A New Lower Bound Technique for Quantum Circuits without Ancillae

We present a technique to derive depth lower bounds for quantum circuits. The technique is based on the observation that in circuits without ancillae, only a few input states can set all the control qubits of a Toffoli gate to 1. This can be used to selectively remove large Toffoli gates from a quantum circuit while keeping the cumulative error low. We use the technique to give another proof that parity cannot be computed by constant depth quantum circuits without ancillæ.

On Finding Sensitivity of Quantum and Classical Gates

We consider a fault model of Boolean gates, both classical and quantum, where some of the inputs may not be connected to the actual gate hardware. This model is somewhat similar to the stuck-at model which is a very popular model in testing Boolean circuits. We consider the problem of detecting such faults; the detection algorithm can query the faulty gate and its complexity is the number of such queries. This problem is related to determining the sensitivity of Boolean functions. We show how quantum parallelism can be used to detect such faults. Specifically, we show that a quantum algorithm can detect such faults more efficiently than a classical algorithm for a Parity gate and an AND gate. We give explicit constructions of quantum detector algorithms and show lower bounds for classical algorithms. We show that the model for detecting such faults is similar to algebraic decision trees and extend some known results from quantum query complexity to prove some of our results.

On the Complexity of Quantum ACC

For any q > 1, let MOD_q be a quantum gate that determines if the number of 1's in the input is divisible by q. We show that for any q,t > 1, MOD_q is equivalent to MOD_t (up to constant depth). Based on the case q=2, Moore has shown that quantum analogs of AC^(0), ACC[q], and ACC, denoted QAC^(0)_wf, QACC[2], QACC respectively, define the same class of operators, leaving q > 2 as an open question. Our result resolves this question, implying that QAC^(0)_wf = QACC[q] = QACC for all q. We also prove the first upper bounds for QACC in terms of related language classes. We define classes of languages EQACC, NQACC (both for arbitrary complex amplitudes) and BQACC (for rational number amplitudes) and show that they are all contained in TC^(0). To do this, we show that a TC^(0) circuit can keep track of the amplitudes of the state resulting from the application of a QACC operator using a constant width polynomial size tensor sum. In order to accomplish this, we also show that TC^(0) can perform iterated addition and multiplication in certain field extensions.

A Neural Model of Surface Perception: Lightness, Anchoring, and Filling-in

This article develops a neural model of how the visual system processes natural images under variable illumination conditions to generate surface lightness percepts. Previous models have clarified how the brain can compute the relative contrast of images from variably illuminate scenes. How the brain determines an absolute lightness scale that "anchors" percepts of surface lightness to us the full dynamic range of neurons remains an unsolved problem. Lightness anchoring properties include articulation, insulation, configuration, and are effects. The model quantatively simulates these and other lightness data such as discounting the illuminant, the double brilliant illusion, lightness constancy and contrast, Mondrian contrast constancy, and the Craik-O'Brien-Cornsweet illusion. The model also clarifies the functional significance for lightness perception of anatomical and neurophysiological data, including gain control at retinal photoreceptors, and spatioal contrast adaptation at the negative feedback circuit between the inner segment of photoreceptors and interacting horizontal cells. The model retina can hereby adjust its sensitivity to input intensities ranging from dim moonlight to dazzling sunlight. A later model cortical processing stages, boundary representations gate the filling-in of surface lightness via long-range horizontal connections. Variants of this filling-in mechanism run 100-1000 times faster than diffusion mechanisms of previous biological filling-in models, and shows how filling-in can occur at realistic speeds. A new anchoring mechanism called the Blurred-Highest-Luminance-As-White (BHLAW) rule helps simulate how surface lightness becomes sensitive to the spatial scale of objects in a scene. The model is also able to process natural images under variable lighting conditions.

SOVEREIGN: A Self-Organizing, Vision, Expectation, Recognition, Emotion, Intelligent, Goal-Oriented Navigation System

Both animals and mobile robots, or animats, need adaptive control systems to guide their movements through a novel environment. Such control systems need reactive mechanisms for exploration, and learned plans to efficiently reach goal objects once the environment is familiar. How reactive and planned behaviors interact together in real time, and arc released at the appropriate times, during autonomous navigation remains a major unsolved problern. This work presents an end-to-end model to address this problem, named SOVEREIGN: A Self-Organizing, Vision, Expectation, Recognition, Emotion, Intelligent, Goal-oriented Navigation system. The model comprises several interacting subsystems, governed by systems of nonlinear differential equations. As the animat explores the environment, a vision module processes visual inputs using networks that arc sensitive to visual form and motion. Targets processed within the visual form system arc categorized by real-time incremental learning. Simultaneously, visual target position is computed with respect to the animat's body. Estimates of target position activate a motor system to initiate approach movements toward the target. Motion cues from animat locomotion can elicit orienting head or camera movements to bring a never target into view. Approach and orienting movements arc alternately performed during animat navigation. Cumulative estimates of each movement, based on both visual and proprioceptive cues, arc stored within a motor working memory. Sensory cues are stored in a parallel sensory working memory. These working memories trigger learning of sensory and motor sequence chunks, which together control planned movements. Effective chunk combinations arc selectively enhanced via reinforcement learning when the animat is rewarded. The planning chunks effect a gradual transition from reactive to planned behavior. The model can read-out different motor sequences under different motivational states and learns more efficient paths to rewarded goals as exploration proceeds. Several volitional signals automatically gate the interactions between model subsystems at appropriate times. A 3-D visual simulation environment reproduces the animat's sensory experiences as it moves through a simplified spatial environment. The SOVEREIGN model exhibits robust goal-oriented learning of sequential motor behaviors. Its biomimctic structure explicates a number of brain processes which are involved in spatial navigation.

SOVEREIGN: An Autonomous Neural System for Incrementally Learning Planned Action Sequence to Navigate Towards a Rewarded Goal

How do reactive and planned behaviors interact in real time? How are sequences of such behaviors released at appropriate times during autonomous navigation to realize valued goals? Controllers for both animals and mobile robots, or animats, need reactive mechanisms for exploration, and learned plans to reach goal objects once an environment becomes familiar. The SOVEREIGN (Self-Organizing, Vision, Expectation, Recognition, Emotion, Intelligent, Goaloriented Navigation) animat model embodies these capabilities, and is tested in a 3D virtual reality environment. SOVEREIGN includes several interacting subsystems which model complementary properties of cortical What and Where processing streams and which clarify similarities between mechanisms for navigation and arm movement control. As the animat explores an environment, visual inputs are processed by networks that are sensitive to visual form and motion in the What and Where streams, respectively. Position-invariant and sizeinvariant recognition categories are learned by real-time incremental learning in the What stream. Estimates of target position relative to the animat are computed in the Where stream, and can activate approach movements toward the target. Motion cues from animat locomotion can elicit head-orienting movements to bring a new target into view. Approach and orienting movements are alternately performed during animat navigation. Cumulative estimates of each movement are derived from interacting proprioceptive and visual cues. Movement sequences are stored within a motor working memory. Sequences of visual categories are stored in a sensory working memory. These working memories trigger learning of sensory and motor sequence categories, or plans, which together control planned movements. Predictively effective chunk combinations are selectively enhanced via reinforcement learning when the animat is rewarded. Selected planning chunks effect a gradual transition from variable reactive exploratory movements to efficient goal-oriented planned movement sequences. Volitional signals gate interactions between model subsystems and the release of overt behaviors. The model can control different motor sequences under different motivational states and learns more efficient sequences to rewarded goals as exploration proceeds.

A Neural Model of 3D Shape-From-Texture: Multiple-Scale Filtering, Boundary Grouping, and Surface Filling-In

A neural model is presented of how cortical areas V1, V2, and V4 interact to convert a textured 2D image into a representation of curved 3D shape. Two basic problems are solved to achieve this: (1) Patterns of spatially discrete 2D texture elements are transformed into a spatially smooth surface representation of 3D shape. (2) Changes in the statistical properties of texture elements across space induce the perceived 3D shape of this surface representation. This is achieved in the model through multiple-scale filtering of a 2D image, followed by a cooperative-competitive grouping network that coherently binds texture elements into boundary webs at the appropriate depths using a scale-to-depth map and a subsequent depth competition stage. These boundary webs then gate filling-in of surface lightness signals in order to form a smooth 3D surface percept. The model quantitatively simulates challenging psychophysical data about perception of prolate ellipsoids (Todd and Akerstrom, 1987, J. Exp. Psych., 13, 242). In particular, the model represents a high degree of 3D curvature for a certain class of images, all of whose texture elements have the same degree of optical compression, in accordance with percepts of human observers. Simulations of 3D percepts of an elliptical cylinder, a slanted plane, and a photo of a golf ball are also presented.

Vector Associative Maps: Unsupervised Real-time Error-based Learning and Control of Movement Trajectories

This article describes neural network models for adaptive control of arm movement trajectories during visually guided reaching and, more generally, a framework for unsupervised real-time error-based learning. The models clarify how a child, or untrained robot, can learn to reach for objects that it sees. Piaget has provided basic insights with his concept of a circular reaction: As an infant makes internally generated movements of its hand, the eyes automatically follow this motion. A transformation is learned between the visual representation of hand position and the motor representation of hand position. Learning of this transformation eventually enables the child to accurately reach for visually detected targets. Grossberg and Kuperstein have shown how the eye movement system can use visual error signals to correct movement parameters via cerebellar learning. Here it is shown how endogenously generated arm movements lead to adaptive tuning of arm control parameters. These movements also activate the target position representations that are used to learn the visuo-motor transformation that controls visually guided reaching. The AVITE model presented here is an adaptive neural circuit based on the Vector Integration to Endpoint (VITE) model for arm and speech trajectory generation of Bullock and Grossberg. In the VITE model, a Target Position Command (TPC) represents the location of the desired target. The Present Position Command (PPC) encodes the present hand-arm configuration. The Difference Vector (DV) population continuously.computes the difference between the PPC and the TPC. A speed-controlling GO signal multiplies DV output. The PPC integrates the (DV)·(GO) product and generates an outflow command to the arm. Integration at the PPC continues at a rate dependent on GO signal size until the DV reaches zero, at which time the PPC equals the TPC. The AVITE model explains how self-consistent TPC and PPC coordinates are autonomously generated and learned. Learning of AVITE parameters is regulated by activation of a self-regulating Endogenous Random Generator (ERG) of training vectors. Each vector is integrated at the PPC, giving rise to a movement command. The generation of each vector induces a complementary postural phase during which ERG output stops and learning occurs. Then a new vector is generated and the cycle is repeated. This cyclic, biphasic behavior is controlled by a specialized gated dipole circuit. ERG output autonomously stops in such a way that, across trials, a broad sample of workspace target positions is generated. When the ERG shuts off, a modulator gate opens, copying the PPC into the TPC. Learning of a transformation from TPC to PPC occurs using the DV as an error signal that is zeroed due to learning. This learning scheme is called a Vector Associative Map, or VAM. The VAM model is a general-purpose device for autonomous real-time error-based learning and performance of associative maps. The DV stage serves the dual function of reading out new TPCs during performance and reading in new adaptive weights during learning, without a disruption of real-time operation. YAMs thus provide an on-line unsupervised alternative to the off-line properties of supervised error-correction learning algorithms. YAMs and VAM cascades for learning motor-to-motor and spatial-to-motor maps are described. YAM models and Adaptive Resonance Theory (ART) models exhibit complementary matching, learning, and performance properties that together provide a foundation for designing a total sensory-cognitive and cognitive-motor autonomous system.