Logarithmic Depth Research Articles

Quantum-classical separations in shallow-circuit-based learning with and without noises

An essential problem in quantum machine learning is to find quantum-classical separations between learning models. However, rigorous and unconditional separations are lacking for supervised learning. Here we construct a classification problem defined by a noiseless constant depth (i.e., shallow) quantum circuit and rigorously prove that any classical neural network with bounded connectivity requires logarithmic depth to output correctly with a larger-than-exponentially-small probability. This unconditional near-optimal quantum-classical representation power separation originates from the quantum nonlocality property that distinguishes quantum circuits from their classical counterparts. We further characterize the noise regimes for demonstrating such a separation on near-term quantum devices under the depolarization noise model. In addition, for quantum devices with constant noise strength, we prove that no super-polynomial classical-quantum separation exists for any classification task defined by Clifford circuits, independent of the structures of the circuits that specify the learning models.

On polytopes with linear rank with respect to generalizations of the split closure

In this paper we study the rank of polytopes contained in the 0-1 cube with respect to t-branch split cuts and t-dimensional lattice cuts for a fixed positive integer t. These inequalities are the same as split cuts when t=1 and generalize split cuts when t>1. For polytopes contained in the n-dimensional 0-1 cube, the work of Balas implies that the split rank can be at most n, and this bound is tight as Cornuéjols and Li gave an example with split rank n. All known examples with high split rank – i.e., at least cn for some positive constant c<1 – are defined by exponentially many (as a function of n) linear inequalities. For any fixed integer t>0, we give a family of polytopes contained in [0,1]n for sufficiently large n such that each polytope has empty integer hull, is defined by O(n) inequalities, and has rank Ω(n) with respect to t-dimensional lattice cuts. Therefore the split rank of these polytopes is Ω(n). It was shown earlier that there exist generalized branch-and-bound proofs, with logarithmic depth, of the nonexistence of integer points in these polytopes. Therefore, our lower bound results on split rank show an exponential separation between the depth of branch-and-bound proofs and split rank.

Depth lower bounds in Stabbing Planes for combinatorial principles

Stabbing Planes (also known as Branch and Cut) is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system. For size lower bounds these are established by monotone circuit arguments, while for depth these are found via communication complexity and protection. As such these bounds apply for lifted versions of combinatorial statements. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas. In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner's Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon's combinatorial Nullenstellensatz. We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.

On the time complexity of achieving optimal throughput in time division multiple access communication networks

&lt;abstract&gt;&lt;p&gt;The fundamental problem of finding transmission schedules for achieving optimal throughput in time division multiple access (TDMA) communication networks is known to be NP-hard. Let $ \mathcal{N} $ be a scheduled $ k $-time slot TDMA network with $ n $ stations and $ m $ links. We showed that an optimal link schedule for $ \mathcal{N} $ can be computed recursively with a recursion tree of logarithmic depth $ \mathcal{O}(\ln m) $ in expectation. Additionally, we showed that optimal link schedules for those TDMA networks, with recursion trees of depth meeting the expectation, can be found in time $ \mathcal{O}(m^{2+\ln k}) $. Likewise, we discuss analogous results for computing optimal station schedules of TDMA networks.&lt;/p&gt;&lt;/abstract&gt;

Image Enhancement Method for Photoacoustic Imaging of Deep Brain Tissue

Photoacoustic imaging (PAI) is an emerging biomedical imaging modality, offering numerous advantages, including high resolution and high contrast. In its application to brain imaging, however, the photoacoustic (PA) signals from brain tissue weaken considerably due to the distortion effects of the skull. This attenuation reduces the resolution and contrast significantly. To address this issue, here we describe a Log-MSR algorithm that combines the logarithmic depth logarithmic enhancement (Log) algorithm and the multi-scale Retinex (MSR) algorithm. In this method, the Log algorithm performs local weighted compensation based on signal attenuation for different depths, while the MSR algorithm improves the contrast of the image. The proposed Log-MSR algorithm was tested and validated using several phantom and in vivo experiments. The enhanced images constructed by the Log-MSR algorithm were qualitatively and quantitatively analyzed in terms of brain structure and function. Our results show that the Log-MSR algorithm may provide a significant enhancement to photoacoustic imaging of deep brain tissue.

3D spatial interpolation of soil heavy metals by combining kriging with depth function trend model

In this study, a hybrid method combining 3D kriging and depth function considering heterogeneity (3DK_DF) was proposed to enhance the accuracy and reliability of 3D spatial interpolation for soil heavy metals. Soil samples collected at varied depth intervals in a mining city in China were used as a case dataset. First, the parameters of a logarithmic depth function model for every horizontal soil sample site were fitted on the basis of the observed values of soil Cd collected at varied depth intervals. Second, the 3D trend of soil Cd was obtained on the basis of the spatial distributions of the parameters of the logarithmic depth function model. Third, 3D kriging was used to generate the 3D spatial distribution of residual Cd. Finally, the 3D spatial distribution of the soil Cd was obtained by combining the 3D trend and residual results. The interpolation accuracy of 3DK_DF improved by 29.71% and 48.9% compared with those of the 3D kriging without a trend analysis and the 3D kriging with a polynomial trend model, respectively. The proposed hybrid 3D interpolation method could be of great significance for the comprehensive assessment of soil heavy metal pollution.

Parameterized Complexity Classes Defined by Threshold Circuits and Their Connection with Sorting Networks

The main complexity classes of the Parameterized Intractability Theory are based on weighted Boolean circuit satisfiability problems and organized into a hierarchy so-called W-hierarchy. The W-hierarchy enables fine-grained complexity analyses of parameterized problems that are unlikely to belong to the FPT class. In this paper, we introduce the Th-hierarchy, a natural generalization of the W-hierarchy defined by unweighted threshold circuit satisfiability problems. Investigating the relationship between Th-hierarchy and W-hierarchy, we discuss the complexity of transforming Threshold circuits into Boolean circuits, and observe that sorting networks are powerful tools to handle such transformations. First, we show that these hierarchies collapse at the last level (W[P][Formula: see text][Formula: see text][Formula: see text]Th[P]). After that, we present a time complexity analysis of an AKS sorting network construction, which supports some of our results. Finally, we prove that Th[[Formula: see text]] [Formula: see text] W[SAT] for every [Formula: see text]. As a by-product, our studies suggest that it is relevant to consider a new class based on logarithmic depth circuits in the W-hierarchy.

Prediction of defect depth in GFRP composite by square-heating thermography

Square-heating thermography was used to predict the depth of glass fiber reinforced plastic (GFRP) flat-bottom holes during its cooling period. Four methods were proposed for depth prediction using a specified characteristic peak time. Square-heating thermography cooling theory was analyzed using numerical methods, the linear relationship between the logarithmic characteristic time and logarithmic depth of the GFRP’s flat-bottom holes was obtained, and the slope and intercept of the linear curve were mainly related to the thermal diffusivity and the heating time. Finally, the flat-bottom hole depth was predicted using experimental cooling data from GFRP samples.

Fault tolerance in qudit circuit design

The efficient decomposition of multicontrolled gates is a significant factor in quantum compiling, both in circuit depth and $T$-gate count. Recent work has demonstrated that qudits have the potential to reduce resource requirements from linear to logarithmic depth and to avoid fractional phase rotations. Here, we argue, based on the scaling of decoherence in high-index states, that circuit depth is not the primary factor, and that both the choice of entangling gate and interaction network topology act together to determine the spread of errors and the ultimate failure rate in a circuit. We further show that for certain linear-depth circuits, additional error mitigation is possible via selective application of resources.

Asymptotically improved circuit for a d -ary Grover's algorithm with advanced decomposition of the n -qudit Toffoli gate

The progress in building quantum computers to execute quantum algorithms has recently been remarkable. Grover's search algorithm in a binary quantum system provides considerable speed-up over classical paradigm. Further, Grover's algorithm can be extended to a $d$-ary (qudit) quantum system for utilizing the advantage of larger state space, which helps to reduce the run-time of the algorithm as compared to the traditional binary quantum systems. In a qudit quantum system, an $n$-qudit Toffoli gate plays a significant role in the accurate implementation of Grover's algorithm. In this article, a generalized $n$-qudit Toffoli gate has been realized using higher dimensional qudits to attain a logarithmic depth decomposition without ancilla qudit. The circuit for Grover's algorithm has then been designed for any $d$-ary quantum system, where $d \ge 2$, with the proposed $n$-qudit Toffoli gate to obtain optimized depth compared to earlier approaches. The technique for decomposing an $n$-qudit Toffoli gate requires access to two immediately higher energy levels, making the design susceptible to errors. Nevertheless, we show that the percentage decrease in the probability of error is significant as we have reduced both gate count and circuit depth as compared to that in state-of-the-art works.

Twin-width II: small classes

The recently introduced twin-width of a graph \(G\) is the minimum integer \(d\) such that \(G\) has a \(d\)-contraction sequence, that is, a sequence of \(\left| V(G) \right|-1\) iterated vertex identifications for which the overall maximum number of red edges incident to a single vertex is at most \(d\), where a red edge appears between two sets of identified vertices if they are not homogeneous in \(G\) (not fully adjacent nor fully non-adjacent). We show that if a graph admits a \(d\)-contraction sequence, then it also has a linear-arity tree of \(f(d)\)-contractions, for some function \(f\). Informally if we accept to worsen the twin-width bound, we can choose the next contraction from a set of \(\Theta(\left| V(G) \right|)\) pairwise disjoint pairs of vertices. This has two main consequences. First it permits to show that every bounded twin-width class is small, i.e., has at most \(n!c^n\) graphs labeled by \([n]\), for some constant \(c\). This unifies and extends the same result for bounded treewidth graphs [Beineke and Pippert, JCT '69], proper subclasses of permutations graphs [Marcus and Tardos, JCTA '04], and proper minor-free classes [Norine et al., JCTB '06]. It implies in turn that bounded-degree graphs, interval graphs, and unit disk graphs have unbounded twin-width. The second consequence is an \(O(\log n)\)-adjacency labeling scheme for bounded twin-width graphs, confirming several cases of the implicit graph conjecture. We then explore the small conjecture that, conversely, every small hereditary class has bounded twin-width. The conjecture passes many tests. Inspired by sorting networks of logarithmic depth, we show that \(\log_{\Theta(\log \log d)}n\)-subdivisions of \(K_n\) (a small class when \(d\) is constant) have twin-width at most \(d\). We obtain a rather sharp converse with a surprisingly direct proof: the \(\log_{d+1}n\)-subdivision of \(K_n\) has twin-width at least \(d\). Secondly graphs with bounded stack or queue number (also small classes) have bounded twin-width. These sparse classes are surprisingly rich since they contain certain (small) classes of expanders. Thirdly we show that cubic expanders obtained by iterated random 2-lifts from \(K_4\) [Bilu and Linial, Combinatorica '06] also have bounded twin-width. These graphs are related to so-called separable permutations and also form a small class. We suggest a promising connection between the small conjecture and group theory. Finally we define a robust notion of sparse twin-width. We show that for a hereditary class \(\mathcal C\) of bounded twin-width the five following conditions are equivalent: every graph in \(\mathcal C\) (1) has no \(K_{t,t}\) subgraph for some fixed \(t\), (2) has an adjacency matrix without a \(d\)-by-\(d\) division with a 1 entry in each of the \(d^2\) cells for some fixed \(d\), (3) has at most linearly many edges, (4) the subgraph closure of \(\mathcal C\) has bounded twin-width, and (5) \(\mathcal C\) has bounded expansion. We discuss how sparse classes with similar behavior with respect to clique subdivisions compare to bounded sparse twin-width.Mathematics Subject Classifications: 68R10, 05C30, 05C48Keywords: Twin-width, small classes, expanders, clique subdivisions, sparsity

An Automaton Group with PSPACE-Complete Word Problem

We construct an automaton group with a PSPACE-complete word problem, proving a conjecture due to Steinberg. Additionally, the constructed group has a provably more difficult, namely EXPSPACE-complete, compressed word problem and acts over a binary alphabet. Thus, it is optimal in terms of the alphabet size. Our construction directly simulates the computation of a Turing machine in an automaton group and, therefore, seems to be quite versatile. It combines two ideas: the first one is a construction used by D’Angeli, Rodaro and the first author to obtain an inverse automaton semigroup with a PSPACE-complete word problem and the second one is to utilize a construction used by Barrington to simulate Boolean circuits of bounded degree and logarithmic depth in the group of even permutations over five elements.

Early Warning of Regional Landslide Disaster and Development of Rural Ecological Industrialization Based on IoT Sensor

Regional landslide disaster is actually the instability caused by the movement of ground or slopes, etc. If a landslide occurs in a habitat area where people live, it can cause a great deal of damage. So, in order to improve the early warning effect of regional geological landslide disaster, the study abandoned the conventional geological probe data, directly used the tilt photography data provided by the fixed camera, introduced the camera clock synchronization control system supported by the high-sensitivity atomic clock timing function, and used the data warehouse hardware and computing host hardware. Combined with the spatial convolution neural network, fuzzy neural network (logarithmic depth iterative regression neural network, polynomial depth iterative regression neural network, transfinite learning machine, and binary neural network) and other technologies finally realize high geological disaster early warning sensitivity and long early warning advance. The system will become an alternative to the previous geological disaster early warning system based on the direct data of geological embedded probes.

Variational quantum algorithms for trace distance and fidelity estimation

Estimating the difference between quantum data is crucial in quantum computing. However, as typical characterizations of quantum data similarity, the trace distance and quantum fidelity are believed to be exponentially-hard to evaluate in general. In this work, we introduce hybrid quantum–classical algorithms for these two distance measures on near-term quantum devices where no assumption of input state is required. First, we introduce the variational trace distance estimation (VTDE) algorithm. We in particular provide the technique to extract the desired spectrum information of any Hermitian matrix by local measurement. A novel variational algorithm for trace distance estimation is then derived from this technique, with the assistance of a single ancillary qubit. Notably, VTDE could avoid the barren plateau issue with logarithmic depth circuits due to a local cost function. Second, we introduce the variational fidelity estimation algorithm. We combine Uhlmann’s theorem and the freedom in purification to translate the estimation task into an optimization problem over a unitary on an ancillary system with fixed purified inputs. We then provide a purification subroutine to complete the translation. Both algorithms are verified by numerical simulations and experimental implementations, exhibiting high accuracy for randomly generated mixed states.

Obfuscating Circuits Via Composite-Order Graded Encoding

We present a candidate obfuscator based on composite-order graded encoding schemes (GES), which are a generalization of multilinear maps. Our obfuscator operates on circuits directly without converting them into formulas or branching programs as was done in previous solutions. As a result, the time and size complexity of the obfuscated program, measured by the number of GES elements, is directly proportional to the circuit complexity of the program being obfuscated. This improves upon previous constructions whose complexity was related to the formula or branching program size. Known instantiations of Graded Encoding Schemes allow us to obfuscate circuit classes of polynomial degree, which include for example families of circuits of logarithmic depth. We prove that our obfuscator is secure against a class of generic algebraic attacks, formulated by a generic graded encoding model. We further consider a more robust model which provides more power to the adversary and extend our results to this setting as well. As a secondary contribution, we define a new simple notion of algebraic security (which was implicit in previous works) and show that it captures standard security relative to an ideal GES oracle.

On Small-depth Frege Proofs for Tseitin for Grids

We prove that a small-depth Frege refutation of the Tseitin contradiction on the grid requires subexponential size. We conclude that polynomial size Frege refutations of the Tseitin contradiction must use formulas of almost logarithmic depth.

On the depth overhead incurred when running quantum algorithms on near-term quantum computers with limited qubit connectivity

This paper addresses the problem of finding the depth overhead that will be incurred when running quantum circuits on near-term quantum computers. Specifically, it is envisaged that near-term quantum computers will have low qubit connectivity: each qubit will only be able to interact with a subset of the other qubits, a reality typically represented by a qubit interaction graph in which a vertex represents a qubit and an edge represents a possible direct 2-qubit interaction (gate). Thus the depth overhead is unavoidably incurred by introducing swap gates into the quantum circuit to enable general qubit interactions. This paper proves that there exist quantum circuits where a depth overhead in Omega(\log n) must necessarily be incurred when running quantum circuits with n qubits on quantum computers whose qubit interaction graph has finite degree, but that such a logarithmic depth overhead is achievable. The latter is shown by the construction of a 4-regular qubit interaction graph and associated compilation algorithm that can execute any quantum circuit with only a logarithmic depth overhead.

Estimating exact energies in quantum simulation without Toffoli gates

Qubitization is a modern approach to estimate Hamiltonian eigenvalues without simulating its time evolution. While in this way approximation errors are avoided, its resource and gate requirements are more extensive: qubitization requires additional qubits to store information about the Hamiltonian, and Toffoli gates to probe them throughout the routine. Recently, it was shown that storing the Hamiltonian in a unary representation can alleviate the need for such gates in one of the qubitization subroutines. Building on that principle, we develop an entirely new decomposition of the entire algorithm: without Toffoli gates, we can encode the Hamiltonian into qubits within logarithmic depth.

Homomorphic Comparison for Point Numbers with User-Controllable Precision and Its Applications

This work is mainly interested in ensuring users’ privacy in asymmetric computing, such as cloud computing. In particular, because lots of user data are expressed in non-integer data types, privacy-enhanced applications built on fully homomorphic encryption (FHE) must support real-valued comparisons due to the ubiquity of real numbers in real-world applications. However, as FHE schemes operate in specific domains, such as that of congruent integers, most FHE-based solutions focus only on homomorphic comparisons of integers. Attempts to overcome this barrier can be grouped into two classes. Given point numbers in the form of approximate real numbers, one class of solution uses a special-purpose encoding to represent the point numbers, whereas the other class constructs a dedicated FHE scheme to encrypt point numbers directly. The solutions in the former class may provide depth-efficient arithmetic (i.e., logarithmic depth in the size of the data), but not depth-efficient comparisons between FHE-encrypted point numbers. The second class may avoid this problem, but it requires the precision of point numbers to be determined before the FHE setup is run. Thus, the precision of the data cannot be controlled once the setup is complete. Furthermore, because the precision accuracy is closely related to the sizes of the encryption parameters, increasing the precision of point numbers results in increasing the sizes of the FHE parameters, which increases the sizes of the public keys and ciphertexts, incurring more expensive computation and storage. Unfortunately, this problem also occurs in many of the proposals that fall into the first class. In this work, we are interested in depth-efficient comparison over FHE-encrypted point numbers. In particular, we focus on enabling the precision of point numbers to be manipulated after the system parameters of the underlying FHE scheme are determined, and even after the point numbers are encrypted. To this end, we encode point numbers in continued fraction (CF) form. Therefore, our work lies in the first class of solutions, except that our CF-based approach allows depth-efficient homomorphic comparisons (more precisely, the complexity of the comparison is O ( log κ + log n ) for a number of partial quotients n and their bit length κ , which is normally small) while allowing users to determine the precision of the encrypted point numbers when running their applications. We develop several useful applications (e.g., sorting) that leverage our CF-based homomorphic comparisons.

Molecular sampling at logarithmic rates for next-generation sequencing.

Next-generation sequencing is a cutting edge technology, but to quantify a dynamic range of abundances for different RNA or DNA species requires increasing sampling depth to levels that can be prohibitively expensive due to physical limits on molecular throughput of sequencers. To overcome this problem, we introduce a new general sampling theory which uses biophysical principles to functionally encode the abundance of a species before sampling, SeQUential depletIon and enriCHment (SQUICH). In theory and simulation, SQUICH enables sampling at a logarithmic rate to achieve the same precision as attained with conventional sequencing. A simple proof of principle experimental implementation of SQUICH in a controlled complex system of ~262,000 oligonucleotides already reduces sequencing depth by a factor of 10. SQUICH lays the groundwork for a general solution to a fundamental problem in molecular sampling and enables a new generation of efficient, precise molecular measurement at logarithmic or better sampling depth.