88 research outputs found
Properties of thermal quantum states: locality of temperature, decay of correlations, and more
We review several properties of thermal states of spin Hamiltonians with
short range interactions. In particular, we focus on those aspects in which the
application of tools coming from quantum information theory has been specially
successful in the recent years. This comprises the study of the correlations at
finite and zero temperature, the stability against distant and/or weak
perturbations, the locality of temperature and their classical simulatability.
For the case of states with a finite correlation length, we overview the
results on their energy distribution and the equivalence of the canonical and
microcanonical ensemble.Comment: v1: 10 pages, 4 figures; v2: minor changes, close to published
versio
Lieb-Robinson bounds and the simulation of time evolution of local observables in lattice systems
This is an introductory text reviewing Lieb-Robinson bounds for open and
closed quantum many-body systems. We introduce the Heisenberg picture for
time-dependent local Liouvillians and state a Lieb-Robinson bound that gives
rise to a maximum speed of propagation of correlations in many body systems of
locally interacting spins and fermions. Finally, we discuss a number of
important consequences concerning the simulation of time evolution and
properties of ground states and stationary states.Comment: 13 pages, 2 figures; book chapte
Improving compressed sensing with the diamond norm
In low-rank matrix recovery, one aims to reconstruct a low-rank matrix from a
minimal number of linear measurements. Within the paradigm of compressed
sensing, this is made computationally efficient by minimizing the nuclear norm
as a convex surrogate for rank.
In this work, we identify an improved regularizer based on the so-called
diamond norm, a concept imported from quantum information theory. We show that
-for a class of matrices saturating a certain norm inequality- the descent cone
of the diamond norm is contained in that of the nuclear norm. This suggests
superior reconstruction properties for these matrices. We explicitly
characterize this set of matrices. Moreover, we demonstrate numerically that
the diamond norm indeed outperforms the nuclear norm in a number of relevant
applications: These include signal analysis tasks such as blind matrix
deconvolution or the retrieval of certain unitary basis changes, as well as the
quantum information problem of process tomography with random measurements.
The diamond norm is defined for matrices that can be interpreted as order-4
tensors and it turns out that the above condition depends crucially on that
tensorial structure. In this sense, this work touches on an aspect of the
notoriously difficult tensor completion problem.Comment: 25 pages + Appendix, 7 Figures, published versio
Guaranteed efficient energy estimation of quantum many-body Hamiltonians using ShadowGrouping
Estimation of the energy of quantum many-body systems is a paradigmatic task
in various research fields. In particular, efficient energy estimation may be
crucial in achieving a quantum advantage for a practically relevant problem.
For instance, the measurement effort poses a critical bottleneck for
variational quantum algorithms.
We aim to find the optimal strategy with single-qubit measurements that
yields the highest provable accuracy given a total measurement budget. As a
central tool, we establish new tail bounds for empirical estimators of the
energy. They are helpful for identifying measurement settings that improve the
energy estimate the most. This task constitutes an NP-hard problem. However, we
are able to circumvent this bottleneck and use the tail bounds to develop a
practical, efficient estimation strategy, which we call ShadowGrouping. As the
name suggests, it combines shadow estimation methods with grouping strategies
for Pauli strings. In numerical experiments, we demonstrate that ShadowGrouping
outperforms state-of-the-art methods in estimating the electronic ground-state
energies of various small molecules, both in provable and practical accuracy
benchmarks. Hence, this work provides a promising way, e.g., to tackle the
measurement bottleneck associated with quantum many-body Hamiltonians.Comment: 14+6 pages, 5+0 figures. v2: revisions in structure of main text and
addition of the schematic figure 1. Presented at TQC 202
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
Time-optimal multi-qubit gates: Complexity, efficient heuristic and gate-time bounds
Multi-qubit interactions are omnipresent in quantum computing hardware, and
they can generate multi-qubit entangling gates. Such gates promise advantages
over traditional two-qubit gates. In this work, we focus on the quantum gate
synthesis with multi-qubit Ising-type interactions and single-qubit gates.
These interactions can generate global ZZ-gates (GZZ gates). We show that the
synthesis of time-optimal multi-qubit gates is NP-hard. However, under certain
assumptions we provide explicit constructions of time-optimal multi-qubit gates
allowing for efficient synthesis. These constructed multi-qubit gates have a
constant gate time and can be implemented with linear single-qubit gate layers.
Moreover, a heuristic algorithm with polynomial runtime for synthesizing fast
multi-qubit gates is provided. Finally, we prove lower and upper bounds on the
optimal GZZ gate-time. Furthermore, we conjecture that any GZZ gate can be
executed in a time O(n) for n qubits. We support this claim with theoretical
and numerical results.Comment: 11+2 pages, 2 figure
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