We consider the problem of quantum state certification, where we are given the description of a mixed state $\sigma \in \mathbb{C}^{d \times d}$, $n$ copies of a mixed state $\rho \in \mathbb{C}^{d \times d}$, and $\varepsilon > 0$, and we are asked to determine whether $\rho = \sigma$ or whether $\| \rho - \sigma \|_1 > \varepsilon$. When $\sigma$ is the maximally mixed state $\frac{1}{d} I_d$, this is known as mixedness testing. We focus on algorithms which use incoherent measurements, i.e. which only measure one copy of $\rho$ at a time. Unlike those that use entangled, multi-copy measurements, these can be implemented without persistent quantum memory and thus represent a large class of protocols that can be run on current or near-term devices. For mixedness testing, there is a folklore algorithm which uses incoherent measurements and only needs $O(d^{3/2} / \varepsilon^2)$ copies. The algorithm is non-adaptive, that is, its measurements are fixed ahead of time, and is known to be optimal for non-adaptive algorithms. However, when the algorithm can make arbitrary incoherent measurements, the best known lower bound is only $\Omega (d^{4/3} / \varepsilon^2)$ [Bubeck-Chen-Li '20], and it has been an outstanding open problem to close this polynomial gap. In this work, 1) we settle the copy complexity of mixedness testing with incoherent measurements and show that $\Omega (d^{3/2} / \varepsilon^2)$ copies are necessary, and 2) we show the instance-optimal bounds for state certification to general $\sigma$ first derived by [Chen-Li-O'Donnell '21] for non-adaptive measurements also hold for arbitrary incoherent measurements. Qualitatively, our results say that adaptivity does not help at all for these problems. Our results are based on new techniques that allow us to reduce the problem to understanding certain matrix martingales, which we believe may be of independent interest.
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This work aims at making a comprehensive contribution in the general area of parametric inference for partially observed diffusion processes. Established approaches for likelihood-based estimation invoke a numerical time-discretisation scheme for the approximation of the (typically intractable) transition dynamics of the Stochastic Differential Equation (SDE) model over finite time periods. The scheme is applied for a step-size that is either a user-selected tuning parameter or determined by the data. Recent research has highlighted the critical effect of the choice of numerical scheme on the behaviour of derived parameter estimates in the setting of hypo-elliptic SDEs. In brief, in our work, first, we develop two weak second order `sampling schemes' (to cover both the hypo-elliptic and elliptic SDE classes) and generate accompanying `transition density schemes' of the SDE (i.e., approximations of the SDE transition density). Then, we produce a collection of analytic results, providing a complete theoretical framework that solidifies the proposed schemes and showcases advantages from their incorporation within SDE calibration methods. We present numerical results from carrying out classical or Bayesian inference, for both elliptic and hypo-elliptic SDE models.
Problem instances of a size suitable for practical applications are not likely to be addressed during the noisy intermediate-scale quantum (NISQ) period with (almost) pure quantum algorithms. Hybrid classical-quantum algorithms have potential, however, to achieve good performance on much larger problem instances. We investigate one such hybrid algorithm on a problem of substantial importance: vehicle routing for supply chain logistics with multiple trucks and complex demand structure. We use reinforcement learning with neural networks with embedded quantum circuits. In such neural networks, projecting high-dimensional feature vectors down to smaller vectors is necessary to accommodate restrictions on the number of qubits of NISQ hardware. However, we use a multi-head attention mechanism where, even in classical machine learning, such projections are natural and desirable. We consider data from the truck routing logistics of a company in the automotive sector, and apply our methodology by decomposing into small teams of trucks, and we find results comparable to human truck assignment.
Many applications, such as system identification, classification of time series, direct and inverse problems in partial differential equations, and uncertainty quantification lead to the question of approximation of a non-linear operator between metric spaces $\mathfrak{X}$ and $\mathfrak{Y}$. We study the problem of determining the degree of approximation of such operators on a compact subset $K_\mathfrak{X}\subset \mathfrak{X}$ using a finite amount of information. If $\mathcal{F}: K_\mathfrak{X}\to K_\mathfrak{Y}$, a well established strategy to approximate $\mathcal{F}(F)$ for some $F\in K_\mathfrak{X}$ is to encode $F$ (respectively, $\mathcal{F}(F)$) in terms of a finite number $d$ (repectively $m$) of real numbers. Together with appropriate reconstruction algorithms (decoders), the problem reduces to the approximation of $m$ functions on a compact subset of a high dimensional Euclidean space $\mathbb{R}^d$, equivalently, the unit sphere $\mathbb{S}^d$ embedded in $\mathbb{R}^{d+1}$. The problem is challenging because $d$, $m$, as well as the complexity of the approximation on $\mathbb{S}^d$ are all large, and it is necessary to estimate the accuracy keeping track of the inter-dependence of all the approximations involved. In this paper, we establish constructive methods to do this efficiently; i.e., with the constants involved in the estimates on the approximation on $\mathbb{S}^d$ being $\mathcal{O}(d^{1/6})$. We study different smoothness classes for the operators, and also propose a method for approximation of $\mathcal{F}(F)$ using only information in a small neighborhood of $F$, resulting in an effective reduction in the number of parameters involved.
Quantum computers promise to enhance machine learning for practical applications. Quantum machine learning for real-world data has to handle extensive amounts of high-dimensional data. However, conventional methods for measuring quantum kernels are impractical for large datasets as they scale with the square of the dataset size. Here, we measure quantum kernels using randomized measurements. The quantum computation time scales linearly with dataset size and quadratic for classical post-processing. While our method scales in general exponentially in qubit number, we gain a substantial speed-up when running on intermediate-sized quantum computers. Further, we efficiently encode high-dimensional data into quantum computers with the number of features scaling linearly with the circuit depth. The encoding is characterized by the quantum Fisher information metric and is related to the radial basis function kernel. Our approach is robust to noise via a cost-free error mitigation scheme. We demonstrate the advantages of our methods for noisy quantum computers by classifying images with the IBM quantum computer. To achieve further speedups we distribute the quantum computational tasks between different quantum computers. Our method enables benchmarking of quantum machine learning algorithms with large datasets on currently available quantum computers.
We introduce a new weight and corresponding metric over finite extension fields for asymmetric error correction. The weight distinguishes between elements from the base field and the ones outside of it, which is motivated by asymmetric quantum codes. We set up the theoretic framework for this weight and metric, including upper and lower bounds, asymptotic behavior of random codes, and we show the existence of an optimal family of codes achieving the Singleton-type upper bound.
We study how the choices made when designing an oracle affect the complexity of quantum property testing problems defined relative to this oracle. We encode a regular graph of even degree as an invertible function $f$, and present $f$ in different oracle models. We first give a one-query QMA protocol to test if a graph encoded in $f$ has a small disconnected subset. We then use representation theory to show that no classical witness can help a quantum verifier efficiently decide this problem relative to an in-place oracle. Perhaps surprisingly, a simple modification to the standard oracle prevents a quantum verifier from efficiently deciding this problem, even with access to an unbounded witness.
This paper proposes a simple, accurate and computationally efficient method to apply the ordinary unscented Kalman filter developed in Euclidean space to systems whose dynamics evolve on manifolds.We use the mathematical theory called stable embedding to make a variant of unscented Kalman filter that keeps state estimates in closeproximity to the manifold while exhibiting excellent estimation performance. We confirm the performance of our devised filter by applying it to the satellite system model and comparing the performance with other unscented Kalman filters devised specifically for systems on manifolds. Our devised filter has a low estimation error, keeps the state estimates in close proximity to the manifold as expected, and consumes a minor amount of computation time. Also our devised filter is simple and easy to use because our filter directly employs the off-the-shelf standard unscented Kalman filter devised in Euclidean space without any particular manifold-structure-preserving discretization method or coordinate transformation.
Measuring the (causal) direction and strength of dependence between two variables (events), Xi and Xj , is fundamental for all science. Our survey of decades-long literature on statistical dependence reveals that most assume symmetry in the sense that the strength of dependence of Xi on Xj exactly equals the strength of dependence of Xj on Xi. However, we show that such symmetry is often untrue in many real-world examples, being neither necessary nor sufficient. Vinod's (2014) asymmetric matrix R* in [-1, 1] of generalized correlation coefficients provides intuitively appealing, readily interpretable, and superior measures of dependence. This paper proposes statistical inference for R* using Taraldsen's (2021) exact sampling distribution of correlation coefficients and the bootstrap. When the direction is known, proposed asymmetric (one-tail) tests have greater power.
Mixed-Integer Linear Programming (MILP) plays an important role across a range of scientific disciplines and within areas of strategic importance to society. The MILP problems, however, suffer from combinatorial complexity. Because of integer decision variables, as the problem size increases, the number of possible solutions increases super-linearly thereby leading to a drastic increase in the computational effort. To efficiently solve MILP problems, a "price-based" decomposition and coordination approach is developed to exploit 1. the super-linear reduction of complexity upon the decomposition and 2. the geometric convergence potential inherent to Polyak's stepsizing formula for the fastest coordination possible to obtain near-optimal solutions in a computationally efficient manner. Unlike all previous methods to set stepsizes heuristically by adjusting hyperparameters, the key novel way to obtain stepsizes is purely decision-based: a novel "auxiliary" constraint satisfaction problem is solved, from which the appropriate stepsizes are inferred. Testing results for large-scale Generalized Assignment Problems (GAP) demonstrate that for the majority of instances, certifiably optimal solutions are obtained. For stochastic job-shop scheduling as well as for pharmaceutical scheduling, computational results demonstrate the two orders of magnitude speedup as compared to Branch-and-Cut (B&C). The new method has a major impact on the efficient resolution of complex Mixed-Integer Programming (MIP) problems arising within a variety of scientific fields.
Due to the noises in crowdsourced labels, label aggregation (LA) has emerged as a standard procedure to post-process crowdsourced labels. LA methods estimate true labels from crowdsourced labels by modeling worker qualities. Most existing LA methods are iterative in nature. They need to traverse all the crowdsourced labels multiple times in order to jointly and iteratively update true labels and worker qualities until convergence. Consequently, these methods have high space and time complexities. In this paper, we treat LA as a dynamic system and model it as a Dynamic Bayesian network. From the dynamic model we derive two light-weight algorithms, LA\textsuperscript{onepass} and LA\textsuperscript{twopass}, which can effectively and efficiently estimate worker qualities and true labels by traversing all the labels at most twice. Due to the dynamic nature, the proposed algorithms can also estimate true labels online without re-visiting historical data. We theoretically prove the convergence property of the proposed algorithms, and bound the error of estimated worker qualities. We also analyze the space and time complexities of the proposed algorithms and show that they are equivalent to those of majority voting. Experiments conducted on 20 real-world datasets demonstrate that the proposed algorithms can effectively and efficiently aggregate labels in both offline and online settings even if they traverse all the labels at most twice.
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