Non-asymptotic Regime Research Articles

Parallelization of Adaptive Quantum Channel Discrimination in the Non-Asymptotic Regime

Parallelization of Adaptive Quantum Channel Discrimination in the Non-Asymptotic Regime

Reconfigurable Adaptive Channel Sensing

This paper proposes an energy-efficient detection scheme, referred to as AdaSense, that is particularly suitable in the sparse regime when events to be detected happen rarely. To minimize energy consumption, AdaSense exploits the dependency between the receiver noise figure (i.e., the receiver added noise) and the receiver power consumption; less noisy channel observations typically imply higher power consumption. AdaSense is duty-cycled and begins each cycle with a few channel observations in a low-power-low-reliability mode. Based on these observations, it makes a first tentative decision on whether or not a message is present. If no message is declared, AdaSense waits till the beginning of the next cycle and starts afresh. If a message is tentatively declared, AdaSense enters a confirmation second phase, takes more samples, but now in a high-power-high-reliability mode. If these observations confirm the tentative decision, AdaSense stops, else AdaSense waits till the beginning of the next cycle and starts afresh in the low-power-low-reliability mode. Compared to prominent detection schemes such as the clear channel assessment algorithm of the Berkeley Media Access Control (BMAC) protocol, AdaSense provides relative energy gains that grow unbounded in the small probability of false-alarm regime, as communication gets sparser. In the non-asymptotic regime, energy gains are 30% to 75% for communication scenarios typically found in the context of wake-up receivers.

Improving exploration strategies in large dimensions and rate of convergence of global random search algorithms

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Only Classical Parameterised States have Optimal Measurements under Least Squares Loss

Measurements of quantum states form a key component in quantum-information processing. It is therefore an important task to compare measurements and furthermore decide if a measurement strategy is optimal. Entropic quantities, such as the quantum Fisher information, capture asymptotic optimality but not optimality with finite resources. We introduce a framework that allows one to conclusively establish if a measurement is optimal in the non-asymptotic regime. Our method relies on the fundamental property of expected errors of estimators, known as risk, and it does not involve optimisation over entropic quantities. The framework applies to finite sample sizes and lack of prior knowledge, as well as to the asymptotic and Bayesian settings. We prove a no-go theorem that shows that only classical states admit optimal measurements under the most common choice of error measurement: least squares. We further consider the less restrictive notion of an approximately optimal measurement and give sufficient conditions for such measurements to exist. Finally, we generalise the notion of when an estimator is inadmissible (i.e. strictly worse than an alternative), and provide two sufficient conditions for a measurement to be inadmissible.

A closed-form bound on the asymptotic linear convergence of iterative methods via fixed point analysis

In many iterative optimization methods, fixed-point theory enables the analysis of the convergence rate via the contraction factor associated with the linear approximation of the fixed-point operator. While this factor characterizes the asymptotic linear rate of convergence, it does not explain the non-linear behavior of these algorithms in the non-asymptotic regime. In this letter, we take into account the effect of the first-order approximation error and present a closed-form bound on the convergence in terms of the number of iterations required for the distance between the iterate and the limit point to reach an arbitrarily small fraction of the initial distance. Our bound includes two terms: one corresponds to the number of iterations required for the linearized version of the fixed-point operator and the other corresponds to the overhead associated with the approximation error. With a focus on the convergence in the scalar case, the tightness of the proposed bound is proven for positively quadratic first-order difference equations.

Fixed-Horizon Active Hypothesis Testing

Two active hypothesis testing problems are formulated. In these problems, the agent can perform a fixed number of experiments and then decide on one of the hypotheses or declare its experiments inconclusive. The first problem is an asymmetric formulation in which the objective is to minimize the probability of incorrectly declaring a particular hypothesis to be true while ensuring that the probability of correctly declaring that hypothesis is moderately high. This formulation is a generalization of the formulation in the Chernoff–Stein lemma to an active setting. The second problem is a symmetric formulation in which the objective is to minimize the misclassification probability while ensuring that the true hypothesis is declared conclusively with moderately high probability. For these problems, lower and upper bounds on the optimal misclassification probabilities are derived and these bounds are shown to be asymptotically tight. Novel experiment selection strategies are provided. It is shown that these strategies are asymptotically optimal and using numerical experiments, it is demonstrated that these strategies have a significantly better performance in the nonasymptotic regime.

One-Shot Yield-Cost Relations in General Quantum Resource Theories

Although it is well known that the amount of resources that can be asymptotically distilled from a quantum state or channel does not exceed the resource cost needed to produce it, the corresponding relation in the non-asymptotic regime hitherto has not been well understood. Here, we establish a quantitative relation between the one-shot distillable resource yield and dilution cost in terms of transformation errors involved in these processes. Notably, our bound is applicable to quantum state and channel manipulation with respect to any type of quantum resource and any class of free transformations thereof, encompassing broad types of settings, including entanglement, quantum thermodynamics, and quantum communication. We also show that our techniques provide strong converse bounds relating the distillable resource and resource dilution cost in the asymptotic regime. Moreover, we introduce a class of channels that generalize twirling maps encountered in many resource theories, and by directly connecting it with resource quantification, we compute analytically several smoothed resource measures and improve our one-shot yield--cost bound in relevant theories. We use these operational insights to exactly evaluate important measures for various resource states in the resource theory of magic states.

Fixed-Horizon Active Hypothesis Testing and Anomaly Detection

Determining the presence of an anomaly or whether a system is safe or not is a problem with wide applicability. The model adopted for this problem is that of verifying whether a multi-component system has anomalies or not. Components can be probed over time individually or as groups in a data-driven manner. The collected observations are noisy and contain information on whether the selected group contains an anomaly or not. The aim is to minimize the probability of incorrectly declaring the system to be free of anomalies while ensuring that the probability of correctly declaring it to be safe is sufficiently large. This problem is modeled as an active hypothesis testing problem in the Neyman-Pearson setting. Asymptotically optimal rates and strategies are characterized. The general strategies are data driven and outperform previously proposed asymptotically optimal methods in the finite sample regime. Furthermore, novel component-selection are designed and analyzed in the non-asymptotic regime. For a specific class of problems admitting a key form of symmetry, strong non-asymptotic converse and achievability bounds are provided which are tighter than previously proposed bounds.

Low Complexity Sequential Search With Size-Dependent Measurement Noise

This paper considers a target localization problem where at any given time an agent can choose a region to query for the presence of the target in that region. The measurement noise is assumed to be increasing with the size of the query region the agent chooses. Motivated by practical applications such as initial beam alignment in array processing, heavy hitter detection in networking, and visual search in robotics, we consider practically important complexity constraints/metrics: time complexity, computational and memory complexity, and the complexity of possible query sets in terms of geometry and cardinality. Two novel search strategy, dyaPM and hiePM, are proposed. Pertinent to the practicality of our solutions, dyaPM and hiePM are of a connected query geometry (i.e. query set is always a connected set) implemented with low computational and memory complexity. Additionally, hiePM has a hierarchical structure and, hence, a further reduction in the cardinality of possible query sets, making hiePM practically suitable for applications such as beamforming in array processing where memory limitations favors a small and predefined query sets. Through a unified analysis with Extrinsic Jensen Shannon (EJS) Divergence, dyaPM is shown to be asymptotically optimal in search time complexity (asymptotic in both resolution (rate) and error (reliability)). On the other hand, hiePM is shown to be near-optimal in rate. In addition, both hiePM and dyaPM are shown to outperform prior work in the non-asymptotic regime.

Hamiltonian simulation algorithms for near-term quantum hardware

The quantum circuit model is the de-facto way of designing quantum algorithms. Yet any level of abstraction away from the underlying hardware incurs overhead. In this work, we develop quantum algorithms for Hamiltonian simulation "one level below” the circuit model, exploiting the underlying control over qubit interactions available in most quantum hardware and deriving analytic circuit identities for synthesising multi-qubit evolutions from two-qubit interactions. We then analyse the impact of these techniques under the standard error model where errors occur per gate, and an error model with a constant error rate per unit time. To quantify the benefits of this approach, we apply it to time-dynamics simulation of the 2D spin Fermi-Hubbard model. Combined with new error bounds for Trotter product formulas tailored to the non-asymptotic regime and an analysis of error propagation, we find that e.g. for a 5 × 5 Fermi-Hubbard lattice we reduce the circuit depth from 1, 243, 586 using the best previous fermion encoding and error bounds in the literature, to 3, 209 in the per-gate error model, or the circuit-depth-equivalent to 259 in the per-time error model. This brings Hamiltonian simulation, previously beyond reach of current hardware for non-trivial examples, significantly closer to being feasible in the NISQ era.

Size-Based Routing Policies: Non-Asymptotic Analysis and Design of Decentralized Systems.

Size-based routing policies are known to perform well when the variance of the distribution of the job size is very high. We consider two size-based policies in this paper: Task Assignment with Guessing Size (TAGS) and Size Interval Task Assignment (SITA). The latter assumes that the size of jobs is known, whereas the former does not. Recently, it has been shown by our previous work that when the ratio of the largest to shortest job tends to infinity and the system load is fixed and low, the average waiting time of SITA is, at most, two times less than that of TAGS. In this article, we first analyze the ratio between the mean waiting time of TAGS and the mean waiting time of SITA in a non-asymptotic regime, and we show that for two servers, and when the job size distribution is Bounded Pareto with parameter , this ratio is unbounded from above. We then consider a system with an arbitrary number of servers and we compare the mean waiting time of TAGS with that of Size Interval Task Assignment with Equal load (SITA-E), which is a SITA policy where the load of all the servers are equal. We show that in the light traffic regime, the performance ratio under consideration is unbounded from above when (i) the job size distribution is Bounded Pareto with parameter and an arbitrary number of servers as well as (ii) for Bounded Pareto distributed job sizes with and the number of servers tends to infinity. Finally, we use the result of our previous work to show how to design decentralized systems with quality of service constraints.

Local characterization of transient chaos on finite times in open systems

To characterize local finite-time properties associated with transient chaos in open dynamical systems, we introduce an escape rate and fractal dimensions suitable for this purpose in a coarse-grained description. We numerically illustrate that these quantifiers have a considerable spread across the domain of the dynamics, but their spatial variation, especially on long but non-asymptotic integration times, is approximately consistent with the relationship that was recognized by Kantz and Grassberger for temporally asymptotic quantifiers. In particular, deviations from this relationship are smaller than differences between various locations, which confirms the existence of such a dynamical law and the suitability of our quantifiers to represent underlying dynamical properties in the non-asymptotic regime.

Amortized channel divergence for asymptotic quantum channel discrimination

It is well known that for the discrimination of classical and quantum channels in the finite, non-asymptotic regime, adaptive strategies can give an advantage over non-adaptive strategies. However, Hayashi [IEEE Trans. Inf. Theory 55(8), 3807 (2009)] showed that in the asymptotic regime, the exponential error rate for the discrimination of classical channels is not improved in the adaptive setting. We extend this result in several ways. First, we establish the strong Stein's lemma for classical-quantum channels by showing that asymptotically the exponential error rate for classical-quantum channel discrimination is not improved by adaptive strategies. Second, we recover many other classes of channels for which adaptive strategies do not lead to an asymptotic advantage. Third, we give various converse bounds on the power of adaptive protocols for general asymptotic quantum channel discrimination. Intriguingly, it remains open whether adaptive protocols can improve the exponential error rate for quantum channel discrimination in the asymmetric Stein setting. Our proofs are based on the concept of amortized distinguishability of quantum channels, which we analyse using data-processing inequalities.

Adaptive Causal Network Coding With Feedback

We propose a novel adaptive and causal random linear network coding (AC-RLNC) algorithm with forward error correction (FEC) for a point-to-point communication channel with delayed feedback. AC-RLNC is adaptive to the channel condition, that the algorithm estimates, and is causal, as coding depends on the particular erasure realizations, as reflected in the feedback acknowledgments. Specifically, the proposed model can learn the erasure pattern of the channel via feedback acknowledgments, and adaptively adjust its retransmission rates using a priori and posteriori algorithms. By those adjustments, AC-RLNC achieves the desired delay and throughput, and enables transmission with zero error probability. We upper bound the throughput and the mean and maximum in order delivery delay of AC-RLNC, and prove that for the point to point communication channel in the non-asymptotic regime the proposed code may achieve more than 90% of the channel capacity. To upper bound the throughput we utilize the minimum Bhattacharyya distance for the AC-RLNC code. We validate those results via simulations. We contrast the performance of AC-RLNC with the one of selective repeat (SR)-ARQ, which is causal but not adaptive, and is a posteriori. Via a study on experimentally obtained commercial traces, we demonstrate that a protocol based on AC-RLNC can, vis-a-vis SR-ARQ, double the throughput gains, and triple the gain in terms of mean in order delivery delay when the channel is bursty. Furthermore, the difference between the maximum and mean in order delivery delay is much smaller than that of SR-ARQ. Closing the delay gap along with boosting the throughput is very promising for enabling ultra-reliable low-latency communications (URLLC) applications.

Distillation of Quantum Steering.

We show-both theoretically and experimentally-that Einstein-Podolsky-Rosen steering can be distilled. We present a distillation protocol that outputs a perfectly correlated system-the singlet assemblage-in the asymptotic infinite-copy limit, even for inputs that are arbitrarily close to being unsteerable. As figures of merit for the protocol's performance, we introduce the assemblage fidelity and the singlet-assemblage fraction. These are potentially interesting quantities on their own beyond the current scope. Remarkably, the protocol works well also in the nonasymptotic regime of few copies, in the sense of increasing the singlet-assemblage fraction. We demonstrate the efficacy of the protocol using a hyperentangled photon pair encoding two copies of a two-qubit state. This represents to our knowledge the first observation of deterministic steering concentration. Our findings are not only fundamentally important but may also be useful for semi-device-independent protocols in noisy quantum networks.

Adaptive Causal Network Coding with Feedback for Multipath Multi-hop Communications

We propose a novel multipath multi-hop adaptive and causal random linear network coding (AC-RLNC) algorithm with forward error correction. This algorithm generalizes our joint optimization coding solution for point-to-point communication with delayed feedback. AC-RLNC is adaptive to the estimated channel condition, and is causal, as the coding adjusts the retransmission rates using a priori and posteriori algorithms. In the multipath network, to achieve the desired throughput and delay, we propose to incorporate an adaptive packet allocation algorithm for retransmission, across the available resources of the paths. This approach is based on a discrete water filling algorithm, i.e., bit-filling, but, with two desired objectives, maximize throughput and minimize the delay. In the multipath multi-hop setting, we propose a new decentralized balancing optimization algorithm. This balancing algorithm minimizes the throughput degradation, caused by the variations in the channel quality of the paths at each hop. Furthermore, to increase the efficiency, in terms of the desired objectives, we propose a new selective recoding method at the intermediate nodes. We derive bounds on the throughput and the mean and maximum in-order delivery delay of AC-RLNC, both in the multipath and multipath multi-hop case. In the multipath case, we prove that in the non-asymptotic regime, the suggested code may achieve more than 90% of the channel capacity with zero error probability under mean and maximum in-order delay constraints, namely a mean delay smaller than three times the optimal genie-aided one and a maximum delay within eight times the optimum. In the multipath multi-hop case, the balancing procedure is proven to be optimal with regards to the achieved rate. Through simulations, we demonstrate that the performance of our adaptive and causal approach, compared to selective repeat (SR)-ARQ protocol, is capable of gains up to a factor two in throughput and a factor of more than three in mean delay and eight in maximum delay. The improvements on the throughput delay trade-off are also shown to be significant with regards to the previously developed singlepath AC-RLNC solution.

Joint Pricing and Inventory Decisions for Substitutable Products

A pervasive problem for retailers is to make assortment and inventory procurement decisions of substitutable products with the goal of maximizing expected profits over a pre-defined selling horizon. Multiple inventory replenishments are not allowed. This problem is challenging because of demand substitution effects that are driven initially by the assortment decision and later by stockouts. We propose intuitive and easy-to-implement heuristic based on the sample average approximation method coupled with the Markov chain approximation to discrete choice modelling. We show that our heuristic is asymptotically optimal, and quite effective even in the non-asymptotic regime. Our heuristic exhibits very small optimality gaps and outperform, sometimes dramatically, traditional newsvendor solutions that ignore dynamic substitution effects. Our numerical analysis also reveals that the newsvendor heuristic may perform quite well when the assortment is properly chosen and demand variability is relatively low. These managerial insights help decide when to employ the traditional newsvendor solution. We also look into endogenous pricing and show that even a simple heuristic can significantly enhance profits if prices can be adjusted sufficiently often. Another finding is that price flexibility mitigates the need for safety stocks to the point that not hedging inventories is often better than hedging them.

Accelerated Gossip in Networks of Given Dimension Using Jacobi Polynomial Iterations

Consider a network of agents connected by communication links, where each agent holds a real value. The gossip problem consists in estimating the average of the values diffused in the network in a distributed manner. We develop a method for solving the gossip problem that depends only on the spectral dimension of the network, that is, in the communication network set-up, the dimension of the space in which the agents live. This contrasts with previous work that required the spectral gap of the network as a parameter, or suffered from slow mixing. Our method shows an important improvement over existing algorithms in the nonasymptotic regime, i.e., when the values are far from being fully mixed in the network. Our approach stems from a polynomial-based point of view on gossip algorithms, as well as an approximation of the spectral measure of the graphs with a Jacobi measure. We show the power of the approach with simulations on various graphs, and with performance guarantees on graphs of known spectral dimension, such as grids and random percolation bonds. An extension of this work to distributed Laplacian solvers is discussed. As a side result, we also use the polynomial-based point of view to show the convergence of the message passing algorithm for gossip of Moallemi and Van Roy on regular graphs. The explicit computation of the rate of the convergence shows that message passing has a slow rate of convergence on graphs with small spectral gap.

Heterogeneous Statistical-QoS Driven Resource Allocation Over mmWave Massive-MIMO Based 5G Mobile Wireless Networks in the Non-Asymptotic Regime

The statistical delay-bounded quality-of-service (QoS) theory has been developed to efficiently support multimedia transmissions over 5G wireless networks. On the other hand, unlike in Shannon’s information-theoretic formalism requiring infinite blocklength, finite blocklength coding (FBC) has recently emerged for error control in the non-asymptotic regime, guaranteeing stringent statistical QoS requirements in terms of both latency and reliability for ultra-reliable low-latency communications (URLLC) in 5G services. Moreover, integrated with FBC, millimeter wave (mmWave) massive multi-input multi-output (m-MIMO) schemes have been designed to significantly improve the performance in guaranteeing delay/error-rate bounded QoS. However, due to the complexity of modeling and solving the optimization problems over mmWave m-MIMO fading channels in the non-asymptotic error-control regime, it is challenging to derive an optimal resource allocation policy for maximizing $\epsilon$ -effective capacity to guarantee statistical delay/error-rate bounded QoS. To overcome the above problems, in this paper we propose heterogeneous statistical-QoS driven resource allocation policies for mmWave m-MIMO based 5G wireless networks in both asymptotic and non-asymptotic regimes. In particular, we develop an mmWave m-MIMO based 5G wireless networks model to optimize the effective capacity for our proposed schemes. Our simulations show that our proposed schemes outperform the existing schemes in guaranteeing heterogeneous statistical delay/error-rate bounded QoS.

Hamiltonian Simulation by Qubitization

We present the problem of approximating the time-evolution operatore−iH^tto errorϵ, where the HamiltonianH^=(⟨G|⊗I^)U^(|G⟩⊗I^)is the projection of a unitary oracleU^onto the state|G⟩created by another unitary oracle. Our algorithm solves this with a query complexityO(t+log⁡(1/ϵ))to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which ared-sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as whereH^is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed anyH^in an invariantSU(2)subspace. A large class of operator functions ofH^can then be computed with optimal query complexity, of whiche−iH^tis a special case.