Estimating the volume of a convex body is a central problem in convex geometry and can be viewed as a continuous version of counting. We present a quantum algorithm that estimates the volume of an ndimensional convex body within multiplicative error ϵ using Õ (n3+n2.5/ϵ) queries to a membership oracle and Õ (n5+n4.5/ϵ) additional arithmetic operations. For comparison, the best known classical algorithm uses Õ (n4+n3/ϵ2) queries and Õ (n6+n5/ϵ2) additional arithmetic operations. To the best of our knowledge, this is the first quantum speedup for volume estimation. Our algorithm is based on a refined framework for speeding up simulated annealing algorithms that might be of independent interest. This framework applies in the setting of "Chebyshev cooling", where the solution is expressed as a telescoping product of ratios, each having bounded variance. We develop several novel techniques when implementing our framework, including a theory of continuousspace quantum walks with rigorous bounds on discretization error.
Quantum algorithms and lower bounds for convex optimization
While recent work suggests that quantum computers can speed up the solution of semidefinite programs, little is known about the quantum complexity of more general convex optimization. We present a quantum algorithm that can optimize a convex function over an n dimensional convex body using O ~ ( n ) queries to oracles that evaluate the objective function and determine membership in the convex body. This represents a quadratic improvement over the bestknown classical algorithm. We also study limitations on the power of quantum computers for general convex optimization, showing that it requires Ω ~ ( n ) evaluation queries and Ω ( n ) membership queries.
 Award ID(s):
 1816695
 Publication Date:
 NSFPAR ID:
 10106373
 Journal Name:
 Quantum
 Volume:
 4
 Page Range or eLocationID:
 221
 ISSN:
 2521327X
 Sponsoring Org:
 National Science Foundation
More Like this


We investigate the approximability of the following optimization problem. The input is an n× n matrix A=(Aij) with real entries and an originsymmetric convex body K⊂ ℝn that is given by a membership oracle. The task is to compute (or approximate) the maximum of the quadratic form ∑i=1n∑j=1n Aij xixj=⟨ x,Ax⟩ as x ranges over K. This is a rich and expressive family of optimization problems; for different choices of matrices A and convex bodies K it includes a diverse range of optimization problems like maxcut, Grothendieck/noncommutative Grothendieck inequalities, small set expansion and more. While the literature studied these special cases using casespecific reasoning, here we develop a general methodology for treatment of the approximability and inapproximability aspects of these questions. The underlying geometry of K plays a critical role; we show under commonly used complexity assumptions that polytime constantapproximability necessitates that K has type2 constant that grows slowly with n. However, we show that even when the type2 constant is bounded, this problem sometimes exhibits strong hardness of approximation. Thus, even within the realm of type2 bodies, the approximability landscape is nuanced and subtle. However, the link that we establish between optimization and geometry of Banach spaces allows usmore »

We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottomlevel gates. Let f be an m bit Boolean function and consider an n bit function F obtained by applying f to conjunctions of possibly overlapping subsets of n variables. If f has quantum query complexity Q ( f ) , we give an algorithm for evaluating F using O ~ ( Q ( f ) ⋅ n ) quantum queries. This improves on the bound of O ( Q ( f ) ⋅ n ) that follows by treating each conjunction independently, and our bound is tight for worstcase choices of f . Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of f .By recursively applying our composition theorems, we obtain a nearly optimal O ~ ( n 1 − 2 − d ) upper bound on the quantum query complexity and approximate degree of linearsize depth d AC 0 circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponentialtime algorithm was not known even for linearsize depth3 AC 0 circuits.As anmore »

Dictionaries remain the most well studied class of data structures. A dictionary supports insertions, deletions, membership queries, and usually successor, predecessor, and extractmin. In a RAM, all such operations take O(log n) time on n elements. Dictionaries are often crossreferenced as follows. Consider a set of tuples {〈ai,bi,ci…〉}. A database might include more than one dictionary on such a set, for example, one indexed on the a ‘s, another on the b‘s, and so on. Once again, in a RAM, inserting into a set of L crossreferenced dictionaries takes O(L log n) time, as does deleting. The situation is more interesting in external memory. On a Disk Access Machine (DAM), Btrees achieve O(logB N) I/Os for insertions and deletions on a single dictionary and Kelement range queries take optimal O(logB N + K/B) I/Os. These bounds are also achievable by a Btree on crossreferenced dictionaries, with a slowdown of an L factor on insertion and deletions. In recent years, both the theory and practice of external memory dictionaries has been revolutionized by write optimization techniques. A dictionary is write optimized if it is close to a Btree for query time while beating Btrees on insertions. The best (and optimal) dictionariesmore »

We present a quasipolynomial time classical algorithm that estimates the partition function of quantum manybody systems at temperatures above the thermal phase transition point. It is known that in the worst case, the same problem is NPhard below this point. Together with our work, this shows that the transition in the phase of a quantum system is also accompanied by a transition in the hardness of approximation. We also show that in a system of n particles above the phase transition point, the correlation between two observables whose distance is at least Ω(logn) decays exponentially. We can improve the factor of logn to a constant when the Hamiltonian has commuting terms or is on a 1D chain. The key to our results is a characterization of the phase transition and the critical behavior of the system in terms of the complex zeros of the partition function. Our work extends a seminal work of Dobrushin and Shlosman on the equivalence between the decay of correlations and the analyticity of the free energy in classical spin models. On the algorithmic side, our result extends the scope of a recent approach due to Barvinok for solving classical counting problems to quantum manybody systems.