Built on

by same author, Random Quantum Circuits are Approximate 2-designs , Communications in Mathematical Physics 291
Original
1919
Earlier work this paper cites.
John Preskill, Reliable quantum computers , Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 454
1969
Earlier work this paper cites.
A. Robert Calderbank, Eric M. Rains, Peter W. Shor, and Neil J. A. Sloane, Quantum Error Correction via Codes over GF(4) , Proceedings of the 1997 IEEE International Symposium on Information Theory, p. 292, IEEE, 1997, arXiv:quant-ph/9608006
1997
Earlier work this paper cites.
Daniel Gottesman, The Heisenberg Representation of Quantum Computers , Group22: Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics, pp. 32–43, 1999, arXiv:quant-ph/9807006
1999
Earlier work this paper cites.
David P. DiVincenzo, Debbie W. Leung, and Barbara M. Terhal, Quantum data hiding , IEEE Transactions on Information Theory 48
2002
Earlier work this paper cites.
Patrick Hayden, Debbie Leung, Peter W. Shor, and Andreas Winter, Randomizing Quantum States: Constructions and Applications , Communications in Mathematical Physics 250
2004
Earlier work this paper cites.
Wolfgang Dür, Marc Hein, J. Ignacio Cirac, and Hans J. Briegel, Standard forms of noisy quantum operations via depolarization , Phys. Rev. A 72
2005
Earlier work this paper cites.
Patrick Hayden, Debbie W. Leung, and Andreas Winter, Aspects of Generic Entanglement , Communications in Mathematical Physics 265
2006
Earlier work this paper cites.
Alex J. Scott, Tight informationally complete quantum measurements , Journal of Physics A: Mathematical and General 39
2006
Earlier work this paper cites.
Andris Ambainis and Joseph Emerson, Quantum t-designs: t-wise independence in the quantum world , Twenty-Second Annual IEEE Conference on Computational Complexity, 2007, pp. 129–140, IEEE, 2007, arXiv:quant-ph/0701126
2007
Earlier work this paper cites.

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David Gross, Koenraad Audenaert, and Jens Eisert, Evenly distributed unitaries: On the structure of unitary designs , Journal of Mathematical Physics 48
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Then

Richard Keung and David Gross, Qubit stabilizer states are complex projective 3-designs , poster presented at the 16th Workshop on Quantum Information Processing, 2013
2013
Later among the works it cites.
Lluís Masanes, Augusto J. Roncaglia, and Antonio Acín, Complexity of energy eigenstates as a mechanism for equilibration , Phys. Rev. E 87
Original
2013
Later among the works it cites.
Jacob M. Farinholt, An ideal characterization of the Clifford operators , Journal of Physics A: Mathematical and Theoretical 47
Original
2014
Later among the works it cites.
Richard Cleve, Debbie Leung, Li Liu, and Chunhao Wang, Near-linear constructions of exact unitary 2-designs , (2015), arXiv:1501.04592
Original
2015
Closest in time.
David Gross, Felix Krahmer, and Richard Kueng, Improved Recovery Guarantees for Phase Retrieval from Coded Diffraction Patterns , Applied and Computational Harmonic Analysis (2015), arXiv:1402.6286

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2015

The Clifford group forms a unitary 3-design

Webb, Zak

Understand

Unitary $k$-designs are finite ensembles of unitary matrices that approximate the Haar distribution over unitary matrices.

Several ensembles are known to be 2-designs, including the uniform distribution over the Clifford group, but no family of ensembles was previously known to form a 3-design.
We prove that the Clifford group is a 3-design, showing that it is a better approximation to Haar-random unitaries than previously expected.
Our proof strategy works for any distribution of unitaries satisfying a property we call Pauli 2-mixing and proceeds without the use of heavy mathematical machinery.

Built on

by same author, Random Quantum Circuits are Approximate 2-designs , Communications in Mathematical Physics 291
Original
1919
Earlier work this paper cites.
John Preskill, Reliable quantum computers , Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 454
1969
Earlier work this paper cites.
A. Robert Calderbank, Eric M. Rains, Peter W. Shor, and Neil J. A. Sloane, Quantum Error Correction via Codes over GF(4) , Proceedings of the 1997 IEEE International Symposium on Information Theory, p. 292, IEEE, 1997, arXiv:quant-ph/9608006
1997
Earlier work this paper cites.
Daniel Gottesman, The Heisenberg Representation of Quantum Computers , Group22: Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics, pp. 32–43, 1999, arXiv:quant-ph/9807006
1999
Earlier work this paper cites.
David P. DiVincenzo, Debbie W. Leung, and Barbara M. Terhal, Quantum data hiding , IEEE Transactions on Information Theory 48
2002
Earlier work this paper cites.
Patrick Hayden, Debbie Leung, Peter W. Shor, and Andreas Winter, Randomizing Quantum States: Constructions and Applications , Communications in Mathematical Physics 250
2004
Earlier work this paper cites.
Wolfgang Dür, Marc Hein, J. Ignacio Cirac, and Hans J. Briegel, Standard forms of noisy quantum operations via depolarization , Phys. Rev. A 72
2005
Earlier work this paper cites.
Patrick Hayden, Debbie W. Leung, and Andreas Winter, Aspects of Generic Entanglement , Communications in Mathematical Physics 265
2006
Earlier work this paper cites.
Alex J. Scott, Tight informationally complete quantum measurements , Journal of Physics A: Mathematical and General 39
2006
Earlier work this paper cites.
Andris Ambainis and Joseph Emerson, Quantum t-designs: t-wise independence in the quantum world , Twenty-Second Annual IEEE Conference on Computational Complexity, 2007, pp. 129–140, IEEE, 2007, arXiv:quant-ph/0701126
2007
Earlier work this paper cites.

Similar

David Gross, Koenraad Audenaert, and Jens Eisert, Evenly distributed unitaries: On the structure of unitary designs , Journal of Mathematical Physics 48
2007
Cited alongside, same era.
Patrick Hayden, Michał Horodecki, Andreas Winter, and Jon Yard, A decoupling approach to the quantum capacity , Open Systems & Information Dynamics 15
2008
Cited alongside, same era.
Patrick Hayden and Andreas Winter, Counterexamples to the Maximal p-Norm Multiplicativity Conjecture for all p > 1 , Communications in Mathematical Physics 284
Original
2008
Cited alongside, same era.
Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine, Exact and approximate unitary 2-designs and their application to fidelity estimation , Phys. Rev. A 80
2009
Cited alongside, same era.
Aram W. Harrow and Richard A. Low, Efficient Quantum Tensor Product Expanders and k-Designs , Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (Irit Dinur, Klaus Jansen, Joseph Naor, and Jose Rolim, eds.), Lecture Notes in Computer Science, vol. 5687, Springer Berlin Heidelberg, 2009, arXiv:0811.2597

Then

Richard Keung and David Gross, Qubit stabilizer states are complex projective 3-designs , poster presented at the 16th Workshop on Quantum Information Processing, 2013
2013
Later among the works it cites.
Lluís Masanes, Augusto J. Roncaglia, and Antonio Acín, Complexity of energy eigenstates as a mechanism for equilibration , Phys. Rev. E 87
Original
2013
Later among the works it cites.
Jacob M. Farinholt, An ideal characterization of the Clifford operators , Journal of Physics A: Mathematical and Theoretical 47
Original
2014
Later among the works it cites.
Richard Cleve, Debbie Leung, Li Liu, and Chunhao Wang, Near-linear constructions of exact unitary 2-designs , (2015), arXiv:1501.04592
Original
2015
Closest in time.
David Gross, Felix Krahmer, and Richard Kueng, Improved Recovery Guarantees for Phase Retrieval from Coded Diffraction Patterns , Applied and Computational Harmonic Analysis (2015), arXiv:1402.6286

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

Original

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