7,299 research outputs found
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
Pseudorandom States, Non-Cloning Theorems and Quantum Money
We propose the concept of pseudorandom states and study their constructions,
properties, and applications. Under the assumption that quantum-secure one-way
functions exist, we present concrete and efficient constructions of
pseudorandom states. The non-cloning theorem plays a central role in our
study---it motivates the proper definition and characterizes one of the
important properties of pseudorandom quantum states. Namely, there is no
efficient quantum algorithm that can create more copies of the state from a
given number of pseudorandom states. As the main application, we prove that any
family of pseudorandom states naturally gives rise to a private-key quantum
money scheme.Comment: 20 page
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.
Efficient approximate unitary t-designs from partially invertible universal sets and their application to quantum speedup
At its core a -design is a method for sampling from a set of unitaries in
a way which mimics sampling randomly from the Haar measure on the unitary
group, with applications across quantum information processing and physics. We
construct new families of quantum circuits on -qubits giving rise to
-approximate unitary -designs efficiently in
depth. These quantum circuits are based on a relaxation of technical
requirements in previous constructions. In particular, the construction of
circuits which give efficient approximate -designs by Brandao, Harrow, and
Horodecki (F.G.S.L Brandao, A.W Harrow, and M. Horodecki, Commun. Math. Phys.
(2016).) required choosing gates from ensembles which contained inverses for
all elements, and that the entries of the unitaries are algebraic. We reduce
these requirements, to sets that contain elements without inverses in the set,
and non-algebraic entries, which we dub partially invertible universal sets. We
then adapt this circuit construction to the framework of measurement based
quantum computation(MBQC) and give new explicit examples of -qubit graph
states with fixed assignments of measurements (graph gadgets) giving rise to
unitary -designs based on partially invertible universal sets, in a natural
way. We further show that these graph gadgets demonstrate a quantum speedup, up
to standard complexity theoretic conjectures. We provide numerical and
analytical evidence that almost any assignment of fixed measurement angles on
an -qubit cluster state give efficient -designs and demonstrate a quantum
speedup.Comment: 25 pages,7 figures. Comments are welcome. Some typos corrected in
newest version. new References added.Proofs unchanged. Results unchange
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
A Numerical Approach for Designing Unitary Space Time Codes with Large Diversity
A numerical approach to design unitary constellation for any dimension and
any transmission rate under non-coherent Rayleigh flat fading channel.Comment: 32 pages, 6 figure
- …