The variational quantum imaginary time-evolution algorithm is efficient in finding the ground state of a quantum Hamiltonian. This algorithm involves solving a system of linear equations in a classical computer and the solution is then used to propagate a quantum wave function. Here, we show that owing to the noisy nature of current quantum processors, such a quantum algorithm or the family of quantum algorithms will require single- and two-qubit gates with very low error probability to produce reliable results. Failure to meet such a condition will result in erroneous quantum data propagation even for a relatively small quantum circuit ansatz. Specifically, we provide the upper bounds on how the relative error in variational parameters' propagation scales with the probability of noise in quantum hardware. We also present an exact expression of how the relative error in variational parameter propagation scales with the probability of partially depolarizing noise.
The variational quantum imaginary time-evolution algorithm is efficient in finding the ground state of a quantum Hamiltonian. This algorithm involves solving a system of linear equations in a classical computer and the solution is then used to propagate a quantum wave function. Here, we show that owing to the noisy nature of current quantum processors, such a quantum algorithm or the family of quantum algorithms will require single- and two-qubit gates with very low error probability to produce reliable results. Failure to meet such a condition will result in erroneous quantum data propagation even for a relatively small quantum circuit ansatz. Specifically, we provide the upper bounds on how the relative error in variational parameters' propagation scales with the probability of noise in quantum hardware. We also present an exact expression of how the relative error in variational parameter propagation scales with the probability of partially depolarizing noise.
Quantum computing not only holds the potential to solve long-standing problems in quantum physics, but also to offer speed-ups across a broad spectrum of other fields. Access to a computational space that incorporates quantum effects, such as superposition and entanglement, enables the derivation of promising quantum algorithms for important tasks, including preparing the ground state of a quantum system or predicting its evolution over time. Successfully tackling these tasks promises insights into significant theoretical and technological questions, such as superconductivity and the design of new materials. The aim of quantum algorithms is to use a series of quantum operations in a quantum circuit to solve a problem beyond the reach of classical computers. However, the noise and limited scale of current quantum computers restricts these circuits to moderate sizes and depths. As a result, many prominent algorithms are currently infeasible to run for problem sizes of practical interest. In response, recent research focused on variational quantum algorithms, which allow the selection of circuits that act within a quantum device's capabilities. Yet, these algorithms can require the execution of a large number of circuits, leading to prohibitively long computation times. This doctoral thesis develops two main techniques to reduce these quantum computational resource requirements, with the goal of scaling up application sizes on current quantum processors. The first approach is based on stochastic approximations of computationally costly quantities, such as quantum circuit gradients or the quantum geometric tensor (QGT). The second method takes a different perspective on the QGT, leading to a potentially more efficient description of time evolution on current quantum computers. Both techniques rely on maintaining available information and only computing necessary corrections, instead of re-computing possibly redundant data. The main focus of application for our algorithms is the simulation of quantum systems, broadly defined as including the preparation of ground and thermal states, and the real- and imaginary-time propagation of a system. The developed subroutines, however, can further be utilized in the fields of optimization or machine learning. Our algorithms are benchmarked on a range of representative models, such as Ising or Heisenberg spin models, both in numerical simulations and experiments on the hardware. In combination with error mitigation techniques, the latter is scaled up to 27 qubits; into a regime that variational quantum algorithms are challenging to scale to on noisy quantum computers without our algorithms.
The development of quantum computers requires not only experimental advances, but also theoretical efforts to tackle the bottlenecks and identify potential applications. This thesis addresses some essential elements in each stage of the development of quantum computers, from the currently available hardware to the long-term devices. The critical factor that hinders the quantum hardware from performing perfect operations is the presence of noise, however, it can be overcome by error correction codes. We first consider a large-scale quantum processor with asymmetrical noise and preferably a modular structure. By concatenating the surface code with an error-detecting code, and utilising the information from detection with a customised decoder, we find very high thresholds for the designed devices. The second topic is small error correction codes which are more realistic to be implemented in the near future. A measure `integrity' is introduced to benchmark the performance of quantum memories to demonstrate beneficial error correction, associated with four milestones. We show integrity is closely related with other commonly-used measures and can be easily assessed by experimentalists. This framework is then studied in detail in the context of an ion-trap quantum device implemented with the seven qubit code. We present a protocol based on detailed sequences of crystal-reconfiguration operations and stabiliser mappings to perform a full error correction cycle. Realistic error models and parameters from reported works are also given. While experimental efforts continue to battle against the noise, some hybrid quantum-classical algorithms have been developed to be implemented on the near-term noisy devices. The variational algorithm is one of the hybrid algorithms that have been widely applied to simulations of condensed matter physics and quantum chemistry. In the next chapter, we introduce a variational quantum compiler to automatically search for the encoding circuits of small error correction codes. By tailoring the ansatz to optimise the circuit for particular hardware systems, we show some novel circuits discovered with the compiler. The final chapter applies the variational algorithm to solve linear algebra problems including linear systems of equations and matrix-vector multiplications. We show that the linear algebra problems can be solved with near-term devices, with circuit depth linear to the condition number and super-linear to the number of qubits.
The role of differential equations (DEs) in science and engineering is of paramount importance, as they provide the mathematical framework for a multitude of natural phenomena. Since quantum computers promise significant advantages over classical computers, quantum algorithms for the solution of DEs have received a lot of attention. Particularly interesting are algorithms that offer advantages in the current noisy intermediate-scale quantum (NISQ) era, characterized by small and error-prone systems. We consider a framework of variational quantum algorithms, quantum circuit learning (QCL), in conjunction with derivation methods, in particular the parameter shift rule, to solve DEs. As these algorithms were specifically designed for NISQ computers, we analyze their applicability on NISQ devices by implementing QCL on an IBM quantum computer. Our analysis of QCL without the parameter shift rule shows that we can successfully learn different functions with three-qubit circuits. However, the hardware errors accumulate with increasing number of qubits, and thus, only a fraction of the qubits available on the current quantum systems can be effectively used. We further show that it is possible to determine derivatives of the learned functions using the parameter shift rule on the IBM hardware. The parameter shift rule results in higher errors, which limits its execution to low-order derivatives. Despite these limitations, we solve a first-order DE on the IBM quantum computer. We further explore the advantages of using multiple qubits in QCL by learning different functions simultaneously and demonstrate the solution of a coupled DE on a simulator.
A recently proposed variational quantum algorithm has expanded the horizon of variational quantum computing to nonlinear physics and fluid dynamics. In this work, we probe the ability of such approach to capture the ground state of the nonlinear Schr\"odinger equation for a range of parameters on real superconducting quantum processors. Specifically, we study the expressivity of real-amplitude, hardware-efficient Ansatz to capture the ground state of this nonlinear system across various interaction regimes and implement different noise scenarios in both simulators and cloud processors. Our investigation reveals that although quantum hardware noise impairs the evaluation of the energy cost function, certain small instances of the problem consistently converge to the ground state. We test and demonstrate a variety of cases on IBM Q superconducting devices and analyze the discrepancies in the energy cost function evaluation due to quantum hardware noise. These discrepancies are absent in the state fidelity estimation because of the shallow state preparation circuit. Our comprehensive analysis offers valuable insights into the practical implementation and advancement of the variational algorithms for nonlinear problems.
Placement, routing, and scheduling are essential tasks for near-optimal performance of programs for noisy quantum processors. Reliable execution of an arbitrary quantum circuit on current devices requires routing methods that overcome connectivity limitations while meeting data locality requirements. However, current devices also express highly variable noise levels in both the quantum gates and quantum registers. This requires any routing algorithm to be adaptive to both the circuit and the operating conditions. We demonstrate near-optimal routing methods of noisy quantum states that minimize the overall error of data movement while also limiting the computational complexity of routing decisions. We evaluate our methods against the noise characteristics of a 20-qubit superconducting quantum processor.
Noise detrimentally affects quantum computations so that they not only become less accurate but also easier to simulate classically as systems scale up. We construct a classical simulation algorithm, lowesa (low weight efficient simulation algorithm), for estimating expectation values of noisy parameterised quantum circuits with a fixed observable. It combines previous results on spectral analysis of parameterised circuits with Pauli back-propagation and recent ideas for simulations of noisy random circuits. We show, under some conditions on the circuits and mild assumptions on noise, that lowesa gives an efficient, polynomial algorithm in the number of qubits (and depth), with approximation error that vanishes exponentially in the physical error rate and a controllable cutoff parameter. This is valid for any expectation value that may be efficiently evaluated on a quantum computer. We discuss the practical limitations of the method for circuit classes with correlated parameters and its scaling with decreasing error rates.