11,030 research outputs found
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.
Kerdock Codes Determine Unitary 2-Designs
The non-linear binary Kerdock codes are known to be Gray images of certain
extended cyclic codes of length over . We show that
exponentiating these -valued codewords by produces stabilizer states, that are quantum states obtained using
only Clifford unitaries. These states are also the common eigenvectors of
commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the
Pauli group. We use this quantum description to simplify the derivation of the
classical weight distribution of Kerdock codes. Next, we organize the
stabilizer states to form mutually unbiased bases and prove that
automorphisms of the Kerdock code permute their corresponding MCS, thereby
forming a subgroup of the Clifford group. When represented as symplectic
matrices, this subgroup is isomorphic to the projective special linear group
PSL(). We show that this automorphism group acts transitively on the Pauli
matrices, which implies that the ensemble is Pauli mixing and hence forms a
unitary -design. The Kerdock design described here was originally discovered
by Cleve et al. (arXiv:1501.04592), but the connection to classical codes is
new which simplifies its description and translation to circuits significantly.
Sampling from the design is straightforward, the translation to circuits uses
only Clifford gates, and the process does not require ancillary qubits.
Finally, we also develop algorithms for optimizing the synthesis of unitary
-designs on encoded qubits, i.e., to construct logical unitary -designs.
Software implementations are available at
https://github.com/nrenga/symplectic-arxiv18a, which we use to provide
empirical gate complexities for up to qubits.Comment: 16 pages double-column, 4 figures, and some circuits. Accepted to
2019 Intl. Symp. Inf. Theory (ISIT), and PDF of the 5-page ISIT version is
included in the arXiv packag
Randomized benchmarking in measurement-based quantum computing
Randomized benchmarking is routinely used as an efficient method for
characterizing the performance of sets of elementary logic gates in small
quantum devices. In the measurement-based model of quantum computation, logic
gates are implemented via single-site measurements on a fixed universal
resource state. Here we adapt the randomized benchmarking protocol for a single
qubit to a linear cluster state computation, which provides partial, yet
efficient characterization of the noise associated with the target gate set.
Applying randomized benchmarking to measurement-based quantum computation
exhibits an interesting interplay between the inherent randomness associated
with logic gates in the measurement-based model and the random gate sequences
used in benchmarking. We consider two different approaches: the first makes use
of the standard single-qubit Clifford group, while the second uses recently
introduced (non-Clifford) measurement-based 2-designs, which harness inherent
randomness to implement gate sequences.Comment: 10 pages, 4 figures, comments welcome; v2 published versio
Permutation Symmetry Determines the Discrete Wigner Function
The Wigner function provides a useful quasiprobability representation of
quantum mechanics, with applications in various branches of physics. Many nice
properties of the Wigner function are intimately connected with the high
symmetry of the underlying operator basis composed of phase point operators:
any pair of phase point operators can be transformed to any other pair by a
unitary symmetry transformation. We prove that, in the discrete scenario, this
permutation symmetry is equivalent to the symmetry group being a unitary
2-design. Such a highly symmetric representation can only appear in odd prime
power dimensions besides dimensions 2 and 8. It suffices to single out a unique
discrete Wigner function among all possible quasiprobability representations.
In the course of our study, we show that this discrete Wigner function is
uniquely determined by Clifford covariance, while no Wigner function is
Clifford covariant in any even prime power dimension.Comment: 5+2 pages, connection with unitary 2-designs added, accepted by Phys.
Rev. Lett. as Editors' Suggestio
Schur-Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations
Schur-Weyl duality is a ubiquitous tool in quantum information. At its heart
is the statement that the space of operators that commute with the tensor
powers of all unitaries is spanned by the permutations of the tensor factors.
In this work, we describe a similar duality theory for tensor powers of
Clifford unitaries. The Clifford group is a central object in many subfields of
quantum information, most prominently in the theory of fault-tolerance. The
duality theory has a simple and clean description in terms of finite
geometries. We demonstrate its effectiveness in several applications:
(1) We resolve an open problem in quantum property testing by showing that
"stabilizerness" is efficiently testable: There is a protocol that, given
access to six copies of an unknown state, can determine whether it is a
stabilizer state, or whether it is far away from the set of stabilizer states.
We give a related membership test for the Clifford group.
(2) We find that tensor powers of stabilizer states have an increased
symmetry group. We provide corresponding de Finetti theorems, showing that the
reductions of arbitrary states with this symmetry are well-approximated by
mixtures of stabilizer tensor powers (in some cases, exponentially well).
(3) We show that the distance of a pure state to the set of stabilizers can
be lower-bounded in terms of the sum-negativity of its Wigner function. This
gives a new quantitative meaning to the sum-negativity (and the related mana)
-- a measure relevant to fault-tolerant quantum computation. The result
constitutes a robust generalization of the discrete Hudson theorem.
(4) We show that complex projective designs of arbitrary order can be
obtained from a finite number (independent of the number of qudits) of Clifford
orbits. To prove this result, we give explicit formulas for arbitrary moments
of random stabilizer states.Comment: 60 pages, 2 figure
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