{\mu}_{A}\otimes\hat{\sigma}^{\nu}_{B}\neq\mathds{1}\otimes\mathds{1}\\}}\mathrm{tr}\bigg{\\{}\left[\hat{U}_{\mathrm{g}}((\hat{\sigma}^{\mu}_{A}\otimes\hat{\sigma}^{\nu}_{B})\oplus\mathbf{0}_{\mathrm{q}^{c}})\hat{U}^{\dagger}_{\mathrm{g}}\right]\mathcal{M}\left[(\hat{\sigma}^{\mu}_{A}\otimes\hat{\sigma}^{\nu}_{B})\oplus\mathbf{0}_{\mathrm{q}^{c}}\right]\bigg{\\}},$
(11)

where $\hat{\sigma}^{\mu}_{A}\otimes\hat{\sigma}^{\nu}_{B}$ (with
$\mu,\nu=\\{0,x,y,z\\}$) are the Pauli matrices acting on qubits $A$ and $B$
within the qubit subspace (namely, the $4\times 4$ block) and
$\mathbf{0}_{\mathrm{q}^{c}}$ is a zero matrix in the nonqubit subspace. Here,

$\hat{U}_{\mathrm{g}}=(\hat{U}_{1,\mathrm{q}}\hat{U}_{\mathrm{G,q}})\oplus\mathds{1}_{\mathrm{q}^{c}}$
(12)

is the ideal target unitary two-qubit gate operation that performs a unitary
$\hat{U}_{\mathrm{G,q}}$ [cf. Eq. (6)] in the qubit subspace up to a trivial
single-qubit phase gate
$\hat{U}_{1,\mathrm{q}}=\mathrm{exp}\left(i\phi_{A}\right)\left(\sigma^{z}_{A}\otimes\mathds{1}_{B}\right)+\mathrm{exp}\left(i\phi_{B}\right)\left(\mathds{1}_{A}\otimes\sigma^{z}_{B}\right)$.
Throughout this paper, the gate fidelity is calculated by optimizing over the
trivial single-qubit phases $\phi_{A}$ and $\phi_{B}$ which is achieved by
choosing the single-qubit $Z$ gates on qubits $A$ and $B$ that maximize the
gate fidelity. From the fidelity, we can calculate the state-averaged gate
error using $\bar{\varepsilon}=1-\bar{F}$.

###  IV.1 Robustness advantages of accelerated geometric gates

To understand the advantages of our accelerated STIRAP two-qubit gate against
parameter and pulse uncertainties, we compare it against the simplest
corresponding gate based on dynamical phases. This latter approach would
simply employ a static $ZZ$ interaction to imprint a dynamical phase on state
$|ee,0\rangle\equiv|11\rangle$. Up to innocuous global and single-qubit
phases, the dynamical $ZZ$ gate can be described using a RWA Hamiltonian

$\hat{H}_{\mathrm{dyn}}^{ZZ}=\frac{\Omega_{0}}{4}\hat{\sigma}_{A}^{z}\otimes\hat{\sigma}_{B}^{z},$
(13)

where $\Omega_{0}/4$ is the $ZZ$ interaction strength.

Figure 5: Comparison of the gate robustness between different CZ gates. (a)
The state-averaged gate errors $\langle\bar{\varepsilon}\rangle$ are averaged
over a 20% uncertainty ($\eta\in[-0.2,0.2]$) in the pulse amplitude. The plots
are for the uncorrected adiabatic protocol (blue curve), SATD-accelerated
adiabatic protocol (red curve), and dynamical gate (purple dot) as a function
of $\Omega_{0}t_{g}/2\pi$. (b) Differential sensitivity of the state-averaged
gate error with respect to small pulse magnitude uncertainty ($\eta$) as a
function of $\Omega_{0}t_{g}/2\pi$. The plots are for the SATD-accelerated
adiabatic protocol (red curve), and dynamical gate (purple dot). For both
panels, $\Omega_{0}t_{g}=\pi$ for a dynamical CZ gate and the green dashed
line denotes the SATD power-optimal protocol
($\Omega_{0}=\Omega_{\mathrm{opt}}$).

An appealing feature of our geometric gate compared to dynamical gates is its
robustness against imperfections in control pulses; this is inherited from the
underlying purely adiabatic gate. For example, imagine a situation where there
is an overall uncertainty in the scale of applied pulse amplitudes (due, e.g.,
to some unknown attenuation). This would result in pulse envelopes
$\Omega_{A,B}(t)\rightarrow\Omega_{A,B}(t)(1+\eta)$, where $\eta$
parameterizes overall uncertainty in the scale of the amplitudes. We could
introduce a corresponding uncertainty in the dynamical gate, by letting
$\Omega_{0}\rightarrow\Omega_{0}(1+\eta)$. Given this uncertainty, we consider
two metrics for characterizing the robustness of a given gate.

  1. 1.

In the limit of small uncertainties $\eta$, the gate error will have a
quadratic dependence on $\eta$. The coefficient of this quadratic term (i.e.,
$d^{2}\bar{\varepsilon}/d\eta^{2}\equiv\xi$) is thus a measure of robustness.

  2. 2.

For larger uncertainties (with $\eta$ being described by some probability
distribution), we could characterize robustness by averaging the gate error
over $\eta$. For concreteness, we take $\eta$ to be described by a uniform
probability distribution over the interval $[-\eta_{0},\eta_{0}]$; the average
gate error is

$\langle\bar{\varepsilon}(\Omega_{0}t_{g})\rangle\equiv\frac{1}{2\eta_{0}}\int_{-\eta_{0}}^{\eta_{0}}d\eta\,\bar{\varepsilon}(\Omega_{0}t_{g},\eta).$
(14)

To calculate the gate performance, we evolve the initial states using the
Schrödinger equation with the RWA Hamiltonian given by Eq. (1) for our STIRAP
gate and by Eq. (13) for the dynamical gate. In this paper, we choose the
lowest-order polynomial [Eq. (32)] that gives a smooth variation of
$\theta(t)$ [Eq. (31)] for our STIRAP gate. The simulations here and
throughout this paper are done numerically using the Python package QuTiP [56,
57].

Figure 5 shows the robustness of CZ gates in the presence of pulse-amplitude
uncertainties. Results are shown for the uncorrected adiabatic (blue curve),
accelerated adiabatic (red curve), and dynamical-gate protocols (purple dot).
For panel (a), we consider large uncertainties, and consider the
$\eta$-averaged gate error with $\eta_{0}=0.2$. We see that the accelerated
STIRAP gate provides, as expected, a robustness advantage over the dynamical
gate. While this improvement is modest if one uses power-optimized STIRAP
(green dashed line), it can be improved significantly by using slightly larger
values of $\Omega_{0}$.

Figure 5(b) shows the differential sensitivity of the gate to small values of
$\eta$. We again see advantages compared to the dynamical gate. The
sensitivity to $\eta$ can be reduced by orders of magnitude compared to the
dynamical gate, by using SATD with power levels slightly higher than the
power-optimal case. For the power-optimal case, the differential sensitivity
of the SATD gate does not change significantly (i.e., $\xi\simeq 0.802$ for
$t_{g}=45$ ns) even if we consider the realistic system as in Sec. V.2 where
we include the $38$ lowest-energy levels with all the non-RWA dynamics.

###  IV.2 Comparison against Raman-type gates

The accelerated STIRAP gate also had advantages over Raman-style two-qubit
gates that use an auxiliary system (coupler) to facilitate virtual
transitions. For the Raman gate, the qubit-auxiliary system couplings are
modulated with frequencies that are detuned from the qubit-auxiliary mode
transition frequencies and the gate operates by using an effective qubit-qubit
coupling obtained via higher-order, coupler-mediated processes. As a result of
being perturbative, the effective interaction here is small, giving rise to a
slower gate compared to our SATD protocol (for a fixed modulation amplitude)
that uses resonant modulations. This ultimately results in the SATD gate being
more tolerant of dissipation than the Raman approach (see, e.g., Appendix H in
Ref. [19] for details on the comparison with the Raman protocol).

###  IV.3 Fundamental limitation of the STIRAP two-qubit gate performance:
$\Lambda_{\rm bad}$ system

The performance of our gate is fundamentally limited by the coherent error
resulting from the undesired driving of the $\Lambda_{\rm bad}$ system. This
error is ultimately determined by the size of the dispersive coupling $\chi$
between qubit $B$ and the auxiliary system (see Fig. 3). As described in Sec
II.2, this nonresonant driving of the $\Lambda_{\rm bad}$ system can cause
both leakage and phase errors. To understand these errors better, we simulate
our gate protocol using the idealized RWA Hamiltonian [Eq. (1)] with only one
additional spurious term: the nonresonant drive of the left arm of the
$\Lambda_{\rm bad}$ system, namely the
$|ge,0\rangle\leftrightarrow|gg,1\rangle$ transition driven by $\Omega_{A}(t)$
(see Fig. 3).

In our simulation, we use the power-optimal SATD protocol
($\Omega_{0}=\Omega_{\mathrm{opt}}$) that minimizes the coherent error due to
the driving of the $\Lambda_{\rm bad}$ system. We do not chirp the modulation
frequency, as that is not effective against bad-$\Lambda$ errors. One simple
error arising from the nonresonant driving of the $\Lambda_{\rm bad}$
transition is a modification of the effective two-qubit phase by
$\phi_{ZZ}\equiv\phi_{00}+\phi_{11}-\phi_{01}-\phi_{10}$ where $\phi_{k}$ is
the phase on the logical qubit state $|k\rangle$. We can correct this error by
simply shifting the geometric phase $\gamma_{0}$ that our gate implements by
$\phi_{ZZ}$, i.e.,

$\gamma_{0}\rightarrow\gamma_{0}-\phi_{ZZ}.$ (15)

Once this phase error is corrected, the dominant remaining error is leakage
from state $|ge,0\rangle$ to $|gg,1\rangle$. In the large dispersive regime
($2\chi t_{g}/2\pi\gtrsim 10$), the gate error as a function of gate time can
be numerically fitted to (not shown)

$\displaystyle\bar{\varepsilon}=\begin{cases}c_{\mathrm{phase}}(\chi
t_{g})^{-2}&\text{without $ZZ$ phase correction},\\\\[10.00002pt]
c_{\mathrm{leak}}(\chi t_{g})^{-4}&\text{with $ZZ$ phase
correction},\end{cases}$ (16)

where $c_{\mathrm{phase}}\simeq 0.05$ and $c_{\mathrm{leak}}\simeq 0.26$. We
note that the above quadratic and quartic scaling of the phase and leakage
errors in the small parameter $\chi t_{g}$ are consistent with the analytic
arguments based on the Magnus expansion of the coherent errors (see Refs. [19,
53]). For the case where the $ZZ$ correction is applied, the numerical fit to
obtain the scaling of the gate error [Eq. (16)] is done by neglecting the fast
oscillations of the gate error with respect to the gate time $t_{g}$ that
arise due to the shuttling of the population between states $|gg,1\rangle$ and
$|ge,0\rangle$.

##  V Fluxonium qubits coupled to a common auxiliary system

###  V.1 Basic setup

We now analyze realizations of our gate in a concrete physical setting: two
fluxonium qubits [41, 42, 43, 44] coupled to a common auxiliary system (such
as a cavity, transmon, fluxonium, or a particular mode of a waveguide). Our
analysis and modeling will consider gate performance in the presence of both
coherent and dissipative error mechanisms. We note that the topic of fluxonium
two-qubit gates has been the subject of considerable recent activity, both
theoretical [58, 59, 60, 61, 62, 63, 64, 65] and experimental [66, 67, 68, 69,
48]. As we discuss below, the combination of parametric modulation and
accelerated adiabatic evolution gives our approach a number of unique
potential advantages over other approaches.

Fluxonium circuits are attractive qubit platforms as they possess large
nonlinearities and long relaxation times (which can reach milliseconds [42,
70, 43, 44, 71]). They can also exhibit first-order insensitivity to $1/f$
flux noise dephasing. A fluxonium qubit consists of a Josephson junction
(typically shunted by a capacitance) that forms a loop with a superinductance
[see Fig. 6]. The Hamiltonian is

$\hat{H}_{j}=4E_{C,j}\hat{n}_{j}^{2}-E_{J,j}\cos\hat{\varphi}_{j}+\frac{1}{2}E_{L,j}(\hat{\varphi}_{j}-2\pi\Phi_{\mathrm{ext},j}/\Phi_{0})^{2},$
(17)

where $E_{C,j}$, $E_{J,j}$, and $E_{L,j}$ are the capacitive, Josephson, and
inductive energies of qubit $j=A,B$; $\hat{n}_{j}$ and $\hat{\varphi}_{j}$ are
the qubit-$j$ charge and phase operators, obeying
$[\hat{\varphi}_{j},\hat{n}_{j}]=i$. The loop formed by the Josephson junction
and the shunting inductance is threaded by an external magnetic flux
$\Phi_{\mathrm{ext},j}$ and $\Phi_{0}=h/2e$ is the flux quantum.

Figure 6: Circuit schematics for systems consisting of two fluxonium qubits
$A$ and $B$, (a) coupled via tunable couplings to an auxiliary system (bus) or
(b) statically coupled to a frequency-modulated auxiliary system (which we
here consider to be a frequency-tunable transmon).

In our analysis below, we consider two scenarios where either (1) the fluxonia
are coupled to a fixed-frequency auxiliary system (e.g., a bus) via directly
tunable couplings, [Fig. 6(a)] or (2) the qubits are statically coupled to a
frequency-modulated auxiliary system (coupler) [Fig. 6(b)]. We model the bus
in the first scheme [Fig. 6(a)] as a harmonic oscillator and the coupler in
the second scheme [Fig. 6(b)] as a weakly anharmonic oscillator (e.g., a
transmon) with Hamiltonian

$\hat{H}_{C}=\omega_{C}\hat{a}^{\dagger}\hat{a}-U\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}\hat{a},$
(18)

where $\hat{a}$ is the coupler lowering operator, $\omega_{C}$ is the harmonic
frequency, and $U$ is the nonlinearity.

The total Hamiltonian of the composite system (fluxonium qubits $A$ and $B$,
auxiliary system $C$) is

$\hat{H}(t)=\hat{H}_{A}+\hat{H}_{B}+\hat{H}_{C}+\hat{H}_{\mathrm{int}}+\hat{H}_{\mathrm{mod}}(t).$
(19)

The static interaction term $\hat{H}_{\mathrm{int}}$ is given by

$\hat{H}_{\mathrm{int}}=\sum_{j=A,B}g^{\mathrm{dc}}_{jC}\hat{n}_{j}(\hat{a}^{\dagger}+\hat{a})+g^{\mathrm{dc}}_{AB}\hat{n}_{A}\hat{n}_{B},$
(20)

where $g^{\mathrm{dc}}_{jC}$ is the static coupling strength between qubit $j$
and the auxiliary system, and $g^{\mathrm{dc}}_{AB}$ is the static qubit-qubit
coupling strength. The term $\hat{H}_{\mathrm{mod}}(t)$ describes the temporal
modulation that is applied to perform the gate (see Fig. 4), and is described
further below. While we consider capacitive couplings here, our basic gate
physics is also applicable to systems with inductive couplings.

Diagonalizing the time-independent Hamiltonian
$\hat{H}_{\mathrm{static}}=\hat{H}(t)-\hat{H}_{\mathrm{mod}}(t)$ results in
dressed eigenstates of the form $|ab,c\rangle$, where the labels $a$, $b$ and
$c$ refer to the states of qubit $A$, qubit $B$, and the auxiliary system,
respectively. We always operate in a strongly detuned regime, where the
hybridization between the auxiliary system and the two qubits is weak (hence,
the dressed eigenstates each have a strong overlap with a single uncoupled
eigenstate). In this paper, we take the logical qubit states to be
$|gg,0\rangle\equiv|00\rangle$, $|ge,0\rangle\equiv|01\rangle$,
$|eg,0\rangle\equiv|10\rangle$, $|ee,0\rangle\equiv|11\rangle$. These four
qubit states together with the two ancillary states
($|ge,1\rangle\equiv|a_{1}\rangle$ and $|gf,0\rangle\equiv|a_{2}\rangle$) of
the $\Lambda$ system form the computational subspace of our STIRAP two-qubit
gate [see Figs. 1(a) and 3(a)].

Each of the three modulation schemes depicted in Fig. 4 corresponds to a
different $\hat{H}_{\mathrm{mod}}(t)$ in Eq. (19). Here, we only analyze the
first two: the direct tunable coupling scheme [method (I)] and the indirect
frequency-modulated auxiliary mode scheme [method (II)]. Method (III) is also
a possible implementation route, i.e., use static qubit-auxiliary system
couplings and modulate the frequencies of both qubits $A$ and $B$.

###  V.2 Method (I): time-dependent couplings, static intermediate auxiliary
mode

We begin by first considering method (I) of Fig. 4, where the needed resonant
interactions between the qubits and the auxiliary mode are induced by
modulating tunable couplings. Such tunable couplings have been implemented in
several transmon-based setups [25, 26, 27, 28, 29]. There are also recent
theoretical studies [72, 60, 65] and an experiment [48] studying tunable
couplings for fluxonium qubits. The focus here is the operation of our gate,
and not the internal workings of a specific tunable coupling device. As such,
we simply model these elements via time-dependent matrix elements, leading to

$\hat{H}_{\mathrm{mod}}(t)=\frac{1}{2}\sum_{j=A,B}(g^{\mathrm{ac}}_{j}(t)e^{-i\int_{0}^{t}\tilde{\omega}_{\mathrm{mod},j}(t^{\prime})dt^{\prime}}+\mathrm{c.c.})\hat{n}_{j}(\hat{a}^{\dagger}+\hat{a}),$
(21)

where we have written the time-dependent coupling between qubit $j$ and the
auxiliary system in terms of a complex envelope function
$g^{\mathrm{ac}}_{j}(t)$, and a chirped modulation frequency
$\tilde{\omega}_{\mathrm{mod},j}(t)$ [Eq. (39)]. The amplitude
$g^{\mathrm{ac}}_{j}(t)$ is directly related to our effective pulse amplitudes
$\tilde{\Omega}_{j}(t)$ [Eqs. (10)] by

$\displaystyle g^{\mathrm{ac}}_{A}(t)$
$\displaystyle=\tilde{\Omega}_{A}(t)/\langle
ge,1|\hat{n}_{A}(\hat{a}^{\dagger}+\hat{a})|ee,0\rangle,$ (22a) $\displaystyle
g^{\mathrm{ac}}_{B}(t)$ $\displaystyle=\tilde{\Omega}_{B}(t)/\langle
ge,1|\hat{n}_{B}(\hat{a}^{\dagger}+\hat{a})|gf,0\rangle,$ (22b)

where the transitions $|ee,0\rangle\leftrightarrow|ge,1\rangle$ and
$|gf,0\rangle\leftrightarrow|ge,1\rangle$ correspond to the left and right
arms of the $\Lambda$ system, respectively [see Fig. 3(a)].

Qubit $A$ | Qubit $B$
---|---
$E_{J,A}/h$ = 5.5 GHz |  $E_{J,B}/h$ = 5.7 GHz
$E_{C,A}/h$ = 1.5 GHz |  $E_{C,B}/h$ = 1.2 GHz
$E_{L,A}/h$ = 1.0 GHz |  $E_{L,B}/h$ = 1.0 GHz
$\Phi_{\mathrm{ext},A}=0.5\Phi_{0}$ | $\Phi_{\mathrm{ext},B}=0.5\Phi_{0}$
$\omega_{A}/2\pi$ = 0.606 GHz |  $\omega_{B}/2\pi$ = 0.354 GHz
Auxiliary system | Static couplings
$\omega_{C}/2\pi$ = 7.5 GHz |  $g^{\mathrm{dc}}_{AC}/h$ = 0
$U$ = 0 |  $g^{\mathrm{dc}}_{BC}/h$ = 0.8 GHz
|  $g^{\mathrm{dc}}_{AB}/h$ = 0
Table 1: Circuit parameters used for simulating the method (I) implementation
of our gate (time-dependent couplings, static intermediate auxiliary mode).
Here $\omega_{A}$, $\omega_{B}$, and $\omega_{C}$ are the frequency
differences between the lowest and second lowest bare energy levels of the
uncoupled qubit $A$, qubit $B$ and auxiliary system, respectively. Parameters
for the two qubits match those used in Ref. [63], while the parameters for the
auxiliary system and static couplings are chosen to mitigate spurious non-RWA
processes.

For our simulations, we choose realistic system parameters that yield
reasonable coupling strengths and avoid spurious resonances; see Table 1. The
resulting transition frequencies and matrix elements for the $\Lambda$ and
$\Lambda_{\rm bad}$ transitions (calculated using the scqubits package [73])
are shown in Table 2. While our modulation tones are relatively high frequency
(i.e., $\omega_{\mathrm{mod},A}/2\pi=6.94$ GHz and
$\omega_{\mathrm{mod},B}/2\pi=2.86$ GHz), similar high-frequency modulations
have been employed in recent experiments with flux-tunable couplings (see
e.g., Ref. [74]).

For the method (I) implementation scheme, we require only a single static
coupling, between qubit $B$ and the auxiliary system (i.e.,
$g^{\mathrm{dc}}_{BC}\neq 0$). This has several desirable features. The
tunable coupling operator $\hat{n}_{j}(\hat{a}^{\dagger}+\hat{a})$ connects
only transitions involving qubit $j$. This implies that there is zero
modulation crosstalk for this scheme (i.e., modulation tones designed to drive
qubit $A$ transitions do not accidentally drive qubit $B$ transitions and vice
versa). Furthermore, the use of only a single static coupling also eliminates
any qubit-qubit $ZZ$ interactions when the gate is inactive.

(a) $\Lambda$ system
---
$|k\rangle$ | $|l\rangle$ | $\omega_{k,l}/2\pi$ | $|\langle k|\hat{n}_{A}(\hat{a}^{\dagger}+\hat{a})|l\rangle|$ | $|\langle k|\hat{n}_{B}(\hat{a}^{\dagger}+\hat{a})|l\rangle|$
$|ee,0\rangle$ | $|ge,1\rangle$ | 6.94 GHz | 0.12 | 0.0
$|gf,0\rangle$ | $|ge,1\rangle$ | 2.86 GHz | 0.0 | 0.55
(b) $\Lambda_{\rm bad}$ system
$|k\rangle$ | $|l\rangle$ | $\omega_{k,l}/2\pi$ | $|\langle k|\hat{n}_{A}(\hat{a}^{\dagger}+\hat{a})|l\rangle|$ | $|\langle k|\hat{n}_{B}(\hat{a}^{\dagger}+\hat{a})|l\rangle|$
$|eg,0\rangle$ | $|gg,1\rangle$ | 6.54 GHz | 0.09 | 0.0
$|ge,0\rangle$ | $|gg,1\rangle$ | 6.80 GHz | 0.0 | 0.04
Table 2: The frequencies ($\omega_{k,l}\equiv\varepsilon_{l}-\varepsilon_{k}$)
and matrix elements [$|\langle
k|\hat{n}_{A,B}(\hat{a}^{\dagger}+\hat{a})|l\rangle|$] of
$|k\rangle\rightarrow|l\rangle$ transitions for (a) $\Lambda$ system and (b)
$\Lambda_{\rm bad}$ system, for the method (I) simulations. The modulation
frequencies of the tunable couplings are $\omega_{\mathrm{mod},A}/2\pi=6.94$
GHz and $\omega_{\mathrm{mod},B}/2\pi=2.86$ GHz. The detuning of the
$\Lambda_{\rm bad}$ system from the $\Lambda$ system are
$2\chi/2\pi\equiv|\omega_{ee0,ge1}-\omega_{eg0,gg1}|/2\pi=0.4$ GHz (for the
left arm) and $|\omega_{gf0,ge1}-\omega_{ge0,gg1}|/2\pi=3.94$ GHz (for the
right arm), respectively.

In the following, we discuss the performance of a CZ gate implemented using
this direct scheme. To this end, we simulated the evolution of the system’s
density matrix, keeping the 38 lowest-energy levels (a number sufficient for
reaching convergence).

####  V.2.1 Gate performance with coherent errors only

We first discuss gate performance by taking into account all coherent errors,
but neglecting dissipation (this is treated in the next section). Coherent
errors are mitigated as discussed above: we use the SATD protocol with power
optimal $\Omega_{0}$, i.e., $\Omega_{0}=\Omega_{\mathrm{opt}}$, (see Appendix
C.1 for details), and chirped tone frequencies as per Eq. (39) (see Appendix
C.2 for details). We also cancel unwanted $ZZ$ phases by optimizing the choice
of the adiabatic phase $\gamma_{0}$ that determines our pulse envelopes; cf.
Eq. (15). Using the parameters in Table 1, we can derive the CZ gate pulse
shapes (i.e., modulation envelopes and chirped frequency profiles) for our
SATD gate (see the inset of Fig. 7 and Appendix D for an example power-optimal
SATD pulse shape).

Fig. 7 shows the state-averaged gate error $\bar{\varepsilon}$ [Eq. (11)] for
a CZ gate as a function of gate time $t_{g}$. As expected, errors decrease
with increasing $t_{g}$, due to the corresponding decrease in instantaneous
power (which reduces the amplitude of non-RWA processes). We find that the
errors here are dominated by leakage in the $\Lambda_{\rm bad}$ system, i.e.,
leakage from $|eg,0\rangle$ to $|gg,1\rangle$ (see Fig. 3). The oscillation of
the coherent error with respect to the gate time is due to the detuned Rabi
oscillations that shuttle the population back and forth between states
$|eg,0\rangle$ and $|gg,1\rangle$. Ignoring the fast oscillation, we show in
Fig. 7 that the scaling of the total gate error of the full system matches
closely the scaling of leakage error $\bar{\varepsilon}=c_{\mathrm{leak}}(\chi
t_{g})^{-4}$ [Eq. (16); dashed gray line] of the reduced model in Sec. IV.3,
where $c_{\mathrm{leak}}\simeq 0.26$ and $\chi$ is the dispersive shift of the
left-arm of the $\Lambda_{\mathrm{bad}}$ system.

Figure 7: State-averaged gate error $\bar{\varepsilon}=1-\bar{F}$ of a CZ gate
as a function of gate time $t_{g}$, in the absence of dissipation, for the
direct tunable coupling scheme [method (I) of Fig. 4]. Results are calculated
using 38 lowest-energy levels, including all non-RWA processes. We use power-
optimized SATD pulses to mitigate both non-RWA and adiabatic errors, as
discussed in the main text (see the inset for an example of the SATD
modulation tone envelope $\Omega_{j}(t)$ at $t_{g}$ = 45 ns). The gate error
of our full multilevel simulations (red curve) follows closely the scaling
$c_{\mathrm{leak}}(\chi t_{g})^{-4}$ [gray dashed line; cf. Eq. (16)] of a
reduced model that considers only one non-RWA process, i.e., the nonresonant
drive of the left arm of the $\Lambda_{\rm{bad}}$ system. This indicates that
this is the dominant coherent error channel. The circuit parameters used are
given in Table 1.

####  V.2.2 Gate performance with $T_{1}$ dissipation

We now add dissipation to our simulations. As our qubits are operated at their
sweet spot, the system is first-order insensitive to dephasing due to $1/f$
flux noise. The most dominant dissipations are thus $T_{1}$ decay processes.
We consider $T_{1}$ relaxation that arises from dielectric loss in the circuit
capacitors, as it is usually the dominant contribution [43, 44]. The $T_{1}$
relaxation rate for the transition $|k\rangle\rightarrow|l\rangle$ due to the
capacitor $C_{j}$ ($j=A,B,C$) is [75, 76]

$1/(T_{1})_{j;kl}=\frac{8\omega_{l,k}E_{C_{j}}}{|\omega_{l,k}|Q_{\mathrm{diel}}}\left[\coth\left(\frac{\omega_{l,k}}{2k_{\mathrm{B}}T}\right)+1\right]|\langle
l|\hat{n}_{j}|k\rangle|^{2}.$ (23)

Here, $Q_{\mathrm{diel}}$ is the dielectric quality factor,
$\omega_{l,k}\equiv\varepsilon_{k}-\varepsilon_{l}$ is the transition energy
between the composite-system eigenstates $|l\rangle$ and $|k\rangle$, and $T$
is the temperature. The total relaxation rate for a given transition is

$1/(T_{1})_{kl}=\sum_{j=A,B,C}1/(T_{1})_{j;kl}.$ (24)

Table 3 shows the most dominant relaxation times involving any one of the six
computational states $|k\rangle$ (four logical qubit states together with two
ancillary states in the $\Lambda$ system). They are calculated for zero
temperature $(T=0)$ and with $Q_{\mathrm{diel}}=10^{6}$.

$|k\rangle$ | $|l\rangle$ |  $(T_{1})_{kl}$ ($\mu$s)
---|---|---
$|ge,1\rangle$ | $|ge,0\rangle$ | 21.51
$|gf,0\rangle$ | $|ge,0\rangle$ | 24.73
Table 3:  The $T_{1}$ relaxation times for the most dominant relaxation
processes involving computational states, for the method (I) simulations. The
relaxation times are calculated at zero temperature ($T=0$) and for a
dielectric quality factor $Q_{\mathrm{diel}}=10^{6}$.

To simulate our gate including the effects of $T_{1}$ relaxation, we use the
Lindblad master equation

$\partial_{t}\hat{\rho}(t)=-i[\hat{H}(t),\hat{\rho}(t)]+\sum_{l<k}\left(\hat{Z}_{kl}\hat{\rho}(t)\hat{Z}_{kl}-\frac{1}{2}\\{\hat{Z}^{2}_{kl},\hat{\rho}(t)\\}\right).$
(25)

Here, $\hat{\rho}(t)$ is the density matrix of our composite qubit-auxiliary
system, $\hat{H}(t)$ is the Hamiltonian used to realize the two-qubit gate
[Eq. (19)], and the $\hat{Z}_{kl}$ are jump operators associated with each
relaxation transition:

$\hat{Z}_{kl}=\sqrt{1/(T_{1})_{kl}}|l\rangle\langle k|.$ (26)

We numerically simulate the above master equation to assess the impact of both
non-RWA errors and dissipation on our gate performance; results are shown in
Fig. 8. The gate error is a nonmonotonic function of the gate time $t_{g}$.
For short $t_{g}$, non-RWA errors limit performance, with the errors
decreasing generally as $t_{g}$ increases. As discussed in Sec. V.2.1, the
coherent errors for the full multilevel model follow closely the gate-time
scaling of leakage errors due to the $\Lambda_{\mathrm{bad}}$ system alone,
i.e., they match the error scaling of the reduced model in Sec. II.2. This
scaling is indicated by a gray dashed line in Fig. 8. For longer $t_{g}$, the
errors are dominated by dissipation and increase linearly with increasing
$t_{g}$. Letting $T_{1,\rm{min}}$ be the shortest relaxation time in the
system, we fit the dissipation-induced error to
$\bar{\varepsilon}_{\mathrm{diss}}=c_{\mathrm{diss}}t_{g}/T_{1,\rm{min}}$, as
shown by the green dash-dot line, where $c_{\mathrm{diss}}\simeq 0.152$.

Figure 8: State-averaged gate error $\bar{\varepsilon}=1-\bar{F}$ of a CZ gate
as a function of gate time $t_{g}$ for the direct tunable coupling scheme
[method (I) of Fig. 4], including both coherent and dissipative error
channels. Simulations treat 38 lowest-energy levels, all non-RWA processes,
and $T_{1}$ relaxation due to dielectric capacitor loss with
$Q_{\mathrm{diel}}=10^{6}$ (cf. Refs. [77, 76]). As before, power-optimized,
chirped SATD pulses are used to realize the gate. In the short-gate-time
regime, the gate error of the full model (red curve) follows the scaling of
leakage error $\bar{\varepsilon}_{\mathrm{leak}}=c_{\mathrm{leak}}(\chi
t_{g})^{-4}$ [gray dashed line; Eq. (16)] of the reduced model in Sec. II.2,
where $c_{\mathrm{leak}}\simeq 0.26$. In the long-gate-time limit, dissipation
dominates the gate error where the error scales with the numerically fitted
results of
$\bar{\varepsilon}_{\mathrm{diss}}=c_{\mathrm{diss}}t_{g}/T_{1,\rm{min}}$
(green dash-dot line), where $c_{\mathrm{diss}}\simeq 0.152$. We find that an
error $\bar{\varepsilon}<10^{-3}$ can be achieved over a range of gate times
$45\lesssim t_{g}\lesssim 60{\,\rm ns}$. Circuit parameters are listed in
Table 1.

As we understand the error scaling with $t_{g}$ in both the coherent and
dissipation limited regimes, we can make estimates for how the minimum error
and optimal gate time scale with system parameters. Note that the coherent
error is controlled by the dispersive coupling $\chi$ (which controls the
detuning of the $\Lambda_{\mathrm{bad}}$ system), whereas the dissipation is
controlled by $T_{1,\rm{min}}$. By adding the two error scalings
$\bar{\varepsilon}_{\mathrm{leak}}$ and $\bar{\varepsilon}_{\mathrm{diss}}$ in
quadrature and then minimizing with respect to $t_{g}$, we find a minimum
error that scales like

$\bar{\varepsilon}_{\rm{min}}=c_{\mathrm{min}}(\chi T_{1,\rm{min}})^{-4/5},$
(27)

at an optimal gate time $t_{g}\simeq 7.96(T_{1,\rm{min}}/\chi^{4})^{1/5}$,
where $c_{\mathrm{min}}\simeq 1.35$. Our numerics reveal that an extremely
high gate fidelity $>0.9995$ can be achieved for gate times in the range
$t_{g}=45$–$60$ ns; this is in good agreement with the above estimate. For the
optimal gate time of $t_{g}=45$ ns, the modulation amplitudes [cf. Eq. (22)]
have maximum values of $g^{\mathrm{ac}}_{A,\mathrm{max}}/2\pi\simeq 478.6$ MHz
and $g^{\mathrm{ac}}_{B,\mathrm{max}}/2\pi\simeq 104.4$ MHz. While the gate
performance is extremely promising, we note that we have not modeled the
internal workings of the modulated tunable couplings. Nonetheless, these
results show that method (I) is an extremely promising implementation
strategy.

###  V.3 Method (II): static couplings, time-dependent intermediate auxiliary
mode

We next consider an alternate implementation strategy for our gate, where one
uses static interactions between the auxiliary system and both qubits, and
time modulates the frequency of the auxiliary system [method (II) of Fig. 4]
[30, 31, 32, 33, 34, 36, 35, 37, 38]. This corresponds to choosing the
modulation term in Eq. (19) to be

$\hat{H}_{\mathrm{mod}}(t)=\delta\omega_{C}(t)\hat{a}^{\dagger}\hat{a},$ (28)

where
$\delta\omega_{C}(t)=\frac{1}{2}\sum_{j=A,B}\left(g^{\mathrm{ac}}_{j}(t)e^{-i\int_{0}^{t}\tilde{\omega}_{\mathrm{mod},j}(t^{\prime})dt^{\prime}}+\textrm{c.c.}\right)$.
For our SATD gate, the modulation envelopes $g^{\mathrm{ac}}_{j}(t)$ are
related to the SATD pulse amplitudes $\tilde{\Omega}_{j}(t)$ [Eqs. (10)] by

$\displaystyle g^{\mathrm{ac}}_{A}(t)$
$\displaystyle=\tilde{\Omega}_{A}(t)/\langle
ge,1|\hat{a}^{\dagger}\hat{a}|ee,0\rangle,$ (29a) $\displaystyle
g^{\mathrm{ac}}_{B}(t)$ $\displaystyle=\tilde{\Omega}_{B}(t)/\langle
ge,1|\hat{a}^{\dagger}\hat{a}|gf,0\rangle,$ (29b)

respectively corresponding to the left and right arms of the $\Lambda$ system
[see Fig. 3(a)].

Qubit $A$ | Qubit $B$
---|---
$E_{J,A}/h$ = 4.5 GHz |  $E_{J,B}/h$ = 3.5 GHz
$E_{C,A}/h$ = 1.8 GHz |  $E_{C,B}/h$ = 1.1 GHz
$E_{L,A}/h$ = 1.5 GHz |  $E_{L,B}/h$ = 1.0 GHz
$\Phi_{\mathrm{ext},A}=0.5\Phi_{0}$ | $\Phi_{\mathrm{ext},B}=0.5\Phi_{0}$
$\omega_{A}/2\pi$ = 1.79 GHz |  $\omega_{B}/2\pi$ = 0.86 GHz
Auxiliary system | Static couplings
$\omega_{C}/2\pi$ = 1.11 GHz |  $g^{\mathrm{dc}}_{AC}/h$ = 0.63 GHz
$U$ = 5 MHz |  $g^{\mathrm{dc}}_{BC}/h$ = 0.6 GHz
|  $g^{\mathrm{dc}}_{AB}/h$ = 0.04 GHz
Table 4: Circuit parameters for the method (II) implementation of our gate
(static couplings, time-dependent auxiliary mode frequency). Here
$\omega_{A}$, $\omega_{B}$, $\omega_{C}$ are the frequency differences between
the lowest and second lowest bare energy levels of the uncoupled qubit $A$,
qubit $B$ and auxiliary system (coupler), respectively.  (a) $\Lambda$ system
---
$|k\rangle$ | $|l\rangle$ | $\omega_{k,l}/2\pi$ | $|\langle k|\hat{a}^{\dagger}\hat{a}|l\rangle|$
$|ee,0\rangle$ | $|ge,1\rangle$ | -0.84 GHz | 0.194
$|gf,0\rangle$ | $|ge,1\rangle$ | -2.43 GHz | 0.111
(b) $\Lambda_{\rm bad}$ system
$|k\rangle$ | $|l\rangle$ | $\omega_{k,l}/2\pi$ | $|\langle k|\hat{a}^{\dagger}\hat{a}|l\rangle|$
$|eg,0\rangle$ | $|gg,1\rangle$ | -0.71 GHz | 0.181
$|ge,0\rangle$ | $|gg,1\rangle$ | 0.31 GHz | 0.366
Table 5: The frequencies ($\omega_{k,l}\equiv\varepsilon_{l}-\varepsilon_{k}$)
and matrix elements ($\langle k|\hat{a}^{\dagger}\hat{a}|l\rangle$) of
$|k\rangle\rightarrow|l\rangle$ transitions for (a) $\Lambda$ system and (b)
$\Lambda_{\rm bad}$ system, for the method (II) simulations. The modulation
frequencies of the auxiliary system are $\omega_{\mathrm{mod},A}/2\pi=0.84$
GHz and $\omega_{\mathrm{mod},B}/2\pi=2.43$ GHz. The detuning of the left arm
of the $\Lambda_{\rm bad}$ system from the left arm of the $\Lambda$ system is
$2\chi/2\pi\equiv||\omega_{ee0,ge1}|-|\omega_{eg0,gg1}||/2\pi=0.13$ GHz. The
detuning of the right arm of the $\Lambda_{\rm bad}$ system from the left arm
of the $\Lambda$ system is $||\omega_{gf0,ge1}|-|\omega_{eg0,gg1}||/2\pi=0.53$
GHz.

For this scheme, we need nonzero static couplings between the auxiliary system
and each qubit, i.e., $g^{\mathrm{dc}}_{AC},g^{\mathrm{dc}}_{BC}\neq 0$. This
will generically lead to a static $ZZ$ interaction between the qubits. While
the resulting phase error $\phi_{ZZ}$ that arises during the gate
implementation can be corrected by simply shifting the phase $\gamma_{0}$ of
our control tones, the bigger issue is that this $ZZ$ interaction remains on
even when the gate is turned off. By judiciously choosing the parameters of
our circuits and introducing a direct static interqubit coupling
($g^{\mathrm{dc}}_{AB}$) with a suitably chosen strength, one can in principle
completely eliminate the static $ZZ$ interaction; analogous strategies have
been used in other systems; see, e.g., Ref. [38]. For our setup (two fluxonium
qubits connected via a frequency-tunable transmon coupler), the $ZZ$
interactions can be eliminated if we pick parameters so that the transmon-
coupler frequency is in between the two qubit frequencies. This motivates our
choice of having asymmetric parameters for qubits $A$ and $B$ (see Table 4).

Optimal gate performance also requires parameter choices that minimize non-RWA
errors. After an approximate numerical optimization, we pick parameters that
mitigate both the static $ZZ$ interaction and non-RWA error channels, while
still remaining compatible with experiments. The resulting parameters are
given in Table 4. For these values, the $ZZ$ interaction strength is
negligible, i.e.,
$(\varepsilon_{11}+\varepsilon_{00}-\varepsilon_{01}-\varepsilon_{10})/h\simeq
6.57$ kHz; it could in principle be minimized even further with more parameter
fine tuning. The frequencies and matrix elements of the $\Lambda$ and
$\Lambda_{\rm bad}$ system transitions for our system parameters are shown in
Table 5.

Figure 9:  State-averaged gate error $\bar{\varepsilon}=1-\bar{F}$ of a CZ
gate as a function of gate time $t_{g}$, for the method (II) implementation of
our gate (cf. Fig. 4), in the absence of dissipation. Results are calculated
using 20 lowest-energy levels and all non-RWA processes. We use power-
optimized SATD pulses with frequency chirping (as discussed in the main text),
and optimize the adiabatic phase $\gamma_{0}$ to cancel spurious $ZZ$
interaction terms. The coherent error has contributions from several
processes, and hence is slightly above the scaling that would be expected if
the only error was $\Lambda_{\mathrm{bad}}$ system leakage, i.e., the scaling
$c_{\mathrm{leak}}(\chi t_{g})^{-4}$ [gray dashed line; see Eq. (16)] Circuit
parameters are given in Table 4.  Figure 10: State-averaged gate error
$\bar{\varepsilon}=1-\bar{F}$ for the CZ gate as a function of gate time
$t_{g}$ for the method (II) implementation of our scheme (cf. Fig. 4),
including both coherent and dissipative errors. Results are calculated using
20 lowest-energy levels, all non-RWA processes, and $T_{1}$ relaxation due to
dielectric capacitor loss (with $Q_{\mathrm{diel}}=10^{6}$). We use power-
optimized SATD pulse shapes with frequency chirping (see the main text), and
correct for unwanted $ZZ$ interaction phase shifts by optimizing the adiabatic
phase shift $\gamma_{0}$ [cf. Eq. (15)]. Circuit parameters are given in Table
4.

Using the optimal parameter set as shown in Table 4, we simulated our
accelerated CZ gate in the Hilbert space spanned by the 20 lowest-energy
levels of the system. Figures 9 and 10 show the gate errors as functions of
gate time, calculated with all non-RWA error channels for the case without and
with $T_{1}$ dissipation, respectively.

We find that there are several equally dominant non-RWA error channels. In
addition to leakage from $|eg,0\rangle$ to $|gg,1\rangle$ (arising from
nonresonant driving of the left arm of the $\Lambda_{\rm bad}$ system), there
is also, e.g., the nonresonant driving of the right arm of the $\Lambda_{\rm
bad}$ system by the modulation tones $\Omega_{A}(t)$. This causes leakage from
$|ge,0\rangle$ to $|gg,1\rangle$. While these coherent errors generally
decrease with increasing gate time, the total gate error no longer follows the
gate-time scaling $c_{\mathrm{leak}}(\chi t_{g})^{-4}$ [gray dashed line; cf.
Eq. (16)] of the gate error obtained from the reduced model that considers
only the nonresonant drive of the left arm of the $\Lambda_{\rm bad}$ system.
Again, this signifies the presence of several non-RWA error channels.

If dissipation is taken into account as in Fig. 10, it will dominate the gate
errors in the long-gate-time regime (see $t_{g}\gtrsim 200$ ns in Fig. 10) and
wash away the oscillations in the gate error. In this dissipation-dominated
regime, the gate error increases linearly with increasing gate time. As shown
in Fig. 10, at an intermediate gate time regime ($t_{g}\approx 130$ ns), where
neither the coherent errors nor dissipation dominates, we can obtain a gate
fidelity of approximately 0.999 (even with the inclusion of all non-RWA error
channels and $T_{1}$ dissipation with dielectric quality factor
$Q_{\mathrm{diel}}=10^{6}$, where the dominant $T_{1}$ relaxation times are
shown in Table 6). For the optimal gate time of $t_{g}=130$ ns, the modulation
amplitudes [cf. Eq. (29)] have maximum values of
$g^{\mathrm{ac}}_{A,\mathrm{max}}/2\pi\simeq 102.5$ MHz and
$g^{\mathrm{ac}}_{B,\mathrm{max}}/2\pi\simeq 179.1$ MHz.

$|k\rangle$ | $|l\rangle$ |  $(T_{1})_{kl}$ ($\mu$s)
---|---|---
$|gf,0\rangle$ | $|ge,0\rangle$ | 39.55
$|ge,1\rangle$ | $|ge,0\rangle$ | 92.67
$|gf,0\rangle$ | $|gg,1\rangle$ | 99.38
$|ee,0\rangle$ | $|ge,0\rangle$ | 103.61
$|eg,0\rangle$ | $|gg,0\rangle$ | 103.98
Table 6: The $T_{1}$ relaxation times for the most dominant processes
involving computational states for the method (II) simulations. These are
calculated for dielectric capacitor loss with a dielectric quality factor
$Q_{\mathrm{diel}}=10^{6}$ and for zero temperature ($T=0$).

##  VI Conclusions

In this work we introduce and analyze a method that harnesses the basic
physics of STIRAP to realize an accelerated adiabatic geometric two-qubit
gate. Our approach is more flexible and resource efficient than previous
proposals for STIRAP-based two-qubit gates in atomic platforms, and is
especially well suited to platforms using an auxiliary system (transmission
line, cavity, qubit) as a coupler. We analyze in detail implementations of our
basic idea in a system of two fluxonium superconducting qubits, considering
both implementations based on direct modulation of coupling amplitudes, and
based on modulation of the auxiliary system frequency. Using realistic
parameters, we find competitive gate fidelities and gate times. We also
discuss how our protocols have an inbuilt robustness to certain kinds of
parameter variations, a feature derived from their connection to a purely
adiabatic protocol.

In future work, it would be interesting to explore how the basic gate
mechanism introduced here could be used in other systems, ranging from
advanced superconducting qubits (e.g., $0$-$\pi$ qubits) to atomic platforms
(e.g., neutral atom systems employing Rydberg levels). It would also be
interesting to study how our approach could be optimally employed in modular
approaches to quantum computing, i.e., to realize remote gates in multiqubit
systems coupled via a common bus mode.

###### Acknowledgements.

This work is financially supported by the Army Research Office under Grant
Number W911NF-19-1-0328. We are grateful to the University of Chicago Research
Computing Center for computing resources to perform the calculations in this
paper.

##  Appendix A Physics of the STIRAP gate

In this section, we discuss the basic STIRAP geometric gate that utilizes a
three-level $\Lambda$ configuration [Fig. 1(a)]. The STIRAP gate was proposed
in Refs. [4, 6] for the adiabatic case and its accelerated version was
subsequently formulated in Ref. [18]; we follow Ref. [18] for the discussion
below.

We begin by noting that Hamiltonian $\hat{H}_{\mathrm{0}}(t)$ [Eq. (1)] of the
$\Lambda$ system has an instantaneous zero-energy dark state

$|\mathrm{d}(t)\rangle=\cos[\theta(t)]|q\rangle-e^{i\gamma(t)}\sin[\theta(t)]|a_{2}\rangle,$
(30)

which is a superposition of a qubit state $|q\rangle$ and the ancillary state
$|a_{2}\rangle$, which are the two lower levels of the $\Lambda$ system. Note
that $|\mathrm{d}(t)\rangle$ is orthogonal to the ancillary state
$|a_{1}\rangle$, i.e., the upper level of the $\Lambda$ system. The STIRAP
gate operates by utilizing a geometric evolution of the zero-energy dark
state. By performing a “double-STIRAP protocol”, one realizes a cyclic
adiabatic evolution $|q\rangle\rightarrow|a_{2}\rangle\rightarrow|q\rangle$.
This results in a geometric phase being imprinted on the qubit state
$|q\rangle$ [4, 6]. If $|q\rangle$ is chosen to be the logical qubit state
$|11\rangle$, the protocol then realizes an arbitrary geometric controlled-
phase gate [4]. Note that such a geometric gate can be performed without
requiring precise timing of the pulses.

The double-STIRAP protocol relies on cyclically varying $\theta(t)$, where
here we take

$\theta(t)=\begin{cases}\displaystyle\frac{\pi}{2}P(t/t_{g}),&\displaystyle
0\leq t\leq\frac{t_{g}}{2}\\\\[8.0pt]
\displaystyle\frac{\pi}{2}\left[1-P\left(\frac{t}{t_{g}}-\frac{1}{2}\right)\right],&\displaystyle\frac{t_{g}}{2}<t\leq
t_{g},\end{cases}$ (31)

with $P(x)$ being a monotonic function that increases from $P(0)=0$ to
$P(1/2)=1$. For a smooth on-and-off switch of the control fields, we pick a
polynomial that satisfies
$\dot{\theta}(0)=\dot{\theta}(t_{g}/2)=\dot{\theta}(t_{g})=\ddot{\theta}(0)=\ddot{\theta}(t_{g}/2)=\ddot{\theta}(t_{g})=0$.
Specifically, we take [18]

$P(x)=6\left(2x\right)^{5}-15\left(2x\right)^{4}+10\left(2x\right)^{3}.$ (32)

To get a geometric phase $\gamma_{0}$, we choose the relative phase
$\gamma(t)$ between the control fields as [18]

$\gamma(t)=\gamma_{0}\Theta\left(t-\frac{t_{g}}{2}\right),$ (33)

where $\Theta(t)$ denotes a Heaviside step function.

For the adiabatic limit [$\dot{\theta}(t)/\Omega_{0}\rightarrow 0$], one can
show that [18] the evolution of the qubit subspace is decoupled from that of
the ancillary-level subspace (the subspace spanned by the ancillary states
$|a_{1}\rangle$ and $|a_{2}\rangle$) and the dark state
$|\mathrm{d}(t)\rangle$ acquires a geometric phase $\gamma_{0}$ at $t=t_{g}$.
The result is a geometric gate that is described by a unitary
$\hat{U}_{\mathrm{G,q}}$ [Eq. (6)] in the qubit subspace. The full unitary
evolution in the adiabatic limit is given by
$\hat{U}_{\mathrm{G}}=\hat{U}_{\mathrm{G,q}}\oplus\hat{U}_{\mathrm{G,anc}}$,
where the $\hat{U}_{\mathrm{G.anc}}$ are the unitary operators in ancillary
subspaces (see Ref. [18]).

##  Appendix B SATD dressing for accelerated protocols

Here we provide a brief overview of the “dressed state” approach of the STA
protocols [49, 18], used for accelerating adiabatic gates. The basic idea is
to let the system evolve following a “dressed” state that coincides with the
adiabatic state at the beginning and end of the protocol. This is accomplished
by choosing a dressing function $\nu(t)$ that goes to zero at $t=0$ and
$t=t_{g}$. As in Refs. [49, 18], we dress the $|\mathrm{d}(t)\rangle$ [Eq.
(4)] as $|\mathrm{d}_{\nu}(t)\rangle$, i.e.,

$|\mathrm{d}_{\nu}(t)\rangle=\exp\left[-i\nu(t)\hat{J}_{x}\right]|\mathrm{d}(t)\rangle,$
(34)

where
$\hat{J}_{x}=(|\mathrm{b}_{+}(t)\rangle\langle\mathrm{d}(t)|+|\mathrm{b}_{-}(t)\rangle\langle\mathrm{d}(t)|+\mathrm{H.c.})/\sqrt{2}$
with $|\mathrm{b}_{\pm}(t)\rangle$ being the bright eigenstates of the
adiabatic Hamiltonian $\hat{H}_{\mathrm{0}}(t)$ having eigenenergies
$\pm\Omega_{0}/2$.

As elaborated in Ref. [18], to have the dark state acquire a purely geometric
phase that is equal to the adiabatic geometric phase $\gamma_{0}$, we have to
impose a constraint $\nu(t_{g}/2)=0$. To fulfill this constraint, we use the
SATD [49, 18] dressing function

$\nu(t)=\nu_{\mathrm{SATD}}(t)\equiv\arctan\left[\frac{2\dot{\theta}(t)}{\Omega_{0}}\right].$
(35)

As shown in Refs. [49, 18], the accelerated protocol obtained using the SATD
dressing is implemented by changing the original pulse sequence following the
_analytical_ formula given in Eq. (10) of the main text.

##  Appendix C Mitigation and cancellation of non-RWA errors

Coherent errors also arise from nonresonant couplings that would be neglected
within the RWA. We can partially mitigate their effects by using the strategy
introduced in Ref. [19]. To understand this approach, we start by writing the
full system Hamiltonian including non-RWA terms as

$\hat{H}(t)=\hat{H}_{\mathrm{0}}(t)+\hat{H}_{\mathrm{err}}(t),$ (36)

where $\hat{H}_{\mathrm{0}}(t)$ is the Hamiltonian for the ideal resonant
(RWA) processes as in Eqs. (1) and $\hat{H}_{\mathrm{err}}(t)$ describes all
the unwanted nonresonant dynamics.

Following Ref. [19], we implement an enhanced version of the SATD protocol
that mitigates the effects of non-RWA errors. This protocol has two key steps.

###  C.1 Step 1: power optimization

Given the degeneracy of perfect STA protocols in the RWA limit, we first
mitigate unwanted effects from non-RWA processes described by
$\hat{H}_{\mathrm{err}}(t)$ [Eq. (36)], by choosing the SATD protocol that
minimizes the rms value of the control field amplitude

$g^{\mathrm{ac}}_{\mathrm{rms}}\equiv\sqrt{\frac{1}{t_{g}}\int_{0}^{t_{g}}(\left|g^{\mathrm{ac}}_{A}(t)\right|^{2}+\left|g^{\mathrm{ac}}_{B}(t)\right|^{2})dt}.$
(37)

Here, $g^{\mathrm{ac}}_{j}(t)$ is the amplitude of the modulation tone $j$
($j=A,B$) (which includes the SATD correction); see Eqs. (22) and (29) for the
expression of $g^{\mathrm{ac}}_{j}(t)$ for the direct tunable couplings and
indirect-modulated auxiliary mode scheme, respectively. This power
minimization strategy can be understood from the fact that coherent errors due
to non-RWA dynamics generally decrease as the pulse amplitude is reduced. One
effect of the non-RWA processes is to induce time-dependent energy shifts of
the computational levels whose magnitudes scale as
$|g^{\mathrm{ac}}_{\mathrm{rms}}|^{2}$. The leading order of the energy shifts
can be derived from second-order perturbation theory or a Magnus-expansion-
based approach [52, 53] as

$\displaystyle\delta\varepsilon_{k}(t)$
$\displaystyle=\sum_{\begin{subarray}{c}j=A,B\\\
\sigma=\pm\end{subarray}}\sum_{l\,|\Delta_{kl,\sigma}^{(j)}\neq
0}\frac{|g^{\mathrm{ac}}_{j}(t)\langle
k|\hat{n}_{j}(\hat{a}^{\dagger}+\hat{a})|l\rangle|^{2}}{4\Delta_{kl,\sigma}^{(j)}},$
(38a) $\displaystyle\delta\varepsilon_{k}(t)$
$\displaystyle=\sum_{\begin{subarray}{c}j=A,B\\\
\sigma=\pm\end{subarray}}\sum_{l\,|\Delta_{kl,\sigma}^{(j)}\neq
0}\frac{|g^{\mathrm{ac}}_{j}(t)\langle
k|\hat{a}^{\dagger}\hat{a}|l\rangle|^{2}}{4\Delta_{kl,\sigma}^{(j)}},$ (38b)

for the direct tunable couplings [method (I) of Fig. 4] and frequency-
modulated auxiliary-mode scheme [Method (II) of Fig. 4], respectively. Here,
$\Delta_{kl,\pm}^{(j)}=\varepsilon_{k}-\varepsilon_{l}\pm\omega_{\mathrm{mod},j}$
is the detuning of the transition $|k\rangle\leftrightarrow|l\rangle$ from the
modulation tone $\omega_{\mathrm{mod},j}$, while
$\hat{n}_{j}(\hat{a}^{\dagger}+\hat{a})$ and $\hat{a}^{\dagger}\hat{a}$ are
the modulation operators for the direct tunable couplings and frequency-
modulated auxiliary-mode scheme, respectively. The sums in Eqs. (38) are
evaluated for all nonresonant processes involving the computational state
$|k\rangle$, which can be any of the four qubit states ($|gg,0\rangle$,
$|ge,0\rangle$, $|eg,0\rangle$, $|ee,0\rangle$) or the two ancillary states in
the $\Lambda$ system ($|ge,1\rangle$, $|gf,0\rangle$). The intermediate states
$|l\rangle$ in Eqs. (38) could be either any of the six computational levels
(i.e., resulting in crosstalk processes) or noncomputational states (i.e.,
“leakage” levels). Since the energy shifts of the computational levels
increase with the pulse amplitude, the SATD protocol that minimizes coherent
errors has the minimum driving power $|g^{\mathrm{ac}}_{\mathrm{rms}}|^{2}$.
As derived in Ref. [19], this power-optimal SATD protocol has a value of
adiabatic pulse magnitude $\Omega_{0}=\Omega_{\mathrm{opt}}$, where
$\Omega_{\mathrm{opt}}/2\pi\simeq 1.135/t_{g}$.

###  C.2 Step 2: cancellation of non-RWA errors via modification of
accelerated protocol pulses

Having minimized the energy shifts $\delta\varepsilon_{k}(t)$ [Eqs. (38)] due
to nonresonant processes by choosing the power-optimal SATD protocol, our next
step to enhance the gate fidelity is to partially cancel the non-RWA errors by
modifying the SATD pulses. To this end, we use an analytic correction strategy
that follows Ref. [19]

Our correction technique is to cancel the undesired time-dependent non-RWA
energy shifts by introducing a time-dependent variation of the two modulation
frequencies $\omega_{\mathrm{mod},j}$ ($j={A,B}$) (i.e., a frequency chirp):

$\displaystyle\omega_{\mathrm{mod},j}$
$\displaystyle\rightarrow\tilde{\omega}_{\mathrm{mod},j}(t)=\omega_{\mathrm{mod},j}+\delta\omega_{\mathrm{mod},j}(t)$
(39)

with $\delta\omega_{j}(t)$ being

$\displaystyle\delta\omega_{\mathrm{mod},A}(t)$
$\displaystyle=\delta\varepsilon_{ge1}(t)-\delta\varepsilon_{ee0}(t),$ (40a)
$\displaystyle\delta\omega_{\mathrm{mod},B}(t)$
$\displaystyle=\delta\varepsilon_{ge1}(t)-\delta\varepsilon_{gf0}(t),$ (40b)

and $\delta\varepsilon_{k}(t)$ given in Eqs. (38). These frequency shifts
partially correct the non-RWA error, as they ensure that at every instant of
time, each modulation tone is resonant (up to the leading order) with the
corresponding transition it is supposed to modulate.

Our correction approach thus results in a two-step modification of the
original pulse in Eqs. (2). For a given gate time $t_{g}$, we first choose a
power-optimal value of $\Omega_{0}$ as discussed in Sec. C.1 and introduce the
SATD correction to the original pulses as in Eqs. (10). Next, we chirp each of
the two central tone frequencies as shown in Eq. (39). Thus, the total
modification of the modulation tone $j$ ($j=A,B$) is given by

$\displaystyle\Omega_{j}(t)$
$\displaystyle\,\mathrm{exp}\left(i\omega_{\mathrm{mod},j}t\right)\rightarrow\tilde{\Omega}_{j}(t)\,\mathrm{exp}\left(i\int_{0}^{t}\tilde{\omega}_{\mathrm{mod},j}(t^{\prime})dt^{\prime}\right),$
(41)

where the $\tilde{\Omega}_{j}(t)$ [see Eqs. (10)] are the SATD envelopes of
the modulation tones, and the $\tilde{\omega}_{\mathrm{mod},j}(t)$ are the
chirped modulation tone frequencies.

##  Appendix D SATD pulse shape

The control field required to implement our two-qubit gate consists of two
modulation tones with complex amplitudes $\tilde{\Omega}_{A}(t)$ and
$\tilde{\Omega}_{B}(t)$ [see Eq. (2) and Eq. (10) for uncorrected and SATD-
corrected pulses, respectively]. Figure 11 shows plots of (a) amplitudes, (b)
frequency chirp-corrections, and (c) Fourier components of the modulation
tones corresponding to the case of the power-optimal SATD dynamics, each
slightly modified by inclusion of pulse smoothing at the beginning and
end—something that would be typically done in experiments.

Figure 11:  Modulation tones for realizing a CZ gate in a realistic
superconducting setup consisting of two fluxonia, both coupled to a shared
auxiliary mode via direct tunable couplings. (a) Time profiles of the pulse
envelopes. (b) Time profiles of the frequency chirp corrections
$\delta\omega_{\mathrm{mod},j}(t)$ [Eqs. (40)]. (c) Fourier transform of
$|g^{\mathrm{ac}}_{j}(f)|$ [Eqs. (22)]. Shown are the SATD pulse sequences
given in Eqs. (42) where the pulses are switched on (off) for a time duration
$t_{\mathrm{ramp}}$ in the beginning (end) of the protocol with a smooth
polynomial ramp function [Eq. (32)]. The pulses are for $t_{g}=45$ ns,
$t_{\mathrm{ramp}}=0.45$ ns, and
$\Omega_{0}/2\pi=\Omega_{\mathrm{opt}}/2\pi\simeq 25.22$ MHz. Circuit
parameters are given in Table 1.

Such modification is implemented by sandwiching the original pulses with a
short ramp up (down), having a duration $t_{\mathrm{ramp}}$, at the beginning
(end) of the protocol, during which $\tilde{\Omega}_{B}(t)$ is smoothly
switched on (off). We can write the full modulation pulses as piecewise,
smooth and continuous functions, split into three time intervals as

$\displaystyle\frac{\tilde{\Omega}_{A}(t)}{\Omega_{0}}$
$\displaystyle=\begin{cases}0&\rm{(I)},\\\
\displaystyle\sin[\theta(t_{-})]+\frac{\cos[\theta(t_{-})]\ddot{\theta}(t_{-})}{\dot{\theta}^{2}(t_{-})+\Omega_{0}^{2}/4}&\rm{(II)},\\\
0&\rm{(III)},\end{cases}$ (42a)
$\displaystyle\frac{\tilde{\Omega}_{B}(t)}{\Omega_{0}}$
$\displaystyle=\begin{cases}\displaystyle
P\left(\frac{t}{2t_{\mathrm{ramp}}}\right)&\rm{(I)},\\\\[10.0pt] \displaystyle
e^{i\gamma(t_{-})}\left\\{\cos[\theta(t_{-})]-\frac{\sin[\theta(t_{-})]\ddot{\theta}(t_{-})}{\dot{\theta}^{2}(t_{-})+\Omega_{0}^{2}/4}\right\\}&\rm{(II)},\\\\[10.0pt]
\displaystyle
1-P\left(\frac{t_{-}-t_{g}}{2t_{\mathrm{ramp}}}\right)&\rm{(III)},\end{cases}$
(42b)

with $t_{-}=t-t_{\mathrm{ramp}}$. Here, we pick $t_{\mathrm{ramp}}=0.01t_{g}$
to be sufficiently shorter than the overall pulse length. This is to ensure
that the dynamics due to the ramp will not substantially modify the dynamics
of the overall protocol, but sufficiently long that there is no abrupt change
in the pulse. The function $P(x)$ in Eq. (42b) can be any smooth function
[here we use a fifth-order polynomial as given in Eq. (32)]. We stress that
the results obtained with the inclusion of these ramps remain largely
unchanged.

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