Moving beyond the transmon: Noise-protected superconducting quantum circuits

Artificial atoms realized by superconducting circuits offer unique opportunities to store and process quantum information with high fidelity. Among them, implementations of circuits that harness intrinsic noise protection have been rapidly developed in recent years. These noise-protected devices constitute a new class of qubits in which the computational states are largely decoupled from local noise channels. The main challenges in engineering such systems are simultaneously guarding against both bit- and phase-flip errors, and also ensuring high-fidelity qubit control. Although partial noise protection is possible in superconducting circuits relying on a single quantum degree of freedom, the promise of complete protection can only be fulfilled by implementing multimode or hybrid circuits. This Perspective reviews the theoretical principles at the heart of these new qubits, describes recent experiments, and highlights the potential of robust encoding of quantum information in superconducting qubits.


I. INTRODUCTION
Superconducting circuits are one of the most promising candidates for large-scale quantum computers, as they can reliably implement strongly interacting artificial atoms that have reduced susceptibility to environmental noise [1][2][3][4][5][6][7][8]. The substantial effort invested in understanding and designing numerous types of superconducting quantum circuits has given rise to small-scale quantum processors that are able to execute proof-of-principle implementations of quantum algorithms [9][10][11][12][13][14]. And yet, despite more than two decades of progress, most superconducting systems today consist simply of arrays of harmonic oscillators -microwave cavities -and slightly anharmonic oscillators -transmon qubits [15]. We have only begun to scratch the surface of the potential versatility of this platform.
Looking ahead, we envision a truly heterogeneous superconducting architecture, where the components that support the varying roles of logic, gates, memory, ancilla, reset, and readout are not necessarily the same. In a heterogeneous architecture, component circuits are optimized for specific functions based on coherence, gate speed, anharmonicity, tunability, interaction strength, noise polarization and more. Given the differing requirements for these functions, it is likely that different types of circuits will play roles in the separate parts of such heterogeneous systems.
How can we ensure that the various blocks of these systems are resilient against environmental noise? There are, quite generally, three strategies to reduce the rate at which errors occur ( Fig. 1): (i) reducing the amplitude of the noise itself (noise filtering) [16][17][18][19][20][21], (ii) boosting the qubits' immunity against environmental noise (hardware-level error protection) [22][23][24][25][26][27][28][29], and (iii) detecting and actively correcting errors (quantum error correction) [30][31][32][33][34][35][36][37][38][39]. These strategies are not mutually exclusive and can benefit from each other when employed in tandem. For example, using more reliable qubits and understanding the noise in these systems is paramount in reducing the physical qubit overhead in quantum error correction. Here, our discussion focuses on these protected qubits that have the potential as building blocks for heterogeneous quantum systems.
The evolution of the transmon itself shows the power of hardware-level noise protection. A critical feature that made this circuit exceptionally successful is its resilience against the most detrimental noise affecting solid-state qubits: charge noise. Its ancestor, the Cooper pair box [40][41][42], which is composed of a Josephson junction shunted by a small capacitance, was used in the very first experiment that demonstrated quantum coherence in a solid-state qubit but was severely limited by charge-noise decoherence. The first stride towards enhanced coherence times was achieved by the quantronium circuit [43]. This qubit showed a dramatic improvement of the coherence times when operated at a special working pointat the sweet spot-with first-order insensitivity to charge fluctuations. By introducing a large shunt capacitor, the transmon qubit extended the quantronium's protection from first to all orders [15], providing excellent coherence but remaining susceptible to relaxation-a mechanism that ultimately limits the coherence of this device. For fifteen years, this qubit has remained ubiquitous, but with gradual improvement earned through the painstaking labor of noise filtering, microwave engineering and advancements in materials.
As the development of the transmon highlights, hardwarelevel protection is a cornerstone of the field of superconducting qubits. But why stop at protection against only dephasing? Can we realize and operate fully noise-protected devices where both bit-flip and phase-flip errors are suppressed? In this Perspective article, we discuss the potential of achieving such intrinsic noise protection in superconducting qubits, starting from the simplest circuits and expanding our discussion towards more complex devices. These next-generation superconducting qubits represent a new paradigm shift with arXiv:2106.10296v1 [quant-ph] 18  emphasis on noise immunity rather than noise filtering. While such paradigm shifts often lead to trade-offs-as it was the case for the transmon with substantially reduced anharmonicity and weaker dispersive shifts-, we expect that future innovations will establish protected qubits as a critical component for heterogeneous superconducting quantum systems.

II. BASICS OF NOISE PROTECTION
To understand the underlying principles of hardware-level protection, we first review the types of errors that affect qubits. If |0 and |1 are the two computational basis states of a qubit, we can encode the quantum information into the superposition of these states, such that |ψ = cos (θ/2) |0 + sin (θ/2) e iφ |1 . Here, tan (θ/2) describes the relative amplitude and φ the relative phase between the basis states. Thus, we need to preserve both the relative phase and amplitude between the basis states to protect the superposition. In the transmon, for example, phase-flip errors are suppressed by design: the frequency of this qubit is insensitive to fluctuations in its control parameters, resulting in little phase-flip errors. However, there is no built-in protection against bit-flip errors as incoherent transitions between the qubit states are not suppressed. On the other hand, the situation is reversed for some variants of the flux qubit which demonstrate robustness against bit-flip errors but little or no protection against phase-flip errors [44][45][46].
There are two different timescales that characterize the qubit's resilience against bit-and phase-flips: the depolarization time T 1 and the pure-dephasing time T ϕ . As the total coherence time of a qubit, T 2 = (1/2T 1 + 1/T ϕ ) −1 , is related to both mechanisms, it is necessary to simultaneously protect against both types of errors. We can quantify the timescale of these error processes with (1) the qubit's sensitivity to the noise and (2) the strength of the noise. To do so, we denote λ as an external parameter entering the circuit Hamiltonian that can tune the energy of the qubit. In superconducting circuits, for example, the most common external knobs are the gate voltage biasing an island of the circuit V gate or the magnetic flux in a loop of the device Φ ext . The qubit can be biased at a valueλ of this control parameter, around which there are fluctuations δλ = λ −λ. If the amplitude of the noise is small enough, it is possible to safely assume that the coupling between a qubit operatorÔ and the noise is linear, H int ∝ δλ ×Ô, whereÔ could be, for example, the charge or phase operator. In this situation, the corresponding dephasing and depolarization rates are [47] where S λ (ω) is the power spectral density of the noise as a function of frequency ω/2π and E 01 is the qubit transition energy. While noise filtering aims at reducing the amplitude of the noise by shielding qubits from its environment [i.e. reduce the power spectral densities in the right-hand side of Eq. (1a) and Eq. (1b)], the key idea behind hardware-level noise protection is to suppress the susceptibility of the qubit to the noise [i.e. reduce the first terms on the right-hand side of Eq. (1a) and Eq. (1b)].
We illustrate the concept of error protection by first considering phase-flip errors [ Fig. 2(a)]. As Eq. (1a) shows, dephasing arises as a consequence of the dispersion of the qubit energy E 01 (λ) with respect to the external parameter λ. Invoking the series expansion E 01 (λ) = E 01 (λ) + ∞ k=1 ∂ k λ E 01 (λ)δλ k /k!, we see that, generally, the first-order term dominates as long as the amplitude of the fluctuation is weak. The first strategy [43,47] to increase the dephasing time is to operate the qubit at a sweet spot where ∂ λ E 01 (λ) = 0, such that the leading-order contribution to the qubit dispersion is of second order [light red lines in Fig. 2(a)]. Further protection against dephasing requires that the higher-order contributions become negligible, which can be guaranteed by exponentially vanishing dispersion of the qubit transition energy [dark red lines in Fig. 2(a)].
On the other hand, bit-flip protection requires the transition matrix element 0|Ô|1 in Eq. (1b) to vanish. Central to this requirement is the concept of disjoint support of the qubit eigenfunctions. Whenever two wavefunctions localize in separate "spatial" regions [ Fig. 2(b)], typical matrix elements between them are dominated by the exponentially small tails and hence significantly reduced in magnitude. This generally leads to long relaxation times [Eq. (1b)]. Useful terminology to describe this situation is the notion of (nearly) disjoint support [48]. Recall that the support of a function f : Principles of noise protection: suppressed energy dispersion and wavefunctions with disjoint support. (a) Energy levels of a qubit as a function of an external parameter λ, such as external flux or offset charge. Dark red lines illustrate the case where the dispersion of the qubit energy level with respect to λ is exponentially small, i.e. the difference between the maximum and minimum transition energies ∆E01 ∝ e −η , where η is related to the circuit parameters of the qubit and can be made large by design. On the same plot, the light red lines show a dispersion with a first-order sweet spot and quadratic dispersion, E01 ≈ c1 + c2λ 2 , where c1, c2 are constants. To ensure that sweet-spot protection is sufficient for reaching highcoherence times, the quadratic term needs to be small compared to the scale of the noise amplitude δλ, such that c2δλ 2 /E01(0) 1. (b) Schematics of the potential landscape of a qubit (black grid) with two quantum degrees of freedom φ and θ. Protection against bit-flip errors can be achieved by localizing the qubit wavefunctions in distinct regions of the potential. Such states with disjoint support have exponentially small transition matrix elements, 0|Ô|1 ∝ e −ζ , where due to the design and parameter choices of the circuit ζ is a large number.
is the subset S ⊂ X on which the function has non-zero values. Outside the support, f vanishes, f (X \ S) = 0. In the case of two wavefunctions ψ 0 (φ) and ψ 1 (φ) localized in separate regions, their respective supports are nearly disjoint, S 0 ∩ S 1 ≈ ∅. The (near-)disjointness can be quantified via the integral dφ |ψ 0 (φ)| 2 |ψ 1 (φ)| 2 1, and results in suppressed matrix elements | 0|Ô|1 | 1 as long asÔ is a local operator like position or momentum operator (as opposed to non-local operators such as displacement operators). Since the common noise channels in superconducting qubits are associated with such local operators, engineering logical qubit wavefunctions with disjoint support is a highly effective strategy to reach bit-flip protection.
Alternatively, symmetries can prohibit qubit transitions when the qubit states are encoded into different eigenspaces of the symmetry operator of the Hamiltonian [22,23]. In this case, the transition matrix elements vanish as long as the noise operators preserve the symmetries of the system.
Circuit-level protection against these two types of error mechanisms shares similarities to the redundant encoding utilized in quantum error correction. Instead of using multiple qubits or multiple energy levels for redundancy, here, large circuit elements or circuits with many degrees of freedom lead to protection against local noise. Accordingly, we consider two possible strategies for realizing noise-protected superconducting circuits: compact protected qubits [49][50][51] and protected multi-junction arrays [22][23][24][25][26][27][28][29]. Compact protected qubits utilize only a few circuit elements which are challenging to realize (for example, large linear inductors so that the inductive energies define the smallest energy scale in the circuit) [ Fig. 3(a)]. In contrast, arrays involve a larger number of elements but in convenient parameter regimes [ Fig. 3(b)]. In this Perspective article, our discussion focuses on the first approach, while the article by Douçot and Ioffe has extensively covered the latter case [28]. Finally, we also note that qubits relying on continuous microwave tones can exploit the combination of hardware-level error protection and autonomous error correction [36,39,[52][53][54][55][56].

III. LIMITATIONS OF SINGLE-MODE SUPERCONDUCTING QUBITS
The simplest superconducting qubits are single-mode devices that contain one or more superconducting elements connected in parallel between two nodes. We dedicate this section to qualitatively study their coherence properties and limitations. We will see that-although their layout is simple and fixed-changing the parameters of their circuit elements can drastically change their coherence properties. These circuits offer protection against only one error channel (either bit-flip or phase-flip processes), and cannot accommodate the requirements for simultaneous protection against both [58]. Larger multimode circuits or hybrid semiconducting-superconducting devices can mitigate the limitations of these simple qubits (Sec. IV and Sec. V).

A. Islands and loops as the charge and flux degrees of freedom
Depending on the geometry of the underlying circuit, superconducting qubits utilize two types of quantum degrees of freedom. On the one hand, two galvanically disconnected pieces of superconductors define a pair of islands, a capacitor [ Fig wire identifies a loop, an inductor [ Fig. 4(b)]. An island can store an integer number of Cooper pairs, while a loop can host an integer number of flux quanta. These elementary charge and flux states of the islands and the loop can be used to store quantum information. However, an additional nonlinear circuit component is necessary to isolate two states in the excitation spectrum of these circuits to serve as the computational states of a qubit. Here, we focus on qubits that use Josephson junctions as the nonlinear elements, which are formed by growing a thin insulating layer between two superconductors, allowing for the tunneling of Cooper pairs or flux quanta through the layer. These two types of circuit geometries define the two elementary superconducting qubits. A pair of superconducting islands with capacitance C connected by a Josephson junction constitutes a charge mode [ Fig. 4(a)]. The Hamiltonian describing this mode iŝ where E C = e 2 /2C is the charging energy, e is the electron charge, E J is the Josephson energy associated with Cooper pair tunneling across the junction, n gate is the induced offset charge on the islands, whilen andθ are the canonical charge and phase operators. The charge mode couples to the environment through the offset charge n gate . This quantity can be used to dc-bias the qubit and manipulate its logical state, but its value is also uncontrollably influenced by the surrounding electrostatic potential. To understand the different parameters regimes of this Hamiltonian, it is useful to interpret Eq.
(2) such that it describes a fictitious particle with position θ and mass C moving in a cosine potential. Similarly, a loop made of a superconducting wire with inductance L and interrupted by a Josephson junction implements a flux mode [ Fig. 4(b)]. The Hamiltonian of the circuit readŝ whereφ is the phase operator, E L = Φ 2 0 /4π 2 L is the inductive energy, Φ 0 = h/2e is the flux quantum, h is the Planck's constant, φ ext is the reduced external flux defined as φ ext = Φ ext /Φ 0 , and Φ ext is the magnetic flux piercing the loop. The flux mode is coupled to the environment through the external flux φ ext , which plays a similar role as the offset gate charge in Eq. (2). With the addition of the inductor, the potential energy of the effective phase particle is now a corrugated harmonic well, illustrated in Fig. 5(b) and (c) for two parameter regimes.

B. High coherence of charge and flux modes
How can we ensure that a charge or a flux mode is protected against environmental noise? Building on the discussion in Sec. II, we first consider the principles of noise protection against phase-flip errors and describe the conditions for realizing offset-charge insensitive charge modes and externalflux insensitive flux modes. To simplify the discussion, here Because of this, reaching high coherence times for charge modes requires exponentially suppressed charge dispersion, while for flux modes even sweet-spot protection can offer high coherence. At the same time, experimentally realizing suppressed flux dispersion is significantly more challenging than achieving charge-noise insensitivity.
we neglect bit-flip errors such as losses due to imperfect capacitors, inductors and Josephson junctions and assume that only the environmental fluctuations of the external bias parameters limit the qubit coherence. By inspecting Eq. (2) and Eq. (3), we can describe the coupling between charge or flux modes and the environment by the HamiltoniansĤ int = −8E Cn × δn gate orĤ int = −E Lφ × δφ ext , respectively. Here δn gate and δφ ext represent fluctuations of the control parameters. Focusing first on charge modes, a strategy to boost the coherence time relies on biasing the qubit atn gate where the expectation value of the qubit's dipole operator is equal for the two logical states, 1|n|1 ≈ 0|n|0 . In this case, the change of the qubit transition frequency under the pertur-bationĤ int , 1|Ĥ int |1 − 0|Ĥ int |0 , vanishes or, equivalently, the first derivative of the energy dispersion is zero. Intuitively, in this situation, dephasing is minimized because the two qubit states cannot be distinguished (by an observer or the environment) from a measurement of the number of charges in the islands. For a charge mode, such sweet spots occur at biases where adjacent charge states are degenerate. Cooper pair tunneling hybridizes these two states leading to an avoided crossing with the usual quadratic dependence of the energy with the bias parameter.
By increasing the Josephson energy with respect to the charging energy-thus enhancing the rate at which Cooper  pairs can tunnel-, we can achieve a more robust protection against dephasing [15]. When E J /E C 1, hybridization occurs over several charge states resulting in qubit's logical states with support over many charges [ Fig. 5(a)]. A consequence of this delocalization over charge states is that the qubit energy levels are now exponentially flat at all bias points. For charge modes, this "sweet-spot-everywhere" condition is the transmon regime [15,70]. In practice, with typical values of E J /E C ∼ 50, transmon qubits feature dephasing times as large as T 2 = 100 − 200 µs with energy relaxation times of T 1 = 200 − 500 µs [20,71].
Coming back to the fictitious phase particle of mass C, we refer to the transmon regime as the heavy regime because it requires a large effective mass. Since flux modes couple to the environment through the phase operator (as opposed to the charge operator), high-coherence requires delocalized eigenfunctions in phase space instead of in charge space. Thus, we arrive at the conclusion that protection against dephasing in a flux mode is realized by working in the opposite, light regime, where E J /E C 1 and E J /E L 1 [72]. In this situation, the phase particle is delocalized over multiple wells of the corrugated harmonic potential leading to insensitivity to flux noise [ Fig. 5(b)]. As a result, while a heavy charge mode needs a large shunting capacitor, a light flux mode requires a large linear inductive element. Crucially, the large inductance should not be accompanied in this case by a large capacitance, something which can be challenging to realize in practice because of the stray capacitance of any superconducting circuit component. Large inductors with small parasitic capacitances are known as superinductors [73,74]. For instance, the experimentally realized light flux mode, the blochnium qubit relies on the Josephson-junction array being released from its substrate to minimize the capacitance to ground [69]. To maximize the qubit coherence time, the superinductor must be engineered such that its self-resonant modes are far detuned from the qubit operation frequency [75,76]. Furthermore, in the case of Josephson-junction-array-based superinductances, special care must be taken to avoid introducing charge dispersion to the qubit transition freqeuncy due to quantum phase slips in the superinductance [60,77].

C. Fundamental asymmetry between charge and flux noise
When identifying a desirable parameter regime of a superconducting qubit, we also need to take into account the amplitude associated with the charge and flux fluctuations [ Fig. 4(c)-(d)]. Experimentally, the amplitude of flux fluctuations (in units of Φ 0 ) is typically about two orders of magnitude smaller than that of charge fluctuations (in units of 2e) [59][60][61][62][63][64][65][66][67][68]. This is the consequence of the lowimpedance of the electromagnetic environment surrounding the qubits [78]. As a result, the scale of the relevant noise fluctuations is smaller in flux modes than in charge modes, relative to the periodicity of their Hamiltonians [ Fig. 4(c)-(d)]. Thus, we expect the flux-mode qubits to have longer coherence times than charge-mode qubits for equal energy dispersion. Thanks to this observation, it is possible to operate flux-mode qubits away from the optimal light regime while maintaining large coherence times. As an example, the fluxonium qubit is defined in the regime E J /E C ∼ 5 − 10 and E L /E C ∼ 0.5 − 1 [73], which is not sufficient to reach exponential flux-noise insensitivity. Still, the coherence time at a magnetic sweet spot has been reported to be longer than in the transmon with best coherence times in the millisecond regime [79][80][81].

D. Bit-flip limit in noise-insensitive single-mode devices
In the noise-insensitive regime of both charge-and fluxmode qubits, the delocalization of the qubit states over charge or flux states results in a large transition matrix element. While this means that these qubits can be strongly coupled to electromagnetic modes to realize fast gates and readout, it also enhances the rate of bit-flip errors [Eq. (1b)]. In these qubits, bit-flip errors can, in fact, become the main contribution to the qubit decoherence with T 2 ∼ 2T 1 . For example, the higher coherence times in the fluxonium compared to the transmon is related to the longer relaxation times in the fluxonium circuit, which is mostly the result of its smaller charge transition matrix element [79,81].
An approach to further improve the situation is to artificially reduce the size of the qubit transition matrix element, essentially trading phase-flip protection for bit-flip protection. For example, in the heavy flux-mode qubit known as the heavy fluxonium [44-46, 76, 80], the larger E J /E C ratio significantly weakens the tunneling between neighboring potential valleys of the corrugated harmonic potential. Away from the flux sweet spot, this leads to long-lived states that are localized in single wells with lifetimes in the several milliseconds regime [ Fig. 5(c)]. As expected, this increase in T 1 is obtained at the expense of reducing the pure dephasing times T ϕ : the heavy fluxonium shows a few microseconds coherence time away from the sweet spot due to the linear flux dispersion.
Moreover, the approach to realizing logical gates on heavy fluxonium highlight an important aspect of protected circuits: the disjoint nature of the qubit states complicates the control of the device. Indeed, while in the transmon, single qubit gates simply rely on driving the qubit at its transition energy, the instead, the qubit is protected against relaxation away from the sweet spot (bit-flip protection). In the case of the charge mode, the situation is reversed as the heavy regime supports offset-charge-noise insensitivity. There is no protection in single charge modes against bit-flip errors. The markers indicate experimentally realized devices: blochnium (triangle) [69], fluxonium (hexagon) [79], heavy fluxonium (diamond) [45] and transmon (square) [15].
suppressed tunneling matrix elements in the heavy fluxonium makes gates based on direct transition between the two logical state impractically long or outright impossible. An alternative approach is to break the protection during the gate, for example, by using Raman-type gates that rely on virtually populating higher lying levels that couple to both qubit states [45,76]. The competition between bit-flip and phase-flip processes in single-mode devices can be illustrated by mapping the transition matrix element and energy dispersion as a function of the circuit parameters (Fig. 6). While charge-mode qubits are protected against charge-noise dephasing in the heavy regime where E J /E C is large, flux modes are insensitive to flux noise in the light regime, where both E J /E C and E L /E C are small. Additionally, flux modes can also be protected against bit-flip processes but only in the heavy regime. This figure summarizes a key point of this discussion: simultaneous protection against energy relaxation and pure dephasing cannot be realized in single-mode qubits.

IV. TOWARDS FULL NOISE PROTECTION WITH MULTIMODE CIRCUITS
An approach to overcome the intrinsic limitations of singlemode qubits is to combine them into multimode circuits. For instance, by connecting two noise-insensitive modes, the resulting circuit can support simultaneous protection against
It is instructive to investigate how charge and flux modes form composite circuits in the case of the 0 -π and bifluxon qubits. Although these two devices have different circuit layouts and operate in distinct parameter regimes, the Hamiltonians describing their qubit degrees of freedom are identical. Both circuits rely on a charge-like mode θ and a fluxlike mode φ, which are strongly and nonlinearly coupled to each other. The charging energies of the two modes are E θ C and E φ C , the inductive energy is E L , while the coupling term between the modes, cosθ cosφ, is proportional to the Josephson energy E J . The Hamiltonian of these qubits takes the formĤ which, in some sense, is the combination of the transmon and fluxonium Hamiltonians. As we will see below, an important difference between the 0 -π and bifluxon qubits is how they physically implement these coupled modes through the circuit layout. When discussing the experimental realization of these ideas it is useful to contrast between two types of parameter regimes. First, there is the theoretically ideal but difficult to realize "hard" regime. Second, there is the "soft" regime, which is experimentally easier to implement but at the price of making compromises on the qubit protection. In practice, much of the difficulty in realizing the hard regime lies in making light flux modes as those require building large inductances and having very little stray capacitances in the circuit. As a result, while in the hard regime flux modes can be exponentially flux-insensitive, in the soft regime protection against flux noise relies on working at sweet spots.
In the hard parameter regime, the Hamiltonian of the circuits can often be approximated by an effective Hamiltonian with an underlying symmetry. For example, in the 0 -π or cos 2θ qubits, the effective Hamiltonian preserves the parity of Cooper-pair-numbers on the islands of the circuit. Similarly, in the bifluxon qubit, the symmetry corresponds to the parity of flux quanta contained in the loop of the circuit. In these cases, the qubit states become nearly degenerate and have odd or even parities. Such symmetry protection satisfies the requirements for high-coherence described in Secs. II and III. First, degenerate levels have vanishing energy dispersion that leads to protection against pure dephasing; second, transitions between states with opposite parity is prohibited as long as the noise respects the symmetry of the system (protection against energy relaxation).
In the soft parameter regime, however, the underlying parity symmetry is not present anymore and the protection originates from qubit wavefunctions with disjoint support and weak energy dispersion, as discussed in Secs. II and III. These devices still offer protection but instead of the simultaneous exponential reduction in bit-flip and phase-flip errors, the logical states might be protected only to first order against these errors.
In this section, we discuss these ideas focusing on two experimentally realized multimode circuits, the bifluxon and the 0 -π qubit. We review both the hard and soft regimes with emphasis on the underlying symmetries and the challenges associated with demonstrating full-noise protection.

A. The bifluxon qubit
As illustrated in Fig. 7(a), the bifluxon qubit is a threenode device composed of a superinductor and two identical Josephson junctions forming a loop [51,85]. The bifluxon circuit has two degrees of freedom, a charge mode θ and a flux mode φ. Correspondingly, the qubit has two control knobs: the offset charge n gate biasing the superconducting island, and the external flux φ ext threading the loop of the device. Both the charge and the flux modes are light where E L is the inductive energy of the superinductor, E J is the Josephson energy, E θ C and E φ C are the charging energies of the θ and φ modes, respectively. We can understand noise protection in this qubit by considering its logical states from the point of view of the flux mode φ at n gate = 0.5. As a consequence of a destructive Aharonov-Casher-type interference [51,85,88], the logical states have opposite fluxon parities: the wavefunctions associated with the |0 and |1 states peak at the phase values φ 2m = 2m × π and φ 2m+1 = (2m + 1) × π, respectively, where m is a small integer [ Fig. 7(b)]. This can be also interpreted as the loop storing an odd or an even number of flux quanta depending on the qubit state. Since logical states with distinct fluxon parity cannot be connected by the phase operator, transitions are prevented. Furthermore, the large zeropoint fluctuations of the φ mode lead to exponential insensitivity against flux noise [ Fig. 7(b)].
The mechanism behind noise protection in this circuit can be further elucidated by focusing on the role of the charge mode θ. As the the charge mode in the bifluxon circuit operates in the Cooper-pair-box regime, the charge component of the low-lying eigenfunctions can be well approximated by a linear combination of only two charge states {|n , |n + 1 } [51]. At the sweet spot n gate = 0.5, the Josephson coupling between these charge states leads to eigenstates with bonding, (|n + |n + 1 )/ √ 2, and antibonding, (|n − |n + 1 )/ √ 2, symmetry in charge space. Effectively, the antibonding state introduces a flux bias of φ ext = 0.5, favoring oddfluxon-number states. The situation is the opposite for the bonding state, which favors even-fluxon-number states.
In the hard parameter regime at n gate = 0.5, the bifluxon qubit is protected against relaxation and displays firstorder insensitivity against charge noise and exponential protection against flux noise. However, because it relies on a large superinductor, reaching the ideal parameter regime of the qubit is experimentally challenging. In the soft parameter regime [51], a heavier flux mode leads to a reduced first-order protection at the flux sweet spot while preserving bit-flip protection because the qubit wavefunctions have disjoint support [ Fig. 7(c)]. In this regime, an order of magnitude enhance-ment of T 1 > 100 µs was demonstrated with respect to the unprotected T 1 value, although qubit coherence was limited to T ϕ = 700 ns. The rather low dephasing time is due to the charge qubit nature of the device.
Similarly to the heavy fluxonium, protection against bitflip errors unavoidably complicates qubit manipulation because direct transitions between the logical states are forbidden. Thus, in the first realization of the bifluxon, control is achieved by breaking the protection while the single qubit gate is applied: the gate bias is quickly detuned from the n gate = 0.5 value to n gate = 0, and tuned back to the protected regime after the end of the gate.

B. The 0 -π qubit
The 0 -π qubit is characterized by protected logical states with exponentially suppressed relaxation and dephasing rates against both charge and flux noise. The device was proposed by Brooks, Preskill and Kitaev [49] building on an earlier design known as the current mirror qubit [57,89].
To realize full noise protection, the 0 -π circuit combines a heavy charge mode θ and a light flux mode φ based on a four-node circuit layout [ Fig. 8(a)]. The wavefunctions are localized along the θ coordinate and delocalized along the φ direction [ Fig. 8(b)]. A first requirement to reach this regime is to ensure that the ratio of charging energies of the two modes, E φ C /E θ C , is large. Moreover, for the φ mode to be in its light regime, the inductive energy E L needs to be small. Finally, the large Josephson energy E J provides the potential barrier between the two valleys along the θ direction, leading to the charge-noise insensitivity and disjoint support of the qubit wavefunctions. When the circuit parameters meet all of these requirements (E L ≈ E θ C E J ≈ E φ C ), the ground and first excited states of the circuit are delocalized along the φ direction while being confined to the θ = 0 and π valleys from which the qubit takes its name.
In the hard parameter regime, it is possible to derive an effective single-mode Hamiltonian for the circuit which corresponds to a capacitively shunted tunneling element for pairs of Cooper pairs [83]. In this case, the cotunneling process can be described by a cos 2θ term, while the single-Cooperpair tunneling term cosθ is highly suppressed. The ground or excited state of this model are even or odd charge states, respectively. The transitions between these states are forbidden because the effective Hamiltonian preserves the Cooper-pairnumber parity. In analogy with the transmon, if the energy associated with the tunneling of pairs of Cooper pairs is large compared to the effective charging energy, the two qubit states are delocalized in charge space and the transition frequency is rendered insensitive to charge noise. Despite the apparent complexity of the full 0 -π circuit, this simple effective model captures the exponential protection against both phase-and bit-flip errors [83].
In addition to the above stringent requirements on the energy scales of the device, it is also important for the pairs of large capacitors and superinductors in the circuit to be as identical as possible. Indeed, since the 0 -π circuit contains Ideal regime Experimentally realized regime four nodes, the circuit inevitably incorporates a third degree of freedom, the ζ mode, which can couple to the qubit modes φ and θ in the presence of circuit-element disorder. Because the energy of this mode is stored in the large inductors and large capacitors of the circuit [ Fig. 8(a)], the ζ mode is a low frequency harmonic mode. In the presence of asymmetries, thermal population of this mode can lead to dephasing of the qubit. While the 0 -π is protected against this noise channel in the hard parameter regime [82,83], the effects of ζmode coupling in the softer parameter regimes can be possibly mitigated with active strategies involving cooling of this mode [83].
Experimentally realizing the hard parameter regime is challenging because of the competition between the conditions that guarantee charge-and flux-noise insensitivity. For example, simultaneously achieving the light and heavy regimes of the flux and charge modes in the same circuit requires a substantial reduction in unwanted cross capacitances. The first experimental implementation of the 0 -π qubit [84] relaxed the requirements of the light flux mode by working with smaller E φ C and larger E L than ideal, leading to considerable flux dispersion of the qubit transition. Importantly, the experimental device retained simultaneous first-order insensitivity to flux noise, exponential protection to charge noise, and the disjoint character of the qubit wavefunctions at φ ext = 0 [ Fig. 8(c)]. Therefore, this implementation demonstrates a degree of noise protection beyond that of the transmon and fluxonium qubits, constituting a blueprint for building a fully noise-protected superconducting qubit. The measured energy decay times of this soft 0 -π qubit is T 1 = 1.6 ms, which is comparable to heavy fluxonium, while the dephasing rate obtained by a spin echo measurement is T 2E = 25 µs.
Single qubit gates on the soft 0 -π circuit also rely on partially lifting the protection [84]. In contrast to the bifluxon, where the system shortly leaves the protected parameter regime during the gate (namely the n gate = 0.5 bias point), here the system rather leaves the protected subspace (the logical |0 and |1 states) by temporarily occupying a higher-lying ancillary state in a Raman-type gate. Since the population of the ancillary state depends on the detuning of the drive tones, the occupation of the unprotected ancillary state can be arbitrary small at the price of increasing drive powers to avoid prohibitively long gate times. Thus, such gates require finding the balance between the amplitude of microwave drives and occupation of unprotected levels [84].

V. HYBRID PROTECTED SEMI-SUPERCONDUCTING QUBITS
In the superconducting qubits discussed so far, the Josephson junctions use aluminum-oxide tunnel barriers between aluminum electrodes. Such superconductor-insulatorsuperconductor junctions result in nonlinear elements with an almost perfectly sinusoidal current-phase relationship. In this case, realizing protected qubits requires fabricating a complex combination of these Josephson junctions, large superinductors and extremely large or tiny capacitors in the same circuit. Semiconductor-based technology circumvents this obstacle by equipping Josephson junctions with non-sinusoidal current-phase behavior [91]. Such unorthodox Josephson junctions open up new possibilities to engineer exotic Hamiltonians that protect quantum information. Although the lower quality of materials currently limits this technology [92], re--π -π/2 0 π/2 π θ  cent advances have enabled the implementation of the first prototypes of protected hybrid qubits. For example, a chargeinsensitive Cooper pair box based on highly-transmissive Josephson junctions [93,94] and a single-mode semiconductor cos 2θ qubit [90] illustrates the potential of this platform.
Here, we focus on the cos 2θ hybrid qubit [90]. To do so, we briefly review the behavior of Josephson junctions from a microscopic viewpoint based on Andreev bound states. When two superconducting electrodes connect through a thin tunnel barrier, quasiparticle bound states form due to multiple Andreev reflections between the two superconducting regions. In most cases, multiple bound states mediate the coupling between the electrodes, and we can express the Josephson energy as the sum over the channels, such thatÛ (θ) = −∆ i 1 − T i sin 2θ /2, where T i is the transmission probability through the tunnel barrier for channel i and ∆ is the superconducting gap [95]. In the case of a large number of weakly transmitting channels-which applies to aluminumoxide based tunnel junctions-we recover the well-known expression for the Josephson energyÛ (θ) = E J (1 − cosθ), with E J = ∆ i T i /4. Weak-link Josephson junctions can have a few highly transmitting channels in semiconductor InAs nanowires with an epitaxially grown superconductor layer [96][97][98][99] or in proximitized two-dimensional electron gasses [92].
In a few-channel superconductor-semiconductor junction, the 2π-periodic Josephson energy can be expressed with its Fourier componentsÛ (θ) = i A k cos(kθ), assuming timereversal symmetry is not broken in the junction [100]. While generally in metallic Josephson junctions the first harmonic term plays the dominant role in the dynamics (A 1 A 2 ...), in semiconductor-based junctions the higher harmonic content can have a significant contribution as well. In a hybrid interferometer circuit formed by two identical junctions [ Fig. 9(a)], the total Josephson energy can be expressed asÛ tot (θ) =Û (θ − φ ext /2) +Û (θ + φ ext /2). When the circuit is frustrated at half-flux quantum (φ ext = 0.5), the odd Fourier components of the two identical junctions cancel each other, while the even contributions constructively interfere, giving rise to a double-well potential with a π period that can be approximated with aÛ tot (θ) ≈ E 2 cos 2θ potential [ Fig. 9(b)], similarly to the low-energy effective model of the 0 -π qubit. Figure 9(c) and (d) show the ground and first excited state of the experimentally realized circuit, which resembles odd and even charge states as expected for the disconnected parity subspaces of a cos 2θ element. Importantly, the qubit serves as a protected circuit only at the half-flux quantum sweet spot [ Fig. 9(e)]. Regarding the charge noise of the device, as long as the mode is heavy (E 2 E C ), the qubit states are exponentially insensitive to the offset charge [ Fig. 9(f)]. We also note that this circuit is the dual element of the bifluxon, as the offset-charge and external-flux dependence of the energy levels are reversed for the two qubits [compare Fig. 7(b) and Fig. 9(e) and (f)]. In the first implementation of this idea, the energy relaxation rate yielded a tenfold improvement compared to the unprotected regime with T 1 = 7 µs [90].

VI. OUTLOOK
Superconducting qubits have progressed in a truly remarkable fashion in the last two decades: Starting with coherence TABLE I. Protection in various superconducting circuits. We compare the different degrees of protection in energy relaxation (T1) and dephasing due to charge (T n gate ϕ ) and flux noise (T φ ext ϕ ). Absent: the qubit is not protected against the error; N/A: the qubit is not coupled to the particular noise; linear: the qubit has sweet-spot protection; exponential: the qubit is insensitive to the type of error; exponential*: for the case of the fluxonium and blochnium qubits, this refers to the exponential insensitivity to offset-charge fluctuations due to coherent quantum phase slips [60].

Circuit name
Ideal case Realized case T1 T n gate ϕ T φ ext ϕ T1 T n gate ϕ T φ ext ϕ Transmon [15,70] absent exponential N/A absent exponential N/A Fluxonium [73] absent exponential* linear absent exponential* linear Blochnium [69] absent exponential* exponential absent exponential* exponential Heavy fluxonium [44][45][46] exponential exponential* absent exponential exponential* absent Bifluxon [51,85] exponential linear exponential exponential linear linear 0 -π qubit [49,84] exponential exponential exponential exponential exponential linear Semiconductor cos 2θ [90] exponential exponential linear exponential exponential linear times of a few nanoseconds, today's typical values are in the hundreds of microseconds. As the evolution of the transmon qubit illustrates, suitable and optimized circuits have enabled significant advancements. The fluxonium circuit is on a similar ascending trajectory, with millisecond coherence times having recently been demonstrated [81]. Their simplicity and partial built-in protection against dominant noise sources is an important reason for their success. Because significant improvements in qubit performances are still necessary to realize the full potential of fault-tolerant quantum computation, it remains essential to not only continue to improve current qubits but also to explore novel qubit designs. Protected qubits play an important role in these efforts as they simultaneously suppress relaxation and dephasing. Although this is obtained at the price of increased circuit complexity, in the long run such circuits have the potential to dramatically reduce the physical qubit count needed for quantum error correction.
In addition to continue the development of protected qubits, it remains important to design single-and two-qubit logical operations on these qubits. Current approaches require leaving the protected regime [51,90] or relying on noncomputational states [45,84]. Finding more fast and robust approaches to control these devices remains a theoretical and experimental challenge.
Whether protected qubits will play the same role as the "mainstream" qubits or be responsible for different tasks in a heterogeneous processor, such as memory or data units, still needs to be decided. Nevertheless, given the formidable challenge that it represents, superconducting quantum processors will likely benefit from the synergy of multiple strategies including quantum error correction, operation in an engineered ultra-low-noise environment, and hardware-level error protection.