Quantum optimization using variational algorithms on near-term quantum devices
Nikolaj Moll, Panagiotis Barkoutsos, Lev S Bishop, Jerry M Chow, Andrew W. Cross, Daniel J Egger, Stefan Filipp, Andreas Fuhrer, Jay M Gambetta, Marc Ganzhorn, Abhinav Kandala, Antonio Mezzacapo, Peter Müller, Walter Riess, Gian Salis, John Smolin, Ivano Tavernelli, Kristan Temme
Universal fault-tolerant quantum computers will require error-free execution of long sequences of quantum gate operations, which is expected to involve millions of physical qubits. Before the full power of such machines will be available, near-term quantum devices will provide several hundred qubits and limited error correction. Still, there is a realistic prospect to run useful algorithms within the limited circuit depth of such devices. Particularly promising are optimization algorithms that follow a hybrid approach: the aim is to steer a highly entangled state on a quantum system to a target state that minimizes a cost function via variation of some gate parameters. This variational approach can be used both for classical optimization problems as well as for problems in quantum chemistry. The challenge is to converge to the target state given the limited coherence time and connectivity of the qubits. In this context, the quantum volume as a metric to compare the power of near-term quantum devices is discussed. With focus on chemistry applications, a general description of variational algorithms is provided and the mapping from fermions to qubits is explained. Coupled-cluster and heuristic trial wave-functions are considered for efficiently finding molecular ground states. Furthermore, simple error-mitigation schemes are introduced that could improve the accuracy of determining ground-state energies. Advancing these techniques may lead to near-term demonstrations of useful quantum computation with systems containing several hundred qubits.
Universal fault-tolerant quantum computers will require error-free execution of long sequences of quantum gate operations, which is expected to involve millions of physical qubits. Before the full power of such machines will be available, near-term quantum devices will provide several hundred qubits and limited error correction. Still, there is a realistic prospect to run useful algorithms within the limited circuit depth of such devices. Particularly promising are optimization algorithms that follow a hybrid approach: the aim is to steer a highly entangled state on a quantum system to a target state that minimizes a cost function via variation of some gate parameters. This variational approach can be used both for classical optimization problems as well as for problems in quantum chemistry. The challenge is to converge to the target state given the limited coherence time and connectivity of the qubits. In this context, the quantum volume as a metric to compare the power of near-term quantum devices is discussed. With focus on chemistry applications, a general description of variational algorithms is provided and the mapping from fermions to qubits is explained. Coupled-cluster and heuristic trial wave-functions are considered for efficiently finding molecular ground states. Furthermore, simple error-mitigation schemes are introduced that could improve the accuracy of determining ground-state energies. Advancing these techniques may lead to near-term demonstrations of useful quantum computation with systems containing several hundred qubits.
Quantum computers exist today that are capable of performing calculations that challenge the largest classical supercomputers. Now the great challenge is to make use of these devices, or their successors, to perform calculations of independent interest. This thesis focuses on a family of algorithms that aim to accomplish this goal, known as variational quantum algorithms, and the challenges that they face. We frame these challenges in terms of two kinds of resources, the number of operations that we can afford to perform in a single quantum circuit, and the overall number of circuit repetitions required. Before turning to our specific contributions, we provide a review of the electronic structure problem, which serves as a central example for our work. We also provide a self-contained introduction to the field of quantum computing and the notion of a variational quantum algorithm.We begin the main body of the thesis by studying a generalization of the standard unitary coupled cluster ansatz. We present a sparse version of unitary coupled cluster and show that it can accurately represent the ground and excited states of small molecular systems using fewer quantum gates than the full unitary coupled cluster. We then introduce a strategy for representing molecular ground states as linear combinations of parameterized wavefunctions, allowing for a tradeoff between the number of operations required for each circuit and the number of circuit repetitions. We provide circuit primitives that allow for the efficient measurement of the required matrix elements between these wavefunctions. Subsequently, we show how the cost of estimating the energy of a quantum chemical wavefunction on a near-term quantum computer can be dramatically reduced by using a factorization of the two-electron integraltensor. Furthermore, we explain how this measurement strategy helps mitigate against errors during state preparation and measurement. We then present a Monte Carlo version of a classical algorithm for calculating the partition function of two-dimensional lattice models that shows a similar kind of tradeoff between two resources, albeit in a different context. Finally, we show how ideas based on tensor networks can inform the design of quantum circuits for machine learning tasks. We argue that this application is especially tolerant to noise and present numerical data substantiating this argument. We conclude by explaining explicitly how our work addresses the problems of the limited resources available to near-term quantum computers and offering some optimism about the future of the field.
Fault-tolerant quantum computers, when realized, guarantee significant speedups for particular computational tasks. Quantum hardware in the near term, however, is limited to low hundreds of noisy qubits that are not error-corrected. To leverage such devices as well as accommodate rapid developments in both hardware and algorithms, variational quantum algorithms (VQAs) were introduced. These algorithms have the potential to be useful in areas such as chemistry, optimization, machine learning, among others. Despite this potential, the largest experiment to date has employed just twelve qubits to implement the Variational Quantum Eigensolver (VQE), a VQA for simulating ground state energies of fermionic systems. In this dissertation, we present several methods for extending and further developing VQAs to step closer towards realizing applications in which quantum computers can provide meaningful advantages over classical computers. In the first part, we present methods for analyzing and optimizing parameterized quantum circuits (PQCs), a core component in VQAs for approximating target quantum states, e.g. ground state wavefunctions in VQE. To compare among PQCs and guide their design and selection, we introduce a set of algorithm-independent descriptors that can be estimated from simulations. Using these descriptors, we rule out circuits with limited capabilities in expressive and entangling powers. Next, we present a strategy for optimizing PQCs with large parameter counts and high redundancy by removing circuit operations that are less important for optimizing the algorithm objective, effectively reducing the search space. We demonstrate that this method enables larger VQE simulations by converging previously difficult optimizations while additionally reducing circuit resources. In the second part, we present methods for developing VQAs that are related to the quantum hardware. We introduce a new algorithm for simulating components of a quantum processor using an existing quantum computer. The goal of this application is to leverage near-term devices and algorithms to aid in the design of more powerful next-generation quantum hardware. Lastly, we present a guide for algorithm prototyping by deploying instances of VQAs on quantum computers via the cloud. We demonstrate instances of two algorithms, the quantum autoencoder and classifier, on an eight-qubit device. While further advancements in hardware and algorithms lie ahead, our work, along with developments from previous studies, represents significant progress towards realizing practical quantum advantage.
Traditional quantum algorithms with guaranteed performance generally require fully coherent quantum computers to operate, making error-correction a necessity. In contrast, variational quantum algorithms seek to remove this requirement by formulating the computation as the approximate optimization of a functional. This optimization is carried out in the space of heuristic functions defined by a parameterized (variational) quantum circuit. Taking advantage of the flexibility in the definition of these heuristics, we can expand the operability of variational algorithms within the regime of coherence times of noisy intermediate-scale quantum devices, avoiding the necessity of error-correction. In this dissertation, we present the application of the variational quantum computing approach to problems in quantum simulation, quantum state preparation, quantum error-correction, and generative modeling.\nIn the first part, we investigate the use of the variational quantum eigensolver (VQE) for simulating the ground state of fermionic systems. We start by studying the implementation of VQE using the unitary coupled cluster (UCC) ansatz and propose strategies to reduce its cost. We use these insights to carry out some of the first experimental quantum computations of molecular energies using simplified versions of this ansatz. We also propose an extension of UCC to study problems in condensed matter physics, along with a new low-depth circuit ansatz for preparing non-gaussian fermionic states on quantum computers. We show the potential of our approach to describe strongly correlated fermionic systems.\nIn the second part, we develop three new variational quantum algorithms for problems in quantum computing and machine learning, namely: 1) the quantum autoencoder, to compress ensembles of quantum states, 2) the variational quantum error corrector, to find device-tailored quantum encoding and recovery circuits for quantum error correction, and 3) the variational quantum generator, to generate classical probability distributions. These techniques offer efficient ways to design new quantum circuits for state preparation, to find more effective error-correcting codes and to perform generative modeling with quantum computers. Our work provides insights into the design of variational quantum algorithms and establishes practical guidelines to implement these methods on near-term quantum computers, as well as some future research directions for this field.
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.
Quantum computers have come a long way since conception, and there is still a long way to go before the dream of universal, fault-tolerant computation is realized. In the near term, quantum computers will occupy a middle ground that is popularly known as the “Noisy, Intermediate-Scale Quantum” (or NISQ) regime. The NISQ era represents a transition in the nature of quantum devices from experimental to computational. There is significant interest in engineering NISQ devices and NISQ algorithms in a manner that will guide the development of quantum computation in this regime and into the era of fault-tolerant quantum computing. In this thesis, we study two aspects of near-term quantum computation. The first of these is the design of device architectures, covered in Chapters 2, 3, and 4. We examine different qubit connectivities on the basis of their graph properties, and present numerical and analytical results on the speed at which large entangled states can be created on nearest-neighbor grids and graphs with modular structure. Next, we discuss the problem of permuting qubits among the nodes of the connectivity graph using only local operations, also known as routing. Using a fast quantum primitive to reverse the qubits in a chain, we construct a hybrid, quantum/classical routing algorithm on the chain. We show via rigorous bounds that this approach is faster than any SWAP-based algorithm for the same problem. The second part, which spans the final three chapters, discusses variational algorithms, which are a class of algorithms particularly suited to near-term quantum computation. Two prototypical variational algorithms, quantum adiabatic optimization (QAO) and the quantum approximate optimization algorithm (QAOA), are studied for the difference in their control strategies. We show that on certain crafted problem instances, bang-bang control (QAOA) can be as much as exponentially faster than quasistatic control (QAO). Next, we demonstrate the performance of variational state preparation on an analog quantum simulator based on trapped ions. We show that using classical heuristics that exploit structure in the variational parameter landscape, one can find circuit parameters efficiently in system size as well as circuit depth. In the experiment, we approximate the ground state of a critical Ising model with long-ranged interactions on up to 40 spins. Finally, we study the performance of Local Tensor, a classical heuristic algorithm inspired by QAOA on benchmarking instances of the MaxCut problem, and suggest physically motivated choices for the algorithm hyperparameters that are found to perform well empirically. We also show that our implementation of Local Tensor mimics imaginary-time quantum evolution under the problem Hamiltonian.
The advancements in classical computational components such as CPUs and GPUs have had a reverberating impact on the democratization of computational power available, with the field of computational chemistry reaping significant benefits from the propulsion of the limits of systems studied. The accurate prediction of chemical properties and reaction dynamics through computational simulation is now a centerpiece of modern chemical research, significantly accelerating discoveries research areas of materials science and drug development, as well as industrially relevant processes such as the emission of aldehydes from soft foams, flame retardants and pyrolysis and the carbonylation of epoxides. These achievements of computational chemistry still are overshadowed in the face of fundamental bottlenecks where classical methods remain constrained by the exponential scaling when treating large and/or strongly correlated quantum many-body systems. This innate computational complexity often constitutes chemically accurate calculations based on first principles impractical, or plain impossible, for systems beyond modest sizes. Quantum computing offers a potential pathway around these limitations, originally formally formulated by David Deutsch as a tool for simulating quantum systems, with the first quantum algorithms indicating speed-ups compared to classical counterparts arising from Peter Shor and Lov Grover for the prime factorization of an integer number and unstructured search respectively. Relying on fundamental theorems by Claude Shannon, it has been demonstrated that through the use of error-correcting codes fault-tolerant quantum computers are able to safeguard information against noise and decoherence, strongly indicating towards exponential speedups in solving central computational chemistry problems, such as determining ground state energies or reaction pathways of chemical systems. Substantial strides in experimental quantum computing over the past decade, including proof-of-principle demonstrations of quantum error correction, mean to bring full-scale fault-tolerant quantum computing within the grasp of humanity. The breakthroughs needed to attain scalable, error-corrected quantum computation with capabilities exceeding those of classical methods in real-world scenarios is likely still some years away. While striving to address the engineering challenges standing in the way of achieving fault tolerant quantum computing, substantial effort has been focused toward developing and benchmarking hybrid algorithms tailored for the Noisy Intermediate- Scale Quantum (NISQ) devices, available now to researchers. These hybrid quantum-classical algorithms, with the most notable being the Variational Quantum Eigensolver (VQE), allow the probing of quantum computing’s utility in computational chemistry even in the face of significant hardware noise and limited coherence times. The work presented in this thesis introduces several key additions the VQE framework, geared especially towards to NISQ hardware, where the mindful use of available resources is imperative. First, the implementation of the Conjugate Model Search optimizer is described, an optimization method that integrates simple models that are surrogate to the original function, combined with conjugate gradient optimization to navigate effectively parameter landscapes typical of variational algorithms, prone to noise. Second, this thesis contains the proposal and extensive study of a hyperparameter tuning procedure employing genetic algorithms, allowing for comprehensive simulation benchmarks that augment the robustness and efficiency of VQE implementations with respect to convergence time through real wall-time models of quantum processors. Continuing, a bridge is presented between quantum simulation and quantum chemistry, through qubit-to-fermion mappings of existing quantum gates, further enabling the direct and accurate modeling of chemical systems on trapped atom quantum hardware with the added arsenal provided by techniques designed for qubit architectures. Additionally, experimental data contained in this thesis include detailed benchmarking data collected from state-of-the-art trapped ion quantum processors available at the Johannes Gutenberg Universität Mainz, demonstrating the actual viability and performance of various methodologies for extracting results. Lastly, the state of VQE is examined, as it served as a springboard for a plethora of advancements in the development of quantum computing hardware, as well as the direction the field of simulation of computational quantum chemistry via quantum devices is headed towards in the coming years.