3.1. Quantum linear algebra solvers

Quantum linear algebra solvers make use of the fact that quantum computation using $ N $ qubits is mathematically described by linear operators in vector spaces of (large) dimension $ {2}^N $ . Linear problems of size $ M $ can be encoded in quantum states and operators using only logarithmically many qubits (in $ M $ ). Provided this encoding, quantum solvers offer exponential improvements in terms of resources needed to solve a problem, requiring only $ \mathcal{O}\left(\mathrm{poly}\log M\right) $ qubits and gates compared to the $ \mathcal{O}\left(\mathrm{poly}M\right) $ operations required for classical algorithms (Harrow et al., Reference Harrow, Hassidim and Lloyd2009). They can therefore significantly speed up large matrix operations, e.g., in the resolution of (partial) differential equations using finite difference/elements methods (Berry, Reference Berry2014; Lloyd et al., Reference Lloyd, Palma, Gokler, Kiani, Liu, Marvian, Tennie and Palmer2020; Li et al., Reference Li, Yin, Wiebe, Chun, Schenter, Cheung and Mülmenstädt2023). This exponential speedup depends on the ability of efficiently (i.e., with costs scaling polynomially in the number of qubits) encoding the problem data in the states and operators on the quantum device, and on the efficient readout of the properties of interest from the quantum state encoding the solution. These challenges, together with the limitations due to noise, restrict the current applicability of these routines to small-scale problems (Cai et al., Reference Cai, Weedbrook, Su, Chen, Gu, Zhu, Li, Liu, Lu and Pan2013; Barz et al., Reference Barz, Kassal, Ringbauer, Lipp, Dakić, Aspuru-Guzik and Walther2014; Zheng et al., Reference Zheng, Song, Chen, Xia, Liu, Guo, Zhang, Xu, Deng, Huang, Wu, Yan, Zheng, Lu, Pan, Wang, Lu and Zhu2017).

3.2. Quantum annealing for optimization problems

Quantum annealing offers a way of solving optimization problems on quantum devices (Das and Chakrabarti, Reference Das and Chakrabarti2008; Albash and Lidar, Reference Albash and Lidar2018). Quantum annealers address combinatorial problems with a discrete solution space. The solution to a given problem is encoded in the ground state of an Ising Hamiltonian (Das and Chakrabarti, Reference Das and Chakrabarti2008). This Hamiltonian is then realized on a quantum device, the annealer, and its ground state prepared by slowly steering an initial state towards it. In this way, the implemented quantum state globally explores the optimization landscape before tunnelling towards the optimal solution. Quantum annealers consisting of thousands of qubits are already available for use in academic and industrial applications (Yarkoni et al., Reference Yarkoni, Raponi, Bäck and Schmitt2022). This approach is potentially scalable to large problems, and the first claims of quantum advantage with quantum annealers have been made (King et al., Reference King, Nocera, Rams, Dziarmaga, Wiersema, Bernoudy, Raymond, Kaushal, Heinsdorf, Harris, Boothby, Altomare, Berkley, Boschnak, Chern, Christiani, Cibere, Connor, Dehn, Deshpande, Ejtemaee, Farré, Hamer, Hoskinson, Huang, Johnson, Kortas, Ladizinsky, Lai, Lanting, Li, MacDonald, Marsden, McGeoch, Molavi, Neufeld, Norouzpour, Oh, Pasvolsky, Poitras, Poulin-Lamarre, Prescott, Reis, Rich, Samani, Sheldan, Smirnov, Sterpka, Clavera, Tsai, Volkmann, Whiticar, Whittaker, Wilkinson, Yao, Yi, Sandvik, Alvarez, Melko, Carrasquilla, Franz and Amin2024). However, noise is an open problem for quantum annealers (also due to the lack of fully fault-tolerant quantum annealing schemes (Pudenz et al., Reference Pudenz, Albash and Lidar2014)).

3.3. Parameterized quantum circuit models

Parameterized quantum circuits (PQCs) are sequences of quantum gates depending on tunable parameters $ \boldsymbol{\theta} $ that are optimized for the quantum device to solve a given problem. Applications include variational quantum eigensolvers (Peruzzo et al., Reference Peruzzo, McClean, Shadbolt, Yung, Zhou, Love, Aspuru-Guzik and O’Brien2014), quantum approximate optimization algorithms (Farhi et al., Reference Farhi, Goldstone and Gutmann2014), and quantum machine learning (QML) (Schuld and Petruccione, Reference Schuld and Petruccione2018; Schuld et al., Reference Schuld, Sinayskiy and Petruccione2015; Biamonte et al., Reference Biamonte, Wittek, Pancotti, Rebentrost, Wiebe and Lloyd2017; Dunjko and Briegel, Reference Dunjko and Briegel2018; Schuld and Petruccione, Reference Schuld and Petruccione2018; Cerezo et al., Reference Cerezo, Verdon, Huang, Cincio and Coles2022). PQCs are NISQ-friendly due to their limited depth, supplemented by the optimization of the parameters $ \boldsymbol{\theta} $ that is achieved iteratively in a hybrid quantum-classical manner (Cerezo et al., Reference Cerezo, Arrasmith, Babbush, Benjamin, Endo, Fujii, McClean, Mitarai, Yuan, Cincio and Coles2021; Bharti et al., Reference Bharti, Cervera-Lierta, Kyaw, Haug, Alperin-Lea, Anand, Degroote, Heimonen, Kottmann, Menke, Mok, Sim, Kwek and Aspuru-Guzik2022): a cost function is measured on the quantum device and fed to a classical routine that proposes new $ \boldsymbol{\theta} $ for the next iteration. This results in shorter and classically optimized circuits that can run within the coherence time of NISQ devices.

In the context of QML, PQCs find applications in regression, classification, and generative modeling tasks (Cerezo et al., Reference Cerezo, Verdon, Huang, Cincio and Coles2022). In regression and classification, PQCs are used as function approximators and are often referred to as quantum neural networks (QNNs) (Farhi and Neven, Reference Farhi and Neven2018). Classical input data $ \boldsymbol{x} $ first is encoded in a quantum state $ \mid \phi \left(\boldsymbol{x}\right)\Big\rangle $ . Then the output is calculated as the expectation value of an observable in the output state $ \mid \psi \left(\boldsymbol{x};\boldsymbol{\theta} \right)\left\rangle ={\hat{U}}_{\boldsymbol{\theta}}\mid \phi \left(\boldsymbol{x}\right)\right\rangle $ , where $ {\hat{U}}_{\boldsymbol{\theta}} $ denotes the action of the PQC. The embedding $ \boldsymbol{x}\to \mid \phi \left(\boldsymbol{x}\right)\Big\rangle $ needs to be carefully chosen as it strongly influences the model performance (Schuld and Petruccione, Reference Schuld and Petruccione2018; Pérez-Salinas et al., Reference Pérez-Salinas, Cervera-Lierta, Gil-Fuster and Latorre2020; Schuld et al., Reference Schuld, Sweke and Meyer2021), and potentially requires compression strategies (Dilip et al., Reference Dilip, Liu, Smith and Pollmann2022). The higher expressivity of QNNs (Du et al., Reference Du, Hsieh, Liu and Tao2020, Reference Du, Hsieh, Liu, You and Tao2021; Yu et al., Reference Yu, Chen, Jiao, Li, Lu, Wang and Yang2023b), makes them interesting ansatzes for ML tasks. Furthermore, several works have investigated their generalization properties, with promising results (Cong et al., Reference Cong, Choi and Lukin2019; Abbas et al., Reference Abbas, Sutter, Zoufal, Lucchi, Figalli and Woerner2021; Caro et al., Reference Caro, Huang, Cerezo, Sharma, Sornborger, Cincio and Coles2022, Reference Caro, Huang, Ezzell, Gibbs, Sornborger, Cincio, Coles and Holmes2023). Regarding their trainability, while suitably designed QNNs show desirable geometrical properties that may lead to faster training (Abbas et al., Reference Abbas, Sutter, Zoufal, Lucchi, Figalli and Woerner2021), in general, the training of PQCs can be hindered by the presence of barren plateaus (Ragone et al., Reference Ragone, Bakalov, Sauvage, Kemper, Marrero, Larocca and Cerezo2024) and potentially requires advanced initialization strategies (Zhang et al., Reference Zhang, Liu, Hsieh, Tao, Koyejo, Mohamed, Agarwal, Belgrave, Cho and Oh2022). However, in quantum convolutional neural networks (QCNNs), barren plateaus do not seem to be a problem (Pesah et al., Reference Pesah, Cerezo, Wang, Volkoff, Sornborger and Coles2021).

The embedding $ \boldsymbol{x}\to \mid \phi \left(\boldsymbol{x}\right)\Big\rangle $ can be thought of as a feature map from the input $ \boldsymbol{x} $ to the (large) Hilbert space of quantum states (Schuld and Killoran, Reference Schuld and Killoran2019; Havlíček et al., Reference Havlíček, Córcoles, Temme, Harrow, Kandala, Chow and Gambetta2019). This observation constitutes the basis of quantum kernel methods (Schuld and Killoran, Reference Schuld and Killoran2019; Havlíček et al., Reference Havlíček, Córcoles, Temme, Harrow, Kandala, Chow and Gambetta2019), where quantum kernels are constructed from inner products $ \left\langle \phi \left(\boldsymbol{x}\right)|\phi \left({\boldsymbol{x}}^{\prime}\right)\right\rangle $ measured on the quantum device. The constructed kernels are then used in subsequent tasks, e.g., in data classification (Havlíček et al., Reference Havlíček, Córcoles, Temme, Harrow, Kandala, Chow and Gambetta2019), or Gaussian process regression (Otten et al., Reference Otten, Goumiri, Priest, Chapline and Schneider2020; Rapp and Roth, Reference Rapp and Roth2024). Despite their advantage on specific datasets (Huang et al., Reference Huang, Broughton, Mohseni, Babbush, Boixo, Neven and McClean2021), quantum kernel methods may also suffer from trainability issues analogous to barren plateaus in QNNs (Thanasilp et al., Reference Thanasilp, Wang, Cerezo and Holmes2022), which highlights the requirement of a careful design.

PQCs also find a natural application as generative models (Amin et al., Reference Amin, Andriyash, Rolfe, Kulchytskyy and Melko2018; Dallaire-Demers and Killoran, Reference Dallaire-Demers and Killoran2018; Coyle et al., Reference Coyle, Mills, Danos and Kashefi2020). The output state of a PQC indeed corresponds to a probability distribution over an exponentially large computational basis, via the relation $ {P}_{\boldsymbol{\theta}}\left(\boldsymbol{\sigma} \right)=\mid \Big\langle \boldsymbol{\sigma} \mid {\hat{U}}_{\boldsymbol{\theta}}{\left|{\phi}_0\Big\rangle \right|}^2 $ , with $ \boldsymbol{\sigma} $ indexing the computational basis states, and $ \mid {\phi}_0\Big\rangle $ a predefined reference state. For generative tasks, the parameters $ \boldsymbol{\theta} $ are optimized for $ {P}_{\boldsymbol{\theta}} $ to approximate the probability distribution underlying a set of training data. The PQC is then used to sample the distribution $ {P}_{\boldsymbol{\theta}} $ . These models have improved representational power over standard generative models (Du et al., Reference Du, Hsieh, Liu and Tao2020), but also require careful design to avoid trainability issues (Rudolph et al., Reference Rudolph, Lerch, Thanasilp, Kiss, Vallecorsa, Grossi and Holmes2023).

We finally remark that despite the several hints to potential advantages on specific tasks, it is not yet clear whether a practical advantage of QML on real-world classical datasets can be demonstrated. Among the practical hurdles are the aforementioned trainability issues, as well as the necessity of efficient coupling with HPC and the inevitable presence of noise.

3.4. Integrated high-performance quantum computing

PQCs are based on the interplay between a quantum and a classical component within a hybrid algorithm. NISQ systems thus need to be integrated with classical computing resources at the hardware level. As quantum circuits become more complex, the classical component in PQCs and thus the required computing resources will grow as well, to the scale of an HPC problem. Given the trend towards heterogeneous development of modern HPC systems, e.g., increasingly using hardware accelerators like graphics processing units (Schulz et al., Reference Schulz, Kranzlmüller, Schulz, Trinitis and Weidendorfer2021), this leads to consider quantum hardware as a new class of accelerators for dedicated tasks within HPC workflows (Humble et al., Reference Humble, McCaskey, Lyakh, Gowrishankar, Frisch and Monz2021; Rüfenacht et al., Reference Rüfenacht, Taketani, Lähteenmäki, Bergholm, Kranzlmüller, Schulz and Schulz2022). The high performance computing—quantum computing (HPCQC) integration is an interdisciplinary challenge with aspects at the hardware, software, programming, and algorithmic level. The hardware integration is crucial to reduce latencies, especially in iterative variational algorithms (Rüfenacht et al., Reference Rüfenacht, Taketani, Lähteenmäki, Bergholm, Kranzlmüller, Schulz and Schulz2022). Moreover, the development of a single software stack for integrated systems, including the offloading of tasks to quantum accelerators and the scheduling of those resources, is necessary for ensuring operation and a smooth user experience (Schulz et al., Reference Schulz, Ruefenacht, Kranzlmuller and Schulz2022).

The HPCQC integration is not a concept limited to the NISQ development phase of quantum hardware, but is also relevant for future, fault-tolerant systems. At that point, we expect that the HPC computational resources will have to take over further tasks related to the operation of the quantum hardware, like circuit compilation and error control (Davenport et al., Reference Davenport, Jones and Thomason2023; Maronese et al., Reference Maronese, Moro, Rocutto and Prati2022).

3.5. Error correction and mitigation

Current NISQ devices are still modest in qubit counts (tens to hundreds), circuit depths (accommodating up to a thousand gate operations), and coherence times (from microseconds to seconds, depending on the platform (Byrd and Ding, Reference Byrd and Ding2023)). The achievable circuit depth and the repetition rate influence the complexity and the practical applicability of QML models, respectively. Hardware noise may negatively impact QML trainability and is also one of the main limiting factors for quantum annealing and for fault-tolerant applications.

Achieving fault-tolerance is the goal of quantum error correction methods, in which one logical qubit is represented using several physical ones, to detect and correct errors at runtime (Devitt et al., Reference Devitt, Munro and Nemoto2013). This is a crucial requirement for quantum algorithms relying on extensive numbers of quantum operations, such as quantum linear algebra subroutines. Given the substantial resource overhead needed for quantum error correction (Davenport et al., Reference Davenport, Jones and Thomason2023), its practical implementation can be considered in its infancy despite recent rapid advances (Google Quantum AI and Collaborators, 2025).

Other techniques to reduce noise-induced biases, without correcting for faulty qubits or operations, go under the name of error mitigation and are based on post-processing measurements collected from a suitably defined ensemble of quantum computation runs (Cai et al., Reference Cai, Babbush, Benjamin, Endo, Huggins, Li, McClean and O’Brien2023). Many mitigation techniques are currently being researched, such as zero-noise extrapolation and probabilistic error cancellation (Cai et al., Reference Cai, Babbush, Benjamin, Endo, Huggins, Li, McClean and O’Brien2023). These are particularly relevant for NISQ-targeted applications, as they can be implemented with little overhead in the number of gates and qubits. Hence, they are promising for “de-noising” the predictions from PQC-based methods.

4.1. Accelerating the resolution of high-dimensional differential equations

Solving differential equations is at the root of climate modelling, especially in the atmosphere and ocean components. Dynamical conservation laws are represented by PDEs, and chemical reactions are described by ordinary differential equations (ODEs) (Alvanos and Christoudias, Reference Alvanos and Christoudias2019; Sander et al., Reference Sander, Kerkweg, Jöckel and Lelieveld2005; Zlatev et al., Reference Zlatev, Dimov, Faragó, Georgiev, Havasi, Lirkov and Margenov2022). The resources needed for solving them grow rapidly with increasing resolution of the models. This calls for methods for efficiently encoding and processing the climate state variables in all the model cells. Several quantum algorithms seeking an exponential reduction of the needed resources have been recently proposed in the context of computational fluid dynamics (Gaitan, Reference Gaitan2020; Steijl and Barakos, Reference Steijl and Barakos2018).

Methods for time-dependent fluid-flow problems have been developed in the framework of gate-based quantum computing (Steijl and Barakos, Reference Steijl and Barakos2018; Mezzacapo et al., Reference Mezzacapo, Sanz, Lamata, Egusquiza, Succi and Solano2015; Gaitan, Reference Gaitan2020) and quantum annealers (Ray et al., Reference Ray, Banerjee, Nadiga and Karra2019). For example, using spatial discretization of the PDEs, Gaitan (Reference Gaitan2020) reduces the one-dimensional Navier–Stokes equations to a system of ODEs

(6) $$ \frac{d\boldsymbol{U}}{dt}=f\left(\boldsymbol{U}\right), $$

with $ \boldsymbol{U} $ being a vector depending on $ \rho $ , $ v $ , and $ T $ , and the driver function $ f\left(\boldsymbol{U}\right) $ depending on the discretization procedure. Kacewicz (Reference Kacewicz2006) showed that an almost optimal quantum algorithm exists for finding an approximate solution to a set of nonlinear ODEs. As shown by Gaitan (Reference Gaitan2020), the solution of Eq. (6) is approximated by discretizing it in both the spatial and time domains, and then determining the approximate solution using quantum amplitude estimation (QAEA) (Brassard et al., Reference Brassard, Høyer, Mosca, Tapp, Lomonaco and Brandt2002). These hybrid approaches for Navier–Stokes equations (Gaitan, Reference Gaitan2020) could thus potentially achieve up to exponential speed-up compared to deterministic classical algorithms, due to the quantum subroutine (Brassard et al., Reference Brassard, Høyer, Mosca, Tapp, Lomonaco and Brandt2002). However, the stability of such algorithms with respect to the spatial discretization still remains an open issue, and these methods inherit the challenges that quantum linear solvers may face, related to the data encoding and readout, and to the implementation of the linear operator defining the problem (Aaronson, Reference Aaronson2015).

Quantum lattice–Boltzmann methods (QLBM) offer alternatives to quantum linear solvers, circumventing solving linear systems directly (Budinski, Reference Budinski2021, Reference Budinski2022; Schalkers and Möller, Reference Schalkers and Möller2022). The lattice–Boltzmann method, a mesoscopic stream-and-collide method for probability densities of fluid particles, lends itself to quantum solution natively and efficiently (Budinski et al., Reference Budinski, Niemimäki, Zamora-Zamora and Lahtinen2023; Li et al., Reference Li, Yin, Wiebe, Chun, Schenter, Cheung and Mülmenstädt2023). QLBM algorithms can be fully quantum (Budinski, Reference Budinski2021), or hybrid, such as Budinski’s Navier–Stokes algorithm (Budinski, Reference Budinski2022), where quantum-classical communication is needed to incorporate non-linearities. Other approaches for non-linearities in QLBM exist, e.g., by using Carleman linearization (Itani and Succi, Reference Itani and Succi2022). Koopman operators can also be used to induce linearity (Bondar et al., Reference Bondar, Gay-Balmaz and Tronci2019). Also, QLBM faces the challenge related to data readout and encoding: for some QLBM models, this must be repeated at all time steps, which makes efficient time-marching a problem of prime importance for QLBM.

Data-driven quantum-assisted approaches are also a possibility to predict the evolution of climate systems, as demonstrated by Jaderberg et al. (Reference Jaderberg, Gentile, Ghosh, Elfving, Jones, Vodola, Manobianco and Weiss2024) in the context of weather models.

In summary, the encoding and readout of classical variables is one of the primary limitations that need to be addressed for applying the aforementioned routines to problems on scales of a climate model. Even though 30 logical qubits could be sufficient to store one billion climate variables (Tennie and Palmer, Reference Tennie and Palmer2023) and to implement their time evolution using quantum circuits of polynomial depth, it is still unclear what type of climate states can be efficiently encoded in quantum states, and which properties of the climate can be efficiently measured. Although full quantum state reconstruction is not experimentally feasible for large quantum systems (it requires a number of measurements growing exponentially in the number of qubits (Struchalin et al., Reference Struchalin, Zagorovskii, Kovlakov, Straupe and Kulik2021)), fundamental insights on the properties that can be efficiently read out from quantum states exist in the framework of shadow tomography (Huang et al., Reference Huang, Kueng and Preskill2020): these results will need to be adapted to the specific use-case of climate simulation and subsequent evaluation.

4.2. QML-based parameterizations

QML models, such as QNNs, can be used to develop data-driven parameterizations by training them with data from short high-resolution simulations, as shown in Figure 5, in analogy to what is currently done with classical machine learning (Gentine et al., Reference Gentine, Eyring, Beucler, Camps-Valls, Tuia, Zhu and Reichstein2021; Eyring et al., Reference Eyring, Collins, Gentine, Barnes, Barreiro, Beucler, Bocquet, Bretherton, Christensen, Dagon, Gagne, Hall, Hammerling, Hoyer, Iglesias-Suarez, Lopez-Gomez, McGraw, Meehl, Molina, Monteleoni, Mueller, Pritchard, Rolnick, Runge, Stier, Watt-Meyer, Weigel, Yu and Zanna2024a). The properties of QML models (Yu et al., Reference Yu, Chen, Jiao, Li, Lu, Wang and Yang2023b; Abbas et al., Reference Abbas, Sutter, Zoufal, Lucchi, Figalli and Woerner2021; Caro et al., Reference Caro, Huang, Cerezo, Sharma, Sornborger, Cincio and Coles2022; Huang et al., Reference Huang, Broughton, Mohseni, Babbush, Boixo, Neven and McClean2021) may lead to highly expressive parameterization models requiring less parameters, hence potentially requiring less training data if matched by good generalization capabilities. These attributes would be crucial for developing stable and reliable long-term climate projections. First quantum-enhanced ML algorithms for the emulation of climate model data have recently been demonstrated by Bazgir and Zhang (Reference Bazgir and Zhang2024) using the ClimSim dataset (Yu et al., Reference Yu, Hannah, Peng, Lin, Bhouri, Gupta, Lütjens, Will, Behrens, Busecke, Loose, Stern, Beucler, Harrop, Hillman, Jenney, Ferretti, Liu, Anandkumar, Brenowitz, Eyring, Geneva, Gentine, Mandt, Pathak, Subramaniam, Vondrick, Yu, Zanna, Zheng, Abernathey, Ahmed, Bader, Baldi, Barnes, Bretherton, Caldwell, Chuang, Han, Huang, Iglesias-Suarez, Jantre, Kashinath, Khairoutdinov, Kurth, Lutsko, Ma, Mooers, Neelin, Randall, Shamekh, Taylor, Urban, Yuval, Zhang, Pritchard, Oh, Naumann, Globerson, Saenko, Hardt and Levine2023a): Using a subset of the dataset, they used three different quantum models to predict target values such as snow and rain rates and various solar fluxes using surface pressure, insolation, and the surface latent and sensible heat fluxes as input. The quantum models they tested were a QCNN, a quantum multilayer perceptron (QMP), and a quantum encoder–decoder (QED), each compared to their classical counterparts. The quantum models typically outperformed the classical models. In another work (Pastori et al., Reference Pastori, Grundner, Eyring and Schwabe2025), some of the authors of the present manuscript recently demonstrated a QML-based parameterization of cloud cover for a climate model. The practical assessment of QML advantages in such tasks requires, however, addressing several open points. Encoding climate variables in quantum states constitutes a critical step, given the efficiency requirements in the NISQ era and the absence of quantum random access memory. Specifically tailored classical data-compression routines before the encoding stage in a QNN could include techniques based on feature selection (Mücke et al., Reference Mücke, Heese, Müller, Wolter and Piatkowski2023), variational encoding sequences (Behrens et al., Reference Behrens, Beucler, Gentine, Iglesias-Suarez, Pritchard and Eyring2022), or tensor networks, which can be naturally translated to quantum circuits (Dilip et al., Reference Dilip, Liu, Smith and Pollmann2022; Cichocki et al., Reference Cichocki, Lee, Oseledets, Phan, Zhao and Mandic2016).

Figure 5.

Hybrid quantum-classical approach for QML-based parameterizations. (a) Offline training: variables from cloud-resolving models are coarse-grained to the scale of the target climate model. The subgrid part of the variables of interest (e.g., fluxes) is calculated and used as a target for training QML-based parameterizations, possibly complemented with ML-based pre- and post-processing steps. The training of the QML model is typically assisted by a classical computer. The result is a replacement for a conventional parameterization and is coupled to the target climate model. (b) Processes occurring while the coarse-scale climate model (summarized by variables $ \boldsymbol{x} $ ) is advanced from time step $ t $ to $ t+\Delta t $ . First, the dynamical core $ \mathcal{D} $ is run, followed in parallel or sequentially by the various parameterizations, denoted here with $ {\mathcal{P}}_2 $ to $ {\mathcal{P}}_4 $ . The QML-parameterization $ \mathcal{QP} $ is run online, replacing the corresponding traditional parameterization $ {\mathcal{P}}_1 $ .

The choice of the QNN structure is problem dependent and needs to match the hardware constraints. Also, it is desirable to encode conservation laws and physical constraints at the level of the underlying PQC, e.g., via the use of equivariant gate sets (Meyer et al., Reference Meyer, Mularski, Gil-Fuster, Mele, Arzani, Wilms and Eisert2023), or by directly encoding the physical laws into the model (Markidis, Reference Markidis2022).

Another point to consider is the probabilistic nature of the measurement outcomes. While the inherent quantum noise could prove to have some benefits for climate simulations (Tennie and Palmer, Reference Tennie and Palmer2023), the accuracy of the model prediction depends on the number of circuit evaluations, thus influencing the runtime. For nowadays’ implementations, the runtime of such QNN parameterizations constitutes a limiting factor in their usages when coupled with the climate model, since they need to be run at each time-step for all model cells. Considering a time requirement of $ 10\;\unicode{x03BC} \mathrm{s} $ per QNN run, and approximately $ 100 $ required measurement runs (Pastori et al., Reference Pastori, Grundner, Eyring and Schwabe2025), the evaluation of a QNN parameterization for a climate model with 1 million grid cells would take $ 1000 $ s per model time-step, which is clearly prohibitive. Expected advances in quantum hardware and its coupling with HPC will also increase the potential for integrating QML routines in climate models.

4.3. Improving climate model tuning

QML models and quantum optimization routines could also be used to improve the accuracy and speed of climate model tuning following an automatic tuning procedure (Figure 6). First, the tuning goals and parameters are chosen. Then, emulators are constructed to approximate the model output and to speed up the calibration process (Bellprat et al., Reference Bellprat, Kotlarski, Lüthi and Schär2012; Watson-Parris et al., Reference Watson-Parris, Williams, Deaconu and Stier2021). Potentially optimal parameter regimes are inferred (Couvreux et al., Reference Couvreux, Hourdin, Williamson, Roehrig, Volodina, Villefranque, Rio, Audouin, Salter, Bazile, Brient, Favot, Honnert, Lefebvre, Madeleine, Rodier and Xu2021; Watson-Parris et al., Reference Watson-Parris, Williams, Deaconu and Stier2021; Allen et al., Reference Allen, Hoppel, Nedoluha, Eckermann and Barton2022; Zhang et al., Reference Zhang, Li, Lin, Xue, Xie, Xu and Huang2015; Cinquegrana et al., Reference Cinquegrana, Zollo, Montesarchio and Bucchignani2023; Bonnet et al., Reference Bonnet, Pastori, Schwabe, Giorgetta, Iglesias-Suarez and Eyring2024), and new climate model outputs are evaluated in the proposed parameter regime. The procedure is iterated until the tuning goals are achieved.

Figure 6.

General steps of an automatic tuning protocol for climate models. In Step (i), the tuning goals and parameters are chosen. Then, in Step (ii) and (iii), emulators are constructed to approximate the model output and to speed up the calibration process. Potentially optimal parameter regimes are inferred in steps (iv) and (v), and new climate model outputs are evaluated in the proposed parameter regime. The procedure is iterated until the tuning goals are achieved.

The main quantum computing applications that we envision here concern the emulator construction and its optimization. Typical choices of emulators in automatic tuning schemes are neural networks and Gaussian processes (Watson-Parris et al., Reference Watson-Parris, Williams, Deaconu and Stier2021). The former choice suggests the use of QNN emulators, while the latter option implies the choice of an underlying kernel function, suggesting the use of quantum kernels in Gaussian process regression (Otten et al., Reference Otten, Goumiri, Priest, Chapline and Schneider2020). For example, Rapp and Roth (Reference Rapp and Roth2024) develop a quantum Bayesian optimization algorithm based on quantum Gaussian processes. They construct a kernel by embedding data into the Hilbert space of a quantum system

(7) $$ \mid \phi \left(\boldsymbol{x};\boldsymbol{\theta} \right)\left\rangle =U\left(\boldsymbol{x};\boldsymbol{\theta} \right)\mid 0\right\rangle . $$

The unitary operator $ U\left(\boldsymbol{x};\boldsymbol{\theta} \right) $ encodes the classical data $ \boldsymbol{x} $ into a quantum state and can depend on trainable parameters $ \boldsymbol{\theta} $ . It implements the quantum feature map $ \phi $ and defines the density matrix $ \rho \left(\boldsymbol{x}\right)=U\left(\boldsymbol{x}\right)\mid 0\left\rangle \right\langle 0\mid {U}^{\dagger}\left(\boldsymbol{x}\right) $ . The quantum kernel then, is given by

(8) $$ k\left(\boldsymbol{x},{\boldsymbol{x}}^{\prime}\right)=\mathrm{Tr}\left[\rho \left(\boldsymbol{x}\right)\rho \left({\boldsymbol{x}}^{\prime}\right)\right]. $$

The corresponding Gram matrix is then substituted as a covariance matrix into a classical Gaussian process, and the feature map is optimized using maximum likelihood estimation. The resulting algorithm is used for a hyperparameter optimization of a machine learning model.

Exploring the potential of QML for constructing emulators is of great interest since highly expressive models, if matched by good generalization properties, could benefit from fewer trainable parameters and training data, and hence fewer climate model simulations. The optimization of the acquisition function resulting from the emulator yields potentially optimal parameter sets and can be tackled with quantum heuristics such as quantum annealing, quantum-approximate optimization algorithms, or variational quantum eigensolvers, upon suitable discretization of the function to be optimized. Several works have recently explored the performance of these methods on continuous variable optimization problems (Izawa et al., Reference Izawa, Kitai, Tanaka, Tamura and Tsuda2022; Koh and Nishimori, Reference Koh and Nishimori2022; Abel et al., Reference Abel, Blance and Spannowsky2022). Furthermore, continuous variable quantum computers, e.g., photonic platforms, could offer alternatives to avoid parameter space discretization (Enomoto et al., Reference Enomoto, Anai, Udagawa and Takeda2023).

4.4. Assisting model analysis and evaluation

Improved climate model analysis and evaluation routines lead to better identification of climate model biases. An important task towards these goals involves learning the probability distributions underlying the models’ outputs. Classical generative methods are able to produce high-fidelity, realistic examples of climate data, e.g., with dynamics consistent with diurnal cycles (Besombes et al., Reference Besombes, Pannekoucke, Lapeyre, Sanderson and Thual2021; Behrens et al., Reference Behrens, Beucler, Gentine, Iglesias-Suarez, Pritchard and Eyring2022). Given the ability of quantum systems to efficiently, i.e., with few parameters, describe complex probability distributions (Du et al., Reference Du, Hsieh, Liu and Tao2020; Wu et al., Reference Wu, Bao, Cao, Chen, Chen, Chen, Chung, Deng, Du, Fan, Gong, Guo, Guo, Guo, Han, Hong, Huang, Huo, Li, Li, Li, Li, Liang, Lin, Lin, Qian, Qiao, Rong, Su, Sun, Wang, Wang, Wu, Xu, Yan, Yang, Yang, Ye, Yin, Ying, Yu, Zha, Zhang, Zhang, Zhang, Zhang, Zhao, Zhao, Zhou, Zhu, Lu, Peng, Zhu and Pan2021; Hangleiter and Eisert, Reference Hangleiter and Eisert2023), quantum generative models are natural candidates for achieving good extrapolating abilities using a small amount of training data. Quantum generative models can be built using PQCs, and trained either by minimizing the divergence between the target and the PQC distribution (Coyle et al., Reference Coyle, Mills, Danos and Kashefi2020) or in an adversarial manner (Dallaire-Demers and Killoran, Reference Dallaire-Demers and Killoran2018). Once trained, new samples following climate variable distributions can be efficiently generated and used for further analysis.

QML could also offer improvements in the classification of climate data, which is useful for improving and complementing observational products to subsequently use them for model evaluation (Kaps et al., Reference Kaps, Lauer, Camps-Valls, Gentine, Gomez-Chova and Eyring2023). PQCs again offer alternative ansatzes to classical ML with potential benefits (Du et al., Reference Du, Hsieh, Liu and Tao2020, Reference Du, Hsieh, Liu, You and Tao2021; Abbas et al., Reference Abbas, Sutter, Zoufal, Lucchi, Figalli and Woerner2021; Caro et al., Reference Caro, Huang, Cerezo, Sharma, Sornborger, Cincio and Coles2022). Existing QML methods for classification include quantum kernels and variants thereof (Schuld and Killoran, Reference Schuld and Killoran2019; Havlíček et al., Reference Havlíček, Córcoles, Temme, Harrow, Kandala, Chow and Gambetta2019; Huang et al., Reference Huang, Broughton, Mohseni, Babbush, Boixo, Neven and McClean2021), and QNNs (Farhi and Neven, Reference Farhi and Neven2018; Abbas et al., Reference Abbas, Sutter, Zoufal, Lucchi, Figalli and Woerner2021; Hur et al., Reference Hur, Kim and Park2022). Among these, quantum convolutional neural networks (Cong et al., Reference Cong, Choi and Lukin2019) have shown remarkable trainability and generalization capabilities when used on quantum data (Cong et al., Reference Cong, Choi and Lukin2019; Pesah et al., Reference Pesah, Cerezo, Wang, Volkoff, Sornborger and Coles2021; Du et al., Reference Du, Hsieh, Liu, You and Tao2021), although recent works have suggested the possibility of their classical simulability (Cerezo et al., Reference Cerezo, Larocca, García-Martín, Diaz, Braccia, Fontana, Rudolph, Bermejo, Ijaz, Thanasilp, Anschuetz and Holmes2023).