We initiate the study of quantum algorithms for optimizing approximately convex functions. Given a convex set K Rn and a function F: Rn Rsuch that there exists a convex function f: K R satisfying supx K|F(x) f(x)| /n, our quantum algorithm finds an x K such that F(x) minx KF(x) using O(n3) quantum evaluation queries to F. This achieves a polynomial quantum speedup compared to the best-known classical algorithms. As an application, we give a quantum algorithm for zeroth-order stochastic convex bandits with O(n5 log2 T) regret, an exponential speedup in T compared to the classical Ω( T) lower bound. Technically, we achieve quantum speedup in nby exploiting a quantum framework of simulated annealing and adopting a quantum version of the hit-and-run walk. Our speedup in T for zeroth-order stochastic convex bandits is due to a quadratic quantum speedup in multiplicative error of mean estimation.

We consider the problem of minimizing a continuous function given given access to a natural quantum generalization of a stochastic gradient oracle. We provide two new methods for the special case of minimizing a Lipschitz convex function. Each method obtains a dimension versus accuracy trade-off which is provably unachievable classically and we prove that one method is asymptotically optimal in low-dimensional settings. Additionally, we provide quantum algorithms for computing a critical point of a smooth non-convex function at rates not known to be achievable classically. To obtain these results we build upon the quantum multivariate mean estimation result of Cornelissen et al. and provide a general quantum variance reduction technique of independent interest.

We consider the problem of minimizing a continuous function given given access to a natural quantum generalization of a stochastic gradient oracle. We provide two new methods for the special case of minimizing a Lipschitz convex function. Each method obtains a dimension versus accuracy trade-off which is provably unachievable classically and we prove that one method is asymptotically optimal in low-dimensional settings. Additionally, we provide quantum algorithms for computing a critical point of a smooth non-convex function at rates not known to be achievable classically. To obtain these results we build upon the quantum multivariate mean estimation result of Cornelissen et al. and provide a general quantum variance reduction technique of independent interest.

We initiate the study of quantum algorithms for optimizing approximately convex functions. This achieves a polynomial quantum speedup compared to the best-known classical algorithms. As an application, we give a quantum algorithm for zeroth-order stochastic convex bandits with \tilde{O}(n {5}\log {2} T) regret, an exponential speedup in T compared to the classical \Omega(\sqrt{T}) lower bound. Technically, we achieve quantum speedup in n by exploiting a quantum framework of simulated annealing and adopting a quantum version of the hit-and-run walk. Our speedup in T for zeroth-order stochastic convex bandits is due to a quadratic quantum speedup in multiplicative error of mean estimation.

NTT Research, Inc., a division of NTT, announced that a scientist from its Cryptography and Information Security (CIS) Lab and a colleague from the NTT Social Informatics Laboratories (SIL) have written a pathbreaking paper on quantum advantage. The paper was selected to be presented at the annual IEEE Symposium on Foundations of Computer Science (FOCS), which is taking place Oct. 31–Nov. The co-authors of the paper, titled "Verifiable Quantum Advantage without Structure," are Dr. Takashi Yamakawa, distinguished researcher at NTT SIL and Dr. Mark Zhandry, senior scientist in the NTT Research CIS Lab. The work was done in part at Princeton University, where Dr. Yamakawa was a visiting research scholar and Dr. Zhandry also serves as an assistant professor of computer science. The topic of quantum advantage (or quantum speedup) relates to the kinds of problems that quantum computers can solve faster than classical, or non-quantum, computers and how much faster they are.

Given a convex function $f\colon\mathbb{R}^{d}\to\mathbb{R}$, the problem of sampling from a distribution $\propto e^{-f(x)}$ is called log-concave sampling. This task has wide applications in machine learning, physics, statistics, etc. In this work, we develop quantum algorithms for sampling log-concave distributions and for estimating their normalizing constants $\int_{\mathbb{R}^d}e^{-f(x)}\mathrm{d} x$. First, we use underdamped Langevin diffusion to develop quantum algorithms that match the query complexity (in terms of the condition number $\kappa$ and dimension $d$) of analogous classical algorithms that use gradient (first-order) queries, even though the quantum algorithms use only evaluation (zeroth-order) queries. For estimating normalizing constants, these algorithms also achieve quadratic speedup in the multiplicative error $\epsilon$. Second, we develop quantum Metropolis-adjusted Langevin algorithms with query complexity $\widetilde{O}(\kappa^{1/2}d)$ and $\widetilde{O}(\kappa^{1/2}d^{3/2}/\epsilon)$ for log-concave sampling and normalizing constant estimation, respectively, achieving polynomial speedups in $\kappa,d,\epsilon$ over the best known classical algorithms by exploiting quantum analogs of the Monte Carlo method and quantum walks. We also prove a $1/\epsilon^{1-o(1)}$ quantum lower bound for estimating normalizing constants, implying near-optimality of our quantum algorithms in $\epsilon$.

For Valeria Saggio to boot up the computer in her former Vienna lab, she needed a special crystal, only as big as her fingernail. Saggio would place it gently into a small copper box, a tiny electric oven, which would heat the crystal to 77 degrees Fahrenheit. Then she would switch on a laser to bombard the crystal with a beam of photons. This crystal, at this precise temperature, would split some of those photons into two photons. One of these would go straight to a light detector, its journey finished; the other would travel into a tiny silicon chip -- a quantum computing processor.

Two physicists, from EPFL and Columbia University, have introduced an approach for simulating the quantum approximate optimization algorithm using a traditional computer. Instead of running the algorithm on advanced quantum processors, the new approach uses a classical machine-learning algorithm that closely mimics the behavior of near-term quantum computers. In a paper published in Nature Quantum Information, EPFL professor Giuseppe Carleo and Matija Medvidović, a graduate student at Columbia University and at the Flatiron Institute in New York, have found a way to execute a complex quantum computing algorithm on traditional computers instead of quantum ones. The specific "quantum software" they are considering is known as Quantum Approximate Optimization Algorithm (QAOA) and is used to solve classical optimization problems in mathematics; it's essentially a way of picking the best solution to a problem out of a set of possible solutions. "There is a lot of interest in understanding what problems can be solved efficiently by a quantum computer, and QAOA is one of the more prominent candidates," says Carleo.

Quantum computers, you might have heard, are magical uber-machines that will soon cure cancer and global warming by trying all possible answers in different parallel universes. For 15 years, on my blog and elsewhere, I've railed against this cartoonish vision, trying to explain what I see as the subtler but ironically even more fascinating truth. I approach this as a public service and almost my moral duty as a quantum computing researcher. Alas, the work feels Sisyphean: The cringeworthy hype about quantum computers has only increased over the years, as corporations and governments have invested billions, and as the technology has progressed to programmable 50-qubit devices that (on certain contrived benchmarks) really can give the world's biggest supercomputers a run for their money. And just as in cryptocurrency, machine learning and other trendy fields, with money have come hucksters.