Here is a list of constant-excitation (CE) quantum codes.
| Code | Description |
|---|---|
| Chuang-Leung-Yamamoto (CLY) code | Bosonic Fock-state code that encodes \(k\) qubits into \(n\) oscillators, with each oscillator restricted to having at most \(N\) excitations. Codewords are superpositions of oscillator Fock states which have exactly \(N\) total excitations, and are either uniform (i.e., balanced) superpositions or unbalanced superpositions. |
| Constant-excitation (CE) code | Code whose codewords lie in an eigenspace of fixed total energy or fixed total excitation number for the underlying quantum system. For qubit codes, such a Hamiltonian is often the total spin Hamiltonian, \(H=\sum_i Z_i\). For spin-\(S\) codes, this generalizes to \(H=\sum_i J_z^{(i)}\), where \(J_z\) is the spin-\(S\) \(Z\)-operator. For bosonic (and, similarly, for fermion) codes, such as Fock-state codes, codewords are often in an eigenspace with eigenvalue \(N>0\) of the total excitation or energy Hamiltonian, \(H=\sum_i \hat{n}_i\). |
| Dual-rail quantum code | Two-mode bosonic code encoding a logical qubit in Fock states with one excitation. The logical-zero state is represented by \(|10\rangle\), while the logical-one state is represented by \(|01\rangle\). This encoding is often realized in temporal or spatial modes, corresponding to a time-bin or frequency-bin encoding. Two different types of photon polarization can also be used. |
| Icosahedral Fock-state code | A constant-excitation Fock-state code designed to realize the \(2I\) group of gates using Gaussian rotations. It is obtained from the corresponding icosahedral spin code via the simplex mapping between spin and constant-excitation Fock spaces [1]. |
| Jump code | A CE code designed to detect and correct AD errors. An \(((n,K))\) jump code is denoted as \(((n,K,t))_w\) (which conflicts with modular-qudit notation), where \(t\) is the maximum number of qubits that can be corrected after each one has undergone a jump error \(|0\rangle\langle 1|\), and where each codeword is a uniform superposition of qubit basis states with Hamming weight \(w\). |
| One-hot quantum code | Encoding of a \(q\)-dimensional qudit into the single-excitation subspace of \(q\) modes. The \(j\)th logical state is the multi-mode Fock state with one photon in mode \(j\) and zero photons in the other modes. This code is useful for encoding and performing operations on qudits in multiple modes [2–6]. |
| Ouyang-Chao constant-excitation PI code | A constant-excitation PI Fock-state code whose construction is based on integer partitions. |
| Two-mode binomial code | Two-mode constant-energy CLY code whose coefficients are square-roots of binomial coefficients. |
| Very small logical qubit (VSLQ) code | A code consisting of two logical codewords \(|\pm\rangle \propto (|0\rangle\pm|2\rangle)(|0\rangle\pm|2\rangle)\), where the total Hilbert space is the tensor product of two transmon qudits (whose ground states \(|0\rangle\) and second excited states \(|2\rangle\) are used in the codewords). Since the code is intended to protect against losses, the qutrits can equivalently be thought of as oscillator Fock-state subspaces. |
| Wasilewski-Banaszek code | Three-oscillator constant-excitation Fock-state code encoding a single logical qubit. |
| \(((8,2,3))\) Plenio-Vedral-Knight CE code | An eight-qubit single-error-correcting code that is the first CE code. Each logical state is a superposition of computational basis states with four excitations. |
| \([[12,1,3]]\) CE CSS code | Twelve-qubit constant-excitation (CE) CSS code that encodes one logical qubit with distance three. It is the smallest CE CSS code that corrects a single-qubit error [7]. Codewords lie in a fixed Hamming-weight subspace, making the code immune to coherent noise in the form of transversal \(Z\)-rotations. |
| \([[14,3,3]]\) CE phantom code | CSS phantom code obtained by concatenating the \([[7,3,(d_X=3,d_Z=2)]]\) punctured hypercube code with the two-qubit phase-flip repetition code. The code is equivalent to the \([[14,3,3]]\) constant-excitation (CE) CSS code obtained by applying dual-rail concatenation to the \([[7,3,2]]\) punctured hypercube code, up to single-qubit Clifford gates, a physical-qubit permutation, and a Pauli frame [7]. |
| \([[4,1,2]]\) Leung-Nielsen-Chuang-Yamamoto (LNCY) code | A four-qubit CSS stabilizer code that is the only qubit CSS code with such parameters. |
List (descendants): All descendants of Constant-excitation (CE) code.
References
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- N. P. D. Sawaya, T. Menke, T. H. Kyaw, S. Johri, A. Aspuru-Guzik, and G. G. Guerreschi, “Resource-efficient digital quantum simulation of d-level systems for photonic, vibrational, and spin-s Hamiltonians”, npj Quantum Information 6, (2020) arXiv:1909.12847 DOI
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- C.-Y. Lai, P.-H. Liou, and Y. Ouyang, “Fault-Tolerant Quantum Error Correction for Constant-Excitation Stabilizer Codes under Coherent Noise”, (2025) arXiv:2507.10395