Cat code[1] 

Also known as Superposition of coherent states (SCS).

Description

Rotation-symmetric bosonic Fock-state code encoding a \(q\)-dimensional qudit into one oscillator which utilizes a constellation of \(q(S+1)\) coherent states distributed equidistantly around a circle in phase space of radius \(\alpha\).

Codewords for a qubit code (\(q=2\)) consist of a coherent state \(|\alpha\rangle\) projected onto a subspace of Fock state number modulo \(2(S+1)\). The logical state \(|\overline{0}\rangle\) is in the \(\{|0\rangle , |2(S+1)\rangle , |4(S+1)\rangle \cdots \}\) Fock-state subspace, while \(|\overline{1}\rangle\) is in the \(\{|(S+1)\rangle, |3(S+1)\rangle , |5(S+1)\rangle , |7(S+1)\rangle \cdots \}\) subspace. These projected coherent states make up generalized cat states [2,3].

Protection

Due to the spacing between sets of Fock states, the distance between two distinct logical states is \(d=S+1\). Hence, this code is able to detect \(S\)-photon loss error.

Encoding

Lindbladian-based dissipative encoding utilizing multi-photon generalization of two-photon absorption [4–7]. Encoding passively protects against modal dephasing, suppressing dephasing noise exponentially with \(|\alpha|^2\) [8].Approximate can states can be prepared using Gaussian operations and photon detectors [9].

Gates

Holonomic gates utilizing the Berry phase of coherent states are universal [10].Universal gates in the microwave setting for the \(S=1\) code can be performed using squeezing operators and a rotation based on the Kerr nonlinearity [8].Error-detecting \(CCZ\) and \(cSWAP\) gates for four-component cat code using three-level ancilla [11].Universal set of error-corrected operations tolerating a single photon loss and an arbitrary ancilla fault [12].

Decoding

Measuring the Fock-state number modulo \(2S\) can be used to determine if photon loss or excitation errors occurred. For \(S=1\), this is the occupation number parity.

Fault Tolerance

Universal set of error-corrected operations tolerating a single photon loss and an arbitrary ancilla fault [12].Linear-optical noise suppression and mitigation scheme [13].

Realizations

Parity-syndrome measurement tested [14] and implemented for the four-component (\(S=1\)) cat code [15] in a microwave cavity coupled to a superconducting circuit. The latter work [15] is the first to reach break-even error-correction, where the lifetime of a logical qubit is on par with the cavity lifetime, despite protection against dephasing not being implemented. A fault-tolerant version of parity measurement has also been realized [16].

Parents

Child

Cousins

  • Number-phase code — In the limit as \(N,S \to \infty\), phase measurement in the cat code has vanishing variance, just like in a number-phase code [17]. Conversely, a cat code can be thought of as an appropriately regularized number-phase code.
  • Phase-shift keying (PSK) code — PSK (cat) codes are used to transmit classical (quantum) information using (superpositions of) single-mode coherent states distributed on a circle over classical (quantum) channels.
  • Polygon code — The \(q(S+1)\)-component cat coherent-state constellation forms the vertices of a \(q(S+1)\)-gon.
  • PSK c-q code — PSK c-q (cat) codes are used to transmit classical (quantum) information using (superpositions of) single-mode coherent states distributed on a circle over quantum channels.
  • Pair-cat code — Cat (pair-cat) codewords are superpositions of coherent (pair-coherent) states. Many cat-code protocols have analogues for the two-mode pair-cat codes.
  • Binomial code — For a fixed \(S\), binomial codes with \(N \to \infty\) coincide with cat codes as \(\alpha \to \infty\) [19].
  • Hybrid cat code — Hybrid cat codewords consist of a bosonic mode in either coherent or cat states.

References

[1]
Z. Leghtas, G. Kirchmair, B. Vlastakis, R. J. Schoelkopf, M. H. Devoret, and M. Mirrahimi, “Hardware-Efficient Autonomous Quantum Memory Protection”, Physical Review Letters 111, (2013) arXiv:1207.0679 DOI
[2]
V. V. Dodonov, I. A. Malkin, and V. I. Man’ko, “Even and odd coherent states and excitations of a singular oscillator”, Physica 72, 597 (1974) DOI
[3]
O. Castaños, R. López-Peña, and V. I. Man’ko, “Crystallized schrödinger cat states”, Journal of Russian Laser Research 16, 477 (1995) DOI
[4]
M. Wolinsky and H. J. Carmichael, “Quantum noise in the parametric oscillator: From squeezed states to coherent-state superpositions”, Physical Review Letters 60, 1836 (1988) DOI
[5]
L. Krippner, W. J. Munro, and M. D. Reid, “Transient macroscopic quantum superposition states in degenerate parametric oscillation: Calculations in the large-quantum-noise limit using the positivePrepresentation”, Physical Review A 50, 4330 (1994) DOI
[6]
E. E. Hach III and C. C. Gerry, “Generation of mixtures of Schrödinger-cat states from a competitive two-photon process”, Physical Review A 49, 490 (1994) DOI
[7]
L. Gilles, B. M. Garraway, and P. L. Knight, “Generation of nonclassical light by dissipative two-photon processes”, Physical Review A 49, 2785 (1994) DOI
[8]
M. Mirrahimi, Z. Leghtas, V. V. Albert, S. Touzard, R. J. Schoelkopf, L. Jiang, and M. H. Devoret, “Dynamically protected cat-qubits: a new paradigm for universal quantum computation”, New Journal of Physics 16, 045014 (2014) arXiv:1312.2017 DOI
[9]
D. Su, C. R. Myers, and K. K. Sabapathy, “Conversion of Gaussian states to non-Gaussian states using photon-number-resolving detectors”, Physical Review A 100, (2019) arXiv:1902.02323 DOI
[10]
V. V. Albert, C. Shu, S. Krastanov, C. Shen, R.-B. Liu, Z.-B. Yang, R. J. Schoelkopf, M. Mirrahimi, M. H. Devoret, and L. Jiang, “Holonomic Quantum Control with Continuous Variable Systems”, Physical Review Letters 116, (2016) arXiv:1503.00194 DOI
[11]
T. Tsunoda, J. D. Teoh, W. D. Kalfus, S. J. de Graaf, B. J. Chapman, J. C. Curtis, N. Thakur, S. M. Girvin, and R. J. Schoelkopf, “Error-detectable bosonic entangling gates with a noisy ancilla”, (2022) arXiv:2212.11196
[12]
Q. Xu, P. Zeng, D. Xu, and L. Jiang, “Fault-Tolerant Operation of Bosonic Qubits with Discrete-Variable Ancillae”, (2023) arXiv:2310.20578
[13]
Y. S. Teo, S. U. Shringarpure, S. Cho, and H. Jeong, “Linear-optical protocols for mitigating and suppressing noise in bosonic systems”, (2024) arXiv:2411.11313
[14]
L. Sun et al., “Tracking photon jumps with repeated quantum non-demolition parity measurements”, Nature 511, 444 (2014) arXiv:1311.2534 DOI
[15]
N. Ofek et al., “Demonstrating Quantum Error Correction that Extends the Lifetime of Quantum Information”, (2016) arXiv:1602.04768
[16]
S. Rosenblum, P. Reinhold, M. Mirrahimi, L. Jiang, L. Frunzio, and R. J. Schoelkopf, “Fault-tolerant detection of a quantum error”, Science 361, 266 (2018) arXiv:1803.00102 DOI
[17]
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
[18]
A. Denys and A. Leverrier, “Quantum error-correcting codes with a covariant encoding”, (2024) arXiv:2306.11621
[19]
M. H. Michael, M. Silveri, R. T. Brierley, V. V. Albert, J. Salmilehto, L. Jiang, and S. M. Girvin, “New Class of Quantum Error-Correcting Codes for a Bosonic Mode”, Physical Review X 6, (2016) arXiv:1602.00008 DOI
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Zoo Code ID: cat

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/oscillators/qsc/cat/cat.yml.