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We present quantum algorithms for sampling from possibly non-logconcave probability distributions expressed as 𝜋(𝑥)∝exp(−𝛽𝑓(𝑥)) as well as quantum algorithms for estimating the partition function for such distributions. We also incorporate a stochastic gradient oracle that implements the quantum walk operators inexactly by only using mini-batch gradients when 𝑓 can be written as a finite sum. One challenge of quantizing the resulting Markov chains is that they do not satisfy the detailed balance condition in general. Consequently, the mixing time of the algorithm cannot be expressed in terms of the spectral gap of the transition density matrix, making the quantum algorithms nontrivial to analyze. We overcame these challenges by first building a reference reversible Markov chain that converges to the target distribution, then controlling the discrepancy between our algorithm’s output and the target distribution by using the reference Markov chain as a bridge to establish the total complexity. Our quantum algorithms exhibit polynomial speedups in terms of dimension or precision dependencies when compared to best-known classical algorithms under similar assumptions. 

A promising avenue for the preparation of Gibbs states on a quantum computer is to simulate the physical thermalization process. The Davies generator describes the dynamics of an open quantum system that is in contact with a heat bath. Crucially, it does not require simulation of the heat bath itself, only the system we hope to thermalize. Using the state-of-the-art techniques for quantum simulation of the Lindblad equation, we devise a technique for the preparation of Gibbs states via thermalization as specified by the Davies generator.In doing so, we encounter a severe technical challenge: implementation of the Davies generator demands the ability to estimate the energy of the system unambiguously. That is, each energy of the system must be deterministically mapped to a unique estimate. Previous work showed that this is only possible if the system satisfies an unphysical 'rounding promise' assumption. We solve this problem by engineering a random ensemble of rounding promises that simultaneously solves three problems: First, each rounding promise admits preparation of a 'promised' thermal state via a Davies generator. Second, these Davies generators have a similar mixing time as the ideal Davies generator. Third, the average of these promised thermal states approximates the ideal thermal state. 

Abstract BackgroundDelta radiomics is a high‐throughput computational technique used to describe quantitative changes in serial, time‐series imaging by considering the relative change in radiomic features of images extracted at two distinct time points. Recent work has demonstrated a lack of prognostic signal of radiomic features extracted using this technique. We hypothesize that this lack of signal is due to the fundamental assumptions made when extracting features via delta radiomics, and that other methods should be investigated. PurposeThe purpose of this work was to show a proof‐of‐concept of a new radiomics paradigm for sparse, time‐series imaging data, where features are extracted from a spatial‐temporal manifold modeling the time evolution between images, and to assess the prognostic value on patients with oropharyngeal cancer (OPC). MethodsTo accomplish this, we developed an algorithm to mathematically describe the relationship between two images acquired at time and . These images serve as boundary conditions of a partial differential equation describing the transition from one image to the other. To solve this equation, we propagate the position and momentum of each voxel according to Fokker–Planck dynamics (i.e., a technique common in statistical mechanics). This transformation is driven by an underlying potential force uniquely determined by the equilibrium image. The solution generates a spatial‐temporal manifold (3 spatial dimensions + time) from which we define dynamic radiomic features. First, our approach was numerically verified by stochastically sampling dynamic Gaussian processes of monotonically decreasing noise. The transformation from high to low noise was compared between our Fokker–Planck estimation and simulated ground‐truth. To demonstrate feasibility and clinical impact, we applied our approach to¹⁸F‐FDG‐PET images to estimate early metabolic response of patients (n = 57) undergoing definitive (chemo)radiation for OPC. Images were acquired pre‐treatment and 2‐weeks intra‐treatment (after 20 Gy). Dynamic radiomic features capturing changes in texture and morphology were then extracted. Patients were partitioned into two groups based on similar dynamic radiomic feature expression via k‐means clustering and compared by Kaplan–Meier analyses with log‐rank tests (p < 0.05). These results were compared to conventional delta radiomics to test the added value of our approach. ResultsNumerical results confirmed our technique can recover image noise characteristics given sparse input data as boundary conditions. Our technique was able to model tumor shrinkage and metabolic response. While no delta radiomics features proved prognostic, Kaplan–Meier analyses identified nine significant dynamic radiomic features. The most significant feature was Gray‐Level‐Size‐Zone‐Matrix gray‐level variance (p = 0.011), which demonstrated prognostic improvement over its corresponding delta radiomic feature (p = 0.722). ConclusionsWe developed, verified, and demonstrated the prognostic value of a novel, physics‐based radiomics approach over conventional delta radiomics via data assimilation of quantitative imaging and differential equations. 

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 ann-dimensional convex body within multiplicative error ε usingÕ(n³+ n^2.5/ε) queries to a membership oracle andÕ(n⁵+n^4.5/ε)additional arithmetic operations. For comparison, the best known classical algorithm usesÕ(n^3.5+n³/ε²)queries andÕ(n^5.5+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 continuous-space quantum walks with rigorous bounds on discretization error. To complement our quantum algorithms, we also prove that volume estimation requiresΩ (√ n+1/ε)quantum membership queries, which rules out the possibility of exponential quantum speedup innand shows optimality of our algorithm in 1/ε up to poly-logarithmic factors. 

We present an efficient quantum algorithm for simulating the dynamics of Markovian open quantum systems. The performance of our algorithm is similar to the previous state-of-the-art quantum algorithm, i.e., it scales linearly in evolution time and poly-logarithmically in inverse precision. However, our algorithm is conceptually cleaner, and it only uses simple quantum primitives without compressed encoding. Our approach is based on a novel mathematical treatment of the evolution map, which involves a higher-order series expansion based on Duhamel’s principle and approximating multiple integrals using scaled Gaussian quadrature. Our method easily generalizes to simulating quantum dynamics with time-dependent Lindbladians. Furthermore, our method of approximating multiple integrals using scaled Gaussian quadrature could potentially be used to produce a more efficient approximation of time-ordered integrals, and therefore can simplify existing quantum algorithms for simulating time-dependent Hamiltonians based on a truncated Dyson series. 

We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang’s breakthrough quantum-inspired algorithm for recommendation systems [STOC’19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyén et al. [STOC’19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffice to generalize all prior results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: ℓ²-norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.