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Register a new userA Novel Algebraic Geometry Compiling Framework for Adiabatic Quantum Computations
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Adiabatic Quantum Computing (AQC) is an attractive paradigm for solving hard integer polynomial optimization problems. Available hardware restricts the Hamiltonians to be of a structure that allows only pairwise interactions. This requires that the original optimization problem to be first converted -- from its polynomial form -- to a quadratic unconstrained binary optimization (QUBO) problem, which we frame as a problem in algebraic geometry. Additionally, the hardware graph where such a QUBO-Hamiltonian needs to be embedded -- assigning variables of the problem to the qubits of the physical optimizer -- is not a complete graph, but rather one with limited connectivity. This problem graph to hardware graph embedding can also be framed as a problem of computing a Groebner basis of a certain specially constructed polynomial ideal. We develop a systematic computational approach to prepare a given polynomial optimization problem for AQC in three steps. The first step reduces an input polynomial optimization problem into a QUBO through the computation of the Groebner basis of a toric ideal generated from the monomials of the input objective function. The second step computes feasible embeddings. The third step computes the spectral gap of the adiabatic Hamiltonian associated to a given embedding. These steps are applicable well beyond the integer polynomial optimization problem. Our paper provides the first general purpose computational procedure that can be used directly as a $translator$ to solve polynomial integer optimization. Alternatively, it can be used as a test-bed (with small size problems) to help design efficient heuristic quantum compilers by studying various choices of reductions and embeddings in a systematic and comprehensive manner. An added benefit of our framework is in designing Ising architectures through the study of $mathcal Y-$minor universal graphs.
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We describe a cohomological framework for measurement based quantum computation, in which symmetry plays a central role. Therein, the essential information about the computational output is contained in topological invariants, namely elements of two cohomology groups. One of those invariants applies to the deterministic case, and the other to the general probabilistic case. The same invariants also witness quantumness in the form of contextuality. In result, they give rise to fundamental algebraic structures underlying quantum computation.
We describe a general methodology for enhancing the efficiency of adiabatic quantum computations (AQC). It consists of homotopically deforming the original Hamiltonian surface in a way that the redistribution of the Gaussian curvature weakens the effect of the anti-crossing, thus yielding the desired improvement. Our approach is not pertubative but instead is built on our previous global description of AQC in the language of Morse theory. Through the homotopy deformation we witness the birth and death of critical points whilst, in parallel, the Gauss-Bonnet theorem reshuffles the curvature around the changing set of critical points. Therefore, by creating enough critical points around the anti-crossing, the total curvature--which was initially centered at the original anti-crossing--gets redistributed around the new neighbouring critical points, which weakens its severity and so improves the speedup of the AQC. We illustrate this on two examples taken from the literature.
We import the tools of Morse theory to study quantum adiabatic evolution, the core mechanism in adiabatic quantum computations (AQC). AQC is computationally equivalent to the (pre-eminent paradigm) of the Gate model but less error-prone, so it is ideally suitable to practically tackle a large number of important applications. AQC remains, however, poorly understood theoretically and its mathematical underpinnings are yet to be satisfactorily identified. Through Morse theory, we bring a novel perspective that we expect will open the door for using such mathematics in the realm of quantum computations, providing a secure foundation for AQC. Here we show that the singular homology of a certain cobordism, which we construct from the given Hamiltonian, defines the adiabatic evolution. Our result is based on E. Wittens construction for Morse homology that was derived in the very different context of supersymmetric quantum mechanics. We investigate how such topological description, in conjunction with Gauss-Bonnet theorem and curvature based reformulation of Morse lemma, can be an obstruction to any computational advantage in AQC. We also explore Conley theory, for the sake of completeness, in advance of any known practical Hamiltonian of interest. We conclude with the instructive case of the ferromagnetic $p-$spin where we show that changing its first order quantum transition (QPT) into a second order QPT, by adding non-stoquastic couplings, amounts to homotopically deform the initial surface accompanied with birth of pairs of critical points. Their number reaches its maximum when the system is fully non-stoquastic. In parallel, the total Gaussian curvature gets redistributed (by the Gauss--Bonnet theorem) around the new neighbouring critical points, which weakens the severity of the QPT.
The design and implementation of parallel algorithms is a fundamental task in computer algebra. Combining the computer algebra system Singular and the workflow management system GPI-Space, we have developed an infrastructure for massively parallel computations in commutative algebra and algebraic geometry. In this note, we give an overview on the current capabilities of this framework by looking into three sample applications: determining smoothness of algebraic varieties, computing GIT-fans in geometric invariant theory, and determining tropicalizations. These applications employ algorithmic methods originating from commutative algebra, sheaf structures on manifolds, local geometry, convex geometry, group theory, and combinatorics, illustrating the potential of the framework in further problems in computer algebra.
A method for compiling quantum algorithms into specific braiding patterns for non-Abelian quasiparticles described by the so-called Fibonacci anyon model is developed. The method is based on the observation that a universal set of quantum gates acting on qubits encoded using triplets of these quasiparticles can be built entirely out of three-stranded braids (three-braids). These three-braids can then be efficiently compiled and improved to any required accuracy using the Solovay-Kitaev algorithm.
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