34 research outputs found
Computational Complexity of interacting electrons and fundamental limitations of Density Functional Theory
One of the central problems in quantum mechanics is to determine the ground
state properties of a system of electrons interacting via the Coulomb
potential. Since its introduction by Hohenberg, Kohn, and Sham, Density
Functional Theory (DFT) has become the most widely used and successful method
for simulating systems of interacting electrons, making their original work one
of the most cited in physics. In this letter, we show that the field of
computational complexity imposes fundamental limitations on DFT, as an
efficient description of the associated universal functional would allow to
solve any problem in the class QMA (the quantum version of NP) and thus
particularly any problem in NP in polynomial time. This follows from the fact
that finding the ground state energy of the Hubbard model in an external
magnetic field is a hard problem even for a quantum computer, while given the
universal functional it can be computed efficiently using DFT. This provides a
clear illustration how the field of quantum computing is useful even if quantum
computers would never be built.Comment: 8 pages, 3 figures. v2: Version accepted at Nature Physics; differs
significantly from v1 (including new title). Includes an extra appendix (not
contained in the journal version) on the NP-completeness of Hartree-Fock,
which is taken from v
Decoherence induced deformation of the ground state in adiabatic quantum computation
Despite more than a decade of research on adiabatic quantum computation
(AQC), its decoherence properties are still poorly understood. Many theoretical
works have suggested that AQC is more robust against decoherence, but a
quantitative relation between its performance and the qubits' coherence
properties, such as decoherence time, is still lacking. While the thermal
excitations are known to be important sources of errors, they are predominantly
dependent on temperature but rather insensitive to the qubits' coherence. Less
understood is the role of virtual excitations, which can also reduce the ground
state probability even at zero temperature. Here, we introduce normalized
ground state fidelity as a measure of the decoherence-induced deformation of
the ground state due to virtual transitions. We calculate the normalized
fidelity perturbatively at finite temperatures and discuss its relation to the
qubits' relaxation and dephasing times, as well as its projected scaling
properties.Comment: 10 pages, 3 figure
Solving Quantum Ground-State Problems with Nuclear Magnetic Resonance
Quantum ground-state problems are computationally hard problems; for general
many-body Hamiltonians, there is no classical or quantum algorithm known to be
able to solve them efficiently. Nevertheless, if a trial wavefunction
approximating the ground state is available, as often happens for many problems
in physics and chemistry, a quantum computer could employ this trial
wavefunction to project the ground state by means of the phase estimation
algorithm (PEA). We performed an experimental realization of this idea by
implementing a variational-wavefunction approach to solve the ground-state
problem of the Heisenberg spin model with an NMR quantum simulator. Our
iterative phase estimation procedure yields a high accuracy for the
eigenenergies (to the 10^-5 decimal digit). The ground-state fidelity was
distilled to be more than 80%, and the singlet-to-triplet switching near the
critical field is reliably captured. This result shows that quantum simulators
can better leverage classical trial wavefunctions than classical computers.Comment: 11 pages, 13 figure
Quantum adiabatic machine learning
We develop an approach to machine learning and anomaly detection via quantum
adiabatic evolution. In the training phase we identify an optimal set of weak
classifiers, to form a single strong classifier. In the testing phase we
adiabatically evolve one or more strong classifiers on a superposition of
inputs in order to find certain anomalous elements in the classification space.
Both the training and testing phases are executed via quantum adiabatic
evolution. We apply and illustrate this approach in detail to the problem of
software verification and validation.Comment: 21 pages, 9 figure
Entanglement scaling in quantum advantage benchmarks
A contemporary technological milestone is to build a quantum device
performing a computational task beyond the capability of any classical
computer, an achievement known as quantum adversarial advantage. In what ways
can the entanglement realized in such a demonstration be quantified? Inspired
by the area law of tensor networks, we derive an upper bound for the minimum
random circuit depth needed to generate the maximal bipartite entanglement
correlations between all problem variables (qubits). This bound is (i) lattice
geometry dependent and (ii) makes explicit a nuance implicit in other proposals
with physical consequence. The hardware itself should be able to support
super-logarithmic ebits of entanglement across some poly() number of
qubit-bipartitions, otherwise the quantum state itself will not possess
volumetric entanglement scaling and full-lattice-range correlations. Hence, as
we present a connection between quantum advantage protocols and quantum
entanglement, the entanglement implicitly generated by such protocols can be
tested separately to further ascertain the validity of any quantum advantage
claim
Reachability Deficits in Quantum Approximate Optimization.
The quantum approximate optimization algorithm (QAOA) has rapidly become a cornerstone of contemporary quantum algorithm development. Despite a growing range of applications, only a few results have been developed towards understanding the algorithm's ultimate limitations. Here we report that QAOA exhibits a strong dependence on a problem instances constraint to variable ratio-this problem density places a limiting restriction on the algorithms capacity to minimize a corresponding objective function (and hence solve optimization problem instances). Such reachability deficits persist even in the absence of barren plateaus and are outside of the recently reported level-1 QAOA limitations. These findings are among the first to determine strong limitations on variational quantum approximate optimization
Community Detection in Quantum Complex Networks
Determining community structure is a central topic in the study of complex
networks, be it technological, social, biological or chemical, in static or
interacting systems. In this paper, we extend the concept of community
detection from classical to quantum systems---a crucial missing component of a
theory of complex networks based on quantum mechanics. We demonstrate that
certain quantum mechanical effects cannot be captured using current classical
complex network tools and provide new methods that overcome these problems. Our
approaches are based on defining closeness measures between nodes, and then
maximizing modularity with hierarchical clustering. Our closeness functions are
based on quantum transport probability and state fidelity, two important
quantities in quantum information theory. To illustrate the effectiveness of
our approach in detecting community structure in quantum systems, we provide
several examples, including a naturally occurring light-harvesting complex,
LHCII. The prediction of our simplest algorithm, semiclassical in nature,
mostly agrees with a proposed partitioning for the LHCII found in quantum
chemistry literature, whereas our fully quantum treatment of the problem
uncovers a new, consistent, and appropriately quantum community structure
Chiral Quantum Walks
Wigner separated the possible types of symmetries in quantum theory into those symmetries that are unitary and those that are antiunitary. Unitary symmetries have been well studied whereas antiunitary symmetries and the physical implications associated with time-reversal symmetry breaking have had little influence on quantum information science. Here we develop a quantum circuits version of time-reversal symmetry theory, classifying time-symmetric and time-asymmetric Hamiltonians and circuits in terms of their underlying network elements and geometric structures. These results reveal that many of the typical quantum circuit networks found across the field of quantum information science exhibit time-asymmetry. We then experimentally implement the most fundamental time-reversal asymmetric process, applying local gates in an otherwise time-symmetric circuit to induce time-reversal asymmetry and thereby achieve (i) directional biasing in the transition probability between basis states, (ii) the enhancement of and (iii) the suppression of these transport probabilities. Our results imply that the physical effect of time-symmetry breaking plays an essential role in coherent transport and its control represents an omnipresent yet essentially untapped resource in quantum transport science