# Protected Solid-State Qubits

Jeroen Danon* Center for Quantum Spintronics, Department of Physics, Norwegian
University of Science and Technology, NO-7491 Trondheim, Norway
<EMAIL_ADDRESS>Anasua Chatterjee  Center for Quantum Devices, Niels Bohr
Institute, University of Copenhagen, 2100 Copenhagen, Denmark  András Gyenis
Department of Electrical, Computer & Energy Engineering, University of
Colorado Boulder, Boulder, Colorado 80309, USA  Ferdinand Kuemmeth  Center for
Quantum Devices, Niels Bohr Institute, University of Copenhagen, 2100
Copenhagen, Denmark

###### Abstract

The implementation of large-scale fault-tolerant quantum computers calls for
the integration of millions of physical qubits, with error rates of physical
qubits significantly below 1%. This outstanding engineering challenge may
benefit from emerging qubits that are protected from dominating noise sources
in the qubits’ environment. In addition to different noise reduction
techniques, protective approaches typically encode qubits in global or local
decoherence-free subspaces, or in dynamical sweet spots of driven systems. We
exemplify such protective qubits by reviewing the state-of-art in protected
solid-state qubits based on semiconductors, superconductors, and hybrid
devices.

††preprint: AIP/123-QED

##  I Protection in Qubit Systems

The extraordinary power of quantum computation stems from the exponentially
large amount of information that can be associated with the entanglement in
multi-qubit systems.Nielsen and Chuang (2010) This implies that quantum
information is in principle only useful if it can stay perfectly coherent for
the whole duration of a computation. However, every physical qubit is embedded
in a noisy environment and the required access to qubit control unavoidably
couples the qubit to this environment. The coupling converts coherent
superposition states into statistical mixtures and this loss of quantum
coherence in the system directly translates to errors in the quantum
information.

Since decoherence is a generic problem, effective error mitigation and
correction are quintessential for the development of reliable quantum
computation in practice.Lidar and Brun (2013) Soon after the first quantum
computation algorithms, powerful algorithms for error correction were
proposed,Shor (1995); Steane (1996); Knill  _et al._ (2001) typically relying
on redundancy in the encoding and the measurement of syndrome operators
followed by active correction of the errors. These active methods can require
a large overhead, both in terms of required information storage capacity and
complexity of the actual algorithms. For this reason, an emerging activity is
on qubits with some kind of passive error resilience, often built into the
system Hamiltonian.

Figure 1: (a) Local fluctuations in the environment can couple to the qubit
subsystem, leading to qubit dephasing and relaxation. Several approaches can
yield a protected qubit. (b) The harmful fluctuations can be removed or
filtered out. (c) The qubit can be encoded into global degrees of freedom that
are insensitive to local fluctuations. (d) The qubit can be smartly encoded
into a subsystem of local degrees of freedom that couple in such a symmetric
way to the environment that the fluctuations do not cause any uncontrollable
qubit dynamics. (e) Strong external driving can in some cases stabilize the
qubit dynamics.

We classify the efforts in this direction into four categories (as illustrated
in Fig. 1):

(i) The most straightforward strategy for protecting a qubit is to identify
the dominating source of noise contributing to decoherence and then tailor the
design and the environment of the qubit to remove or reduce that particular
noise before it reaches the qubit [Fig. 1(b)]. This noise-filtering approach
is very effective, but not always easy to realize in practice.

(ii) Since environmental fluctuations are typically local in nature, one can
encode qubits into _global_ degrees of freedom of the host system that are
insensitive to local fluctuations [Fig. 1(c)]. This is the main idea behind
topological qubits,Stanescu (2017) where the qubit control mechanisms couple
in a topologically different way to the system than the noise sources.

Figure 2: Different strategies to mitigate magnetic noise in semiconductor
spin qubits. Local DFS-based approaches rely on multi-spin encodings, which
often come with significant spin-charge hybridization, making the qubit
susceptible to charge noise. Tuning the qubit to a sweet spot, where the two
states couple to linear order in the same way to the (charge) noise results in
a protection against dephasing similar to the mechanism of collective
dephasing in two-spin encoded qubits.

(iii) One can encode the qubits into local degrees of freedom, but use a
subspace of the Hilbert space that couples to the fluctuations of concern such
that the environment cannot distinguish the different qubit states [Fig.
1(d)]. These subspaces are commonly referred to as _decoherence-free
subspaces_ ,Lidar, Chuang, and Whaley (1998) and can offer various degrees of
protection depending on the complexity of the encoding. In general, this type
of protection works best for long-wavelength noise, when fluctuations are
uniform on the physical scale corresponding to the subspace used for the
encoding.

(iv) Encoding qubits into time-dependent states is another emergent approach
towards increasing coherence [Fig. 1(e)]. In this approach, the qubits are
subject to strong external driving fields that can compete with the noise and
stabilize the qubit states, for example through autonomous error
correctionGertler  _et al._ (2021) or by creating dynamical sweet spots.Huang
_et al._ (2021); Mundada  _et al._ (2020)

In this article, we discuss the progress and current state of the art in the
implementation of protected qubit systems in three solid-state platforms:
semiconductor spin qubits, superconducting qubits and Majorana zero-mode-based
qubits.

##  II Protected semiconductor spin qubits

Due to their small size and fast operation times, gate-defined semiconductor
spin qubits are considered an attractive candidate platform for massively
scaling quantum information processing.Chatterjee  _et al._ (2021) A primary
source of decoherence for the first generation of spin qubits was the
hyperfine interaction between the localized electron spins and the randomly
fluctuating nuclear spins of the semiconductor, effectively leading to
magnetic noise that causes unwanted qubit rotations.Hanson  _et al._ (2007) So
far, the most effective mitigation strategy for this noise has been one of
_noise reduction_ : Using group-IV semiconductors such as Ge and Si, instead
of GaAs or InAs, allows to work with isotopically purified nuclear-spin-free
samples,Zwanenburg  _et al._ (2013); Scappucci  _et al._ (2020) which resulted
in an extension of coherence times by more than four orders of magnitude.Stano
and Loss (2021) Apart from this effective and conceptually simple, though
technically challenging approach, also more sophisticated (yet similarly
passive) spin-qubit protection strategies have been proposed and implemented
over the years, a few of which we will highlight below, see also Fig. 2.

As one attractive feature of spin qubits is their small size, protection
schemes based on _local_ encodings in decoherence-free subspaces (DFSs) are
particularly interesting, because these encodings do not require a qualitative
scale-up in qubit size, as opposed to global (e.g., topological) encodings.

A local DFS is often defined in a subspace of the Hilbert space spanned by a
few qubits.Lidar and Whaley (2003) The simplest multi-qubit (in this case
multi-spin) encoding that provides some built-in protection against magnetic
noise is the singlet-triplet qubit, which is defined in the $S_{z}^{\rm
tot}=0$ subspace of two spins localized in two quantum dots.Levy (2002); Petta
_et al._ (2005) The two states in this subspace couple in exactly the same way
to a magnetic field that is uniform on the scale of the double quantum dot,
and the subspace is thus insensitive to (long-wavelength) magnetic noise along
the quantization axis $z$. This type of protection is more generally known as
_collective dephasing_ Palma, Suominen, and Ekert (1996); Duan and Guo (1998)
and boils down to the following observation. If two qubits suffer the same
pure dephasing, i.e.,
$\alpha|{0}\rangle+\beta|{1}\rangle\to\alpha|{0}\rangle+e^{i\varphi}\beta|{1}\rangle$,
then any state in the two-dimensional space spanned by
$\\{|{10}\rangle,|{01}\rangle\\}$ only acquires an unimportant overall phase
factor. However, since the nuclear magnetic noise is short-wavelength (not
strongly correlated across different qubits) and not uniaxial, it does not
result in collective dephasing and the two-spin encoding does not provide
significant protection against it.

Figure 3: Examples of protected superconducting qubits. (a) The rhombi
superconducting circuit (top), which consists of four identical Josephson
junctions in a single loop, can encode a qubit into quasi-degenerate quantum
states, but these states are extremely sensitive to flux noise. By building an
array of unprotected qubits one can increase protection against dephasing due
to flux noise (bottom). (b) The $0$–$\pi$ circuit and the bifluxon qubit are
two examples of locally encoded protected superconducting circuits with only a
few degrees of freedom. (c) The Floquet-fluxonium qubit is a qubit whose
protection is the result of periodically modulating the flux through the loop
of the device, which creates dynamical sweet spots.

It is worth pointing out that collective dephasing is conceptually similar to
the mechanism that underlies improved qubit coherence at so-called _sweet
spots_. In multi-dot multi-spin qubits the interdot tunnel coupling leads to
spin-charge hybridization. While this allows fast electrical control of the
qubits, it makes the qubit sensitive to ubiquitous fluctuations of the
electric field, coming from the substrate and/or the gate electrodes. One
approach to mitigate such charge noise, without compromising the electric
control over the qubit, is to operate the qubit at sweet spots, where the
qubit splitting $E_{10}$ is to lowest order insensitive to the dominating
electric fluctuations. Small fluctuations then translate to an irrelevant
overall phase factor for the qubit subspace, much like the effect of
collective dephasing. The sweet-spot approach simply amounts to choosing
optimal working points in parameter space and has proven to be capable of
significantly increasing the charge-noise-resilience of multi-spin
qubits.Martins  _et al._ (2016); Reed  _et al._ (2016)

A more advanced local DFS, providing protection against magnetic noise along
all three axes, can be formed by combining three spins:Knill, Laflamme, and
Viola (1999); Filippo (2000); Yang and Gea-Banacloche (2001) A three-spin
system hosts one $S=\frac{3}{2}$ spin-quadruplet and two $S=\frac{1}{2}$ spin-
doublets. A qubit encoded into two _identical_ spin-$\frac{1}{2}$ states but
belonging to _different_ doublets evolves trivially under the Hamiltonian
$\hat{H}={\bf B}(t)\cdot\hat{\bf S}^{\rm tot}$ (with $\hat{\bf S}^{\rm tot}$
the total spin operators) and is thus protected against all long-wavelength
magnetic noise. This protection is better than that of the two-spin qubit, but
still inefficient against short-wavelength nuclear magnetic noise.

Nevertheless, three-spin qubits implemented in triple quantum dots have drawn
much attention for a different reason: The two independently controllable
exchange couplings between neighboring dots translate into _two_ electrically
tunable control axes on the qubit’s Bloch sphere.DiVincenzo  _et al._ (2000);
Laird  _et al._ (2010); Medford  _et al._ (2013a, b); Russ and Burkard (2017)
This so-called exchange-only qubit thus offers all-electric control, at the
price of an increased sensitivity to charge noise. As for the two-spin case,
operation of this qubit at a (multiple) sweet spot can efficiently reduce the
effects of charge noise.Fei  _et al._ (2015); Russ, Ginzel, and Burkard
(2016); Shim and Tahan (2016); Zhang, Yang, and Wang (2017).

Finally, using four qubits one can create a “true” local DFS,Lidar, Chuang,
and Whaley (1998) which is probably most easily understood in terms of spin:
The four-spin Hilbert space contains _two_ $S=0$ spin-singlet subspaces, and a
qubit encoded into the two corresponding spinless states does not couple to
any uniform magnetic field. The important advantage over the three-spin
encoding is that the singlet-only qubit also provides some protection against
_local_ magnetic fluctuations:Lidar, Chuang, and Whaley (1998) The effect of
these fluctuations becomes higher-order and is suppressed by a factor
$B/\Delta$, where $B$ characterizes the magnitude of the fluctuating fields
and $\Delta$ the typical energy splitting between the singlets and the other
states. Recently, several quantum-dot-based implementations of the singlet-
only spin qubit were proposed,Sala and Danon (2017); Russ, Petta, and Burkard
(2018); Sala, Qvist, and Danon (2020) all of which offer the same exchange-
based all-electric control as the three-spin exchange-only qubit.

##  III Protected superconducting qubits

Since the first realizations of superconducting qubits, the idea to create
quantum states that are robust against environmental noise has fueled intense
research activity.Kjaergaard  _et al._ (2020); Gyenis  _et al._ (2021a)
Although most superconductor-based quantum devices are constructed from only
three basic circuit elements, (capacitors, inductors and aluminum-oxide-based
Josephson junctions), the combination of these building blocks offers
outstanding flexibility. For example, the number of nodes in the circuit
determines the dimension of the quantum system, the size of inductors and
Josephson junctions fixes the potential, and the capacitance values define the
kinetic energy of the quantum state.Vool and Devoret (2017) The ability to
independently control these three aspects of a quantum state (dimension,
kinetic and potential energy) allows to construct superconducting circuits
that satisfy the requirements for intrinsic noise protection.

The development and design of protected superconducting qubits have evolved
along three main paths: Josephson-junction-array-based multimode
circuits,Kitaev (2003); Ioffe  _et al._ (2002); Ioffe and Feigel’man (2002);
Douçot and Vidal (2002); Douçot  _et al._ (2005); Gladchenko  _et al._ (2008);
Douçot and Ioffe (2012); Bell  _et al._ (2014) compact few-mode
circuits,Brooks, Kitaev, and Preskill (2013); Smith  _et al._ (2020);
Kalashnikov  _et al._ (2020) and driven systemsOfek  _et al._ (2016);
Mirrahimi  _et al._ (2014); Puri, Boutin, and Blais (2017); Campagne-Ibarcq
_et al._ (2020); Grimm  _et al._ (2020); Huang  _et al._ (2021); Mundada  _et
al._ (2020) (see Fig. 3). In the first approach, the logical qubit is
constructed by concatenating a set of noisy circuit unit cells to reduce the
total effect of local errors along the chain (a type of global encoding). This
approach to noise protection is rooted in the concept of a topological ground
state degeneracy. On the other hand, compact protected circuits are
freestanding qubits with only a few degrees of freedom, and are examples of
qubits in local decoherence-free subspaces. Generally, to achieve noise
protection, these circuits need to meet extreme requirements, such as large
inductors and a strong reduction of parasitic capacitances. Protection in
these compact qubits often arises because the low-energy behavior of the
circuit can be approximated by an effective Hamiltonian with parity symmetry
such as Cooper-pair or fluxon symmetry. Finally, systems under intense
continuous driving can also show protection against various noise sources. In
this case, the rich interaction of the driving field and the qubit leads to
advantageous properties. Below we discuss examples for each of these
approaches.

We first discuss the case of the Josephson-junction-array qubits [Fig. 3(a)].
The basic building block of such arrays is often referred to as the “$\cos
2\phi$” circuit. There are multiple ways to realize these $\cos 2\phi$
circuits, but a common feature is that they have potential landscapes with two
minima as a function of a phase bias $\phi$ of the superconductor order
parameter (effectively a double-well potential) and discrete charge states.
The qubit states are quasi-degenerate and can be described by an effective
Cooper-pair parity: the ground state wavefunction contains an even number of
Cooper pair states, while the excited state has an odd number. Such a $\cos
2\phi$ element, in principle, could be a protected qubit by itself:
transitions between the qubit states are not allowed—as long as the noise does
not break the parity symmetry—and dephasing is suppressed because of the
degeneracy of the levels. But it is important to highlight that the realized
circuits behave as $\cos 2\phi$ elements only at a very narrow range of
external parameters and when the circuit elements are highly symmetric.Douçot
and Vidal (2002); Gladchenko  _et al._ (2008); Bell  _et al._ (2014); Smith
_et al._ (2020) For example, flux noise can drive the qubit out of the $\cos
2\phi$ regime, easily destroying the protection. The idea behind the
concatenation is to combat the flux noise: when multiple of these units are
connected to each other, the effect of flux noise is reduced, and the chain is
less sensitive to the parity breaking effects of local flux noise. As a
result, the logical qubit is more robust, and its sensitivity to flux
fluctuations is suppressed approximately as an exponential function of the
number of unit cells used in the chain.

Regarding the second approach,Gyenis  _et al._ (2021a) the prototypical
examples for compact protected qubits are the $0$–$\pi$ qubitKitaev (2006);
Brooks, Kitaev, and Preskill (2013); Dempster  _et al._ (2014); Groszkowski
_et al._ (2018); Paolo  _et al._ (2019); Gyenis  _et al._ (2021b) and the
bifluxon qubitKalashnikov  _et al._ (2020) [Fig. 3(b)]. Focusing on the
$0$–$\pi$ qubit here, this circuit has only three degrees of freedom, and when
the qubit parameters meet strict requirements, the Hamiltonian of the device
can be approximated with an effective $\cos 2\phi$ Hamiltonian. In contrast to
the noisy $\cos 2\phi$ elements discussed above, the $0$–$\pi$ qubit behaves
as a $\cos 2\phi$ qubit at all external parameters and it has exponentially
reduced sensitivity to noise. Thus, there is no need to concatenate multiple
of these qubits to secure the protection: the qubit is protected by itself.
The challenge of realizing this qubit is that the requirements for the circuit
parameters is beyond the currently available materials and fabrication
procedures. In the experimentally realized soft-$0$–$\pi$ qubit version of the
device, the protection against flux noise is reduced from exponential
protection to first-order sweet-spot protection.Gyenis  _et al._ (2021b)

The third common approach towards noise protection in superconducting qubits
is to expose the quantum circuit to periodically modulated external fields
[see Fig. 3(c)]. For example, in a recent work, the flux-sensitivity of a
superconducting qubit, the fluxonium, was significantly reduced by strongly
modulating the external flux at frequencies close to the qubit transition
frequency.Huang  _et al._ (2021); Mundada  _et al._ (2020) The interplay of
the field and the qubit levels leads to dynamical sweet spots in certain
parameter regimes of the drive, where the Floquet-fluxonium qubit becomes
first-order protected against flux noise.

Figure 4: Topological quantum computation: (a) Qubits are encoded globally in
low-energy modes resulting in a decoupling from local fluctuations. (b) Some
qubit gates could be performed by braiding the localized modes, the outcome of
which is independent of the local details of the trajectories followed by the
modes.

##  IV Topologically protected qubits

In recent years topological qubit encodings, as mentioned under point (ii) in
the introduction, gained much attention and topology became almost a synonym
for protection. The idea that hybrid semiconductor-superconductor devices,
based on new materials and fabrication techniques, could be tuned to a
topologically non-trivial phase that could be used for protected qubit
encodingsSato, Takahashi, and Fujimoto (2009); Sau  _et al._ (2010); Alicea
(2010) fueled an explosion of research in this direction, both theoretical and
experimental.

Most notably, a quasi-one-dimensional semiconductor with strong spin–orbit
coupling and a large Landé g-factor could, when proximitized by an $s$-wave
superconductor, effectively become a topological superconductor with
exponentially localized Majorana zero-modes (MZMs) at its ends.Oreg, Refael,
and von Oppen (2010); Lutchyn, Sau, and Sarma (2010); Hell, Leijnse, and
Flensberg (2017) These MZMs are decoupled from local fluctuations and are
predicted to obey non-Abelian anyonic exchange statistics. Therefore, if
successful, qubits encoded into a space spanned by such MZMs would not only
have a decoherence that is exponentially suppressed in the physical separation
between the MZMs [as illustrated in Fig. 4(a)], but they would also allow to
perform gate operations through braiding, which is topologically protected
against local fluctuations [Fig. 4(b)].

Such built-in topological protection is indeed very appealing, justifying the
intense research efforts in this direction. However, so far a clear
experimental demonstration of the existence of non-Abelian low-energy modes in
solid-state devices is still lacking. Additionally, due to the Ising-like
nature of their braiding statistics, MZMs do not provide a universal set of
topologically protected quantum gates.Stanescu (2017) For a recent review of
this fast-evolving field we refer the reader to Ref. Lutchyn  _et al._ , 2018.

##  V Outlook

Despite recent progress, creating a full-scale quantum computer remains very
challenging, not in the least because of the large ratio of physical to
logical qubits needed for quantum error correction. Qubits with inherent
protection against specific errors, as discussed in this article, have the
potential to strongly relax this requirement, and implementations of such
qubits are therefore being pursued around the world. In addition, the alluring
concepts of inherent and topological protection have a lot to add to our
understanding of decoherence, entanglement and the quantum dynamics of
embedded qubits.

When designing protected qubits, their possible drawbacks as well as the
required overhead for implementing quantum codes have to be kept in mind: two-
qubit gates could be less feasible, the number of leakage channels out of the
computational subspace may increase, or the qubit splitting and coupling to
control pulses can become impractically small near (protective) symmetry
points. In the future, one may therefore expect quantum processors based on a
heterogeneous qubit system: quantum memories implemented in long-lived
protected qubits (ideally with a low overhead of classical computing power),
gate operations performed in more controllable qubits, and reliable transfer
of quantum information from one type of qubit to another.

###### Acknowledgements.

JD acknowledges support via FRIPRO-project 274853, which is funded by the
Research Council of Norway (RCN).

## References

  * Nielsen and Chuang (2010) M. A. Nielsen and I. L. Chuang,  _Quantum Computation and Quantum Information_ (Cambridge University Press, 2010).
  * Lidar and Brun (2013) D. A. Lidar and T. A. Brun, eds.,  _Quantum Error Correction_ (Cambridge University Press, 2013).
  * Shor (1995) P. W. Shor, “Scheme for reducing decoherence in quantum computer memory,” Phys. Rev. A 52, R2493 (1995).
  * Steane (1996) A. M. Steane, “Error Correcting Codes in Quantum Theory,” Phys. Rev. Lett. 77, 793 (1996).
  * Knill  _et al._ (2001) E. Knill, R. Laflamme, R. Martinez, and C. Negrevergne, “Benchmarking Quantum Computers: The Five-Qubit Error Correcting Code,” Phys. Rev. Lett. 86, 5811 (2001).
  * Stanescu (2017) T. D. Stanescu,  _Introduction to Topological Quantum Matter & Quantum Computation_ (CRC Press, Taylor & Francis Group, 2017).
  * Lidar, Chuang, and Whaley (1998) D. A. Lidar, I. L. Chuang, and K. B. Whaley, “Decoherence-Free Subspaces for Quantum Computation,” Phys. Rev. Lett. 81, 2594 (1998).
  * Gertler  _et al._ (2021) J. M. Gertler, B. Baker, J. Li, S. Shirol, J. Koch, and C. Wang, “Protecting a bosonic qubit with autonomous quantum error correction,” Nature 590, 243 (2021).
  * Huang  _et al._ (2021) Z. Huang, P. S. Mundada, A. Gyenis, D. I. Schuster, A. A. Houck, and J. Koch, “Engineering dynamical sweet spots to protect qubits from $1/f$ noise,” Phys. Rev. Applied 15, 034065 (2021).
  * Mundada  _et al._ (2020) P. S. Mundada, A. Gyenis, Z. Huang, J. Koch, and A. A. Houck, “Floquet-engineered enhancement of coherence times in a driven fluxonium qubit,” Phys. Rev. Applied 14, 054033 (2020).
  * Chatterjee  _et al._ (2021) A. Chatterjee, P. Stevenson, S. D. Franceschi, A. Morello, N. P. d. Leon, and F. Kuemmeth, “Semiconductor qubits in practice,” Nat. Rev. Phys. 3, 157 (2021).
  * Hanson  _et al._ (2007) R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. Vandersypen, “Spins in few-electron quantum dots,” Rev. Mod. Phys. 79, 1217 (2007).
  * Zwanenburg  _et al._ (2013) F. A. Zwanenburg, A. S. Dzurak, A. Morello, M. Y. Simmons, L. C. L. Hollenberg, G. Klimeck, S. Rogge, S. N. Coppersmith, and M. A. Eriksson, “Silicon quantum electronics,” Rev. Mod. Phys. 85, 961 (2013).
  * Scappucci  _et al._ (2020) G. Scappucci, C. Kloeffel, F. A. Zwanenburg, D. Loss, M. Myronov, J.-J. Zhang, S. D. Franceschi, G. Katsaros, and M. Veldhorst, “The germanium quantum information route,” Nat. Rev. Mat. 6, 926 (2020).
  * Stano and Loss (2021) P. Stano and D. Loss, “Review of performance metrics of spin qubits in gated semiconducting nanostructures,” arXiv:2107.06485 (2021).
  * Lidar and Whaley (2003) D. A. Lidar and K. B. Whaley, “Decoherence-Free Subspaces and Subsystems,” in  _Irreversible Quantum Dynamics_ , edited by F. Benatti and R. Floreanini (Springer Berlin, 2003) pp. 83–120.
  * Levy (2002) J. Levy, “Universal Quantum Computation with Spin-1/2 Pairs and Heisenberg Exchange,” Phys. Rev. Lett. 89, 147902 (2002).
  * Petta  _et al._ (2005) J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science 309, 2180 (2005).
  * Palma, Suominen, and Ekert (1996) G. M. Palma, K.-A. Suominen, and A. Ekert, “Quantum computers and dissipation,” Proc. R. Soc. Lond. A. 452, 567 (1996).
  * Duan and Guo (1998) L.-M. Duan and G.-C. Guo, “Reducing decoherence in quantum-computer memory with all quantum bits coupling to the same environment,” Phys. Rev. A 57, 737 (1998).
  * Martins  _et al._ (2016) F. Martins, F. K. Malinowski, P. D. Nissen, E. Barnes, S. Fallahi, G. C. Gardner, M. J. Manfra, C. M. Marcus, and F. Kuemmeth, “Noise Suppression Using Symmetric Exchange Gates in Spin Qubits,” Phys. Rev. Lett. 116, 116801 (2016).
  * Reed  _et al._ (2016) M. D. Reed, B. M. Maune, R. W. Andrews, M. G. Borselli, K. Eng, M. P. Jura, A. A. Kiselev, T. D. Ladd, S. T. Merkel, I. Milosavljevic, E. J. Pritchett, M. T. Rakher, R. S. Ross, A. E. Schmitz, A. Smith, J. A. Wright, M. F. Gyure, and A. T. Hunter, “Reduced Sensitivity to Charge Noise in Semiconductor Spin Qubits via Symmetric Operation,” Phys. Rev. Lett. 116, 110402 (2016).
  * Knill, Laflamme, and Viola (1999) E. Knill, R. Laflamme, and L. Viola, “Theory of Quantum Error Correction for General Noise,” Phys. Rev. Lett. 84, 2525 (1999).
  * Filippo (2000) S. D. Filippo, “Quantum computation using decoherence-free states of the physical operator algebra,” Phys. Rev. A 62, 052307 (2000).
  * Yang and Gea-Banacloche (2001) C.-P. Yang and J. Gea-Banacloche, “Three-qubit quantum error-correction scheme for collective decoherence,” Phys. Rev. A 63, 022311 (2001).
  * DiVincenzo  _et al._ (2000) D. P. DiVincenzo, D. Bacon, J. Kempe, G. Burkard, and K. B. Whaley, “Universal quantum computation with the exchange interaction,” Nature 408, 339 (2000).
  * Laird  _et al._ (2010) E. A. Laird, J. M. Taylor, D. P. DiVincenzo, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent spin manipulation in an exchange-only qubit,” Phys. Rev. B 82, 075403 (2010).
  * Medford  _et al._ (2013a) J. Medford, J. Beil, J. M. Taylor, E. I. Rashba, H. Lu, A. C. Gossard, and C. M. Marcus, “Quantum-Dot-Based Resonant Exchange Qubit,” Phys. Rev. Lett. 111, 050501 (2013a).
  * Medford  _et al._ (2013b) J. Medford, J. Beil, J. M. Taylor, S. D. Bartlett, A. C. Doherty, E. I. Rashba, D. P. DiVincenzo, H. Lu, A. C. Gossard, and C. M. Marcus, “Self-consistent measurement and state tomography of an exchange-only spin qubit,” Nat. Nano. 8, 654 (2013b).
  * Russ and Burkard (2017) M. Russ and G. Burkard, “Three-electron spin qubits,” J. Phys.: Cond. Matt. 29, 393001 (2017).
  * Fei  _et al._ (2015) J. Fei, J.-T. Hung, T. S. Koh, Y.-P. Shim, S. N. Coppersmith, X. Hu, and M. Friesen, “Characterizing gate operations near the sweet spot of an exchange-only qubit,” Phys. Rev. B 91, 205434 (2015).
  * Russ, Ginzel, and Burkard (2016) M. Russ, F. Ginzel, and G. Burkard, “Coupling of three-spin qubits to their electric environment,” Phys. Rev. B 94, 165411 (2016).
  * Shim and Tahan (2016) Y.-P. Shim and C. Tahan, “Charge-noise-insensitive gate operations for always-on, exchange-only qubits,” Phys. Rev. B 93, 121410 (2016).
  * Zhang, Yang, and Wang (2017) C. Zhang, X.-C. Yang, and X. Wang, “Leakage and sweet spots in triple-quantum-dot spin qubits: A molecular-orbital study,” Phys. Rev. A 97, 042326 (2017).
  * Sala and Danon (2017) A. Sala and J. Danon, “Exchange-only singlet-only spin qubit,” Phys. Rev. B 95, 241303 (2017).
  * Russ, Petta, and Burkard (2018) M. Russ, J. R. Petta, and G. Burkard, “Quadrupolar Exchange-Only Spin Qubit,” Phys. Rev. Lett. 121, 177701 (2018).
  * Sala, Qvist, and Danon (2020) A. Sala, J. H. Qvist, and J. Danon, “Highly tunable exchange-only singlet-only qubit in a GaAs triple quantum dot,” Phys. Rev. Research 2, 012062 (2020).
  * Kjaergaard  _et al._ (2020) M. Kjaergaard, M. E. Schwartz, J. Braumüller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver, “Superconducting qubits: Current state of play,” Annu. Rev. Condens. Matter Phys. 11, 369–395 (2020).
  * Gyenis  _et al._ (2021a) A. Gyenis, A. Di Paolo, J. Koch, A. Blais, A. A. Houck, and D. I. Schuster, “Moving beyond the transmon: Noise-protected superconducting quantum circuits,” PRX Quantum 2, 030101 (2021a).
  * Vool and Devoret (2017) U. Vool and M. Devoret, “Introduction to quantum electromagnetic circuits,” Int. J. Circ. Theor. App. 45, 897 (2017).
  * Kitaev (2003) A. Kitaev, “Fault-tolerant quantum computation by anyons,” Annals of Physics 303, 2 – 30 (2003).
  * Ioffe  _et al._ (2002) L. B. Ioffe, M. V. Feigel’man, A. Ioselevich, D. Ivanov, M. Troyer, and G. Blatter, “Topologically protected quantum bits using Josephson junction arrays,” Nature 415, 503 (2002).
  * Ioffe and Feigel’man (2002) L. B. Ioffe and M. V. Feigel’man, “Possible realization of an ideal quantum computer in Josephson junction array,” Phys. Rev. B 66, 224503 (2002).
  * Douçot and Vidal (2002) B. Douçot and J. Vidal, “Pairing of Cooper pairs in a fully frustrated Josephson-junction chain,” Phys. Rev. Lett. 88, 227005 (2002).
  * Douçot  _et al._ (2005) B. Douçot, M. V. Feigel’man, L. B. Ioffe, and A. S. Ioselevich, “Protected qubits and Chern-Simons theories in Josephson junction arrays,” Phys. Rev. B 71, 024505 (2005).
  * Gladchenko  _et al._ (2008) S. Gladchenko, D. Olaya, E. Dupont-Ferrier, B. Douçot, L. B. Ioffe, and M. E. Gershenson, “Superconducting nanocircuits for topologically protected qubits,” Nat. Phys. 5, 48 (2008).
  * Douçot and Ioffe (2012) B. Douçot and L. B. Ioffe, “Physical implementation of protected qubits,” Rep. Prog. Phys. 75, 072001 (2012).
  * Bell  _et al._ (2014) M. T. Bell, J. Paramanandam, L. B. Ioffe, and M. E. Gershenson, “Protected Josephson rhombus chains,” Phys. Rev. Lett. 112, 167001 (2014).
  * Brooks, Kitaev, and Preskill (2013) P. Brooks, A. Kitaev, and J. Preskill, “Protected gates for superconducting qubits,” Phys. Rev. A 87, 052306 (2013).
  * Smith  _et al._ (2020) W. C. Smith, A. Kou, X. Xiao, U. Vool, and M. H. Devoret, “Superconducting circuit protected by two-Cooper-pair tunneling,” npj Quantum Information 6, 8 (2020).
  * Kalashnikov  _et al._ (2020) K. Kalashnikov, W. T. Hsieh, W. Zhang, W.-S. Lu, P. Kamenov, A. Di Paolo, A. Blais, M. E. Gershenson, and M. Bell, “Bifluxon: Fluxon-parity-protected superconducting qubit,” PRX Quantum 1, 010307 (2020).
  * Ofek  _et al._ (2016) N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas, B. Vlastakis, Y. Liu, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Extending the lifetime of a quantum bit with error correction in superconducting circuits,” Nature 536, 441 (2016).
  * Mirrahimi  _et al._ (2014) M. Mirrahimi, Z. Leghtas, V. V. Albert, S. Touzard, R. J. Schoelkopf, L. Jiang, and M. H. Devoret, “Dynamically protected cat-qubits: a new paradigm for universal quantum computation,” New J. Phys. 16, 045014 (2014).
  * Puri, Boutin, and Blais (2017) S. Puri, S. Boutin, and A. Blais, “Engineering the quantum states of light in a Kerr-nonlinear resonator by two-photon driving,” npj Quantum Information 3, 18 (2017).
  * Campagne-Ibarcq  _et al._ (2020) P. Campagne-Ibarcq, A. Eickbusch, S. Touzard, E. Zalys-Geller, N. E. Frattini, V. V. Sivak, P. Reinhold, S. Puri, S. Shankar, R. J. Schoelkopf, L. Frunzio, M. Mirrahimi, and M. H. Devoret, “Quantum error correction of a qubit encoded in grid states of an oscillator,” Nature 584, 368 (2020).
  * Grimm  _et al._ (2020) A. Grimm, N. E. Frattini, S. Puri, S. O. Mundhada, S. Touzard, M. Mirrahimi, S. M. Girvin, S. Shankar, and M. H. Devoret, “Stabilization and operation of a Kerr-cat qubit,” Nature 584, 205 (2020).
  * Kitaev (2006) A. Kitaev, “Protected qubit based on a superconducting current mirror,” arXiv:0609441 (2006).
  * Dempster  _et al._ (2014) J. M. Dempster, B. Fu, D. G. Ferguson, D. I. Schuster, and J. Koch, “Understanding degenerate ground states of a protected quantum circuit in the presence of disorder,” Phys. Rev. B 90, 094518 (2014).
  * Groszkowski  _et al._ (2018) P. Groszkowski, A. D. Paolo, A. L. Grimsmo, A. Blais, D. I. Schuster, A. A. Houck, and J. Koch, “Coherence properties of the 0–$\pi$ qubit,” New J. Phys. 20, 043053 (2018).
  * Paolo  _et al._ (2019) A. D. Paolo, A. L. Grimsmo, P. Groszkowski, J. Koch, and A. Blais, “Control and coherence time enhancement of the 0–$\pi$ qubit,” New J. Phys. 21, 043002 (2019).
  * Gyenis  _et al._ (2021b) A. Gyenis, P. S. Mundada, A. Di Paolo, T. M. Hazard, X. You, D. I. Schuster, J. Koch, A. Blais, and A. A. Houck, “Experimental realization of a protected superconducting circuit derived from the $0$–$\pi$ qubit,” PRX Quantum 2, 010339 (2021b).
  * Sato, Takahashi, and Fujimoto (2009) M. Sato, Y. Takahashi, and S. Fujimoto, “Non-Abelian Topological Order in s-Wave Superfluids of Ultracold Fermionic Atoms,” Phys. Rev. Lett. 103, 020401 (2009).
  * Sau  _et al._ (2010) J. D. Sau, R. M. Lutchyn, S. Tewari, and S. D. Sarma, “Generic New Platform for Topological Quantum Computation Using Semiconductor Heterostructures,” Phys. Rev. Lett. 104, 040502 (2010).
  * Alicea (2010) J. Alicea, “Majorana fermions in a tunable semiconductor device,” Phys Rev. B 81, 125318 (2010).
  * Oreg, Refael, and von Oppen (2010) Y. Oreg, G. Refael, and F. von Oppen, “Helical Liquids and Majorana Bound States in Quantum Wires,” Phys. Rev. Lett. 105, 177002 (2010).
  * Lutchyn, Sau, and Sarma (2010) R. M. Lutchyn, J. D. Sau, and S. D. Sarma, “Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures,” Phys. Rev. Lett. 105, 077001 (2010).
  * Hell, Leijnse, and Flensberg (2017) M. Hell, M. Leijnse, and K. Flensberg, “Two-Dimensional Platform for Networks of Majorana Bound States,” Phys. Rev. Lett. 118, 107701 (2017).
  * Lutchyn  _et al._ (2018) R. M. Lutchyn, E. P. Bakkers, L. P. Kouwenhoven, P. Krogstrup, C. M. Marcus, and Y. Oreg, “Majorana zero modes in superconductor–semiconductor heterostructures,” Nat. Rev. Mat. 3, 52 (2018).