Here is a list of codes related to 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 excited-state eigenspace of a Hamiltonian governing the total energy or total number of excitations of 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, \(H=\sum_i \hat{n}_i\). |
| Constant-weight block code | A block code whose codewords all have the same number of nonzero coordinates. Code constructions exist for codes over fields [1] or rings [2]. |
| 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. |
| Fermion code | Finite-dimensional quantum error-correcting code encoding a logical qudit or fermionic Hilbert space into a physical Fock space of fermionic modes. Codes are typically described using Majorana operators, which are linear combinations of fermionic creation and annihilation operators [3]. Majorana operators may either be considered individually or paired in various ways into creation and annihilation operators to yield fermionic modes. They form a Clifford algebra and can be interpreted as Ising anyons in certain contexts. |
| Icosahedral Fock-state code | A constant-excitation Fock-state code designed to realize the \(2I\) group of gates using Gaussian rotations. |
| 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 qubits [4–8]. |
| Ouyang-Chao constant-excitation PI code | A constant-excitation PI Fock-state code whose construction is based on integer partitions. |
| Permutation-invariant (PI) code | Block quantum code such that any permutation of the subsystems leaves any codeword invariant. In other words, the automorphism group of the code contains the symmetric group \(S_n\). |
| Quantum parity code (QPC) | A \([[m_1 m_2,1,\min(m_1,m_2)]]\) CSS code family obtained from concatenating an \(m_1\)-qubit phase-flip repetition code with an \(m_2\)-qubit bit-flip repetition code. |
| Qubit CSS code | An \([[n,k,d]]\) stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type Pauli strings. Codes can be defined from two classical codes and/or chain complexes over \(\mathbb{Z}_2\) per the qubit CSS-to-homology correspondence below. |
| Qubit stabilizer code | An \(((n,2^k,d))\) qubit stabilizer code is denoted as \([[n,k]]\) or \([[n,k,d]]\), where \(d\) is the code’s distance. Logical subspace is the joint eigenspace of commuting Pauli operators forming the code’s stabilizer group \(\mathsf{S}\). Traditionally, the logical subspace is the joint \(+1\) eigenspace of a set of \(2^{n-k}\) commuting Pauli operators which do not contain \(-I\). The distance is the minimum weight of a Pauli string that implements a nontrivial logical operation in the code. |
| Two-mode binomial code | Two-mode constant-energy CLY code whose coefficients are square-roots of binomial coefficients. |
| Very small logical qubit (VSLQ) code | The two logical codewords are \(|\pm\rangle \propto (|0\rangle\pm|2\rangle)(|0\rangle\pm|2\rangle)\), 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. |
| \([[5,1,3]]\) Five-qubit perfect code | Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error. |
References
- [1]
- Fang-Wei Fu, A. J. Han Vinck, and Shi-Yi Shen, “On the constructions of constant-weight codes”, IEEE Transactions on Information Theory 44, 328 (1998) DOI
- [2]
- J. A. Wood, “The structure of linear codes of constant weight”, Transactions of the American Mathematical Society 354, 1007 (2001) DOI
- [3]
- S. B. Bravyi and A. Yu. Kitaev, “Fermionic Quantum Computation”, Annals of Physics 298, 210 (2002) arXiv:quant-ph/0003137 DOI
- [4]
- R. D. Somma, “Quantum Computation, Complexity, and Many-Body Physics”, (2005) arXiv:quant-ph/0512209
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- M. R. Geller, J. M. Martinis, A. T. Sornborger, P. C. Stancil, E. J. Pritchett, H. You, and A. Galiautdinov, “Universal quantum simulation with prethreshold superconducting qubits: Single-excitation subspace method”, (2015) arXiv:1505.04990
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- S. McArdle, A. Mayorov, X. Shan, S. Benjamin, and X. Yuan, “Digital quantum simulation of molecular vibrations”, Chemical Science 10, 5725 (2019) arXiv:1811.04069 DOI
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- N. P. D. Sawaya and J. Huh, “Quantum Algorithm for Calculating Molecular Vibronic Spectra”, The Journal of Physical Chemistry Letters 10, 3586 (2019) arXiv:1812.10495 DOI
- [8]
- 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