Alternative names: \(\mathbb{C}P^N\) code, Packing in \(\mathbb{C}P^N\).
Description
Encodes \(K\) states (codewords) into a complex projective space \(\mathbb{C}P^N\), the space of lines in complex space. The space for \(N=2\) is called the complex projective plane.Notes
Review and tables of packings in complex projective space [1].Cousins
- Finite-dimensional quantum error-correcting code— Quantum states in an \(N\)-dimensional Hilbert space are parameterized by points in the complex projective space \(\mathbb{C}P^N\). As such, (classical) complex projective codes can be associated with subsets of quantum states.
- \(t\)-design— Quantum states in an \(N\)-dimensional Hilbert space are parameterized by points in the complex projective space \(\mathbb{C}P^N\). As such, complex projective designs are designs on the space of all quantum states [2–4]. Symmetric informationally complete quantum measurements (SIC-POVMs) [2,5] and mutually unbiased bases (MUBs) [6–11] are important examples of such designs.
- Kerdock code— Kerdock codes correspond to cluster states, and the corresponding Clifford-group automorphisms of this set form a particular group [12] that is a unitary 2-design on \(U(2^n)\) [13]. As such, cluster states form complex projective on 2-designs \(\mathbb{C}P^{2^n}\). These are useful in matrix-vector multiplication [14].
- Clifford group— Stabilizer states on \(n\) qubits form 3-designs on complex projective spaces \(\mathbb{C}P^{2^n}\) [15], while the Clifford group is a unitary 3-design on \(U(2^n)\) [16,17]. The \([[2m,2m-2,2]]\) code for \(2m\) being a multiple of four obstructs the Clifford group from being a 4-design [18].
- \(3_{21}\) polytope code— Antipodal pairs of points of the \(3_{21}\) polytope code correspond to the 28 bitangent lines of a general quartic plane curve in the complex project plane [19–22].
- Hessian polyhedron code— The (antipodal pairs of) points of the (double) Hessian polyhedron correspond to the 27 lines on a smooth cubic surface in the complex projective plane [19–22].
- Witting polytope code— Antipodal pairs of points of the Witting polytope code correspond to the 120 tritangent planes of a canonical sextic curve in \(\mathbb{C}P^3\) [19–22].
- Cluster-state code— Kerdock codes correspond to cluster states, and the corresponding Clifford-group automorphisms of this set form a particular group [12] that is a unitary 2-design on \(U(2^n)\) [13]. As such, cluster states form complex projective 2-designs on \(\mathbb{C}P^{2^n}\). These are useful in matrix-vector multiplication [14].
- Qubit stabilizer code— Stabilizer states on \(n\) qubits form 3-designs on complex projective spaces \(\mathbb{C}P^{2^n}\) [15], while the Clifford group is a unitary 3-design on \(U(2^n)\) [16,17].
- Modular-qudit stabilizer code— Stabilizer states on \(n\) prime-dimensional qubits form 2-designs on complex projective spaces \(\mathbb{C}P^{p^n}\) [15], while the prime-qudit Clifford group is a unitary 2-design on \(U(p^n)\) [23].
- Galois-qudit stabilizer code— Stabilizer states on \(n\) Galois qubits form 2-designs on complex projective spaces \(\mathbb{C}P^{p^{mn}}\) [24].
Member of code lists
Primary Hierarchy
Parents
Complex projective spaces \(\mathbb{C}P^N\) are complex Grassmannians \(G/H\) for \(G = U(N+1)\) and \(H = U(N)\times U(1)\).
Complex projective space code
References
- [1]
- J. Jasper, E. J. King, and D. G. Mixon, “Game of Sloanes: Best known packings in complex projective space”, (2019) arXiv:1907.07848
- [2]
- J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements”, Journal of Mathematical Physics 45, 2171 (2004) arXiv:quant-ph/0310075 DOI
- [3]
- A. Ambainis and J. Emerson, “Quantum t-designs: t-wise independence in the quantum world”, (2007) arXiv:quant-ph/0701126
- [4]
- I. Bengtsson and K. Życzkowski, Geometry of Quantum States (Cambridge University Press, 2017) DOI
- [5]
- Zauner, G. (1999). Grundzüge einer nichtkommutativen Designtheorie. Ph. D. dissertation, PhD thesis.
- [6]
- A. Klappenecker and M. Roetteler, “Constructions of Mutually Unbiased Bases”, (2003) arXiv:quant-ph/0309120
- [7]
- A. Klappenecker and M. Roetteler, “Mutually Unbiased Bases are Complex Projective 2-Designs”, (2005) arXiv:quant-ph/0502031
- [8]
- A. J. Scott, “Optimizing quantum process tomography with unitary2-designs”, Journal of Physics A: Mathematical and Theoretical 41, 055308 (2008) arXiv:0711.1017 DOI
- [9]
- T. DURT, B.-G. ENGLERT, I. BENGTSSON, and K. ŻYCZKOWSKI, “ON MUTUALLY UNBIASED BASES”, International Journal of Quantum Information 08, 535 (2010) arXiv:1004.3348 DOI
- [10]
- H. Zhu, “Mutually unbiased bases as minimal Clifford covariant 2-designs”, Physical Review A 91, (2015) arXiv:1505.01123 DOI
- [11]
- D. McNulty and S. Weigert, “Mutually Unbiased Bases in Composite Dimensions – A Review”, (2024) arXiv:2410.23997
- [12]
- A. Calderbank, P. Cameron, W. Kantor, and J. Seidel, “Z\({}_{\text{4}}\) -Kerdock Codes, Orthogonal Spreads, and Extremal Euclidean Line-Sets”, Proceedings of the London Mathematical Society 75, 436 (1997) DOI
- [13]
- T. Can, N. Rengaswamy, R. Calderbank, and H. D. Pfister, “Kerdock Codes Determine Unitary 2-Designs”, IEEE Transactions on Information Theory 66, 6104 (2020) arXiv:1904.07842 DOI
- [14]
- T. Fuchs, D. Gross, F. Krahmer, R. Kueng, and D. G. Mixon, “Sketching with Kerdock’s crayons: Fast sparsifying transforms for arbitrary linear maps”, (2021) arXiv:2105.05879
- [15]
- R. Kueng and D. Gross, “Qubit stabilizer states are complex projective 3-designs”, (2015) arXiv:1510.02767
- [16]
- H. Zhu, “Multiqubit Clifford groups are unitary 3-designs”, Physical Review A 96, (2017) arXiv:1510.02619 DOI
- [17]
- Z. Webb, “The Clifford group forms a unitary 3-design”, (2016) arXiv:1510.02769
- [18]
- H. Zhu, R. Kueng, M. Grassl, and D. Gross, “The Clifford group fails gracefully to be a unitary 4-design”, (2016) arXiv:1609.08172
- [19]
- P. du Val, “On the Directrices of a Set of Points in a Plane”, Proceedings of the London Mathematical Society s2-35, 23 (1933) DOI
- [20]
- Arnold, V. I. (1999). Symplectization, complexification and mathematical trinities. The Arnoldfest, 23-37.
- [21]
- H. Cohn and A. Kumar, “Universally optimal distribution of points on spheres”, Journal of the American Mathematical Society 20, 99 (2006) arXiv:math/0607446 DOI
- [22]
- Y.-H. He and J. McKay, “Sporadic and Exceptional”, (2015) arXiv:1505.06742
- [23]
- M. A. Graydon, J. Skanes-Norman, and J. J. Wallman, “Clifford groups are not always 2-designs”, (2021) arXiv:2108.04200
- [24]
- H. F. Chau, “Unconditionally Secure Key Distribution in Higher Dimensions by Depolarization”, IEEE Transactions on Information Theory 51, 1451 (2005) arXiv:quant-ph/0405016 DOI
Page edit log
- Victor V. Albert (2025-10-27) — most recent
Cite as:
“Complex projective space code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/complex_projective