Recent years have seen a proliferation of research proclaiming the utility of quantum machine learning (QML) algorithms for analyzing classical data in many sectors, e.g. finance, cybersecurity, logistics, pharmaceuticals, energy, minerals, and healthcare. With the increasing digitization of health data, the growth of electronic health and medical records1 paves the way for the use of algorithmic techniques – quantum or classical – for analyzing this data. Potential digital health applications could include clinical decision support, clinical predictive health and health monitoring, public health applications and improving health services delivery and data fusion2,3,4,5. The potential for use-case discovery for QML in healthcare6 and biomedical7 applications is found to be compelling in previous systematic reviews. Other broader reviews on quantum computing for health, biology and lifesciences8,9,10,11,12,13,14,15 hypothesize the potential utility of QML algorithms or quantum subroutines in health, but none of these works are rigorous systematic reviews (and thus reproducible). Indeed, across all of these standard and systematic reviews, we find that the strength of the current evidence base even under mildly realistic operating conditions is not examined.
Characterizing the role of QML algorithms applied to real-world classical data is nuanced and a challenging question in applications development but also in fundamental QML theory16,17. Quantum advantage refers to asymptotic reduction in computational resources (or some other metric18) required by quantum algorithms when compared to classical counterparts, i.e. resources are saved as problem size scales to infinity. Empirical quantum advantage19 colloquially refers to finite-sized simulations or experiments using quantum over classical algorithms to perform a task, where one assumes any desired resource savings will scale to larger problems, e.g. in qubit number, high-dimensional or highly structured datasets. However, for classical datasets of arbitrary structure such as those encountered in healthcare settings, there is no known theoretically provable quantum advantage18. Instead, the field relies on mostly empirical analysis of QML performance for a variety of pseudo-real-world data, where performance differentials between quantum and classical methods on these smaller problems constitute evidence for testing empirical quantum advantage. Most computational analysis of scaling behavior assumes ideal operating conditions and it is unknown if QML methods will retain any benefits in realistic operating settings, such as on near-term noisy quantum hardware. In some cases, the role of quantum algorithms for solving inference tasks has been entirely replaced by equivalent classical capability, in a process known as dequantization (e.g.20,21).
In this work, we undertake a systematic literature review of QML applications in digital health between 2015 and 2024. As typical in medical research settings, a systematic literature review is a standard methodological approach for assessing the strength of evidence for proposed interventions in clinical contexts and public health22. Based on existing evidence in literature, we use the SPICE framework23 to ask: In developing digital health technologies, could quantum machine learning algorithms potentially outperform existing classical methods in efficacy or efficiency? A systematic review was conducted in line with the PRISMA (Preferred Reporting Items for Systematic Reviews and Meta-Analyses)24 (Supplementary Note 2) detailed in Methods. Our methodology assesses the strength of the evidence and dominant trends associated with using QML algorithms for digital health, including assessing the extent to which performance robustness of proposed QML algorithms has been characterized.
Our current-state analysis reveals that the empirical evidence for QML in digital health cannot conclusively address our research question. We find that numerous studies had to be excluded due to a lack of technical rigor in their analysis of QML algorithms. The majority of eligible studies use only ideal simulations of QML algorithms, thereby excluding the resource overhead incurred for error-mitigated or error-corrected algorithms required for noisy quantum hardware. Of high quality studies, nearly all QML algorithms are found to be linear quantum models, and therefore represent a small subset of general QML. Most use-cases in digital health focussed on providing clinical support, and no studies considered health service delivery or public health applications. Only two synthesized studies used electronic health records for quantum machine learning applications, while the remaining studies repeatedly gravitated towards a handful of open-source health databases. Finally, 13 studies used quantum hardware demonstrations and separated into two classes: either algorithms for a gate-based, universal quantum computer using up to 20 qubits, or quantum annealers using O(100) qubits. Whether potential advantages of QML can be retained in the presence of noise is largely unaddressed in all studies.
We devote the remainder of this Introduction to providing an overview of quantum machine learning for those unfamiliar with this domain. We will also briefly discuss performance metrics, properties of different families of quantum machine learning algorithms, techniques for encoding data into quantum states, and data pre-processing. Quantum algorithms refer to a broad category of algorithms, for which it is desired that quantum computing hardware will be required to perform some of the computations. We distinguish these quantum algorithms from quantum-inspired classical algorithms that use insights from quantum mechanics to perform computations on classical computers. Quantum machine learning algorithms are a subset of quantum algorithms. For the scope of this review, a quantum machine learning algorithm takes as input a classical dataset, and an inference problem is defined on the classical dataset.
Much of the literature we encountered in our review discussed the potential benefits of using quantum machine learning techniques in lieu of classical methods to analyze health data. The terminology used to communicate these benefits is often ill-defined, e.g. quantum ‘speed-up’, ‘utility’ or ‘advantage’ are used interchangeably. In QML, computational ‘advantage’ accrues when a QML algorithm can reduce the number of operations required to solve this inference problem as the size of the problem becomes asymptotically large. Here, the problem size is typically associated with features of the input data e.g. with input data dimension. From a computer science perspective, algorithms can either improve on the number of queries or samples required (sample complexity) or the number of parallelizable quantum operations (time complexity, or runtime). When quantum algorithms enable improvements in complexity, this is sometimes referred to as ‘quantum advantage’, while ‘speed-up’ is often reserved only for reduction in time complexity. An additional metric of memory complexity quantifies the size or type of data structures required to efficiently store and recall intermediary information during computation. While memory complexity is typically not discussed in the literature for quantum algorithms, subroutines such as QRAM may play an analogous role. A comparison of computational costs of selected classical vs. quantum algorithms for ideal mathematical regimes can be found in ref. 11, but these were not encountered for real-world health data in our review.
Quantum algorithms separate into two different categories in this review: gate-based quantum models, or quantum annealing. This categorization can broadly reflect the difference between digital and universal vs. analog and non-universal quantum computing. While we provide a high-level summary of classes of quantum algorithms that were encountered in this review, it cannot be construed as a comprehensive overview of quantum machine learning (see for example, refs. 25,26). Background quantum notation and a fuller discussion is provided in Supplementary Note 1. The majority of studies in the review focussed on quantum algorithms designed for gate-based universal quantum computers. These algorithms include quantum kernel methods (including quantum support vector machines), quantum neural networks, quantum convolutional neural networks, and quantum deep learning. We summarize the quantum computational steps in these protocols by considering how outputs are generated from inputs in Fig. 1 by representing these steps as quantum circuits.
Fig. 1: Common models in quantum machine learning.
Quantum circuit depictions of linear vs. non-linear embedding in quantum models27. Horizontal wires represent qubits where input states are shown as ket \(| \cdot \left.\right\rangle\) symbols; temporal order of computations progresses from left to right. Boxed quantum gates (blue, orange) are reversible rotations, or `unitary gates’, of quantum states, i.e. U† = U−1. If data encoding (blue) is separable from variational gates (orange), then the quantum model is linear. ‘Circuit size’ refers to the number of qubits, while ‘circuit depth’ represents the number of time-steps required to run the full circuit assuming that quantum operations on disjoint qubits have been parallelized. a–c Measurements (msmts.) are pushed to the end; quantum circuit can be summarized by a unitary operation. d Mid-circuit measurement outcomes change quantum operations ‘on the fly’ (pink). Circuits with tunable θ (blue gates) can be broadly referred to as variational (VQC) or parameterized (PQC) quantum circuits.
In the circuit visualization of Fig. 1, inputs to a quantum algorithm are qubit states denoted with ket-notation \(| \cdot \left.\right\rangle\) and boxed operations denote quantum gates. These gates are associated with reversible, logical operations performed on quantum states. The circuit is terminated with measurements of a quantum state which yield probabilistic outcomes, ‘0’ or ‘1’, where probabilities are determined by the quantum circuit. Suppose for some input quantum state, ρ0, the average output of a quantum computation is given by f(x, θ) where (x, θ) define classical inputs to a quantum algorithm. Here, ρ0 represents an input state, such as all qubits in their ground (zero) state; x represents one sample of real data with dimension d, \(x\in {{\mathbb{R}}}^{d}\) for a dataset with N samples, and tunable free parameters, θ, that parameterize the circuit. One encodes data x into quantum states using a parameterized quantum gate, denoted U(x). Meanwhile, free parameters, θ, implement classically optimized or trained quantum gates, V(θ). With these assumptions, the desired output information required from the circuit is typically given with reference to an observable quantity, \(\hat{O}\). This output information is inherently statistical, i.e. one must infer average information about \(\hat{O}\) from a statistical ensemble of ‘0’ or ‘1’ measurements obtained by repeatedly preparing and measuring the same quantum circuit Ns number of times. Therefore to extract information about \(\hat{O}\), we build up an ensemble of quantum measurements by repeatedly running a quantum circuit Ns number of times for a single instance of x, and repeating for different choices of x.
A quantum machine learning algorithm typically consists of input data (x-dependent) and tunable (θ-dependent) quantum operations. Using Supplementary Note 1, we can write the general output of a QML algorithm as,
$$f(x,\theta ):= {\rm{Tr}}\left[U(x,\theta ){\rho }_{0}{U}^{\dagger }(x,\theta )\hat{O}\right]=\langle {\rho }_{x,\theta },\hat{O}\rangle ,$$
(1)
where the data (x-dependent) and tunable (θ-dependent) components of the quantum state ρx,θ cannot be separated. In equation (1), U(x, θ) represents a parameterized quantum gate which depends on data x and tunable parameters θ. The output of a QML algorithm thus computes the overlap between information in the quantum state ρx,θ = U(x, θ)ρ0U†(x, θ), and the desired output \(\hat{O}\), using an inner product. In contrast, linear quantum models allow us to separate the x-dependent quantum operations and θ-dependent quantum operations within the inner product27. In these models, we perform data encoding operations followed by tunable gates V(θ). As shown in Fig. 1(a), a linear quantum neural network (QNN) can be expressed by,
$$f(x,\theta ):= {\rm{Tr}}\left[V(\theta )U(x){\rho }_{0}{U}^{\dagger }(x){V}^{\dagger }(\theta )\hat{O}\right]=\langle {\rho }_{x},{\hat{O}}_{\theta }\rangle .$$
(2)
In equation (2), θ can take the form of any other classical parameters that are not x; data encoding is expressed by ρx ≔ U(x)ρ0U†(x), and the parameterized neural net is expressed as \({\hat{O}}_{\theta }:= {V}^{\dagger }(\theta )\hat{O}V(\theta )\). We note that the embedding U(x) can be nonlinear transformation of the input data, x. However, the terminology ‘linear’ quantum model refers to the linearity of the model with respect to the embedding, i.e. data-dependent and parameterized components of the quantum algorithm can be separated as shown above27.
With this structure, we can additionally describe many other types of quantum machine learning algorithms. For example, we can omit θ entirely, and recover sophisticated algorithms that focus on data encoding procedures. In quantum kernel methods (QKMs), θ is replaced by training data, and the algorithm output f during prediction represents a linear combination of all training samples. Sometimes the action of ρ, U(x) or V(θ) is non-trivially restricted to some subset of quantum states. Quantum convolutional neural networks (QCNNs), quantum generative adversarial networks, quantum causal modeling, quantum transformers, and quantum deep reinforcement learning all have regimes in which they reduce to linear quantum models of the form in Eq. (2) as discussed in Supplementary Note 1.
Meanwhile, quantum annealing algorithms assume a very specific type of quantum computing hardware, namely adiabatic computers, (e.g. D-Wave) to solve specific learning tasks. Adiabatic quantum computers can approximately solve computationally hard (i.e. ‘NP-hard’) problems28 including approximately solving combinatorial optimization problems. The main class of problems encountered in this review relates to quadratic unconstrained binary optimization (QUBO). Examples of QUBO optimization problems include regression, classification, and data compression tasks. Classical, quantum and hybrid annealers can all approximately solve QUBO optimization problems29, or be used to draw samples from particular types of probability distributions (e.g. Boltzmann distributions)30. While more general forms of adiabatic quantum computing than annealing techniques do exist, we did not encounter any within our included literature, and for this reason have not included a discussion of this form of learning.
Quantum algorithms for QUBO formulations have provable advantage over classical counterparts in some regimes. Quantum QUBO algorithms for optimizing support vector machines (SVMs) and balanced k-means clustering have better computational complexities compared to classical counterparts, while quantum algorithms for QUBO formulations of regression have equivalent computational complexity to classical algorithms28. For this limited class of problems, quantum adiabatic computers, such as D-Wave 2X processors, can access ≈ 1000 qubits, which is an order of magnitude larger than O(100) qubit processors for universal non-annealing quantum computers developed by IBM and Google. We also note that it is possible to realize quantum annealing tasks on gate-based quantum computers, e.g. ref. 31, and therefore our classification represents one choice of a non-exclusive method for framing the discussion of QML algorithms.
So far we have introduced quantum machine learning algorithms in generality without reference to the dataset under consideration. However, characteristics of classical data and the representation of this data in a quantum algorithm can affect potential attainability of computational advantage in solving inference tasks32,33. Data encoding describes the process of representing classical data as quantum states, such as the choice of a data encoder, U(x), in Fig. 1. Data encoding is required for both annealing and non-annealing quantum algorithms. Ideally, data encoders must be efficient in computational complexity in both circuit size (number of qubits) and circuit depth (number of parallel operations). There are a number of ways to embed classical data x in quantum states, as summarized in Table 1. For continuous variable inputs, one may use binary representation of data to finite precision τ and encode using discrete methods such as basis encoding, as reported in Table 1. The growth of the number of computations required for encoding is mathematically expressed in \({\mathcal{O}}(g(n))\)-notation to express an upper bound g(n) on the number of operations as the argument n goes to infinity, ignoring constant multiplicative or additive factors. As an example from Table 1, angle-encoding can be prepared in constant depth but scales linearly with number of qubits. The trade-off is switched for amplitude encoding, which in general scales linearly with runtime and logarithmically with qubit number.
Table 1 Comparison of data encoding strategies
Hardware-specific considerations can change implementation details of a quantum algorithm. The decomposition of required quantum operations to the native set of quantum gates available on hardware may change the number of operations, e.g. replacing one 2-qubit gate with a decomposition involving several single and 2-qubit gates. Similarly, hardware implementation of any continuous variable often also incurs finite precision. In most cases, these changes are multiplicative or additive with problem size. These multiplicative or additive changes do not affect the overall asymptotic scaling behavior of the encoder. Some data encoders are not intended as a near-term, implementable strategy. For example, quantum random access memories (QRAM)34 use an additional m = O(dτ) ancillary qubits to randomly access superpositions of basis-encoded states in favorable logarithmic \(O(\log (m))\) time. However, robust QRAMs remain extremely challenging to implement on hardware35. Finally, the parallel unary encoder assumes specific hardware capabilities that affect the complexity of data encoding, and we return to this point in the Discussion.
Distinct from the choice of data encoding strategy, data pre-processing is concerned with using classical techniques to clean up, rescale, compress, or transform data. Since data encoding is expensive in quantum resources, and may impact performance, raw data is pre-processed before encoding data in quantum states. Data pre-processing can have many goals, e.g. to compress raw data, identify key features, or address missing values. For most near-term demonstrations of QML, it is well known that dimensionality reduction of classical datasets is often required to encode data into small or intermediate scale quantum circuits. However, the potential impact of data pre-processing on comparisons of quantum vs. classical algorithm performance is not addressed in literature.
We now turn to presenting our key results and the methodology for our review. The structure of this document is as follows. We begin by conducting meta-analysis and synthesizing the empirical evidence for all eligible studies in Results. We comment on the extent to which this evidence base addresses our research questions, and discuss limitations and future outlook in Discussion. Details of our systematic review methodology, including a study quality assessment framework and comprehensive search and screening criteria, is outlined in Methods.