Description
Rotation-symmetric bosonic Fock-state code encoding a \(q\)-dimensional qudit into one oscillator. 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. An alternative basis, valid for for general \(q\) and \(\alpha\neq 0\), consists of \(q\) coherent states distributed equidistanctly around a circle in phase space of radius \(\alpha\).
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. Two-legged cat codes (\(S=0\)) do not protect against loss events, but there exist modifications based on sign alternation [3] or squeezing [4] that add such protection.
Encoding
Lindbladian-based dissipative encoding [5] utilizing multi-photon absorption [6][7][8][9]. Encoding passively protects against cavity dephasing, suppressing dephasing noise exponentially with \(\alpha^2\) [5].Hamiltonian-based 'Kerr-cat' encoding utilizing the Kerr effect [10] (see also Ref. [11]).Combined dissipative and Hamiltonian-based encoding utilizing two-photon exchange for \(S=0\) codes [12].
Gates
For \(S=0\), universal gates in the quantum optical setting can be performed using teleportation, Bell measurements, displacements, and rotations [13].For \(S=0\), universal gates in the microwave setting can be performed using displacement operators and a rotation based on the Kerr nonlinearity [5]. For \(S=1\), squeezing replaces displacements.Holonomic gates utilizing the Berry phase of coherent states are universal [14].Bias-preserving Hamiltonian-based CNOT gate utilizing an \(X\) gate with a topological Berry phase [15][16].
Decoding
Measurement in the Fock basis. For a \(2(S+1)\) cat code, a number measurement returns outcome \(2(S+1)k\), if \(k\) is even, then it corresponds to logical 0 state; if \(k\) is odd, then it corresponds to logical 1 state.
Fault Tolerance
Bias-preserving Hamiltonian-based CNOT gate [16] is part of a universal bias-preserving gate set that can be made fault tolerant using concatenation [15][16].Ancilla qubits encoded in cat codes yield fault-tolerant syndrome extraction circuits [17].
Realizations
Two-legged (\(S=0\)) Lindbladian-based [18][19] and Hamiltonian-based 'Kerr-cat' encoding [20] has been achieved in superconducting circuit devices by the Devoret group; Ref. [19] also demonstrated a displacement-based gate. The Lindbladian-based scheme has further achieved a suppression of bit-flip errors that is exponential in the average photon number [21] up to a bit-flip time of 1ms. A bit-flip time of 1s has been achieved in a similar system in the classical bit regime [22].Four-legged (\(S=1\)) cat code has been realized in a superconducting circuit device [23]. This paper is the first to reach break-even error-correction, where the lifetime of a logical qubit is on par with the lifetime of the noisiest constituent of the system.Approximate version of the \(S=0\) code realized in a superconducting circuit device by the Wang group [24].
Notes
Pedagogical introduction to cat codes in the context of microwave cavities can be found in Refs. [25][26], and in the context of optical systems in books [27][28].
Parent
- Bosonic rotation code — The cat code is a bosonic rotation code whose primitive state is the coherent state \(|\alpha\rangle\) [29].
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 [29].
- Hamiltonian-based code — Two-legged cat codewords form ground-state subspace of a Kerr Hamiltonian [10].
- Binomial code — For a fixed \(S\), binomial codes with \(N \to \infty\) coincide with cat codes as \(\alpha \to \infty\) [30].
- 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.
References
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- R. Lescanne et al., “Exponential suppression of bit-flips in a qubit encoded in an oscillator”, Nature Physics 16, 509 (2020). DOI; 1907.11729
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- C. Berdou et al., “One hundred second bit-flip time in a two-photon dissipative oscillator”. 2204.09128
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- Nissim Ofek et al., “Demonstrating Quantum Error Correction that Extends the Lifetime of Quantum Information”. 1602.04768
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Page edit log
- Victor V. Albert (2022-07-03) — most recent
- Alexander Grimm (2022-07-03)
- Victor V. Albert (2022-01-11)
- Joseph T. Iosue (2021-12-19)
- Yijia Xu (2021-12-14)
Zoo code information
Cite as:
“Cat code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/cat