We draw the current landscape of quantum algorithms, by classifying about 130 quantum algorithms, according to the fundamental mathematical problems they solve, their real-world applications, the main subroutines they employ, and several other relevant criteria. The primary objectives include revealing trends of algorithms, identifying promising fields for implementations in the NISQ era, and identifying the key algorithmic primitives that power quantum advantage.

Keywords: Quantum Algorithms, Overview, Dependencies, NISQ, LSQ, Heuristics, Oracular, Promise, Sampling, Computational Model.

The selection of the algorithms was by no means random, it was the result of a literature review-based approach, governed by the goal to create an all-inclusive overview that captures the overall landscape of quantum algorithms. The algorithm selection process of course prioritized those algorithms that are highly recognized within the quantum-computing community, ensuring the inclusion of foundational algorithms that have significantly contributed to the advancement of the field over the years. In addition to these most well-known algorithms, one of the several criteria for choosing new ones was their popularity, based on citation frequency. Subsequently, the classification table was updated to try and capture recent developments within current research trends. Digging into each of these topics, we then took the liberty to include further results that appeared to us as quite significant scientifically. Eventually, a last factor that was considered, was the consistency of the classification table itself, e.g., making sure that all main subroutines were present, all application domains inhabited, etc.. Again, although we tried to capture the main algorithms using objective and scientific criteria to minimize bias, it is inevitable that some degree of arbitrariness remains in the selection process.

We now outline the different classification criteria that were chosen in order to explore and enclose the various dimensions of quantum algorithms. Examples of algorithms will be given for each criterion.

2.2.1 Main subroutine

2.2.2 Fundamental mathematical problem

2.2.3 Applications

2.2.4 Heuristic versus Proven algorithms

2.2.5 NISQ versus LSQ algorithms

2.2.6 Oracular (versus Non-oracular) algorithms

2.2.7 Promise algorithms

2.2.8 Sampling algorithms

2.2.9 Partial dequantization

2.2.10 Computational model

2.2.11 Citations

Whilst the area of quantum algorithms was, two or three decades ago, dominated by a handful of seminal algorithms like Shor’s and Grover’s, it has now expanded into a rich landscape of numerous novel algorithms. However, a legitimate question is whether the new algorithms of this field are made from combinations of the old ones used as subroutines, or whether they are fundamentally novel altogether. Analyzing the “Main subroutine” criterion of the classification table, can give us an answer to this question.

3.1.1 Dependency network

3.1.2 Algorithmic-primitives analysis

Going back to the first column “Main subroutine” of our classification table, and focusing on the Primitives, one immediately sees that, beyond 8 usual “suspects”, many other primitives, or non-primitive popular subroutines (not necessarily used by other algorithms) could be listed. Those findings are summarized in Table 1. In the first column of that table, we listed the 8 usual primitives mentioned above. In the second column, we listed other, non-primitive popular subroutines, either already old, or new ones, more or less half of which are already indeed used as subroutines of other algorithms. In the last column, we listed the remaining primitives.

Table 1. Table of primitives and non-primitive popular subroutines (none of the subroutines of the second column is, indeed, a primitive). Note that in order to be intuitive, all HS algorithms are clustered as “HSs”, and all QLSA algorithms are clustered as “QLSAs”. The same convention applies for Fig4, which uses the second column only, and for Fig5, which uses the first column only.

Popular Primitives	Popular Subroutines	Other Primitives
QFT [31]	QPE [32]	QITE [33]
AA [34] QAE [35]	QLanczos [36]
QA [37]	Grover [38]	Variational QSVD [39]
VQE [40]	QLSAs [41, 42, 30, 43, 44]	Bayesian QPE [45]
HSs [46, 47, 48, 49, 50, 51] QSVT [52]	Hadamard Test [53]
Block Encoding [49]	QSP [54]	Hilbert-Schmidt Test [55]
BS [56]	Szegedy’s QW [57]	QHierarchical Clustering [58]
Swap Test [59]	SKW’s QW Search [60]	QDistance Classifier [61]
	Adiabatic Search [62]	Q.-Enhanced Markov-Chain Monte Carlo [63]
	D& HMinimization [64]	Simon’s Algo[65]
	QCounting [66]	Fermionic QFT [67]
	QMonte-Carlo Integral [68]	QCA for QElectrodynamics [69]
	Gaussian BS [70]	QFinal State Shower [71]
	Ising-QUBO [72]	Free-Q.-Field-Theory Ground State [73]
	QAOA [74]	QFock Space Dynamics [75]
	QSDP [76]	Dirac-Equation QW [77]
	Dihedral HSP Algo[78]
	Shor [79]

Let us briefly speculate about which primitives or popular subroutines may be most dominant in the near or further future. It seems likely that the Quantum Singular-Value Transformation (QSVT) will play a prominent role in the future. Indeed, QSVT is famously a unifying algorithm, that subsumes most of the other prominent algorithms as special cases [80, 81]. Moreover, QSVT also plays the role of a framework that captures the “modern” approach to quantum-algorithm design, namely, using quantum computers to manipulate the eigenvalues or singular values of exponentially large matrices – applied to some suitable input. This idea is extremely powerful, and the first hint of it actually came in the Harrow-Hassidim-Lloyd (HHL) algorithm, for which one of the primitives is QPE. Indeed, unlike most other algorithms that use QPE, rather than just extracting a particular eigenvalue phase, HHL manipulates the quantum state by accessing the several phases of eigenvalues in superposition in order to reach the desired result.

An interesting development is Tang’s [13] methodology for extracting quantum-inspired classical algorithms from existing quantum algorithms, which are efficient for a certain range of parameters. This methodology is useful mainly for the HHL, QPE, and QLSA algorithms (a.k.a. qBLAS algorithms [82]), with impact on linear-algebra problems and machine-learning and data-science applications, amongst others. But the methodology can also be used for QSVT. Again, given the unifying nature of the latter, it is at this stage rather difficult to gauge the potential ramifications of dequantization that may ensue from a use of Tang’s methodology on QSVT.

After thoroughly filling, for each algorithm, the specifics (i) for the “Fundamental mathematical problem” criterion/column of our classification table, and (ii) for the “Applications” criterion/column, we realized the need to take a step back to get to the bigger picture.

For the “Fundamental mathematical problem” criterion, we identified 6 classes, described below.

• Hidden-Subgroup Problems: This class of problems is about identifying a subgroup from a function that is constant “on” each coset of that subgroup and that takes distinct values on different cosets of this subgroup. Let us give a formal definition. First of all, let us define what a coset of a subgroup is. A left coset of some subgroup \( S\) of \( G\) is a set \( gS\) for some \( g \in G\) . The definition for right cosets is analogous. A fundamental result about cosets is that two left cosets or two right cosets are either the same or disjoint. Let us now define the standard Hidden-Subgroup Problem (HSP). Given a group \( G\) , a finite set \( X\) and a function \( f : G \rightarrow X\) , the goal is to identify a hidden subgroup \( H \subsetneqq G\) of the group \( G\) such that \( f(x)\) is constant “on” each coset of \( H\) , which means that, for all \( g_1, g_2 \in G\) , \( f(g_1) = f(g_2)\) if and only if \( g_1H = g_2H\) [1]. Shor’s algorithm, which solves the problems of prime-numbers factorization and of computing discrete logarithms, can be generalized as an HSP for finite Abelian groups [2].
• Linear Algebra: Linear-algebra problems deal with linear equations, linear transformations and their representations using matrices. For example, the Quantum Singular-Value Decomposition (QSVD) algorithm [3] produces the singular-value decomposition of a bipartite pure state, which is indeed a linear-algebra problem.
• Dynamical Systems: A dynamical system is a system whose state changes over time. This could be either (i) in discrete time, via finite-difference equations in either only time, or both time and space, or (ii) in continuous time, with a partial derivative with respect to time at least, and, if we are also in continuous space, another partial derivative, with respect to space this time, leading to a partial differential equation (otherwise it is an ordinary differential equation with respect to time, with finite differences in space). Quantum algorithms that belong to this class seek to predict and analyze the system’s time evolution, e.g., by preparing an initial state, encoding information therein, evolving it, or evaluating key features of this evolution. For instance, the Quantum Lanczos algorithm [4] computes the ground state of some Hamiltonian, which is a key feature of the evolution of the system described by the Hamiltonian, so that this algorithm belongs to the class “Dynamical Systems”.
• Stochastic Processes & Statistics: A stochastic process, also called a random process, is a mathematical concept used to model and describe the evolution of variables/quantities that evolve randomly over time (or some other parameter), using probabilistic rules. Statistics is a branch of mathematics that studies the collection, analysis, interpretation, and presentation of data, so as to make inferences or decisions based on them. Typically, the stochastic process is what generates the data that statistics analyze. In the context of this work, an example of an algorithm that solves a stochastic-processes problem is the Gaussian-Boson-Sampling (GBS) Matrix-Point Process [5] – which, in addition, is a sampling algorithm, as defined above. An example of an algorithm that solves a statistics problem is the Quantum k-means algorithm [6].
• Optimization: “Optimization” refers to the class of problems that deal with finding the best solution amongst a given set of candidates. The “best” solution is determined by minimizing or maximizing some target function while satisfying a set of constraints. Optimization includes numerical optimization and combinatorial optimization problems. When an algorithm is of the combinatorial-optimization type, we assigned it both the “Optimization” class described here, and the “Combinatorics” class described below; otherwise (algorithm of the numerical-optimization type), we just assigned it the “Optimization” class. An example is the Quantum Linear Regression algorithm [7], whose fundamental mathematical problem is linear regression using least-squares optimization, which is convex optimization (which is a a subclass of numerical optimization).
• Combinatorics: Combinatorial problems involve finding, for a discrete finite set of objects that satisfy some given conditions, either a grouping, or an ordering, or an assignment. They can be divided into tree basic types, that are, enumeration problems, existence problems, and optimization problems. An algorithm classified under “Combinatorics” in the classification table, focuses on solving problems like graph-theoretical problems, search tasks, counting tasks, etc.. For example, Grover’s search algorithm has been extended to Quantum Counting [8], and both can be classified as combinatorial.

Regarding the “Applications” criterion, we extracted 6 application domains, described below.

• Cryptanalysis: Cryptanalysis refers to the study of cryptographic systems with the aim of understanding their strengths and weaknesses. Most works in this domain study the applicability of quantum algorithms to break classical cryptographic schemes. The obvious, early example is Shor’s factorization algorithm, whose LSQ implementation would break the RSA public-key encryption system [88].
• Machine Learning & Data Science: This domain involves (i) performing tasks like classification, regression, clustering, etc., and (ii) applications of quantum algorithms in the fields of data analysis, statistical modeling, or predictive analytics for managing large datasets. For instance, the Quantum Algorithm for Recommendation Systems [89] has applications in both these areas.
• First-Principle Quantum Simulation: “First-Principle (also called “ab initio”) Quantum Qimulation” refers to using quantum computing devices in order to simulate and predict physical properties of a system, based upon its fundamental quantum mechanical description – without any detour via empirical models or parameter fittings. Indeed, although classical algorithms for matrix arithmetics have only polynomial complexity in the matrix dimension, the Hilbert-space dimension of multi-particle systems/many-body problems is exponential in the system’s size. Thus, quantum algorithms are bound to quickly overcome their classical competitors [90]. In fact, using programmable quantum systems to simulate quantum physical systems was Feynman’s original motivation for inventing the very concept of a quantum computer [91]. Typically, first-principle quantum simulations involve solving particle-structure problems, electronic structures [92], atomic-nucleus structures [93], or material properties. We distinguish this application domain from other scientific computing approaches, which we refer to as “classical” below.
• Classical Scientific Computing: This domain is used to accommodate scientific-computing applications that are not first-principle quantum simulations, including empirical-modeling applications and data/signal-processing applications. Quantum algorithms of this application domain may outperform their classical counterparts for specific tasks, including classical-mechanics simulations [94], empirical/semi-empirical numerical model prediction and optimization [95, 96], and experimental data processing/mining [97].
• Operational Research: “Operational Research” involves employing analytical methods, like modeling, statistics, optimization, in order to arrive at optimal or near-optimal solutions to decision-making problems, in complex scenarios like logistics, supply-chain management, resource allocation, etc.. Within the context of this work, mathematical programming, optimization, and game-theory algorithms are classified as “Operational Research”. As an example, let us cite the Quantum Algorithm for CVaR [98], which has a specific application in finance, for risk management.
• Quantum Enablement: This domain covers the particular quantum algorithms, closer to the hardware part, that underlie the experimental implementation of other quantum algorithms, closer to the software part, or belong to the construction of the quantum technical stack itself. Typically, they have no direct applications by themselves, although some may be applied in quantum sensing or metrology [99, 76], or to demonstrate quantum advantage [100, 101, 102]. They often represent general algorithmic-design ideas on which many more specific quantum algorithms are based [103, 31], e.g., quantum test algorithms are classified in this domain [104, 105, 106].

Having coarse-grained fundamental mathematical problems into classes, and applications into domains, helps identifying correlations and trends.

3.3.1 Correlations between mathematical classes and application domains, related trends, and correlations with important subroutines

In Fig2, we plot the number of algorithms for each of the 6 application domains, over time. Some trends become apparent. Indeed, the application domains in which there is an increasing number of new algorithms in the last years (here we roughly merge the acceleration of this number and its absolute value when we say “increasing number”, we do not disambiguate between both features at this stage), are, by decreasing increasing number, “First-Principle Quantum Simulation”, “Machine Learning & Data Science”, “Operational Research”, and, to a lesser extent “Classical Scientific Computing” and “Quantum Enablement”. The “Cryptanalysis” domain is the one that evolves slowest in terms of acceleration of the number of new algorithms.

A similar diagram, focusing this time on the mathematical classes, is presented in Fig3. It shows that an important proportion of new algorithms fall into the “Dynamical Systems” class. This trend aligns with the previously noted growth in “First-Principle Quantum Simulation” applications, as these applications often require solving dynamical-system problems. Similarly, an important proportion of new algorithms fall into the “Linear Algebra” and “Optimization” classes, which again aligns with the previously noted growth in “Machine Learning & Data Science” and “Operational Research” applications, since the latter require solving a combination of linear algebra, statistical, and optimization problems. The “Hidden-Subgroup Problems” class is stable, correlated with the stability of the “Cryptanalysis” application.

The previous trend analyses clearly suggest correlations between mathematical classes and their application domains, through the algorithms serving them. Let us evaluate this more precisely. We have computed these correlations and visualized [107] the result in Fig4 . Notice how these findings were enabled by the dependency analysis of Subsec3.1, allowing us to focus on main subroutines for increased robustness: indeed, the x-axes of the two diagrams of Fig4, which are both the same, correspond to the list of Popular Subroutines identified thanks to the dependency network, and which appears in the second column of Table 1. Hopefully these two diagrams can help trace back the root of some quantum advantage, according to one’s objective.

3.3.2 Correlations between algorithmic primitives, mathematical classes, and application domains

Let us dig in the correlations between mathematical classes and application domains, via the algorithmic primitives that serve them both. Those are represented in Fig. 5. Notice that the results are relatively elaborate and do not immediately follow from common sense. We are going to comment on them progressively as we advance in this subsubsection. We are going to start by making a number of observations, organized per primitive, and partly based on both Figs4 and 5, but also on general knowledge acquired during our investigation.

\( \bullet \) About the use of Grover’s algorithm – based on the primitive AA:

Amplitude Amplification (AA) is a major primitive, on which Grover’s algorithm [9] depends {logically} (not chronologically). Based on Grover’s algorithm, which uses the quantum Circuit Model of computation, many quantum search algorithms have been developed with other computational models such as Adiabatic Quantum Computation [62] and Quantum Walks  [60, 108] . The ability to search faster leads to advantages in solving problems in the “Combinatorics”  [109, 110, 111, 112, 113, 114, 64, 115, 57, 116] , { “Hidden-Subgroup Problems” [117, 118] and “Stochastic Processes & statistics” [119, 120, 121, 122, 123] classes.

\( \bullet \) About the use of QPE – based on the primitive QFT – and of the QLSAs:

As we can see on Fig4, QPE is one of the most popular subroutines. When aiming at solving problems in the “Dynamical Systems” class, QPE provides a fundamental method for estimating the ground-state energy. When aiming at solving problems in the “Linear Algebra” class, QPE served as a subroutine of early QLSAs (i.e., HHL).

Further QLSAs escape QPE and make use of other primitives and subroutines instead, such as AA, QA, and QSVT [42, 124, 125, 30]. Still, whatever the QLSA that we use, and since many problems can be formulated in terms of systems of linear equations, QLSAs help to solve problems in the “Linear Algebra”, “Optimization”, “Stochastic Processes & Statistics”, and “Dynamical Systems” classes.

\( \bullet \) About QAE and, in general, the mixing of QPE and AA:

QPE and Grover’s algorithm are often used in conjuction in quantum walk-based search algorithms for solving problems in the “Combinatorics” class.

QAE was also first obtained by combining QPE with AA [35]. QAE allows one to estimate the amplitude of a state within a superposition [35]. It acts as a fundamental primitive for those algorithms that use amplitude encoding and require retrieving the information encoded by the amplitude. The first use that comes to mind is in counting and matching, i.e., usually in the “Combinatorics” class. But QAE is also used for solving stochastic differential equations and some quantum machine-learning tasks. Moreover, further progress achieved QAE without QPE [126, 127]. This, in turn, is used by other algorithms to perform numerical calculations and solve problems of the “Linear Algebra” [128], “Optimization” [129, 130], “Stochastic Processes & Statistics” [68, 131], and “Dynamical Systems” [132, 133, 134, 46, 47] classes.

\( \bullet \) About QSVT and QSP:

The eventual impact of the QSVT and QSP algorithms is expected to be greater than what Fig4 indicates, not only because many of the earlier subroutine usages of HHL can be directly replaced by advanced QLSAs that use QSVT as a subroutine, but also because using QSVT for Hamiltonian Simulation (HS) could become more popular, as it can sometimes provide optimal complexity, for example for the BQP-complete problem [135] of Ref[30], in the LSQ sense. Moreover, powerful features such as eigenvalue transformation and filtering (of arbitrary Chebyshev polynomials) are likely to be found more uses as the exploration goes.

\( \bullet \) About HS:

Hamiltonian Simulation (HS), as a fundamental tool for encoding information and constructing the time evolution of the system under study, has bred a variety of approaches [49, 47, 136, 137, 138, 139]. Besides the multi-purpose subroutines such as QSVT and QSP [140, 80], which are not confined exclusively to HS, and excluding the Suzuki-Trotter decomposition [141], which we have not considered as a quantum algorithm, in Fig5 we have put all the HS algorithms under the umbrella of “Hamiltonian Simulations” (exactly as we did in Table 1 above, which is mentioned in the caption). Among all of these algorithms, Block Encoding emerges as the most significant primitive in our dependency network, so that we have isolated it from other HS primitives in Fig5; we see in Fig5 that it is mostly used to solve problems in the “Linear Algebra” class. Generally speaking, Fig5 shows that HS algorithms, in addition to serving problems in the “Linear Algebra” class, also serve problems in the “Dynamical Systems” class, as we may expect, most often with applications in the “First-Principle Quantum Simulation” domain.

\( \bullet \) About VQAs:

The Variational Quantum Eigensolver (VQE) [142] is an algorithm designed to find the eigenvalue of an operator and it is quite often used to find the ground-state energy of a quantum system. Many quantum algorithms use the VQE as their main subroutine to solve problems in the “Dynamical Systems” class, with applications in chemistry or materials science, e.g., for simulating the electronic structure of a molecule [143, 144, 145, 146]. As a consequence, there is a large proportion of VQE uses in the “First-Principle Quantum Simulation” application domain, as we can see on Fig5. However, we can also notice a good proportion of VQE uses in the “Operational Research” application domain, due to the growth of variational quantum algorithms for problems in the “Combinatorics” and “Optimization” classes. It is noteworthy that following the development of the VQE, numerous Variational Quantum Algorithms (VQAs) based on the variational principle have been developed. These algorithms are capable of solving a broader range of problems in the (numerical) “Optimization” and “Linear Algebra” classes, beyond the computation of ground-state energies.

\( \bullet \) More applications of the QFT (apart from the original QPE):

The Quantum Fourier Transform (QFT) is one of the major primitives among quantum algorithms. With the generalization of the Fourier sampling technique [147], algorithms could rely on QFT to solve “Hidden-Subgroup Problems” [148, 149, 150], which have many applications in the Cryptanalysis domain. QFT is also used as a primitive to solve problems in the “Dynamical Systems” class [151, 152], e.g., for Hamiltonian simulation [51, 50, 47]. Algorithms that solve problems in the “Linear Algebra” [135, 52, 153, 41, 44] and “Optimization” [154, 155, 156, 157, 158, 159] classes also use QFT as subroutine, with applications in the “Machine Learning & Data Science” and “Classical Scientific Computing” domains.

\( \bullet \) About QA and QUBO:

Again in the context of our dependency and correlation analyses, we have put all the Quantum Adiabatic (QA)-type algorithms under the same umbrella [37], especially in Fig5, and we will refer to them in the body of this paper as “QA algorithms”. As for solving problems in the “Linear Algebra” and (numerical) “Optimization” classes, one of the crucial usages of QA algorithms is state preparation [30, 160], which is often used as a subroutine among QLSAs. Another branch of usage of QA algorithms is (heuristically) solving problems in the “Combinatorics” class, where the Ising-QUBO formulation [72], as well as the the QAOA [74] (in the quantum Circuit Model context), give a uniform way of encoding solutions of many NP problems onto the ground state of an Ising-type spin Hamiltonian. With ingeniously designed adiabatic time scheduling, the minimum energy gap of the adiabatic Hamiltonian can be reasonably manipulated, which enables a time-complexity advantage [19]. Notice that, due to the capacities of QA algorithms in terms of combinatorial-problem encoding and time-evolution implementation, QA algorithms give rise to some NISQ experimental implementations (i.e., Quantum Annealing, or Coherent Ising Machines) [161, 162, 163, 164, 165, 166], which claimed to have shown evidence of quantum advantage.

\( \bullet \) About BS:

Again from the perspective of NISQ devices, Boson Sampling (BS)-based algorithms [167, 100], including Gaussian BS-based algorithms, also claimed to have shown evidence of quantum advantage in solving problems in the “Combinatorics” and “Stochastic Processes & Statistics” classes [70, 95, 168, 169, 170, 171, 172].

\( \bullet \) About the variety of quantum test algorithms:

Finally, quantum test algorithms form an indispensable toolbox for quantum-algorithm research. The Swap Test is used to compare two states and can be used for fidelity checks [105]. This solves problems in the “Linear Algebra” class. The Hadamard Test is used to sample the expectation of acting with a given operator on some input state, which enables to tackle problems in the “Stochastic Processes & Statistics” class [104]. In addition, the Hadamard Test is used to calculate the distance between two operators, which is meaningful for evaluating some quantum algorithms that update gate parameters iteratively and solve problems in the “Linear Algebra” class.

\( \bullet \) Discussing Fig5:

Overall, it is noteworthy that the quantum subroutines do not entirely dictate the relationship between application domains and mathematical classes. Instead, formulating application-domain problems into quantum-solvable mathematical problems by modeling and reduction is a crucial part of exploring practical applications of quantum computing. Moreover, a given, general field of application, actually often spans across several application domains: for instance, the task of drug design spans across application domains such as “Operational Research”, “First-Principle Quantum Simulation”, and “Machine Learning and Data Science”; indeed and hence – since we now merge (i) this large spectrum of applications with (ii) its influence on the precise mathematical problems to be solved, just to give an example of the fact that a strict distinction between both is not always that relevant –, depending on the scale the chemical properties of a drug can be modeled and analyzed either from first principles, semi-empirically, or entirely numerically, and the pharmacological and toxicological properties of the drug could also be optimized along these paths [173]. A counter example example is the Cryptanalysis application domain. The relevance of this application domain to mathematical problems is quite singular: quantum algorithms for solving hidden-subgroup problems are almost exclusively applied in this field, and they are based on QFT. All that being said, despite the complexity of Fig5, when faced with computationally intensive tasks, our typology could provide an effective perspective, offering potential quantum-computing users a path to find quantum speed-up solutions.

\( \bullet \) NISQ/LSQ criterion with respect to the two main criteria

Our analysis of the “NISQ/LSQ” criterion resulted in two figures illustrating the percentage of NISQ/LSQ algorithms categorized by mathematical classes (Fig6) or application domains (Fig. 7). Judging by Fig7, it seems that the most promising application domain is “First-Principle Quantum Simulation”, likely due to the variational algorithms that have been developed lately for this purpose, although full quantum simulation will still require LSQ devices. The “Classical Scientific Computing” application domain also seems promising, which is primarily due to the development of variational linear-algebra algorithms (also called QLSAs), as well as of other algorithms tackling dynamical-system problems. In contrast, there is a limited number of NISQ algorithms in the “Cryptanalysis” application domain, which is due to the fact that this domain prominently depends on the QFT. The large percentage of “Stochastic Processes & Statistics” algorithms falling into the NISQ category is due to the probabilistic nature of stochastic process problems, which are often tackled with sampling algorithms and seem more compatible with noisy devices, showing great promise in achieving quantum-computational advantages.

\( \bullet \) Other criteria

Let us close this analysis section by commenting about some correlations unraveled by the classification table, and involving secondary criteria such a being “Heuristic/Proven”, “Oracular”, “Promise”, or the subject of “Partial dequantization”.

We notice that NISQ is correlated with being a -heuristic algorithm. This is as one would expect. The general philosophy behind NISQ algorithm design is to obtain results in spite of errors and limited resources – which, in particular, unfortunately implies that any advantage is going to be hard to prove in this context. Conversely, LSQ is correlated with Proven algorithms, showing that whenever researchers are working in this more idealized setting, they do strive for proofs of quantum advantage. Similarly, the correlation between being Oracular or Promise, which plant an even more idealized setting, with being Proven, comes to reinforce this understanding. Let us close this loop by noticing that there are almost no Oracular NISQ algorithms – again, because the NISQ rationale to come up with things that work right now on present-day hardware, is somewhat orthogonal to the mathematically inclined considerations of there existing some abstract oracle.

Within LSQ, the mathematical class “Combinatorics” is correlated with the algorithm being Oracular. This is likely explained by the fact that the AA algorithm (and search algorithms in general) are Oracular. The mathematical class “Dynamical Systems”, on the other hand, is anti-correlated with being Oracular, likely reflecting the importance of the QPE and Non-oracular VQE algorithms in this field.

Partial dequantization mainly affects Proven LSQ algorithms, and mainly from the mathematical class “Linear Algebra” but not only (some “Optimization”, “Combinatorics” and “Dynamical Systems” problems are impacted), and mainly from the “Machine Learning & Data Science” application domain but not only (some “Operational Research” and “First-Principle Quantum Simulation” applications are affected). This is mostly explained by the underlying use of the QPE and HHL algorithms.

The “Sampling” criterion does not show any significant correlation.

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