Many promising applications of quantum computing with a provable speedup center around the HHL algorithm. Due to restrictions on the hardware and its significant demand on qubits and gates in known implementations, its execution is prohibitive on near-term quantum computers. Aiming to facilitate such NISQ-implementations, we propose a novel circuit approximation technique that enhances the arithmetic subrou-tines in the HHL, which resemble a particularly resource-demanding component in small-scale settings. For this, we provide a description of the algorithmic implementation of space-efficient rotations of polynomial functions that do not demand explicit arithmetic calculations inside the quantum circuit. We show how these types of circuits can be reduced in depth by providing a simple and powerful approximation technique. Moreover, we provide an algorithm that converts lookup-tables for arbitrary function rotations into a structure that allows an application of the approximation technique. This allows implementing approximate rotation circuits for many polynomial and non-polynomial functions. Experimental results obtained for realistic early-application dimensions show significant improve-ments compared to the state-of-the-art, yielding small circuits while achieving good approximations.
Many promising applications of quantum computing with a provable speedup center around the HHL algorithm. Due to restrictions on the hardware and its significant demand on qubits and gates in known implementations, its execution is prohibitive on near-term quantum computers. Aiming to facilitate such NISQ-implementations, we propose a novel circuit approximation technique that enhances the arithmetic subrou-tines in the HHL, which resemble a particularly resource-demanding component in small-scale settings. For this, we provide a description of the algorithmic implementation of space-efficient rotations of polynomial functions that do not demand explicit arithmetic calculations inside the quantum circuit. We show how these types of circuits can be reduced in depth by providing a simple and powerful approximation technique. Moreover, we provide an algorithm that converts lookup-tables for arbitrary function rotations into a structure that allows an application of the approximation technique. This allows implementing approximate rotation circuits for many polynomial and non-polynomial functions. Experimental results obtained for realistic early-application dimensions show significant improve-ments compared to the state-of-the-art, yielding small circuits while achieving good approximations.
In this paper, we present Q# implementations for arbitrary single-variabled fixed-point arithmetic op-erations for a gate-based quantum computer based on lookup tables (LUTs). In general, this is an inefficent way of implementing a function since the number of inputs can be large or even infinite. However, if the input domain can be bounded and there can be some error tolerance in the output (both of which are often the case in practical use-cases), the quantum LUT implementation of certain quantum arithmetic functions can be more efficient than their corresponding reversible arithmetic implementations. We discuss the implementation of the LUT using Q# and its approximation errors. We then show examples of how to use the LUT to implement quantum arithmetic functions and compare the resources required for the implementation with the current state-of-the-art bespoke implementations of some commonly used arithmetic functions. The implementation of the LUT is designed for use by practitioners to use when implementing end-to-end quantum algorithms. In addition, given its well-defined approximation errors, the LUT implementation makes for a clear benchmark for evaluating the efficiency of bespoke quantum arithmetic circuits.
To assess whether a gate-based quantum algorithm can be executed successfully on a noisy intermediate-scale quantum (NISQ) device, both complexity and actual value of quantum resources should be considered carefully. Based on quantum phase estimation, we implemente arbitrary controlled rotation of quantum algorithms with a proposed modular method. The proposed method is not limited to be used as a submodule of the HHL algorithm and can be applied to more general quantum machine learning algorithms. Compared with the polynomial-fitting function method, our method only requires the least ancillas and the least quantum gates to maintain the high fidelity of quantum algorithms. The method theoretically will not influence the acceleration of original algorithms. Numerical simulations illustrate the effectiveness of the proposed method. Furthermore, if the corresponding diagonal unitary matrix can be effectively decomposed, the method is also polynomial in time cost.
Quantum access to arbitrary classical data encoded in unitary black-box oracles underlies interesting data-intensive quantum algorithms, such as machine learning or electronic structure simulation. The feasibility of these applications depends crucially on gate-efficient implementations of these oracles, which are commonly some reversible versions of the boolean circuit for a classical lookup table. We present a general parametrized architecture for quantum circuits implementing a lookup table that encompasses all prior work in realizing a continuum of optimal trade-offs between qubits, non-Clifford gates, and error resilience, up to logarithmic factors. Our architecture assumes only local 2D connectivity, yet recovers results, with the appropriate parameters, polylogarithmic error scaling. We also identify regimes, such as simultaneous sublinear scaling, in all parameters. These results enable tailoring implementations of the commonly used lookup table primitive to any given quantum device with constrained resources.
Quantum computing faces challenges such as noise, short coherence time, and limited qubit connections. These challenges worsen as quantum circuits become more complex. One major issue is the increasing depth of quantum circuits. This research proposes an optimization framework targeting depth and gate count reduction in quantum circuits, specifically for Noisy Intermediate-Scale Quantum (NISQ) devices. The proposed approach combines unitary merging of single-qubit gates, CNOT cancellation, gate commuting, and rotation gate rewriting strategies. Consecutive gates acting on the same qubit, such asRx(θ1)⋅Rx(θ2), are algebraically merged into a single rotation, while pairs of redundant CNOT gates are eliminated based on gate identity relations. The technique is implemented using Qiskit and validated across five diverse circuits including complex, random, and multi-qubit configurations. Experimental results show an average depth reduction of 33.33% and gate count reduction of 32.14%, with runtime improvement of up to 25%. For instance, an input circuit with a depth of 7 and gate count of 11 was reduced to a depth of 2 and 4 gates. All optimized circuits preserve functional correctness with a fidelityF≥0.99. High-resolution circuit diagrams are presented to visually demonstrate improvements before and after optimization. Additionally, global phase shifts such aseiπ/4are preserved or analytically characterized where relevant. This work enhances the viability of quantum computations on near-term hardware and opens pathways for future AI-driven quantum optimizations.
The rapid advancement of quantum computing has created new opportunities for solving computationally intensive problems across diverse fields such as optimization, physical system simulation, and machine learning. However, practical implementation on Noisy Intermediate-Scale Quantum (NISQ) devices remains challenging due to hardware limitations including noise, decoherence, and restricted qubit connectivity. A fundamental constraint further complicating this effort is that all quantum operations must be unitary, while many real-world problems are naturally expressed in non-unitary forms. This discrepancy necessitates novel techniques for transforming non-unitary mathematical structures into unitary or Hamiltonian representations compatible with quantum hardware.This thesis proposes a unified framework for converting structured, sparse, or otherwise non-unitary classical matrices into unitary forms that can be executed efficiently on NISQ devices. The methodology introduces general transformation strategies such as linear combinations of unitaries, structured embeddings, and resource-aware circuit construction, enabling the deployment of quantum algorithms without incurring prohibitive overhead. Each technique is developed with an emphasis on preserving problem structure, minimizing circuit depth, and ensuring compatibility with IBM’s NISQ hardware. The proposed frameworks are validated through theoretical analysis and quantum simulations, with performance benchmarks based on gate count, circuit fidelity, convergence behavior, and scalability. This work demonstrates how non-unitary formulations arising in various computational contexts can be systematically adapted for quantum execution, advancing the practical utility of NISQ-era devices.
Quantum computing, leveraging the principles of superposition and entanglement, holds the potential to revolutionize computational problem-solving across a wide array of fields, from cryptography to complex system simulations. However, unlocking the full power of quantum computing requires the development of advanced quantum compilation techniques that can efficiently map high-level algorithms onto physical quantum hardware. This paper delves into state-of-the-art optimization strategies within quantum compilers, with a particular focus on methods such as Relaxed Peephole Optimization (RPO), gate reordering techniques for the Quantum Approximate Optimization Algorithm (QAOA), and reinforcement learning (RL)-driven frameworks. It also explores innovative approaches like distributed quantum compilation, parity quantum optimization, and the Paulihedral framework, all designed to minimize gate counts, reduce circuit depth, and optimize overall resource utilization. A key contribution of this work is the introduction of error mitigation techniques that enhance the fidelity and scalability of quantum circuits, particularly on noisy intermediate-scale quantum (NISQ) devices. Through comprehensive empirical evaluations across a variety of quantum algorithms, the paper demonstrates significant improvements in critical performance metrics such as circuit depth, gate count, and execution efficiency, providing a promising direction for optimizing quantum circuit design tailored to diverse quantum hardware platforms.