680 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.
Phase Retrieval Using Unitary 2-Designs
We consider a variant of the phase retrieval problem, where vectors are
replaced by unitary matrices, i.e., the unknown signal is a unitary matrix U,
and the measurements consist of squared inner products |Tr(C*U)|^2 with unitary
matrices C that are chosen by the observer. This problem has applications to
quantum process tomography, when the unknown process is a unitary operation.
We show that PhaseLift, a convex programming algorithm for phase retrieval,
can be adapted to this matrix setting, using measurements that are sampled from
unitary 4- and 2-designs. In the case of unitary 4-design measurements, we show
that PhaseLift can reconstruct all unitary matrices, using a near-optimal
number of measurements. This extends previous work on PhaseLift using spherical
4-designs.
In the case of unitary 2-design measurements, we show that PhaseLift still
works pretty well on average: it recovers almost all signals, up to a constant
additive error, using a near-optimal number of measurements. These 2-design
measurements are convenient for quantum process tomography, as they can be
implemented via randomized benchmarking techniques. This is the first positive
result on PhaseLift using 2-designs.Comment: 21 pages; v3: minor revisions, to appear at SampTA 2017; v2:
rewritten to focus on phase retrieval, with new title, improved error bounds,
and numerics; v1: original version, titled "Quantum Compressed Sensing Using
2-Designs
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
Recovering Quantum Gates from Few Average Gate Fidelities
Characterizing quantum processes is a key task in the development of quantum technologies, especially at the noisy intermediate scale of today’s devices. One method for characterizing processes is randomized benchmarking, which is robust against state preparation and measurement errors and can be used to benchmark Clifford gates. Compressed sensing techniques achieve full tomography of quantum channels essentially at optimal resource efficiency. In this Letter, we show that the favorable features of both approaches can be combined. For characterizing multiqubit unitary gates, we provide a rigorously guaranteed and practical reconstruction method that works with an essentially optimal number of average gate fidelities measured with respect to random Clifford unitaries. Moreover, for general unital quantum channels, we provide an explicit expansion into a unitary 2-design, allowing for a practical and guaranteed reconstruction also in that case. As a side result, we obtain a new statistical interpretation of the unitarity—a figure of merit characterizing the coherence of a process
Conjugate weight enumerators and invariant theory
The Galois group of a finite field extension defines a grading on the
symmetric algebra of the -space which we use to introduce the notion
of homogeneous conjugate invariants for subgroups G\leq \GL_v(K). If the
Weight Enumerator Conjecture holds for a finite representation then the
genus- conjugate complete weight enumerators of self-dual codes generate the
corresponding space of conjugate invariants of the associated genus-
Clifford-Weil group {\mathcal C}_m(\rho ) \leq \GL_{v^m}(K). This
generalisation of a paper by Bannai, Oura and Da Zhao provides new examples of
Clifford-Weil orbits that form projective designs
Low rank matrix recovery from rank one measurements
We study the recovery of Hermitian low rank matrices from undersampled measurements via nuclear norm minimization. We
consider the particular scenario where the measurements are Frobenius inner
products with random rank-one matrices of the form for some
measurement vectors , i.e., the measurements are given by . The case where the matrix to be recovered
is of rank one reduces to the problem of phaseless estimation (from
measurements, via the PhaseLift approach,
which has been introduced recently. We derive bounds for the number of
measurements that guarantee successful uniform recovery of Hermitian rank
matrices, either for the vectors , , being chosen independently
at random according to a standard Gaussian distribution, or being sampled
independently from an (approximate) complex projective -design with .
In the Gaussian case, we require measurements, while in the case
of -designs we need . Our results are uniform in the
sense that one random choice of the measurement vectors guarantees
recovery of all rank -matrices simultaneously with high probability.
Moreover, we prove robustness of recovery under perturbation of the
measurements by noise. The result for approximate -designs generalizes and
improves a recent bound on phase retrieval due to Gross, Kueng and Krahmer. In
addition, it has applications in quantum state tomography. Our proofs employ
the so-called bowling scheme which is based on recent ideas by Mendelson and
Koltchinskii.Comment: 24 page
Duality theory for Clifford tensor powers
The representation theory of the Clifford group is playing an increasingly
prominent role in quantum information theory, including in such diverse use
cases as the construction of protocols for quantum system certification,
quantum simulation, and quantum cryptography. In these applications, the tensor
powers of the defining representation seem particularly important. The
representation theory of these tensor powers is understood in two regimes. 1.
For odd qudits in the case where the power t is not larger than the number of
systems n: Here, a duality theory between the Clifford group and certain
discrete orthogonal groups can be used to make fairly explicit statements about
the occurring irreps (this theory is related to Howe duality and the
eta-correspondence). 2. For qubits: Tensor powers up to t=4 have been analyzed
on a case-by-case basis. In this paper, we provide a unified framework for the
duality approach that also covers qubit systems. To this end, we translate the
notion of rank of symplectic representations to representations of the qubit
Clifford group, and generalize the eta correspondence between symplectic and
orthogonal groups to a correspondence between the Clifford and certain
orthogonal-stochastic groups. As a sample application, we provide a protocol to
efficiently implement the complex conjugate of a black-box Clifford unitary
evolution.Comment: 47 page
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
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