6 research outputs found
Scalable reconstruction of density matrices
Recent contributions in the field of quantum state tomography have shown
that, despite the exponential growth of Hilbert space with the number of
subsystems, tomography of one-dimensional quantum systems may still be
performed efficiently by tailored reconstruction schemes. Here, we discuss a
scalable method to reconstruct mixed states that are well approximated by
matrix product operators. The reconstruction scheme only requires local
information about the state, giving rise to a reconstruction technique that is
scalable in the system size. It is based on a constructive proof that generic
matrix product operators are fully determined by their local reductions. We
discuss applications of this scheme for simulated data and experimental data
obtained in an ion trap experiment.Comment: 9 pages, 5 figures, replaced with published versio
Efficient and feasible state tomography of quantum many-body systems
We present a novel method to perform quantum state tomography for
many-particle systems which are particularly suitable for estimating states in
lattice systems such as of ultra-cold atoms in optical lattices. We show that
the need for measuring a tomographically complete set of observables can be
overcome by letting the state evolve under some suitably chosen random circuits
followed by the measurement of a single observable. We generalize known results
about the approximation of unitary 2-designs, i.e., certain classes of random
unitary matrices, by random quantum circuits and connect our findings to the
theory of quantum compressed sensing. We show that for ultra-cold atoms in
optical lattices established techniques like optical super-lattices, laser
speckles, and time-of-flight measurements are sufficient to perform fully
certified, assumption-free tomography. Combining our approach with tensor
network methods - in particular the theory of matrix-product states - we
identify situations where the effort of reconstruction is even constant in the
number of lattice sites, allowing in principle to perform tomography on
large-scale systems readily available in present experiments.Comment: 10 pages, 3 figures, minor corrections, discussion added, emphasizing
that no single-site addressing is needed at any stage of the scheme when
implemented in optical lattice system
Efficient tomography of a quantum many-body system
Quantum state tomography (QST) is the gold standard technique for obtaining an estimate for the state of small quantum systems in the laboratory [1]. Its application to systems with more than a few constituents (e.g. particles) soon becomes impractical as the e ff ort required grows exponentially with the number of constituents. Developing more e ffi cient techniques is particularly pressing as precisely-controllable quantum systems that are well beyond the reach of QST are emerging in laboratories. Motivated by this, there is a considerable ongoing e ff ort to develop new state characterisation tools for quantum many-body systems [2–11]. Here we demonstrate Matrix Product State (MPS) tomography [2], which is theoretically proven to allow the states of a broad class of quantum systems to be accurately estimated with an e ff ort that increases e ffi ciently with constituent number. We use the technique to reconstruct the dynamical state of a trapped-ion quantum simulator comprising up to 14 entangled and individually-controlled spins (qubits): a size far beyond the practical limits of QST. Our results reveal the dynamical growth of entanglement and description complexity as correlations spread out during a quench: a necessary condition for future beyond-classical performance. MPS tomography should therefore find widespread use to study large quantum many-body systems and to benchmark and verify quantum simulators and computers
Quantum Magnetism of Spin-Ladder Compounds with Trapped-Ion Crystals
The quest for experimental platforms that allow for the exploration, and even
control, of the interplay of low dimensionality and frustration is a
fundamental challenge in several fields of quantum many-body physics, such as
quantum magnetism. Here, we propose the use of cold crystals of trapped ions to
study a variety of frustrated quantum spin ladders. By optimizing the trap
geometry, we show how to tailor the low dimensionality of the models by
changing the number of legs of the ladders. Combined with a method for
selectively hiding of ions provided by laser addressing, it becomes possible to
synthesize stripes of both triangular and Kagome lattices. Besides, the degree
of frustration of the phonon-mediated spin interactions can be controlled by
shaping the trap frequencies. We support our theoretical considerations by
initial experiments with planar ion crystals, where a high and tunable
anisotropy of the radial trap frequencies is demonstrated. We take into account
an extensive list of possible error sources under typical experimental
conditions, and describe explicit regimes that guarantee the validity of our
scheme
Permutationally invariant state reconstruction
Feasible tomography schemes for large particle numbers must possess, besides
an appropriate data acquisition protocol, also an efficient way to reconstruct
the density operator from the observed finite data set. Since state
reconstruction typically requires the solution of a non-linear large-scale
optimization problem, this is a major challenge in the design of scalable
tomography schemes. Here we present an efficient state reconstruction scheme
for permutationally invariant quantum state tomography. It works for all common
state-of-the-art reconstruction principles, including, in particular, maximum
likelihood and least squares methods, which are the preferred choices in
today's experiments. This high efficiency is achieved by greatly reducing the
dimensionality of the problem employing a particular representation of
permutationally invariant states known from spin coupling combined with convex
optimization, which has clear advantages regarding speed, control and accuracy
in comparison to commonly employed numerical routines. First prototype
implementations easily allow reconstruction of a state of 20 qubits in a few
minutes on a standard computer.Comment: 25 pages, 4 figues, 2 table