Binomial code[1] 


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 [2].

A simple example of a binomial code is the "0-2-4" qubit code with codewords \begin{align} \begin{split} |\overline{0}\rangle&=\frac{1}{\sqrt{2}}\left(|0\rangle+|4\rangle\right)\\ |\overline{1}\rangle&=|2\rangle~, \end{split} \tag*{(1)}\end{align} constructed out of binomial states [3].

General \(q\)-dimensional qudit \((N, S)\) binomial codeword states are \(\{|\overline{i}\rangle\mid i\in \mathbb Z_q \}\), where \begin{align} |\overline{i}\rangle = \frac{1}{\sqrt{q^N}} \sum_{\substack{p=0\\p\equiv i \pmod{q}}}^{(q-1)(N+1)} \sqrt{\binom{N+1}{p}_q} \ket{p(S+1)}. \tag*{(2)}\end{align} The set \( \ket{i} \mid i \in \mathbb{N}\) is the set of Fock states. Also, \(\binom{N+1}{p}_q\) are extended binomial coefficients, or polynomial coeffiients, defined recursively as \begin{align} \binom{n}{m}_1 \equiv 1,\quad \binom{n}{m}_q \equiv \sum_{k=0}^n \binom{n}{k}\binom{k}{m-k}_{q-1}. \tag*{(3)}\end{align} The extended binomial coefficients \( \binom{n}{m}_q \) are also the coefficients of \( x^m \) in the polynomial \( (1 + x + \cdots + x^{q-1})^n \).


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.


Error-detecting \(CCZ\) and \(cSWAP\) gates for "0-2-4" code using three-level ancilla [4].


Photon loss and dephasing errors can be detected by measuring the phase-space rotation \(\exp\left(2\pi\mathrm{i} \hat{n} / (S+1)\right)\) and the check operator \(J_x/J\) in the spin-coherent state language, where \(J\) is the total angular momentum and \(J_x\) is the angular momentum in the \(x\) direction [2]. This type of error correction fails for errors that are products of photon loss/gain and dephasing errors. However, for certain \((N,S)\) instances of the binomial code, detection of these types of errors can be done.Recovery can be done via projective measurements and unitary operations [1,2].


Microwave cavities coupled to superconducting circuits: state transfer between a binomial codeword to another system [5], error-correction protocol nearly reaching break-even [6], and a teleported CNOT gate [7]. A realization of the "0-2-4" encoding is the first to go beyond break-even error-correction and yields a logical lifetime that exceeds the cavity lifetime by \(16\%\) [8] (see also [9]).


The mean occupation number, or average Fock-state number in maximally-mixed state of the code, is \((N+1)(S+1)(q-1)/2 \), where \(q\) is the qudit dimension.


  • Bosonic rotation code — One can verify by direct calculation that the logical states are eigenstates of the discrete rotation operator. One has freedom in the exact form of the primitive state to choose; see Appendix B.2 of Ref. [10].


  • Cat code — For a fixed \(S\), binomial codes with \(N \to \infty\) coincide with cat codes as \(\alpha \to \infty\) [1].
  • Number-phase code — In the limit as \(N,S \to \infty\), phase measurement in the binomial code has vanishing variance, just like in a number-phase code [10].
  • Chuang-Leung-Yamamoto (CLY) code — Two-mode version of binomial codes correspond to two-mode "0-2-4" CLY codes (see Sec. IV.A of Ref. [1]).
  • Chebyshev code — Chebyshev codes resemble binomial codes, and a class of binomial codes have similar error-correcting properties [11].
  • \([[4,2,2]]\) CSS code — \([[4,1,2]]\) subcode consisting of \(\{|\overline{00}\rangle\) and any other codeword reduces to the \(0,2,4\) binomial code when the basis labels in each codeword are written as in base-ten. Such a mapping can be generalized [12].
  • GNU permutation-invariant code — Binomial codes and GNU codes related via the Holstein-Primakoff mapping [13] (see also [14]). A qudit generalization of GNU codes can be obtained from qudit binomial codes [2; Appx. C].


M. H. Michael et al., “New Class of Quantum Error-Correcting Codes for a Bosonic Mode”, Physical Review X 6, (2016) arXiv:1602.00008 DOI
V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018) arXiv:1708.05010 DOI
D. Stoler, B. E. A. Saleh, and M. C. Teich, “Binomial States of the Quantized Radiation Field”, Optica Acta: International Journal of Optics 32, 345 (1985) DOI
T. Tsunoda et al., “Error-detectable bosonic entangling gates with a noisy ancilla”, (2022) arXiv:2212.11196
C. J. Axline et al., “On-demand quantum state transfer and entanglement between remote microwave cavity memories”, Nature Physics 14, 705 (2018) arXiv:1712.05832 DOI
L. Hu et al., “Quantum error correction and universal gate set operation on a binomial bosonic logical qubit”, Nature Physics 15, 503 (2019) arXiv:1805.09072 DOI
Y. Xu et al., “Demonstration of Controlled-Phase Gates between Two Error-Correctable Photonic Qubits”, Physical Review Letters 124, (2020) arXiv:1810.04690 DOI
Z. Ni et al., “Beating the break-even point with a discrete-variable-encoded logical qubit”, Nature 616, 56 (2023) arXiv:2211.09319 DOI
V. V. Sivak et al., “Real-time quantum error correction beyond break-even”, Nature 616, 50 (2023) arXiv:2211.09116 DOI
A. L. Grimsmo, J. Combes, and B. Q. Baragiola, “Quantum Computing with Rotation-Symmetric Bosonic Codes”, Physical Review X 10, (2020) arXiv:1901.08071 DOI
D. Layden et al., “Ancilla-Free Quantum Error Correction Codes for Quantum Metrology”, Physical Review Letters 122, (2019) arXiv:1811.01450 DOI
Linshu Li, private communication, 2018
T. Holstein and H. Primakoff, “Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet”, Physical Review 58, 1098 (1940) DOI
C. D. Cushen and R. L. Hudson, “A quantum-mechanical central limit theorem”, Journal of Applied Probability 8, 454 (1971) DOI
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Zoo Code ID: binomial

Cite as:
“Binomial code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
  title={Binomial code},
  booktitle={The Error Correction Zoo},
  editor={Albert, Victor V. and Faist, Philippe},
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“Binomial code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.