Description
A unitary \(t\)-design is a subset of the unitary group that reproduces Haar averages of polynomials over the group up to degree \(t\) [3; Def. 1]. The design conditions can defined using the \(t\)th tensor product of the group’s adjoint representation.Protection
Bounds on design size have been computed [5].Cousins
- Kerdock code— Kerdock codes correspond to cluster states, and the corresponding Clifford-group automorphisms of this set form a particular group [6] that is a unitary 2-design on \(U(2^n)\) [7]. As such, cluster states form complex projective 2-designs. These are useful in matrix-vector multiplication [8].
- Hamiltonian-based code— Evolving with a Hamiltonian that has constant locality does not yield a unitary 2-design, but increasing the locality slightly overcomes this and yields a design [9].
- Haar-random qubit code— Approximating the random projections through \(t\)-designs is necessary in order to make the Haar-random qubit protocol practical. Replacing with random Clifford gates is especially convenient since the Clifford group forms a unitary 2-design and produces stabilizer codes.
- Local Haar-random circuit qubit code— Local Haar-random circuits of polynomial depth form approximate unitary designs [10].
- Qubit stabilizer code— Stabilizer states on \(n\) qubits form 3-designs on complex projective spaces \(\mathbb{C}P^{2^n}\) [11], while the Clifford group is a unitary 3-design on \(U(2^n)\) [12,13]. The \([[2m,2m-2,2]]\) code for \(2m\) being a multiple of four obstructs the Clifford group from being a 4-design [14].
- Kitaev surface code— Unitary \(t\)-designs can be generated via coherent errors, syndrome extraction, and correction [15].
- Modular-qudit code— The prime-qudit Pauli group is a unitary 1-design.
- Modular-qudit stabilizer code— Stabilizer states on \(n\) prime-dimensional qubits form 2-designs on complex projective spaces \(\mathbb{C}P^{p^n}\) [11], while the prime-qudit Clifford group is a unitary 2-design on \(U(p^n)\) [16].
- Twisted \(1\)-group code— Twisted unitary \(t\)-groups [17] generalize the idea of unitary \(t\)-groups [18–20], which are subgroups of the unitary group that form unitary \(t\)-designs.
Member of code lists
Primary Hierarchy
Parents
Unitary \(t\)-designs are designs on the unitary group \(U(N)\).
Unitary \(t\)-designs are designs on the unitary group \(U(N)\).
Unitary \(t\)-design
References
- [1]
- C. Dankert, “Efficient Simulation of Random Quantum States and Operators”, (2005) arXiv:quant-ph/0512217
- [2]
- C. Dankert, R. Cleve, J. Emerson, and E. Livine, “Exact and approximate unitary 2-designs and their application to fidelity estimation”, Physical Review A 80, (2009) arXiv:quant-ph/0606161 DOI
- [3]
- D. Gross, K. Audenaert, and J. Eisert, “Evenly distributed unitaries: On the structure of unitary designs”, Journal of Mathematical Physics 48, (2007) arXiv:quant-ph/0611002 DOI
- [4]
- T. Schuster, J. Haferkamp, and H.-Y. Huang, “Random unitaries in extremely low depth”, (2025) arXiv:2407.07754
- [5]
- A. Roy, “Bounds for codes and designs in complex subspaces”, (2008) arXiv:0806.2317
- [6]
- 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
- [7]
- 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
- [8]
- 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
- [9]
- L. Cui, T. Schuster, L. Mao, H.-Y. Huang, and F. Brandao, “Random unitaries from Hamiltonian dynamics”, (2025) arXiv:2510.08434
- [10]
- F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki, “Local Random Quantum Circuits are Approximate Polynomial-Designs”, Communications in Mathematical Physics 346, 397 (2016) arXiv:1208.0692 DOI
- [11]
- R. Kueng and D. Gross, “Qubit stabilizer states are complex projective 3-designs”, (2015) arXiv:1510.02767
- [12]
- H. Zhu, “Multiqubit Clifford groups are unitary 3-designs”, Physical Review A 96, (2017) arXiv:1510.02619 DOI
- [13]
- Z. Webb, “The Clifford group forms a unitary 3-design”, (2016) arXiv:1510.02769
- [14]
- H. Zhu, R. Kueng, M. Grassl, and D. Gross, “The Clifford group fails gracefully to be a unitary 4-design”, (2016) arXiv:1609.08172
- [15]
- Z. Cheng, E. Huang, V. Khemani, M. J. Gullans, and M. Ippoliti, “Emergent Unitary Designs for Encoded Qubits from Coherent Errors and Syndrome Measurements”, PRX Quantum 6, (2025) arXiv:2412.04414 DOI
- [16]
- M. A. Graydon, J. Skanes-Norman, and J. J. Wallman, “Clifford groups are not always 2-designs”, (2021) arXiv:2108.04200
- [17]
- E. Kubischta and I. Teixeira, “Quantum Codes from Twisted Unitary t -Groups”, Physical Review Letters 133, (2024) arXiv:2402.01638 DOI
- [18]
- R. M. Guralnick and P. H. Tiep, “Decompositions of Small Tensor Powers and Larsen’s Conjecture”, (2005) arXiv:math/0502080
- [19]
- A. Roy and A. J. Scott, “Unitary designs and codes”, Designs, Codes and Cryptography 53, 13 (2009) arXiv:0809.3813 DOI
- [20]
- E. Bannai, G. Navarro, N. Rizo, and P. H. Tiep, “Unitary t-groups”, (2018) arXiv:1810.02507
Page edit log
- Victor V. Albert (2025-10-27) — most recent
Cite as:
“Unitary \(t\)-design”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/unitary_design