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].
Realizations
Parity-syndrome measurement tested [13] and implemented for the four-component (\(S=1\)) cat code [14] in a microwave cavity coupled to a superconducting circuit. The latter work [14] 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 [15].
Parents
- Quantum spherical code (QSC) — Cat codes are QSCs on the one-dimensional complex sphere.
- Bosonic rotation code — The cat code is a bosonic rotation code whose primitive state is the coherent state \(|\alpha\rangle\) [16].
- Group-representation code — Cat codes are group representation codes with \(G\) being a cyclic group [17].
Child
- Two-component cat code — The cat code reduces to its two-component version for \(S=0\).
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 [16].
- 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 \(\alpha \to \infty\) [18].
- Hybrid cat code — Hybrid cat codewords consist of a bosonic mode in either coherent or cat states.
References
- [1]
- Z. Leghtas et al., “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 et al., “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 et al., “Holonomic Quantum Control with Continuous Variable Systems”, Physical Review Letters 116, (2016) arXiv:1503.00194 DOI
- [11]
- T. Tsunoda et al., “Error-detectable bosonic entangling gates with a noisy ancilla”, (2022) arXiv:2212.11196
- [12]
- Q. Xu et al., “Fault-Tolerant Operation of Bosonic Qubits with Discrete-Variable Ancillae”, (2023) arXiv:2310.20578
- [13]
- L. Sun et al., “Tracking photon jumps with repeated quantum non-demolition parity measurements”, Nature 511, 444 (2014) arXiv:1311.2534 DOI
- [14]
- N. Ofek et al., “Demonstrating Quantum Error Correction that Extends the Lifetime of Quantum Information”, (2016) arXiv:1602.04768
- [15]
- S. Rosenblum et al., “Fault-tolerant detection of a quantum error”, Science 361, 266 (2018) arXiv:1803.00102 DOI
- [16]
- 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
- [17]
- A. Denys and A. Leverrier, “Quantum error-correcting codes with a covariant encoding”, (2024) arXiv:2306.11621
- [18]
- 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 (2022-11-06) — most recent
- Victor V. Albert (2022-07-03)
- Alexander Grimm (2022-07-03)
- Victor V. Albert (2022-01-11)
- Joseph T. Iosue (2021-12-19)
- Yijia Xu (2021-12-14)
Cite as:
“Cat code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/cat