Dissipative processes have long been proposed as a means of performing computational tasks on quantum computers that may be intrinsically more robust to noise. In this work, we prove two main results concerning the error-resilience capabilities of two types of dissipative algorithms: dissipative ground state preparation in the form of the dissipative quantum eigensolver (DQE), and dissipative quantum computation (DQC). The first result is that under circuit-level depolarizing noise, a version of the DQE algorithm applied to the geometrically local, stabilizer-encoded Hamiltonians that arise naturally when fermionic Hamiltonians are represented in qubits, can suppress the additive error in the ground space overlap of the final output state exponentially in the code distance. This enables us to get closer to fault-tolerance for this task without the associated overhead. In contrast, for computation as opposed to ground state preparation, the second result proves that DQC is no more robust to noise than the standard quantum circuit model.
Dissipative processes have long been proposed as a means of performing computational tasks on quantum computers that may be intrinsically more robust to noise. In this work, we prove two main results concerning the error-resilience capabilities of two types of dissipative algorithms: dissipative ground state preparation in the form of the dissipative quantum eigensolver (DQE), and dissipative quantum computation (DQC). The first result is that under circuit-level depolarizing noise, a version of the DQE algorithm applied to the geometrically local, stabilizer-encoded Hamiltonians that arise naturally when fermionic Hamiltonians are represented in qubits, can suppress the additive error in the ground space overlap of the final output state exponentially in the code distance. This enables us to get closer to fault-tolerance for this task without the associated overhead. In contrast, for computation as opposed to ground state preparation, the second result proves that DQC is no more robust to noise than the standard quantum circuit model.
In recent years, variational quantum algorithms have gained significant attention due to their adaptability and efficiency on near-term quantum hardware. They have shown potential in a variety of tasks, including linear algebra, search problems, Gibbs, and ground state preparation. Nevertheless, the presence of noise in current day quantum hardware severely limits their performance. In this work, we introduce dissipative variational quantum algorithms (D-VQAs) by incorporating dissipative operations, such as qubit RESET and stochastic gates, as an intrinsic part of a variational quantum circuit. We argue that such dissipative variational algorithms possess some natural resilience to dissipative noise. We demonstrate how such algorithms can prepare Gibbs states over a wide range of quantum many-body Hamiltonians and temperatures, while significantly reducing errors due to both coherent and noncoherent noise. An additional advantage of our approach is that no ancilla qubits are need. Our results highlight the potential of D-VQAs to enhance the robustness and accuracy of variational quantum computations on noisy intermediate-scale quantum (NISQ) devices.
This thesis is concerned with certain theoretical problems that arise naturally in the context of fault-tolerant quantum computation. Fault-tolerance can be defined as the art of building reliable devices from unreliable components, and is of particular importance for quantum computers that aim to precisely control the dynamics of extremely sensitive quantum systems. A model of quantum computation is a specification of the basic building blocks by which a quantum computation is implemented. The best known model is the quantum circuit model, where computations are implemented by means of unitary quantum gates that are applied to two-level quantum systems known as qubits. In a physical implementation of a quantum circuit, the gates and qubits will inevitably be affected by noise. Fault-tolerant quantum circuits are designed to be resilient against the effects of this noise, provided that it is not too strong. Fault-tolerance in the quantum circuit model is well developed thanks to the theory of quantum error-correcting codes. These codes allow for the correction of small numbers of errors introduced by a variety of noise processes. In a fault-tolerant quantum circuit, qubits are replaced with encoded qubits, and quantum gates with encoded gates that are immediately followed by special quantum circuits for error correction. Provided the rate at which errors occur is below a constant threshold value, the accumulation of errors can be checked so that the correct output of the computation correctly determined. The threshold acts as both a measure of how good the design of a quantum circuit is, and also as a target for experimenters aiming to implement quantum circuits. Much current research in quantum computation is aimed at designing quantum circuits that increase the noise threshold, hopefully to the point where it becomes within the reach of experimenters. A fault-tolerant encoded quantum gate must limit the propagation of errors so that the code’s corrective capabilities are not overwhelmed. It is not so easy to design encoded gates that satisfy this property, and to date only a finite handful of such gates are known. In a faulttolerant quantum circuit, all quantum gates must be decomposed or compiled in terms of those that may be implemented fault-tolerantly. In the first part of this thesis, we present two results that may be applied to this problem of gate compilation. The first is a generic method based on the Solovay-Kitaev theorem that may be applied to all quantum gates, but is most effective for those that act on single qubits. We present the Solovay-Kitaev theorem in its simples known form as an algorithm, together with novel constructions that can be used to implement it. Following this we give a two specialized methods for two-qubit gate decomposition, based on the Cartan decomposition of the Lie group SU(4). The cluster-state model of computation is an alternative to the quantum circuit model, and makes use of quantum measurements and highly entangled cluster-states to implement a quantum computation. The fault-tolerant techniques developed for quantum circuits are not immediately applicable in this model, so in order for it to be a realistic candidate for performing computations we prove that such techniques are possible. In the second part of the thesis we prove that constant fault-tolerance thresholds may be achieved in the cluster-state model, and in particular in an adaption to an optical implementation. Following this we design a complete error correction scheme for optical cluster-state computation, and numerically determine the threshold of this model in the face of the dominant noise models likely to affect such an implementation.
This thesis is concerned with fault-tolerant quantum information processing using quantum error-correcting codes. It contains two major pieces of work. The first is a study of coherent noise in the context of stabilizer error-correcting codes. The second is a proposed scheme for a universal set of fault-tolerant logical gates in a particular code family built out of the 3D toric code. Chapter 1 provides an introduction to quantum computation and fault tolerance. Many basic concepts in error-correcting codes are defined. Special attention is paid to the set of code properties that are most likely to determine how easily a given fault-tolerant scheme might be implemented on a physical device. These include the fault-tolerant noise threshold and the overhead. In Chapters 2 and 3 we study the effectiveness of quantum error correction against coherent noise. Coherent errors (for example, unitary noise) can interfere constructively, so that in some cases the average infidelity of a quantum circuit subjected to coherent errors may increase quadratically with the circuit size; in contrast, when errors are incoherent (for example, depolarizing noise), the average infidelity increases at worst linearly with circuit size. We consider the performance of quantum stabilizer codes against a noise model in which a unitary rotation is applied to each qubit, where the axes and angles of rotation are nearly the same for all qubits. In Chapter 2 we introduce coherent noise and incoherent noise and a number of methods that are useful for the study of coherent noise. We study the repetition code as a basic example, and we also study a correlated noise model. In Chapter 3 we show that for the toric code subject to such independent coherent noise, and for minimal-weight decoding, the logical channel after error correction becomes increasingly incoherent as the length of the code increases, provided the noise strength decays inversely with the code distance. A similar conclusion holds for weakly correlated coherent noise. Our methods can also be used for analyzing the performance of other codes and fault-tolerant protocols against coherent noise. However, our result does not show that the coherence of the logical channel is suppressed in the more physically relevant case where the noise strength is held constant as the code block grows, and we recount the difficulties that prevented us from extending the result to that case. Nevertheless our work supports the idea that fault-tolerant quantum computing schemes will work effectively against coherent noise, providing encouraging news for quantum hardware builders who worry about the damaging effects of control errors and coherent interactions with the environment. Chapter 4 is connected to another aspect of fault tolerance, fault-tolerant logical gates. The toric code is a promising candidate for fault-tolerant quantum computation because of its high threshold and low-weight stabilizers. A universal gate set in the toric code generally requires magic state distillation, which can incur a significant qubit overhead. In this work we construct an error-correcting code in three dimensions based on the toric code that features a fault-tolerant T gate with no magic state distillation required. We further describe a subsystem version of our code which supports a universal set of fault-tolerant gates. This code can be converted into the stabilizer version using gauge-fixing. We also argue that our code can be converted to a (2+1)-D protocol, where a 2D lattice undergoes a measurement-based protocol over time. In this way, a fault-tolerant logical T gate can be realized in a 2D toric code structure.
Usual scenarios of fault-tolerant computation are concerned with the fault-tolerant realization of quantum algorithms that compute classical functions, such as Shor's algorithm for factoring. In particular, this means that input and output to the quantum algorithm are classical. In contrast to stand-alone single-core quantum computers, in many distributed scenarios, quantum information might have to be passed on from one quantum information processing system to another one, possibly via noisy quantum communication channels with noise levels above fault-tolerant thresholds. In such situations, quantum information processing devices will have quantum inputs, quantum outputs or even both, which pass qubits among each other. Working in the fault-tolerant framework of [Kitaev, 1997], we show that any quantum circuit with quantum input and output can be transformed into a fault-tolerant circuit that produces the ideal circuit with some controlled noise applied at the input and output. The framework allows the direct composition of the statements, enabling versatile future applications. We illustrate this with a concrete application, namely, communication over a noisy channel with faulty encoding and decoding operations [Christandl and M{ü}ller-Hermes, 2024]. For communication codes with linear minimum distance, we construct fault-tolerant encoders and decoders for general noise (including coherent errors). For the weaker, but standard, model of local stochastic noise, we obtain fault-tolerant encoders and decoders for any communication code that can correct a constant fraction random errors.
Usual scenarios of fault-tolerant computation are concerned with the fault-tolerant realization of quantum algorithms that compute classical functions, such as Shor’s algorithm for factoring. In particular, this means that input and output to the quantum algorithm are classical. In contrast to stand-alone single-core quantum computers, in many distributed scenarios, quantum information might have to be passed on from one quantum information processing system to another one, possibly via noisy quantum communication channels with noise levels above fault-tolerant thresholds. In such situations, quantum information processing devices will have quantum inputs, quantum outputs or even both, which pass qubits among each other. Working in the fault-tolerant framework of [Kitaev, 1997], we show that any quantum circuit with quantum input and output can be transformed into a fault-tolerant circuit that produces the ideal circuit with some controlled noise applied at the input and output. The framework allows the direct composition of the statements, enabling versatile future applications. We illustrate this with a concrete application, namely, communication over a noisy channel with faulty encoding and decoding operations [Christandl and Müller-Hermes, 2024]. For communication codes with linear minimum distance, we construct fault-tolerant encoders and decoders for general noise (including coherent errors). For the weaker, but standard model of local stochastic noise, we obtain fault-tolerant encoders and decoders for any communication code that can correct a constant fraction of random errors.
We simulate the effects of different types of noise in state preparation circuits of variational quantum algorithms. We first use a variational quantum eigensolver to find the ground state of a Hamiltonian in the presence of noise and adopt two quality measures in addition to the energy, namely, fidelity and concurrence. We then extend the task to the one of constructing, with a layered quantum circuit ansatz, a set of general random target states. We determine the optimal circuit depth for different types and levels of noise, and observe that the variational algorithms mitigate the effects of noise by adapting the optimized parameters. We find that the inclusion of redundant parameterized gates makes the quantum circuits more resilient to noise. For such overparameterized circuits, different sets of parameters can result in the same final state in the noiseless case, which we denote as parameter degeneracy. Numerically, we show that this degeneracy can be lifted in the presence of noise, with some states being significantly more resilient to noise than others. We also show that the average deviation from the target state is linear in the noise level, as long as this is small compared to a circuit-dependent threshold. In this region, the deviation is well described by a stochastic model. Above the threshold, the optimization can converge to states with largely different physical properties from the true target state, so for practical applications it is critical to ensure that noise levels are below this threshold.