The q-qubit Clifford group, that is, the normalizer of the q-qubit Pauli group in U(2q), is a fundamental structure in quantum information with a wide variety of applications. We characterize all irreducible subrepresentations of the two-copy representation φ⊗2 of the Clifford group on the two-fold tensor product of the space of linear operators M2q⊗2⁠. In the companion paper [Helsen et al., e-print arXiv:1701.04299 (2017)], we apply this result to improve the statistics of randomized benchmarking, a method for characterizing quantum systems.

Topics
Group theory, Operator theory, Representation theory, Functions and mappings, Isospin, Quantum mechanical systems and processes, Hilbert space, Quantum computing, Quantum information, Tomography
1.
D.
 
Gottesman
, “
An introduction to quantum error correction and fault-tolerant quantum computation
,”
Proc. Symp. Appl. Math.
 
68
,
13
(
2010
).
https://doi.org/10.1090/psapm/068/2762145
Google Scholar
Crossref
Search ADS
 
2.
D.
 
Gross
,
Y.-K.
 
Liu
,
S. T.
 
Flammia
,
S.
 
Becker
, and
J.
 
Eisert
, “
Quantum state tomography via compressed sensing
,”
Phys. Rev. Lett.
 
105
(
15
),
150401
(
2010
).
https://doi.org/10.1103/PhysRevLett.105.150401
Google Scholar
Crossref
Search ADS
PubMed
 
3.
D. P.
 
DiVincenzo
,
D. W.
 
Leung
, and
B. M.
 
Terhal
, “
Quantum data hiding
,”
IEEE Trans. Inf. Theory
 
48
(
3
),
580
–
599
(
2002
); e-print arXiv:quant-ph/0103098.
https://doi.org/10.1109/18.985948
Google Scholar
Crossref
Search ADS
 
4.
S.
 
Aaronson
 and
D.
 
Gottesman
, “
Improved simulation of stabilizer circuits
,”
Phys. Rev. A
 
70
,
052328
(
2004
).
https://doi.org/10.1103/PhysRevA.70.052328
Google Scholar
Crossref
Search ADS
 
5.
C.
 
Dankert
,
R.
 
Cleve
,
J.
 
Emerson
, and
E.
 
Livine
, “
Exact and approximate unitary 2-designs and their application to fidelity estimation
,”
Phys. Rev. A
 
80
(
1
),
012304
(
2009
).
https://doi.org/10.1103/PhysRevA.80.012304
Google Scholar
Crossref
Search ADS
 
6.
E.
 
Magesan
,
J. M.
 
Gambetta
, and
J.
 
Emerson
, “
Scalable and robust randomized benchmarking of quantum processes
,”
Phys. Rev. Lett.
 
106
(
18
),
180504
(
2011
).
https://doi.org/10.1103/PhysRevLett.106.180504
Google Scholar
Crossref
Search ADS
PubMed
 
7.
D.
 
Gross
,
K.
 
Audenaert
, and
J.
 
Eisert
, “
Evenly distributed unitaries: On the structure of unitary designs
,”
J. Math. Phys.
 
48
(
5
),
052104
(
2007
).
https://doi.org/10.1063/1.2716992
Google Scholar
Crossref
Search ADS
 
8.
R.
 
Goodman
 and
N. R.
 
Wallach
,
Symmetry, Representations, and Invariants
, Graduate Texts in Mathematics (
Springer
,
2009
), ISBN: 9780387798516.
Google Scholar
Crossref
Search ADS
 
9.
H.
 
Zhu
, “
Multiqubit Clifford groups are unitary 3-designs
,”
Phys. Rev. A
 
96
(
6
),
062336
(
2017
).
https://doi.org/10.1103/PhysRevA.96.062336
Google Scholar
Crossref
Search ADS
 
10.
Z.
 
Webb
, “
The Clifford group forms a unitary 3-design
,”
Quantum Info. Comput.
 
16
(
15–16
),
1379
–
1400
(
2016
).
Google Scholar
 
11.
J.
 
Joel Wallman
 and
S. T.
 
Flammia
, “
Randomized benchmarking with confidence
,”
New J. Phys.
 
16
(
10
),
103032
(
2014
).
https://doi.org/10.1088/1367-2630/16/10/103032
Google Scholar
Crossref
Search ADS
 
12.
J.
 
Helsen
,
J. J.
 
Wallman
,
S. T.
 
Flammia
, and
S.
 
Wehner
. “
Multi-qubit randomized benchmarking using few samples
,” e-print arXiv:1701.04299 (
2017
).
Google Scholar
 
13.
W.
 
Fulton
 and
J.
 
Harris
,
Representation Theory: A First Course
, Readings in Mathematics (
Springer-Verlag
 
New York
,
2004
), ISBN: 978-1-4612-0979-9.
Google Scholar
Crossref
Search ADS
 
14.
J. M.
 
Farinholt
, “
An ideal characterization of the Clifford operators
,”
J. Phys. A: Math. Theor.
 
47
,
305303
(
2014
).
https://doi.org/10.1088/1751-8113/47/30/305303
Google Scholar
Crossref
Search ADS
 
15.
R. A.
 
Low
, “
Large deviation bounds for k-designs
,”
Proc. R. Soc. London, Ser. A
 
465
,
3289
–
3308
(
2009
).
https://doi.org/10.1098/rspa.2009.0232
Google Scholar
Crossref
Search ADS
 
16.
H.
 
Zhu
,
R.
 
Kueng
,
M.
 
Grassl
, and
D.
 
Gross
, “
The Clifford group fails gracefully to be a unitary 4-design
,” e-print arXiv:1609.08172 (
2016
).
Google Scholar
 
17.
R.
 
Kueng
,
H.
 
Zhu
, and
D.
 
Gross
. “
Distinguishing quantum states using Clifford orbits
,” e-print arXiv:1609.08595 (
2016
).
Google Scholar
 
18.
R.
 
Kueng
,
H.
 
Zhu
, and
D.
 
Gross
, “
Low rank matrix recovery from Clifford orbits
,” e-print arXiv:1610.08070 (
2016
).
Google Scholar
 
19.
E.
 
Knill
,
D.
 
Leibfried
,
R.
 
Reichle
,
J.
 
Britton
,
R. B.
 
Blakestad
,
J. D.
 
Jost
,
C.
 
Langer
,
R.
 
Ozeri
,
S.
 
Seidelin
, and
D. J.
 
Wineland
, “
Randomized benchmarking of quantum gates
,”
Phys. Rev. A
 
77
(
1
),
012307
(
2008
).
https://doi.org/10.1103/PhysRevA.77.012307
Google Scholar
Crossref
Search ADS
 
20.

Technically the character inner product of the representation C⊗4 rather than C ⊗ C* ⊗ C ⊗ C* is calculated in Ref. 9, but it can be easily seen that the character inner product is invariant under complex conjugation of some or all tensor factors of the representation.

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