How symmetric is too symmetric for large quantum speedups?

• URL: http://arxiv.org/abs/2001.09642v1
• Date: Mon, 27 Jan 2020 09:34:01 GMT
• Title: How symmetric is too symmetric for large quantum speedups?
• Authors: Shalev Ben-David and Supartha Podder
• Abstract summary: We make steps towards understanding the group actions $G$ which are "quantum intolerant" We show that a "well-shuffling" property of group actions -- which happens to be preserved by several natural transformations -- does not allow a quantum speedup. This means no graph property testing problems can have super-polynomial quantum speedups.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Suppose a Boolean function $f$ is symmetric under a group action $G$ acting on the $n$ bits of the input. For which $G$ does this mean $f$ does not have an exponential quantum speedup? Is there a characterization of how rich $G$ must be before the function $f$ cannot have enough structure for quantum algorithms to exploit? In this work, we make several steps towards understanding the group actions $G$ which are "quantum intolerant" in this way. We show that sufficiently transitive group actions do not allow a quantum speedup, and that a "well-shuffling" property of group actions -- which happens to be preserved by several natural transformations -- implies a lack of super-polynomial speedups for functions symmetric under the group action. Our techniques are motivated by a recent paper by Chailloux (2018), which deals with the case where $G=S_n$. Our main application is for graph symmetries: we show that any Boolean function $f$ defined on the adjacency matrix of a graph (and symmetric under relabeling the vertices of the graph) has a power $6$ relationship between its randomized and quantum query complexities, even if $f$ is a partial function. In particular, this means no graph property testing problems can have super-polynomial quantum speedups, settling an open problem of Ambainis, Childs, and Liu (2011).

Related papers

• A one-query lower bound for unitary synthesis and breaking quantum cryptography [7.705803563459633]
  The Unitary Synthesis Problem asks whether any $n$qubit unitary $U$ can be implemented by an efficient quantum $A$ augmented with an oracle that computes an arbitrary Boolean function $f$. In this work, we prove unitary synthesis as an efficient challenger-ad game, which enables proving lower bounds by analyzing the maximum success probability of an adversary $Af$.
  arXiv  Detail & Related papers  (2023-10-13T05:39:42Z)
• On the universality of $S_n$-equivariant $k$-body gates [0.0]
  We study how the interplay between symmetry and $k$-bodyness in the QNN generators affect its expressiveness. Our results bring us a step closer to better understanding the capabilities and limitations of equivariant QNNs.
  arXiv  Detail & Related papers  (2023-03-01T18:42:14Z)
• Quantum and classical low-degree learning via a dimension-free Remez inequality [52.12931955662553]
  We show a new way to relate functions on the hypergrid to their harmonic extensions over the polytorus. We show the supremum of a function $f$ over products of the cyclic group $exp(2pi i k/K)_k=1K$. We extend to new spaces a recent line of work citeEI22, CHP, VZ22 that gave similarly efficient methods for learning low-degrees on hypercubes and observables on qubits.
  arXiv  Detail & Related papers  (2023-01-04T04:15:40Z)
• Monogamy of entanglement between cones [68.8204255655161]
  We show that monogamy is not only a feature of quantum theory, but that it characterizes the minimal tensor product of general pairs of convex cones. Our proof makes use of a new characterization of products of simplices up to affine equivalence.
  arXiv  Detail & Related papers  (2022-06-23T16:23:59Z)
• Beyond the Berry Phase: Extrinsic Geometry of Quantum States [77.34726150561087]
  We show how all properties of a quantum manifold of states are fully described by a gauge-invariant Bargmann. We show how our results have immediate applications to the modern theory of polarization.
  arXiv  Detail & Related papers  (2022-05-30T18:01:34Z)
• Exponential Separation between Quantum and Classical Ordered Binary Decision Diagrams, Reordering Method and Hierarchies [68.93512627479197]
  We study quantum Ordered Binary Decision Diagrams($OBDD$) model. We prove lower bounds and upper bounds for OBDD with arbitrary order of input variables. We extend hierarchy for read$k$-times Ordered Binary Decision Diagrams ($k$-OBDD$) of width.
  arXiv  Detail & Related papers  (2022-04-22T12:37:56Z)
• Uncertainties in Quantum Measurements: A Quantum Tomography [52.77024349608834]
  The observables associated with a quantum system $S$ form a non-commutative algebra $mathcal A_S$. It is assumed that a density matrix $rho$ can be determined from the expectation values of observables. Abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables.
  arXiv  Detail & Related papers  (2021-12-14T16:29:53Z)
• The Parallel Reversible Pebbling Game: Analyzing the Post-Quantum Security of iMHFs [6.305950347749109]
  We introduce a new parallel reversible pebbling game which captures additional restrictions imposed by the No-Deletion Theorem in Quantum Computing. We apply our new reversible pebbling game to analyze the reversible space-time complexity of several important graphs.
  arXiv  Detail & Related papers  (2021-10-08T15:24:31Z)
• Quantum double aspects of surface code models [77.34726150561087]
  We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double $D(G)$ symmetry. We show how our constructions generalise to $D(H)$ models based on a finite-dimensional Hopf algebra $H$.
  arXiv  Detail & Related papers  (2021-06-25T17:03:38Z)
• Symmetries, graph properties, and quantum speedups [3.4253416336476246]
  We show that hypergraph symmetries in the adjacency matrix model allow at most a separation between randomized and quantum query complexities. We also show that, remarkably, permutation groups constructed out of these symmetries are essentially the only permutation groups that prevent super-polynomial quantum speedups.
  arXiv  Detail & Related papers  (2020-06-23T05:00:15Z)
• Can graph properties have exponential quantum speedup? [5.101801159418223]
  In particular, it is natural to ask whether exponential speedup is possible for (partial) graph properties. We show that the answer to this question depends strongly on the input model.
  arXiv  Detail & Related papers  (2020-01-28T18:45:48Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.