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

Wrack Writing (Selections)

How does one write—about bodies, sensations, the more-than-human world—in the midst of, and in response to, the mounting devastation that settler colonial capitalism continues to wreak on lands, waters and relationships? Theodor Adorno’s (1983 [1967], p. 34) diversely interpreted statement that ‘to write poetry after Auschwitz is barbaric’ resonates strongly at the current moment: what does it mean to write, and especially to write beautifully, in conditions that are permeated with colonial violence and capitalist devastation? How do we, as feminist writers, imagine our words as witnessing, or even as politicising, these violences? How can feminist lyrical writing sharpen our longing for justice rather than serve as an alibi for continued dispossession and commodification (including the commodification of our writing in the neoliberal university)? Are there practices of writing self-consciously ‘in the wrack zone’ that might help us develop new forms, processes and conversations to inspire and narrate reflection and resistance

The Role of PI3K Isoforms in Regulating Bone Marrow Microenvironment Signaling Focusing on Acute Myeloid Leukemia and Multiple Myeloma

Despite the development of novel treatments in the past 15 years, many blood cancers still remain ultimately fatal and difficult to treat, particularly acute myeloid leukaemia (AML) and multiple myeloma (MM). While significant progress has been made characterising small-scale genetic mutations and larger-scale chromosomal translocations that contribute to the development of various blood cancers, less is understood about the complex microenvironment of the bone marrow (BM), which is known to be a key player in the pathogenesis of chronic lymphocytic leukaemia (CLL), AML and MM. This niche acts as a sanctuary for the cancerous cells, protecting them from chemotherapeutics and encouraging clonal cell survival. It does this by upregulating a plethora of signalling cascades within the malignant cell, with the phosphatidylinositol-3-kinase (PI3K) pathway taking a critical role. This review will focus on how the PI3K pathway influences disease progression and the individualised role of the PI3K subunits. We will also summarise the current clinical trials for PI3K inhibitors and how these trials impact the treatment of blood cancers