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C. Cîrstoiu, K. Korzekwa, and D. Jennings, Robustness of Noether’s principle: Maximal disconnects between conservation laws and symmetries in quantum theory, Physical Review X 10
Original
1908
Earlier work this paper cites.
M. T. Quintino, Q. Dong, A. Shimbo, A. Soeda, and M. Murao, Probabilistic exact universal quantum circuits for transforming unitary operations, Physical Review A 100
Original
1909
Earlier work this paper cites.
C. Itzykson and M. Nauenberg, Unitary Groups: Representations and Decompositions, Rev. Mod. Phys. 38
1966
Earlier work this paper cites.
E. Noether, Invariant variation problems, Transport theory and statistical physics 1
1971
Earlier work this paper cites.
W. K. Wootters and W. H. Zurek, A single quantum cannot be cloned, Nature 299
1982
Earlier work this paper cites.
D. Dieks, Communication by EPR devices, Physics Letters A 92
1982
Earlier work this paper cites.
N. J. Vilenkin and A. U. Klimyk, Representations of Lie groups and special functions (Springer, 1995)
1995
Earlier work this paper cites.
H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B. Schumacher, Noncommuting mixed states cannot be broadcast, Physical Review Letters 76
1996
Earlier work this paper cites.
V. Bužek and M. Hillery, Quantum copying: Beyond the no-cloning theorem, Physical Review A 54
1996
Earlier work this paper cites.
N. Gisin and S. Massar, Optimal quantum cloning machines, Physical review letters 79
1997
Earlier work this paper cites.
D. Bruß, D. P. DiVincenzo, A. Ekert, C. A. Fuchs, C. Macchiavello, and J. A. Smolin, Optimal universal and state-dependent quantum cloning, Physical Review A 57
1998
Earlier work this paper cites.
R. F. Werner, Optimal cloning of pure states, Physical Review A 58
1998
Earlier work this paper cites.
M. Keyl and R. F. Werner, Optimal cloning of pure states, testing single clones, Journal of Mathematical Physics 40
1999
Earlier work this paper cites.
N. J. Cerf, Asymmetric quantum cloning in any dimension, Journal of Modern Optics 47
2000
Earlier work this paper cites.
R. F. Werner and A. S. Holevo, Counterexample to an additivity conjecture for output purity of quantum channels, Journal of Mathematical Physics 43
2002
Earlier work this paper cites.
T. Eggeling, D. Schlingemann, and R. F. Werner, Semicausal operations are semilocalizable, Europhysics Letters 57
2002
Earlier work this paper cites.
G. M. D’Ariano and C. Macchiavello, Optimal phase covariant cloning for qubits and qutrits, Physical Review A 67
2003
Earlier work this paper cites.
C. King, The capacity of the quantum depolarizing channel, IEEE Transactions on Information Theory 49
2003
Earlier work this paper cites.
A. W. Harrow, Applications of coherent classical communication and the Schur transform to quantum information theory , Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, USA (2005), arXiv:quant-ph/0512255
2005
Earlier work this paper cites.
V. Scarani, S. Iblisdir, N. Gisin, and A. Acín, Quantum cloning, Reviews of Modern Physics 77
2005
Earlier work this paper cites.
A. Holevo, Additivity conjecture and covariant channels, International Journal of Quantum Information 3
2005
Earlier work this paper cites.
N. Datta, A. S. Holevo, and Y. Suhov, Additivity for transpose depolarizing channels, International Journal of Quantum Information 4
2006
Earlier work this paper cites.
D. Bacon, I. L. Chuang, and A. W. Harrow, Efficient quantum circuits for schur and clebsch-gordan transforms, Physical review letters 97
2006
Earlier work this paper cites.
D. Bacon, I. L. Chuang, and A. W. Harrow, The quantum schur transform: I. efficient qudit circuits, in Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms (2007) pp. 1235–1244, arXiv:quant-ph/0601001
2007
Earlier work this paper cites.

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2025

Sequential quantum processes with group symmetries

Grinko, Dmitry, Yoshida, Satoshi, Murao, Mio et al.

Understand

Symmetry plays a crucial role in the design and analysis of quantum protocols.

This result shows a canonical circuit decomposition of a $(G\times H)$-invariant quantum comb for compact groups $G$ and $H$ using the corresponding Clebsch--Gordan transforms, which naturally extends to the $G$-covariant quantum comb.
By using this circuit decomposition, we propose a parametrized quantum comb with group symmetry, and derive the optimal quantum comb which transforms an unknown unitary operation $U\in \mathrm{SU}(d)$ into its inverse $U^\dagger$ or transpose $U^\top$.
From numerics, we find a deterministic and exact unitary transposition protocol for $d=3$ with $7$ queries to $U$.

Built on

C. Cîrstoiu, K. Korzekwa, and D. Jennings, Robustness of Noether’s principle: Maximal disconnects between conservation laws and symmetries in quantum theory, Physical Review X 10
Original
1908
Earlier work this paper cites.
M. T. Quintino, Q. Dong, A. Shimbo, A. Soeda, and M. Murao, Probabilistic exact universal quantum circuits for transforming unitary operations, Physical Review A 100
Original
1909
Earlier work this paper cites.
C. Itzykson and M. Nauenberg, Unitary Groups: Representations and Decompositions, Rev. Mod. Phys. 38
1966
Earlier work this paper cites.
E. Noether, Invariant variation problems, Transport theory and statistical physics 1
1971
Earlier work this paper cites.
W. K. Wootters and W. H. Zurek, A single quantum cannot be cloned, Nature 299
1982
Earlier work this paper cites.
D. Dieks, Communication by EPR devices, Physics Letters A 92
1982
Earlier work this paper cites.
N. J. Vilenkin and A. U. Klimyk, Representations of Lie groups and special functions (Springer, 1995)
1995
Earlier work this paper cites.
H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B. Schumacher, Noncommuting mixed states cannot be broadcast, Physical Review Letters 76
1996
Earlier work this paper cites.
V. Bužek and M. Hillery, Quantum copying: Beyond the no-cloning theorem, Physical Review A 54
1996
Earlier work this paper cites.
N. Gisin and S. Massar, Optimal quantum cloning machines, Physical review letters 79
1997
Earlier work this paper cites.
D. Bruß, D. P. DiVincenzo, A. Ekert, C. A. Fuchs, C. Macchiavello, and J. A. Smolin, Optimal universal and state-dependent quantum cloning, Physical Review A 57
1998
Earlier work this paper cites.
R. F. Werner, Optimal cloning of pure states, Physical Review A 58
1998
Earlier work this paper cites.
M. Keyl and R. F. Werner, Optimal cloning of pure states, testing single clones, Journal of Mathematical Physics 40
1999
Earlier work this paper cites.
N. J. Cerf, Asymmetric quantum cloning in any dimension, Journal of Modern Optics 47
2000
Earlier work this paper cites.
R. F. Werner and A. S. Holevo, Counterexample to an additivity conjecture for output purity of quantum channels, Journal of Mathematical Physics 43
2002
Earlier work this paper cites.
T. Eggeling, D. Schlingemann, and R. F. Werner, Semicausal operations are semilocalizable, Europhysics Letters 57
2002
Earlier work this paper cites.
G. M. D’Ariano and C. Macchiavello, Optimal phase covariant cloning for qubits and qutrits, Physical Review A 67
2003
Earlier work this paper cites.
C. King, The capacity of the quantum depolarizing channel, IEEE Transactions on Information Theory 49
2003
Earlier work this paper cites.
A. W. Harrow, Applications of coherent classical communication and the Schur transform to quantum information theory , Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, USA (2005), arXiv:quant-ph/0512255
2005
Earlier work this paper cites.
V. Scarani, S. Iblisdir, N. Gisin, and A. Acín, Quantum cloning, Reviews of Modern Physics 77
2005
Earlier work this paper cites.
A. Holevo, Additivity conjecture and covariant channels, International Journal of Quantum Information 3
2005
Earlier work this paper cites.
N. Datta, A. S. Holevo, and Y. Suhov, Additivity for transpose depolarizing channels, International Journal of Quantum Information 4
2006
Earlier work this paper cites.
D. Bacon, I. L. Chuang, and A. W. Harrow, Efficient quantum circuits for schur and clebsch-gordan transforms, Physical review letters 97
2006
Earlier work this paper cites.
D. Bacon, I. L. Chuang, and A. W. Harrow, The quantum schur transform: I. efficient qudit circuits, in Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms (2007) pp. 1235–1244, arXiv:quant-ph/0601001
2007
Earlier work this paper cites.