Quantum computing is considered a promising platform for next-generation large-scale computing. Although fully scalable quantum computers are still under development, quantum accelerators already offer a practical means to test, evaluate, and better understand quantum computation. Existing simulators provide valuable tools, but they typically rely on deterministic arithmetic and often incur significant computational overhead. To address these limitations, we developed a quantum computing simulator based on bipolar stochastic arithmetic, designed with two goals in mind. First, stochastic computation enables flexible, on-demand control of simulation precision and runtime. Second, stochastic circuits generally require significantly less hardware area than their deterministic counterparts. In our approach, quantum states and operations are encoded using bipolar stochastic bitstreams, allowing quantum circuits to be simulated using lightweight stochastic logic. We implement a representative set of quantum circuits and evaluate the simulator with respect to accuracy and required stream length. The results demonstrate that stochastic simulation yields satisfactory accuracy for small quantum circuits in software, and with improved error-correction techniques, it has the potential to serve as an efficient alternative to conventional quantum simulators.
Quantum computing is considered a promising platform for next-generation large-scale computing. Although fully scalable quantum computers are still under development, quantum accelerators already offer a practical means to test, evaluate, and better understand quantum computation. Existing simulators provide valuable tools, but they typically rely on deterministic arithmetic and often incur significant computational overhead. To address these limitations, we developed a quantum computing simulator based on bipolar stochastic arithmetic, designed with two goals in mind. First, stochastic computation enables flexible, on-demand control of simulation precision and runtime. Second, stochastic circuits generally require significantly less hardware area than their deterministic counterparts. In our approach, quantum states and operations are encoded using bipolar stochastic bitstreams, allowing quantum circuits to be simulated using lightweight stochastic logic. We implement a representative set of quantum circuits and evaluate the simulator with respect to accuracy and required stream length. The results demonstrate that stochastic simulation yields satisfactory accuracy for small quantum circuits in software, and with improved error-correction techniques, it has the potential to serve as an efficient alternative to conventional quantum simulators.
Various emerging technologies promise advantages with respect to integration density, performance or power consumption, at the cost of approximate or probabilistic behavior. Approximate computing, where limited computational inaccuracies are tolerated at the system or application level is therefore of increasing interest. This paper investigates the use of stochastic computing (SC) as a tool for approximate simulation of probabilistic behavior. SC has the advantage of processing probabilities directly at very low hardware cost. It also allows accuracy to be traded for run-time in a natural way (progressive precision). AS a target technology to be simulated, we choose quantum computing circuits, whose behavior is inherently probabilistic and cannot be efficiently simulated by conventional (classical) means. We show how complex operations such as superposition and entanglement can be handled by SC. Finally, we report experimental results on software-based simulation of representative quantum circuits, both stand-alone and FPGA-supported. The results show that the SC implementations are orders of magnitude more compact than those based on classical circuits. Accurate results may require very long simulation runs, but run-times can be reduced by exploiting SC's progressive precision property.
Quantum computing is an emerging technology that has the potential to change the perspectives and applications of computing in general. A wide range of applications are enabled: from faster algorithmic solutions of classically still difficult problems to theoretically more secure communication protocols. A quantum computer uses the quantum mechanical effects of particles or particle-like systems, and a major similarity between quantum and classical computers consists of both being abstracted as information processing machines. Whereas a classical computer operates on classical digital information, the quantum computer processes quantum information, which shares similarities with analog signals. One of the central differences between the two types of information is that classical information is more fault-tolerant when compared to its quantum counterpart. Faults are the result of the quantum systems being interfered by external noise, but during the last decades quantum error correction codes (QECC) were proposed as methods to reduce the effect of noise. Reliable quantum circuits are the result of designing circuits that operate directly on encoded quantum information, but the circuit’s reliability is also increased by supplemental redundancies, such as sub-circuit repetitions. Reliable quantum circuits have not been widely used, and one of the major obstacles is their vast associated resource overhead, but recent quantum computing architectures show promising scalabilities. Consequently the number of particles used for computing can be more easily increased, and that the classical control hardware (inherent for quantum computation) is also more reliable. Reliable quantum circuits haev been investigated for almost as long as general quantum computing, but their limited adoption (until recently) has not generated enough interest into their systematic design. The continuously increasing practical relevance of reliability motivates the present thesis to investigate some of the first answers to questions related to the background and the methods forming a reliable quantum circuit design stack. The specifics of quantum circuits are analysed from two perspectives: their probabilistic behaviour and their topological properties when a particular class of QECCs are used. The quantum phenomena, such as entanglement and superposition, are the computational resources used for designing quantum circuits. The discrete nature of classical information is missing for quantum information. An arbitrary quantum system can be in an infinite number of states, which are linear combinations of an exponential number of basis states. Any nontrivial linear combination of more than one basis states is called a state superposition. The effect of superpositions becomes evident when the state of the system is inferred (measured), as measurements are probabilistic with respect to their output: a nontrivial state superposition will collapse to one of the component basis states, and the measurement result is known exactly only after the measurement. A quantum system is, in general, composed from identical subsystems, meaning that a quantum computer (the complete system) operates on multiple similar particles (subsystems). Entanglement expresses the impossibility of separating the state of the subsystems from the state of the complete system: the nontrivial interactions between the subsystems result into a single indivisible state. Entanglement is an additional source of probabilistic behaviour: by measuring the state of a subsystem, the states of the unmeasured subsystems will probabilistically collapse to states from a well defined set of possible states. Superposition and entanglement are the building blocks of quantum information teleportation protocols, which in turn are used in state-of-the-art fault-tolerant quantum computing architectures. Information teleportation implies that the state of a subsystem is moved to a second subsystem without copying any information during the process. The probabilistic approach towards the design of quantum circuits is initiated by the extension of classical test and diagnosis methods. Quantum circuits are modelled similarly to classical circuits by defining gate-lists, and missing quantum gates are modelled by the single missing gate fault. The probabilistic approaches towards quantum circuits are facilitated by comparing these to stochastic circuits, which are a particular type of classical digital circuits. Stochastic circuits can be considered an emulation of analogue computing using digital components. A first proposed design method, based on the direct comparison, is the simulation of quantum circuits using stochastic circuits by mapping each quantum gate to a stochastic computing sub-circuit. The resulting stochastic circuit is compiled and simulated on FPGAs. The obtained results are encouraging and illustrate the capabilities of the proposed simulation technique. However, the exponential number of possible quantum basis states was translated into an exponential number of stochastic computing elements. A second contribution of the thesis is the proposal of test and diagnosis methods for both stochastic and quantum circuits. Existing verification (tomographic) methods of quantum circuits were targeting the reconstruction of the gate-lists. The repeated execution of the quantum circuit was followed by different but specific measurement at the circuit outputs. The similarities between stochastic and quantum circuits motivated the proposal of test and diagnosis methods that use a restricted set of measurement types, which minimise the number of circuit executions. The obtained simulation results show that the proposed validation methods improve the feasibility of quantum circuit tomography for small and medium size circuits. A third contribution of the thesis is the algorithmic formalisation of a problem encountered in teleportation-based quantum computing architectures. The teleportation results are probabilistic and require corrections represented as quantum gates from a particular set. However, there are known commutation properties of these gates with the gates used in the circuit. The corrections are not applied as dynamic gate insertions (during the circuit’s execution) into the gate-lists, but their effect is tracked through the circuit, and the corrections are applied only at circuit outputs. The simulation results show that the algorithmic solution is applicable for very large quantum circuits. Topological quantum computing (TQC) is based on a class of fault-tolerant quantum circuits that use the surface code as the underlying QECC. Quantum information is encoded in lattice-like structures and error protection is enabled by the topological properties of the lattice. The 3D structure of the lattice allows TQC computations to be visualised similarly to knot diagrams. Logical information is abstracted as strands and strand interactions (braids) represent logical quantum gates. Therefore, TQC circuits are abstracted using a geometrical description, which allows circuit input-output transformations (correlations) to be represented as geometric sub-structures. TQC design methods were not investigated prior to this work, and the thesis introduces the topological computational model by first analysing the necessary concepts. The proposed TQC design stack follows a top-down approach: an arbitrary quantum circuit is decomposed into the TQC supported gate set; the resulting circuit is mapped to a lattice of appropriate dimensions; relevant resulting topological properties are extracted and expressed using graphs and Boolean formulas. Both circuit representations are novel and applicable to TQC circuit synthesis and validation. Moreover, the Boolean formalism is broadened into a formal mechanism for proving circuit correctness. The thesis introduces TQC circuit synthesis, which is based on a novel logical gate geometric description, whose formal correctness is demonstrated. Two synthesis methods are designed, and both use a general planar representation of the circuit. Initial simulation results demonstrate the practicality and performance of the methods. An additional group of proposed design methods solves the problem of automatic correlation construction. The methods use validity criteria which were introduced and analysed beforehand in the thesis. Input-output correlations existing in the circuit are inferred using both the graph and the Boolean representation. The thesis extends the TQC state-of-the-art by recognising the importance of correlations in the validation process: correlation construction is used as a sub-routine for TQC circuit validation. The presented cross-layer validation procedure is useful when investigating both the QECC and the circuit, while a second proposed method is QECC-independent. Both methods are scalable and applicable even to very large circuits. The thesis completes with the analysis of TQC circuit identities, where the developed Boolean formalism is used. The proofs of former known circuit identities were either missing or complex, and the presented approach reduces the length of the proofs and represents a first step towards standardising them. A new identity is developed and detailed during the process of illustrating the known circuit identities. Reliable quantum circuits are a necessity for quantum computing to become reality, and specialised design methods are required to support the quest for scalable quantum computers. This thesis used a twofold approach towards this target: firstly by focusing on the probabilistic behaviour of quantum circuits, and secondly by considering the requirements of a promising quantum computing architecture, namely TQC. Both approaches resulted in a set of design methods enabling the investigation of reliable quantum circuits. The thesis contributes with the proposal of a new quantum simulation technique, no
Simulating the evolution of quantum circuits is a computationally intensive task, fundamentally limited by the need to explore the exponentially large Hilbert space associated with quantum systems. This work introduces a novel approach utilizing probabilistic programming to efficiently model and simulate quantum circuit dynamics. The core idea involves representing quantum circuit operations as probabilistic functions, enabling the use of Monte Carlo methods for estimating circuit evolution with significantly reduced computational overhead. This approach provides a flexible and scalable framework for exploring complex quantum systems, particularly those where traditional deterministic simulation methods become intractable. The method's effectiveness is demonstrated through theoretical analysis and a conceptual outline of implementation, highlighting potential benefits and future research directions. Key mathematical concepts employed include probability theory, Markov chains, and Monte Carlo integration. The primary goal is to offer a practical alternative to standard quantum simulation techniques, particularly for scenarios requiring approximate solutions or exploration of large parameter spaces.
Quantum circuit simulation is usually evaluated through final numerical accuracy, while the dynamics of stochastic execution itself are less explicitly characterized. This work presents a precision-elastic persistent stochastic execution framework based on integral stochastic computing (ISC), where execution behavior is controlled by stream length N and ISC multiplicity m, where m represents the number of aggregated stochastic sub-streams per cycle (standard SC corresponds to m = 1). We focus on correlation-sensitive propagation under persistent reuse, and show that this regime produces circuit-dependent stochastic behavior and execution uncertainty patterns that are not captured by stage-wise re-encoded execution alone. To characterize this behavior, we use high-m tail descriptors, including the circuit-dependent coefficient γc, as compact indicators of persistent stochastic sensitivity. Across benchmark circuits, persistent execution exhibits reproducible tail regimes and structured cross-circuit variability, while deterministic reduced-precision baselines are used only as trend-consistency references. We further demonstrate adaptive stochastic precision scheduling, where circuit-dependent (N,m) settings satisfy a target fidelity with reduced stochastic workload. These results position persistent ISC as a configurable stochastic execution framework for analyzing execution-induced uncertainty propagation and correlation-sensitive behavior in quantum circuits.
Developing state-of-the-art classical simulators of quantum circuits is of utmost importance to test and evaluate early quantum technology and understand the true potential of full-blown error-corrected quantum computers. In the past few years, multiple theoretical and numerical advances have continuously pushed the boundary of what is classically simulable, hence the development of a plethora of tools which are often limited to a specific purpose or designed for a particular hardware (e.g. CPUs vs. GPUs). Moreover, such tools are typically developed using tailored languages and syntax, which makes it hard to compare results from, and create hybrid approaches using, different simulation techniques. To support unified and optimized use of these techniques across platforms, we developed HybridQ , a highly extensible platform designed to provide a common framework to integrate multiple state-of-the-art techniques to run on a variety of hardware. The philosophy behind its development has been driven by three main pillars: Easy to Use, Easy to Extend, and Use the Best Available Technology. The powerful tools of HybridQ allow users to manipulate, develop, and extend noiseless and noisy circuits for different hardware architectures. HybridQ supports large-scale high-performance computing (HPC) simulations, automatically balancing workload among different processor nodes and enabling the use of multiple backends to maximize parallel efficiency. Everything is then glued together by a simple and expressive language that allows seamless switching from one technique to another as well as from one hardware to the next, without the need to write lengthy translations, thus greatly simplifying the development of new hybrid algorithms and techniques.
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.