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

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

The complexity of translationally-invariant low-dimensional spin lattices in 3D

In this theoretical paper, we consider spin systems in three spatial dimensions and consider the computational complexity of estimating the ground state energy, known as the local Hamiltonian problem, for translationally invariant Hamiltonians. We prove that the local Hamiltonian problem for 3D lattices with face-centered cubic unit cells and 4-local translationally invariant interactions between spin-3/2 particles and open boundary conditions is QMAEXP-complete, where QMAEXP is the class of problems which can be verified in exponential time on a quantum computer. We go beyond a mere embedding of past hard 1D history state constructions, for which the local spin dimension is enormous: even state-of-the-art constructions have local dimension 42. We avoid such a large local dimension by combining some different techniques in a novel way. For the verifier circuit which we embed into the ground space of the local Hamiltonian, we utilize a recently developed computational model, called a quantum ring machine, which is especially well suited for translationally invariant history state constructions. This is encoded with a new and particularly simple universal gate set, which consists of a single 2-qubit gate applied only to nearest-neighbour qubits. The Hamiltonian construction involves a classical Wang tiling problem as a binary counter which translates one cube side length into a binary description for the encoded verifier input and a carefully engineered history state construction that implements the ring machine on the cubic lattice faces. These novel techniques allow us to significantly lower the local spin dimension, surpassing the best translationally invariant result to date by two orders of magnitude (in the number of degrees of freedom per coupling). This brings our models on par with the best non-translationally invariant constructio