and mitigated during each of these steps. Additionally, to build qubit architectures comprising dissimilar materials/chemistries (e.g., in hybrid quantum devices), heterogeneous interfaces between molecules and host materials need to be studied carefully and controlled. While current physical qubit proposals cover a spectrum of physical systems including molecular and solid-state spin centers (Miao et al. 2020), trapped ions (Bruzewicz et al. 2019; Georgescu 2020), superconducting (Devoret, Wallraff, and Martinis 2004), and topological systems (Freedman et al. 2003), all have decoherence and scaling issues that critically affect their potential as scalable, fault-tolerant systems. Exquisite control over constituent chemical environments at an atomic scale is central to understanding and controlling coherence and scaling in any qubit system. Chemists are needed at each step of this process:

1. Synthesis or fabrication of high-quality, chemically precise molecules and materials that have near-optimal or tunable quantum properties across the range of quantum architectures;
2. Development of a fundamental understanding of decoherence processes that arise from integrating molecular qubits into larger coherent quantum architectures;
3. Design and synthesis of chemical environments that are robust to decoherence that can decouple the fundamental degrees of freedom of qubits from unwanted environment-induced noise;
4. Bottom-up and top-down fabrication of heterogeneous material platforms that can transduce quantum information coherently from one chemical environment to another by leveraging surface chemistry and single-digit nanofabrication; and
5. Design and fabrication of routes to couple physical qubit systems, especially molecules, that retain their quantum coherence but that can be patterned or arranged in repeatable, identical arrays.

demonstrated precise quantum control of individual electron and nuclear spins, robust entanglement, electron–nuclear quantum registers, and increased control of the spin–optical interface at the level of single photons (Anderson et al. 2022). Moreover, driven by predictive theoretical work, several defect-based states have been engineered for practical applications including quantum memories and optical emission (Singh et al. 2022) in the telecommunications regime. Impressive quantum sensing of electric, magnetic, and strain fields has been shown along with dramatic improvements in small-volume and single-spin nuclear magnetic resonance (Morello et al. 2010). However, these material systems face considerable challenges for QIS and engineering, including the creation of scalable and precise quantum states and the mitigation of the impact of strain (Hruszkewycz et al. 2018; Rose et al. 2018), isotopic variations (Balasubramanian et al. 2009; Bourassa et al. 2020), decoherence-driven charge fluctuations from interfaces (Bluvstein et al. 2019; Sangtawesin et al. 2019), and unintentional dopants. Chemistry can play a key role in resolving many of these challenges, especially by exploiting the broad atomic-scale synthesis capabilities for a target qubit, the host matrix, and their mutual interactions. For example, an electron spin qubit interaction with a selected proximal nuclear spin (Graham, Krzyaniak, et al. 2017; Graham, Yu, et al. 2017; Yu et al. 2016) may be optimized through ligand design between weak and strong coupling regimes. In addition, spin-lattice relaxation (Amdur et al. 2022) and the operating temperature of a spin qubit may be engineered by modifying the chemical structure of the ligand and thereby the degree of vibronic coupling at the qubit frequency. Chemistry also provides an opportunity to enhance qubit coherence by controlling the symmetry of the qubit itself or its packing within its host environment (Bayliss et al. 2022; Headley et al. 2016).

The selection of appropriate hosts and defects, and their mutual interactions with one another, remains both a challenge and an opportunity for scientific research. In particular, opportunities that build upon the success of density functional theory (DFT) and coupled cluster expansion techniques in predicting coherent properties of solid-state spin defects (Seo et al. 2017; Weber et al. 2010) may enable the discovery of robust spin qubit candidates. There have been considerable advances in understanding the mechanisms of spin decoherence in semiconductors and solid-state nanostructures (Kanai et al. 2022). State-of-the-art synchrotron spectroscopies, coherent Bragg diffraction imaging, and high-resolution magnetic resonance techniques have proven to be powerful tools to improve the quality of host materials. In addition, improvements in first-principles approaches to understanding and predicting properties (e.g., DFT) have led to a deeper physical understanding of quantum decoherence and successful predictions for mitigating many sources of noise, from dynamical decoupling to unique pulse sequences. Furthermore, there have been successful efforts aimed at harnessing the capabilities of today’s electronic technologies for quantum-state control, including electrical control and readout of single quantum states (Sangtawesin et al. 2019), the creation of decoherence-free subspaces to locally enhance coherence (Miao et al. 2020), and the integration of photonics to both enhance the efficiency of quantum emitters and entangle nearby qubits (Dibos et al. 2018; Lukin, Guidry, and Vučković 2020; Wan et al. 2020). Recently, rare-earth ions in oxide semiconductor hosts—including silicon-compatible materials—have emerged with encouraging properties as single quantum memories, with impressively long coherence times and single-shot photonic readout.

4.2.1c Chemical Design of Superconducting Qubits

Chemistry also offers a potential path toward improving the coherence times of superconducting qubits. While currently one of the most scalable platforms (Arute et al. 2019), superconducting qubits are still plagued with hardware coherence issues that prohibit the realization of fault-tolerant systems (Sheridan et al. 2021; Siddiqi 2021). In superconducting systems (comprising qubits and a resonator), the central hurdle is mitigating decoherence processes that significantly suppress coherence times to well below their theoretical limits. Decoherence processes in superconducting systems currently are dominated by materials-based decoherence through the inevitable inhomogeneities present in multicomponent materials platforms such as defects and interfaces (Krantz et al. 2019). Therefore, reducing materials-based losses in superconducting qubits requires extensive knowledge of the chemical and structural makeup of such complex chemical systems, and predictive routes to their control and mitigation via physical and chemical means. The former requires advanced spectroscopies and microscopies, in tandem with theory, to probe and uncover the structure–property relationships of these heterogeneous nanoscale systems (both their bulk and interfacial properties), often at the forefront of nanoscale characterization. For example, many superconducting qubit systems are made up of amorphous superconductors, for which conventional diffraction

and spectroscopies that rely on reciprocal space cannot be used; novel techniques such as fluctuation electron microscopy (Kennedy et al. 2020) can be used to probe these noncrystalline systems to elucidate information on the bonding environment. Current characterization needs also include nondestructive interfacial probes that can give information on the structure–property relationships of buried interfaces that are often nanometers in extent.

Defect-based decoherence in superconducting qubits is the known Achilles’ heel for superconducting qubits—these comprise primarily parasitic “two-level systems” (TLSs) that are commonly formed at the interfaces between different superconducting components (Martinis et al. 2005). As shown in Figure 4-3, in both Al- and Nb-based superconducting qubits (the most commonly employed materials platforms to date), native oxides form at the interfaces of these elemental systems, resulting in trapped structural configurations that make up the TLS. These metastable chemical configurations can tunnel between equivalent configurations, resulting in an oscillating dipole that electromagnetically couples to the qubit—resulting in decoherence (Phillips 1987). Chemistry can play a unique role in understanding and designing a more favorable chemical environment to reduce, or even eliminate, the chemical motifs associated with TLS noise and other defect-based decoherence. Understanding, reducing, and/or entirely circumventing these parasitic TLSs is a central challenge for developing superconducting circuits. For instance, TLS noise is directly related to the local bonding environment in amorphous suboxides—these suboxides can be passivated with appropriate selective chemical treatment of lossy surfaces (Altoé et al. 2022), and/or modified through doping and/or alloying to reduce the prevalence of the most deleterious TLS by controlling local chemical motifs/coordination (Hamdan, Trinastic, and Cheng 2014). Oxygen off-stoichiometry has been identified as a key cause of parasitic magnetic-based decoherence in Nb oxide thin films (Sheridan et al. 2021), providing a chemical indicator of decoherence in these systems. Another emerging area is the exploration of novel superconducting materials and chemistries as qubits and resonators that are more robust to deleterious TLS formation and/or are comprised of stacked van der Waals bonded two-dimensional (2D) materials, which can circumvent lossy interfaces between dissimilar three-dimensional elements (Altoé et al. 2022).

4.2.1d Chemical Design of Trapped Molecule/Ion Qubits

Ions and molecules that are electromagnetically confined by electromagnetic fields in ultrahigh vacuum are “trapped” ions and molecules. Upon laser-cooling to their motional ground states, they can be used as multiqubit systems (Bollinger et al. 1991). Decoherence for trapped ions is related to both the electric field noise of the laser/trap system and materials-based losses from the electrode surfaces, which are shown to reduce multiqubit

gate fidelities (Brown et al. 2021). Understanding the microscopic origins of these decoherence channels is an ongoing challenge requiring exquisite knowledge and control of the electrode composition—possible sources include vibrations (Wineland et al. 1998), excitations, surface diffusion of chemisorbed atoms/molecules (Kim et al. 2017), and fluctuating “patch charges” on inhomogeneous electrode surfaces (Hite et al. 2013). Current and future challenges for trapped ion/molecule scaling will require surface chemistry approaches (Figure 4-4) to understand lossy electrode surfaces and to design robust, noise-tolerant components and surface treatments that can be cycled for several operations.

4.2.1e Chemical Aspects for the Design of Topological Qubits

While still in their infancy with regard to their demonstrated potential for QIS systems, topological qubits present an exciting possibility for robust, scalable qubits (Kitaev 2003). Owing to its fundamental nature (e.g., the topology of the material), the topological qubit should be inherently robust to certain external perturbations such as defects, strain, and interfaces—in stark contrast to other qubit platforms—and does not require complex error correction. However, while these perturbations often do not destroy the topology of the material, they can make identifying reciprocal-space features extremely difficult by inducing mid-gap states. This, in particular, is the case for both identifying and manipulating Majorana zero modes—the fundamental topological qubit—in physical systems (Nayak et al. 2008). While several routes to achieving Majorana zero modes have been theoretically proposed, no experiments to date have shown definitive evidence of their appearance and braiding (Kayyalha et al. 2020; P. Yu et al. 2021), which motivates materials and chemical approaches to achieving these highly sought-after emergent particles (Figure 4-5; Flensberg, von Oppen, and Stern 2021). Current chemical and materials approaches

to combining strong spin–orbit coupling, superconductivity, and appropriate time-reversal symmetry breaking to achieve Majorana zero modes include semiconductor/superconducting interfaces and molecular/superconducting interfaces (see Figure 4-6). Huge surface chemistry efforts are required to make clean, precisely controlled surfaces with deterministic molecular placement and/or thin nanowires to pass the topological protocol.

4.2.1f Designing Hybrid Quantum Architectures That Mutually Enhance Each Other’s Quantum Properties

Hybrid quantum computers synergistically combine the strengths of classical computing with the opportunities of quantum computing. For instance, a classical optimization algorithm can be used to guide the quantum circuit parameters, while the quantum components solve subtasks more efficiently than would be possible classically.

Further information is detailed below about hybrid quantum–classical algorithms, such as the variational quantum eigensolver (VQE), which have been applied to a range of problems in quantum chemistry. However, for their practical implementation, hybrid quantum computers must additionally combine the hardware requirements of quantum computers (i.e., scalability, coherence, and low error rates) with traditional classical computing components. This will require efficient and low-loss integration of qubit systems with conventional semiconductor technologies such as complementary metal-oxide semiconductor circuits.

4.2.2 Developing Noise Models and Quantum Error-Mitigation Techniques for Individual Qubits, Systems of Qubits, and Quantum Architectures That Can Be Experimentally Validated

No physical system is able to store and operate on information perfectly without errors. Indeed, one of the biggest challenges for quantum information protocols is protection from noise. Error-mitigation techniques that reduce noise thus have to be applied to qubits just as they are routinely applied to classical bits. In contrast to error-correction strategies, error-mitigation strategies involve developing noise models and do not necessitate making and manipulating copies of qubit information, which often makes them more practical in the context of chemistry. Such techniques could accelerate the adoption of moderately imperfect qubits and may enable the use of larger arrays of qubits at the same error rates as smaller arrays without correction, paving the way to larger, multiqubit technologies (Kandala et al. 2019).

But, to apply error-mitigation techniques, detailed models of qubit errors that depend upon the details of the specific qubit technology must be developed. If error models are known, they can be used to analytically reduce errors. Often the origin of this noise stems from molecular processes, meaning that first principles of computational modeling can play a key role in determining the source of the noise and analyzing its form to perform corrections. For example, substantial research has been conducted on quantitatively developing error models for superconducting qubits (Burnett et al. 2019) and ion traps (Foulon et al. 2022) based on the theory of TLSs. Research points to chemical defects at the surface of the oxide and other materials that make up superconducting qubits (Altoé et al. 2022) and the presence of adsorbates on the surface of the electrodes of a trapped-ion quantum computer as leading to anomalous 1/f noise that causes related qubits to decohere rapidly (Bruno et al. 2015; Foulon et al. 2022; Noel et al. 2019; Sedlacek et al. 2018; Wang et al. 2015). First-principles computational simulations of the chemistry at these surfaces enable one to predict how this noise varies based on defect and adsorbate concentrations, adsorbate type, temperature, and frequency, thus yielding models that can be applied directly to measurements to reduce noise and related errors (Aliferis and Preskill 2008; Georgopoulos, Emary, and Zuliani 2021; Tuckett, Bartlett, and Flammia 2018).

Similar models could be developed for molecular qubits. Simple proof-of-principle quantum algorithms for quantum error mitigation have been demonstrated in nuclear spin-based qubits (qudits), where qudits are d-dimensional (d > 2) quantum systems. Molecular strategies for this typically involve single-ion magnets, in which the electron and nuclear spin states are weakly anisotropic or exchange-coupled, leading to multiple spin states that are low in energy. For suitably engineered systems, unequal energy spacings allow addressing via microwave-resonant pulses (Aguilà, Roubeau, and Aromí 2021; Chicco et al. 2021; Gimeno et al. 2021; Hussain et al. 2018; Jenkins et al. 2017; Luis et al. 2011; Moreno-Pineda et al. 2017, 2018). Molecular electron–nuclear spin-based qudits (Ferrando-Soria et al. 2016; Godfrin et al. 2017; Luis et al. 2020) have been utilized for the implementation of quantum mitigation codes, where the molecular systems function as noisy intermediate-scale quantum (NISQ) units (Chiesa et al. 2020, 2021; Macaluso et al. 2020) and the electronic structure is “protected from decoherence.” Taking advantage of nuclear spin structures with greater than two quantum levels (d > 2) in qudits leads to the possibility for long coherence times, due to isolation of the system from the environment, but, consequently, long manipulation times. Strategies to shorten the manipulation times for gate operations involve taking advantage of electron–nuclear coupling (hyperfine interactions) to perform operations on nuclear spin states at rates much shorter than the decoherence times (Castro et al. 2022; Chizzini et al. 2022; Hussain et al. 2018). A combination of further experiments and theory will be necessary to realize these nascent error-mitigation strategies (Castro et al. 2022; Chizzini et al. 2022; Hussain et al. 2018). One should distinguish important problems related to calculating excited-state energies from more formidable challenges of calculating entanglement or coherence in molecules with similar approaches. While the committee realizes that both are very important to chemical understanding, the calculations and approaches should specify the problems to be solved and the potential outcomes initially.

systems combined with the typical desire to understand the decoherence processes involving many electronic and phononic degrees of freedom at play in these materials makes directly determining their electronic structure a formidable challenge (Philbin and Narang 2021). While algorithmic advances, including linear scaling and tensor contraction techniques (Hohenstein, Parrish, and Martínez 2012), continue to be made that accelerate the direct modeling of these systems, scaling challenges are often mitigated by different forms of divide-and-conquer techniques: methods that divide the system into smaller portions, which can be modeled with highly accurate theories that can either be stitched back together to recreate the electronic structure of the entire system (Pruitt et al. 2014) or integrated into lower-accuracy descriptions of larger portions of the system.

One of the most productive divide-and-conquer strategies for qubits is embedding (Huang, Pavone, and Carter 2011; Sun and Chan 2016). In embedding theories, a smaller portion of a system that necessitates high-accuracy quantum simulations using techniques such as multireference perturbation theories (Shavitt and Bartlett 2009), CC (Shavitt and Bartlett 2009), QMC (Foulkes et al. 2001), or configuration interaction (Szabo and Ostlund 1989) is modeled surrounded by a larger portion of the system that is treated using lower-accuracy quantum or classical methods. In the context of quantum information systems, the atoms, vacancies, or complexes that constitute the system’s physical qubits are typically modeled with high accuracy, while their hosts are modeled with lower accuracy.

In regard to the various theoretical mechanisms for coupling defect center–based spin qubits to other qubit platforms, potential interface coupling mechanisms such as superconducting qubits, other defect centers, and photons were discussed Wang and colleagues (2021) and are graphically illustrated in Figure 4-7. For the case of gigahertz platforms, dipole-, phonon-, and magnon-mediated mechanisms have been demonstrated. In the case of a dipole-mediated process, challenges remain in the selectivity of the interface mechanism due to the requirement for strict distance and orientation dependence between qubits. While the cavity phonon–mediated dipole coupling may overcome these limitations, other limiting factors present further challenges for the cavity-phonon mechanism. For example, in this mechanism, dipole selection rules limit the use of certain qubit states. Also, further

investigation could be undertaken to determine the usefulness of this mechanism, as there may be limitations in the ability to tune two qubits, on demand into resonance for fast entanglement. Another solid-state interface coupling mechanism is that of phonon-mediated quantum interfaces. This approach seems to have the potential to couple qubits over arbitrary distances as long as the qubit states exhibit strain susceptibility. Wang and colleagues (2021) suggest further investigation and development at the nanoscale regarding the systems engineering of the cavity, the materials used, as well as a deeper understanding of the phonon-mediated mechanism. Finally, a more recent approach utilizes a magnon-mediated interaction to effectively couple defect center–based spin qubits with other qubits. This approach offers an enhanced coupling rate in comparison to phonon-mediated interactions. Wang and colleagues (2021) stressed that nanomagnonic systems offer a unique opportunity for nanotechnologists and may have a significant impact on quantum device innovation and translation.

An alternative divide-and-conquer approach to modeling, particularly disordered quantum systems, is through sampling. Disorder is a leading cause of decoherence in many materials and thin films of use in QIS applications (Foulon et al. 2022; Harrelson et al. 2021; Ray et al. 2019). For example, in recent work, researchers studied decoherence in superconducting Nb thin films by sampling over forms of disorder (Harrelson et al. 2021; Sheridan et al. 2021). In sampling techniques, different portions of a typically heterogeneous material are randomly selected to represent the whole, are modeled using high-accuracy methods, and then are averaged to recreate the electronic structure of the larger system. The sampling can be performed according to a uniform distribution or, if information is known about the system, to a more complicated distribution through importance sampling. Alternatively, samples can be generated from snapshots of molecular dynamics simulations or replica exchange Monte Carlo simulations (Frenkel and Smit 2002). Even more recently, experimental data have been used to train machine learning models to learn forms of disorder present over length scales not easily attainable within simulations (Kilgour and Simine 2022). Given sufficient experimental data on surfaces, this presents a promising path forward while emphasizing the need for more detailed experimental characterization of quantum materials for quantum information.

These strategies are but a few of many possible strategies for capturing the complexity of quantum information systems without invoking descriptions that may be more expensive or detailed than necessary. Given these techniques’ inherent approximations, continued research could determine their accuracy, further improve their efficiency, and understand for which materials and properties they perform best. Properly modeling decoherence will moreover necessitate increased exploration into the most efficient ways to integrate descriptions of phonons and electromagnetism into the ab initio modeling of molecules and materials (Philbin and Narang 2021). The committee concludes that the computational modeling of large, heterogenous systems can be accelerated via embedding and sampling approaches on classical systems. Currently, the suite of embedding and other reduced-scaling techniques is underdeveloped. Improving these tools will ultimately lead to a deeper understanding of QIS challenges.

4.2.3c Theoretical and Computational Approaches for Modeling Entanglement

Even if scaling hurdles to simulating large quantum systems on a classical computer can be surmounted, quantifying entanglement and coherence in these systems remains a costly proposition because calculating these quantities requires an accurate accounting of multielectron interactions—often in the presence of phonons and other non-electronic interactions, and usually within a time-dependent framework.

In recent years, several new modeling techniques have emerged for computing entanglement (typically in terms of nth-order Rényi entanglement entropies) in interacting systems, including density matrix renormalization group (DMRG), tensor network, and QMC techniques. Nonetheless, these techniques scale steeply as high-polynomial powers with the size and number of electrons in the system. Even within these frameworks, the computation of the entanglement itself necessitates more time-consuming algorithms than the computation of more traditional quantities like energies and charge densities. To mitigate these steep scalings, attention has turned toward using embedding algorithms, which partition a system into a region that is treated with near-exact levels of theory and a surrounding region that is treated with less accuracy. In the context of quantum information, high-accuracy regions may, for example, be placed around transition metal or lanthanide atoms that host different spin states, while surrounding organic groups may be treated with lower levels of theory. Such embedding methods, including DFT-based embeddings, dynamical mean field theory, and density matrix embedding theory, enable only the

most physically/chemically important regions of a system to be treated, thus substantially reducing the expense of quantum calculations at the cost of modest approximations. In their original forms, these methods were overwhelmingly developed to treat electrons alone. In part inspired by QIS needs, more recent investigations have begun to incorporate phonons/vibrations (Sandhoefer and Chan 2016) into these treatments because of the important role that they assume in decoherence (Sheng et al. 2022). In sum, the committee concludes that computational or theoretical tools for modeling the entanglement and coherence within quantum information processors remain costly. The high cost to use classical computational tools for modeling quantum phenomena is inhibiting key research activities such as direct simulation of large quantum information processors.

4.2.3d Validating the Modeling of Quantum Information Systems

The synergy between experiment and theory can help uncover the molecular and electronic structure that can further develop and improve qubits. As simulations become capable of accurately determining essential properties such as T₁ and T₂ relaxation times (Ariciu et al. 2019) and the optical response in optically addressable qubits (Goh, Pandharkar, and Gagliardi 2022), models will continue to play an important role in understanding and predicting the feasibility of using a system in a QIS application. Validating computational models is often hindered by the lack of experiments on systematically altered systems. Graham, Yu, and colleagues (2017) demonstrated the power of systematic modifications to vanadyl complexes with increasing distance between vanadium(IV) and a propyl moiety that demonstrates the effect on decoherence as a function of nuclear and electronic spin distance. The growing infrastructure of automated and robotic systems (Seifrid et al. 2022) that synthesize and analyze holds great promise in generating both systematically modified systems and data that can hasten validations.

4.2.3e Leveraging and Developing Machine Learning, Chemical Informatics, Chemical Databases, Molecular Simulations, and Quantum Computing Algorithms to Inform and Facilitate Qubit Design

Materials-informatics approaches are used broadly across the fields of chemistry, materials science, and physics to accelerate the discovery of novel materials and molecules. The explosion of computing power over the past two decades along with improvements in theoretical and computational approaches enable predictive calculations of many compounds, which can be collected into databases. The Materials Genome Initiative kickstarted the foundation of several materials databases with properties predicted from first-principles calculations with a range of applications and properties targeted. For instance, the Materials Project was originally developed for the discovery of battery materials but has expanded to have success across a range of applications including novel photovoltaics, multiferroics, and topological materials. It has recently been extended into the molecular space to include an MOF database (Rosen et al. 2021, 2022). Equivalent databases are starting to become available for molecular compounds (Organic Materials Database) and more complex systems (Automated Interactive Infrastructure and Database for Computational Science and Materials Cloud Two-dimensional Crystals Database).

The use of such materials and cheminformatics approaches for QIS applications is in its infancy yet presents an enormous opportunity to accelerate both the discovery of novel qubits and the understanding of their control in a systematic fashion. High-throughput searches have predicted new topological materials (Frey et al. 2020), and “low-throughput” calculations have been used to identify new spin defects in hexagonal boron nitride (Bhang et al. 2021). First-principles approaches such as these were used to search large combinations of possible defects in Si for new spin defect qubit candidates, with the resulting 400 combinations available in the Quantum Defects Genome database (Xiong et al. 2023). The first-principles approach includes careful benchmarking of the theoretical approaches with experimental measurements, which will improve the predictive power of such approaches. These tools can enable discovery through the exploration of compounds that are synthesizable and have yet to be tested for potential use in QIS technologies (Lilienfeld, Müller, and Tkatchenko 2020). Cheminformatics and machine learning have also been used to determine synthesis routes and to improve synthesis conditions through, for example, the design of experiments methodology (Weissman and Anderson 2015). Keeping the databases fully open and available would advance the field quickly.

computable methods are especially viable when the calculations do not require high accuracy and when systems are not “strongly correlated.” A strong correlation (high entanglement) of electrons tends to occur near bond-breaking transitions, in free radicals, and around transition metal centers. Much work remains in understanding how and for what problems quantum computers might provide otherwise inaccessible new chemical insights. First, even if one has an oracle for efficient and completely accurate energy calculations, this does not immediately resolve all chemically relevant questions. An insightful chemist is needed to interrogate the quantum computer thoughtfully to derive a chemical understanding. Second, quantum computers might scale polynomially, but that does not mean the calculations are free (or even as efficient as DFT); indeed, the simulation of chemical Hamiltonians still takes considerable quantum computing resources. Thus, currently, we remain limited in how many electrons can be explicitly treated on a quantum computer; we will need improved methods of embedding the most strongly correlated subsystems inside quantum computations, with the environment treated at lower levels of theory on classical computers.

Finally, as classical methods continue to improve, we face the prospect that the “truly hard” chemistry problems that require a quantum computer will become increasingly esoteric. The problem does not lie in the class of nondeterministic polynomial time (NP) but rather the challenge lies in the fact that the proposed problems cannot be verified efficiently by a classical witness. A generalization of this challenge is Quantum Merlin Arthur (QMA)Hard, which is the quantum version of the class NP; thus, any problem in NP could be solved in polynomial time. This generalization implies that some quantum chemistry problems exist, although of unknown measure, that would be inefficient to solve even on a quantum computer. However, it is still unclear if natural molecular or material Hamiltonians exist that are expected to be found in their ground states in nature that would qualify as QMA-Hard. If that is the case, questions arise about how those systems get into their ground states since any quantum dynamics can be efficiently simulated by a quantum computer (Schuch and Verstraete 2009). Furthermore, see Goings and colleagues (2022) for a case study proposing a target that would be appropriate for a quantum computer and benchmarking the best classical methods against the projected quantum costs. The following section discusses how various chemistry research problems could be vetted to determine their viability for quantum computing applications.

4.3.1a Evaluation of Chemistry Use Cases with Quantum Computers

Below is a set of criteria that could be used to evaluate research proposals that identify chemistry problems that could be addressed using a quantum computer:

1. The challenges the investigator is aiming to study need either to have significant scientific merit or industrial relevance or to address a fundamental chemistry question.
2. An explanation for why these quantities cannot be obtained using classical methods needs to be provided.
3. A compilation is involved of the proposed quantum algorithm to an error-corrected instruction set that is compatible with a scalable, fault-tolerant error-correcting code such as the surface code.
4. Estimation of computing time and resources of the best classical methods are included in the proposal. Any research will benchmark these more precisely.
5. The proposed work aims to answer the following question: what quantities are being measured and to what precision?

These investigations will serve as case studies that clearly target quantum computing efforts and their role in facilitating our understanding of chemistry. Given the complexity involved between ongoing developments of the quantum computer’s architecture and the need to have appropriate chemical problems to study with these new technologies, the committee has provided Recommendation 4-2, summarized at the end of the chapter.

4.3.2 Studying How Quantum Computing Algorithms for Dynamics Can Be Used to Accelerate Chemistry and Spectroscopy

The most promising problems to solve on quantum computers are electronic structures, which can have exponential computational complexity and for which there are several known quantum algorithms to provide exponential speedup. Electronic structure is the foundation for understanding chemical properties and reactivities. For

example, electronic structure calculations have been used often to help assign peaks for complicated experimental spectroscopy; the molecular orbitals from electronic structure calculations are often used to predict and interpret the reactivity of chemical species.

Although challenges are evident, one of the most significant promises of quantum computing is that it can provide exponential speedup for a certain class of problems. The electronic structure problem is the problem of solving for the ground state of electrons interacting with each other through the Coulomb potential and with the point charges arising from the nuclei. Such ground states characterize the electronic wave functions of molecular systems and materials.

Subspace full configuration interaction (FCI) is not commonly used in routine chemistry applications due to the very restricted size (small molecules) of the affordable one-particle basis and because reliable relative energies may not require ultimate accuracy of total energies (Liu et al. 2022). In the spirit of FCI, many lower-scaling algorithms have been developed for chemistry applications. These traditional algorithms represent the state of the art for solving the electronic Schrödinger equation with high accuracy (Eriksen et al. 2020; Motta et al. 2017; Williams et al. 2020). The core idea is still to solve the eigenvalue problem either by using predefined restrictions of the many-electron basis (coupled cluster with a full treatment of singles and doubles (CCSD(T)), or Davidson correction (MRCI+Q)) or through an iterative construction of the basis-set expansion (DMRG, FCIQMC). The restriction to a selected finite set of active orbitals in all FCI-type approaches generates an artificial distinction of electronic correlations into two groups: (1) static electronic correlations, which are typically characterized by orbitals that occur in determinants with large weight in the wave function expansion; and (2) dynamical electronic correlations, which refers to orbitals present in determinants with small to vanishing weights. This artificial split into static and dynamical electronic correlations can be overcome if a routine numerical approach is used to obtain results of FCI quality on a one-particle basis of one to a few thousand orbitals. There are several FCI-type approaches to address static and dynamical electronic correlations. Even though it is difficult to rigorously assess the error in the energy after a fixed number of optimization cycles with predefined parameters, these traditional methods likely will remain the reference methods for most quantum chemistry calculations, at least in the near term.

Certain critical chemical problems and processes have electronic structures that are too complicated to be handled by any classical methods. FeMo cofactor (FeMoCo) of the nitrogenase enzyme responsible for nitrogen fixation has been discussed frequently and likely represents the most complicated electronic structure (Lee et al. 2021; Li et al. 2019; Reiher et al. 2017). Metalloporphyrin, which often plays an important role in biological processes, is another example of a complex electronic structure (but simpler and more typical than FeMoCo) that supports the need for quantum computers. Even though the active space size in both examples is similar, due to the nature of the electronic structure, the estimated run time on quantum computers is different. Thus, the high-accuracy treatment of strong correlation offered by quantum computers is relevant for understanding mechanisms in inorganic catalysis, for example. Solving the modern electronic structure problem will provide deeper insight into the properties and behaviors of other strongly correlated systems, such as transition metal and heavy metal complexes and biradicals. As demonstrated in Chapter 3 these classes of molecules also have potential utility in a variety of quantum applications. Therefore, gaining more accurate details about the nature of their chemical processes like bond breaking, excited-state transitions, and magnetic properties will be critical for streamlining the development of functional materials. However, to reach high levels of accurate predictions, progress first needs to be made toward solving the modern electronic structure problem.

The committee was presented with expert information regarding the possibility of utilizing quantum computers with new strategies for accurate calculations of highly correlated systems. The committee found that chemistry could play a role in the development of new approaches and algorithms specific to the use of a quantum computer in chemistry. One such idea, of many, involves a method of moments coupled cluster (MMCC) formulism. Some success has been reported on utilizing this method to effectively correct the energies of approximate coupled cluster formulations using moments that are not used to determine approximate cluster amplitudes (Peng and Kowalski 2022). Reports have suggested that particular quantum algorithms for computing MMCC ground-state energies offer a clear advantage over classical computing approaches. While great work remains to be performed to illustrate the robustness of such an approach, this example demonstrates a quantum computing approach that may provide the utility to calculate a strongly correlated system, which was previously thought to be impossible

with classical approaches. This and other approaches on the horizon are important in the development of our understanding of the true advantage of the quantum computing approach. Researchers in this area, in all settings, should be encouraged in their use of new approaches with quantum computers in chemistry that illustrate a strong advantage (McArdle et al. 2020).

4.3.2a Developing More Efficient Methods of Encoding Chemical Systems on Quantum Computers

Calculating total electronic energies with known accuracy is the basis for any theoretical description of molecular systems. As encoding the exact determinant space for large systems having many orbitals (>18) is very challenging on classical hardware, a key goal of all these novel methods is to approximate the full determinant space. Quantum computing holds promise to accomplish this goal, provided that a sufficiently large quantum computer can be built. The most studied approach to solving the electronic structure problem is quantum phase estimation (QPE), which is a fully quantum algorithm that can solve exact solutions of the chemical wave functions and energies. Bayesian phase difference estimation (Sugisaki et al. 2022) and other methods (Arute et al. 2019) are also used; however, the remaining discussion will be focused on QPE. QPE provides an appealing quantum speedup compared to classical rivalries like FCI. For classical computers, as explained in Box 4-1, FCI is a classical algorithm that solves the electronic structure problem; however, it uses exponential scaling.

In QPE, one chooses a trial state, a target error in the eigenvalue estimate, and a desired success probability. The algorithm then returns an estimate of a randomly selected eigenstate. Importantly, the eigenvalue is sampled with some probability. If ground-state energies are desired, then the trial state should be chosen to make the overlap with the true ground state reasonably large. Many ways exist to prepare initial states including using matrix product states, adiabatic-state preparation, and unitary coupled clusters. Different research groups are developing procedures to increase the efficiency of preparing the initial states (Lee et al. 2023). Importantly, if an initial state is difficult to establish, as is the case in strongly correlated systems, then the initial-state overlap will be weak.

There are concerns that sometimes the dependence on the initial-state overlap can lead to poor scaling in the quantum algorithm, especially when the initial overlap is weak (Aspuru-Guzik et al. 2005; Lee et al. 2023). Furthermore, this algorithm measures the eigenvalue of unitary matrices. Quantum circuits can be used to synthesize unitary matrices that share an eigenbasis with the molecular Hamiltonian—for example, via realizing time evolution. Quantum walks are also a popular alternative (Babbush et al. 2018). Developments are under way to increase the efficiencies of these quantum algorithm approaches and lower their costs. Finally, a major setback of using QPE, at the moment, is that it requires quantum hardware that may not be completely accessible in the near future.

4.3.2b VQE to Solve for Electronic Structure

An alternative to a fully quantum algorithm like QPE is VQE. VQE is considered a meta-algorithm, or algorithmic framework, for a quantum–classical hybrid approach where the quantum computer does some work and the classical computer does some work (Chen et al. 2021). VQE is attractive for studying quantum chemistry problems using a popular quantum computer known as the NISQ (McClean et al. 2016).

Running quantum algorithms on real hardware is essential for understanding their strengths and limitations, especially in the NISQ. Simulations assume a perfect system and do not effectively take into account the decoherence errors, loss of accuracy, and time-outs that can occur on real systems (Yamamoto et al. 2022). Quantum computers have evolved from exploratory physics experiments into cloud-accessible hardware as early stage commercial products that are available to a broad audience of chemists and those in other disciplines (Endo, Benjamin, and Li 2018). NISQ-era computers can experience a wide range of complex errors. Many of these errors result from noise in or miscalibrations of the lowest-level components in the system, quantum gate operations, and decoherence in the qubits themselves. The Quantum Economic Development Consortium sponsored a quantum benchmarking project that resulted in an open-source suite of bench tools to run on various quantum hardware systems (Lubinski et al. 2023). This work could be extended to benchmark real chemical structure calculations and molecular energy measurement algorithms on real hardware (Gulka et al. 2021).

VQE is attractive for NISQ because it does not require long coherent circuits like QPE (O’Malley et al. 2016). However, the number of circuit repetitions required is dramatically more for VQE, which can eventually

lead to other challenges such as reaching the limits of NISQ supporting capabilities (Peruzzo et al. 2014; Wecker, Hastings, and Troyer 2015).

The compatibility between VQE and NISQ is exemplified in the simulation of electronic structures, where VQE can be modified or constrained to find the electronic state with a specific number of electrons, electron spin, or other properties (Ryabinkin, Genin, and Izmaylov 2019). Another significant simulation problem addressed by VQE is the dynamic correlated states in a quantum system. The state characterizes the dynamics of quantum particles through the (correlated) system and provides insight into its resulting optical, magnetic, and transport properties. The state also allows access to many static observables, notably the total energy of the system. The ability to understand the dynamic correlated-state system can apply to the understanding of high-temperature semiconductors although the observation will be limited by circuit depth (Chen et al. 2021).

VQE uses the output of a parameterized quantum circuit as an “ansatz” (i.e., a mathematical assumption about the form of an unknown function that is made to find a solution to an equation or other problem) or a trial for the true ground state. The energy of the ground state is calculated using the quantum computer. Then, the classical computer optimizes the parameters of the quantum circuit to lower the energy of the ansatz (Grimsley et al. 2019; Peruzzo et al. 2014).

Furthermore, the committee received expert information from Professor Giulia Galli (see Appendix B), who has made progress in the use of quantum computers for chemical calculations. Galli’s work involves using quantum calculations to predict and design novel molecular and nano-qubits. The long-term goal of the research is to create new simulations on quantum computers for greater accuracy and speed. As illustrated in Figure 4-8, many of the chemistry problems explored using VQE on NISQ are on very simple chemical systems, like diatomic molecules (H₂, LiH, NaH, and others), and still require significant error correction to the raw data (see graph in Figure 4-8).

VQE has also been implemented to simulate two intermediate-scale chemistry problems: the binding energy of hydrogen chains, as large as H₁₂, and the isomerization mechanism of diazene (Arute et al. 2020). The simulation was performed using a circuit consisting of 12 qubits and up to 72 qubit gates. However, the authors reiterate the 50-qubit barrier (known as the classically intractable regime), where the key building blocks of the proposed VQE algorithm are potentially scalable to larger systems that cannot be simulated classically (Arute et al. 2020; O’Brien et al. 2022). Many NISQ VQE chemistry experiments have only been 12 qubits, and data suggest that scaling all up to 50 or more qubits to surpass the classically intractable regime will be challenging.

4.3.2c Exploring How Quantum Machine Learning Can Be Used to Accelerate Chemical Research by Processing Quantum Data from Entangled Sensor Arrays or Quantum Simulations of Chemistry

Another widely anticipated application area for quantum computing is machine learning. Quantum machine learning (QML) (Biamonte et al. 2017) consists of two distinct branches: (1) using quantum computers to accelerate classical machine learning as performed by classical computers and (2) using the quantum advantage of quantum computers to construct fully nonclassical quantum neural networks. While quantum computers hold promise for accelerating classical machine learning algorithms, here we instead focus on the potential for quantum computers to analyze data more rapidly. Perhaps the most pressing challenge in the field of QML is to answer the following question: on what types of data sets do we expect QML to provide a quantum advantage? Currently, the compelling answer to this question is that QML is particularly promising for modeling “quantum data” (Huang et al. 2021). Quantum data are data that consist of quantum states rather than classical information. Examples of quantum data include wave function output by quantum simulations as well as data that are obtained from quantum sensors and then transduced to quantum computers. Section 4.2.3a discusses using classical computers to simulate “quantum data”; however, this technique is limited in providing “enough” data due to computational power and cost.

Recent work has shown that when learning from quantum data is applied, it is possible to achieve an exponential advantage in learning. Quantum data can be generated from performing quantum simulations of chemical systems, and learning is applied on top of those data (Huang et al. 2022). This exponential advantage arises because exponentially fewer data are needed if the data are “quantum data” and entangled. Furthermore, because this is a query advantage (i.e., needing less data) as opposed to a conventional quantum speedup (i.e., an algorithm with faster run time), it is possible to reach a quantum advantage with few qubits, especially in the context where data are limited. For example, this advantage could help in understanding topological order in systems, which is a

global property of the wave function that cannot be observed by local measurements. Another potential impact of this area on chemistry is that quantum sensors (either molecular sensors or sensors using a different technology), where the state of the sensor can be transduced to a quantum computer, might allow new tools for investigations within a chemical laboratory. This direction has many synergies with molecular sensors and molecular qubits in the context of quantum signal transduction, and both technologies would likely need to be further developed to make this a reality.

4.3.3 Developing More Efficient Quantum Algorithms for Fault-Tolerant Quantum Computers to Simulate Molecular Systems

Quantum simulations, like classical theory, are facing accuracy issues in predicting large molecules. Currently, the most studied model for such systems is medium-sized inorganic catalyst molecules, which are a subject of homogeneous catalysis research. The challenge is to find an application that is insurmountable by current quantum

computers. And, the system is large enough to be predicted accurately with a quantum computer without the need for gross oversimplifications. Because the architecture of quantum computers is still premature, the technology is understandably plagued with defects that ultimately cause data to be highly erroneous. Having an understanding of the basic architecture of quantum computers will help the chemist determine which types of chemistry problems will be suitable for a quantum computer versus a classical computer. Box 4-3 elaborates further on quantum digital gates.

Two major architectures are competing for popular acceptance: fault-tolerant quantum computing (FTQC) and, as described earlier, NISQ. NISQ exists now, and is small scale and non–error corrected. NISQ may use digital or analog paradigms, and current devices are based on both approaches. For FTQC, the digital (gate) approach to error correction for fault tolerance is better developed as a concept, hence the belief that it will be the most likely useful, far-future solution. Whether it would be possible to realize a fault-tolerant analog quantum computer remains unknown.

The ultimate goal for quantum computing systems is to design platforms capable of eliminating the amount and intensity of noise generated in the data, which is directly related to the overall quality of analysis. Two

main approaches have been executed in the development process to address this problem: error correction and error mitigation.

Unlike with classical computers, one cannot simply copy quantum information and use a “repetition code” to correct errors as a consequence of the quantum no-cloning theorem (Wootters and Zurek 1982). Instead, a theory of quantum error correction is now very developed that is based on topology (Kitaev 2006). Topology is a global property of objects. For example, one cannot tell if a donut and a coffee cup are topologically the same or different just by examining a small piece of one of those objects; one must examine the entire object. Hence, topological information is nonlocal. By contrast, errors that occur in individual two-qubit gates or errors that occur from the system interacting with the environment (in most cases) are local. Thus, quantum error correction is based on the idea that if quantum information is encoded in global topological properties of quantum states, it will be robust to errors. A well-developed theory of FTQC now exists.

With FTQC codes, as one increases the number of physical qubits used to store a single “logical qubit,” the error rate decreases exponentially. The most popular error-correcting code that can be made fully fault tolerant (meaning that one can exponentially suppress errors in all aspects of the code, including logical operations, measurements, decoding, and encoding, so long as error rates are “low enough”) is the surface code. The surface code is a topological code (a 2D variant of the Toric code) composed to work for qubits on a planar lattice (Fowler et al. 2012). The surface code is the most popular code because it has the lowest “threshold” of any known code involving only 2D connectivity. 2D connectivity is attractive because devices with higher connectivity pose extreme engineering challenges. The surface code, or close variants like the honeycomb code (Haah and Hastings 2022), is the plan of record for most efforts to build an error-corrected quantum computer, including the leading ion trap, photonics, and superconducting qubit efforts.

The best two-qubit gate fidelities for quantum hardware today that have scaled to more than a few qubits have roughly f = 0.99 (Egan et al. 2021; Google AI n.d.). This value is near the error threshold for the surface code. At those error rates, one would require at least several thousand physical qubits for each logical qubit. Thus, a quantum computer with a few hundred logical qubits would require nearly one million physical qubits. The overheads decrease somewhat if the error rates come down further. For example, with 1e⁻⁴ or 1e⁻⁵ error rates, the total number of physical qubits for a processor with a few hundred logical qubits might be reduced to closer to 100,000. But it is considered unlikely that error rates as low as 1e⁻⁵ will be reached with our current paradigms.

One possibility for getting physical error rates that low is topological quantum computing (Nayak et al. 2008), but so far not even a single topological qubit has been produced. If we were to use a platform with f = 0.999 gate fidelities as a NISQ platform (this would be an extremely accurate NISQ device), then already for L = 1e⁵ gates (a relatively “small” quantum circuit) the total success probability would be on the order of 4e⁻⁵. This means that the quantum computer would need to be run 1/(4e⁻⁵) times, or about 20,000 times, just to see a single realization that is not errant. How plausible that is depends on the hardware. Superconducting qubits have gate times on the order of tens of nanoseconds, so this might take only tens of seconds. But ion traps, for example, are several orders of magnitude slower. And it can be very difficult to determine when an error has occurred. Strategies for detecting when errors have occurred so that one can postselect on error-free outcomes, but which do not actually remove the exponential scaling associated with these errors, are referred to as “error mitigation” (which is quite a different topic entirely from “error correction,” despite similar sounding names; see Section 2.2.3 for further details) (Cai et al. 2022).

Most of the time, FTQC algorithms are studied formally. For example, scientists aim to quantify (either asymptotically or in terms of finite resources like constant factors) the number of logical or physical qubits and quantum gates required to perform a precise task. An example of this task is simulating a wave function, starting in its initial state under the molecular Hamiltonian for time (t) and having it evolve within a controlled error (ε). The parameters considered in this system would include the number of particles, number of basis functions, size of particles, and other variables that could affect the behavior of the system.

Work to improve the cost of these algorithms has been ongoing. Su and colleagues (2021) and Lee and colleagues (2021) attempt to show how the cost of quantum algorithms for chemistry in both plane wave and molecular orbital basis sets has improved over the years. Note that while these studies contain very detailed analyses of the costings, the costings are sometimes difficult to compare.

• Develop more efficient methods of encoding chemical systems on quantum computers (e.g., better basis sets, quantization, fermion mappings, embedding theories).
• Develop more efficient quantum algorithms for fault-tolerant quantum computers to simulate molecular systems, including those with QIS relevance.
• Study how quantum computing algorithms for dynamics can be used to accelerate chemistry and spectroscopy.
• Explore how quantum machine learning can be used to accelerate chemical research by processing quantum data from entangled sensor arrays or quantum simulations of chemistry.

The committee makes Recommendation 4-1 to address issues concerning open-access databases and the importance of sharing data across relevant research entities to move the field of QIS and chemistry forward. The related discussion included ways to improve the organization and types of relevant experimental and theoretical data that would be useful for the research community. The committee puts forth Recommendation 4-2 to ensure that efforts in QIS research involving quantum-accelerated calculations keep chemistry-related problems at the forefront of the research questions to solve.

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