47 research outputs found
Unitary designs and codes
A unitary design is a collection of unitary matrices that approximate the
entire unitary group, much like a spherical design approximates the entire unit
sphere. In this paper, we use irreducible representations of the unitary group
to find a general lower bound on the size of a unitary t-design in U(d), for
any d and t. We also introduce the notion of a unitary code - a subset of U(d)
in which the trace inner product of any pair of matrices is restricted to only
a small number of distinct values - and give an upper bound for the size of a
code of degree s in U(d) for any d and s. These bounds can be strengthened when
the particular inner product values that occur in the code or design are known.
Finally, we describe some constructions of designs: we give an upper bound on
the size of the smallest weighted unitary t-design in U(d), and we catalogue
some t-designs that arise from finite groups.Comment: 25 pages, no figure
Rolling quantum dice with a superconducting qubit
One of the key challenges in quantum information is coherently manipulating
the quantum state. However, it is an outstanding question whether control can
be realized with low error. Only gates from the Clifford group -- containing
, , and Hadamard gates -- have been characterized with high
accuracy. Here, we show how the Platonic solids enable implementing and
characterizing larger gate sets. We find that all gates can be implemented with
low error. The results fundamentally imply arbitrary manipulation of the
quantum state can be realized with high precision, providing new practical
possibilities for designing efficient quantum algorithms.Comment: 8 pages, 4 figures, including supplementary materia
Qubit flip game on a Heisenberg spin chain
We study a quantum version of a penny flip game played using control
parameters of the Hamiltonian in the Heisenberg model. Moreover, we extend this
game by introducing auxiliary spins which can be used to alter the behaviour of
the system. We show that a player aware of the complex structure of the system
used to implement the game can use this knowledge to gain higher mean payoff.Comment: 13 pages, 3 figures, 3 table
Multiqubit Clifford groups are unitary 3-designs
Unitary -designs are a ubiquitous tool in many research areas, including
randomized benchmarking, quantum process tomography, and scrambling. Despite
the intensive efforts of many researchers, little is known about unitary
-designs with in the literature. We show that the multiqubit
Clifford group in any even prime-power dimension is not only a unitary
2-design, but also a 3-design. Moreover, it is a minimal 3-design except for
dimension~4. As an immediate consequence, any orbit of pure states of the
multiqubit Clifford group forms a complex projective 3-design; in particular,
the set of stabilizer states forms a 3-design. In addition, our study is
helpful to studying higher moments of the Clifford group, which are useful in
many research areas ranging from quantum information science to signal
processing. Furthermore, we reveal a surprising connection between unitary
3-designs and the physics of discrete phase spaces and thereby offer a simple
explanation of why no discrete Wigner function is covariant with respect to the
multiqubit Clifford group, which is of intrinsic interest to studying quantum
computation.Comment: 7 pages, published in Phys. Rev.
Clifford group dipoles and the enactment of Weyl/Coxeter group W(E8) by entangling gates
Peres/Mermin arguments about no-hidden variables in quantum mechanics are
used for displaying a pair (R, S) of entangling Clifford quantum gates, acting
on two qubits. From them, a natural unitary representation of Coxeter/Weyl
groups W(D5) and W(F4) emerges, which is also reflected into the splitting of
the n-qubit Clifford group Cn into dipoles Cn . The union of the
three-qubit real Clifford group C+ 3 and the Toffoli gate ensures a orthogonal
representation of the Weyl/Coxeter group W(E8), and of its relatives. Other
concepts involved are complex reflection groups, BN pairs, unitary group
designs and entangled states of the GHZ family.Comment: version revised according the recommendations of a refere