Quantum information processing with bosonic qubits in circuit QED

In section 2, we begin by outlining the basic principles of QEC as well as bosonic codes. Here, we highlight their nature by comparing a bosonic code with a multi-qubit code in the presence of similar errors. In section 3, we present various bosonic encoding schemes proposed in literature and compare their respective strengths and limitations. In particular, we wish to emphasize crucial considerations for constructing these codes and evaluating their performance in the presence of naturally-occurring errors.

In section 4, we introduce the key hardware building blocks required for cQED implementations of bosonic encoding schemes. In this section, we also consolidate the progress made in improving the intrinsic quality factors of superconducting microwave cavities over the last decade. Subsequently, in section 5, we explore the latest developments in implementing robust universal control on bosonic qubits encoded in superconducting cavities, both in terms of single-mode gates as well as novel two-mode operations. We then describe the different strategies for detecting and correcting quantum errors on bosonic logical qubits encoded in superconducting cavities.

In section 6, we discuss the concept of fault-tolerance and how that might be realized with protected bosonic qubits. Here, we also feature some novel schemes that concatenate bosonic codes with other QEC codes to protect against quantum errors more comprehensively. Finally, in section 7, we provide some perspectives for achieving QEC on a larger scale. We conclude by remarking on the appeal of the modular architecture, which offers a promising path for practical and robust quantum information processing with individually protected bosonic logical elements.

The class of rotation-symmetric bosonic codes [40] refers to encodings that remain invariant under a set of discrete rotations in phase space. Encoding schemes with rotation-symmetry, such as the cat and binomial codes, are stabilized by a photon number super-parity operator ${\hat{{\Pi}}}_{N}={\mathrm{e}}^{\mathrm{i}\left(2\pi /N\right)\hat{n}}\enspace \left(N\in \left\{2,3,\cdots \enspace \right\}\right)$, which is equivalent to the 360/N° rotation operator. As such, these codes are, by design, capable of detecting N − 1 photon loss events.

Furthermore, these rotation-symmetric codes can also be made robust against dephasing errors in addition to photon loss errors [41]. For instance, the kitten code may be modified to have the following logical states:

$$\begin{equation}\vert {0}_{\mathrm{L}}\rangle =\frac{1}{2}\left(\vert 0\rangle +\sqrt{3}\vert 4\rangle \right),\hspace{25.0pt}\vert {1}_{\mathrm{L}}\rangle =\frac{1}{2}\left(\sqrt{3}\vert 2\rangle +\vert 6\rangle \right).\end{equation} \tag{ 8 }$$

This version of the binomial code is also stabilized by the parity operator and is invariant under the 180° rotation, thus making it robust against single photon loss errors. However, the modified code is now also higher in energy as its logical states have 3 photons on average, as opposed to 2 in the case of the kitten code. This additional redundancy makes the modified binomial code robust against single dephasing events as well. In other words, the logical states of the modified binomial code satisfy the Knill–Laflamme condition for an extended error set $\left\{\hat{I},\hat{a},{\hat{a}}^{{\dagger}}\hat{a}\right\}\;\text{where}\;{\hat{a}}^{{\dagger}}\hat{a}$ describes single dephasing errors in the cavity. This means that the two logical code words also have the same second moment of the photon number probability distribution besides having the same average photon number. Note that when more moments of the photon number probability distribution are the same for the two code words, distinguishing them becomes more difficult for the environment, which enhances the error-correcting capability of the code.

Another way to generalize rotation-symmetric codes is by considering different rotation angles. For instance, we can design codes that are invariant under a 120° rotation instead of a 180° rotation and are thus stabilized by the super-parity modulo 3 operator ${\hat{{\Pi}}}_{3}={\mathrm{e}}^{\mathrm{i}\left(2\pi /3\right)\hat{n}}$. By taking advantage of a larger spacing in the photon number basis, 120°-rotation-symmetric codes can detect two-photon loss events as well as single photon loss events, thus providing enhanced protection to the logical information. For instance, a variant of the binomial code with the logical states

$$\begin{equation}\vert {0}_{\mathrm{L}}\rangle =\frac{1}{2}\left(\vert 0\rangle +\sqrt{3}\vert 6\rangle \right),\hspace{25.0pt}\vert {1}_{\mathrm{L}}\rangle =\frac{1}{2}\left(\sqrt{3}\vert 3\rangle +\vert 9\rangle \right),\end{equation} \tag{ 9 }$$

is invariant under the 120° rotation, or equivalently, has photon numbers that are integer multiples of 3. Moreover, the particular coefficients associated with each photon number state in the code words, which are derived from the binomial coefficients (hence the name 'binomial' code), ensure that the code satisfies the Knill–Laflamme condition for an error set $\left\{\hat{I},\hat{a},{\hat{a}}^{2},{\hat{a}}^{{\dagger}}\hat{a}\right\}$. This indicates that the code is robust against both single and two-photon loss events, as well as single dephasing errors. The binomial code can be further generalized to protect against higher-order effects of loss and dephasing errors [25] as well as used for autonomous QEC [42].

Cat codes are another important example of rotation-symmetric bosonic codes. The four-component cat codes (or four-cat codes), which are composed of four coherent states |±α⟩, |±iα⟩, are the simplest variants of the cat codes that are robust against photon loss errors [43, 44]. In this encoding, the logical states are defined by superpositions of coherent states:

$$\begin{align}\hfill \vert {0}_{\mathrm{L}}\rangle & \propto \vert \alpha \rangle +\vert \mathrm{i}\alpha \rangle +\vert -\alpha \rangle +\vert -\mathrm{i}\alpha \rangle \propto \sum\limits _{n=0}^{\infty }\frac{{\alpha }^{4n}}{\sqrt{\left(4n\right)!}}\vert 4n\rangle ,\hfill \\ \hfill \vert {1}_{\mathrm{L}}\rangle & \propto \vert \alpha \rangle -\vert \mathrm{i}\alpha \rangle +\vert -\alpha \rangle -\vert -\mathrm{i}\alpha \rangle \propto \sum\limits _{n=0}^{\infty }\frac{{\alpha }^{4n+2}}{\sqrt{\left(4n+2\right)!}}\vert 4n+2\rangle .\hfill \end{align} \tag{ 10 }$$

Note that the amplitude of the coherent states |α| determines the size of the cat code. Similar to the kitten code (equation (2)), the logical states of the four-cat code have an even number of photons. Thus, the four-cat code is invariant under the 180° rotation and is able to detect single photon loss events. Here, we reinforce that the parity (or rotation-symmetry) alone does not ensure the recoverability of logical information against single photon loss errors, which is only guaranteed when the Knill–Laflamme condition is satisfied for the error set $\left\{\hat{I},\hat{a}\right\}$. For even parity codes, as explained in section 2, recoverability requires the two logical states to have the same number of photons on average.

For large cat codes with |α| ≫ 1, the average photon number is approximately given by $\bar{n}\simeq \vert \alpha {\vert }^{2}$ for both logical states, and the Knill–Laflamme condition is approximately fulfilled. More importantly, the average photon numbers of the two logical states are exactly the same for certain values of |α| (also known as 'sweet spots' [45]) such that:

$$\begin{equation}\mathrm{tan}\vert \alpha {\vert }^{2}=-\mathrm{tanh}\vert \alpha {\vert }^{2}.\end{equation} \tag{ 11 }$$

The smallest such |α| is given by |α| = 1.538, which corresponds to the average photon number $\bar{n}=2.324$. By increasing the size of the cat codes, we can construct logical states that are robust against both single photon loss and dephasing errors. In particular, cat codes with a large average photon number |α|² satisfy the Knill–Laflamme condition for the dephasing error set $\left\{\hat{I},\hat{n},{\hat{n}}^{2},\cdots \right\}$ approximately modulo an inaccuracy that scales as ${\mathrm{e}}^{-2\vert \alpha {\vert }^{2}}$ [46]. However, an extra error-correcting mechanism, such as a multi-photon engineered dissipation [44], is required to exploit the intrinsic error-correcting capability of the cat codes against dephasing errors. Moreover, increasing the number of coherent state components in the cat code introduces further protection [36, 47]. For instance, with six coherent state components, the code words become 120°-rotation-symmetric and can thus detect up to two-photon loss events. Generalizations of cat codes and analyses of their performance can be found in reference [45], while a multi-mode generalization is available in reference [48].

Another class of bosonic codes are translation-symmetric, with the GKP codes [49] being a prominent example. The simplest variant of the GKP codes is the square lattice GKP code, which encodes a logical qubit in the phase space of a harmonic oscillator stabilized by two commuting displacement operators:

$$\begin{equation}{\hat{S}}_{q}\equiv \mathrm{exp}\left[\mathrm{i}2\sqrt{\pi }\hat{q}\right]=\hat{D}\left(\mathrm{i}\sqrt{2\pi }\right),\hspace{25.0pt}{\hat{S}}_{p}\equiv \mathrm{exp}\left[-\mathrm{i}2\sqrt{\pi }\hat{p}\right]=\hat{D}\left(\sqrt{2\pi }\right),\end{equation} \tag{ 12 }$$

where $\hat{q}$ and $\hat{p}$ are the position and momentum operators respectively.

A key motivation for choosing these stabilizers is to circumvent the Heisenberg uncertainty principle, which dictates that the position and momentum operators cannot be measured simultaneously as they do not commute. Since the two displacement operators in equation (12) commute with each other, we can measure them simultaneously. Measuring the displacement operators $\mathrm{exp}\left[\mathrm{i}2\sqrt{\pi }\hat{q}\right]$ and $\mathrm{exp}\left[-\mathrm{i}2\sqrt{\pi }\hat{p}\right]$ is equivalent to measuring their phases $2\sqrt{\pi }\hat{q}$ and $-2\sqrt{\pi }\hat{p}$ modulo 2π, which is in turn the same as simultaneously measuring $\hat{q}$ and $\hat{p}$ modulo $\sqrt{\pi }$. Therefore, the square lattice GKP code is, by design, capable of addressing two non-commuting quadrature operators by simultaneously measuring them within a unit cell of a square lattice. The uncertainty now lies in the fact that we do not know which unit cell the state is in.

The two logical states of the square lattice GKP code are explicitly given by:

$$\begin{align}\hfill \vert {0}_{\mathrm{L}}\rangle & \propto \sum\limits _{n\in \mathbb{Z}}\vert \hat{q}=\left(2n\right)\sqrt{\pi }\rangle \propto \sum\limits _{n\in \mathbb{Z}}\vert \hat{p}=n\sqrt{\pi }\rangle ,\hfill \\ \hfill \vert {1}_{\mathrm{L}}\rangle & \propto \sum\limits _{n\in \mathbb{Z}}\vert \hat{q}=\left(2n+1\right)\sqrt{\pi }\rangle \propto \sum\limits _{n\in \mathbb{Z}}{\left(-1\right)}^{n}\vert \hat{p}=n\sqrt{\pi }\rangle ,\hfill \end{align} \tag{ 13 }$$

and satisfy $\hat{q}=\hat{p}=0$ modulo $\sqrt{\pi }$, thus clearly illustrating that a periodic simultaneous quadrature measurement is indeed possible if the spacing is chosen appropriately. Additionally, the code states are invariant under discrete translations of length $2\sqrt{\pi }$ in both the position and momentum directions, which makes the code symmetric under translations.

Conceptually, ideal GKP code states consist of infinitely many infinitely squeezed states where each component is described by a Dirac delta function. However, precisely implementing these ideal states is not feasible in realistic quantum systems. In practice, only approximate GKP states can be realized where each position or momentum eigenstate is replaced by a finitely squeezed state and large position and momentum components are suppressed by a Gaussian envelope [49–51]. Many proposals to realize approximate GKP states in various physical platforms [49, 50, 52–65] and ways to simulate approximate states efficiently [66–71] have been explored in the field. The quality of such states can be characterized by the degree of squeezing in both the position and momentum quadratures. As the squeezing, and hence the average photon number, increases, the approximate state will converge toward an ideal code state. Recently, approximate states of squeezing 5.5–9.5 dB have been realized in trapped ion [72, 73] and cQED [74] systems. The cQED realization is discussed further in section 5.

With a discrete translation-symmetry, GKP codes are naturally robust against random displacement errors as long as the size of the displacement is small compared to the separation between distinct logical states. For instance, the square lattice GKP code is robust against any displacements of size less than $\sqrt{\pi }/2$ as they can be identified via the quadrature measurements modulo $\sqrt{\pi }$ and then countered accordingly. For moderately squeezed approximate GKP states that contain a small number of photons, photon loss errors can be decomposed as small shift errors and therefore can be effectively addressed by the code [49, 50]. On the other hand, for large approximate GKP states that are highly squeezed, even a tiny fraction of photon loss results in large shift errors which cannot be corrected by the code. Thus, naively using the standard GKP error correction protocol to decode does not work for large GKP states under photon loss errors. Nevertheless, studies have observed that if an optimal decoding scheme is adopted, excellent performance against photon loss errors can be achieved even with large GKP states [46, 75]. This improvement happens because photon loss errors can be converted into random displacement errors via amplification. This implies that for highly squeezed large GKP codes, a suitable decoding strategy is to first amplify the contracted states and then correct the resulting random shift errors by measuring the quadrature operators modulo $\sqrt{\pi }$ [76]. We note that at present, no analogous techniques are known for dephasing errors. Thus, highly squeezed GKP codes are not robust against dephasing errors since even a small random rotation can result in large shift errors [50].

While the preparation of GKP states is challenging, implementing logical operations on GKP states is relatively straightforward. Any logical Pauli or Clifford operation on GKP states can be realized by a displacement or Gaussian operation (via a linear drive or a bilinear coupling). Moreover, magic states [77] encoded in the GKP code, which are necessary for implementing non-Clifford operations, can be prepared with only Gaussian operations and GKP states [50, 78]. Thus, the preparation of code words is the only required non-Gaussian operation for performing universal quantum computation with the GKP code. Non-Clifford operations can be directly enacted on GKP qubits [49], although the cubic phase gate suggested in the original proposal has been recently shown to perform poorly for this purpose [79].

Furthermore, GKP states can be defined over lattices other than the square lattice. For instance, hexagonal GKP codes can correct any shift errors of size less than ${\left(2/\sqrt{3}\right)}^{1/2}\sqrt{\pi }/2\simeq 1.07\sqrt{\pi }/2$, which is larger than the size of shifts that are correctable by the square lattice GKP code [49, 75]. A recent work has also explored the rectangular GKP code [80]. More generally, a multi-mode GKP code can be defined over any symplectic lattice [81]. Lastly, the GKP code may also be used for building robust quantum repeaters for long-distance quantum communication [82, 83].

Recently, there has been growing interest in biased-noise bosonic qubits, where a quantum system is engineered to have one type of error occur with a much higher probability than other types of errors. This noise bias can simplify the next layer of error correction [84]. For instance, we can design a biased-noise code that suppresses bit-flips and then correct the dominant phase-flip errors by using repetition codes [85, 86]. Alternatively, we can also tailor the surface code to leverage the advantages of biased-noise models to increase fault-tolerance thresholds and reduce resource overheads [87–90].

A promising candidate for biased-noise bosonic qubits is the two-component cat code [91–95], or two-cat code, whose logical code words are given by:

$$\begin{align}\hfill \vert {+}_{\mathrm{L}}\rangle & \propto \vert \alpha \rangle +\vert -\alpha \rangle \propto \sum\limits _{n=0}^{\infty }\frac{{\alpha }^{2n}}{\sqrt{\left(2n\right)!}}\vert 2n\rangle ,\hfill \\ \hfill \vert {-}_{\mathrm{L}}\rangle & \propto \vert \alpha \rangle -\vert -\alpha \rangle \propto \sum\limits _{n=0}^{\infty }\frac{{\alpha }^{2n+1}}{\sqrt{\left(2n+1\right)!}}\vert 2n+1\rangle .\hfill \end{align} \tag{ 14 }$$

If α is large enough, two-cat codes are capable of correcting dephasing errors. One way to implement two-cat codes is by autonomously stabilizing them via an engineered dissipation of the form ${\kappa }_{2}\mathcal{D}\left[{\hat{a}}^{2}-{\alpha }^{2}\right]$ [44]. This engineered two-photon dissipation can exponentially suppress the logical bit-flip error in the code states due to dephasing in superconducting cavities $\left({\kappa }_{\phi }\mathcal{D}\left[{\hat{a}}^{{\dagger}}\hat{a}\right]\right)$, i.e.,

$$\begin{equation}{\gamma }_{\text{bit}-\text{flip}}\simeq 2{\kappa }_{\phi }\vert \alpha {\vert }^{2}\enspace {\mathrm{e}}^{-2\vert \alpha {\vert }^{2}}\quad \mathrm{i}\mathrm{f}\quad {\kappa }_{\phi }\ll {\kappa }_{2},\end{equation} \tag{ 15 }$$

where |α|² is the size of the cat code, κ₂ is the two-photon dissipation rate, and κ_ϕ is the dephasing rate.

However, the dominant error source in superconducting cavities is photon loss. While dephasing of the cavity does not change the photon number parity, a single photon loss can directly flip the photon number parity of the state. In other words, a single photon loss maps an even parity state |+_L⟩ to an odd parity state |−_L⟩, thus causing a phase-flip error. As such, phase-flip errors due to single photon loss cannot be mitigated by the two-cat code. Nevertheless, bit-flip errors due to single photon loss can be suppressed exponentially in the photon number α² provided that the two-photon dissipation is strong enough. Once bit-flip errors are countered, the next layer of QEC will only need to tackle the phase-flip errors, which can be induced by spurious transitions of the ancilla states. In particular, in the stabilized two-cat codes, thermal excitation of the ancilla during idle times can result in a significant rotation of the cavity state and complete dephasing of the encoded qubit. Currently, such events cannot be corrected by the code and are observed to be a limiting factor for the logical lifetime of the encoded qubit [39].

Biased-noise two-cat qubits can also be realized by using an engineered Hamiltonian $\hat{H}=-K\left({\hat{a}}^{{\dagger}}-{\left({\alpha }^{{\ast}}\right)}^{2}\right)\left({\hat{a}}^{2}-{\alpha }^{2}\right)$, which has two coherent states |±α⟩ as degenerate eigenstates [96–98]. A crucial general consideration for these biased-noise bosonic codes is that the asymmetry in the noise must be preserved during the implementation of gates. The theoretical framework to achieve a universal gate set on the two-cat codes in a bias-preserving manner has been proposed recently [85, 98].

In summary, rotation-symmetric codes (e.g., four-cat codes and binomial codes) are robust against both photon loss and dephasing errors. Translation-symmetric codes such as the GKP codes can be made highly robust against photon loss errors but are susceptible to dephasing errors. Thus, translation-symmetric codes are suited for loss dominated systems. In contrast, two-cat codes can correct dephasing errors well if they are stabilized by an engineered two-photon dissipation, but are not capable of correcting photon loss errors. Hence, two-cat codes are naturally suited for dephasing dominated systems. However, as discussed in section 3.3, two-cat codes can also be useful in the loss dominated regime as their large noise bias can simplify any higher-level error correction schemes. In figure 2(e), we provide a qualitative schematic that represents different regimes of photon loss and dephasing where each code is designed to perform well. Note that if the loss and dephasing error probabilities are too high, the quantum capacity [99, 100] of the corresponding quantum channel will vanish. When this happens, encoding logical quantum information in a reliable way becomes impossible even with an optimal QEC code [101–105].

While photon loss is the dominant error channel in typical cQED implementations, excitation gain events do also occur, although at a much lower rate. In general, codes robust against photon loss errors tend to be robust against photon gain as well. For rotation-symmetric codes, parity (or super-parity) measurements can detect both photon loss and photon gain events. In the case of translation-symmetric GKP codes, photon gain can be converted into a random shift error by applying a suitable photon loss channel and hence be monitored with the modular quadrature measurement. On the other hand, since the two-cat codes are not robust against photon loss, they are consequently susceptible to photon gain too, which results in phase-flip errors. Nevertheless, in the presence of engineered dissipation, the stabilized two-cat codes can still preserve their noise bias under both photon loss and gain errors.

Until now, we have focused only on the intrinsic error-correcting capability of various bosonic codes without considering practical imperfections. In the following sections, we will discuss the implementation of bosonic QEC in cQED systems and examine errors that occur in practical situations.

cQED explores light–matter interactions by confining quantized electromagnetic fields in precisely engineered compositions of superconducting inductors and capacitors [15]. These superconducting circuit elements can be tailor-made by conventional fabrication techniques [112] and controlled by commercially available microwave electronics and dilution refrigerators [113] to access strong coupling regimes [114–116]. These characteristics make cQED a compelling platform for universal quantum computation, as noted by several review articles [14, 117–123].

Quantum devices in the cQED framework are typically built by coupling two components, a linear oscillator mode and an anharmonic mode, in different configurations [15]. The anharmonic modes are discrete few-level systems that can be implemented using Josephson junctions. They can be designed to interact with one or more harmonic modes, akin to atoms in an optical field in the cavity QED framework [124]. Unlike naturally occurring atoms, the parameters of these anharmonic oscillators, or artificial atoms, can be precisely engineered. For instance, they may be made to have a fixed resonance frequency in devices with a single Josephson junction or have tunable frequencies by integrating multiple junctions in the presence of an external magnetic flux. The lowest two or three energy levels of these artificial atoms, typically in the form of transmons [125, 126], can be used to effectively encode and process quantum information, as demonstrated by successful realizations of various NISQ era [127] processors [128–134].

Linear oscillators are typically realized in cQED by superconducting microwave cavities. Commonly used architectures include coplanar waveguide (CPW) [114], three-dimensional (3D) rectangular [135] and cylindrical co-axial [136], and micromachined [137] cavities. These quantum harmonic oscillators have well-defined but degenerate energy transitions. Therefore, to selectively address their transitions, we must introduce some non-linearities in them. In cQED, non-linearities are introduced by coupling superconducting cavities to artificial atoms in either the resonant or dispersive regimes.

Under resonant coupling, the transition frequencies of the artificial atom and the cavity coincide, which allows the direct exchange of energy from one mode to the other. In this configuration, cavities can act as an on-demand single photon source [138] or a quantum bus that mediates operations between two isolated artificial atoms by sequentially interacting with each of them [139–141]. Dispersive coupling is achieved by detuning the frequencies of the cavity, ω_a , and the artificial atom, ω_b , such that the detuning is much larger than the direct interaction strength between them. In this regime, there is no resonant energy exchange between the modes. Instead, the coupling translates into a state-dependent frequency shift, which can be described by the following Hamiltonian:

$$\begin{equation}\hat{H}={\omega }_{a}{\hat{a}}^{{\dagger}}\hat{a}+{\omega }_{b}{\hat{b}}^{{\dagger}}\hat{b}-{\chi }_{ab}{\hat{b}}^{{\dagger}}\hat{b}{\hat{a}}^{{\dagger}}\hat{a}-\frac{\alpha }{2}{\hat{b}}^{{\dagger}2}{\hat{b}}^{2}-\frac{K}{2}{\hat{a}}^{{\dagger}2}{\hat{a}}^{2},\end{equation} \tag{ 16 }$$

where $\hat{a},\hat{b}$ are the annihilation operators associated with cavity and artificial atom respectively, χ_ab is the dispersive coupling strength between them, α is the anharmonicity of the artificial atom, and K is the non-linearity of the cavity inherited from the atom. Equation (16) further illustrates that the coupling between the two modes is symmetric. In other words, the frequency of the cavity shifts conditioned on the state of the artificial atom, and vice versa.

Superconducting cavities dispersively coupled to an artificial atom constitute a highly versatile tool that can be employed to fulfill many different roles for quantum information processing. For instance, cavities that are strongly coupled to a transmission line are useful for the efficient readout of the quantum state of the artificial atom [114, 142, 143]. Conversely, cavities weakly coupled to the environment can be used as quantum memories for storing information coherently [136]. Moreover, when the linewidth of the cavity is narrow compared to the dispersive shift χ_ab , we can selectively address the individual energy levels of the cavity via the artificial atom [115]. In this case, the artificial atoms act as non-linear ancillae whose role is to enable conditional operations and perform efficient tomography of the cavity state [144]. This configuration where multi-photon states of the cavity encode logical information and the ancilla affords universal control [145] has become an increasingly prevalent choice for implementing bosonic QEC.

Superconducting microwave cavities may be realized in several geometries, with each having their respective advantages. Typically, cavities are constructed in two main architectures, which are two-dimensional (2D) and 3D, based on the dimensionality of the electric field distribution. A third possibility combines the advantages of both the 2D and 3D designs to realize a compact and highly coherent '2.5D' cavity structure, for instance, using micromachining techniques [137, 146].

In the 2D architecture, such as the CPW, the cavity is defined by gaps between circuit elements printed on a substrate which is typically made from silicon or sapphire (figure 3(a)). In such planar structures, the energy is mostly stored in the substrate, surfaces, and interfaces, all of which suffer from losses due to spurious two-level systems in the resonator dielectrics [147–150]. Hence, the Q_int of 2D cavities is currently limited to ∼10⁵ to 10⁶ but can potentially be improved with more sophisticated cavity design [151–154], materials selection [155–158], and surface treatment [159–162]. Despite their limited coherence properties, 2D cavities are widely featured in cQED, and especially in NISQ processors, as they have a small footprint and a straightforward fabrication process. Furthermore, in these processors, these cavities are typically used as readout or bus modes, which do not require long coherence times.

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In contrast, 3D cavities (figure 3(c)) achieve a higher Q_int of ∼10⁷ to 10⁸, at the cost of a much larger footprint than their 2D counterparts, by storing energy in the vacuum between the walls of a superconducting box. Among 3D designs, 3D co-axial cavities machined out of high-purity aluminum (≃99.999%) have shown a Q_int as high as 1.1 × 10⁸ [163]. This is achieved by significantly suppressing the dissipation due to current flowing along the seams of the superconducting walls and imperfections on the inner surfaces of the structure. Additionally, this design is particularly effective as a platform for implementing bosonic codes as one or more ancillae and readout modes can be conveniently integrated into the platform [164].

One strategy to take advantage of both the long lifetimes of 3D designs and the small footprint and scalable fabrication of 2D geometries is to construct compact 2.5D cavities [137, 146, 164–168]. In particular, the micromachining technique provides a promising method to fabricate lithographically defined 2.5D cavities etched out of silicon wafers [137]. In this configuration, the energy can be stored primarily in the vacuum but the depth of the cavity is much smaller compared to the other dimensions (figure 3(b)). Here, the key challenge is to realize a high-quality contact between the top wall and the etched region. With carefully optimized indium bump-bonding methods, internal quality factors surpassing 300 million have recently been achieved in the 2.5D architecture [169]. Moreover, the integration of a transmon ancilla in this design has also been demonstrated [146], thus making these 2.5D cavities promising candidates for realizing large-scale quantum devices based on bosonic modes.

Regardless of the architecture, the effective implementation of bosonic QEC schemes requires a careful balance between the need for isolation and coherence as well as the ability to effectively manipulate and characterize the cavity. The various loss channels [175, 176] as well as lossy interfaces [177–179] of superconducting microwave cavities have been extensively studied and their intrinsic coherence properties have been improving significantly over the last 15 years [176]. In figure 3(d), we compile a non-exhaustive summary of the internal quality factors of cavities demonstrated in various geometries over the last decade. Besides enhancing the Q_int of the cavities, integrating them with ancillary mode(s) is crucial for realizing bosonic logical qubits. However, note that introducing an ancilla results in the degradation of the Q_int, as the best ancilla coherence times (∼50–100 μs) are typically about 10–20 times lower than those of the state-of-the-art superconducting cavities. Hence, while comparing the performance of the cavities in figure 3(d), we have only included demonstrations that are compatible with being coupled to non-linear ancillae in the cQED architecture. From the figure, the 3D co-axial cavities emerge as the leading design currently used to realize bosonic qubits.

For processing quantum information encoded in multi-photon states of superconducting cavities, we must be able to perform effective operations on and characterization of the cavity states. The only operation available for a standalone cavity mode is a displacement, $\hat{D}\left(\alpha \right)={\mathrm{e}}^{\alpha {\hat{a}}^{{\dagger}}-{\alpha }^{{\ast}}\hat{a}}$, which displaces the position and/or momentum of the harmonic oscillator depending on the value of the complex number α. Real values of α correspond to pure position displacements, while imaginary values of α correspond to pure momentum ones [181]. Displacements can only result in the generation of coherent states from vacuum without the possibility to selectively address individual photon number states in the cavity. Therefore, non-trivial operations on these bosonic logical qubits are implemented by dispersively coupling the bosonic mode to a non-linear ancilla in combination with simple displacements.

A key capability enabled by this natural dispersive coupling is a controlled phase shift (CPS). CPS is a unitary operation that imparts a well-defined ancilla state-dependent phase on arbitrary cavity states, and is governed by ${\hat{U}}_{\text{CPS}}\left(t\right)=\vert g\rangle \langle g\vert \otimes \hat{I}+\vert e\rangle \langle e\vert \otimes {\mathrm{e}}^{\mathrm{i}\hat{n}\chi t}$, where $\hat{n}$ is the cavity photon number operator, χ is the dispersive coupling strength, and t is the evolution time. With this unitary, we can efficiently implement conditional phase operations by simply adjusting the evolution time. In particular, when t = π/χ, all the odd photon number states acquire an overall π-phase while the even states get none. This allows us to effectively map the photon number parity of the bosonic mode onto the state of the ancilla (figure 4(a)).

Note that the time required for the parity-mapping operation scales inversely with the dispersive coupling strength, χ. Naively, one might want to minimize the operation time by engineering a large χ. However, increasing χ can also result in a stronger inherited non-linearity in the cavity (also known as the Kerr effect), which distorts the encoded information. Therefore, the coupling strength between the cavity and its ancilla, as well as the type of the non-linear ancillary mode, such as the transmon or the SNAIL design [182], are usually carefully optimized for each system to produce the desired Hamiltonian configuration.

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With ${\hat{U}}_{\text{CPS}}$, we can deterministically create complex bosonic encodings using analytically designed protocols such as the qcMAP operation, which maps an arbitrary qubit state onto a superposition of coherent states in the cavity [183]. This scheme is useful for preparing four-cat states in both single [172] and multiple [184] cavities. Furthermore, ${\hat{U}}_{\text{CPS}}\left(\pi /\chi \right)$ is also employed as the parity-mapping operation which is crucial for the characterization and tomography of the encoded bosonic qubits and the engineered gates on these qubits.

In general, the quantum states encoded in a cavity can be fully characterized by probing the cavity's quasi-probability distributions, which is commonly achieved by performing Wigner tomography. The Wigner function can be defined as the expectation value of the displaced photon number parity operator, $W\left(\beta \right)=\frac{2}{\pi }\enspace \mathrm{Tr}\left[\hat{D}{\left(\beta \right)}^{{\dagger}}\rho \hat{D}\left(\beta \right)\hat{P}\right]$. In cQED, the Wigner functions of arbitrary quantum states can be measured precisely with a well-defined sequence that uses only the cavity displacement, ancilla rotation, and CPS operations [15]. From the results of the Wigner tomography, we can reconstruct the full density matrix and characterize the action enacted on the cavity states in either the Pauli transfer matrix [185] or the process matrix [24].

Another crucial operation arising from the natural dispersive coupling is the non-linear selective number-dependent arbitrary phase (SNAP) gate [145, 186]. An SNAP gate (figure 4(b)), defined as ${\hat{S}}_{n}\left({\theta }_{n}\right)={\mathrm{e}}^{\mathrm{i}{\theta }_{n}\left(\vert n\rangle \langle n\vert \right)}$, selectively imparts a phase θ_n to the number state |n⟩. Due to the energy-preserving nature of this operation, we can simultaneously perform ${\hat{S}}_{n}\left({\theta }_{n}\right)$ on multiple number states. By numerically optimizing the linear displacement gates and the phases applied to each photon number state in the cavity, we can effectively cancel out the undesired Fock components via destructive interference to obtain the intended target state.

While schemes like qcMAP and SNAP are sufficient to realize universal control on the cavity state, they quickly become impractical for handling more complex bosonic states. For instance, an operation on n photons requires O(n²) gates using the SNAP protocol. To address this challenge, a fully numerical approach using optimal control theory (OCT) has been developed and widely adopted in recent years. The OCT framework provides an efficient general-purpose tool to implement arbitrary operations. In particular, the gradient ascent pulse engineering method [187, 188] has been successfully deployed in other physical systems [189, 190] to implement robust quantum control. By constructing an accurate model of the time-dependent Hamiltonian of the system in the presence of arbitrary control fields, we can apply this technique to cQED systems to realize high-fidelity universal gate sets on any bosonic qubit encoded in cavities, as demonstrated in reference [17]. A typical example of a set of pulses obtained through the gradient-based OCT framework is shown in figure 4(c).

More recently, various concepts from classical machine learning, such as automatic differentiation [191] and reinforcement learning [192, 193], have been applied to enhance the efficiency of the numerical optimization. Crucially, the success of these techniques does not only rely on the robustness of the algorithms, but also depends on the choice of boundary conditions. Knowledge of these boundary conditions requires comprehensive investigations of the physics of the quantum system and the practical constraints of the control and measurement apparatus.

Apart from robust single-mode operations, universal quantum computation using bosonic qubits also requires at least one entangling gate between two modes. Realizing such an operation can be challenging due to the lack of a natural coupling between cavities. Moreover, the individual coherence of each bosonic qubit must be maintained while maximizing the rate of interactions between them. One promising strategy to tackle this issue is to use the non-linear frequency conversion capability of the Josephson junction to provide a driven coupling between two otherwise isolated cavities [194]. Such operations are fully activated by external microwave drives which can be tuned on and off on-demand without modifying the hardware. This arrangement ensures that the individual bosonic modes remain well-isolated during idle times and undergo the engineered interaction only when an operation is enacted.

Using this strategy, a CNOT gate was the first logical gate enacted on two bosonic qubits [195]. This gate is facilitated by a parametrically-driven sideband transition between the ancilla and the control mode together with a carefully chosen conditional phase gate between the ancilla and the target cavity. By achieving a gate fidelity above 98%, this study showcases the potential of such engineered quantum gates between bosonic modes. More recently, a controlled-phase gate has been demonstrated between two binomial logical qubits in reference [20]. Here, the microwave drives are tailored to induce a geometric phase that depends on the joint state of the two bosonic modes. However, these two types of operations are both customized for a selective set of code words and do not yet generalize readily to other bosonic encoding schemes.

A code-independent coupling mechanism between two otherwise isolated bosonic modes is a crucial ingredient for realizing universal control on these logical qubits. The isolated cavities must be sufficiently detuned from each other to ensure the coherence of each mode and the absence of undesired cross-talk. Reference [180] demonstrated how the four-wave mixing process in a Josephson junction can provide a frequency converting bilinear coupling of the form ${H}_{\text{int}}\left(t\right)/\hslash =g\left(t\right)\left({\mathrm{e}}^{\mathrm{i}\varphi }\hat{a}{\hat{b}}^{{\dagger}}+{\mathrm{e}}^{-\mathrm{i}\varphi }{\hat{a}}^{{\dagger}}\hat{b}\right)$. Here, $\hat{a},\hat{b}$ are the annihilation operations associated with each of the cavities, the time-dependent coefficient g(t) is the coupling strength, and φ is the relative phase between the two microwave drives. The coefficient g(t) depends on the effective amplitudes of the drives, which satisfy the frequency matching condition |ω₂ − ω₁| = |ω_a − ω_b |. Most notably, this coupling can be programmed to implement an identity, a 50:50 beamsplitter, or a full SWAP operation between the stationary microwave fields in the cavities by simply adjusting the duration of the evolution. Such an engineered coupling provides a powerful tool for implementing programmable interferometry between cavity states [180], which is a key building block for realizing various continuous-variable information processing tasks such as boson sampling [196], simulation of vibrational quantum dynamics of molecules [197–200], and distributed quantum sensing [201–204].

Moreover, this bilinear coupling is also a valuable resource for enacting gates on two logical elements encoded in GKP states [49]. As mentioned in section 3, GKP encodings rely on the non-linearity of the code words and only require linear or bilinear operations for universal control [78]. Therefore, this engineered bilinear coupling provides a simple and effective strategy for implementing a deterministic entangling operation for the GKP code.

For other bosonic codes, this bilinear interaction is not alone sufficient to generate a universal gate set, which requires at least one entangling gate. In this case, the exponential SWAP (eSWAP) operation can be designed to provide deterministic and code-independent entanglement [205]. The eSWAP operation, akin to an exchange operation between spins, implements a programmable unitary of the form:

$$\begin{equation}\hat{U}\left(\theta \right)=\mathrm{cos}\left(\theta \right)\hat{I}+\mathrm{sin}\left(\theta \right)\mathrm{S}\mathrm{W}\mathrm{A}\mathrm{P},\end{equation} \tag{ 17 }$$

where θ is the rotation angle on the ancilla and $\hat{I}$ is the identity operation. Intuitively, this unitary implements a weighted superposition of the identity and SWAP operations between two bosonic modes regardless of their specific encodings. The eSWAP unitary has been realized in reference [19] between two bosonic modes housed in 3D co-axial cavities bridged by an ancilla. In this demonstration, an additional ancilla is introduced to one of the cavity modes and the resultant dispersive coupling is used to enact a ${\hat{U}}_{\text{CPS}}$ operation to provide the tunable rotation necessary for the eSWAP unitary. The eSWAP unitary is then enacted on several encoding schemes in the Fock, coherent, and binomial basis. The availability of such a deterministic and code-independent entangling operation is a crucial step toward universal quantum computation using bosonic logical qubits. A recent study has shown that universal control and operations on tens of bosonic qubits can be achieved in a novel architecture comprising a single transmon coupled simultaneously to a multi-mode superconducting cavity [206].

In general, implementations of QEC with bosonic codes suffer from an initial rise in error rates due to the presence of and need to control multi-photon states. Hence, the main experimental challenge is to achieve an enhancement in the lifetime of the encoded qubit despite this initial penalty. In the context of bosonic codes, the break-even point is defined relative to the |0, 1⟩ Fock states, which are the longest-lived physical elements in the cavity without error correction. Beyond the break-even point, we can be confident that the chosen QEC protocol does not introduce more errors into the system, thus attaining an improvement in the lifetime of the logical qubit. Till date, three studies have approached [74, 173] or achieved [16] the 'break-even point' for QEC with bosonic codes in cQED devices without any post-selection.

Broadly, QEC schemes fall into three categories based on how they afford protection to the logical qubit. In active QEC, error syndromes are repeatedly measured during the state evolution of the logical qubit and any errors detected are subsequently corrected based on the measurement outcomes. In autonomous QEC, errors are removed by tailored dissipation or by coupling to an auxiliary system, without repeatedly probing the logical qubit. In passive QEC, the logical information is intrinsically protected from decoherence because of specifically designed physical symmetries [207]. In this discussion, we have opted to distinguish between autonomous and passive QEC to highlight potential differences in system design. In the autonomous approach, the environment is intentionally engineered to suppress or mitigate errors. Whereas in the passive case, the system Hamiltonian itself is tailored to be robust against certain errors, which often involves constructing a unique physical element in the hardware [208–210].

Typically, active QEC requires robust measurements of the error syndrome and real-time feedback. In superconducting cavities, the dominant source of error is single photon loss. For encoding schemes with rotational symmetries, such as the cat and binomial codes, single photon loss results in a flip in the parity of the code words. Therefore, measuring the parity operator tells us whether a photon jump has taken place, thus allowing us to detect the error syndrome of these logical qubits [211]. In reference [16], which corrected a four-component cat state under photon loss (figure 5), the correct logical state was recovered by making appropriate adjustments in the decoding step based on the number of parity flips detected with real-time feedback. The corrected logical qubit showed an improved lifetime compared to both the uncorrected state and the |0, 1⟩ Fock state encoding, thereby achieving the break-even point. However, this error correction strategy is not suitable for protecting bosonic qubits for timescales exceeding the intrinsic cavity lifetime as the loss of energy from the system is accounted for but not physically rectified. Thus, such energy attenuation should be physically compensated to significantly surpass the break-even point.

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In contrast, reference [173] admits a photon pumping operation to achieve QEC on a logical qubit encoded in the binomial code. The errors are detected by photon number parity measurements, as in reference [16]. The errors occurring on the logical state are then corrected by an appropriate recovery operation as soon as they are detected. An approximated recovery operation is still required in case no errors are detected as the system evolves under the no-parity-jump operator [25]. In this experiment, the lifetime of the logical qubit was greater than that of the uncorrected binomial code state, but was marginally below that of the |0, 1⟩ Fock state.

In addition to the cat and binomial encodings, active QEC has also been recently demonstrated on a high-quality GKP state stored in a superconducting cavity [74]. As explained in section 3.2, errors occurring on a moderately squeezed GKP state simply manifest as displacements of the cavity state, and are revealed and mitigated by measuring the displacement stabilizers. While mitigating errors is relatively straightforward, experimental challenges in implementing the GKP code lie in preparing finitely squeezed approximate GKP states and performing modulo quadrature measurement of stabilizers. The QEC protocol used in reference [74] suppresses all logical errors for a GKP state by alternating between two peak-sharpening and envelope-trimming rounds, each consisting of different conditional displacements on the cavity. The magnitude of these conditional displacements is dependent on the measured state of the ancilla onto which displacement stabilizers are mapped. This sharpen-and-trim technique can be generalized based on the second-order Trotter formula [65].

An alternative strategy for QEC that does not require real-time feedback based on measurement outcomes involves engineering the dynamics of the system to correct errors autonomously. In practice, autonomous QEC can be achieved by introducing a tailored dissipation [212] such that the logical subspace is stabilized via controlled interactions with the environment. In references [212, 213], the desired dissipation process subtracts photons in pairs from the bosonic mode via four-wave mixing in the Josephson junction of the ancilla. This process defines a stabilized manifold of steady states spanned by superpositions of coherent states (two-component cat states). The relative strength between the engineered two-photon dissipation and the intrinsic single photon loss of the cavity defines the separation of the two components of the cat state, which determines the extent of protection against bit-flip errors. In a more recent study, significant improvements in the suppression of bit-flip errors were demonstrated by using a tailored multi-junction ancilla to enact the two-photon dissipation process [39].

Similarly, an engineered four-photon dissipation achieved by an eight-wave mixing process, can protect four-component cat codes against dephasing [214]. However, this strategy only accounts for dephasing errors and does not recover logical errors caused by photon loss. In contrast, in another autonomous QEC implementation with a truncated four-cat state [215], a photon was added to the cavity upon detecting a photon loss using a synthetic dissipation operator. This work effectively corrects photon loss events but does not account for dephasing errors. Similarly, another study performed autonomous QEC against photon loss with the kitten code by triggering an engineered jump operation [174]. This operation recovered the code states whenever the system entered an error state without the need for any external probing. Although these current state-of-the-art demonstrations of autonomous QEC techniques only address one type of errors, they provide convincing evidence for the viability of the respective bosonic encoding schemes, thus paving the way for higher-order QEC protocols that can protect logical information against both photon loss and dephasing errors.

Finally, QEC with bosonic codes can also be realized using a passive approach, for instance, by designing logical qubits with highly biased-noise channels to provide intrinsic protection without the need for probing any error syndromes. Very recently, two studies [39, 209] verified that the bias between phase and bit-flip errors increases exponentially with α, the size of the coherent state components of the two-cat code. This strong asymmetry between the different types of errors can be exploited to significantly reduce the hardware overhead for fault-tolerant quantum computation [87–89].