# Binomial code[1]

## Description

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 \( (1 + x + \cdots + x^{q-1})^n \).

## Protection

## Gates

## Decoding

## Realizations

## Notes

## Parent

- 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. [13].

## Cousins

- 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 [13].
- Two-mode binomial code
- Chebyshev code — Chebyshev codes resemble binomial codes, and a class of binomial codes have similar error-correcting properties [14].
- Asymmetric quantum code — Binomial code parameters against loss/gain errors and dephasing can be tuned.
- Four-qubit single-deletion code — The four-qubit single-deletion code can be obtained from the "0-2-4" single-mode binomial code by substituting Fock states with Dicke states.
- GNU PI code — Binomial codes and GNU codes related via the Holstein-Primakoff mapping [15] (see also [16]). A qudit generalization of GNU codes can be obtained from qudit binomial codes [2; Appx. C].
- \([[4,2,2]]\) 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 [17].
- Æ code — Many well-performing Æ codes can be mapped into shifted versions of binomial codes via the Holstein-Primakoff mapping.

## References

- [1]
- 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
- [2]
- V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018) arXiv:1708.05010 DOI
- [3]
- 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
- [4]
- T. Tsunoda et al., “Error-detectable bosonic entangling gates with a noisy ancilla”, (2022) arXiv:2212.11196
- [5]
- Y. Ouyang, “Robust projective measurements through measuring code-inspired observables”, (2024) arXiv:2402.04093
- [6]
- 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
- [7]
- 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
- [8]
- Y. Xu et al., “Demonstration of Controlled-Phase Gates between Two Error-Correctable Photonic Qubits”, Physical Review Letters 124, (2020) arXiv:1810.04690 DOI
- [9]
- Z. Ni et al., “Beating the break-even point with a discrete-variable-encoded logical qubit”, Nature 616, 56 (2023) arXiv:2211.09319 DOI
- [10]
- V. V. Sivak et al., “Real-time quantum error correction beyond break-even”, Nature 616, 50 (2023) arXiv:2211.09116 DOI
- [11]
- M. Kudra et al., “Robust Preparation of Wigner-Negative States with Optimized SNAP-Displacement Sequences”, PRX Quantum 3, (2022) arXiv:2111.07965 DOI
- [12]
- V. G. Matsos et al., “Robust and Deterministic Preparation of Bosonic Logical States in a Trapped Ion”, (2023) arXiv:2310.15546
- [13]
- 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
- [14]
- D. Layden et al., “Ancilla-Free Quantum Error Correction Codes for Quantum Metrology”, Physical Review Letters 122, (2019) arXiv:1811.01450 DOI
- [15]
- T. Holstein and H. Primakoff, “Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet”, Physical Review 58, 1098 (1940) DOI
- [16]
- C. D. Cushen and R. L. Hudson, “A quantum-mechanical central limit theorem”, Journal of Applied Probability 8, 454 (1971) DOI
- [17]
- Linshu Li, private communication, 2018

## Page edit log

- Victor V. Albert (2022-07-03) — most recent
- Thomas Wrona (2022-05-18)
- Victor V. Albert (2021-12-30)
- Joseph T. Iosue (2021-12-19)

## Cite as:

“Binomial code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/binomial