Bosonic rotation code[1]

Description

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 \(\{|2\rangle,|6\rangle,|10\rangle,\cdots\}\).

Encoding of a \(q\)-dimensional logical qudit admit a basis whose elements are eigenstates of the rotation operator \(\exp\left(\mathrm{i} 2\pi \hat{n}/qN \right)\), where \(\hat{n}\) is the number operator diagonal in the Fock basis. Basis elements are of the form \(\sum_{j=0}^\infty c_j |(kq+j)N \rangle\) for some coefficients \(c_j\) and \(k\). This is because the rotation acting on a general Fock-state superposition \(\sum_n a_n |n\rangle\) yields \(\sum_n a_n \exp\left(\mathrm{i} 2\pi n / qN \right) |n\rangle\). In order for a codeword to be an eigenvector of this operation, \(a_n\) must be zero whenever \(n \neq (kq+j)N\) for some \(k\).

Codewords can be uniquely specified by choosing a primitive state \(|\Theta\rangle\). To ensure valid (orthogonal and nonzero) codewords, \(|\Theta\rangle\) must satisfy the following requirement: for each \(j \in \mathbb{Z}_q\), \(|\Theta\rangle\) must have support on the Fock state \(|(k_j q+j)N\rangle\) for at least one \(k_j \in \mathbb{N}_0\). From such a primitive, the codewords \(\{|\overline{j}\rangle \mid j \in \mathbb{Z}_q\}\) are constructed as \begin{align} |\overline{j}\rangle \propto \sum_{m=0}^{2N-1} \mathrm{e}^{2\pi \mathrm{i} m (\hat n / N + j) / q} |\Theta\rangle~. \end{align}

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.

Encoding

The optimal way to prepare codewords depends on the exact rotation code in question [1].

Gates

The logical Pauli-\(Z\) gate can be the discrete rotation operator \(\mathrm{e}^{\mathrm{i} \pi \hat n /N}\), and the logical Pauli-\(X\) gate can be the (note: non-unitary) number-translation operator \(\sum_{n=0}^\infty |n\rangle\bra{n+N}\).For qubit codes, a logical phase gate is \(S = \mathrm{e}^{\pi \mathrm{i} \hat n^2 / 2N^2}\).The \(T = \mathrm{diag}(1,\exp(\mathrm{i}\pi/4))\) gate can be done via gate teleportation and a resource state \(\vert 0_N\rangle + \exp(\mathrm{i}\pi/4) \vert 1_N \rangle\).A controlled-rotation gate between an order \(N\) rotation code and an order \(M\) rotation code is \(\mathrm{CROT}_{NM} = \mathrm{e}^{(2\pi\mathrm{i} / qNM) \hat n \otimes \hat n}\).

Decoding

One can distinguish (destructively) the codewords by performing a Fock-state number measurement. If a Fock state state \(|n\rangle\) is measured, then one rounds to the nearest integer of the form \((kq+j)/N\), and deduces that the true state was \(|\overline{j}\rangle\).One can distinguish states in the dual basis by performing phase estimation on \(\mathrm{e}^{\mathrm{i} \theta \hat n}\). One then rounds the resulting \(\theta\) to the nearest number \(2\pi j / qN\) in order to determine which dual basis state \(j \in \mathbb Z_q\) it came from.Autonomous quantum error correction schemes for \(S=1\) codes [2].

Parent

Children

  • Binomial 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. [1].
  • Cat code — The cat code is a bosonic rotation code whose primitive state is the coherent state \(|\alpha\rangle\) [1].
  • Number-phase code — Number-phase codes are bosonic rotation codes with the primitive state is a Pegg-Barnett phase state [3].

Cousin

Zoo code information

Internal code ID: bosonic_rotation

Your contribution is welcome!

on github.com (edit & pull request)

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Zoo Code ID: bosonic_rotation

Cite as:
“Bosonic rotation code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/bosonic_rotation
BibTeX:
@incollection{eczoo_bosonic_rotation, title={Bosonic rotation code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/bosonic_rotation} }
Permanent link:
https://errorcorrectionzoo.org/c/bosonic_rotation

References

[1]
A. L. Grimsmo, J. Combes, and B. Q. Baragiola, “Quantum Computing with Rotation-Symmetric Bosonic Codes”, Physical Review X 10, (2020). DOI; 1901.08071
[2]
Sangil Kwon, Shohei Watabe, and Jaw-Shen Tsai, “Autonomous Quantum Error Correction in a Four-Photon Kerr Parametric Oscillator”. 2203.09234
[3]
S. M. Barnett and D. T. Pegg, “Phase in quantum optics”, Journal of Physics A: Mathematical and General 19, 3849 (1986). DOI

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/oscillators/fock_state/bosonic_rotation.yml.