Large-scale quantum computers that can execute practical quantum algorithms have the potential to solve complex problems that are currently challenging for classical computers. This involves converting these problems into a form that can be processed by quantum circuits, a crucial process that requires minimizing quantum resources like qubit count, gate count, and circuit depth. Our work focuses on implementing and optimizing the foundational task of quantum finance, known as option pricing, as a quantum circuit. This enables the utilization of quantum computing benefits, within the financial domain. Specifically, we implement and optimize the function fK(S) = max(S−K, 0). Taking into consideration the significant trade-offs between qubit count and circuit depth, we have developed quantum circuits for the optimized implementation of the fK(S). Our work incorporates various optimization techniques for the circuit, such as selecting the optimal adder, optimizing the S−K operation, parallelization, and qubit reuse. Furthermore, we offer various versions of our quantum circuits for the fK(S), each featuring different adders and Toffoli decompositions, thereby providing flexibility for a wide range of use cases.
Large-scale quantum computers that can execute practical quantum algorithms have the potential to solve complex problems that are currently challenging for classical computers. This involves converting these problems into a form that can be processed by quantum circuits, a crucial process that requires minimizing quantum resources like qubit count, gate count, and circuit depth. Our work focuses on implementing and optimizing the foundational task of quantum finance, known as option pricing, as a quantum circuit. This enables the utilization of quantum computing benefits, within the financial domain. Specifically, we implement and optimize the function fK(S) = max(S−K, 0). Taking into consideration the significant trade-offs between qubit count and circuit depth, we have developed quantum circuits for the optimized implementation of the fK(S). Our work incorporates various optimization techniques for the circuit, such as selecting the optimal adder, optimizing the S−K operation, parallelization, and qubit reuse. Furthermore, we offer various versions of our quantum circuits for the fK(S), each featuring different adders and Toffoli decompositions, thereby providing flexibility for a wide range of use cases.
Options are financial instruments whose payoffs are determined by the value of an underlying asset. These underlying assets are typically stocks, interest rates, or exchange rates, but can also include commodities such as energy or agricultural products. Options serve diverse purposes, including hedging against price volatility and speculating on market movements. Additionally, they play a critical role in assessing certain positions on the balance sheets of insurance companies, particularly in the context of solvency capital requirement calculations. As an option’s payoff lies in the future and is stochastic in nature, finding a fair price today is a non-trivial task. The focus of this work is finding quantum algorithms for calculating these prices and compare them to algorithms designed for classical computers. This work follows two different directions on how to improve on classical methods for option pricing. The first direction is improving on Monte-Carlo (MC) methods and the second one is adapting Fourier based pricing algorithms. The Amplitude Estimation (AE) algorithm promises a quadratic speedup compared to classical MC. This fact makes this algorithm an interesting candidate for improvements by quantum algorithms. It was shown in the literature, that simple options with very crude discretization of the underlying stochastic models, can be implemented on a quantum device and evaluated with AE. One big challenge is the efficient implementation of the option’s payoff profile. To solve this problem colleagues and I have introduced a multi-objective genetic algorithm to automatically find well performing and efficient circuits to implement the payoff of non-path-dependent European options on one or more underlyings. Another strategy for pricing options is based on the fact, that in many models the characteristic function of the underlying is numerically more tractable than its distribution function. There are methods that make use of this fact and the performance of the Fast Fourier Transform (FFT) to efficiently calculate prices for European call options for many strikes in one go. The quantum version of the FFT, the Quantum Fourier Transform (QFT), is in some sense, even faster than its classical counterpart. We have adapted the existing classical method to make use of QFT, to benefit from its performance. Finally, the different strategies are benchmarked and compared regarding their accuracy, their potential for scaling and their adaptability to different styles of derivatives and models. The result of this comparison are requirements on the quantum hardware for the methods to be viable as well as use cases where each method has its benefits over the others.
In recent decades, quantum computing has witnessed remarkable progress, achieved through various advancements across the technology stack, such as quantum error-correcting codes and quantum algorithms. It is noteworthy that several promising algorithms incorporate quantum arithmetic circuits as fundamental building blocks. However, the inherent properties of quantum circuits challenge the efficiency of quantum arithmetic circuits, prompting creative solutions. While several quantum arithmetic circuits have been proposed, there is still a substantial gap in quantum arithmetic circuits optimization. For example, methods for integrating classical techniques such as high radix strategy into quantum designs still remain underexplored, highlighting a significant space for further advancement. This thesis aims to improve the performance of quantum arithmetic designs by integrating classical arithmetic circuit design techniques. Specifically, this thesis contains the following research: (1) The utilization of classical higher radix strategy to enhance quantum addition, reducing overall cost by lowering T-depth and T-count. The proposed design outperforms existing quantum carry-lookahead adders. (2) The application of Ling structure, achieving a reduced Toffoli-depth of 4log(n/2). (3) The exploration of the Kogge-Stone architecture, resulting in the lowest Toffoli-depth of log(n) for quantum adders. (4) The development of an efficient modular addition framework using carry-save architecture, demonstrating a 33.3% improvement in Toffoli depth compared to existing designs. (5) The design of an optimized quantum non-restoring divider framework. (6) A comprehensive overview of current advancements in quantum arithmetic designs, offering insights into potential research directions. These proposed designs are mapped to real-world quantum hardware architectures, including Noisy Intermediate-Scale Quantum systems, to validate their functionality and evaluate efficiency. Their applications in quantum machine learning and cryptography further underscore the broader impact of these improvements. Overall, this thesis makes significant advancements in reducing quantum circuit depth and improving resource efficiency, contributing to the development of scalable and efficient large-scale quantum circuits. By adapting classical design strategies to quantum circuits, it not only enhances various quantum circuit designs but also fosters further exploration at the intersection of classical and quantum computing.
Quantum computing (Q-computing) is a revolutionary technology that applies quantum mechanics (QM) to implement calculations significantly quicker than traditional computers. Q-computing employs Q-bits (qubits) rather than classical bits, which can take on a superposition state, making it possible to represent both 0 and 1 at once. This distinctive property allows qubits to perform multiple calculations simultaneously, making Q computers exponentially more powerful as more qubits are added, compared to the linear scaling of classical computers. However, challenges such as qubit instability and error rates remain significant obstacles to fully realizing this potential. A vital aspect of this technology is the development of Q arithmetic circuits, which act as the foundational blocks for efficiently executing Q algorithms. Q half-adder and full-adder circuits are integral to performing fundamental arithmetic operations within Q computers. In contrast to classical adders, Q adders rely on Q gates such as Controlled NOT (CNOT) and Toffoli gates. These gates exploit Q phenomena like entanglement and superposition, allowing for more efficient computations. The Q half-adder consists of two qubits and adds two single bits, producing a carry and sum output. It utilizes the CNOT gate to compute the sum and the Toffoli gate (controlled-controlled-NOT [CCNOT]) to compute the carry. The Q full-adder builds on this, adding three qubits to produce a sum and a carry output, further increasing computational capacity. Through the entanglement of qubits, these circuits allow for more parallelism and speed compared to classical adders. Q carry-look-ahead adders (QCLAs) and Q ripple-carry adders (QRCAs) are used to perform more complex arithmetic operations based on these half-adder and full-adder structures. Q arithmetic circuits have broad applications, boosting secure key generation in cryptography, speeding up matrix operations for faster model training in machine learning, and enabling more precise, efficient image transformations in Q image processing, benefiting fields like medical imaging and computer vision.
A possible approach to tackling computationally demanding issues beyond the capabilities of conventional algorithms is the use of quantum circuits, which provide a fundamental foundation for utilizing the power of quantum computation. This abstract examines how quantum circuits are implemented, emphasizing important topics such as circuit optimization, gate operations, and qubit manipulation. We explore different methods of encoding quantum information into topological qubits, trapped ions, and superconducting circuits, among other physical qubit formats. Furthermore, we go over the function of gate operations in quantum circuits, emphasizing crucial gates like phase, Hadamard, and CNOT gates, which serve as the foundation for quantum algorithms. Additionally, we look at methods for maximizing the efficiency and dependability of quantum processing by designing quantum circuits to minimize gate counts, depth, and error rates. To enhance the capabilities of quantum computing, this abstract presents a novel synthesis of quantum neural networks (QNNs) and quantum parameterized circuits. Through the use of parameterized quantum circuits' inherent flexibility, our approach makes it easier to design and optimize quantum circuits with flexible parameters, providing a flexible foundation for a variety of quantum algorithms. Using this amalgamation, we unleash the capacity of quantum computers to address intricate optimization and machine learning assignments with unparalleled effectiveness and expandability. Our solution bridges the gap between traditional machine learning approaches and quantum computing paradigms by enabling the smooth integration of classical data processing techniques with quantum computation. We use quantum states as data representations in our implementation of quantum neural networks, and we optimize trainable parameters to meet predetermined goals. This new paradigm opens up the investigation of the inherent features of quantum states and provides new perspectives on their processing power compared to their classical counterparts. We show, by thorough testing and analysis, how effective quantum parameterized circuits and QNNs are at solving practical problems in a variety of fields. Our work sets the ground for revolutionary breakthroughs in quantum information processing by pushing the boundaries of quantum computation and machine learning integration, offering hitherto unseen insights and solutions to challenging challenges in science, engineering, and beyond.
Prior studies have largely focused on quantum algorithms, often reducing parallel computing designs to abstract models or overly simplified circuits. This has contributed to the misconception that most applications are feasible only through VLSI circuits and cannot be implemented using quantum circuits. To challenge this view, we present a mixed-signal quantum circuit framework incorporating three novel methods that reduce circuit complexity and improve noise tolerance. In a 12 qubit case study comparing our design with JP Morgan's option pricing circuit, we reduced the gate count from 4095 to 392, depth from 2048 to 6, and error rate from 25.86\% to 1.64\%. Our design combines analog simplicity with digital flexibility and synthesizability, demonstrating that quantum circuits can effectively leverage classical VLSI techniques, such as those enabled by Synopsys Design Compiler to address current quantum design limitations.
Efficient quantum realizations of arithmetic operations, such as multiplication/division, addition/subtraction, square root, and arcsine — are of utmost importance for several quantum algorithms of practical importance. In the literature, such implementations are reported with the objective of minimizing qubit count, Toffoli/T/CNOT gate count, logical depth, quantum volume, and other commonly used metrics. By extending the realm of quantum states to higher dimensions, it has been demonstrated recently that intermediate qutrits can be leveraged to reduce ancilla qubits, thereby reducing qubit count. In this paper, we have incorporated this approach to prepare efficient implementation for several quantum arithmetic operations, obtaining significantly lowered qubit count. As an exemplary application, we study derivative pricing, in which, quantum arithmetic circuits are necessary for path loading using the re-parameterization method, as well as the payoff calculation. The intermediate qutrit approach requires to access higher energy levels, making the design prone to errors. Nevertheless, we show that the overall decrease in the error probability of error is significant owing to the fact that we achieve circuit robustness compared to the qubit-only approach in this NISQ era. We also study how we may achieve possible effectiveness with fewer gate counts of the proposed approach in the fault-tolerant setting.