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 \(\{|2\rangle,|6\rangle,|10\rangle,\cdots\}\).
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 at least one Fock state \(|(k_j q+j)N\rangle\) for some \(k_j \in \mathbb{N}_0\). The logical codewords are then obtained by projecting \(|\Theta\rangle\) onto the \(q\) eigenspaces of the discrete rotation operator, equivalently by summing the \(qN\) rotated copies of \(|\Theta\rangle\) with appropriate phases.
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 Susskind–Glogower phase 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
For qubit rotation codes, one can distinguish the computational-basis codewords destructively by performing a Fock-state number measurement. If a Fock state \(|n\rangle\) is measured, then one rounds to the nearest multiple of \(N\) and infers the logical value from the parity of that multiple [1].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 QEC for \(S=1\) codes [2].Decoder [1] based on measuring in the phase-state basis and using Knill error correction (a.k.a. telecorrection [3]), which is based on teleportation [4,5].Performance under non-Markovian noise has been investigated [6].Fault Tolerance
Decoder based on measuring in the phase-state basis and using Knill error correction [1] is fault-tolerant under circuit-level noise [7].Cousin
- Random quantum code— Random bosonic rotation codes can outperform cat and binomial codes when loss rate is large relative to dephasing rate [8].
Primary Hierarchy
References
- [1]
- 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
- [2]
- S. Kwon, S. Watabe, and J.-S. Tsai, “Autonomous quantum error correction in a four-photon Kerr parametric oscillator”, npj Quantum Information 8, (2022) arXiv:2203.09234 DOI
- [3]
- C. M. Dawson, H. L. Haselgrove, and M. A. Nielsen, “Noise thresholds for optical cluster-state quantum computation”, Physical Review A 73, (2006) arXiv:quant-ph/0601066 DOI
- [4]
- E. Knill, “Quantum computing with realistically noisy devices”, Nature 434, 39 (2005) arXiv:quant-ph/0410199 DOI
- [5]
- E. Knill, “Scalable Quantum Computation in the Presence of Large Detected-Error Rates”, (2004) arXiv:quant-ph/0312190
- [6]
- A. Udupa, T. Hillmann, R. G. Ahmed, A. Smirne, and G. Ferrini, “Performance of rotation-symmetric bosonic codes in the presence of random telegraph noise”, (2025) arXiv:2505.08670
- [7]
- L. D. H. My, S. Qin, and H. K. Ng, “Circuit-level fault tolerance of cat codes”, Quantum 9, 1810 (2025) arXiv:2406.04157 DOI
- [8]
- S. Totey, A. Kyle, S. Liu, P. J. Barge, N. Lordi, and J. Combes, “The performance of random bosonic rotation codes”, (2023) arXiv:2311.16089
- [9]
- S. M. Barnett and D. T. Pegg, “Phase in quantum optics”, Journal of Physics A: Mathematical and General 19, 3849 (1986) DOI
Page edit log
- Victor V. Albert (2021-12-30) — most recent
- Joseph T. Iosue (2021-12-19)
Cite as:
“Bosonic rotation code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/bosonic_rotation