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arXiv:2205.15822 · International Journal of Quantum Information

Intermediate Qutrit-based improved quantum arithmetic operations with application on financial derivative pricing

Amit Saha, Turbasu Chatterjee, Anupam Chattopadhyay, Amlan Chakrabarti

Published Nov 4, 2025• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1142/s0219749925500364

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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.
Pricing financial derivatives on quantum computers typically includes quantum arithmetic components which contribute heavily to the quantum resources required by the corresponding circuits. In this manuscript, we introduce a method based on Quantum Signal Processing (QSP) to encode financial derivative payoffs directly into quantum amplitudes, alleviating the quantum circuits from the burden of costly quantum arithmetic. Compared to current state-of-the-art approaches in the literature, we find that for derivative contracts of practical interest, the application of QSP significantly reduces the required resources across all metrics considered, most notably the total number of T-gates by ∼ 16 x and the number of logical qubits by ∼ 4 x. Additionally, we estimate that the logical clock rate needed for quantum advantage is also reduced by a factor of ∼ 5 x. Overall, we find that quantum advantage will require 4.7 k logical qubits, and quantum devices that can execute 10 9 T-gates at a rate of 45 MHz. While in this work we focus specifically on the payoff component of the derivative pricing process where the method we present is most readily applicable, similar techniques can be employed to further reduce the resources in other applications, such as state preparation.
Arithmetic operations are an important component of many quantum algorithms. Optimizing quantum circuits for these operations therefore leads to more efficient implementations of the corresponding algorithms. In this paper, we develop new fault-tolerant quantum circuits for various integer division algorithms (both reversible and non-reversible). These circuits, when implemented in the Clifford+T gate set, achieve improvements in the asymptotic T-count and CNOT-count from O(n2)O(n^2)O(n2) to O(nm)O(nm)O(nm), and in the asymptotic T-depth from O(n2)O(n^2)O(n2) to O(nlog⁡m)O(n \log m)O(nlogm), where nnn and mmm are the bit lengths of the dividend and divisor, respectively. The qubit counts are significantly lower than those in previous works. We achieve this through careful modifications of the division algorithms and by expressing them in terms of a primitive we call COMP-N-SUB that compares two integers and conditionally subtracts them. We show that this primitive can be implemented at a cost, in terms of both Clifford and non-Clifford gates, comparable to that of one addition. By contrast, performing comparison and conditional subtraction separately would cost roughly as much as a controlled addition plus a regular addition.
Quantum computation is traditionally expressed in terms of quantum bits, or qubits. In this work, we instead consider three-level qu trits . Past work with qutrits has demonstrated only constant factor improvements, owing to the log 2 (3) binary-to-ternary compression factor. We present a novel technique, intermediate qutrits, to achieve sublinear depth decompositions of the Generalized Toffoli and other arithmetic circuits using no additional ancilla—a significant improvement over linear depth for the best qubit-only equivalents. For example, our Generalized Toffoli construction features a 70× improvement in two-qudit gate count over a qubit-only decomposition. This results in circuit cost reductions for important algorithms like quantum neurons, Grover search, and even Shor’s algorithm. Using a previously developed simulator with near-term noise models, we demonstrate for these models over 90% mean reliability (fidelity) for the Toffoli construction, versus under 30% for the qubit-only baseline. For our other constructions, such as the Incrementer, the A + B adder and the +K adder, we demonstrate the power of intermediate qutrits in producing asymptotic depth improvements with no additional ancilla. Together, these results suggest qutrits offer a promising path toward scaling quantum computation.
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.
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.
We advocate for a fundamentally different way to perform quantum computation by using three-level qutrits instead of qubits. In particular, we substantially reduce the resource requirements of quantum computations by exploiting a third state for temporary variables (ancilla) in quantum circuits. Past work with qutrits has demonstrated only constant factor improvements, owing to the log2(3) binary-to-ternary compression factor. We present a novel technique using qutrits to achieve a logarithmic runtime decomposition of the Generalized Toffoli gate using no ancilla---an exponential improvement over the best qubit-only equivalent. Our approach features a 70x improvement in total two-qudit gate count over the qubit-only decomposition. This results in improvements for important algorithms for arithmetic and QRAM. Simulation results under realistic noise models indicate over 90% mean reliability (fidelity) for our circuit, versus under 30% for the qubit-only baseline. These results suggest that qutrits offer a promising path toward extending the frontier of quantum computers.