††thanks: This review paper was written before K. Noh joined AWS Center for
Quantum Computing.

# Quantum Information Processing with Bosonic Qubits in Circuit QED

Atharv Joshi  Centre for Quantum Technologies, National University of
Singapore  Kyungjoo Noh  AWS Center for Quantum Computing, Pasadena, CA,
91125, USA  Yvonne Y. Gao<EMAIL_ADDRESS>Centre for Quantum
Technologies, National University of Singapore

###### Abstract

The unique features of quantum theory offer a powerful new paradigm for
information processing. Translating these mathematical abstractions into
useful algorithms and applications requires quantum systems with significant
complexity and sufficiently low error rates. Such quantum systems must be made
from robust hardware that can coherently store, process, and extract the
encoded information, as well as possess effective quantum error correction
(QEC) protocols to detect and correct errors. Circuit quantum electrodynamics
(cQED) provides a promising hardware platform for implementing robust quantum
devices. In particular, bosonic encodings in cQED that use multi-photon states
of superconducting cavities to encode information have shown success in
realizing hardware-efficient QEC. Here, we review recent developments in the
theory and implementation of quantum error correction with bosonic codes and
report the progress made towards realizing fault-tolerant quantum information
processing with cQED devices.

###### Contents

  1. I Introduction
    1. I.1 Organization of the article
  2. II Concepts of Bosonic Quantum Error Correction
  3. III Performance of Bosonic Codes for Loss and Dephasing Errors
    1. III.1 Rotation-symmetric codes: Binomial and cat codes
    2. III.2 Translation-symmetric codes: GKP codes
    3. III.3 Biased-noise bosonic qubits: Two-component cat codes
    4. III.4 Comparison of various bosonic codes for loss and dephasing errors
  4. IV The cQED Hardware for Bosonic Codes
    1. IV.1 Components of the cQED architecture
    2. IV.2 Coherence of superconducting cavities
  5. V Realization of Bosonic Logical Qubits
    1. V.1 Operations on single bosonic modes
    2. V.2 Operations on multiple bosonic modes
    3. V.3 Implementations of QEC
  6. VI Towards Fault-tolerant Quantum Computation with Bosonic Modes
  7. VII Outlook and Perspectives

##  I Introduction

A quantum computer harnesses unique features of quantum theory, such as
superposition and entanglement, to tackle classically challenging tasks. To
perform faithful computation, quantum information must be protected against
errors due to decoherence mechanisms and operational imperfections. While
these errors are relatively insignificant individually, they can quickly
accumulate to completely scramble the information.

To protect quantum information from scrambling, the theoretical frameworks of
quantum error correction (QEC) shor1995_scheme ; steane1996_error  and fault-
tolerant quantum computation preskill1998_reliable  were developed in the
early days of quantum computing. Essentially, these frameworks devise
encodings which map a collection of physical elements onto a single ‘logical’
bit of quantum information. Such a logical qubit is endowed with cleverly
chosen symmetry properties that allow us to extract error syndromes and enact
error correction without disturbing the encoded information.

An important metric for evaluating the effectiveness of QEC implementations is
the break-even point, which is achieved when the lifetime of a logical qubit
exceeds that of the best single physical element in the system. Achieving the
break-even point entails that additional physical elements and operations
introduced to a QEC process do not cause more degradation than the protection
they afford. Hence, reaching the break-even point is a critical pre-requisite
for implementing fault-tolerant gates and eventually performing robust quantum
information processing on a large scale.

In the conventional approach to QEC, the physical elements are realized by
discrete two-level systems. In this approach, even a simple QEC scheme
designed to correct single errors, such as the Steane code steane1996_error ,
requires tens of two-level systems, ancillary qubits, and measurement
elements. Constructing physical devices that contain these many interconnected
elements can be a significant engineering challenge. More crucially, having
many interconnected elements often degrades the device performance and
introduces new uncorrectable errors such as cross-talk due to undesired
couplings between the elements. Over the last decade, many proof-of-principle
demonstrations of QEC schemes have been realized with encoding schemes based
on two-level systems chiaverini2004_realization ; reed2012_realization ;
taminiau2014_universal ; waldherr2014_quantum ; linke2017_faulttolerant ;
andersen2020_repeated ; erhard2021_entangling . However, given the practical
challenges described above, these demonstrations have not deterministically
extended the performance of the logical qubits beyond that of the best
available physical qubit in the system.

A promising alternative with the potential to realize robust universal quantum
computing with effective QEC beyond the break-even point involves encoding
logical qubits in continuous variables braunstein1998_quantum ;
bartlett2002_universal ; bartlett2002_quantum . In particular, superconducting
microwave cavities coupled to one or more anharmonic elements in the circuit
quantum electrodynamics (cQED) architecture provide a valuable resource for
the hardware-efficient encoding of logical qubits devoret2013_superconducting
; blais2020_circuit . These cavities have a large Hilbert space for encoding
information in multi-photon states compactly and thus form a logical qubit in
a single piece of hardware. The anharmonic element, typically in the form of a
transmon and henceforth referred to as the ‘ancilla’, provides the necessary
non-linearity to control and measure cavity states. This strategy of using
multi-photon states of superconducting cavities to encode logical information
is also known as bosonic codes. Implementations of bosonic codes in cQED have
thus far not only demonstrated QEC at the break-even point ofek2016_extending
, but also robust operations heeres2017_implementing ; chou2018_deterministic
; gao2019_entanglement ; xu2020_demonstration ; reinhold2020_error-corrected
and fault-tolerant measurement of error syndromes rosenblum2018_fault-tolerant
, thus making rapid progress in recent years.

In this article, we review the recent developments in bosonic codes in the
cQED setting. In particular, we highlight the progress made in demonstrating
effective QEC and information processing with logical elements implemented
using bosonic codes. These recent works provide compelling evidence for the
vast potential of bosonic codes in cQED as the fundamental building blocks for
robust universal quantum computing.

###  I.1 Organization of the article

In Sec. II, 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 Sec. III, 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 Sec. IV, 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 Sec.
V, 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 Sec. VI, 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 Sec. VII, 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.

##  II Concepts of Bosonic Quantum Error Correction

The general principle of QEC is to encode logical quantum information
redundantly in a large Hilbert space with certain symmetry properties, which
can be used to detect errors. In particular, logical code states are designed
such that they can be mapped onto orthogonal subspaces under distinct errors.
Crucially, the logical information can be recovered faithfully only if the
mapping between the logical and the error states does not distort the code
words.

Mathematically, these requirements can be succinctly described by the Knill-
Laflamme condition knill1997_theory , which states that an error-correcting
code $\mathcal{C}$ can correct any error operators in the span of an error set
$\mathcal{E}\equiv\\{\hat{E}_{1},\cdots,\hat{E}_{|\mathcal{E}|}\\}$ if and
only if it satisfies:

$\displaystyle\hat{P}_{\mathcal{C}}\hat{E}_{\ell}^{\dagger}\hat{E}_{\ell^{\prime}}\hat{P}_{\mathcal{C}}=\alpha_{\ell\ell^{\prime}}\hat{P}_{\mathcal{C}}$
(1)

for all $\ell,\ell^{\prime}\in\\{1,\cdots,|\mathcal{E}|\\}$, where
$|\mathcal{E}|$ is the size of the error set, $\hat{P}_{\mathcal{C}}$ is the
projection operator to the code space $\mathcal{C}$, and
$\alpha_{\ell\ell^{\prime}}$ are matrix elements of a Hermitian and positive
semi-definite matrix. A derivation of the Knill-Laflamme condition is given in
Ref. nielsen2000_quantum .

Bosonic codes achieve these requirements by cleverly configuring the
excitations in a harmonic oscillator mode. For instance, in the cQED
architecture, information is encoded in multi-photon states of superconducting
microwave cavities. We will illustrate how bosonic QEC codes work by
describing the simplest binomial code michael2016_new , also known as the
‘kitten’ code. We will then compare this code with its multi-qubit cousin, the
‘4-qubit code’ (or the $[[4,1,2]]$ code or the distance-$2$ surface code)
leung1997_approximate , in the presence of similar errors in the cQED
architecture. This comparison, adapted from Ref. girvin2019_quantum ,
emphasizes the hardware-efficient nature of the bosonic QEC approach.

Figure 1: Hardware comparison between the kitten code and the 4-qubit code.
(a) Encoding a single logical qubit using multiple energy levels of a single
harmonic oscillator, where the only dominant error channel is photon loss. (b)
The 4-qubit scheme based on a collection of two-level systems. Individually,
each two-level system can experience errors due to spontaneous absorption and
emission. Moreover, additional errors can be introduced due to undesired
couplings between these systems. (c) The typical hardware for implementing a
bosonic logical qubit. The logical information is encoded in a single
superconducting cavity (orange) in the kitten code, which can be fully
controlled by a single non-linear ancilla (green) and read out via another
cavity (gray). (d) Example of a system designed to implement the 4-qubit code,
where the logical information is spread out across four data qubits (orange)
checked by three ancillary ones (green). The interactions between each two-
level system are mediated by coupling cavities (teal), and their respective
states are read out via seven planar cavities (gray).

The kitten code is designed to correct only single photon loss events, or
$\hat{a}$, which are the dominant error channel in superconducting cavities.
This scheme encodes logical information in the even photon number parity
subspace of a harmonic oscillator (Fig. 1a):

$\displaystyle|0_{L}\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|4\rangle),\quad|1_{L}\rangle=|2\rangle.$
(2)

With these code words, a single photon loss event maps an even parity logical
state $|\psi_{L}\rangle=\alpha|0_{L}\rangle+\beta|1_{L}\rangle$ to an odd
parity error state
$\hat{a}|\psi_{L}\rangle=\sqrt{2}(\alpha|3\rangle+\beta|1\rangle)$. Because of
the parity difference, the code space and the error space are mutually
orthogonal. Therefore, single photon loss events can be detected by measuring
the photon number parity operator $\hat{\Pi}_{2}=(-1)^{\hat{n}}$. Importantly,
since the error states $\hat{a}|0_{L}\rangle$ and $\hat{a}|1_{L}\rangle$ have
the same normalization constant,

$\displaystyle\langle 0_{L}|\hat{a}^{\dagger}\hat{a}|0_{L}\rangle=\langle
1_{L}|\hat{a}^{\dagger}\hat{a}|1_{L}\rangle=2,$ (3)

or equivalently, the logical states $|0_{L}\rangle$ and $|1_{L}\rangle$ have
the same average photon number, a single photon loss event does not distort
the encoded information. In other words, by mapping the normalized error
states $|3\rangle$ and $|1\rangle$ back to the original code states
$|0_{L}\rangle$ and $|1_{L}\rangle$, we can recover the input logical state up
to an overall normalization constant:

$\displaystyle\hat{a}|\psi_{L}\rangle$
$\displaystyle=\sqrt{2}(\alpha|3\rangle+\beta|1\rangle)$
$\displaystyle\xrightarrow{\textrm{recovery}}\sqrt{2}(\alpha|0_{L}\rangle+\beta|1_{L}\rangle)\propto|\psi_{L}\rangle.$
(4)

Note that we cannot faithfully recover logical information if the logical
states have different average photon numbers. For instance, with
$|0_{L}\rangle=\frac{1}{2}|0\rangle+\frac{\sqrt{3}}{2}|4\rangle$, which has
$3$ photons on average, as opposed to $2$ photons for
$|1_{L}\rangle=|2\rangle$, a single photon loss and recovery event yields a
state which is not proportional to $|\psi_{L}\rangle$:

$\displaystyle\hat{a}|\psi_{L}\rangle$
$\displaystyle=\sqrt{3}\alpha|3\rangle+\sqrt{2}\beta|1\rangle$
$\displaystyle\xrightarrow{\textrm{recovery}}\sqrt{3}\alpha|0_{L}\rangle+\sqrt{2}\beta|1_{L}\rangle.$
(5)

Specifically, the error and recovery process distorts the relative phase
between $|0_{L}\rangle$ and $|1_{L}\rangle$. An intuitive way to understand
such a phase distortion is by considering the role of the environment. When
$|0_{L}\rangle$ and $|1_{L}\rangle$ have different average photon numbers, the
environment gains partial information on whether $|\psi_{L}\rangle$ was in
$|0_{L}\rangle$ or in $|1_{L}\rangle$. In the example above, $|0_{L}\rangle$
has more photons than $|1_{L}\rangle$ and hence has a higher probability of
losing a photon. When one photon is lost, the environment knows that the
encoded state was more likely to be $|0_{L}\rangle$ than $|1_{L}\rangle$ and
the weights in Eq. (5) are therefore adjusted accordingly. Alternatively, we
can say that the environment performs a weak measurement on the logical state
in the $|0_{L}/1_{L}\rangle$ basis, thus leading to the dephasing of
$|\psi_{L}\rangle$ in the $|0_{L}/1_{L}\rangle$ basis.

Hence, while the even parity structure allows the detection of single photon
loss events, it does not guarantee the recoverability of the logical
information without any distortion. Faithful recovery is ensured by selecting
$|0_{L}\rangle$ and $|1_{L}\rangle$ to have the same average photon number. In
the case of the kitten code, average photon numbers are matched by choosing
equal coefficients for the vacuum and the four photon state components in the
logical zero state.

More generally, a code $\mathcal{C}$ can correct single photon loss events if
it satisfies the Knill-Laflamme condition for the error set
$\\{\hat{I},\hat{a}\\}$. That is, the projection operator to the code space
should satisfy
$\hat{P}_{\mathcal{C}}\hat{x}\hat{P}_{\mathcal{C}}\propto\hat{P}_{\mathcal{C}}$
for all
$\hat{x}\in\\{\hat{I},\hat{a},\hat{a}^{\dagger},\hat{a}^{\dagger}\hat{a}\\}$.
The first condition with $\hat{x}=\hat{I}$ is trivially satisfied for any
code. For even parity codes, which are composed of logical states of even
photon number parity, the second and the third conditions with
$\hat{x}=\hat{a}$ and $\hat{x}=\hat{a}^{\dagger}$ are satisfied due to the
parity structure. This implies that even parity codes are capable of
detecting, but not necessarily correcting, any single photon loss and gain
events. Single photon loss events can be corrected if the fourth condition is
met, that is,
$\hat{P}_{\mathcal{C}}\hat{a}^{\dagger}\hat{a}\hat{P}_{\mathcal{C}}\propto\hat{P}_{\mathcal{C}}$,
or equivalently, if all logical states have the same average photon number.

Now, using an example inspired by Ref. girvin2019_quantum , we compare the
kitten code with its multi-qubit cousin, the 4-qubit code, whose logical
states consist of 4 distinct two-level systems (Fig. 1b):

$\displaystyle|0_{L}\rangle=\frac{1}{\sqrt{2}}(|0000\rangle+|1111\rangle),$
$\displaystyle|1_{L}\rangle=\frac{1}{\sqrt{2}}(|0101\rangle+|1010\rangle).$
(6)

The $4$-qubit code is stabilized by three stabilizers:
$\hat{S}_{1}=\hat{Z}_{1}\hat{Z}_{3}$, $\hat{S}_{2}=\hat{Z}_{2}\hat{Z}_{4}$,
and $\hat{S}_{3}=\hat{X}_{1}\hat{X}_{2}\hat{X}_{3}\hat{X}_{4}$. This scheme is
capable of detecting any arbitrary single-qubit errors. Moreover, it can
correct single excitation loss errors,
$\\{\hat{\sigma}^{-}_{1},\hat{\sigma}^{-}_{2},\hat{\sigma}^{-}_{3},\hat{\sigma}^{-}_{4}\\}$,
via approximate QEC leung1997_approximate , which happens when the Knill-
Laflamme condition is approximately satisfied only to a certain low order in
the error parameters. This capability is comparable to the protection afforded
by the kitten code against single photon loss errors.

Despite their comparable error-correcting capability, the $4$-qubit code and
the kitten code incur significantly different hardware overheads. A cQED
implementation of the kitten code (Fig. 1c) requires a single bosonic mode to
store logical information, a single ancilla (typically a transmon) to measure
and control the cavity state, and a single readout cavity mode to measure the
ancilla state. In contrast, a cQED realization of the $4$-qubit code (Fig. 1d)
uses $4$ data qubits to encode logical information, $3$ ancillae to measure
the three stabilizers, and additional cavity modes to connect and measure all
$7$ physical qubits. Apart from the pure complexity of realizing such a
device, the presence of additional elements introduces other error channels
such as cross-talk arising from spurious couplings between the physical
qubits. While these effects can be calibrated and mitigated on a small scale
with clever techniques sheldon2016_procedure ; rol2019_fast , they can quickly
become intractable for more complex devices. This comparison thus illustrates
the advantage of using the multiple levels of a single bosonic mode over using
multiple two-level systems as a redundant resource for QEC.

While we focus on bosonic codes that encode a qubit in a single bosonic mode,
there are proposals for encoding a qubit in many bosonic modes, namely
permutation-invariant codes ouyang2014_permutationinvariant ;
ouyang2016_permutationinvariant ; ouyang2017_permutationinvariant ;
ouyang2020_permutationinvariant . These codes are tailored for excitation loss
errors and generalize the simple 4-qubit code. Other codes of a similar nature
have also been studied in Refs. wasilewski2007_protecting ;
kapit2016_hardwareefficient ; bergmann2016_quantum ;
kapit2018_errortransparent .

##  III Performance of Bosonic Codes for Loss and Dephasing Errors

In general, bosonic modes in cQED systems typically undergo both photon loss
and dephasing errors. Photon loss is considered to be the dominant error-
channel. The rate of photon loss is determined by the internal quality factor
($Q_{\mathrm{int}}$) of the superconducting cavity. Intrinsic dephasing is
usually insignificant for such cavities reagor2013_reaching .

However, as the cavity is dispersively coupled to a non-linear ancilla, if the
ancilla experiences undesired absorption ($\Gamma_{\uparrow}$) or emission
($\Gamma_{\downarrow}$) of excitations due to stray radiation, then the
encoded logical information undergoes induced dephasing. The rate of this
dephasing depends on $\Gamma_{\uparrow}$, $\Gamma_{\downarrow}$, and the
coupling strength $\chi$. In general, a transition of the ancilla state leads
to a rotation on the logical qubit by an angle $\sim\chi t$, where $t$ is the
time the ancilla spends in the resulting state after the transition. In the
limit where this rotation is small, we can apply the Markovian approximation
and use the following Lindblad master equation to describe the evolution of
the encoded qubit:

$\displaystyle\frac{d\hat{\rho}(t)}{dt}=\big{(}\kappa\mathcal{D}[\hat{a}]+\kappa_{\phi}\mathcal{D}[\hat{a}^{\dagger}\hat{a}]\big{)}\hat{\rho}(t),$
(7)

where $\kappa$ and $\kappa_{\phi}$ are the photon loss and dephasing rates
respectively, and
$\mathcal{D}[\hat{A}](\hat{\rho})\equiv\hat{A}\hat{\rho}\hat{A}^{\dagger}-\frac{1}{2}\\{\hat{A}^{\dagger}\hat{A},\hat{\rho}\\}$
is the dissipation superoperator. Note that loss and dephasing errors are
generated by the jump operators $\hat{a}$ and
$\hat{n}=\hat{a}^{\dagger}\hat{a}$ respectively. We say that a system is loss
dominated if $\kappa\gg\kappa_{\phi}$ and dephasing dominated if
$\kappa\ll\kappa_{\phi}$.

Figure 2: Visualization and performance of different bosonic codes. (a) - (d)
Wigner functions of the code words for the four-component cat, binomial
kitten, GKP square lattice, and two-component cat code, respectively. (e) A
qualitative description of the robustness of various classes of bosonic codes
against dephasing and photon loss errors. Note that for the four-component cat
code and the binomial kitten code, the logical states in the $|0/1_{L}\rangle$
basis are invariant under a $90\degree$ rotation, up to a global phase.
However, a general code state, that is, an arbitrary superposition of
$|0_{L}\rangle$ and $|1_{L}\rangle$), is invariant only under the $180\degree$
rotation.

Importantly, in typical experimental regimes, dephasing errors induced by
transitions of the ancilla state are generally non-Markovian. This limits the
performance of some current implementations lescanne2020_exponential  of
bosonic QEC codes.

In this section, we review the various single-mode bosonic codes, namely, cat
(Fig. 2a) and binomial (Fig. 2b) codes (rotation-symmetric), Gottesman-Kitaev-
Preskill (GKP) codes (translation-symmetric) (Fig. 2c), and two-component cat
codes (biased-noise bosonic qubits) (Fig. 2d), and discuss their error-
correcting capability against both photon loss and dephasing errors.

###  III.1 Rotation-symmetric codes: Binomial and cat codes

The class of rotation-symmetric bosonic codes grimsmo2020_quantum  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
$\hat{\Pi}_{N}=e^{i(2\pi/N)\hat{n}}$ ($N\in\\{2,3,\cdots\\}$), which is
equivalent to the $360/N\degree$ 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 ouyang2020_tradeoffs . For
instance, the kitten code may be modified to have the following logical
states:

$\displaystyle|0_{L}\rangle$
$\displaystyle=\frac{1}{2}(|0\rangle+\sqrt{3}|4\rangle),$
$\displaystyle|1_{L}\rangle$
$\displaystyle=\frac{1}{2}(\sqrt{3}|2\rangle+|6\rangle).$ (8)

This version of the binomial code is also stabilized by the parity operator
and is invariant under the $180\degree$ 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
$\\{\hat{I},\hat{a},\hat{a}^{\dagger}\hat{a}\\}$ 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
$120\degree$ rotation instead of a $180\degree$ rotation and are thus
stabilized by the super-parity modulo $3$ operator
$\hat{\Pi}_{3}=e^{i(2\pi/3)\hat{n}}$. By taking advantage of a larger spacing
in the photon number basis, $120\degree$-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

$\displaystyle|0_{L}\rangle$
$\displaystyle=\frac{1}{2}(|0\rangle+\sqrt{3}|6\rangle),$
$\displaystyle|1_{L}\rangle$
$\displaystyle=\frac{1}{2}(\sqrt{3}|3\rangle+|9\rangle),$ (9)

is invariant under the $120\degree$ 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
$\\{\hat{I},\hat{a},\hat{a}^{2},\hat{a}^{\dagger}\hat{a}\\}$. 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
michael2016_new  as well as used for autonomous QEC lihm2018_implementation .

Cat codes are another important example of rotation-symmetric bosonic codes.
The four-component cat codes (or 4-cat codes), which are composed of four
coherent states $|\pm\alpha\rangle$, $|\pm i\alpha\rangle$, are the simplest
variants of the cat codes that are robust against photon loss errors
leghtas2013_hardware-efficient ; mirrahimi2014_dynamically . In this encoding,
the logical states are defined by superpositions of coherent states:

$\displaystyle|0_{L}\rangle$
$\displaystyle\propto|\alpha\rangle+|i\alpha\rangle+|-\alpha\rangle+|-i\alpha\rangle$
$\displaystyle\propto\sum_{n=0}^{\infty}\frac{\alpha^{4n}}{\sqrt{(4n)!}}|4n\rangle,$
$\displaystyle|1_{L}\rangle$
$\displaystyle\propto|\alpha\rangle-|i\alpha\rangle+|-\alpha\rangle-|-i\alpha\rangle$
$\displaystyle\propto\sum_{n=0}^{\infty}\frac{\alpha^{4n+2}}{\sqrt{(4n+2)!}}|4n+2\rangle.$
(10)

Note that the amplitude of the coherent states $|\alpha|$ determines the size
of the cat code. Similar to the kitten code (Eq. (2)), the logical states of
the 4-cat code have an even number of photons. Thus, the 4-cat code is
invariant under the $180\degree$ 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 $\\{\hat{I},\hat{a}\\}$. For even parity codes,
as explained in Sec II, recoverability requires the two logical states to have
the same number of photons on average.

For large cat codes with $|\alpha|\gg 1$, the average photon number is
approximately given by $\bar{n}\simeq|\alpha|^{2}$ 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 $|\alpha|$ (also known as ‘sweet spots’ li2017_cat ) such
that:

$\displaystyle\tan|\alpha|^{2}=-\tanh|\alpha|^{2}.$ (11)

The smallest such $|\alpha|$ is given by $|\alpha|=1.538$, 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 $|\alpha|^{2}$ satisfy the Knill-Laflamme condition for
the dephasing error set $\\{\hat{I},\hat{n},\hat{n}^{2},\cdots\\}$
approximately modulo an inaccuracy that scales as $e^{-2|\alpha|^{2}}$
albert2018_performance . However, an extra error-correcting mechanism, such as
a multi-photon engineered dissipation mirrahimi2014_dynamically , 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 albert2016_holonomic ;
bergmann2016_quantum . For instance, with six coherent state components, the
code words become $120\degree$-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 Ref. li2017_cat , while a multi-mode
generalization is available in Ref. albert2019_pair-cat .

###  III.2 Translation-symmetric codes: GKP codes

Another class of bosonic codes are translation-symmetric, with the Gottesman-
Kitaev-Preskill (GKP) codes gottesman2001_encoding  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:

$\displaystyle\hat{S}_{q}$
$\displaystyle\equiv\exp[i2\sqrt{\pi}\hat{q}]=\hat{D}(i\sqrt{2\pi}),$
$\displaystyle\hat{S}_{p}$
$\displaystyle\equiv\exp[-i2\sqrt{\pi}\hat{p}]=\hat{D}(\sqrt{2\pi}),$ (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 Eq. (12) commute with each other, we
can measure them simultaneously. Measuring the displacement operators
$\exp[i2\sqrt{\pi}\hat{q}]$ and $\exp[-i2\sqrt{\pi}\hat{p}]$ is equivalent to
measuring their phases $2\sqrt{\pi}\hat{q}$ and $-2\sqrt{\pi}\hat{p}$ modulo
$2\pi$, 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:

$\displaystyle|0_{L}\rangle$
$\displaystyle\propto\sum_{n\in\mathbb{Z}}|\hat{q}=(2n)\sqrt{\pi}\rangle\propto\sum_{n\in\mathbb{Z}}|\hat{p}=n\sqrt{\pi}\rangle,$
$\displaystyle|1_{L}\rangle$
$\displaystyle\propto\sum_{n\in\mathbb{Z}}|\hat{q}=(2n+1)\sqrt{\pi}\rangle\propto\sum_{n\in\mathbb{Z}}(-1)^{n}|\hat{p}=n\sqrt{\pi}\rangle,$
(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 gottesman2001_encoding ; terhal2016_encoding ;
matsuura2019_equivalence . Many proposals to realize approximate GKP states in
various physical platforms gottesman2001_encoding ; travaglione2002_preparing
; pirandola2004_constructing ; pirandola2006_generating ;
vasconcelos2010_alloptical ; terhal2016_encoding ; motes2017_encoding ;
weigand2018_generating ; arrazola2019_machine ; shi2019_fault-tolerant ;
su2019_conversion ; eaton2019_nongaussian ; hastrup2019_measurementfree ;
weigand2020_realizing ; hastrup2020_improved ; royer2020_stabilization  and
ways to simulate approximate states efficiently wan2019_memoryassisted ;
tzitrin2020_progress ; terhal2020_scalable ; pantaleoni2020_modular ;
walshe2020_continuousvariable ; mensen2020_phasespace  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 towards an ideal code state. Recently, approximate states of
squeezing $5.5$ – $9.5$ dB have been realized in trapped ion
fluhmann2019_encoding ; deNeeve2020_error  and cQED campagne-
ibarcq2019_quantum  systems. The cQED realization is discussed further in Sec.
V.

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 gottesman2001_encoding ; terhal2016_encoding
. 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 albert2018_performance ;
noh2019_quantum . 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 $\sqrt{\pi}$
gottesman2003_secure . 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 terhal2016_encoding .

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
bravyi2005_universal  encoded in the GKP code, which are necessary for
implementing non-Clifford operations, can be prepared with only Gaussian
operations and GKP states terhal2016_encoding ; baragiola2019_allgaussian .
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
gottesman2001_encoding , although the cubic phase gate suggested in the
original proposal has been recently shown to perform poorly for this purpose
hastrup2020_cubic .

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 $(2/\sqrt{3})^{1/2}\sqrt{\pi}/2\simeq 1.07\sqrt{\pi}/2$, which
is larger than the size of shifts that are correctable by the square lattice
GKP code gottesman2001_encoding ; noh2019_quantum . A recent work has also
explored the rectangular GKP code hanggli2020_enhanced . More generally, a
multi-mode GKP code can be defined over any symplectic lattice
harrington2001_achievable . Lastly, the GKP code may also be used for building
robust quantum repeaters for long-distance quantum communication
rozpedek2020_quantum ; fukui2020_alloptical .

###  III.3 Biased-noise bosonic qubits: Two-component cat codes

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 aliferis2008_faulttolerant . 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
guillaud2019_repetition ; guillaud2020_error . 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
tuckett2018-ultrahigh ; tuckett2019_tailoring ; tuckett2020_faulttolerant ;
chamberland2020_building .

A promising candidate for biased-noise bosonic qubits is the two-component cat
code cochrane1999_macroscopically ; jeong2002_efficient ; ralph2003_quantum ;
glancy2004_transmission ; lund2008_faulttolerant , or 2-cat code, whose
logical code words are given by:

$\displaystyle|+_{L}\rangle$
$\displaystyle\propto|\alpha\rangle+|-\alpha\rangle\propto\sum_{n=0}^{\infty}\frac{\alpha^{2n}}{\sqrt{(2n)!}}|2n\rangle,$
$\displaystyle|-_{L}\rangle$
$\displaystyle\propto|\alpha\rangle-|-\alpha\rangle\propto\sum_{n=0}^{\infty}\frac{\alpha^{2n+1}}{\sqrt{(2n+1)!}}|2n+1\rangle.$
(14)

If $\alpha$ is large enough, 2-cat codes are capable of correcting dephasing
errors. One way to implement 2-cat codes is by autonomously stabilizing them
via an engineered dissipation of the form
$\kappa_{2}\mathcal{D}[\hat{a}^{2}-\alpha^{2}]$ mirrahimi2014_dynamically .
This engineered two-photon dissipation can exponentially suppress the logical
bit-flip error in the code states due to dephasing in superconducting cavities
($\kappa_{\phi}\mathcal{D}[\hat{a}^{\dagger}\hat{a}]$), i.e.,

$\displaystyle\gamma_{\textrm{bit-flip}}\simeq
2\kappa_{\phi}|\alpha|^{2}e^{-2|\alpha|^{2}}\,\,\,\textrm{if}\,\,\,\kappa_{\phi}\ll\kappa_{2},$
(15)

where $|\alpha|^{2}$ is the size of the cat code, $\kappa_{2}$ is the two-
photon dissipation rate, and $\kappa_{\phi}$ 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 $|+_{L}\rangle$ to
an odd parity state $|-_{L}\rangle$, thus causing a phase-flip error. As such,
phase-flip errors due to single photon loss cannot be mitigated by the 2-cat
code. Nevertheless, bit-flip errors due to single photon loss can be
suppressed exponentially in the photon number $\alpha^{2}$ 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 2-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 lescanne2020_exponential .

Biased-noise 2-cat qubits can also be realized by using an engineered
Hamiltonian
$\hat{H}=-K(\hat{a}^{\dagger}-(\alpha^{*})^{2})(\hat{a}^{2}-\alpha^{2})$,
which has two coherent states $|\pm\alpha\rangle$ as degenerate eigenstates
puri2017_engineering ; puri2019_stabilized ; puri2020_bias-preserving . 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 2-cat codes
in a bias-preserving manner has been proposed recently guillaud2019_repetition
; puri2020_bias-preserving .

###  III.4 Comparison of various bosonic codes for loss and dephasing errors

In summary, rotation-symmetric codes (e.g., 4-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, 2-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, 2-cat codes are naturally suited for dephasing dominated
systems. However, as discussed in Sec. III.3, 2-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 Fig. 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 lloyd1997_capacity ;
devetak2005_private  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 holevo2001_evaluating ;
wolf2007_quantum ; wilde2012_quantum ; wilde2018_energyconstrained ;
arqand2020_quantum .

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 2-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 2-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.

##  IV The cQED Hardware for Bosonic Codes

In the preceding sections, we have discussed the concepts and merits of
various bosonic QEC codes. These ideas are brought to reality by developing
robust quantum hardware with both coherent harmonic modes for encoding
information as well as non-linear ancillae for effective control and
tomography of the encoded information. Such systems have been realized with
the motional degree of freedom in trapped ions fluhmann2019_encoding ;
deNeeve2020_error , electromagnetic fields of microwave
deleglise2008_reconstruction  and optical cavities hacker2019_deterministic ,
Rydberg atom arrays omran2019_generation , and flying photons
ourjoumtsev2006_generating , and there are proposals that use other physical
systems hou2016_generation ; bulutay2017_cat-state  as well. Among the various
platforms, the cQED architecture consisting of superconducting microwave
cavities and Josephson junction-based non-linear ancillae have enabled many
prominent experimental milestones towards realizing QEC using bosonic codes.
In this section, we will introduce the key hardware building blocks necessary
for the successful realization of bosonic codes in cQED.

###  IV.1 Components of the cQED architecture

Circuit quantum electrodynamics (cQED) explores light-matter interactions by
confining quantized electromagnetic fields in precisely engineered
compositions of superconducting inductors and capacitors blais2020_circuit .
These superconducting circuit elements can be tailor-made by conventional
fabrication techniques frunzio2005_fabrication  and controlled by commercially
available microwave electronics and dilution refrigerators
schoelkopf2008_wiring  to access strong coupling regimes wallraff2004_strong ;
schuster2007_resolving ; niemczyk2010_circuit . These characteristics make
cQED a compelling platform for universal quantum computation, as noted by
several review articles girvin2009_circuit ; ladd2010_quantum ;
devoret2013_superconducting ; wendin2017_quantum ; krantz2019_quantum ;
kjaergaard2019_superconducting ; blais2020_quantum ; cai2020_bosonic .

Quantum devices in the cQED framework are typically built by coupling two
components, a linear oscillator mode and an anharmonic mode, in different
configurations blais2020_circuit . 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 haroche2006_exploring . 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 koch2007_charge-
insensitive ; schreier2008_suppressing , can be used to effectively encode and
process quantum information, as demonstrated by successful realizations of
various NISQ era preskill2018_quantum  processors xiang2017_experimental ;
otterbach2017_unsupervised ; kandala2017_hardware ; arute2019_quantum ;
jurcevic2020_demonstration ; google2020_hartree-fock ; lacroix2020_improving .

Linear oscillators are typically realized in cQED by superconducting microwave
cavities. Commonly used architectures include coplanar waveguide (CPW)
wallraff2004_strong , three-dimensional (3D) rectangular paik2011_observation
and cylindrical co-axial reagor2016_quantum , and micromachined
brecht2015_demonstration  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 houck2007_generating  or a quantum bus that mediates operations
between two isolated artificial atoms by sequentially interacting with each of
them majer2007_coupling ; sillanpaa2007_coherent ; dicarlo2009_demonstration .
Dispersive coupling is achieved by detuning the frequencies of the cavity,
$\omega_{a}$, and the artificial atom, $\omega_{b}$, 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:

$\displaystyle\hat{H}$
$\displaystyle=\omega_{a}\hat{a}^{\dagger}\hat{a}+\omega_{b}\hat{b}^{\dagger}\hat{b}-\chi_{ab}\hat{b}^{\dagger}\hat{b}\hat{a}^{\dagger}\hat{a}-\frac{\alpha}{2}\hat{b}^{\dagger
2}\hat{b}^{2}-\frac{K}{2}\hat{a}^{\dagger 2}\hat{a}^{2},$ (16)

where $\hat{a},\hat{b}$ are the annihilation operators associated with cavity
and artificial atom respectively, $\chi_{ab}$ is the dispersive coupling
strength between them, $\alpha$ is the anharmonicity of the artificial atom,
and $K$ is the non-linearity of the cavity inherited from the atom. Eq. (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 wallraff2004_strong ;
wallraff2005_approaching ; johansson2006_vacuum . Conversely, cavities weakly
coupled to the environment can be used as quantum memories for storing
information coherently reagor2016_quantum . Moreover, when the linewidth of
the cavity is narrow compared to the dispersive shift $\chi_{ab}$, we can
selectively address the individual energy levels of the cavity via the
artificial atom schuster2007_resolving . 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 blais2004_cavity . This
configuration where multi-photon states of the cavity encode logical
information and the ancilla affords universal control krastanov2015_universal
has become an increasingly prevalent choice for implementing bosonic QEC.

###  IV.2 Coherence of superconducting cavities

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 2-dimensional (2D) and 3-dimensional
(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 brecht2015_demonstration ; brecht2017_micromachined
.

In the 2D architecture, such as the coplanar waveguide (CPW), the cavity is
defined by gaps between circuit elements printed on a substrate which is
typically made from silicon or sapphire (Fig. 3a). 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 martinis2005_decoherence ; oconnell2008_microwave ;
gao2008_experimental ; muller2019_towards . Hence, the $Q_{\mathrm{int}}$ of
2D cavities is currently limited to $\sim{10^{5}-10^{6}}$ but can potentially
be improved with more sophisticated cavity design sage2011_study ;
vissers2012_reduced ; geerlings2012_improving ; calusine2018_analysis ,
materials selection wang2009_improving ; barends2010_minimal ; vissers2010_low
; place2020_new , and surface treatment megrant2012_planar ; sandberg2012_etch
; bruno2015_reducing ; melville2020_comparison . 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.

In contrast, 3D cavities (Fig. 3c) achieve a higher $Q_{\mathrm{int}}$ of
$\sim{10^{7}-10^{8}}$, 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 ($\simeq{99.999}\%$) have shown a $Q_{\mathrm{int}}$ as
high as $1.1\times 10^{8}$ kudra2020_high . 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 axline2016_architecture .

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 minev2013_planar ; brecht2015_demonstration ;
brecht2016_multilayer ; minev2016_planar ; axline2016_architecture ;
brecht2017_micromachined ; zoepfl2017_characterization . In particular, the
micromachining technique provides a promising method to fabricate
lithographically defined 2.5D cavities etched out of silicon wafers
brecht2015_demonstration . 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 (Fig. 3b). 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
lei2020_high . Moreover, the integration of a transmon ancilla in this design
has also been demonstrated brecht2017_micromachined , thus making these 2.5D
cavities promising candidates for realizing large-scale quantum devices based
on bosonic modes.

Figure 3: Designs of superconducting cavities and their coherence (a)-(c)
Illustrations of 2D CPW, 2.5D micromachined, and 3D co-axial cavities
respectively. (d) A selected set of $Q_{\mathrm{int}}$ for four commonly used
cavity designs in cQED extracted from the literature. Overall, the coherence
properties of superconducting cavities have been steadily increasing over the
last decade, with the 3D co-axial cavities currently being the main
architecture for realizing bosonic QEC. The references for each cavity design,
in chronological order, are: 2D cavities: Refs. hofheinz2009_synthesizing ,
leek2010_cavity , megrant2012_planar , axline2016_architecture ,
calusine2018_analysis ; 3D rectangular cavities (3DR): Refs.
paik2011_observation , vlastakis2013_deterministically , ofek2016_extending ,
hu2019_quantum ; 3D cylindrical co-axial cavities (3DC): Refs.
reagor2016_quantum , reinhold2020_error-corrected , ma2020_error-transparent ,
kudra2020_high ; 2.5D cavities: Refs. minev2016_planar ,
brecht2017_micromachined , zoepfl2017_characterization , lei2020_high . We
have highlighted studies that successfully integrated one or more non-linear
ancillae with an asterisk (*).

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 oliver2013_materials ;
mcrae2020_materials  as well as lossy interfaces wenner2011_surface ;
wang2015_surface ; woods2019_determining  of superconducting microwave
cavities have been extensively studied and their intrinsic coherence
properties have been improving significantly over the last 15 years
mcrae2020_materials . In Fig. 3d, we compile a non-exhaustive summary of the
internal quality factors of cavities demonstrated in various geometries over
the last decade. Besides enhancing the $Q_{\mathrm{int}}$ 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 $Q_{\mathrm{int}}$, as the best ancilla coherence times
($\sim{50-100}{\mu}$s) 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 Fig. 3d, 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.

##  V Realization of Bosonic Logical Qubits

The remarkable improvements in the performance of cQED hardware components
highlighted in Sec. IV have made realizing protected logical qubits using
bosonic codes a realistic goal. Studies that encode a single logical mode and
protect it against dominant error channels have been reported for the four-
component cat ofek2016_extending , binomial kitten hu2019_quantum , and square
and hexagonal GKP campagne-ibarcq2019_quantum  codes. Moreover, robust
operations on both single heeres2017_implementing ; ma2020_error-transparent ;
reinhold2020_error-corrected  and two bosonic modes gao2018_programmable ;
gao2019_entanglement ; xu2020_demonstration  have also been explored, thus
paving the way towards building a fault-tolerant universal quantum computer
based on bosonic logical qubits. In this section, we highlight recent efforts
to implement universal control of as well as error correction protocols with
bosonic modes.

###  V.1 Operations on single bosonic modes

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,
$\hat{D}(\alpha)=e^{\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a}}$, which
displaces the position and/or momentum of the harmonic oscillator depending on
the value of the complex number $\alpha$. Real values of $\alpha$ correspond
to pure position displacements, while imaginary values of $\alpha$ correspond
to pure momentum ones glauber1963_coherent . 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
$\hat{U}_{\mathrm{CPS}}(t)=|g\rangle\langle g|\otimes\hat{I}+|e\rangle\langle
e|\otimes e^{i\hat{n}\chi t}$, where $\hat{n}$ is the cavity photon number
operator, $\chi$ 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
$t=\pi/\chi$, all the odd photon number states acquire an overall $\pi$-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 (Fig. 4a).

Note that the time required for the parity-mapping operation scales inversely
with the dispersive coupling strength, $\chi$. Naively, one might want to
minimize the operation time by engineering a large $\chi$. However, increasing
$\chi$ 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 frattini2017_3-wave , are usually carefully optimized for each
system to produce the desired Hamiltonian configuration.

Figure 4: Operations on single bosonic modes. (a) The sequence for mapping the
photon number parity of a bosonic mode to the states of its ancilla. (b)
Creation of arbitrary quantum states via repeat SNAP operations on the bosonic
mode. (c) Implementing universal control on a single bosonic mode by
concurrently driving the cavity and its ancilla with numerically optimized
pulses.

With $\hat{U}_{\mathrm{CPS}}$, 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 leghtas2013_deterministic . This scheme is useful for preparing
4-cat states in both single vlastakis2013_deterministically  and multiple
wang2016_schrodinger  cavities. Furthermore,
$\hat{U}_{\mathrm{CPS}}(\pi/\chi)$ 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,
$W(\beta)=\frac{2}{\pi}\mathrm{Tr}[\hat{D}(\beta)^{\dagger}\rho\hat{D}(\beta)\hat{P}]$.
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 controlled phase shift operations blais2020_circuit .
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 chow2012_universal  or the process matrix
nielsen2000_quantum .

Another crucial operation arising from the natural dispersive coupling is the
non-linear selective number-dependent arbitrary phase (SNAP) gate
krastanov2015_universal ; heeres2015_cavity . A SNAP gate (Fig. 4b), defined
as $\hat{S}_{n}(\theta_{n})=e^{i\theta_{n}(|n\rangle\langle n|)}$, selectively
imparts a phase $\theta_{n}$ to the number state $|n\rangle$. Due to the
energy-preserving nature of this operation, we can simultaneously perform
$\hat{S}_{n}(\theta_{n})$ 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 $O(n^{2})$
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 (GRAPE) method khaneja2005_optimal ;
defouquieres2011_second  has been successfully deployed in other physical
systems dolde2014_high-fidelity ; anderson2015_accurate  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 Ref.
heeres2017_implementing . A typical example of a set of pulses obtained
through the gradient-based OCT framework is shown in (Fig. 4c).

More recently, various concepts from classical machine learning, such as
automatic differentiation abdelhafez2019_gradient-based  and reinforcement
learning niu2019_universal ; zhang2019_reinforcement , 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.

###  V.2 Operations on multiple bosonic modes

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 bergeal2010_phase-preserving
. 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 rosenblum2018_cnot . 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 Ref. xu2020_demonstration . 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.
Ref. gao2018_programmable  demonstrated how the four-wave mixing process in a
Josephson junction can provide a frequency converting bilinear coupling of the
form
$H_{\mathrm{int}}(t)/\hbar=g(t)(e^{i\varphi}\hat{a}\hat{b}^{\dagger}+e^{-i\varphi}\hat{a}^{\dagger}\hat{b})$.
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 $\varphi$ 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
$|\omega_{2}-\omega_{1}|=|\omega_{a}-\omega_{b}|$. 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 gao2018_programmable , which is a key building block for
realizing various continuous-variable information processing tasks such as
boson sampling aaronson2011_computational , simulation of vibrational quantum
dynamics of molecules huh2015_boson ; sparrow2018_simulating ;
clements2018_approximating ; wang2020_efficient , and distributed quantum
sensing zhuang2018_distributed ; zhuang2020_distributed ;
xia2020_demonstration ; noh2020_encoding .

Moreover, this bilinear coupling is also a valuable resource for enacting
gates on two logical elements encoded in GKP states gottesman2001_encoding .
As mentioned in Sec. III, GKP encodings rely on the non-linearity of the code
words and only require linear or bilinear operations for universal control
baragiola2019_allgaussian . 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 lau2016_universal . The eSWAP
operation, akin to an exchange operation between spins, implements a
programmable unitary of the form:

$\hat{U}(\theta)=\cos{(\theta)}\hat{I}+\sin{(\theta)}\mathrm{SWAP},$ (17)

where $\theta$ 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
Ref. gao2019_entanglement  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 $\hat{U}_{\mathrm{CPS}}$ 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 towards 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
chakram2020_seamless .

###  V.3 Implementations of QEC

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
$|0,1\rangle$ 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 campagne-ibarcq2019_quantum ;
hu2019_quantum  or achieved ofek2016_extending  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 terhal2015_quantum . 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 gyenis2019_experimental ;
grimm2020_stabilization ; smith2020_magnifying .

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 sun2014_tracking . In Ref. ofek2016_extending
, which corrected a four-component cat state under photon loss (Fig. 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 $|0,1\rangle$ 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.

Figure 5: Four-component cat code under photon loss. Every photon loss event
($\hat{a}$) changes not only the parity of the basis states, but also changes
the phase relationship between them. The encoded state cycles between the even
(logical) and odd (error) parity subspaces, while also rotating about the
Z-axis by $\pi/2$. The decoding sequence must take both these effects into
account to correctly recover the logical information.

In contrast, Ref. hu2019_quantum  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 Ref. ofek2016_extending . 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 michael2016_new . 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
$|0,1\rangle$ 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 campagne-ibarcq2019_quantum . As explained in Sec. III.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 Ref. campagne-
ibarcq2019_quantum  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
royer2020_stabilization .

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 leghtas2015_confining  such that the
logical subspace is stabilized via controlled interactions with the
environment. In Refs. leghtas2015_confining ; touzard2018_coherent , 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 lescanne2020_exponential .

Similarly, an engineered four-photon dissipation achieved by an eight-wave
mixing process, can protect four-component cat codes against dephasing
mundhada2019_experimental . 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 4-cat state
gertler2020_protecting , 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 ma2020_error-
transparent . 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 lescanne2020_exponential ; grimm2020_stabilization
verified that the bias between phase and bit-flip errors increases
exponentially with $\alpha$, the size of the coherent state components of the
2-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 tuckett2018-ultrahigh ; tuckett2019_tailoring ;
tuckett2020_faulttolerant .

##  VI Towards Fault-tolerant Quantum Computation with Bosonic Modes

A crucial consideration for a successful QEC implementation is that even the
attempts to correct errors can themselves be erroneous. Thus, a key
requirement for realizing ‘fault-tolerant’ quantum computation is to carefully
design error correction circuits such that faults that occur during the
execution of a QEC protocol are also tolerated gottesman2009_introduction . A
simple example of fault-tolerant circuit design is the transversal
implementation of logical operations in the conventional paradigm of quantum
error correction with multiple two-level systems. For instance, by
implementing logical gates transversally for distance-$3$ codes that can
correct any single qubit errors, one can ensure that a single fault in the
circuit induces at most a single qubit error in any logical code block which
remains correctable by the code aliferis2006_quantum .

One key challenge in achieving fault-tolerant operations on bosonic qubits in
the cat and binomial encodings is the propagation of uncorrectable errors due
to ancilla decoherence ofek2016_extending ; hu2019_quantum . The ancilla,
often in the form of a transmon, is employed during state preparation, gates,
measurements, and error correction on the encoded qubit via the natural
dispersive coupling. Spontaneous emission of the ancilla during these
processes induces dephasing on the logical qubit, thus irreversibly corrupting
the encoded information. Furthermore, two-photon loss and Kerr effects also
limit the effectiveness of these QEC implementations. One technique used to
mitigate these imperfections is to optimize the cadence of the parity
measurements ofek2016_extending . Frequent parity measurements allow more
accurate tracking of the error syndrome but increase the likelihood of the
ancilla relaxing during measurement and thereby dephasing the cavity. On the
other hand, a larger interval between two measurements not only increases the
probability of two-photon loss, but also results in a higher phase
accumulation due to self-Kerr interactions. As the self-Kerr operator does not
commute with the annihilation operator, photon jumps cause cavity dephasing.
In Ref. ofek2016_extending , the phase accumulated due to the self-Kerr effect
was first estimated and then accounted for by adjusting the phase of
subsequent drive pulses.

Several approaches have been devised to reduce the propagation of
uncorrectable errors from the ancilla to the encoded information. In Ref.
rosenblum2018_fault-tolerant , the third energy level of the ancilla is
employed in the parity-mapping process to suppress the dephasing of the
bosonic mode. A similar strategy can also be applied to general cavity
operations, as shown in Ref. reinhold2020_error-corrected . In these schemes,
additional levels of the ancilla as well as carefully engineered cavity-
ancilla interactions are introduced to prevent incoherent evolution of the
bosonic mode in the event of ancilla decay ma2020_pathindependent .
Alternatively, error-transparent gates have been realized for binomial
encodings ma2020_error-transparent  by cleverly engineering the evolution of
the bosonic qubit in both the code and error space.

For GKP codes, early discussions on fault-tolerance were focused on whether a
small shift error occurring in the middle of an error correction circuit
remains small throughout the entire circuit gottesman2001_encoding ;
glancy2006_error . Recently, numerous studies have approached fault-tolerant
quantum computation with the GKP code by combining the GKP code with
conventional error correction schemes based on multiple two-level systems that
are already shown to be fault-tolerant. In performing such a concatenation of
codes, additional analog information acquired during the error correction
protocol can significantly boost the performance of a next-level QEC scheme
based on multiple error-corrected GKP qubits fukui2017_analog ;
fukui2018_tracking . For instance, studies have shown that the fault-tolerance
thresholds of the circuit-based toric code and the surface code can be
increased by using the additional analog information vuillot2019_quantum ;
noh2020_faulttolerant . Another way to realize fault-tolerant quantum
computing with the GKP code is to combine the GKP code with the idea of fault-
tolerant measurement-based quantum computation using cluster states, for which
the additional analog information also plays an important role
menicucci2014_faulttolerant ; fukui2018_highthreshold ; walshe2019_robust ;
fukui2019_highthreshold ; yamasaki2020_polylog ; bourassa2020_blueprint ;
larsen2021_faulttolerant . From the circuit-based computing studies, we know
that fault-tolerant quantum computing with the GKP code is possible if the
squeezing of the GKP states is larger than $11$dB. Whereas from the
measurement-based computing scenarios, a more favorable threshold value of
$8$dB can be obtained by taking advantage of post-selection. However, as the
measurement-based schemes use post-selection, they require a higher resource
overhead. Recently, Ref. shi2019_fault-tolerant  made the phase estimation
protocol to prepare GKP states in Ref. terhal2016_encoding  fault-tolerant by
using the concept of flag qubits chao2018_quantum ; chamberland2018_flag .

There have also been general studies on the fault-tolerance of biased-noise
qubits aliferis2008_faulttolerant ; tuckett2018-ultrahigh ;
tuckett2019_tailoring ; tuckett2020_faulttolerant . These studies highlight
the potential of achieving higher fault-tolerance thresholds and lower
resource overheads by taking advantage of the noise bias. Refs.
guillaud2019_repetition ; guillaud2020_error  suggested pieceable fault-
tolerant yoder2016_universal  Toffoli circuits while Ref. puri2020_bias-
preserving  offered a magic state distillation scheme tailored to biased-noise
models webster2015_reducing  as a means of achieving fault-tolerant universal
quantum computation with biased-noise 2-cat qubits. A detailed roadmap for
realizing biased-noise 2-cat qubits using acoustic nanomechanical resonators
as well as alternative fault-tolerant universal quantum computation schemes
were recently proposed chamberland2020_building .

##  VII Outlook and Perspectives

Continuous variables have long been recognized as promising candidates for the
efficient encoding and processing of quantum information lloyd1999_quantum .
Bosonic codes leverage the advantages afforded by continuous variable quantum
devices to realize hardware-efficient QEC and pave the way towards the
eventual realization of a fault-tolerant quantum computer. In this review, we
have presented some of the most compelling results that attest to the
viability of the cQED platform. In particular, many milestones, from reaching
the break-even point to implementing logical operations between bosonic
logical qubits, have thus far been demonstrated using 3D cQED devices. An
important next milestone would be to demonstrate the break-even point for
logical operations, i.e., to achieve gate fidelities with error-correctable
bosonic qubits that are higher than the highest gate fidelity attainable with
the best physical element without error correction. Achieving the break-even
point in both QEC and single or two-mode logical operations forms a critical
foundation upon which fault-tolerant gates and algorithms can be constructed.

We further emphasize that the techniques and methodologies developed in these
3D cQED implementations of bosonic codes are agnostic to the hardware
architecture. They can therefore be readily adapted to more compact designs,
such as planar or 2.5D devices, as their performance improves with further
developments in materials engineering and new fabrication techniques
place2020_new .

Crucially, encoding logical information in bosonic codes is naturally
compatible with a modular approach to scalability. In the modular
architecture, each logical element can be encoded, protected, and optimized
individually, and then connected to other elements via on-demand communication
channels jiang2007_distributed ; kimble2008_quantum ; monroe2014_large-scale .
In contrast to directly wiring up an increasing number of physical elements,
modularity has the benefit of reduced cross-talk, robustness against local
failure modes, and enhanced re-configurability. Encoding and manipulating
individual logical qubits encoded in cavities is a crucial primitive for
eventually realizing a large-scale modular quantum device. In addition, recent
experiments have also demonstrated other key elements of a modular
architecture, including programmable quantum communication and entanglement
generation between two distant bosonic modes axline2018_on-demand ; campagne-
ibarcq2018_deterministic ; burkhart2020_error-detected  as well as a
teleported CNOT gate chou2018_deterministic .

As we continue to improve the performance and scale of these bosonic quantum
systems on the hardware level, we must also expand our technological
repertoire to effectively control and characterize these more complex devices.
This expansion may involve, for instance, the development of more
sophisticated measurement electronics and protocols as well as new theoretical
frameworks to capture the signatures of multipartite entanglement. The small-
scale bosonic devices developed so far provide an indispensable platform to
test and refine these crucial elements.

In recent years, cQED devices have become a workhorse for the implementations
of quantum error correction and quantum information processing. Alongside the
developments in cQED, we are also witnessing remarkable progress with other
platforms based on trapped-ions landsman2019_two-qubit ; egan2020_fault-
tolerant , silicon spins veldhorst2015_two-qubit ; watson2018_programmable ,
neutral atoms levine2019_parallel , etc, from both academic and industrial
initiatives. While the ultimate hardware to achieve fault-tolerance may not
consist of millions of 3D cavities, the insights gained from and the technical
achievements demonstrated by the results summarized in this article will serve
as crucial building blocks for the eventual realization of a scalable and
robust universal quantum computer.

## Acknowledgments

The authors thank S. M. Girvin and L. Sun for fruitful discussions and
comments on the manuscript. Yvonne Y. Gao acknowledges funding from the
Singapore National Research Foundation Fellowship (Class of 2020). This work
is also supported by the Ministry of Education in Singapore.

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