44 research outputs found

### The complexity of antiferromagnetic interactions and 2D lattices

Estimation of the minimum eigenvalue of a quantum Hamiltonian can be
formalised as the Local Hamiltonian problem. We study the natural special case
of the Local Hamiltonian problem where the same 2-local interaction, with
differing weights, is applied across each pair of qubits. First we consider
antiferromagnetic/ferromagnetic interactions, where the weights of the terms in
the Hamiltonian are restricted to all be of the same sign. We show that for
symmetric 2-local interactions with no 1-local part, the problem is either
QMA-complete or in StoqMA. In particular the antiferromagnetic Heisenberg and
antiferromagnetic XY interactions are shown to be QMA-complete. We also prove
StoqMA-completeness of the antiferromagnetic transverse field Ising model.
Second, we study the Local Hamiltonian problem under the restriction that the
interaction terms can only be chosen to lie on a particular graph. We prove
that nearly all of the QMA-complete 2-local interactions remain QMA-complete
when restricted to a 2D square lattice. Finally we consider both restrictions
at the same time and discover that, with the exception of the antiferromagnetic
Heisenberg interaction, all of the interactions which are QMA-complete with
positive coefficients remain QMA-complete when restricted to a 2D triangular
lattice.Comment: 35 pages, 11 figures; v2 added reference

### 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

### Universal Quantum Hamiltonians

Quantum many-body systems exhibit an extremely diverse range of phases and
physical phenomena. Here, we prove that the entire physics of any other quantum
many-body system is replicated in certain simple, "universal" spin-lattice
models. We first characterise precisely what it means for one quantum many-body
system to replicate the entire physics of another. We then show that certain
very simple spin-lattice models are universal in this very strong sense.
Examples include the Heisenberg and XY models on a 2D square lattice (with
non-uniform coupling strengths). We go on to fully classify all two-qubit
interactions, determining which are universal and which can only simulate more
restricted classes of models. Our results put the practical field of analogue
Hamiltonian simulation on a rigorous footing and take a significant step
towards justifying why error correction may not be required for this
application of quantum information technology.Comment: 78 pages, 9 figures, 44 theorems etc. v2: Trivial fixes. v3: updated
and simplified proof of Thm. 9; 82 pages, 47 theorems etc. v3: Small fix in
proof of time-evolution lemma (this fix not in published version

### The classical limit of Quantum Max-Cut

It is well-known in physics that the limit of large quantum spin $S$ should
be understood as a semiclassical limit. This raises the question of whether
such emergent classicality facilitates the approximation of computationally
hard quantum optimization problems, such as the local Hamiltonian problem. We
demonstrate this explicitly for spin-$S$ generalizations of Quantum Max-Cut
($\mathrm{QMaxCut}_S$), equivalent to the problem of finding the ground state
energy of an arbitrary spin-$S$ quantum Heisenberg antiferromagnet
($\mathrm{AFH}_S$). We prove that approximating the value of $\mathrm{AFH}_S$
to inverse polynomial accuracy is QMA-complete for all $S$, extending previous
results for $S=1/2$. We also present two distinct families of classical
approximation algorithms for $\mathrm{QMaxCut}_S$ based on rounding the output
of a semidefinite program to a product of Bloch coherent states. The
approximation ratios for both our proposed algorithms strictly increase with
$S$ and converge to the Bri\"et-Oliveira-Vallentin approximation ratio
$\alpha_{\mathrm{BOV}} \approx 0.956$ from below as $S \to \infty$.Comment: 19+4 page

### A Quantum Search Decoder for Natural Language Processing

Funder: Science and Engineering Research Board; doi: https://doi.org/10.13039/501100001843Funder: Cambridge Commonwealth Trust; doi: https://doi.org/10.13039/501100003342Probabilistic language models, e.g. those based on an LSTM, often face the
problem of finding a high probability prediction from a sequence of random
variables over a set of tokens. This is commonly addressed using a form of
greedy decoding such as beam search, where a limited number of
highest-likelihood paths (the beam width) of the decoder are kept, and at the
end the maximum-likelihood path is chosen. In this work, we construct a quantum
algorithm to find the globally optimal parse (i.e. for infinite beam width)
with high constant success probability. When the input to the decoder is
distributed as a power-law with exponent $k>0$, our algorithm has runtime $R^{n
f(R,k)}$, where $R$ is the alphabet size, $n$ the input length; here $f<1/2$,
and $f\rightarrow 0$ exponentially fast with increasing $k$, hence making our
algorithm always more than quadratically faster than its classical counterpart.
We further modify our procedure to recover a finite beam width variant, which
enables an even stronger empirical speedup while still retaining higher
accuracy than possible classically. Finally, we apply this quantum beam search
decoder to Mozilla's implementation of Baidu's DeepSpeech neural net, which we
show to exhibit such a power law word rank frequency

### NADPH oxidase-2 derived superoxide drives mitochondrial transfer from bone marrow stromal cells to leukemic blasts

Improvements in the understanding of the metabolic cross-talk between cancer and its micro-environment are expected to lead to novel therapeutic approaches. Acute myeloid leukemia (AML) cells have increased mitochondria compared to non-malignant CD34+ hematopoietic progenitor cells. Furthermore, contrary to the Warburg hypothesis, (AML) relies on oxidative phosphorylation to generate ATP. Here we report that in human AML, NOX2 generates superoxide which stimulates bone marrow stromal cells (BMSC) to AML blast transfer of mitochondria through AML derived tunnelling nanotubes. Moreover, inhibition of NOX2 was able to prevent mitochondrial transfer, increase AML apoptosis and improve NSG AML mouse survival. Although mitochondrial transfer from BMSC to non-malignant CD34+ cells occurs in response to oxidative stress, NOX2 inhibition had no detectable effect on non-malignant CD34+ cell survival. Taken together we identify tumor-specific dependence on NOX2 driven mitochondrial transfer as a novel therapeutic strategy in AML