Near-term quantum computers are likely to have small depths due to short coherence time and noisy gates. A natural approach to leverage these quantum computers is interleaving them with classical computers. Understanding the capabilities and limits of this hybrid approach is an essential topic in quantum computation. Most notably, the quantum Fourier transform can be implemented by a hybrid of logarithmic-depth quantum circuits and a classical polynomial-time algorithm. Therefore, it seems possible that quantum polylogarithmic depth is as powerful as quantum polynomial depth in the presence of classical computation. Indeed, Jozsa conjectured that “Any quantum polynomial-time algorithm can be implemented with only O(log n) quantum depth interspersed with polynomial-time classical computations.” This can be formalized as asserting the equivalence of
In this work, we manage to prove Aaronson’s conjecture and in the meantime prove that Jozsa’s conjecture, relative to an oracle, is false. In fact, we prove a stronger statement that for any depth parameter d, there exists an oracle that separates quantum depth d and 2d+1 in the presence of classical computation. Thus, our results show that relative to oracles, doubling the quantum circuit depth does make the hybrid model more powerful, and this cannot be traded by classical computation.
We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank 3):
(a) | The number of extreme points in an n-point order type, chosen uniformly at random from all such order types, is on average 4+o(1). For labeled order types, this number has average (4- mbox{$frac{8}{n^2 - n +2}$}) and variance at most 3. | ||||
(b) | The (labeled) order types read off a set of n points sampled independently from the uniform measure on a convex planar domain, smooth or polygonal, or from a Gaussian distribution are concentrated, i.e., such sampling typically encounters only a vanishingly small fraction of all order types of the given size. | ||||
For the unlabeled case in (a), we prove that for any antipodal, finite subset of the two-dimensional sphere, the group of orientation preserving bijections is cyclic, dihedral, or one of A4, S4, or A5 (and each case is possible). These are the finite subgroups of SO(3) and our proof follows the lines of their characterization by Felix Klein.
We present an O(nm) algorithm for all-pairs shortest paths computations in a directed graph with n nodes, m arcs, and nonnegative integer arc costs. This matches the complexity bound attained by Thorup [31] for the all-pairs problems in undirected graphs. The main insight is that shortest paths problems with approximately balanced directed cost functions can be solved similarly to the undirected case. The algorithm finds an approximately balanced reduced cost function in an O(m√ n log n) preprocessing step. Using these reduced costs, every shortest path query can be solved in O(m) time using an adaptation of Thorup’s component hierarchy method. The balancing result can also be applied to the ℓ∞-matrix balancing problem.
We study search problems that can be solved by performing Gradient Descent on a bounded convex polytopal domain and show that this class is equal to the intersection of two well-known classes: PPAD and PLS. As our main underlying technical contribution, we show that computing a Karush-Kuhn-Tucker (KKT) point of a continuously differentiable function over the domain [0,1]2 is PPAD ∩ PLS-complete. This is the first non-artificial problem to be shown complete for this class. Our results also imply that the class CLS (Continuous Local Search) – which was defined by Daskalakis and Papadimitriou as a more “natural” counterpart to PPAD ∩ PLS and contains many interesting problems – is itself equal to PPAD ∩ PLS.
Finite-domain constraint satisfaction problems are either solvable by Datalog or not even expressible in fixed-point logic with counting. The border between the two regimes can be described by a universal-algebraic minor condition. For infinite-domain constraint satisfaction problems (CSPs), the situation is more complicated even if the template structure of the CSP is model-theoretically tame. We prove that there is no Maltsev condition that characterizes Datalog already for the CSPs of first-order reducts of (ℚ;<); such CSPs are called temporal CSPs and are of fundamental importance in infinite-domain constraint satisfaction. Our main result is a complete classification of temporal CSPs that can be expressed in one of the following logical formalisms: Datalog, fixed-point logic (with or without counting), or fixed-point logic with the mod-2 rank operator. The classification shows that many of the equivalent conditions in the finite fail to capture expressibility in Datalog or fixed-point logic already for temporal CSPs.
Given a separation oracle SO for a convex function f defined on ℝn that has an integral minimizer inside a box with radius R, we show how to find an exact minimizer of f using at most
O(n (n log log (n)/log (n) + log (R))) calls to SO and poly (n, log (R)) arithmetic operations, or
O(n log (nR) calls to SO and exp (O(n)) ⋅ poly (log (R)) arithmetic operations.
When the set of minimizers of f has integral extreme points, our algorithm outputs an integral minimizer of f. This improves upon the previously best oracle complexity of O(n2 (n + log (R))) for polynomial time algorithms and O(n2 log (nR) for exponential time algorithms obtained by [Grötschel, Lovász and Schrijver, Prog. Comb. Opt. 1984, Springer 1988] over thirty years ago. Our improvement on Grötschel, Lovász and Schrijver’s result generalizes to the setting where the set of minimizers of f is a rational polyhedron with bounded vertex complexity.
For the Submodular Function Minimization problem, our result immediately implies a strongly polynomial algorithm that makes at most O(n3 log log (n)/log (n)) calls to an evaluation oracle, and an exponential time algorithm that makes at most O(n2 log (n)) calls to an evaluation oracle. These improve upon the previously best O(n3 log2(n)) oracle complexity for strongly polynomial algorithms given in [Lee, Sidford and Wong, FOCS 2015] and [Dadush, Végh and Zambelli, SODA 2018], and an exponential time algorithm with oracle complexity O(n3 log (n)) given in the former work.
Our result is achieved via a reduction to the Shortest Vector Problem in lattices. We show how an approximately shortest vector of an auxiliary lattice can be used to effectively reduce the dimension of the problem. Our analysis of the oracle complexity is based on a potential function that simultaneously captures the size of the search set and the density of the lattice, which we analyze via tools from convex geometry and lattice theory.
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