01616nas a2200121 4500008004100000245005900041210005800100260001400158520124400172100002301416700001801439856003701457 2020 eng d00aCan graph properties have exponential quantum speedup?0 aCan graph properties have exponential quantum speedup c1/28/20203 a

Quantum computers can sometimes exponentially outperform classical ones, but only for problems with sufficient structure. While it is well known that query problems with full permutation symmetry can have at most polynomial quantum speedup -- even for partial functions -- it is unclear how far this condition must be relaxed to enable exponential speedup. In particular, it is natural to ask whether exponential speedup is possible for (partial) graph properties, in which the input describes a graph and the output can only depend on its isomorphism class. We show that the answer to this question depends strongly on the input model. In the adjacency matrix model, we prove that the bounded-error randomized query complexity R of any graph property P has R(P)=O(Q(P)6), where Q is the bounded-error quantum query complexity. This negatively resolves an open question of Montanaro and de Wolf in the adjacency matrix model. More generally, we prove R(P)=O(Q(P)3l) for any l-uniform hypergraph property P in the adjacency matrix model. In direct contrast, in the adjacency list model for bounded-degree graphs, we exhibit a promise problem that shows an exponential separation between the randomized and quantum query complexities.

1 aChilds, Andrew, M.1 aWang, Daochen uhttps://arxiv.org/abs/2001.1052001730nas a2200145 4500008004100000245005400041210005400095260001500149490000900164520131300173100002301486700001901509700001901528856003701547 2019 eng d00aCircuit Transformations for Quantum Architectures0 aCircuit Transformations for Quantum Architectures c02/25/20190 v135 3 a

Quantum computer architectures impose restrictions on qubit interactions. We propose efficient circuit transformations that modify a given quantum circuit to fit an architecture, allowing for any initial and final mapping of circuit qubits to architecture qubits. To achieve this, we first consider the qubit movement subproblem and use the routing via matchings framework to prove tighter bounds on parallel routing. In practice, we only need to perform partial permutations, so we generalize routing via matchings to that setting. We give new routing procedures for common architecture graphs and for the generalized hierarchical product of graphs, which produces subgraphs of the Cartesian product. Secondly, for serial routing, we consider the token swapping framework and extend a 4-approximation algorithm for general graphs to support partial permutations. We apply these routing procedures to give several circuit transformations, using various heuristic qubit placement subroutines. We implement these transformations in software and compare their performance for large quantum circuits on grid and modular architectures, identifying strategies that work well in practice.

1 aChilds, Andrew, M.1 aSchoute, Eddie1 aUnsal, Cem, M. uhttps://arxiv.org/abs/1902.0910200953nas a2200157 4500008004100000245006400041210006400105260001500169300000900184490000700193520050200200100002300702700001800725700001400743856003800757 2016 eng d00aComplexity of the XY antiferromagnet at fixed magnetization0 aComplexity of the XY antiferromagnet at fixed magnetization c2016/01/01 a1-180 v163 a We prove that approximating the ground energy of the antiferromagnetic XY model on a simple graph at fixed magnetization (given as part of the instance specification) is QMA-complete. To show this, we strengthen a previous result by establishing QMA-completeness for approximating the ground energy of the Bose-Hubbard model on simple graphs. Using a connection between the XY and Bose-Hubbard models that we exploited in previous work, this establishes QMA-completeness of the XY model. 1 aChilds, Andrew, M.1 aGosset, David1 aWebb, Zak uhttp://arxiv.org/abs/1503.07083v101064nas a2200145 4500008004100000245008600041210006900127260001500196300001200211490000700223520060700230100002100837700002300858856003700881 2014 eng d00aThe computational power of matchgates and the XY interaction on arbitrary graphs0 acomputational power of matchgates and the XY interaction on arbi c2014/09/01 a901-9160 v143 a Matchgates are a restricted set of two-qubit gates known to be classically simulable when acting on nearest-neighbor qubits on a path, but universal for quantum computation when the qubits are arranged on certain other graphs. Here we characterize the power of matchgates acting on arbitrary graphs. Specifically, we show that they are universal on any connected graph other than a path or a cycle, and that they are classically simulable on a cycle. We also prove the same dichotomy for the XY interaction, a proper subset of matchgates related to some implementations of quantum computing. 1 aBrod, Daniel, J.1 aChilds, Andrew, M. uhttp://arxiv.org/abs/1308.1463v101598nas a2200157 4500008004100000245007300041210006900114260001500183300001100198490000600209520112600215100002301341700001501364700002401379856003701403 2014 eng d00aConstructing elliptic curve isogenies in quantum subexponential time0 aConstructing elliptic curve isogenies in quantum subexponential  c2014/01/01 a1 - 290 v83 a Given two elliptic curves over a finite field having the same cardinality and endomorphism ring, it is known that the curves admit an isogeny between them, but finding such an isogeny is believed to be computationally difficult. The fastest known classical algorithm takes exponential time, and prior to our work no faster quantum algorithm was known. Recently, public-key cryptosystems based on the presumed hardness of this problem have been proposed as candidates for post-quantum cryptography. In this paper, we give a subexponential-time quantum algorithm for constructing isogenies, assuming the Generalized Riemann Hypothesis (but with no other assumptions). Our algorithm is based on a reduction to a hidden shift problem, together with a new subexponential-time algorithm for evaluating isogenies from kernel ideals (under only GRH), and represents the first nontrivial application of Kuperberg's quantum algorithm for the hidden shift problem. This result suggests that isogeny-based cryptosystems may be uncompetitive with more mainstream quantum-resistant cryptosystems such as lattice-based cryptosystems. 1 aChilds, Andrew, M.1 aJao, David1 aSoukharev, Vladimir uhttp://arxiv.org/abs/1012.4019v201458nas a2200145 4500008004100000245005700041210005600098260001500154520102700169100002301196700001801219700002101237700001701258856003701275 2010 eng d00aCharacterization of universal two-qubit Hamiltonians0 aCharacterization of universal twoqubit Hamiltonians c2010/04/093 a Suppose we can apply a given 2-qubit Hamiltonian H to any (ordered) pair of qubits. We say H is n-universal if it can be used to approximate any unitary operation on n qubits. While it is well known that almost any 2-qubit Hamiltonian is 2-universal (Deutsch, Barenco, Ekert 1995; Lloyd 1995), an explicit characterization of the set of non-universal 2-qubit Hamiltonians has been elusive. Our main result is a complete characterization of 2-non-universal 2-qubit Hamiltonians. In particular, there are three ways that a 2-qubit Hamiltonian H can fail to be universal: (1) H shares an eigenvector with the gate that swaps two qubits, (2) H acts on the two qubits independently (in any of a certain family of bases), or (3) H has zero trace. A 2-non-universal 2-qubit Hamiltonian can still be n-universal for some n >= 3. We give some partial results on 3-universality. Finally, we also show how our characterization of 2-universal Hamiltonians implies the well-known result that almost any 2-qubit unitary is universal. 1 aChilds, Andrew, M.1 aLeung, Debbie1 aMancinska, Laura1 aOzols, Maris uhttp://arxiv.org/abs/1004.1645v2