26,036 research outputs found
The Complexity of Simulating Local Measurements on Quantum Systems
An important task in quantum physics is the estimation of local quantities for ground states of local Hamiltonians. Recently, Ambainis defined the complexity class P^QMA[log], and motivated its study by showing that the physical task of estimating the expectation value of a local observable against the ground state of a local Hamiltonian is P^QMA[log]-complete. In this paper, we continue the study of P^QMA[log], obtaining the following results.
The P^QMA[log]-completeness result of Ambainis requires O(log n)-local observ- ables and Hamiltonians. We show that simulating even a single qubit measurement on ground states of 5-local Hamiltonians is P^QMA[log]-complete, resolving an open question of Ambainis. We formalize the complexity theoretic study of estimating two-point correlation functions against ground states, and show that this task is similarly P^QMA[log]-complete.
P^QMA[log] is thought of as "slightly harder" than QMA. We justify this formally by exploiting the hierarchical voting technique of Beigel, Hemachandra, and Wechsung to show P^QMA[log] subseteq PP. This improves the containment QMA subseteq PP from Kitaev and Watrous. A central theme of this work is the subtlety involved in the study of oracle classes in which the oracle solves a promise problem. In this vein, we identify a flaw in Ambainis\u27 prior work regarding a P^UQMA[log]-hardness proof for estimating spectral gaps of local Hamiltonians. By introducing a "query validation" technique, we build on his prior work to obtain P^UQMA[log]-hardness for estimating spectral gaps under polynomial-time Turing reductions
Tensor Norms and the Classical Communication Complexity of Nonlocal Quantum Measurement
We initiate the study of quantifying nonlocalness of a bipartite measurement
by the minimum amount of classical communication required to simulate the
measurement. We derive general upper bounds, which are expressed in terms of
certain tensor norms of the measurement operator. As applications, we show that
(a) If the amount of communication is constant, quantum and classical
communication protocols with unlimited amount of shared entanglement or shared
randomness compute the same set of functions; (b) A local hidden variable model
needs only a constant amount of communication to create, within an arbitrarily
small statistical distance, a distribution resulted from local measurements of
an entangled quantum state, as long as the number of measurement outcomes is
constant.Comment: A preliminary version of this paper appears as part of an article in
Proceedings of the the 37th ACM Symposium on Theory of Computing (STOC 2005),
460--467, 200
Optimal classical simulation of state-independent quantum contextuality
Simulating quantum contextuality with classical systems requires memory. A
fundamental yet open question is what is the minimum memory needed and,
therefore, the precise sense in which quantum systems outperform classical
ones. Here, we make rigorous the notion of classically simulating quantum
state-independent contextuality (QSIC) in the case of a single quantum system
submitted to an infinite sequence of measurements randomly chosen from a finite
QSIC set. We obtain the minimum memory needed to simulate arbitrary QSIC sets
via classical systems under the assumption that the simulation should not
contain any oracular information. In particular, we show that, while
classically simulating two qubits tested with the Peres-Mermin set requires
bits, simulating a single qutrit tested with the
Yu-Oh set requires, at least, bits.Comment: 7 pages, 4 figure
Quantum computing and the entanglement frontier - Rapporteur talk at the 25th Solvay Conference
Quantum information science explores the frontier of highly complex quantum states,
the "entanglement frontier". This study is motivated by the observation (widely believed
but unproven) that classical systems cannot simulate highly entangled quantum systems
efficiently, and we hope to hasten the day when well controlled quantum systems can
perform tasks surpassing what can be done in the classical world. One way to achieve
such "quantum supremacy" would be to run an algorithm on a quantum computer which
solves a problem with a super-polynomial speedup relative to classical computers, but
there may be other ways that can be achieved sooner, such as simulating exotic quantum
states of strongly correlated matter. To operate a large scale quantum computer reliably
we will need to overcome the debilitating effects of decoherence, which might be done
using "standard" quantum hardware protected by quantum error-correcting codes, or by
exploiting the nonabelian quantum statistics of anyons realized in solid state systems,
or by combining both methods. Only by challenging the entanglement frontier will we
learn whether Nature provides extravagant resources far beyond what the classical world
would allow
Oracle Complexity Classes and Local Measurements on Physical Hamiltonians
The canonical problem for the class Quantum Merlin-Arthur (QMA) is that of
estimating ground state energies of local Hamiltonians. Perhaps surprisingly,
[Ambainis, CCC 2014] showed that the related, but arguably more natural,
problem of simulating local measurements on ground states of local Hamiltonians
(APX-SIM) is likely harder than QMA. Indeed, [Ambainis, CCC 2014] showed that
APX-SIM is P^QMA[log]-complete, for P^QMA[log] the class of languages decidable
by a P machine making a logarithmic number of adaptive queries to a QMA oracle.
In this work, we show that APX-SIM is P^QMA[log]-complete even when restricted
to more physical Hamiltonians, obtaining as intermediate steps a variety of
related complexity-theoretic results.
We first give a sequence of results which together yield P^QMA[log]-hardness
for APX-SIM on well-motivated Hamiltonians: (1) We show that for NP, StoqMA,
and QMA oracles, a logarithmic number of adaptive queries is equivalent to
polynomially many parallel queries. These equalities simplify the proofs of our
subsequent results. (2) Next, we show that the hardness of APX-SIM is preserved
under Hamiltonian simulations (a la [Cubitt, Montanaro, Piddock, 2017]). As a
byproduct, we obtain a full complexity classification of APX-SIM, showing it is
complete for P, P^||NP, P^||StoqMA, or P^||QMA depending on the Hamiltonians
employed. (3) Leveraging the above, we show that APX-SIM is P^QMA[log]-complete
for any family of Hamiltonians which can efficiently simulate spatially sparse
Hamiltonians, including physically motivated models such as the 2D Heisenberg
model.
Our second focus considers 1D systems: We show that APX-SIM remains
P^QMA[log]-complete even for local Hamiltonians on a 1D line of 8-dimensional
qudits. This uses a number of ideas from above, along with replacing the "query
Hamiltonian" of [Ambainis, CCC 2014] with a new "sifter" construction.Comment: 38 pages, 3 figure
Introduction to Quantum Algorithms for Physics and Chemistry
In this introductory review, we focus on applications of quantum computation
to problems of interest in physics and chemistry. We describe quantum
simulation algorithms that have been developed for electronic-structure
problems, thermal-state preparation, simulation of time dynamics, adiabatic
quantum simulation, and density functional theory.Comment: 44 pages, 5 figures; comments or suggestions for improvement are
welcom
Non-classicality of temporal correlations
The results of space-like separated measurements are independent of distant
measurement settings, a property one might call two-way no-signalling. In
contrast, time-like separated measurements are only one-way no-signalling since
the past is independent of the future but not vice-versa. For this reason
temporal correlations that are formally identical to non-classical spatial
correlations can still be modelled classically. We define non-classical
temporal correlations as the ones which cannot be simulated by propagating in
time a classical information content of a quantum system. We first show that
temporal correlations between results of any projective quantum measurements on
a qubit can be simulated classically. Then we present a sequence of POVM
measurements on a single -level quantum system that cannot be explained by
propagating in time -level classical system and using classical computers
with unlimited memory.Comment: 6 pages, 1 figur
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