Welcome to the Bosonic Kingdom.

Bosonic code Also called an oscillator or a continuous-variable (CV) code. Encodes logical Hilbert space, finite- or infinite-dimensional, into a physical Hilbert space that contains at least one oscillator (a.k.a. bosonic mode or qumode). States of a single oscillator are elements of the Hilbert space of $$\ell^2$$-normalizable functions on $$\mathbb{R}$$). Ideal codewords may not be normalizable because the space is infinite-dimensional, so approximate versions have to be constructed in practice. Protection: An error set relevant to bosonic stabilizer codes is the set of displacement operators, a bosonic analogue of the Pauli string basis for qubit codes. For a single mode, its elements are products of exponentials of the mode's position and momentum operators, acting on the mode's position states $$|y\rangle$$ for $$y\in\mathbb{R}$$ as \begin{align} e^{-iq\hat{p}}\left|y\right\rangle =\left|y+q\right\rangle \,\,\text{ and }\,\,e^{iq\hat{x}}\left|y\right\rangle =e^{iq y}\left|y\right\rangle ~, \end{align} where $$q\in\mathbb{R}$$. For multiple modes, error set elements are tensor products of elements of the single-qudit error set, characterized by the vector of coefficients $$\xi\in\mathbb{R}^{2n}$$. Parents: Quantum error-correcting code (QECC). Cousin of: Fermionic code, Group-based quantum code.
Also known as a continuous-variable (CV) stabilizer code. Bosonic code whose codespace is defined as the common $$+1$$ eigenspace of a group of mutually commuting displacement operators. Displacements form the stabilizers of the code, and have continuous eigenvalues, in contrast with the discrete set of eigenvalues of qubit stabilizers. As a result, exact codewords are non-normalizable, so approximate constructions have to be considered. Protection: Protective properties can be delineated in terms of the nullifiers or displacements, and the most natural noise model for such codes is displacement noise. If an error operator does not commute with a stabilizer group element, then that error is detectable. Oscillator-into-oscillator stabilizer codes protect against erasures of a subset of modes, while GKP codes protect against sufficiently small displacements in any number of modes. Parents: Stabilizer code, Bosonic code. Parent of: Analog stabilizer code, Multi-mode GKP code. Cousin of: Group GKP code, Number-phase code.
Coherent-state constellation quantum code Qudit-into-oscillator code whose codewords can succinctly be expressed as superpositions of a countable set of coherent states that is called a constellation. Some useful constellations form a group (see gkpcat or $$2T$$-qutrit codes) while others make up a Gaussian quadrature rule [2][3]. Parents: Bosonic code. Parent of: 2T-qutrit code, Cat code, Multi-mode GKP code.
Hybrid qudit-oscillator code Encodes a $$K$$-dimensional logical Hilbert space into $$n_1$$ qudits of dimension $$q$$ and $$n_2 \neq 0$$ oscillators, i.e., the Hilbert space of $$\ell^2$$-normalizable functions on $$\mathbb{Z}_q^{n_1} \times \mathbb{R}^{n_2}$$. Parents: Bosonic code. Parent of: Very small logical qubit (VSLQ) code. Cousins: Qudit-into-oscillator code.
Also called an analog quantum code. Encodes $$k$$ bosonic modes into $$n$$ bosonic modes. Parents: Bosonic code. Parent of: Analog stabilizer code, GKP-stabilizer code.
Qudit-into-oscillator code Encodes $$K$$-dimensional Hilbert space into $$n$$ bosonic modes. Parents: Bosonic code. Cousin of: Hybrid qudit-oscillator code.
Analog stabilizer code Also known as a linear or Gaussian stabilizer code. Oscillator-into-oscillator stabilizer code encoding $$k$$ logical modes into $$n$$ physical modes. An $$((n,k,d))_{\mathbb{R}}$$ analog stabilizer code is denoted as $$[[n,k,d]]_{\mathbb{R}}$$, where $$d$$ is the code's distance. Protection: Protect against erasures of at most $$d-1$$ modes, or arbitrarily large dispalcements on those modes. If an error operator does not commute with a nullifier, then that error is detectable. Protection of logical modes against small displacements cannot be done using only Gaussian resources [6][7] (see also [8][9]). There are no such restrictions for non-Gaussian noise [10]. Cousins: GKP-stabilizer code.
Generalization of the GKP code to $$n$$ bosonic modes whose stabilizer group is an infinite countable group of oscillator displacement operators. Protection: The level of protection against displacement errors is quantified by the Euclidean code distance $$\Delta=\min_{x\in {\mathcal{L}}^{\perp}\setminus {\mathcal{L}}} \|x\|_2$$ [13].
Multi-mode code encoding logical qubits into a cluster-state stabilizer code concatenated with a single-mode GKP code. Provides a way to perform a continuous-variable (CV) analogue of fault-tolerant measurement-based qubit computation. Cousins: Concatenated quantum code.
Stub. Cousin of: Analog stabilizer code.
2T-qutrit code Two-mode qutrit code constructed out of superpositions of coherent states whose amplitudes make up the binary tetrahedral group $$2T$$. The codespace is a particular three-dimensional subspace of the 24-dimensional two-mode coherent-state subspace, \begin{align} \mathrm{Span}( \{|\sqrt{2} e^{i (2k+1) \pi/4} \alpha\rangle |0\rangle, |0\rangle |\sqrt{2} e^{i (2k+1) \pi/4} \alpha\rangle, |e^{i k\pi/2} \alpha\rangle |e^{i \ell \pi/2} \alpha\rangle \: : \: 0\leq k, \ell \leq 3\}) \end{align} for any $$\alpha \geq 0$$. A basis can be constructed whose elements are equal superpositions of coherent states whose amplitudes make up cosets of the quaternion subgroup $$Q$$ in $$2T$$. Cousins: Chuang-Leung-Yamamoto (CLY) code.
Rotation-symmetric bosonic Fock-state code encoding a $$q$$-dimensional qudit into one oscillator which utilizes a constellation of $$q(S+1)$$ coherent states distributed equidistantly around a circle in phase space of radius $$\alpha$$. Protection: Due to the spacing between sets of Fock states, the distance between two distinct logical states is $$d=S+1$$. Hence, this code is able to detect $$S$$-photon loss error. Parent of: Two-component cat code. Cousins: Number-phase code.
The two logical codewords are $$|\pm\rangle \propto (|0\rangle\pm|2\rangle)(|0\rangle\pm|2\rangle)|0\rangle|0\rangle$$, where the total Hilbert space is the tensor product of two qudits (whose ground states $$|0\rangle$$ and second excited states $$|2\rangle$$ are used in the codewords) and two oscillators. In the original proposal for implementation, the single logical qubit is given by the two lowest energy states of a circuit composed of two transmons coupled to two lossy resonators, but the resonators can also be thought of as qubits since only a few low-lying Fock states are used by the code. Protection: Passively protects against single photon loss. Parents: Hybrid qudit-oscillator code. Cousins: Quantum repetition code.
Fock-state bosonic code Qudit-into-oscillator code whose protection against amplitude damping (i.e., photon loss) stems from the use of disjoint sets of Fock states for the construction of each code basis state. The simplest example is the dual-rail code, which has codewords consisting of single Fock states $$|10\rangle$$ and $$|01\rangle$$. This code can detect a single loss error since a loss operator in either mode maps one of the codewords to a different Fock state $$|00\rangle$$. More involved codewords consist of several well-separated Fock states such that multiple loss events can be detected and corrected. Protection: Code distance $$d$$ is the minimum distance (assuming some metric) between any two labels of Fock states corresponding to different code basis states. For a single mode, $$d$$ is the minimum absolute value of the difference between any two Fock-state labels; such codes can detect up to $$d-1$$ loss events. Multimode distances can be defined analogously; see, e.g., Chuang-Leung-Yamamoto codes. Parents: Qudit-into-oscillator code. Cousins: Binary code, Qubit code. Cousin of: Fusion-based quantum computing (FBQC) code.
Single-mode bosonic code Encodes $$K$$-dimensional Hilbert space into a single bosonic mode. Parents: Qudit-into-oscillator code.
Bosonic Fock-state code that encodes $$k$$ qubits into $$n$$ oscillators, with each oscillator restricted to having at most $$N$$ excitations. Codewords are superpositions of oscillator Fock states which have exactly $$N$$ total excitations, and are either uniform (i.e., balanced) superpositions or unbalanced superpositions. Codes can be denoted as $$[[N,n,2^k,d]]$$, which conflicts with stabilizer code notation. Protection: Protects against amplitude damping for up to $$t = d-1$$ excitation losses. Defining the spacing between two Fock states $$|u_1\cdots u_n\rangle$$ and $$|v_1\cdots v_n\rangle$$, \begin{align} \text{Spacing}(u,v) = \frac{1}{2}\sum_{i=1}^n |u_i - v_i|, \end{align} the code distance $$d$$ can be defined as the minimial spacing between Fock states making up the codewords. Cousin of: 2T-qutrit code, Binomial code.
Single-mode bosonic Fock-state code that can be used for error-corrected sensing of a signal Hamiltonian $${\hat n}^s$$, where $${\hat n}$$ is the occupation number operator. Codewords for the $$s$$th-order Chebyshev code are \begin{align} \begin{split} \ket{\overline 0} &=\sum_{k \text{~even}}^{[0,s]} \tilde{c}_k \Ket{\left\lfloor M\sin^2\left( k\pi/{2s}\right) \right\rfloor},\\ \ket{\overline 1} &= \sum_{k \text{~odd}}^{[0,s]} \tilde{c}_k \Ket{\left\lfloor M\sin^2 \left(k\pi/{2s}\right) \right\rfloor}, \end{split} \end{align} where $$\tilde{c}_k>0$$ can be obtained by solving a system of order $$O(s^2)$$ linear equations, and where $$\lfloor x \rfloor$$ is the floor function. The code approaches optimality for sensing the signal Hamiltonian as $$M$$ increases. Protection: The $$s$$th-order code corrects errors from the set $$\{I,a,a^{\dagger},{\hat n},{\hat n}^2,\cdots,{\hat n}^{s-1}\}$$. Cousins: Binomial code.
Also called a continuous-variable (CV) surface code. An analog CSS version of the Kitaev surface code. Parents: Analog stabilizer code. Cousins: Kitaev surface code.
A $$[[5,1,3]]_{\mathbb{R}}$$ analog stabilizer version of the five-qubit perfect code. Parents: Analog stabilizer code. Cousins: Five-qubit perfect code.
An $$[[n,1]]_{\mathbb{R}}$$ Gaussian CSS code defined using homological structres associated with an $$n-1$$ simplex. Relevant to the study of spacetime replication of quantum information [22]. Protection: Protects against certain types of erasure errors (depending on the specific dimension). Certain constructions also protect arbitrary sized errors on multiple photon states. Parents: Analog stabilizer code.
A $$[[9,1,3]]_{\mathbb{R}}$$ analogue CSS version of Shor's nine-qubit code. Parents: Analog stabilizer code. Cousins: $$[[9,1,3]]$$ Shor code.
Bosonic qudit-into-oscillator code whose stabilizers are oscillator displacement operators $$\hat{S}_q(2\alpha)=e^{-2i\alpha \hat{p}}$$ and $$\hat{S}_p(2\beta)=e^{2i\beta \hat{x}}$$. To ensure $$\hat{S}_q(2\alpha)$$ and $$\hat{S}_p(2\beta)$$ generate a stabilizer group that is Abelian, there is another constraint that $$\alpha\beta=2k\pi$$ where $$k$$ is an integer. Codewords can be expressed as equal weight superpositions of coherent states on an infinite lattice, such as a square lattice in phase space with spatial period $$2\sqrt{\pi}$$. The exact GKP state is non-normalizable, so approximate constructs have to be considered. Protection: For stabilizer $$\hat{S}_q(2\alpha),\hat{S}_p(2\beta)$$, code can correct displacement errors up to $$\frac{\alpha}{2}$$ in the $$q$$-direction and $$\frac{\beta}{2}$$ at $$p$$-direction. Approximately protects against photon loss errors [23][24], outperforming most other codes designed to explicitly protect against loss [24]. Very sensitive to dephasing errors [25]. A biased-noise GKP error correcting code can be prepared by choosing $$\alpha\neq \beta$$.
Code whose codespace is spanned by two coherent states $$\left|\pm\alpha\right\rangle$$ for nonzero complex $$\alpha$$. An orthonormal basis for the codespace consists of the bosonic cat states \begin{align} |\overline{\pm}\rangle=\frac{\left|\alpha\right\rangle \pm\left|-\alpha\right\rangle }{\sqrt{2\left(1\pm e^{-2|\alpha|^{2}}\right)}} \end{align} for any complex $$\alpha$$. Protection: Two-legged cat codes for large $$\alpha$$ provide protection against modal dephasing, i.e., diffusion of the angular degree of freedom of the mode. They do not protect against photon loss events, but there exist modifications based on sign alternation [27], squeezing [28][29][29], and detuning [30] that can add such protection. Parents: Cat code. Cousins: Hamiltonian-based code. Cousin of: Binary PSK (BPSK) code.
A single-mode Fock-state bosonic code whose codespace is preserved by a phase-space rotation by a multiple of $$2\pi/N$$ for some $$N$$. The rotation symmetry ensures that encoded states have support only on every $$N^{\textrm{th}}$$ Fock state. For example, single-mode Fock-state codes for $$N=2$$ encoding a qubit admit basis states that are, respectively, supported on Fock state sets $$\{|0\rangle,|4\rangle,|8\rangle,\cdots\}$$ and $$\{|2\rangle,|6\rangle,|10\rangle,\cdots\}$$. Protection: Losses or gains less than $$N$$ are detectable. Dephasing rotations $$\exp(\mathrm{i}\theta \hat{n})$$ can be detected whenever $$\theta$$ is roughly less than $$\pi/N$$. To get precise bounds on $$\theta$$, one needs to analyze the particular bosonic rotation code. Parent of: Binomial code, Cat code, Chebyshev code, Number-phase code.
Two- or higher-mode extension of cat codes whose codewords are right eigenstates of powers of products of the modes' lowering operators. Many gadgets for cat codes have two-mode pair-cat analogues, with the advantage being that such gates can be done in parallel with a dissipative error-correction process. Protection: The occupation-number differences form the syndromes, as opposed to the photon number parity for the single-mode cat code. Any loss even combination that changes the relative differences of photons between modes is a detectable error. The two-mode two-component paircat code can detect arbitrary single-mode losses, but cannot detect simultaneous photon loss in both modes. An $$n$$-mode code can detect any loss errors of at most $$n-1$$ weight. Higher numbers of legs correspond to more pair-coherent state present in the codewords, and allow for protection against simulataneous losses. Parents: Fock-state bosonic code. Cousins: Cat code, Hamiltonian-based code.
Bosonic Fock-state code obtained from a numerical minimization procedure, e.g., from enforcing error-correction criteria against some number of losses while minimizing average occupation number. Protection: Number phase codes protect from a finite number of loss events. However, unlike Fock-state codes, their protection does not stem from a Fock-state spacing. Parents: Single-mode bosonic code.
Bosonic rotation codes designed to approximately protect against errors consisting of powers of raising and lowering operators up to some maximum power. Binomial codes can be thought of as spin-coherent states embedded into an oscillator [24]. Protection: An $$(N, S)$$ binomial code protects against $$L$$ boson losses, $$G$$ boson gains, and dephasing up to $$\hat{n}^{D}$$, where $$S=L+G$$ and $$N = \mathrm{max}(L,G,2D)$$. Binomial codes approximately protect against continuous-time amplitude damping, boson loss and gain, and dephasing. Parents: Bosonic rotation code.
Bosonic rotation code consisting of superpositions of Pegg-Barnett phase states [35], \begin{align} |\phi\rangle \equiv \frac{1}{\sqrt{2\pi}}\sum_{n=0}^{\infty} \mathrm{e}^{\mathrm{i} n \phi} \ket{n}. \end{align} Since phase states and thus the ideal codewords are not normalizable, approximate versions need to be constructed. The codes' key feature is that, in the ideal case, phase measurement has zero uncertainty, making it a good canditate for a syndrome measurement. Protection: Number-phase codes of order $$N$$ detect up to $$N$$ photon loss or gain errors, and dephasing up to $$\theta = \pi/N$$. Parents: Bosonic rotation code. Cousin of: Binomial code, Cat code.
Two-mode code encoding a logical qubit in Fock states with one excitation. The logical-zero state is represented by $$|01\rangle$$, while the logical-one state is represented by $$|10\rangle$$. The two modes of the encoding can represent temporal or spatial modes, corresponding to a time-bin or frequency-bin encoding. Protection: This is an error-detecting code against one photon loss event; it is often used in photonic quantum devices because of its ease of realization. A single loss event can be detected because, after the loss occurs, the output state $$|00\rangle$$ is orthogonal to the codespace. Recovery is not possible, so a successful run of a quantum circuit is conditioned on not losing a photon during the circuit.
Three-oscillator Fock-state code encoding a single logical qubit using codewords \begin{align} \begin{split} |\overline{0}\rangle &= \frac{1}{\sqrt{3}}(|003\rangle+|030\rangle+|300\rangle)\\ |\overline{1}\rangle &= |111\rangle \end{split}. \end{align} Protection: Protects against single photon loss in any one mode.

## References

[1]
Richard L. Barnes, “Stabilizer Codes for Continuous-variable Quantum Error Correction”. quant-ph/0405064
[2]
F. Lacerda, J. M. Renes, and V. B. Scholz, “Coherent-state constellations and polar codes for thermal Gaussian channels”, Physical Review A 95, (2017). DOI; 1603.05970
[3]
F. Lacerda, J. M. Renes, and V. B. Scholz, “Coherent state constellations for Bosonic Gaussian channels”, 2016 IEEE International Symposium on Information Theory (ISIT) (2016). DOI
[4]
S. Lloyd and J.-J. E. Slotine, “Analog Quantum Error Correction”, Physical Review Letters 80, 4088 (1998). DOI; quant-ph/9711021
[5]
S. L. Braunstein, “Error Correction for Continuous Quantum Variables”, Physical Review Letters 80, 4084 (1998). DOI; quant-ph/9711049
[6]
J. Niset, J. Fiurášek, and N. J. Cerf, “No-Go Theorem for Gaussian Quantum Error Correction”, Physical Review Letters 102, (2009). DOI; 0811.3128
[7]
C. Vuillot et al., “Quantum error correction with the toric Gottesman-Kitaev-Preskill code”, Physical Review A 99, (2019). DOI; 1810.00047
[8]
J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian States with Gaussian Operations is Impossible”, Physical Review Letters 89, (2002). DOI; quant-ph/0204052
[9]
G. Giedke and J. Ignacio Cirac, “Characterization of Gaussian operations and distillation of Gaussian states”, Physical Review A 66, (2002). DOI; quant-ph/0204085
[10]
Peter van Loock, “A note on quantum error correction with continuous variables”. 0811.3616
[11]
D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator”, Physical Review A 64, (2001). DOI; quant-ph/0008040
[12]
J. Harrington and J. Preskill, “Achievable rates for the Gaussian quantum channel”, Physical Review A 64, (2001). DOI; quant-ph/0105058
[13]
J. Conrad, J. Eisert, and F. Arzani, “Gottesman-Kitaev-Preskill codes: A lattice perspective”, Quantum 6, 648 (2022). DOI; 2109.14645
[14]
N. C. Menicucci, “Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States”, Physical Review Letters 112, (2014). DOI; 1310.7596
[15]
K. Noh, S. M. Girvin, and L. Jiang, “Encoding an Oscillator into Many Oscillators”, Physical Review Letters 125, (2020). DOI; 1903.12615
[16]
Z. Leghtas et al., “Hardware-Efficient Autonomous Quantum Memory Protection”, Physical Review Letters 111, (2013). DOI; 1207.0679
[17]
E. Kapit, “Hardware-Efficient and Fully Autonomous Quantum Error Correction in Superconducting Circuits”, Physical Review Letters 116, (2016). DOI
[18]
I. L. Chuang, D. W. Leung, and Y. Yamamoto, “Bosonic quantum codes for amplitude damping”, Physical Review A 56, 1114 (1997). DOI
[19]
D. Layden et al., “Ancilla-Free Quantum Error Correction Codes for Quantum Metrology”, Physical Review Letters 122, (2019). DOI; 1811.01450
[20]
J. Zhang et al., “Anyon statistics with continuous variables”, Physical Review A 78, (2008). DOI; 0711.0820
[21]
P. Hayden et al., “Spacetime replication of continuous variable quantum information”, New Journal of Physics 18, 083043 (2016). DOI; 1601.02544
[22]
P. Hayden and A. May, “Summoning information in spacetime, or where and when can a qubit be?”, Journal of Physics A: Mathematical and Theoretical 49, 175304 (2016). DOI; 1210.0913
[23]
B. M. Terhal and D. Weigand, “Encoding a qubit into a cavity mode in circuit QED using phase estimation”, Physical Review A 93, (2016). DOI; 1506.05033
[24]
V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018). DOI; 1708.05010
[25]
A. L. Grimsmo and S. Puri, “Quantum Error Correction with the Gottesman-Kitaev-Preskill Code”, PRX Quantum 2, (2021). DOI; 2106.12989
[26]
P. T. Cochrane, G. J. Milburn, and W. J. Munro, “Macroscopically distinct quantum-superposition states as a bosonic code for amplitude damping”, Physical Review A 59, 2631 (1999). DOI; quant-ph/9809037
[27]
L. Li et al., “Phase-engineered bosonic quantum codes”, Physical Review A 103, (2021). DOI; 1901.05358
[28]
David S. Schlegel, Fabrizio Minganti, and Vincenzo Savona, “Quantum error correction using squeezed Schrödinger cat states”. 2201.02570
[29]
Qian Xu et al., “Autonomous quantum error correction and fault-tolerant quantum computation with squeezed cat qubits”. 2210.13406
[30]
Diego Ruiz et al., “Two-photon driven Kerr quantum oscillator with multiple spectral degeneracies”. 2211.03689
[31]
A. L. Grimsmo, J. Combes, and B. Q. Baragiola, “Quantum Computing with Rotation-Symmetric Bosonic Codes”, Physical Review X 10, (2020). DOI; 1901.08071
[32]
V. V. Albert et al., “Pair-cat codes: autonomous error-correction with low-order nonlinearity”, Quantum Science and Technology 4, 035007 (2019). DOI; 1801.05897
[33]
Y. Ouyang and R. Chao, “Permutation-Invariant Constant-Excitation Quantum Codes for Amplitude Damping”, IEEE Transactions on Information Theory 66, 2921 (2020). DOI; 1809.09801
[34]
M. H. Michael et al., “New Class of Quantum Error-Correcting Codes for a Bosonic Mode”, Physical Review X 6, (2016). DOI; 1602.00008
[35]
S. M. Barnett and D. T. Pegg, “Phase in quantum optics”, Journal of Physics A: Mathematical and General 19, 3849 (1986). DOI
[36]
I. L. Chuang and Y. Yamamoto, “Simple quantum computer”, Physical Review A 52, 3489 (1995). DOI
[37]
W. Wasilewski and K. Banaszek, “Protecting an optical qubit against photon loss”, Physical Review A 75, (2007). DOI; quant-ph/0702075