Quantum lattice code 

Description

Bosonic stabilizer code on \(n\) bosonic modes whose stabilizer group is an infinite countable group of oscillator displacement operators which implement lattice translations in phase space.

Displacement operators on \(n\) modes can be written as \begin{align} D(\xi) = \exp \left\{-i \sqrt{2\pi} {\xi}^\mathrm{T} J \hat{q} \right\} , \quad \xi \in \mathbb{R}^{2n}~, \tag*{(1)}\end{align} where \(\hat{q}\) is a \(2n\)-dimensional vector position and momentum operators of the modes, the symplectic form \begin{align} J = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \otimes I_n = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}~, \tag*{(2)}\end{align} and \(I_n\) is the identity matrix. A group generated by a set of independent displacement operators is given by a lattice \({\mathcal{L}}\) \begin{align} \langle D(\xi_1) ,\dots, D(\xi_{m}) \rangle = \{ e^{ i \phi_M (\xi) } D(\xi) ~\vert~ \xi \in {\mathcal{L}} \} \tag*{(3)}\end{align} and becomes a valid stabilizer group when every symplectic inner product between lattice vectors yields an integer. In other words, the corresponding lattice is symplectically integral, corresponding to an integer-valued symplectic Gram matrix \(A\), \begin{align} A_{ij}={\xi}^T_i J \xi_j \in \mathbb{Z}~. \tag*{(4)}\end{align} The \(m=2n\) case yields multimode GKP codes encoding a finite-dimensional logical subspace, while removing some displacements yields oscillator-into-oscillator GKP codes encoding an infinite-dimensional logical subspace. Codes defined on a hyper-rectangular lattice are CSS GKP codes, and more general lattices, obtained by Gaussian transformations, yield non-CSS codes.

Notes

Quantum lattice states have been introduced in quantum foundations research defining modular conjugate variables [1] and in coherent-state theory associated with the Heisenberg-Weyl group [2,3][4; Sec. 1.5 and 3.2]. They are featured in the proof of hardness of LWE [5; pg. 12]. The basis formed by quantum lattice states is known as the Zak basis, Weil-Brezin transform, or \(kq\) representation in condensed-matter physics [6] and signal processing [7; Ch. 1][8; Eq. (1.112)].

Parents

  • Bosonic stabilizer code — Quantum lattice codes are bosonic stabilizer codes with a countably infinite stabilizer group, corresponding to modular constraints on positions and momenta.
  • Coherent-state constellation code — Quantum lattice codewords can be written as superpositions of coherent states lying on a lattice in phase space [9,10].

Children

  • Oscillator-into-oscillator GKP code — Oscillator-into-oscillator GKP codes are \(n\)-mode quantum lattice codes with less than \(2n\) stabilizers, i.e., constructed using a degenerate lattice (see Appx. A of Ref. [11]).
  • Gottesman-Kitaev-Preskill (GKP) code — GKP codes are \(n\)-mode quantum lattice codes with \(2n\) stabilizers, i.e., constructed using a non-degenerate lattice.

Cousins

  • Lattice-based code — Quantum lattice codes can be thought of as quantum lattice codes because they store information in quantum superpositions of points on a lattice in quantum phase space.
  • Calderbank-Shor-Steane (CSS) stabilizer code — Quantum lattice codes defined on rectangular lattices are CSS codes. There is no known relation to chain complexes for such codes. More general lattices, obtained from rectangular lattices by Gaussian transformations, yield non-CSS codes.

References

[1]
Y. Aharonov, H. Pendleton, and A. Petersen, “Modular variables in quantum theory”, International Journal of Theoretical Physics 2, 213 (1969) DOI
[2]
Cartier, Pierre. "Quantum mechanical commutation relations and theta functions." Proc. Sympos. Pure Math. Vol. 9. 1966.
[3]
A. M. Perelomov, “Coherent states and theta functions”, Functional Analysis and Its Applications 6, 292 (1973) DOI
[4]
A. Perelomov, Generalized Coherent States and Their Applications (Springer Berlin Heidelberg, 1986) DOI
[5]
O. Regev, “On lattices, learning with errors, random linear codes, and cryptography”, Journal of the ACM 56, 1 (2009) DOI
[6]
J. Zak, “Finite Translations in Solid-State Physics”, Physical Review Letters 19, 1385 (1967) DOI
[7]
H. G. Feichtinger and T. Strohmer, editors , Gabor Analysis and Algorithms (Birkhäuser Boston, 1998) DOI
[8]
G. B. Folland, Harmonic Analysis in Phase Space. (AM-122) (Princeton University Press, 1989) DOI
[9]
D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator”, Physical Review A 64, (2001) arXiv:quant-ph/0008040 DOI
[10]
V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018) arXiv:1708.05010 DOI
[11]
J. Conrad, J. Eisert, and F. Arzani, “Gottesman-Kitaev-Preskill codes: A lattice perspective”, Quantum 6, 648 (2022) arXiv:2109.14645 DOI
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Zoo Code ID: quantum_lattice

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/oscillators/stabilizer/lattice/quantum_lattice.yml.