arXiv subjects

Arjan Cornelissen

Publications and source records attributed to Arjan Cornelissen.

20 records · Page 2Linked to original sources

arXiv2020

Span programs and quantum time complexity

Span programs are an important model of quantum computation due to their tight correspondence with quantum query complexity. For any decision problem $f$, the minimum complexity of a span program for $f$ is equal, up to a constant factor, to the quantum query complexity of $f$. Moreover, this correspondence is constructive. A span program for $f$ with complexity $C$ can be compiled into a bounded error quantum algorithm for $f$ with query complexity $O(C)$, and vice versa. In this work, we prove an analogous connection for quantum time complexity. In particular, we show how to convert a quantum algorithm for $f$ with time complexity $T$ into a span program for $f$ such that it compiles back into a quantum algorithm for $f$ with time complexity $\widetilde{O}(T)$. While the query complexity of quantum algorithms obtained from span programs is well-understood, it is not generally clear how to implement certain query-independent operations in a time-efficient manner. We show that for span programs derived from algorithms with a time-efficient implementation, we can preserve the time efficiency when implementing the span program. This means in particular that span programs not only fully capture quantum query complexity, but also quantum time complexity. One practical advantage of being able to convert quantum algorithms to span programs in a way that preserves time complexity is that span programs compose very nicely. We demonstrate this by improving Ambainis's variable-time quantum search result using our construction through a span program composition for the OR function.

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arXiv2019

Quantum gradient estimation of Gevrey functions

Gradient-based numerical methods are ubiquitous in optimization techniques frequently applied in industry to solve practical problems. Often times, evaluating the objective function is a complicated process, so estimating the gradient of a function with as few function evaluations as possible is a natural problem. We investigate whether quantum computers can perform $\ell^{\infty}$-approximate gradient estimation of multivariate functions $f : \mathbb{R}^d \to \mathbb{R}$ with fewer function evaluations than classically. Following previous work by Jordan [Jor05] and Gilyén et al. [GAW18], we prove that one can calculate an $\ell^{\infty}$-approximation of the gradient of $f$ with a query complexity that scales sublinearly with $d$ under weaker smoothness conditions than previously considered. Furthermore, for a particular subset of smoothness conditions, we prove a new lower bound on the query complexity of the gradient estimation problem, proving essential optimality of Gilyén et al.'s gradient estimation algorithm in a broader range of parameter values, and affirming the validity of their conjecture [GAW18]. Moreover, we improve their lower bound qualitatively by showing that their algorithm is also optimal for functions that satisfy the imposed smoothness conditions globally instead of locally. Finally, we introduce new ideas to prove lower bounds on the query complexity of the $\ell^p$-approximate gradient estimation problem where $p \in [1,\infty)$, and prove that lifting Gilyén et al.'s algorithm to this domain in the canonical manner is essentially optimal.

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