It is one of the most fundamental objectives in quantum information science to understand the boundary between the computational power of classical and quantum computers. One possible avenue to explore this boundary is to identify classes of quantum circuits that can be efficiently simulated on a classical computer. Instead of simulating a general quantum circuit with a classical device, new schemes have recently emerged to simulate them on a quantum device that is restricted in some manner. As such, these techniques allow us to study how the restrictions impact the computational power of the device. One such technique is called quasiprobability simulation (QPS) and it estimates the result of a quantum circuit with a Monte Carlo procedure that randomly replaces circuit elements with ones that can be executed on the restricted quantum device. The main focus of this thesis is dedicated to the QPS-based simulation of nonlocal quantum computation using local quantum operations. On the practical side, this enables the simulation of large quantum circuits using multiple smaller quantum devices - a procedure that is sometimes called circuit knitting. We uncover a rich mathematical formalism with many connections to the resource theory of entanglement. We characterize the optimal simulation overhead for a broad range of practically relevant nonlocal states and channels and we explicitly provide achieving protocols. Moreover, we also investigate the utility of classical communication between the local parties. Our results address both the single-shot and asymptotic regime. We frame QPS in a quantum resource theoretic framework, which highlights similarities that arise in the different instantiations of the technique. Furthermore, we study the importance of classical side information in the QPS procedure and how it impacts the overhead and expressibility of QPS.
It is one of the most fundamental objectives in quantum information science to understand the boundary between the computational power of classical and quantum computers. One possible avenue to explore this boundary is to identify classes of quantum circuits that can be efficiently simulated on a classical computer. Instead of simulating a general quantum circuit with a classical device, new schemes have recently emerged to simulate them on a quantum device that is restricted in some manner. As such, these techniques allow us to study how the restrictions impact the computational power of the device. One such technique is called quasiprobability simulation (QPS) and it estimates the result of a quantum circuit with a Monte Carlo procedure that randomly replaces circuit elements with ones that can be executed on the restricted quantum device. The main focus of this thesis is dedicated to the QPS-based simulation of nonlocal quantum computation using local quantum operations. On the practical side, this enables the simulation of large quantum circuits using multiple smaller quantum devices - a procedure that is sometimes called circuit knitting. We uncover a rich mathematical formalism with many connections to the resource theory of entanglement. We characterize the optimal simulation overhead for a broad range of practically relevant nonlocal states and channels and we explicitly provide achieving protocols. Moreover, we also investigate the utility of classical communication between the local parties. Our results address both the single-shot and asymptotic regime. We frame QPS in a quantum resource theoretic framework, which highlights similarities that arise in the different instantiations of the technique. Furthermore, we study the importance of classical side information in the QPS procedure and how it impacts the overhead and expressibility of QPS.
The scarcity of qubits is a major obstacle to the practical usage of quantum computers in the near future. To circumvent this problem, various circuit knitting techniques have been developed to partition large quantum circuits into subcircuits that fit on smaller devices, at the cost of a simulation overhead. In this work, we study a particular method of circuit knitting based on quasiprobability simulation of nonlocal gates with operations that act locally on the subcircuits. We investigate whether classical communication between these local quantum computers can help. We provide a positive answer by showing that for circuits containingnnonlocal CNOT gates connecting two circuit parts, the simulation overhead can be reduced fromO(9n)toO(4n)if one allows for classical information exchange. Similar improvements can be obtained for general Clifford gates and, at least in a restricted form, for other gates such as controlled rotation gates.
Circuit knitting, a method for connecting quantum circuits across multiple processors to simulate nonlocal quantum operations, is a promising approach for distributed quantum computing. While various techniques have been developed for circuit knitting, we uncover fundamental limitations to the scalability of this technology. We prove that the sampling overhead of circuit knitting is exponentially lower bounded by the exact entanglement cost of the target bipartite dynamic, even with the asymptotic amount of resources using the parallel cut strategy. Specifically, we prove that the regularized sampling overhead assisted with local operations and classical communication of any bipartite quantum channel is lower bounded by the exponential of its exact entanglement cost under separable preserving operations. Furthermore, we develop the faithful lower bounds for the regularized sampling overhead based on channels' $\ensuremath{\kappa}$ entanglement and max-Rains information, providing efficiently computable benchmarks. Our work reveals a profound connection between virtual quantum information processing via quasiprobability decomposition and quantum Shannon theory, highlighting the critical role of entanglement in distributed quantum computing.
One of the core research questions in the theory of quantum computing is to find out to what precise extent the classical simulation of a noisy quantum circuits is possible and where potential quantum advantages can set in. In this work, we introduce a unified framework for the classical simulation of quantum circuits based on frame theory, encompassing and generalizing a broad class of existing simulation strategies. Within this framework, the computational cost of a simulation algorithm is determined by the one-norm of an associated quasi-probability distribution, providing a common quantitative measure across different simulation approaches. This enables a comprehensive perspective on common methods for the simulation of noisy circuits based on different quantum resources, such as entanglement or non-stabilizerness. It further provides a clear scheme for generating novel classical simulation algorithms. Indeed, by exploring different choices of frames within this formalism and resorting to tools of convex optimization, we are able not only to obtain new insights and improved bounds for existing methods -- such as stabilizer state simulation or Pauli back-propagation -- but also to discover a new approach with an improved performance based on a generalization of the Pauli frame. We, thereby, show that classical simulation techniques can directly benefit from a perspective -- that of frames -- that goes beyond the traditional classification of quantum resources.
Circuit knitting, a method for connecting quantum circuits across multiple processors to simulate nonlocal quantum operations, is a promising approach for distributed quantum computing. While various techniques have been developed for circuit knitting, we uncover fundamental limitations to the scalability of this technology. We prove that the sampling overhead of circuit knitting is exponentially lower bounded by the exact entanglement cost of the target bipartite dynamic, even for asymptotic overhead in the parallel cut regime. Specifically, we prove that the regularized sampling overhead assisted with local operations and classical communication (LOCC), of any bipartite quantum channel is lower bounded by the exponential of its exact entanglement cost under separable preserving operations. Furthermore, we show that the regularized sampling overhead for simulating a general bipartite channel via LOCC is lower bounded by κ-entanglement and max-Rains information, providing efficiently computable benchmarks. Our work reveals a profound connection between virtual quantum information processing via quasi-probability decomposition and quantum Shannon theory, highlighting the critical role of entanglement in distributed quantum computing.
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.
Circuit knitting is the process of partitioning large quantum circuits into smaller subcircuits such that the result of the original circuits can be deduced by only running the subcircuits. Such techniques will be crucial for near-term and early fault-tolerant quantum computers, as the limited number of qubits is likely to be a major bottleneck for demonstrating quantum advantage. One typically distinguishes between gate cuts and wire cuts when partitioning a circuit. The cost for any circuit knitting approach scales exponentially in the number of cuts. One possibility to realize a cut is via the quasiprobability simulation technique. In fact, we argue that all existing rigorous circuit knitting techniques can be understood in this framework. Furthermore, we characterize the optimal overhead for wire cuts where the subcircuits can exchange classical information or not. We show that the optimal cost for cuttingnwires without and with classical communication between the subcircuits scales asO(16n) andO(4n), respectively.