Description
An oscillator-into-oscillator stabilizer code encoding \(k\) logical modes into \(n\) physical modes. An \(((n,k,d))_{\mathbb{R}}\) analog stabilizer code is denoted as \([[n,k,d]]_{\mathbb{R}}\), where \(d\) is the code's distance. Any analog stabilizer state can be thought of as a pure Gaussian state that has been infinitely squeezed on all modes [2].
Analog stabilizer codes admit continuous stabilizer group of displacements. This group can equivalently be defined in terms of its Lie algebra. The codespace is equivalently the common \(0\)-eigenvalue eigenspace of the Lie algebra generators, which are mutually commuting linear combinations of oscillator position and momentum operators called nullifiers [3] or annihilators. An analog stabilizer code admitting a set of nullifiers such that each nullifier consists of either position or momentum operators is called an analog CSS code.
Protection
Protects against erasures of or any errors on at most \(d-1\) modes. If an error operator does not commute with a nullifier, then that error is detectable. There are conditions on the encoding circuit which guarantee that the code can correct errors [4]. The code can be further optimized to increase resolution between syndrome spaces for certain noise [4].
Protection of logical modes against small displacements or other errors acting in every physical mode cannot be done using only Gaussian resources [5,6] (see also [7,8]). There are no such restrictions for non-Gaussian noise [4,9].
Encoding
Gaussian circuit applied to \(k\) modes storing logical information and \(n-k\) modes initialized in a position state [6].Decoding
Homodyne measurement of nullifiers yields real-valued syndromes, and recovery can be performed by displacements conditional on the syndromes.Realizations
One-sided device-independent QKD [10].Cousins
- Oscillator-into-oscillator GKP code— Analog stabilizer codes protect logical modes against arbitrarily large displacements on a few modes, while oscillator-into-oscillator GKP codes protect a finite-dimensional logical space against sufficiently small displacements in any number of modes. Encoding in analog-stabilizer (oscillator-into-oscillator GKP) codes can be done by a Gaussian operation acting on a tensor product of an arbitrary state in the first mode and position states (GKP states) on the remaining modes. Protection of logical modes against small displacements cannot be done using only Gaussian resources [5–7], so oscillator-into-oscillator GKP codes can be thought of as analog stabilizer encodings utilizing non-Gaussian GKP resource states.
- Modular-qudit stabilizer code— Prime-qudit stabilizer codes can be converted into analog stabilizer codes whose distance is at least as large as that of the original code [11].
- Galois-qudit stabilizer code— Prime-qudit stabilizer codes can be converted into analog stabilizer codes whose distance is at least as large as that of the original code [11].
- \([[8, 3, 3]]\) Eight-qubit Gottesman code— The eight-qubit Gottesman code has been extended to an analog stabilizer code [1].
- Perfect-tensor code— Analog stabilizer states are generically CV AME [12], and explicit constructions exist for any number of modes [2]. The codespace of an analog stabilizer code with pure distance \(d_{\textnormal{pure}}\) is a \((d_{\textnormal{pure}}-1)\)-uniform space [2]. Normalizable finitely squeezed versions of infinitely squeezed Gaussian states are locally thermal, up to corrections in the squeezing parameter [13,14].
- Real block code— Analog stabilizer codes are quantum versions of real block codes.
- EA analog stabilizer code— EA analog stabilizer codes utilize additional ancillary modes in a pre-shared entangled state, but reduce to ordinary analog stabilizer codes when said modes are interpreted as noiseless physical modes.
- Qudit cubic code— The qudit cubic code can be generalized to oscillators [15].
Primary Hierarchy
References
- [1]
- R. L. Barnes, “Stabilizer Codes for Continuous-variable Quantum Error Correction”, (2004) arXiv:quant-ph/0405064
- [2]
- J. I. Kwon, A. J. Brady, and V. V. Albert, “Most continuous-variable cluster states are too entangled to be useless”, (2025) arXiv:2503.15698
- [3]
- M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters”, Physical Review A 79, (2009) arXiv:0903.3233 DOI
- [4]
- T. A. Walker and S. L. Braunstein, “Five-wave-packet linear optics quantum-error-correcting code”, Physical Review A 81, (2010) DOI
- [5]
- J. Niset, J. Fiurášek, and N. J. Cerf, “No-Go Theorem for Gaussian Quantum Error Correction”, Physical Review Letters 102, (2009) arXiv:0811.3128 DOI
- [6]
- C. Vuillot, H. Asasi, Y. Wang, L. P. Pryadko, and B. M. Terhal, “Quantum error correction with the toric Gottesman-Kitaev-Preskill code”, Physical Review A 99, (2019) arXiv:1810.00047 DOI
- [7]
- J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian States with Gaussian Operations is Impossible”, Physical Review Letters 89, (2002) arXiv:quant-ph/0204052 DOI
- [8]
- G. Giedke and J. Ignacio Cirac, “Characterization of Gaussian operations and distillation of Gaussian states”, Physical Review A 66, (2002) arXiv:quant-ph/0204085 DOI
- [9]
- P. van Loock, “A note on quantum error correction with continuous variables”, (2008) arXiv:0811.3616
- [10]
- E. Culf, T. Vidick, and V. V. Albert, “Group coset monogamy games and an application to device-independent continuous-variable QKD”, (2022) arXiv:2212.03935
- [11]
- L. G. Gunderman, “Stabilizer Codes with Exotic Local-dimensions”, Quantum 8, 1249 (2024) arXiv:2303.17000 DOI
- [12]
- J. Zhang, G. Adesso, C. Xie, and K. Peng, “Quantum Teamwork for Unconditional Multiparty Communication with Gaussian States”, Physical Review Letters 103, (2009) arXiv:0901.1488 DOI
- [13]
- P. Facchi, G. Florio, G. Parisi, and S. Pascazio, “Maximally multipartite entangled states”, Physical Review A 77, (2008) arXiv:0710.2868 DOI
- [14]
- P. Facchi, G. Florio, C. Lupo, S. Mancini, and S. Pascazio, “Gaussian maximally multipartite-entangled states”, Physical Review A 80, (2009) arXiv:0908.0114 DOI
- [15]
- V. V. Albert, S. Pascazio, and M. H. Devoret, “General phase spaces: from discrete variables to rotor and continuum limits”, Journal of Physics A: Mathematical and Theoretical 50, 504002 (2017) arXiv:1709.04460 DOI
- [16]
- N. C. Menicucci, S. T. Flammia, and P. van Loock, “Graphical calculus for Gaussian pure states”, Physical Review A 83, (2011) arXiv:1007.0725 DOI
- [17]
- C. González-Arciniegas, P. Nussenzveig, M. Martinelli, and O. Pfister, “Cluster States from Gaussian States: Essential Diagnostic Tools for Continuous-Variable One-Way Quantum Computing”, PRX Quantum 2, (2021) arXiv:1912.06463 DOI
Page edit log
- Victor V. Albert (2022-07-05) — most recent
Cite as:
“Analog stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/analog_stabilizer