**1. Introduction**

This chapter was inspired by and dedicated to our friend and colleague Wasyl J. Malyj, PhD (April 30, 1947–Oct. 6, 2014), shown in **Figure 1**. Wasyl was a scientist, bioinformatics pioneer, and early adopter of the central importance of heart rate variability (HRV) in health and wellness. Wasyl's achievements spanned fields as diverse as computer science, bioinformatics, genomics, nutritional science, and most relevant here, the significant role of heart rate variability in human health and performance. His role as Founding Director of the Medical Informatics Program at the University of California, Davis, and School of Medicine brought together his expertise in biomedicine, genomics, and advanced computational and network analysis skills, which he applied productively to the genomic understanding of nutritional science. We are in awe of his intellectual contributions and pay tribute to his role as a mentor and supporter of developing scientists and engineers, including ourselves. As his career matured, Wasyl became increasingly involved in developing applied technologies to harness HRV for health care research and practice, and examples of his contributions will be cited in the latter part of this chapter.

Integrative Management is a subcategory of i4P Health [1], which identifies the centrality of integrative, personalized, prescriptive, preventive, and participatory principles and practices in safe and effective health promotion, care and maintenance. Integrative Management emphasizes that the relationship between healthcare practitioner and patient is central to achieving an outcome of improved health and wellness. This is often referred to as "empowering the patient", who is seen as a central member of the treatment team. In this paper, we explore the advantages of using the integrative approach to managing chronic stress, and how new and emerging technologies clearly lead to successful outcomes in this area of health promotion. Integrative health and medicine focus on the whole person and make use of all appropriate assessment and therapeutic approaches that are informed by evidence. Integrative health care is inherently transdisciplinary. Inter-professional and traditional allopathic medicine

*New and Emerging Technologies for Integrative Ambulatory Autonomic Assessment… DOI: http://dx.doi.org/10.5772/intechopen.104092*

is but one of several aspects of holistic health care. The Integrative Management framework depends on the diagnosis, treatment, and prevention of disease provided by a team of allied health professionals that includes the health-seeking individual, making optimal personal health and not just simply medical disease management, the central focus. Thus, this approach broadens the focus to include a comprehensive set of independent (diagnostic) and dependent (outcome) measures.

### **2. Autonomic nervous system functioning**

While the important role of the autonomic nervous system (ANS) has been historically underappreciated, recently there has been a rapid proliferation of empirical, methodological, and theoretical progress in our more detailed understanding of the ANS. Previous more simplistic models of the role of the ANS using the construct of homeostasis have been enhanced by the use of the construct of allostasis and a wide variety of technological innovations including wearable and implantable biosensors have led to improved understanding of both basic and applied knowledge. This chapter will explore in particular, heart rate variability (HRV) as a rich and complex variable that has generated extensive literature, beginning with predicting all-cause mortality, but now encompassing a wide variety of disease and illness states; cognitive, affective, and behavioral processes and performance optimization. A critical analysis of HRV from the perspective of complex adaptive systems and non-linear processes will be included and innovative future uses of HRV will be described.

Normal ANS function reflects an adaptive level of interplay between the parasympathetic nervous system (PNS) and sympathetic nervous system (SNS) that produces a response to physical and psychophysiological challenge and stress. The PNS produces cardiac deceleration ("rest and digest" or "tend and befriend") and the SNS produces cardiac acceleration ("fight or flight" and stress response), while extreme stress elicits the "freeze" or "deer in the headlight" response, technically termed death feigning and can actually lead to death. However, chronic stress develops into hyper-arousal of the SNS, a process referred to as "HPA overdrive" because of the involvement of the hypothalamic-pituitary-adrenal axis. HPA overdrive causes excess glucocorticoid signaling, receptor downregulation, an end to normal negative feedback regulation of the stress response, and proliferation of peripheral pro-inflammatory cytokines by catecholamines. Thus, chronic stress reinforces more stress responding in a feed-forward cycle accompanied by a neuro-modulator presentation that is similar to depression. Psychological catastrophizing and rumination further augment the prolonged stress response and are core aspects of the expression of chronic stress. The work of Bruce McEwen of Rockefeller University and associates on allostasis and allostatic load identifies adaptive and maladaptive outcomes in the stress and coping process.

The negative health effects of chronic stress can be reduced by Autonomic Self-Regulation (ASR) because ASR dampens HPA hyper-arousal, calms the SNS, stimulates robust PNS activity, and restores normal ANS function. ASR further empowers patients to overcome the psychological sources of stress that accompany chronic nociceptive pain and self-regulate their emotions. ASR is defined as the technique of Heart Rate Variability (HRV) Biofeedback (HRVB) that incorporates (1) paced resonant frequency breathing (RFB), (2) focused attention or Mindfulness, and (3) positive emotional cognitions including those such as acceptance, compassion, gratitude, prayer, and love. ASR can rehabilitate the ANS that has been dysregulated by sensitized chronic nociceptive

pain. Although HRV can be quite simply defined as a variation in the time interval between heartbeats recorded either from the ECG or a plethysmographic (pulse) sensor, this simple definition belies the complexity that exists in both the quantitative analysis of inter-beat interval (IBI) data and the fundamental systemic physiological processes that underlie HRV. Healthy HRV contains a regular pattern of increasing and decreasing IBI's between consecutive beats that increases HRV, while unhealthy HRV is relatively low due to either little variation between IBI's or random, unorganized differences between consecutive beats. In the late 1970s, low HRV was found to be a powerful clinical predictor of sudden cardiac mortality after myocardial infarction [2, 3]. By the late1980's, research revealed that adaptive cognitive performance was related to high HRV [4] and certain forms of mental disorder were related to low HRV [5].

### **3. Biofeedback**

Biofeedback in general is simply defined as the process of gaining greater awareness of physiological functions using instruments that provide information on the activity of those same systems, with the goal of being able to control them volitionally. In addition to HR, physiological processes that can be controlled with biofeedback include electroencephalogram electromyogram and skin conductance. Clinically accurate measurement of IBI dates back to the beginning of the twentieth century, but it was the electronic digitization of computer software and increased computing power that made the quantitative analysis of HR and calculation of HRV easy and simple, accounting for the proliferation of interest in HRV. While breath or breathing training is an ancient practice with numerous forms, the production of HRV Coherence depends critically on RFB which paced breathing around 6 breaths per minute. The response of the ANS to RFB increases the amplitude of HRV rhythmic variation because 6 breath cycles/minute (= 10 seconds/cycle = 0.1 cycle/second =0.1 HZ) is the resonant frequency of the entire cardiovascular system (respiration, heart rate, baroreflex, and vasomotor tone) and parasympathetic outflow peaks [6]. Today, HRVB is widely and popularly taught and learned globally, building on the work of Gewirtz and Lehrer. Continual improvements in software algorithms and hardware have produced tools that are more efficient, more sensitive, more adaptable, more meaningful, and better visualized for collection and analysis of HRV data.

## **4. Heart rate variability**

While HRV monitoring in the past has been done as a static 'snapshot' of HRV status with sensors cabled to a desktop or laptop computer, the future is ambulatory, real-time, and dynamic in naturalistic settings. Development of different platforms is being solicited by small business development grants by NIH/ARPA-H, DoD/DARPA, and NSF, in the private entrepreneur market, and university research to create wearable systems that are effective for reliable measurement of IBI in naturalistic environments, including home, employment, and battle settings. But many issues remain to be resolved before this movement gets going full steam into clinical practice and utility for clinical pain management: HRV does not have an accepted "gold standard" definition; not all devices have bridged from research quality to FDA approved; questions of privacy, confidentiality, and HIPAA compliance for HRV data are being confronted; third party reimbursement is poor to non-existent.

### *New and Emerging Technologies for Integrative Ambulatory Autonomic Assessment… DOI: http://dx.doi.org/10.5772/intechopen.104092*

Fitness watches continuously track HR and can transfer data to a software dashboard that can compute HRV. Recent clinical research with an Apple watch app tracked people with epilepsy and found that seizures are often related to stress and missed sleep. Small chest patches with electrodes contain highly miniaturized fully-featured circuits for ECG detection. Vests are available that have HR electrode sensors and include additional sensors such as 3-axis accelerometers, respiration, skin conductance, and even more sophisticated physiological measures such as skin and ambient temperatures, "pulse-transit-time" (an indirect measure of systolic blood pressure), and EMG. The ultimate goal in signal acquisition is an entirely unobtrusive and longlasting electrode array and signal transmission system which connects with networked devices since belts, cords, and watches are frequently not used reliably over long durations by many users. The rapidly advancing technology of biofilms with electronic monitoring capabilities will certainly assist in creating an unobtrusive, durable, and reliable HR acquisition technology. In addition to engineering fixes to this problem, the Quantitative Self Movement is making this type of assessment more culturally acceptable, with some individuals even having hardware installed in their bodies.

Ambulatory HRV monitoring has become a player in the health informatics "big data" movement. What is envisioned is having wireless transmission of HR data through processing algorithms in the cloud or through separate servers. The largescale application of this plan falls into data mining protocols, from which new and important insights about basic HRV properties can be extracted. On the individualized level, transmitted HR data can be analyzed for comparability with normal healthy individuals and known physical and mental clinical populations as has been demonstrated by Jarczok et al. [1, 20].

Beginning in 1984, Wasyl Malyj's work anticipated the current groundswell of interest in heart rate variability and the use of biocomputation in the analysis of complex data sets. Since his initial work, the tiny literature has grown exponentially and now includes over 21,000 citations in a recent PubMed search of the term heart rate variability or HRV. Malyj's patented "Trainable adaptive focused replicator network for analyzing data" classifies signal patterns using array elements that "learn" to replicate predetermined sub-groups. This advanced wireless signal processor inputs physiological measures through large-scale databases and Malyj's patented FFT/neural network and pattern recognition algorithm. The result is fast matching of patient data to (1) provide predictive warning of acute health crises, and (2) real-time evaluation of diagnostic & treatment options for complex patient needs, using matched clinical records from other, similar cases. This is the bridge to individualized medical care plus a way to fill gaps in patient-doctor communication, as opposed to the averaged approaches dictated by today's dominant insurance/corporate models for efficient health care this, then, represents a very distinct form of personalized medicine.

Surely, one futuristic method of HR monitoring which is now a reality is remote real-time detection of pulse. Researchers have successfully deployed several different systems that measure pulse with as much accuracy as ECG: near field radar embedded in a smartphone camera programmed to display pulses as micro-movements invisible to the eye; video processing algorithm magnifying subtle changes in color reflecting redness due to pulse pressure on skin; microwave Doppler radar and more speculatively, satellite measurement of carotid artery pulsations. This is an age when the science-fictional Star Trek medical tricorder for whole-body scanning examination is no longer apocryphal (see discussion of "Berkeley Tricorder" that follows). And the acquisition, algorithmic analysis, real-time therapeutic feedback, and actionable information based on HRV are within technological reach.

Unfortunately, despite the optimistic outlook that follows from the recounting so far of opportunities for individualized and personalized integrative management of health and wellness, we must acknowledge that the United States healthcare care system suffers from significant conceptual and operational shortcomings. Theoretical and conceptual limitations of the traditional biomedical model are gradually being addressed and a fuller range of independent and dependent variables are being used, which include both individual factors and environmental issues. Because of these limitations, the United States with perhaps the highest per capita expenditure of health funds has both lower quality of outcomes and safety of the modern industrial nations. These relatively poor results come from both business practices by both the pharmaceutical and healthcare industries and inadequate governmental and regulatory These multiple factors are poorly integrated conceptually but this chapter proposes an initial synthesis of emerging technologies that can contribute useful, practical, and inexpensive indicators of health status and outcome, both for clinical and population health interventions. Such technologies can also be used for interventions such as biofeedback and patient education and self-regulation. It is axiomatic that from the scientific standpoint which can guide rational policy, without reliable and valid metrics, our understanding and ability to act is severely limited. While far from definitively addressing all of the multifarious issues of health care, a wedding of advanced technologies will catalyze progress in integrating consilient scientific knowledge. The potential role of HRV as a catalyst emerges as a practical way to improve this condition since it is a highly sensitive indicator of a wide variety of pathological conditions, diseases, and health-related phenomena. This venture is in the early development stage, but the concept has been proven and demonstrated and awaits the applications of appropriate resources to advance to operational capability.

Historically, major improvements in human health have come from public health interventions that target technological factors such as creating sanitary and salubrious environments. John Snow's removing the Broad Street pump handle terminated the London cholera epidemic and temporarily shutting down their coal-fired power plants ended a killer smog. As technology has continued to evolve, the concept of "smart technology" has emerged and led to the term Internet of Things (IoT). It is now reasonable to propose an Internet of Healthy Things (IoHT), which should be conceptualized as a public utility, rather than a consumer marketed commodity. Capra and Luisi's [7] complex adaptive systems framework can be applied to use sensor acquired bio information with networked cloud computed deep learning algorithms to produce significant improvements in health for both individuals and populations. We will now describe an emerging opportunity to use remotely acquired and network processed HRV to catalyze such a development.

## **5. HRV application**

While the understandable major concern of most individuals is their own health status and that of their primary support network, understanding of population health and its dynamics are essential to informed policy development and practice guidelines. This is extremely timely since the current ongoing global SARS Cov 2 pandemic has devastated the health and lives of millions and the lack of monitoring and testing has been a major determinant of the unfortunate course of the multiple waves of variants of concern. Several examples of previous work that set the stage for the proposed

*New and Emerging Technologies for Integrative Ambulatory Autonomic Assessment… DOI: http://dx.doi.org/10.5772/intechopen.104092*

HRV model will be described as preconditions for a "perfect storm" of technological evolution [8, 9].

The first example is described by the National Institute of Standards as "Analysis as a Service" (AaaS) [10] and was developed by IBM [11]. Watson Analytics (WA) carries out multiple cloud-based data analyses and displays them in multiple formats as shown in **Figure 2**.

Guidi et al [12] used WA to demonstrate proof of concept for a cloud-based HRV data acquisition and analysis system which can make accurate clinical diagnostic decisions differentiating patients with Heart Failure from normal individuals on the basis of HRV. As illustrated in **Figure 3**, the process involves data acquisition using the PhysioBank and PhysioNet [13] to obtain and categorize standardized ECG data sets into the appropriate format of R to R intervals using the PhysioNet HRV Toolkit. The accuracy of prediction using HRV is displayed in **Figure 4** and compared to data from published literature, These statistics were compared to the data available in the current literature and predictive accuracy of 90% was derived. This study demonstrates proof of concept that cloud computing can generate accurate HRV data. **Figure 4** shows the results concerning accuracy of prediction using the Total Power HRV (TOT\_PWR) statistic with predictive accuracy data.

The second example is the work of King et al. [14], who showed that on-scene accident triage decisions using a brief remotely obtaining HRV sample produced superior decisions to those of on-scene EMTs when requesting expensive but potentially life-saving helicopter evacuation. They used the standard deviation of non-normal intervals (SDNN), one of the candidates for broader use of HRV, and showed a sensitivity of 80% and specificity of 75%. Both the King nad Giudi studies provide proof of concept for the important role that HRV can play in healthcare settings, including triage and other diagnostic processes. It has also been demonstrated [8] that HRV can detect septicemia well before any clinical symptoms or signs emerge and that COVID-19 can be detected seven to nine days before symptoms emerge [1]. Beyond identifying pathological states and conditions, HRV has also been used to study important psychosocial functions such as executive functioning and resilience [1, 15].

As has been suggested in my previous work [8], technical developments in biosensors, microelectronics, computer networking, algorithmic data analysis such as deep learning, psychological self-regulation, and control allow a synergistic confluence which allows multiparameter continuous individual or population data

### **Figure 2.**

*IBM's Watson Analytics multiply modalities. Reproduced from Guidi et al. [12].*

**Figure 4.**

*WA Results using HRV Total Power as the predictor in the Guidi et al study. Reproduced from Guidi et al. [12].*

for both assessment and intervention by means of a miniature electronic device. Interestingly, such a device, shown in **Figure 5**, originally dubbed by Dr. Malyj the "Berkeley Tricorder" was indeed loosely described in the prescient science fiction of Robert Heinlein in his masterpiece *Starship Troopers* and popularized in *Star Trek.* The therapeutic use of the device was described by Drury et al. [16]. The ECG data acquired can be easily analyzed by both linear and nonlinear HRV statistics. Using such devices in a networked fashion through secure encrypted data processing would realize the fictional ability Heinlein described where military team members would be in constant nonverbal communication and awareness of the functional state of each of their fellow combat team members. Such technology could easily be created to use algorithmic analysis of the aggregated HRV, respiration, and accelerometry data to indicate categorical personnel status indications: fully operational, impaired,

*New and Emerging Technologies for Integrative Ambulatory Autonomic Assessment… DOI: http://dx.doi.org/10.5772/intechopen.104092*

### **Figure 5.**

*Demonstrating the use of a wireless multi-parameter biosensor in conditions of rest, exercise, and recovery which was transmitted in real time to a laptop computer via Bluetooth Reproduced from Drury et al. [16].*

disabled or dead, which would allow mission sensitive special operations personnel to be instantly and continuously aware of overall team functional ability and allow for fine-grained command and control on scene and at higher levels of command for decision support.

A similar technological approach could be taken to routinely monitor individual and population health status and would facilitate the early identification of deviations from healthy health parameters. Rather than waiting for the emergence of symptoms

necessitating intensive, heroic, and highly expensive inpatient ICU treatment, this approach would constitute a less expensive Extensive Care System (ECS) which would blend population health, epidemiological methods with ipsative clinical intervention which could range from health promotion and disease prevention to multidimensional clinical treatment interventions. Such an application should be deemed Digital Epidemiology or Smart Health. Since this type of system would not require verbal input from patients, it could be used in assessing the health status of those who traditionally underreport symptoms, such as the elderly, and act as a check on possible overutilization because of the extensive baseline data available for individuals. Given the potential for Bluetooth enhanced bidirectional voice communication, if desired, verbal health promotion prompts and instructions could be easily delivered as well. Further, the use of smartphones with apps, which are widely used worldwide, even when little or no more conventional infrastructure exists, is practical and scalable.

This type of performance capability monitoring would also be valuable in vocational settings where fatigue and exhaustion are factors since the system described here could be enhanced with a continuous performance task, which would detect increasing signal detection errors, a sensitive measure of fatigue. The same technology could be configured to assess the ongoing ability of elders and vulnerable populations for independent living and detect the early onset of symptoms and HRV, a biomarker of disease, disability, and functional status. A similar data acquisition system using Bluetooth connection to cell devices could function as a digital epidemiology tool that would be particularly valuable in developing countries where cellphones are a primary means of communication. With the addition of EEG and EMG sensors, this device would be fully capable of conducting all-night polysomnography (PSG) in the patients' home, thereby surpassing the "gold standard" sleep laboratory PSG, with a "platinum" PSG in the natural sleep environment, eliminating the well documented "first-night effect" of sleeping in the foreign setting of a sleep laboratory, and enhancing the ecological validity of the field of polysomnography.

These uses are being facilitated and expanded by the rapid development of advanced miniaturized sensors and data acquisition materials. For example, Blaschke et al. [17] have described the use of flexible graphene-containing solution gated field-effect transistors to acquire high fidelity EEG signals in a noninvasive and unobtrusive manner. Similarly, Coleman and colleagues [18], with support of the Gates Foundation have described the use of ultra-thin stretchable and flexible devices which include adhesive peeled attachment nodes for long-term continuous monitoring of electrophysiological data. Thus, the field of advanced materials is progressing rapidly and can play an integral role in the development of iP4 Health, as can developments in genomics, data mining, cloud computing, regenerative medicine, and microbiomics [18], which have high synergistic potential.

An example of this synergistic potential is the use of HRV and other ANS techniques and concepts in the area of stem-cell and regenerative medicine. Gogolu et al. [19] summarize literature demonstrating the viability of using pluripotent human stem cells in generating enteric nervous system progenitors, while Major et al. [20] outline the step-wise differentiation of forebrain late oligodendrocyte progenitor cells (OPCs) from human pluripotent stem cells in defined chemical in vitro culture conditions. The enteric nervous system (ENS) is a key component of our enhanced view of the ANS, described as the Central Autonomic System by Thayer and Lane [21] and Benarroch [22]. Recently, Liu [23] has summarized the important and complex relationship between the microbiome, stress, and HRV. The close relationship between

*New and Emerging Technologies for Integrative Ambulatory Autonomic Assessment… DOI: http://dx.doi.org/10.5772/intechopen.104092*

the nervous system, stem cell biology, and the microbiome is highly significant as an area of great importance for further research. In particular, stem cell interventions may allow modification and repair of key anatomical and physiological structures and processes.

## **6. Conclusion**

Given the importance of global health highlighted by the Gates Foundation's Grand Challenges and others [24], a great advantage could be obtained by the use of the rapidly proliferating cellular networks that have leapfrogged traditional wired telephony and made higher computing power available through smartphones. The multi-parameter data acquisition, processing, and analysis system described above could be easily integrated into existing cellular networks and provide extensive health status monitoring in less developed and poor areas of the world. The great advantage of HRV is its high sensitivity to a very wide and diverse inventory of disorders and conditions, although it is not highly specific in identifying discrete pathology. This makes it ideal for ongoing primary health surveillance and screening in the natural environment, while follow-up evaluation is focused on specific identification and treatment of the condition. Given the digital nature of the HRV signal (interbeat intervals), it also streamlines algorithmic analysis and case identification to health personnel.

We now have the opportunity to apply new HRV technologies and algorithms in a dynamic way for a modest cost to yield powerful gains in research and development of individualized i4P health enhancement. One starting point is the use of technologies for ambulatory self-monitoring, with reliance on point-of-service medical service resources reduced, lowering costs with fewer side effects. The approaches described here represent an inflection point for translational research and development which may advance health care significantly. Despite the clearly inadequate conceptualization and deployment of the current "healthcare" system (actually a "cost containment, chronic disease management" system), the bottom line proposed here is using HRV with a suite of sister technologies as a catalyst for better health, safety and quality of life and more efficient allocation of expensive healthcare resources in an accessible manner to achieve truly smart health and wellness(IoHT).


*<IP_ADDRESS> Cocktail effect*

Cocktail effect affects the properties of the system designed. It includes the compositional mixtures of elements where the consequence of the blend is both unpredictable and unexpected because of the distinctive properties the individual element provides. Adding an element to the mixture with properties realized will enhance the combination through the cocktail effect. High entropy alloys may exhibit properties dependent on the reaction between elements in the system. On the off chance that a high-temperature high entropy alloy is desired, elements with elevated temperature strength ought to be used.


**Mauro Mezzini**

Mauro Mezzini has been an Assistant Professor of Computer Science at the Roma Tre University since 2012. Previously, he was Adjunct Professor at La Sapienza University (2005–2011) and Project Manager at Telecom Italia (1990–2012). Some of his works have been published in SIAM Journals on computing, aCm transaction on information and system security, discrete applied mathematics and theoretical computer science. He has been a Guest Editor for the Special Issues "Quantum Computing Algorithms and Computational Complexity" and "Quantum Computing Algorithms and Quantum Computing Simulators", both belonging to the Mathematics and Computer Science sections of the Mathematics journal. His research interests include quantum computing, machine learning, graph theory and algorithm, databases and discrete mathematics.

## *Editorial* **Preface to the Special Issue on "Quantum Computing Algorithms and Computational Complexity"**

**Fernando L. Pelayo 1,\*and Mauro Mezzini 2,\***

#### **1. Call for Papers**

In 1982, Richard Feynman stated that in order to simulate quantum systems, we would rather go for a sort of brand-new powered quantum processor instead of a classical one. Since then, Quantum Computation has been growing in terms of both architectural issues associated with such quantum computers and the algorithms that can be run with them. All this has attracted much interest from the computer science community.

Just to mention some facts, it is obvious that the intrinsic parallelism that comes with the superposition of quantum states together with interference features provides us with a very good perspective to deal with heavy computational problems, such as encrypting/decrypting tasks or studying quantum issues of matter.

Quantum computing is a hot field of research at the intersection of mathematics, computer science, and physics that promises to significantly revolutionise many aspects of the technology industry such as medicine, machine learning, artificial intelligence, cryptography, and operations research to name a few. Investors and governments from all over the world promote its development considering that it is crucial and of strategic importance for countries, companies and, therefore, society as a whole. The huge investments in resources to develop quantum computing by countries such as China, India, the United States, Russia, and so on only confirms this reality.

This Special Issue was mainly concerned with quantum algorithms, the mathematics underlying them, and those complexity issues arising from them.

## **2. Published Papers**

This is a Special Issue of *Mathematics* belonging to the section "Mathematics and Computer Science", which was closed on 30 June 2022.

A total of 13 papers were submitted to it, of which 7 have been accepted. This represents an acceptance ratio of 53.8%. The average time for accepted papers to be published is 43.4 days.

The main contributions of these seven papers are the following:

Two of these seven papers are focused on improving the performance by means of quantum algorithms over the best instances of classical ones:

**Citation:** Pelayo, F.L.; Mezzini, M. Preface to the Special Issue on "Quantum Computing Algorithms and Computational Complexity". *Mathematics* **2022**, *10*, 4032. https://doi.org/10.3390/ math10214032

Received: 19 October 2022 Accepted: 21 October 2022 Published: 30 October 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

•

1

k", "the most frequent string search problem", and "searching intersection of two sequences of strings". Based on the quantum procedure for comparing two strings of length *k* in *O*( √*k*) queries, they are able to reduce time complexities, thus moving the factor *k* to √*k* in all its instances as parameters.

Another four papers deal with former quantum algorithms (or part of them) for which some improvements or different perspectives have been addressed:

Finally, there is a paper that researches a classical Operational Research problem by means of quantum annealing:

• Saul Gonzalez-Bermejo, Guillermo Alonso-Linaje, and Parfait Atchade-Adelomou, in their paper "GPS: A New TSP Formulation for Its Generalizations Type QUBO" [7], propose a new Quadratic Unconstrained Binary Optimization (QUBO) formulation of the Travelling Salesman Problem (TSP) with a smaller number of necessary variables, together with a thorough study of the constraints and their management. This study includes a practical test over D-wave quantum annealers platform.

As Guest Editors of this Special Issue, we would like to thank all the authors who make contributions on these quite conceptually similar fields of research.

We also would like to thank all the reviewers for their big effort in developing so constructive reports that contribute to improve the quality and quantity of the results provided within this Special Issue on "Quantum Computing Algorithms and Computational Complexity".

We hope that the research papers published in this Special Issue promote more extensive research and lend further support to quantum computing. We are believers of the wide and crucial effect that quantum computing can have in our society, from the domain of energy-efficient computing, through to high-performance computing, up to the management of many of the most challenging problems which still remain open.

**Author Contributions:** All authors have contributed equally. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.