Project-Based Learning in Introductory Quantum Computing Courses: A Case Study on Quantum Algorithms for Medical Imaging

TL;DR

This paper explores how Project-Based Learning enhances understanding of quantum algorithms in introductory courses by involving students in real-world interdisciplinary projects, demonstrated through a case study on quantum image reconstruction.

Contribution

It introduces a PBL approach for teaching quantum computing, integrating real-world applications and interdisciplinary collaboration to improve student engagement and comprehension.

Findings

PBL improves conceptual understanding of quantum algorithms.

Students gain practical experience and research skills.

Interdisciplinary projects enhance engagement and learning outcomes.

Abstract

Quantum computing introduces abstract concepts and non-intuitive behaviors that can be challenging for students to grasp through traditional lecture-based instruction alone. This paper demonstrates how Project-Based Learning (PBL) can be leveraged to bridge that gap. This can be done by engaging students in a real-world, interdisciplinary task that combines quantum computing with their field of interest. As part of a similar assignment, we investigated the application of the Harrow-Hassidim-Lloyd (HHL) algorithm for computed tomography (CT) image reconstruction and benchmarked its performance against the classical Algebraic Reconstruction Technique (ART). Through implementing and analyzing both methods on a small-scale problem, we gained practical experience with quantum algorithms, critically evaluated their limitations, and developed technical writing and research skills. The experience demonstrated that Project-Based Learning not only enhances conceptual understanding but also encourages students to engage deeply with emerging technologies through research, implementation, and reflection. We recommend the integration of similar PBL modules in introductory quantum computing courses. The assignment also works better if students are required to write and submit a conference-style paper, supported by mentorship from faculty across the different fields. In such course interdisciplinary, real-world problems can transform abstract theory into meaningful learning experiences and better prepare students for future advancements in quantum technologies.

Equations

$$A\vec{x}=\vec{b}$$

$$x^{(k+1)}=x^{(k)}+\lambda \frac{b_i-a_i{x}^{(k)}}{\Vert a_i{\Vert }^2}{a}_i^T$$

$$\sum _j{\beta }_j|v_j⟩|{\lambda }_j⟩$$

$$\sum _j{\beta }_j\frac{1}{{\lambda }_j}|v_j⟩|1⟩$$

$$|b⟩=\sum _{i=1}^n{b}_i|i⟩$$

$$|x⟩=\sum _{i=1}^n{x}_i|i⟩$$

$$(A^{\mathsf{T}}A+I)\mathbf{x}=A^{\mathsf{T}}\mathbf{b},$$

Full text

As part of a similar assignment, we investigated the application of the Harrow-Hassidim-Lloyd (HHL) algorithm for computed tomography (CT) image reconstruction and benchmarked its performance against the classical Algebraic Reconstruction Technique (ART). Through implementing and analyzing both methods on a small-scale problem, we gained practical experience with quantum algorithms, critically evaluated their limitations, and developed technical writing and research skills.

The experience demonstrated that Project-Based Learning not only enhances conceptual understanding but also encourages students to engage deeply with emerging technologies through research, implementation, and reflection. We recommend the integration of similar PBL modules in introductory quantum computing courses. The assignment also works better if students are required to write and submit a conference-style paper, supported by mentorship from faculty across the different fields. In such course interdisciplinary, real-world problems can transform abstract theory into meaningful learning experiences and better prepare students for future advancements in quantum technologies.

I Introduction

I-A Project-Based Learning

Project-based learning (PBL) [1] [2] is an inquiry driven, student-centred approach that challenges learners to tackle authentic problems and produce concrete outcomes. It is on John Dewey’s philosophy of “learning by doing” [3] and William Heard Kilpatrick’s early ”project method [4] which held that powerful learning grows from activities students genuinely care about [5]. Contemporary PBL rests on some constructivist principles [6]:

Context-specific learning [7]: Knowledge is constructed in real-world situations rather than through abstract drills. 2. 2.

Active involvement [8]: Students lead the process, posing driving questions, designing investigations, gathering and analyzing data, and iterating on solutions. 3. 3.

Social construction of understanding [9]: Learning emerges through collaboration, discussion, and the shared refinement of ideas.

This approach helps learners grasp complex concepts, cultivates positive attitudes toward the subject matter, and sharpens reasoning and critical-thinking skills. In practice, students move cyclically through problem identification, research, data collection, analysis, strategy development, and product creation. Each phase is tightly integrated with coursework and includes coordinated individual, group, and classroom activities aimed at fostering high-level thinking skills. [10] [11]

For beginners without a background in quantum mechanics, quantum computing may seem notoriously abstract. Students must juggle linear-algebraic formalisms, complex amplitudes, and non-intuitive circuit behavior that can feel detached from concrete experience. Recent research in quantum education shows that after traditional lecture-based instruction, many learners still cling to faulty “reasoning primitives” [12] (e.g. assuming an $N$-qubit computer simply upgrades every factor of $N$ by $2^{N}$ as compared to a classical computer) and struggle to visualize the vast state space available to a superposition.

We suggest that students in introductory courses on quantum computing may benefit from PBL assignments that require both individual research and technical writing. This paper provides an example of such an assignment used in a medical imaging course. We, the authors, found this to be enriching and recommend it for use in introductory Quantum Computing courses.

I-B Objectives

Guided by the challenges and opportunities outlined above, this study pursues a dual agenda: pedagogical and technical, framed by project-based learning:

Carry out a PBL module that bridges quantum computing and an arbitrary research field.

Develop a scaffolded learning sequence in which students can build and test a Quantum Algorithm or apply knowledge of Quantum Computing to a task related to an arbitrary research field. 2. 2.

Evaluate educational impact.

Assess the success of the PBL task by noting down any changes in the students’

1. (a)

conceptual understanding of the specific quantum computing primitives, 2. (b)

confidence in mapping domain problems to quantum formulations, and 3. (c)

ability to critique classical versus quantum solvers.

II Example of PBL Integration in a Course: Image Reconstruction in Medical Imaging

In this paper, we present as a model project, an exploration of Quantum Computing concepts applied to image processing. Specifically, we focus on an image reconstruction technique based on a quantum approach developed by Harrow, Hassidim, and Loyd named the Harrow-Hassidim-Loyd (HHL) algorithm. [13] We demonstrate image reconstruction based on the HHL technique and benchmark it against the Algebraic Reconstruction Technique (ART) [14] technique. (for a toy problem). These calculations reveal that Noisy Intermediate-Scale Quantum (NISQ) [15] era quantum computers are not yet capable of handling industrial scale image reconstruction problems. We discuss advances in technology required to make HHL a viable tool for MI applications. Finally, we conclude by highlighting the value of such a PBL assignment in the context of introductory quantum computing course. The assignment also included a technical writing component requiring students to prepare a formal conference-style paper and submit it to a conference of their choice.

II-A Introduction of the Project

The Algebraic Reconstruction Technique (ART) [14] is a widely used iterative method in Computed Tomography (CT) [16] for image reconstruction. ART reconstructs images by solving a system of linear equations [17] derived from X-ray projection measurements:

[TABLE]

Here, $x$ is the vector of unknown pixel or voxel intensities, $b$ is the measured projection data, and $A$ is the system matrix that encodes the geometry of the CT scanner and the imaging process. ART updates an initial image estimate by iteratively reducing the discrepancy between measured and simulated projections:

[TABLE]

where $a_{i}$ is the $i^{\text{th}}$ row of matrix $A$, $b_{i}$ is the corresponding measurement, and $\lambda$ is a relaxation parameter that controls the convergence behavior. ART is particularly effective in handling incomplete or noisy data. However, its iterative nature can be computationally intensive for large-scale imaging tasks. [14] [18]

Quantum computers leverage the principles of quantum mechanics to process information in fundamentally different ways than classical computers. They also offer exponential speedups for specific computational problems. In 2009, Harrow, Hassidim, and Lloyd introduced the HHL algorithm. [13] It is a quantum algorithm capable of solving linear systems with exponentially faster time complexity than classical methods.

HHL encodes the input vector $\vec{b}$ into a quantum state using amplitude encoding. This allows a $2^{N}\times 1$ vector to be represented using only $N$ qubits. Compared to other quantum algorithms like Quantum Phase Estimation (QPE) [19] and the Variational Quantum Eigensolver (VQE) [20], HHL achieves an exponential reduction in qubit requirements. This efficiency may make it attractive for large-scale applications such as medical imaging. [13]

The HHL algorithm presents a distinct and promising possibility of replacing ART in computed tomography. It has the ability to process large-scale linear systems efficiently. As quantum hardware improves, this opens the door to integrating HHL into real-world application for CT image reconstruction. This will also result in reducing computation time and improving the handling of high-resolution imaging data. While ART remains the current standard, HHL is presented as a candidate with the potential to replace CT in a quantum-enabled future.

HHL consists of three main quantum subroutines as presented in Fig. 1 and Fig. 2:

Quantum Phase Estimation (QPE): The eigenvalues $\lambda_{j}$ of $A$ are encoded into a quantum register using phase estimation on the unitary operator $e^{iAt}$. The input state $|b\rangle$, expressed in the eigenbasis of $A$, evolves into:

[TABLE] 2. 2.

Controlled Rotation [21]: An ancilla qubit is rotated based on $1/\lambda_{j}$. This step encodes the inverse of the eigenvalues into amplitudes:

[TABLE]

where success depends on measuring the ancilla in the $|1\rangle$ state. 3. 3.

Inverse-QPE and Post-selection [22]: Phase estimation is reversed (uncomputed), and the system collapses to a quantum state approximating the normalized solution $|\tilde{x}\rangle\propto A^{-1}|b\rangle$.

III Comparative Analysis of ART and HHL for CT Reconstruction

III-A Core Mathematical Foundation and Applicability

Both ART and HHL algorithms fundamentally solve the system of linear equations related to $Ax=b$. This represents the essential mathematical model in computed tomography. In this context, $x$ corresponds to the unknown voxel intensities of the medical image, $b$ is the measured projection data, and $A$ encodes the system geometry. ART is widely recognized for its capability in handling large, sparse matrices efficiently and performs robustly with noisy data that is commonly encountered in real-world CT scans. [23] Similarly, the quantum-based HHL algorithm is theoretically well-suited to large sparse systems. The condition for it is that the coefficient matrix is Hermitian  [13].

III-B Input Format and Preprocessing

ART operates directly with classical data obtained from projection measurements. The input vector $b$ is straightforwardly loaded into classical computational memory. Each component $b_{i}$ corresponds directly to an X-ray measurement and can be accessed individually or sequentially using standard computational resources. ART does not require special preprocessing or complex data structures making it highly flexible and practically efficient. [14]

In contrast, the HHL algorithm requires a fundamentally different way to input data. Rather than conventional arrays, HHL requires the vector $b=(b_{1},b_{2},\dots,b_{n})$ to be encoded directly into the amplitudes of quantum states. Specifically, HHL prepares the quantum state:

[TABLE]

where $|b\rangle$ is a quantum state consisting of $\log_{2}(n)$ qubits, with amplitudes encoding the classical input data $b$. This fact is shown in Fig. 3 In theory, encoding vector $b$ into quantum memory can be achieved by quantum RAM, capable of storing classical data values $b_{i}$ and loading them simultaneously into quantum superposition. The efficiency of encoding and subsequent usage within the HHL algorithm is sensitive to the distribution of entries in $b$. If the vector $b$ contains a few elements $b_{i}$ significantly larger than others, efficiently encoding and loading it into a quantum state becomes difficult. If quantum RAM is unavailable or impractical, quantum state preparation may rely on explicit classical preprocessing or quantum gate sequences. If preparation of the quantum state $|b\rangle$ requires $n^{c}$ computational steps for some constant $c$ then the exponential speedup promised by HHL effectively disappears at the initial input preparation stage itself.  [24]  [25]

III-C Runtime and Computational Complexity

ART typically exhibits a worst-case computational complexity of $\mathcal{O}(n^{3})$ particularly when large-scale iterative updates are involved. In practical implementations variants such as SART (Simultaneous ART) or regularized solvers reduce this burden. But ART still scales poorly with high-resolution volumetric data. The HHL algorithm offers a theoretical exponential speedup by solving linear systems in $\mathcal{O}(\log n)$ time under ideal conditions. According to Zaman, Morrell and Wong  [48], HHL accomplishes this speedup by leveraging quantum subroutines such as Quantum Phase Estimation (QPE) and Hamiltonian simulation which enable eigenvalue encoding and solution retrieval in the amplitude of quantum states.

III-D Output Format and Interpretability

In ART, the output format is straightforward and immediately interpretable. After reconstruction ART directly yields the solution vector $x=(x_{1},x_{2},\dots,x_{n})$, where each entry $x_{i}$ represents a voxel intensity within the reconstructed medical image. ART operates entirely within classical computing paradigms. So, the reconstructed vector $x$ can be easily visualized, interpreted, and utilized directly in clinical decision-making. Visualization tools, diagnostics, and analyses can immediately be applied to the output without additional complexity or significant post-processing. [26]

The HHL algorithm does not directly yield the classical solution vector $x$. Instead, it produces a quantum state:

[TABLE]

encoded within $\log_{2}(n)$ qubits. The amplitudes of this quantum state approximate the entries of the solution vector $x$. Thus, the algorithm’s solution exists only implicitly within quantum amplitudes and not explicitly as classical data. The user can only extract limited statistical information from this quantum state through quantum measurement. For example, measurements can identify positions of unusually large entries in the solution vector or estimate inner products $\langle x|z\rangle$ with predetermined vectors $z$. However, retrieving the precise numerical value of a specific entry $x_{i}$ generally requires repeated executions of the HHL algorithm for $n$ repetitions to achieve reliable estimates. This requirement for repetition effectively reduces the exponential computational advantage initially promised by the quantum algorithm. [24]

III-E Noise Tolerance and Robustness

ART is inherently robust to noise due to its iterative nature. In computed tomography noise arises from various sources including photon statistics and electronic fluctuations. ART addresses this by iteratively refining the solution vector $x$ to minimize the discrepancy between the measured projections and the projections calculated from the current estimate of $x$. This process allows ART to reduce the effects of noise in the data. ART can also incorporate prior knowledge and regularization techniques to enhance noise tolerance. Constraints such as non-negativity or smoothness can be applied to the solution to improve the quality of the reconstructed image in the presence of noise.[27] However, it’s important to note that excessive noise can still impact the convergence and accuracy of ART. This requires careful selection of relaxation parameters and stopping criteria. [28]

The HHL algorithm operates within the quantum computing paradigm where noise is generated differently compared to classical systems. Quantum noise such as decoherence and gate errors poses significant challenges to the practical implementation of HHL. The algorithm’s reliance on QPE and controlled rotations makes it particularly sensitive to such noise as these operations require high fidelity to maintain the integrity of the quantum state. Recent studies have explored the resilience of HHL to noise on Noisy Intermediate-Scale Quantum (NISQ)[29] devices. Findings indicate that current noise mitigation techniques are insufficient to fully counteract the impact of quantum noise on HHL’s performance. The algorithm’s sensitivity to noise requires the development of fault-tolerant quantum computing architectures and advanced error correction methods to realize its theoretical advantages in practical applications. [30]

III-F Hardware Readiness and Implementation

ART is well-established within classical computational frameworks. Its implementation relies on readily available computing hardware. GPUs have significantly accelerated ART by parallelizing matrix-vector computations, iterative updates, and back-projection operations. Modern computing infrastructures and widely accessible software libraries ensure that ART is both practical and scalable for routine clinical deployment. This is true even in high-resolution, large-scale CT imaging scenarios. ART faces minimal hardware limitations and has benefitted from decades of optimization in computational resources and technological advancements. [26]

In contrast, quantum computing hardware is currently at the early stages of development.[31] This can be seen in the significant limitations in qubit count, coherence times, and error rates. Real-world CT reconstruction tasks typically involve large-scale datasets with millions of variables and so require quantum computers with thousands to millions of logical qubits. Current quantum hardware is limited to around a hundred noisy physical qubits. This makes practical-scale CT reconstruction unfeasible at present. The HHL algorithm also requires precise quantum operations which demand highly reliable quantum gates and extended coherence times. [32] While significant research is ongoing in quantum hardware development and quantum error correction, substantial technological breakthroughs remain necessary before practical medical imaging applications can become viable.

III-G Flexibility and Customizability

ART is highly flexible and customizable. This makes it well-suited for diverse clinical and research applications. ART allows straightforward integration of constraints, and regularization techniques to enhance reconstruction quality. Parameters such as relaxation factors, stopping criteria, and regularization terms can be easily tuned to optimize performance for different imaging conditions and clinical scenarios. ART can also accommodate modifications tailored specifically for parse or noisy datasets which enables practitioners to adjust the reconstruction algorithm according to their specific clinical or research objectives. This flexibility ensures ART remains adaptable and effective across a wide range of CT imaging contexts. [60]

The flexibility and customizability of the HHL algorithm are currently limited by the underlying quantum hardware and algorithmic constraints. HHL could incorporate custom-designed quantum subroutines and tailored unitary transformations to handle specific linear system characteristics. However, practical limitations imposed by current quantum hardware significantly restrict its adaptability. [33] The requirement that the matrix $A$ be sparse, Hermitian, and well-conditioned places strict constraints on the types of problems that can be effectively solved by HHL. Customizing HHL for different problems also often requires developing specialized quantum circuits and advanced quantum state encoding methods which are currently challenging due to hardware constraints and quantum noise. [34]

III-H Future Potential and Research Directions

The future potential of ART revolves around incremental improvements rather than transformative breakthroughs. Research directions include further optimization of computational efficiency, enhancement of noise robustness, and refinement of regularization strategies. [35] Advanced machine learning methods, such as deep neural networks integrated with ART [36], represent promising directions.

While, the HHL algorithm presents transformative future potential driven by ongoing advancements in quantum computing. Key research directions include the development of fault-tolerant quantum computers capable of handling large-scale clinical datasets and mitigating quantum noise. Another critical direction involves creating efficient quantum data encoding and decoding techniques to bridge the gap between classical data acquisition and quantum processing. Hybrid quantum-classical frameworks also represent a promising research area. Such frameworks could leverage quantum algorithms like HHL for specific computational bottlenecks complemented by classical methods like ART [37] to handle more conventional tasks. Quantum-assisted feature extraction, quantum-enhanced regularization, and quantum-driven data compression also represent potential avenues through which HHL could enhance medical imaging significantly. [38]

IV Implementing the HHL algorithm for a small-scale image reconstruction

To examine how the promise of HHL translates into the CT domain, we built an end-to-end miniature experiment that places a classical ART solver and an HHL-based solver side-by-side on the smallest meaningful case [40]: reconstructing a $2\times 2$ phantom image [39]. A phantom is a synthetic image with known ground truth. The phantom image used in this test is shown in Fig. 5 The problem was run on the IBM Qiskit [41] [42] [43] environment in Python using a noiseless AerSimulator [44] backend.

In CT each pixel is never measured directly. Detectors record line-integral projections. Combined projection data collected at rotation angles are arranged into a two-dimensional array called a sinogram [45]. Each row records detector readings for one angle, so points in the object trace sinusoidal paths across the array. The sinogram is the raw dataset from which algorithms back-project and reconstruct cross-sectional CT images. The sinogram for the test is represented in Fig. 6 An established ART algorithm is used to reconstruct the images from this dataset, which is quite successful.

The implementation of the HHL algorithm was not so straightforward. A direct HHL application requires $A$ to be Hermitian and well-conditioned. Neither is true, so we solve the normal equations

[TABLE]

which are Hermitian, positive-definite, and exhibit required eigenvalues. Adding $I$ keeps eigenvalues away from 0. This improves robustness at the cost of a slight downward bias and removes the divide by 0 error we may face in HHL implementation (we encode $1/\lambda$ in the controlled rotation gates). The text block below shows the input matrices and how it is initialized in the HHL algorithm.

⬇

1 Projection matrix A (4x4):

2 [[1. 0. 1. 0.]

3 [0. 1. 0. 1.]

4 [1. 1. 0. 0.]

5 [0. 0. 1. 1.]]

6

7 Transpose A^T:

8 [[1. 0. 1. 0.]

9 [0. 1. 1. 0.]

10 [1. 0. 0. 1.]

11 [0. 1. 0. 1.]]

12

13 b (4x1):

14 [[4.]

15 [6.]

16 [3.]

17 [7.]]

18

19 Hermitian matrix A_herm = A4^T A4 + I:

20 [[3. 1. 1. 0.]

21 [1. 3. 0. 1.]

22 [1. 0. 3. 1.]

23 [0. 1. 1. 3.]]

24

25 b_herm = A^T b:

26 [[ 7.]

27 [ 9.]

28 [11.]

29 [13.]]

30

31 Normalized vector b_norm (used in state initialize):

32 [[0.34156503]

33 [0.43915503]

34 [0.53674504]

35 [0.63433505]]

HHL’s Quantum Phase Estimation (QPE) needs controlled unitaries [46] $e^{\mathrm{i}At}$. For the matrix the exponential is computed numerically, and embedded in controlled gates.

The final circuit implementation of HHL contains one ancilla qubit for eigenvalue reciprocals, three ”clock” qubits for QPE, and two system qubits for the four-pixel state vector. This is shown by qiskit’s circuit representation in fig. 7.

The algorithm then followed the following steps:

initializes the system qubits in the normalised vector, 2. 2.

performs QPE to entangle eigen-phases, 3. 3.

conditionally rotates the ancilla to encode $1/\lambda_{j}$, 4. 4.

reverses QPE to disentangle phases, 5. 5.

measures the ancilla, post-selecting successful inversions.

A state-vector simulation [47] then extracts the unmeasured amplitudes of the system register. The image is then reconstructed using the amplitudes of the system register.

A comparison of the original image, its ART reconstruction and the HHL reconstruction is shown in fig. 8. The reconstructed pattern is correct for ART and almost correct for HHL. However, this level of inaccuracy is not suitable for medical application as CT image reconstruction requires a very high level of precision that is not possible with modern day quantum hardware.

V Key Challenges Identified in the Practical Implementation of HHL

After a small-scale implementation of the HHL, the following key challenges have been identified:

V-A Quantum Data Encoding Complexity

Unlike ART, which operates directly on classical projection data, the HHL workflow first demands that the linear system be mapped to a quantum-compatible form. The coefficient matrix $A$ must be embedded in a Hermitian operator by forming the normal equations $(A^{\mathsf{T}}A+I)\mathbf{x}=A^{\mathsf{T}}\mathbf{b}$. Simultaneously, the right-hand-side vector must be amplitude-encoded on the system qubits as the circuit’s initial state. Engineering these controlled unitaries and preparing the input state generally requires $\mathcal{O}(n^{c})$ gate operations for realistic problem sizes, a scaling overhead that can erode HHL’s theoretical exponential speed-up when applied to large datasets. [48]

V-B Quantum Output Interpretability

HHL outputs solutions implicitly encoded in quantum states and each solution entry is represented as quantum amplitudes. Direct extraction of specific voxel intensities requires multiple repeated quantum measurements and algorithm executions. To accurately retrieve individual solution values approximately $n$ repetitions of the algorithm are necessary which significantly undermines the exponential advantage promised by quantum algorithms for classical data retrieval. In our toy $2\times 2$ study we sidestepped the issue by using a state-vector simulator, which grants direct access to the full wavefunction in a single run. Such global readout is infeasible on current quantum hardware.

V-C Quantum Noise and Robustness

The sensitivity of the HHL algorithm to quantum noise presents a substantial challenge. Quantum computing currently suffers from decoherence and gate operation errors which severely affects the performance of quantum subroutines needed for HHL. The reliance of HHL on precise quantum state manipulation magnifies its vulnerability to noise. This necessitates fault-tolerant quantum architectures and advanced error-correction techniques not yet available on practical scales.

V-D Hardware Limitations

Practical CT imaging tasks require quantum processors capable of handling millions of data points which translates to millions of logical qubits. Current quantum hardware is restricted to roughly a hundred noisy physical qubits [49] with limited coherence times. Moreover, the quantum operations integral to HHL (like controlled rotations and QPE) require high fidelity and low error rates which are currently unattainable on existing quantum hardware platforms.

V-E Algorithmic Flexibility Constraints

HHL’s practical adaptability is currently limited due to stringent algorithmic constraints. The requirement that the coefficient matrix $A$ must be sparse, Hermitian, and well-conditioned restricts the algorithm’s applicability to a relatively narrow range of CT scenarios. Customizing HHL for varied clinical applications involves specialized quantum circuits and complex encoding strategies which remain difficult under present hardware and noise constraints.

VI Potential Roles and Future Integration of HHL in Computed Tomography

Addressing the challenges through continued research and technological advancement is essential to realize the transformative promise of quantum computing in medical imaging. The HHL algorithm holds substantial promise for enhancing ART in computed tomography. While immediate full replacement of ART is improbable, targeted integration and specialized applications of HHL offer viable pathways for innovation and improvement within medical imaging workflows. The following section explores specific potential roles and integration scenarios for HHL in computed tomography despite the limitations.

VI-A Hybrid Quantum-Classical Reconstruction Models

Hybrid quantum-classical reconstruction models have emerged as a promising approach to integrating quantum computing capabilities into classical computational workflows. These are particularly relevant for computationally demanding tasks. These hybrid frameworks leverage the advantages of both classical and quantum computation and addresses the limitations imposed by current quantum hardware constraints.

Yalovetzky, Minssen, Herman, and Pistoia [50] introduced the Hybrid HHL++ algorithm, an advancement of the HHL quantum algorithm specifically tailored for execution on NISQ hardware. Hybrid HHL++ is designed to efficiently solve linear systems by combining classical preprocessing and optimization steps with quantum processing. Classical methods are utilized to determine optimal scaling factors and to perform circuit compression. This reduces the quantum resource overhead significantly. By doing this the hybrid approach mitigates challenges associated with qubit limitations, decoherence, and operational noise inherent in current quantum systems. Consequently, Hybrid HHL++ enhances the practical feasibility of quantum linear solvers, making it possible to address larger-scale linear systems than would be achievable using quantum computing alone.

Building upon a similar hybrid computational paradigm, Ye et al. [51] presented a quantum-classical framework designed for computational fluid dynamics (CFD) [52] applications. CFD may differ in application context from CT but the underlying computational methodology shares critical similarities including the requirement to efficiently solve large-scale linear systems. Their approach utilizes quantum algorithms to tackle specific computational subproblems within the classical iterative reconstruction pipeline. In this model quantum algorithms, including variants of HHL, solve key sub-tasks involving linear algebra operations that are computationally intensive for classical solvers. Classical algorithms subsequently handle iterative refinement, convergence control, and result interpretation. This hybridization approach capitalizes on quantum advantages and also ensures robustness and practicality by combining classical and quantum strengths.

Integrating these insights into computed tomography reconstruction indicates a viable and beneficial direction. Hybrid quantum-classical methods could significantly improve reconstruction performance by selectively employing quantum algorithms to rapidly solve critical linear systems derived from projection data. Classical components of the reconstruction pipeline would continue managing data preprocessing, noise handling, iterative refinement, and image interpretation. This complementary integration can potentially yield faster convergence, higher reconstruction accuracy, and more efficient handling of large datasets common in medical imaging applications.

VI-B Efficient Representation of Classical Data in Quantum Computation

Encoding classical data into quantum states efficiently is essential for the practical implementation of HHL. As discussed by Mitarai, Kitagawa, and Fujii [53], classical data can be encoded in quantum systems via two approaches: analog and digital encoding. In analog encoding, data is stored as amplitude coefficients of basis states that allow for compact representation of large vectors using fewer qubits. This method is used in HHL to encode the input vector $\vec{b}$ and enables exponential compression. Digital encoding stores classical data as binary strings across quantum registers. This format is more suitable for performing arithmetic operations and is widely used in quantum optimization and machine learning applications. Algorithms often require transformation between these encoding types to leverage both advantages.

They propose two key techniques for this purpose. Quantum Digital-to-Analog Conversion (QDAC) [53] transforms a digitally encoded quantum state into an analog format using controlled rotations. This approach is probabilistic and can be amplified through amplitude amplification. Quantum Analog-to-Digital Conversion (QADC) converts amplitude information from an analog state into digital bitstrings using swap tests, phase estimation, and quantum arithmetic. This process is deterministic and enables the digital extraction of the real, imaginary, or absolute values of amplitudes. These conversions are crucial for enabling nonlinear transformations on quantum states and are necessary for advanced quantum algorithms and quantum machine learning. They also enable flexible preprocessing of classical input data for use in quantum linear solvers like HHL. Such encoding allows efficient loading and manipulation of high-dimensional projection data within quantum circuits and helps in improving integration between quantum and classical workflows.

VI-C Enhancement through Quantum Error Correction (QEC) [54]

The deployment of HHL on NISQ hardware is hindered by sensitivity to quantum noise. QPE and Hamiltonian simulation are particularly vulnerable to decoherence and gate imperfections. In their detailed analysis, Phillips [55] examine how eigenvalue approximation errors introduced during QPE propagate through the algorithm. Their results demonstrate that small errors in estimating eigenvalues $\lambda_{j}$ lead to significant entanglement between the phase and flag registers and reduce the fidelity of the post-selected solution state. The deviation from the ideal output state is bounded by $\mathcal{O}(\kappa/t_{0})$, where $\kappa$ is the condition number and $t_{0}$ is the simulation duration. This relationship underscores the precision-resource tradeoff in practical implementations.

QEC provides a robust framework to overcome these limitations. By encoding logical qubits in error-resilient subspaces, QEC can protect the HHL circuit from bit-flip, phase-flip, and more general errors. When combined with fault-tolerant quantum gates this enhances the reliability of deeper circuits required for fine-grained eigenvalue resolution. QEC can also enable larger clock register sizes $T$ and longer simulation times $t_{0}$. Both of them are crucial for reducing approximation error in the QPE step. As quantum hardware evolves, the integration of QEC into hybrid quantum-classical HHL pipelines will likely be essential for realizing the algorithm’s full potential in scientific computing and imaging applications.

VI-D Rapid Advancement in Quantum Computation

Quantum computing has rapidly evolved from theoretical constructs into an experimental and increasingly practical discipline and pushing the boundaries of what is computationally possible. As outlined by Memon, Al Ahmad, and Pecht [56], this field is undergoing transformative advancements across hardware platforms, enabling scientists and engineers to inch closer toward quantum advantage—solving problems intractable for classical computers. Key breakthroughs are being made across various quantum hardware platforms. Superconducting qubits which are one of the most mature architectures have seen improvements in scalability, error mitigation, and cryogenic infrastructure. Companies like IBM [42] and Google [57] have demonstrated multi-qubit systems that maintain coherence for longer durations enhancing circuit depth and algorithmic fidelity. Trapped-ion qubits [58] offer another promising path. They boast exceptionally high-fidelity gates and long coherence times. Recent innovations have minimized the complexity of their control systems, improving their practicality for larger-scale implementations. Photonic qubits are also being explored for room-temperature quantum computing. [31] The convergence of AI, quantum simulation, and quantum machine learning [59] is shaping the development of new applications in this sector. Such advancements in quantum computation are essential for implementing algorithms like HHL in medical imaging, a domain that is both computationally intensive and demands high accuracy with minimal tolerance for error.

VII Educational Insights and Recommendations

By implementing the complete HHL algorithm for a small scale CT reconstruction, we turned abstract quantum-computing ideas into concrete tasks. We wrote, debugged, and benchmarked every subroutine ourselves. This allowed us to shift the basic concepts of quantum computing, specifically the HHL algorithm, from theory to experience. We also learned problem solving by learning to dive into primary literature when documentation failed. This also allowed us to judge if the advertised quantum speed-up is neutralized by data-loading overhead.

For an introductory quantum computing course, introducing the assignment early in the semester seems to be highly beneficial. This approach allows students to engage with fundamental concepts in the context of their own meaningful, practical project. This can also help to reinforce ideas as they are learned. It also provides ample time for iteration, reflection, and deeper understanding. This also keeps students motivated by connecting theoretical knowledge with hands-on application throughout the course. For instance, our own project choice enabled us to focus specifically on essential quantum computing concepts such as Quantum Phase Estimation (QPE) and the Quantum Fourier Transform (QFT) which are critical subroutines within the HHL algorithm. Qiskit remains the most accessible platform because it pairs a Python interface with exact state-vector simulation and a growing set of pedagogical notebooks.

We noted that the assignment works best when presented as a narrative rather than a checklist. Framing the project as “using quantum algorithm for CT scans” instantly motivates the students to push through any hardships that follows. Providing a minimal but functioning skeleton code lowers the activation energy. One of the main hurdles we faced was a lack of existing code, not even in a small-scale. Providing a basic code allows students to spend their cognitive capital on conceptual hurdles.

We recommend incorporating a conference-style paper writing component too into project-based assignments, as in our case. For graduate students, it provides structured practice in communicating complex technical ideas clearly in formal academic applications. It also introduces them to peer-reviewed formatting standards, citation practices, and the process of preparing work for submission to a real conference. For undergraduate students, the experience serves as an introduction to academic research and scientific writing. It fosters ownership of their learning and familiarizes them with the expectations of scholarly communication. In our experience, we found that the writing component not only reinforced the conceptual understanding but also deepened our appreciation of the broader implications and relevance of the work.

We also emphasize the need for a good interdisciplinary mentorship in project-based assignments that span multiple domains. In our project, since the topic bridged two different disciplines, support from faculty across both domains was absolutely necessary. Beyond technical instruction, mentors from both fields should ideally provide support with academic writing, especially regarding how to adapt technical content for a research audience. This dual-mentorship model not only enhances student outcomes but also prepares them for the collaborative, interdisciplinary nature of real-world scientific research.

In sum, the assignment’s blend of project implementation, and technical writing activates a range of skills that traditional lecture-based courses rarely address. Deployed at the right moment in the semester, with scaffolding that balances guidance and autonomy, it can become a signature learning experience that anchors an introductory quantum computing curriculum.

VIII Conclusion

This paper highlights the value of project-based learning (PBL) as an effective approach for teaching quantum computing concepts. By engaging with a real-world problem, we were able to move beyond theoretical understanding and actively explore how quantum algorithms perform in practice. The process of researching, implementing, and critically evaluating a quantum solution within a familiar context fostered deeper conceptual understanding, problem-solving skills, and technical communication abilities.

The assignment, using HHL for image reconstruction, demonstrated how PBL encourages learners to confront authentic challenges, from handling quantum data encoding to interpreting algorithmic limitations. More importantly, it provided a framework where students could develop critical thinking by assessing when quantum approaches are advantageous and when classical methods remain superior. We learned that ART remains a robust, practical method for CT reconstruction. However, quantum computing, especially the HHL algorithm, offers transformative potential through faster linear system solutions. Despite current hardware and implementation challenges, hybrid quantum-classical approaches may soon enhance CT pipelines. As quantum technology matures it promises to advance medical imaging’s computational foundations.

Our experience suggests that integrating structured PBL modules into introductory quantum computing courses can transform passive learning into active learning. By framing assignments around interdisciplinary applications educators can provide context-driven education. This enhances engagement and retention among students. Moreover, tasks that combine project implementation, and technical writing cultivate a broad skill set rarely achieved through traditional lecture formats. Additionally, we recommend that educators consider incorporating a formal writing and submission component alongside interdisciplinary mentorship, as these elements greatly enhance students’ technical communication skills and expose them to authentic research practices.

In conclusion, this project serves as a model for how PBL can be leveraged to teach quantum computing more effectively. We recommend that educators adopt similar approaches, where students are challenged to apply quantum principles to real-world problems, encouraging both technical mastery and critical evaluation.