169 research outputs found
Reliable recovery of hierarchically sparse signals for Gaussian and Kronecker product measurements
We propose and analyze a solution to the problem of recovering a block sparse
signal with sparse blocks from linear measurements. Such problems naturally
emerge inter alia in the context of mobile communication, in order to meet the
scalability and low complexity requirements of massive antenna systems and
massive machine-type communication. We introduce a new variant of the Hard
Thresholding Pursuit (HTP) algorithm referred to as HiHTP. We provide both a
proof of convergence and a recovery guarantee for noisy Gaussian measurements
that exhibit an improved asymptotic scaling in terms of the sampling complexity
in comparison with the usual HTP algorithm. Furthermore, hierarchically sparse
signals and Kronecker product structured measurements naturally arise together
in a variety of applications. We establish the efficient reconstruction of
hierarchically sparse signals from Kronecker product measurements using the
HiHTP algorithm. Additionally, we provide analytical results that connect our
recovery conditions to generalized coherence measures. Again, our recovery
results exhibit substantial improvement in the asymptotic sampling complexity
scaling over the standard setting. Finally, we validate in numerical
experiments that for hierarchically sparse signals, HiHTP performs
significantly better compared to HTP.Comment: 11+4 pages, 5 figures. V3: Incomplete funding information corrected
and minor typos corrected. V4: Change of title and additional author Axel
Flinth. Included new results on Kronecker product measurements and relations
of HiRIP to hierarchical coherence measures. Improved presentation of general
hierarchically sparse signals and correction of minor typo
General guarantees for randomized benchmarking with random quantum circuits
In its many variants, randomized benchmarking (RB) is a broadly used
technique for assessing the quality of gate implementations on quantum
computers. A detailed theoretical understanding and general guarantees exist
for the functioning and interpretation of RB protocols if the gates under
scrutiny are drawn uniformly at random from a compact group. In contrast, many
practically attractive and scalable RB protocols implement random quantum
circuits with local gates randomly drawn from some gate-set. Despite their
abundance in practice, for those non-uniform RB protocols, general guarantees
under experimentally plausible assumptions are missing. In this work, we derive
such guarantees for a large class of RB protocols for random circuits that we
refer to as filtered RB. Prominent examples include linear cross-entropy
benchmarking, character benchmarking, Pauli-noise tomography and variants of
simultaneous RB. Building upon recent results for random circuits, we show that
many relevant filtered RB schemes can be realized with random quantum circuits
in linear depth, and we provide explicit small constants for common instances.
We further derive general sample complexity bounds for filtered RB. We show
filtered RB to be sample-efficient for several relevant groups, including
protocols addressing higher-order cross-talk. Our theory for non-uniform
filtered RB is, in principle, flexible enough to design new protocols for
non-universal and analog quantum simulators.Comment: 77 pages, 3 figures. Accepted for a talk at QIP 202
Compressive gate set tomography
Flexible characterization techniques that identify and quantify experimental
imperfections under realistic assumptions are crucial for the development of
quantum computers. Gate set tomography is a characterization approach that
simultaneously and self-consistently extracts a tomographic description of the
implementation of an entire set of quantum gates, as well as the initial state
and measurement, from experimental data. Obtaining such a detailed picture of
the experimental implementation is associated with high requirements on the
number of sequences and their design, making gate set tomography a challenging
task even for only two qubits.
In this work, we show that low-rank approximations of gate sets can be
obtained from significantly fewer gate sequences and that it is sufficient to
draw them randomly. Such tomographic information is needed for the crucial task
of dealing with coherent noise. To this end, we formulate the data processing
problem of gate set tomography as a rank-constrained tensor completion problem.
We provide an algorithm to solve this problem while respecting the usual
positivity and normalization constraints of quantum mechanics by using
second-order geometrical optimization methods on the complex Stiefel manifold.
Besides the reduction in sequences, we demonstrate numerically that the
algorithm does not rely on structured gate sets or an elaborate circuit design
to robustly perform gate set tomography and is therefore more broadly
applicable than traditional approaches.Comment: 14+12 pages, several figures and diagram
Closed-form analytic expressions for shadow estimation with brickwork circuits
Properties of quantum systems can be estimated using classical shadows, which
implement measurements based on random ensembles of unitaries. Originally
derived for global Clifford unitaries and products of single-qubit Clifford
gates, practical implementations are limited to the latter scheme for moderate
numbers of qubits. Beyond local gates, the accurate implementation of very
short random circuits with two-local gates is still experimentally feasible
and, therefore, interesting for implementing measurements in near-term
applications. In this work, we derive closed-form analytical expressions for
shadow estimation using brickwork circuits with two layers of parallel
two-local Haar-random (or Clifford) unitaries. Besides the construction of the
classical shadow, our results give rise to sample-complexity guarantees for
estimating Pauli observables. We then compare the performance of shadow
estimation with brickwork circuits to the established approach using local
Clifford unitaries and find improved sample complexity in the estimation of
observables supported on sufficiently many qubits.Comment: 15+12 pages, several figures; v2: small improvements and new
examples. Close to published versio
Emergent statistical mechanics from properties of disordered random matrix product states
The study of generic properties of quantum states has led to an abundance of
insightful results. A meaningful set of states that can be efficiently prepared
in experiments are ground states of gapped local Hamiltonians, which are well
approximated by matrix product states. In this work, we introduce a picture of
generic states within the trivial phase of matter with respect to their
non-equilibrium and entropic properties: We do so by rigorously exploring
non-translation-invariant matrix product states drawn from a local i.i.d.
Haar-measure. We arrive at these results by exploiting techniques for computing
moments of random unitary matrices and by exploiting a mapping to partition
functions of classical statistical models, a method that has lead to valuable
insights on local random quantum circuits. Specifically, we prove that such
disordered random matrix product states equilibrate exponentially well with
overwhelming probability under the time evolution of Hamiltonians featuring a
non-degenerate spectrum. Moreover, we prove two results about the entanglement
Renyi entropy: The entropy with respect to sufficiently disconnected subsystems
is generically extensive in the system-size, and for small connected systems
the entropy is almost maximal for sufficiently large bond dimensions.Comment: 11 page
Semi-device-dependent blind quantum tomography
Extracting tomographic information about quantum states is a crucial task in
the quest towards devising high-precision quantum devices. Current schemes
typically require measurement devices for tomography that are a priori
calibrated to a high precision. Ironically, the accuracy of the measurement
calibration is fundamentally limited by the accuracy of state preparation,
establishing a vicious cycle. Here, we prove that this cycle can be broken and
the fundamental dependence on the measurement devices significantly relaxed. We
show that exploiting the natural low-rank structure of quantum states of
interest suffices to arrive at a highly scalable blind tomography scheme with a
classically efficient post-processing algorithm. We further improve the
efficiency of our scheme by making use of the sparse structure of the
calibrations. This is achieved by relaxing the blind quantum tomography problem
to the task of de-mixing a sparse sum of low-rank quantum states. Building on
techniques from model-based compressed sensing, we prove that the proposed
algorithm recovers a low-rank quantum state and the calibration provided that
the measurement model exhibits a restricted isometry property. For generic
measurements, we show that our algorithm requires a close-to-optimal number
measurement settings for solving the blind tomography task. Complementing these
conceptual and mathematical insights, we numerically demonstrate that blind
quantum tomography is possible by exploiting low-rank assumptions in a
practical setting inspired by an implementation of trapped ions using
constrained alternating optimization.Comment: 22 pages, 8 Figure
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