Power of Uninitialized Qubits in Shallow Quantum Circuits

We study the computational power of shallow quantum circuits with O(log n) initialized and n^{O(1)} uninitialized ancillary qubits, where n is the input length and the initial state of the uninitialized ancillary qubits is arbitrary. First, we show that such a circuit can compute any symmetric function on n bits that is classically computable in polynomial time. Then, we regard such a circuit as an oracle and show that a polynomial-time classical algorithm with the oracle can estimate the elements of any unitary matrix corresponding to a constant-depth quantum circuit on n qubits. Since it seems unlikely that these tasks can be done with only O(log n) initialized ancillary qubits, our results give evidences that adding uninitialized ancillary qubits increases the computational power of shallow quantum circuits with only O(log n) initialized ancillary qubits. Lastly, to understand the limitations of uninitialized ancillary qubits, we focus on near-logarithmic-depth quantum circuits with them and show the impossibility of computing the parity function on n bits

Quantum Convolutional Neural Networks for Multi-Channel Supervised Learning

As the rapidly evolving field of machine learning continues to produce incredibly useful tools and models, the potential for quantum computing to provide speed up for machine learning algorithms is becoming increasingly desirable. In particular, quantum circuits in place of classical convolutional filters for image detection-based tasks are being investigated for the ability to exploit quantum advantage. However, these attempts, referred to as quantum convolutional neural networks (QCNNs), lack the ability to efficiently process data with multiple channels and therefore are limited to relatively simple inputs. In this work, we present a variety of hardware-adaptable quantum circuit ansatzes for use as convolutional kernels, and demonstrate that the quantum neural networks we report outperform existing QCNNs on classification tasks involving multi-channel data. We envision that the ability of these implementations to effectively learn inter-channel information will allow quantum machine learning methods to operate with more complex data. This work is available as open source at https://github.com/anthonysmaldone/QCNN-Multi-Channel-Supervised-Learning

Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits

Recently, Bravyi, Gosset, and K\"{o}nig (Science, 2018) exhibited a search problem called the 2D Hidden Linear Function (2D HLF) problem that can be solved exactly by a constant-depth quantum circuit using bounded fan-in gates (or QNC^0 circuits), but cannot be solved by any constant-depth classical circuit using bounded fan-in AND, OR, and NOT gates (or NC^0 circuits). In other words, they exhibited a search problem in QNC^0 that is not in NC^0. We strengthen their result by proving that the 2D HLF problem is not contained in AC^0, the class of classical, polynomial-size, constant-depth circuits over the gate set of unbounded fan-in AND and OR gates, and NOT gates. We also supplement this worst-case lower bound with an average-case result: There exists a simple distribution under which any AC^0 circuit (even of nearly exponential size) has exponentially small correlation with the 2D HLF problem. Our results are shown by constructing a new problem in QNC^0, which we call the Relaxed Parity Halving Problem, which is easier to work with. We prove our AC^0 lower bounds for this problem, and then show that it reduces to the 2D HLF problem. As a step towards even stronger lower bounds, we present a search problem that we call the Parity Bending Problem, which is in QNC^0/qpoly (QNC^0 circuits that are allowed to start with a quantum state of their choice that is independent of the input), but is not even in AC^0[2] (the class AC^0 with unbounded fan-in XOR gates). All the quantum circuits in our paper are simple, and the main difficulty lies in proving the classical lower bounds. For this we employ a host of techniques, including a refinement of H{\aa}stad's switching lemmas for multi-output circuits that may be of independent interest, the Razborov-Smolensky AC^0[2] lower bound, Vazirani's XOR lemma, and lower bounds for non-local games

On Algorithms, Separability and Cellular Automata in Quantum Computing

In Part I of this thesis, we present a new model of quantum cellular automata (QCA) based on local unitary operations. We will describe a set of desirable properties for any QCA model, and show that all of these properties are satisfied by the new model, while previous models of QCA do not. We will also show that the computation model based on Local Unitary QCA is equivalent to the Quantum Circuit model of computation, and give a number of applications of this new model of QCA. We also present a physical model of classical CA, on which the Local Unitary QCA model is based, and Coloured QCA, which is an alternative to the Local Unitary QCA model that can be used as the basis for implementing QCA in actual physical systems. In Part II, we explore the quantum separability problem, where we are given a density matrix for a state over two quantum systems, and we are to determine whether the state is separable with respect to these systems. We also look at the converse problem of finding an entanglement witness, which is an observable operator which can give a verification that a particular quantum state is indeed entangled. Although the combined problem is known to be NP-hard in general, it reduces to a convex optimization problem, and by exploiting specific properties of the set of separable states, we introduce a classical algorithm for solving this problem based on an Interior Point Algorithm introduced by Atkinson and Vaidya in 1995. In Part III, we explore the use of a low-depth AQFT (approximate quantum Fourier transform) in quantum phase estimation. It has been shown previously that the logarithmic-depth AQFT is as effective as the full QFT for the purposes of phase estimation. However, with sub-logarithmic depth, the phase estimation algorithm no longer works directly. In this case, results of the phase estimation algorithm need classical post-processing in order to retrieve the desired phase information. A generic technique such as the method of maximum likelihood can be used in order to recover the original phase. Unfortunately, working with the likelihood function analytically is intractable for the phase estimation algorithm. We develop some computational techniques to handle likelihood functions that occur in phase estimation algorithms. These computational techniques may potentially aid in the analysis of certain likelihood functions

Charge State Dynamics and Quantum Sensing with Defects in Diamond

In recent years, defect centers in wide band gap semiconductors such as diamond, have received significant attention. Defects offer great utility as single photon emitters, nanoscale sensors, and quantum memories and registers for quantum computation. Critical to the utility of these defects, is their charge state. In this dissertation, experiments surrounding the charge state dynamics and the carrier dynamics are performed and analyzed. Extensive studies of the ionization and recombination processes of defects in diamond, specifically, the Nitrogen Vacancy (NV) center, have been performed. Diffusion of ionized charge carriers has been imaged indirectly through the recapture of said carriers by optically active defects such as the NV center and the Silicon Vacancy (SiV) center. With proper understanding of the carrier dynamics, diamond stands to be a strong competitor in the field of spintronics for quantum information processing. Additionally, the understanding of these charge state dynamics is utilized in a novel proof of principle experiment, showing that the NV center defect’s charge state could serve as an ultra-dense 3D memory platform. The NV center has also been used as a nanoscale magnetometer. The high degree of spin polarization and the ease of manipulation of the NV allows us to transfer this polarization to other spins in the diamond and assists in the detection of spins outside of the diamond. Ensembles of Nitrogen Vacancy centers can be used to perform NMR spectroscopy of sample volumes not achievable through traditional methods. The operating mechanisms of the magnetic resonance aspects of the NV center will be discussed in depth. Work surrounding the control and polarization of the NV center’s host nitrogen spin will be covered. The topic of sensitivity and methods to improve that sensitivity will be covered as well

Computer Aided Verification

This open access two-volume set LNCS 13371 and 13372 constitutes the refereed proceedings of the 34rd International Conference on Computer Aided Verification, CAV 2022, which was held in Haifa, Israel, in August 2022. The 40 full papers presented together with 9 tool papers and 2 case studies were carefully reviewed and selected from 209 submissions. The papers were organized in the following topical sections: Part I: Invited papers; formal methods for probabilistic programs; formal methods for neural networks; software Verification and model checking; hyperproperties and security; formal methods for hardware, cyber-physical, and hybrid systems. Part II: Probabilistic techniques; automata and logic; deductive verification and decision procedures; machine learning; synthesis and concurrency. This is an open access book

Tools and Algorithms for the Construction and Analysis of Systems

This open access two-volume set constitutes the proceedings of the 26th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The total of 60 regular papers presented in these volumes was carefully reviewed and selected from 155 submissions. The papers are organized in topical sections as follows: Part I: Program verification; SAT and SMT; Timed and Dynamical Systems; Verifying Concurrent Systems; Probabilistic Systems; Model Checking and Reachability; and Timed and Probabilistic Systems. Part II: Bisimulation; Verification and Efficiency; Logic and Proof; Tools and Case Studies; Games and Automata; and SV-COMP 2020