# 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]. The \(q\)-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)}.
\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}.
\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

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.

## Decoding

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].

## Realizations

Realized in microwave cavities coupled to superconducting circuits [3].

## Notes

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.

## 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. [4].

## 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 [4].
- Chuang-Leung-Yamamoto code — Two-mode version of binomial codes correspond to two-mode CLY codes (see Sec. IV.A of Ref. [1]).
- GNU permutation-invariant code — Binomial codes and GNU codes are both related to spin-coherent states, and a qudit generalization can be obtained from qudit binomial codes ([2], Appx. C).

## Zoo code information

## References

- [1]
- M. H. Michael et al., “New Class of Quantum Error-Correcting Codes for a Bosonic Mode”, Physical Review X 6, (2016). DOI; 1602.00008
- [2]
- V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018). DOI; 1708.05010
- [3]
- L. Hu et al., “Quantum error correction and universal gate set operation on a binomial bosonic logical qubit”, Nature Physics 15, 503 (2019). DOI
- [4]
- A. L. Grimsmo, J. Combes, and B. Q. Baragiola, “Quantum Computing with Rotation-Symmetric Bosonic Codes”, Physical Review X 10, (2020). DOI; 1901.08071

## Cite as:

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