Analog stabilizer code[1]
Alternative names: Gaussian stabilizer code, Linear stabilizer code, Symplectic stabilizer code, Wavepacket code.
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.
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 [2] 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
Protect against erasures of at most \(d-1\) modes, or arbitrarily large dispalcements on those modes. If an error operator does not commute with a nullifier, then that error is detectable. Protection of logical modes against small displacements cannot be done using only Gaussian resources [3,4] (see also [5,6]). There are no such restrictions for non-Gaussian noise [7].Encoding
Gaussian circuit applied to \(k\) modes storing logical information and \(n-k\) modes initialized in a position state.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 [8].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 [3–5], 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 [9].
- \([[8, 3, 3]]\) Eight-qubit Gottesman code— The eight-qubit Gottesman code has been extended to an analog stabilizer code [1].
- Real-number block code— Analog stabilizer codes are quantum versions of real-number 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 [10].
Primary Hierarchy
Parents
Analog stabilizer codes are bosonic stabilizer codes with a continuous stabilizer group, corresponding to linear constraints on positions and momenta.
Analog stabilizer code
Children
Analog-cluster-state codes are particular analog stabilizer codes. Relaxing the real weighted adjacency matrix of an analog cluster state to be complex yields a description of a general analog (i.e., Gaussian) stabilizer code state [11]. Pure Gaussian states, which are normalizable approximate versions of analog stabilizer states, are not equivalent to finitely squeezed analog cluster states via Gaussian local unitaries [12].
References
- [1]
- R. L. Barnes, “Stabilizer Codes for Continuous-variable Quantum Error Correction”, (2004) arXiv:quant-ph/0405064
- [2]
- 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
- [3]
- 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
- [4]
- 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
- [5]
- 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
- [6]
- 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
- [7]
- P. van Loock, “A note on quantum error correction with continuous variables”, (2008) arXiv:0811.3616
- [8]
- 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
- [9]
- L. G. Gunderman, “Stabilizer Codes with Exotic Local-dimensions”, Quantum 8, 1249 (2024) arXiv:2303.17000 DOI
- [10]
- 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
- [11]
- 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
- [12]
- 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