Number-phase code[1] 

Also known as Single-mode translationally invariant Fock-state code.


Bosonic rotation code consisting of superpositions of Pegg-Barnett phase states [2], \begin{align} |\phi\rangle \equiv \frac{1}{\sqrt{2\pi}}\sum_{n=0}^{\infty} \mathrm{e}^{\mathrm{i} n \phi} \ket{n}. \tag*{(1)}\end{align} Since phase states and thus the ideal codewords are not normalizable, approximate versions need to be constructed. The codes' key feature is that, in the ideal case, phase measurement has zero uncertainty, making it a good canditate for a syndrome measurement.

Logical states of an order-\(N\) number-phase qubit encoding are \(|\overline{0}\rangle= \sum_{m=0}^{2N-1} |\phi = m\pi/N\rangle\) and \(|\overline{1}\rangle = \sum_{m=0}^{2N-1} (-1)^m |\phi=m\pi/N\rangle\). By performing the summation over \(m\), one finds that \(|\overline{0}\rangle\) is supported on Fock states \(|2kN\rangle\), while \(|\overline{1}\rangle\) is supported on states \(|(2k+1)N\rangle\), for \(k \geq 0\).


Number-phase codes of order \(N\) detect up to \(N\) photon loss or gain errors, and dephasing up to \(\theta = \pi/N\). However, the code is only approximately error-correcting due to the non-orthogonality of Pegg-Barnett phase states [2], which act as the angular position states in the number-phase interpretation of the oscillator.


Modular phase measurement done in the logical \(X\), or dual, basis has zero uncertainty in the case of ideal number phase codes. This is equivalent to a quantum measurement of the spectrum of the Susskind–Glogower phase operator. Approximate number-phase codes are characterized by vanishing phase uncertainty. Such measurements can be utilized for Knill error correction (a.k.a. telecorrection [3]), which is based on teleportation [4,5]. This type of error correction avoids the complicated correction procedures typical in Fock-state codes, but requires a supply of clean codewords [1]. Performance of this method was analyzed in Ref. [6].Number measurement can be done by extracting modular number information using a CROT gate \(\mathrm{e}^{(2\pi \mathrm{i} / NM) \hat n \otimes \hat n}\) and performing phase measurements [7,8] on an ancillary mode. See Section 4.B.1 of Ref. [1].

Fault Tolerance

Fault-tolerant computation schemes with number-phase codes have been proposed based on concatenation with Bacon-Shor subsystem codes [1].


  • Bosonic rotation code — Number-phase codes are bosonic rotation codes with the primitive state is a Pegg-Barnett phase state [2].


  • Rotor GKP code — Number-phase codes are a manifestation of planar-rotor GKP codes in an oscillator. Both codes protect against small shifts in angular degrees of freedom.
  • Bosonic stabilizer code — Number-phase codewords span the joint right eigenspace of the \(N\)th power of the Susskind–Glogower phase operator and the bosonic rotation operator [1]. These operators no longer form a group since the phase operator is not unitary.
  • Square-lattice GKP code — Square-lattice GKP codes utilize translational symmetry in phase space, while number-phase codes utilize rotational symmetry. The two are related via a mapping [9].
  • Cat 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 [1].
  • Binomial code — In the limit as \(N,S \to \infty\), phase measurement in the binomial code has vanishing variance, just like in a number-phase code [1].
  • Homological number-phase code — Homological number-phase codes and number-phase codes are both manifestations of certain rotor codes, namely, the homological rotor codes and rotor GKP codes, respectively.


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
S. M. Barnett and D. T. Pegg, “Phase in quantum optics”, Journal of Physics A: Mathematical and General 19, 3849 (1986) DOI
C. M. Dawson, H. L. Haselgrove, and M. A. Nielsen, “Noise thresholds for optical cluster-state quantum computation”, Physical Review A 73, (2006) arXiv:quant-ph/0601066 DOI
E. Knill, “Quantum computing with realistically noisy devices”, Nature 434, 39 (2005) arXiv:quant-ph/0410199 DOI
E. Knill, “Scalable Quantum Computation in the Presence of Large Detected-Error Rates”, (2004) arXiv:quant-ph/0312190
T. Hillmann et al., “Performance of Teleportation-Based Error-Correction Circuits for Bosonic Codes with Noisy Measurements”, PRX Quantum 3, (2022) arXiv:2108.01009 DOI
Carl W. Helstrom. Quantum Detection and Estimation Theory. Elsevier, 1976.
A. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Edizioni della Normale, 2011) DOI
A. D. C. Tosta, T. O. Maciel, and L. Aolita, “Grand Unification of continuous-variable codes”, (2022) arXiv:2206.01751
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Zoo Code ID: number_phase

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“Number-phase code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_number_phase, title={Number-phase code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Number-phase code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.