Chuang-Leung-Yamamoto (CLY) code

Chuang-Leung-Yamamoto (CLY) code[1]

Description

Bosonic Fock-state code that encodes \(k\) qubits into \(n\) oscillators, with each oscillator restricted to having at most \(N\) excitations. Codewords are superpositions of oscillator Fock states which have exactly \(N\) total excitations, and are either uniform (i.e., balanced) superpositions or unbalanced superpositions. Codes can be denoted as \([[N,n,2^k,d]]\), which conflicts with stabilizer code notation.

A simple example of a CLY code is a two-mode "0-2-4" qubit code with codewords \begin{align} \begin{split} |\overline{0}\rangle&=\frac{1}{\sqrt{2}}\left(|40\rangle+|04\rangle\right)\\ |\overline{1}\rangle&=|22\rangle~. \end{split} \tag*{(1)}\end{align}

Protection

Protects against amplitude damping for up to \(t = d-1\) excitation losses. Defining the spacing between two Fock states \(|u_1\cdots u_n\rangle\) and \(|v_1\cdots v_n\rangle\), \begin{align} \text{Spacing}(u,v) = \frac{1}{2}\sum_{i=1}^n |u_i - v_i|, \tag*{(2)}\end{align} the code distance \(d\) can be defined as the minimial spacing between Fock states making up the codewords.

Rate

Code rate is \(\frac{k}{n \log_2(N+1)}\). To correct the loss of up to \(t\) excitations with \(K+1\) codewords, a code exists with scaling \(N \sim t^3 K/2\).

Encoding

Photon Fock state input into a network of beamsplitters, phase shifters, and Kerr media. These operations all preserve total photon number. Beamsplitters and phase shifters take annihilation operators to linear combinations of annihilation operators, and the transformation matrix is unitary. The operations corresponding to Kerr nonlinear media are diagonal in the Fock basis, but they implement phases that in general depend nonlinearly on the number of photons in each mode. State preparation may require ancillary modes and be conditioned on photon-number measurement results.

Decoding

Destructive decoding with a photon number measurement on each mode.State can be decoded with a network of beamsplitters, phase shifters, and Kerr media.

Parents

Fock-state bosonic code — Chuang-Leung-Yamamoto code are multi-mode Fock-state codes.
Constant-excitation (CE) code — Chuang-Leung-Yamamoto codewords are constructed out of Fock states with the same total excitation number.

Child

One-hot quantum code

Cousins

2T-qutrit code — The \(2T\)-qutrit code reduces to the two-mode "0-2-4" CLY code as \(\alpha\to 0\).
Binomial code — Two-mode version of binomial codes correspond to two-mode "0-2-4" CLY codes (see Sec. IV.A of Ref. [2]).

References

[1]: I. L. Chuang, D. W. Leung, and Y. Yamamoto, “Bosonic quantum codes for amplitude damping”, Physical Review A 56, 1114 (1997) DOI
[2]: 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

Page edit log

Victor V. Albert (2022-03-02) — most recent
Dhruv Devulapalli (2021-12-17)
Jonathan Kunjummen (2021-12-07)

Cite as:

“Chuang-Leung-Yamamoto (CLY) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/chuang-leung-yamamoto

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/oscillators/fock_state/constant_excitation/chuang-leung-yamamoto.yml.