Symbolic Quantum Circuit Simplification in SymPy

In the field of quantum information science, one can design a series of quantum logic operations known as a circuit. Circuits are the basis for quantum computations in quantum computing. As circuits will most likely be designed from a logical standpoint, there could exist mathematical redundancies which will lead to a larger circuit than necessary. These redundancies are computationally expensive, and there is a need for them to be found and eliminated to simplify the circuit. We present our research on finding the rules for simplifying circuits and its implementation in SymPy

Applications and hardware considerations for quantum computing

Quantum computers are expected to one day be able to solve a set of problems which are practically impossible with classical super computers, even with their projected continued improvement. As the field of quantum computing has continued to evolve the somewhat disparate research areas of algorithms and hardware have improved in their integration. Fully optimizing quantum algorithms requires a solid understanding of the quantum hardware, and metering experimental hardware priorities requires an understanding of the general algorithm requirements. This thesis initially provides an overview of quantum hardware and applications and discusses their interplay in both the near term and fault tolerant regime. A greater focus is placed on trapped ion architectures in this thesis and in particular the shuttling based approach of the Ion Quantum Technology group at the University of Sussex. A routing algorithm is provided which can efficiently enable all to all connectivity for the shuttling based trapped ion design without positional swaps. A simulation tool was created and used to develop and characterize routing algorithms. The cost of enabling connectivity in Noisy-Intermediate-Scale-Quantum devices is an important factor in determining computational power. The core ideas of this routing algorithm are currently being integrated into a software compiler stack that will control real quantum hardware. An error model for the shuttling based design is presented which makes use of the time cost for connectivity results from the simulation tool. The error model is used to estimate the computational power (quantum volume) of the design as a function of experimental parameters. The error model can be used to help meter experimental priorities by identifying the most impactful parameters across particular regimes. A comparison is performed using metrics such as Quantum Volume, between the shuttling based trapped ion design and a superconducting grid which uses logical swaps to enable connectivity, and it is found that the trapped ion design has a substantially lower cost associated with connectivity. Large scale trapped ion devices are considered and the total time required to enable all to all connectivity is estimated for both the modular shuttling approach and for the approach that uses small scale modules connected via photonic interconnects. A review of fault tolerant methods for quantum chemistry is presented. Resource estimations are provided all the way down to the required wall-clock time and number of physical qubits, for ground state energy calculations for molecules across different basis set sizes. The basis set size at which a quantum computer can meaningfully outperform a classical supercomputer is estimated. Determining the point at which a quantum advantage may be realised can help the field progress by setting realistic expectations and by having a device size to aim for. The impact of hardware considerations such as the code cycle time is investigated by including a wider range of possible surface code error correction configurations. Two distinct methods are investigated which allow one to incrementally speed up the rate of computation until the time optimal limit is reached by introducing additional qubits. The number of physical qubits required to reach a desirable run time is estimated as a function of the hardware's code cycle time, for problems such as the ground state estimation of the FeMoco molecule, and for breaking the encryption of the Bitcoin network. It is found that for the quantum advantage problems investigated in this work, hardware with considerably slower code cycle times than the more usually considered 1µs of superconducting qubits, will still be able to reach desirable run times provided enough physical qubits are available

Deterministic entangling gates with nonlinear quantum photonic interferometers

The quantum computing paradigm in photonics currently relies on the multi-port interference in linear optical devices, which is intrinsically based on probabilistic measurements outcome and thus non-deterministic. Devising a fully deterministic, universal, and practically achievable quantum computing platform based on integrated photonic circuits is still an open challenge. Here we propose to exploit weakly nonlinear photonic devices to implement deterministic entangling quantum gates, following the definition of dual rail photonic qubits. It is shown that a universal set of single- and two-qubit gates can be designed by a suitable concatenation of few optical interferometric elements, with optimal fidelities arbitrarily close to 100% theoretically demonstrated through a bound constrained optimization algorithm. The actual realization would require the concatenation of a few tens of elementary operations, as well as on-chip optical nonlinearities that are compatible with some of the existing quantum photonic platforms, as it is finally discussed

Using the qubus for quantum computing

In this thesis I explore using the qubus for quantum computing. The qubus is an architecture of quantum computing, where a continuous variable ancilla is used to generate operations between matter qubits. I concentrate on using the qubus for two purposes - quantum simulation, and generating cluster states. Quantum simulation is the idea of using a quantum computer to simulate a quantum system. I focus on conducting a simulation of the BCS Hamiltonian. I demonstrate how to perform the necessary two qubit operations in a controlled fashion using the qubus. In particular I demonstrate an O(N3) saving over an implementation on an NMR computer, and a factor of 2 saving over a naıve technique. I also discuss how to perform the quantum Fourier transform on the qubus quantum computer. I show that it is possible to perform the quantum Fourier transform using just, 24⌊N/2⌋ + 7N − 6, this is an O(N) saving over a naıve method. In the second part of the thesis, I move on, and consider generating cluster states using the qubus. A cluster state, is a universal resource for one-way or measurement-based computation. In one-way computation, the pre-generated, entangled resource is used to perform calculations, which only require local corrections and measurement. I demonstrate that the qubus can generate cluster states deterministically, and in a relatively short time. I discuss several techniques of cluster state generation, one of which is optimal, given the physical architecture we are using. This can generate an n × m cluster in only 3nm − 2n − 2m + 4 operations. The alternative techniques look at generating a cluster using layers or columns, allowing it to be built dynamically, while the cluster is used to perform calculations. I then move on, and discuss problems with error accumulation in the generation process

Extending ancilla driven universal quantum computation beyond stepwise determinism

A major research goal in the field of quantum computation is the construction of the universal quantum computer (UQC): a device that can implement any quantum algorithm. Several theoretical schemes for implementing UQC have been developed which require different sets of resources and capabilities with varying implications for the optimum experimental implementations. The ancilla driven quantum computation scheme (ADQC) comprises two subsystems: a memory register of qubits on which information is retained and processed and an ancilla system of qubits which couple to the register. This coupling is represented in the ADQC scheme by a fixed quantum gate.By preparing the ancilla in selected states before applying this gate and then measuring it in selected measurement basis afterwards, quantum gates are enacted on the register qubits. ADQC is deterministic in that the probability of the outcome after performing the entire procedure is 1 but we have to apply corrections to the procedure at each step that depend on the probabilistic outcome of the ancilla measurement. An important resource in this model is the availability of a maximally entangling two-qubit gate between the ancilla and register qubits because if the gate is not maximally entangling,the resulting gates on the register can not be selected with stepwise determinism.It is proven in this thesis that in fact ADQC with non-maximally entangling interaction gates is universal. This requires showing that single- and two-qubit unitary gates can be effciently implemented probabilistically. We also show a relationship between the expected time of the probabilistic implementation of a gate and the ability to control the ancilla. In the ADQC model, the ancilla is controlled with single qubit unitary gates just before interacting with the register and just before measurement.We show that the increase in time caused by a loss of maximally entangling two-qubit gates can be counteracted by control over the ancilla. This needs not be the ability to perform any single qubit unitary to the ancilla but just the ability to perform a specific small finite set of operations.This is important because the resource requirements described by a scheme affect the properties of possible experimental implementations. The ADQC scheme was originally designed to be used with physical implementations of quantum computing that involves qubits coming from different physical systems that have different properties.This may restrict the availability of couplings between the register and ancilla systems equivalent to maximally entangling quantum gates. By further focusing on the model under specific restrictions, such as minimal control of the ancilla system or long distance separation between register qubits, we find certain properties of the physical implementation that may best suit it for ADQC beyond stepwise determinism. Minimal control appears best suited for symmetric ancilla-register interactions; use overlong distances suits a transmitter going to an unknown receiver with possible small errors in the receiver's interaction with the ancilla.A major research goal in the field of quantum computation is the construction of the universal quantum computer (UQC): a device that can implement any quantum algorithm. Several theoretical schemes for implementing UQC have been developed which require different sets of resources and capabilities with varying implications for the optimum experimental implementations. The ancilla driven quantum computation scheme (ADQC) comprises two subsystems: a memory register of qubits on which information is retained and processed and an ancilla system of qubits which couple to the register. This coupling is represented in the ADQC scheme by a fixed quantum gate.By preparing the ancilla in selected states before applying this gate and then measuring it in selected measurement basis afterwards, quantum gates are enacted on the register qubits. ADQC is deterministic in that the probability of the outcome after performing the entire procedure is 1 but we have to apply corrections to the procedure at each step that depend on the probabilistic outcome of the ancilla measurement. An important resource in this model is the availability of a maximally entangling two-qubit gate between the ancilla and register qubits because if the gate is not maximally entangling,the resulting gates on the register can not be selected with stepwise determinism.It is proven in this thesis that in fact ADQC with non-maximally entangling interaction gates is universal. This requires showing that single- and two-qubit unitary gates can be effciently implemented probabilistically. We also show a relationship between the expected time of the probabilistic implementation of a gate and the ability to control the ancilla. In the ADQC model, the ancilla is controlled with single qubit unitary gates just before interacting with the register and just before measurement.We show that the increase in time caused by a loss of maximally entangling two-qubit gates can be counteracted by control over the ancilla. This needs not be the ability to perform any single qubit unitary to the ancilla but just the ability to perform a specific small finite set of operations.This is important because the resource requirements described by a scheme affect the properties of possible experimental implementations. The ADQC scheme was originally designed to be used with physical implementations of quantum computing that involves qubits coming from different physical systems that have different properties.This may restrict the availability of couplings between the register and ancilla systems equivalent to maximally entangling quantum gates. By further focusing on the model under specific restrictions, such as minimal control of the ancilla system or long distance separation between register qubits, we find certain properties of the physical implementation that may best suit it for ADQC beyond stepwise determinism. Minimal control appears best suited for symmetric ancilla-register interactions; use overlong distances suits a transmitter going to an unknown receiver with possible small errors in the receiver's interaction with the ancilla

A Programmable Five Qubit Quantum Computer Using Trapped Atomic Ions

Quantum computers can solve certain problems more efficiently compared to conventional classical methods. In the endeavor to build a quantum computer, several competing platforms have emerged that can implement certain quantum algorithms using a few qubits. However, the demonstrations so far have been done usually by tailoring the hardware to meet the requirements of a particular algorithm implemented for a limited number of instances. Although such proof of principal implementations are important to verify the working of algorithms on a physical system, they further need to have the potential to serve as a general purpose quantum computer allowing the flexibility required for running multiple algorithms and be scaled up to host more qubits. Here we demonstrate a small programmable quantum computer based on five trapped atomic ions each of which serves as a qubit. By optically resolving each ion we can individually address them in order to perform a complete set of single-qubit and fully connected two-qubit quantum gates and alsoperform efficient individual qubit measurements. We implement a computation architecture that accepts an algorithm from a user interface in the form of a standard logic gate sequence and decomposes it into fundamental quantum operations that are native to the hardware using a set of compilation instructions that are defined within the software. These operations are then effected through a pattern of laser pulses that perform coherent rotations on targeted qubits in the chain. The architecture implemented in the experiment therefore gives us unprecedented flexibility in the programming of any quantum algorithm while staying blind to the underlying hardware. As a demonstration we implement the Deutsch-Jozsa and Bernstein-Vazirani algorithms on the five-qubit processor and achieve average success rates of 95 and 90 percent, respectively. We also implement a five-qubit coherent quantum Fourier transform and examine its performance in the period finding and phase estimation protocol. We find fidelities of 84 and 62 percent, respectively. While maintaining the same computation architecture the system can be scaled to more ions using resources that scale favorably (O(N^2)) with the number of qubits N

Quantum bits with Josephson junctions

Recent demonstrations of macroscopic quantum coherence in Josephson junction based electronic circuits have opened an entirely new dimension for research and applications in the established field of Josephson electronics. In this article we discuss basic Josephson circuits for qubit applications, methods of quantum description of these circuits, and circuit solutions for qubit couplings. Principles of manipulation and readout of superconducting qubits are reviewed and illustrated with recent experiments using various qubit types

Quantum entanglement

All our former experience with application of quantum theory seems to say: {\it what is predicted by quantum formalism must occur in laboratory}. But the essence of quantum formalism - entanglement, recognized by Einstein, Podolsky, Rosen and Schr\"odinger - waited over 70 years to enter to laboratories as a new resource as real as energy. This holistic property of compound quantum systems, which involves nonclassical correlations between subsystems, is a potential for many quantum processes, including ``canonical'' ones: quantum cryptography, quantum teleportation and dense coding. However, it appeared that this new resource is very complex and difficult to detect. Being usually fragile to environment, it is robust against conceptual and mathematical tools, the task of which is to decipher its rich structure. This article reviews basic aspects of entanglement including its characterization, detection, distillation and quantifying. In particular, the authors discuss various manifestations of entanglement via Bell inequalities, entropic inequalities, entanglement witnesses, quantum cryptography and point out some interrelations. They also discuss a basic role of entanglement in quantum communication within distant labs paradigm and stress some peculiarities such as irreversibility of entanglement manipulations including its extremal form - bound entanglement phenomenon. A basic role of entanglement witnesses in detection of entanglement is emphasized.Comment: 110 pages, 3 figures, ReVTex4, Improved (slightly extended) presentation, updated references, minor changes, submitted to Rev. Mod. Phys

Quantum logic and entanglement by neutral Rydberg atoms: methods and fidelity

Quantum gates and entanglement based on dipole-dipole interactions of neutral Rydberg atoms are relevant to both fundamental physics and quantum information science. The precision and robustness of the Rydberg-mediated entanglement protocols are the key factors limiting their applicability in experiments and near-future industry. There are various methods for generating entangling gates by exploring the Rydberg interactions of neutral atoms, each equipped with its own strengths and weaknesses. The basics and tricks in these protocols are reviewed, with specific attention paid to the achievable fidelity and the robustness to the technical issues and detrimental innate factors.Comment: 57 pages, 10 figure

Distributed Simulation of Statevectors and Density Matrices

Classical simulation of quantum computers is an irreplaceable step in the design of quantum algorithms. Exponential simulation costs demand the use of high-performance computing techniques, and in particular distribution, whereby the quantum state description is partitioned between a network of cooperating computers - necessary for the exact simulation of more than approximately 30 qubits. Distributed computing is notoriously difficult, requiring bespoke algorithms dissimilar to their serial counterparts with different resource considerations, and which appear to restrict the utilities of a quantum simulator. This manuscript presents a plethora of novel algorithms for distributed full-state simulation of gates, operators, noise channels and other calculations in digital quantum computers. We show how a simple, common but seemingly restrictive distribution model actually permits a rich set of advanced facilities including Pauli gadgets, many-controlled many-target general unitaries, density matrices, general decoherence channels, and partial traces. These algorithms include asymptotically, polynomially improved simulations of exotic gates, and thorough motivations for high-performance computing techniques which will be useful for even non-distributed simulators. Our results are derived in language familiar to a quantum information theory audience, and our algorithms formalised for the scientific simulation community. We have implemented all algorithms herein presented into an isolated, minimalist C++ project, hosted open-source on Github with a permissive MIT license, and extensive testing. This manuscript aims both to significantly improve the high-performance quantum simulation tools available, and offer a thorough introduction to, and derivation of, full-state simulation techniques.Comment: 56 pages, 18 figures, 28 algorithms, 1 tabl