Cover artwork: Experiments using specially shaped pulses of light demonstrate that 

quantum states can be guided along “superadiabatic” trajectories, allowing for 

coherent manipulations with both high speed and robustness.

Image courtesy of David Awschalom, Brian Zhou, and Peter Allen, University of Chicago 

and Argonne National Laboratory
Report of the Basic Energy Sciences Roundtable 

Opportunities for Quantum Computing in 

Chemical and Materials Sciences

October 31–November 1, 2017

Gaithersburg, Maryland

Chair: Joel E. Moore, Lawrence Berkeley National Laboratory and University of 

California–Berkeley

Co-Chair: Alán Aspuru-Guzik, Harvard University

Participants:

Bela Bauer, Microsoft

Sue Coppersmith, University of Wisconsin

Wibe (Bert) de Jong, Lawrence Berkeley National Laboratory

Thomas Devereaux, SLAC National Accelerator Laboratory

Marivi Fernandez-Serra, Stony Brook University

Giulia Galli, University of Chicago/Argonne National Laboratory

Robert Harrison, Brookhaven National Laboratory/Stony Brook University

Peter Love, Tufts University

Thomas Maier, Oak Ridge National Laboratory

Antonio Mezzacapo, IBM

Jarrod McClean, Google

Chris Monroe, University of Maryland, IonQ

Joel Moore, Lawrence Berkeley National Laboratory/University of California–Berkeley

John Preskill, California Institute of Technology

Gus Scuseria, Rice University

Birgitta Whaley, University of California–Berkeley 

James Whitfield, Dartmouth University

Norman Yao, Lawrence Berkeley National Laboratory/University of California–Berkeley

Dominika Zgid, University of Michigan

Basic Energy Science (BES) and Advanced Scientific Computing Research (ASCR)

Team:

Bruce Garrett (BES) Jim Davenport (BES) George Maracas (BES)

Linda Horton (BES) Matthias Graf (BES) Tom Russell (BES)

Jim Murphy (BES) Jeff Krause (BES) Ceren Susut-Bennett (ASCR)

Katie Runkles, BES Administrative Lead

Oak Ridge National Laboratory Publications Team: Brenda Wyatt, Deborah Counce, and Kathy 

Jones

https://science.energy.gov/bes/community-resources/reports/

iii

Contents

Abbreviations, Acronyms, and Initialisms ...............................................................................................v

Executive Summary..................................................................................................................................vii

1. Introduction.........................................................................................................................................1

2. Priority Research Opportunities .......................................................................................................5

PRO 1: Controlling the Quantum Dynamics of Nonequilibrium Chemical and Materials 

Systems....................................................................................................................................5

PRO 2: Unraveling the Physics and Chemistry of Strongly Correlated Electron Systems ................13

PRO 3: Embedding Quantum Hardware in Classical Frameworks....................................................19

PRO 4: Bridging the Classical–Quantum Computing Divide ............................................................25

3. Crosscutting Computational Themes and Challenges...................................................................29

4. References..........................................................................................................................................31

Appendix A: Workshop Participants......................................................................................................37

Appendix B: Workshop Agenda..............................................................................................................41

v

Abbreviations, Acronyms, and Initialisms

1D one-dimensional

2D two-dimensional

3D three-dimensional

BES Office of Basic Energy Sciences

CASPT2 complete active space perturbation theory second order

DFT density functional theory

DMFT dynamical mean-field theory

DMRG density matrix renormalization group

DOE Department of Energy

GHZ Greenberger–Horne–Zeilinger

LHC light-harvesting complex

NEVPT2 n-electron valence perturbation theory second order

PRO Priority Research Opportunity

QMC quantum Monte Carlo

SEET self-energy embedding theory

VQE variational quantum eigensolver

vii

Executive Summary

Fundamental transformations in the basic logic of computing are few and far between. Since the invention

of digital computers in the early 1940s, the logic underlying computation has remained the same, even as 

computing hardware evolved from vacuum tubes to silicon transistors. With the advent of quantum 

computation, a fundamental transformation is near. Quantum computation is based on a different type of 

logic: rather than being in one of the two states of a classical bit, a quantum bit or qubit can be in a 

superposition of two states simultaneously. Operations and measurements on these qubits obey the 

constraints of quantum mechanics. It is now understood that quantum computers have great power in 

principle to go beyond classical computers, but that not every application is well suited to implementation 

on quantum computers.

For reasons explained in more detail in the Introduction, scientific problems in chemical and materials 

sciences are uniquely suited to take advantage of quantum computing in the relatively near future. Indeed, 

quantum computing offers the best hope to solve many of the most important and difficult problems in 

this field. For example, quantum materials, such as superconductors and complex magnetic materials, 

show novel kinds of ordered phases that are natural from the point of view of quantum mechanics but 

difficult to access via computation on classical computers. Quantum sensors based on solid materials are 

already widely used but could be greatly improved with insight from quantum computations, as could 

materials for information technologies. Quantum chemical dynamics is another example of a problem that 

is intrinsically well suited to studies on quantum computers. Applications of quantum chemical dynamics 

include catalysis, artificial photosynthesis, and other industrially important processes.

Quantum computers exist in the laboratory and are beginning to exceed 50 qubits, which is roughly the 

size beyond which their behavior cannot be predicted or emulated on present-day classical 

supercomputers. While a quantum computer of 50 qubits is almost certainly not powerful enough to 

tackle the major scientific challenges in chemical and materials sciences, some of these major challenges 

start to become accessible with a few hundred qubits if error rates can be kept small. This roundtable was 

convened to ask how emerging quantum computers can be applied to major scientific problems in 

chemical and materials sciences, in light of Basic Energy Sciences’s leading role in these fields and the 

Department of Energy’s leading role in high-performance scientific computation more generally. The 

main outcome of the roundtable was a consensus that there are scientific problems of great importance on

which emerging quantum computers have the potential for disruptive impact, and where comparable 

progress is unlikely to occur by other means.

These scientific problems are grouped into four Priority Research Opportunities that form the bulk of this 

report. Controlling the quantum dynamics of nonequilibrium chemical and materials systems will 

enable remarkable progress in major application areas such as catalysis and quantum sensing, and enable 

an understanding of nonequilibrium systems in which few basic principles are available to guide theory.

Unraveling the physics and chemistry of strongly correlated electron systems is essential for progress 

in many basic and applied fields because strong correlations between electrons cause the standard 

approximations used in current approaches to break down, even when the structure of a material is 

relatively simple. Embedding quantum hardware in classical frameworks requires the development of 

algorithms that, as the computation is performed, efficiently pass information between classical methods 

for the “easy” parts of a problem and quantum hardware for the most computationally challenging parts.

Such embedding methods could greatly increase the applicability of quantum hardware in the medium 

term by directing the limited quantum resources to the aspects of a molecule or material where they are 

most needed. Bridging the classical–quantum computing divide relies on the use of quantum 

algorithms to improve the key inputs of classical algorithms or capture key quantum processes. The latter 

is particularly important for quantum mechanical processes in open systems when they occur within 

decoherence-inducing environments such as electromagnetic fields and heat baths (e.g., liquid water).
viii

The report closes with a summary of crosscutting challenges and some comments on how the different 

communities represented at this roundtable could continue to interact as useful quantum computation 

becomes a reality. The applications of quantum computing to chemical and materials sciences are still in 

their infancy, but visionary work has already illuminated the potential, in principle, of quantum computers 

to revolutionize these areas of science. Identifying specific challenges and the necessary steps to start 

addressing them, as this roundtable sought to do, is intended to aid progress in this difficult but 

worthwhile endeavor.
1

1. Introduction

This introduction seeks to give a brief explanation of why advances in quantum computation can be 

expected to impact the chemical and materials sciences. Historically, the invention of modern computers 

transformed theoretical chemical and materials sciences into predictive disciplines for some classes of 

material and molecular systems. For many important classes of materials and molecules, standard 

approximations such as density functional theory (DFT) are able, when implemented on highperformance computers, to make usefully accurate predictions about energetics and other key properties.

However, other classes of important materials and molecules remain inaccessible to even the most 

powerful existing computers and are likely to remain so for the foreseeable future. There are also large 

classes of phenomena, such as those involving dynamics of excited states, for which reliable approximate 

methods are not available.

As outlined in this introduction, quantum computation promises to make these problems accessible.

Additional background and information on the current state of quantum computation, including the 

various hardware options currently being explored, are available in the Quantum Information Science 

Factual Status Document available from the Department of Energy (DOE) website.* For the purposes of 

the present report, the essential properties of a quantum computer are just its computational parameters, 

such as the number of qubits and the speed and fidelity of gate operations. Quantum “gates” or gate 

operations are the basic quantum logical operations from which more complicated computations can be 

built, in the standard gate-based model of quantum computation.

Quantum computers operate on a fundamentally different basis from the classical Boolean logic of 

standard computers. At the lowest level, a standard digital computer applies logical operations (e.g., 

AND, OR, or NOT) to logical bits, each of which can take two possible values. Of course it is now 

understood how to formulate many more complicated questions, including the mathematical problems 

that arise in physics and chemistry, in terms of these classical bits. Although computers have come 

remarkably far since the early days of electromechanical relays and vacuum tubes, and many 

mathematical operations are nearly instantaneous, the mathematical problems that arise in systems of 

strongly interacting quantum particles are in general difficult or impossible because the time required for 

an accurate computation scales exponentially according to the number of particles.

Many years of effort have already gone into understanding how quantum computers—which apply 

unitary operations to a set of quantum two-level systems, or “qubits”†

—are in principle enormously more 

efficient than classical computers in addressing some problems. One representation of a qubit, the “Bloch 

sphere,” is shown in Figure 1. The north and south poles on the sphere correspond to the values of a 

classical bit, 0 or 1, or up and down in the language of electron spins. The equator of the sphere contains 

many different quantum states that are equal parts 0 and 1, but their quantum phase is distinguishable by 

other measurements.

Consequently a single qubit contains more information than a classical bit, at least as long as the qubit’s 

coherence (the details of the superposition of 0 and 1 states) can be maintained. An example of a problem 

that can be solved rapidly by a quantum computer is the factoring of a very large integer into its prime 

components, which is currently, and widely expected to remain, exponentially difficult for classical 

 * https://science.energy.gov/bes/community-resources/reports

† There are other formulations of a quantum computer that lead to computational power that is roughly equal to the gate-based 

model mentioned here. For yet other approaches, such as quantum annealing with the transverse Ising model, the computational 

power is possibly less than the standard gate-based model and in general not as well understood. The roundtable did not discuss 

these, nor did it cover analog quantum computers or quantum emulators that target individual problems.
2

computers. Hundreds or possibly thousands of 

logical qubits are required for many problems 

originating in computer science, such as factoring.

Furthermore, in the presence of errors, each of these

logical qubits may require a number of physical 

qubits for the correction of errors. This difficulty 

raises the question of whether other kinds of 

problems might be more promising for the first 

applications of quantum computers, which are 

currently limited to fewer than one hundred qubits.

Why is quantum computation relevant to chemical 

and materials sciences?

As originally explained by the well-known American 

theoretical physicist Richard Feynman, and justified 

in more detail by rigorous theorems, quantum 

computers are particularly well suited to find 

quantum-mechanical solutions to problems in 

chemical and materials sciences —at least in 

principle. The Schrödinger equation that describes 

time evolution in atomic and molecular systems can 

be efficiently simulated on a quantum computer; and 

quantum algorithms have been developed to evaluate 

properties of physically relevant systems such as 

energies, excitation gaps, and correlations. The 

scaling of computational effort with particle number 

is no longer exponential but only polynomial. For

some of these algorithms, current estimates suggest 

that they will already be able to exceed the 

performance of the best classical computers when 

running on as few as 100 qubits if the qubits are 

sufficiently free from error.

The first goal of this roundtable, in informal terms, 

was to go beyond the “in principle” statement that 

quantum computers should be good for theoretical 

chemical and materials sciences by identifying important problems upon which quantum computers might 

have their first dramatic impact. The second goal was to identify challenges that need to be overcome to 

realize the potential of quantum computers to solve these science problems. Quantum computers of 

various types already exist in laboratories around the world. An overview of the current state of the art of 

the field was given by John Preskill (of California Institute of Technology) at the beginning of the 

roundtable.

How large do quantum computers need to be to achieve useful results?

Several current or near-term quantum computing machines have more than 50 physical qubits. Given 

their ability to execute high-fidelity gate operations, this size would place them beyond the current ability 

to simulate the computation accurately on even a very large classical computer. For comparison, a recent 

large-scale classical simulation of quantum computing at the National Energy Research Scientific 

Computing Center was able to model 45 qubits,

1 with a further increase to 49+ qubits, albeit with a small 

Figure 1. The Bloch sphere, a representation of 

the possible states of a single qubit (quantum 

bit). The north and south poles are the states that 

correspond to classical bit values 0 and 1. Other 

states on the surface of the sphere are 

distinguishable quantum states that “superpose” 

0 and 1 in different proportions, and even states 

with the same proportions of 0 and 1 need not 

be the same quantum state. For example, all 

states on the sphere’s equator are equal-weight 

superpositions of 0 and 1, but with different 

relative phases. The coherence in such a 

superposition leads to many more different states 

than for a classical bit, and quantum 

computation uses the ability to make coherent 

superpositions of the states of many qubits. |

Image reproduced under the terms and conditions of 

the Creative Commons license
3

number (27) of gate operations performed, as reported in a preprint.

2 Although these quantum computing 

machines do not yet provide the required number of qubits and gate fidelities to solve the major challenge 

problems identified, they are large enough to conduct detailed studies of how existing and future 

algorithms scale with the number of qubits (e.g., updated versions of the 2005 estimates3 shown in 

Figure 2) and whether they achieve robustness to errors. A sustained effort, closely connected to science 

questions, to improve quantum algorithms for chemical and materials sciences, as they function on realworld quantum hardware, has the potential to make major progress on problems that otherwise appear to 

be intractable.

History suggests that a sustained

effort coupled to quantum 

hardware is important: very few 

of the common algorithms 

currently used on classical 

computers existed before 

algorithms could be tested on 

physical hardware. Even the 

programmable von Neumann 

architecture associated with the 

word “computer” grew only out 

of practical experience with the 

first machines. Although 

quantum algorithms in a general 

sense have been considered for 

many years—mostly in the 

theoretical computer science 

community—the effort has been 

concentrated on standard 

computer science problems rather 

than on chemical and materials 

sciences. In part because of the 

absence of hardware, the focus 

has been on algorithms whose 

speeds can be demonstrated with 

mathematical rigor. The quantum 

computing community would like to motivate more scientists to turn their interest to chemical and 

materials sciences problems and to focus on how well algorithms perform for difficult problems of 

practical importance, which may be better than mathematical worst-case bounds. Indeed, this is the case 

for many important classical algorithms, especially in the simulation context. A classic example is the 

density matrix renormalization group (DMRG). It has been widely used since the 1990s based on 

empirical evidence for its effectiveness, but a rigorous upper bound on its scaling was only recently 

established.

4 Perhaps more important than mathematical rigor is a serious effort, respected by the 

community, to perform unbiased comparisons of algorithms and estimate their speed and robustness.

Fortunately, there is a rapidly growing community of scientists developing quantum hardware and 

software, first for chemistry and now also for materials problems. This community is not limited to 

academia and national laboratories; a highlight of the roundtable for many participants was the morning 

session with presentations from researchers at three major companies (Google, IBM, and Microsoft) and 

from one of the many startups in this area (IonQ). All three companies have substantial efforts in quantum 

software and hardware, including quantum chemistry efforts that interface closely with the problems 

discussed in this roundtable. It is encouraging that quantum chemical and materials sciences are already 

Figure 2. Illustration of linear scaling of the number of qubits with 

number of electronic orbits used in electronic structure calculations. 

The molecules corresponding to each calculation are indicated, and 

some molecules appear more than once because calculations were 

performed with different numbers of orbitals. | Graph based on 

information from Rinton Press, QIC 15, Improving quantum algorithms for 

quantum chemistry, M. B. Hastings, D. Wecker, B. Bauer, and M. Troyer, 

copyright 2015. Used by permission
4

recognized by most or all of the major industrial players in quantum computing as viable and important 

applications for emerging quantum hardware.

How does this area connect to the DOE mission?

The considerable progress in quantum computing hardware to date has been enabled in large part by 

federal funding via multiple agencies and programs, as well as industrial research and development.

Building on this progress to achieve dramatic advances in chemical and materials sciences is a logical 

goal for DOE’s Basic Energy Sciences (BES) office, which is the largest single funder of chemical and 

materials sciences research in the United States. BES support includes long-standing programs in a wide 

variety of computational approaches to chemical and materials sciences. Indeed, many of the scientific 

problems discussed in the following sections were previously identified as BES goals in previous reports, 

such as the 2016 report Basic Research Needs Workshop on Quantum Materials for Energy Relevant 

Technology. DOE also has considerable experience in integrating high-performance computation with 

scientific goals. Finally, many of the demanding experimental projects supported by the Office of Science 

would benefit from the possible advances in chemical and materials sciences discussed in this report.

Roundtable participants identified the following four Priority Research Opportunities (PROs), which are 

described in detail in the following sections:

1. Controlling the quantum dynamics of nonequilibrium chemical and materials systems

2. Unraveling the physics and chemistry of strongly correlated electron systems

3. Embedding quantum hardware in classical frameworks

4. Bridging the classical–quantum computing divide

The path to substantive applications of quantum computers such as these will not be an easy road. New 

hardware challenges will emerge as the size of quantum machines grows, and it could well be that the 

scale of future quantum computers does not show the exponentially rapid growth (i.e., Moore’s Law) 

familiar from the history of classical computing. Fortunately, for some of the scientific problems 

identified, the hardware size needed for progress is not very much larger than the sizes of quantum 

computers expected to exist in the next few years—although the number of qubits alone is not very 

meaningful without a quantification of errors. Decades of experience with classical computers have given 

the chemistry/materials community some insight into which problems need a truly new approach such as 

quantum computing. An exciting aspect of the roundtable, noted by several of the experienced 

computational scientists present, was the glimpse of a pathway by which these truly fundamental and 

challenging problems might be solved.
5

2. Priority Research Opportunities

PRO 1: Controlling the Quantum Dynamics of Nonequilibrium Chemical and 

Materials Systems

Contributors: Giulia Galli, Norman Yao, Wibe (Bert) de Jong, Thomas Devereaux, and Jarrod McClean

Overview 

Understanding, predicting, and controlling the quantum dynamics of nonequilibrium systems has the 

potential to accelerate key science and engineering applications ranging from quantum chemistry and 

metrology to materials design. Although the past half-century has seen tremendous progress in capturing 

the low-energy landscape of equilibrium quantum systems, it has become increasingly clear that (1) 

nonequilibrium systems can exhibit fundamentally richer behavior than their static counterparts, and (2) 

understanding such behavior will be essential to discover novel catalytic pathways, topological materials, 

quantum sensors, and the like. To solve those questions, one must move beyond a powerful set of 

classical computational tools that, while ideal for calculating equilibrium low-energy properties, are 

wholly unsuited for calculating the complex nonequilibrium dynamics of chemical and materials systems. 

To this end, a near-term quantum information processor can enable important advances in simulating and 

predicting the behavior of nonequilibrium systems, well beyond what is achievable via classical 

computations. The diversity associated with such nonequilibrium systems stems, in part, from the 

multitude of ways in which such behavior can emerge; for example, out-of-equilibrium dynamics are 

found in optically driven systems, chemical reactions, frustrated glasses, catalytic pathways, and 

molecular potentials. The discussion that follows focuses on four scientific challenges at the interface of 

chemistry, physics, and materials engineering in which a near-term quantum computer can provide 

transformative insights via simulations of nonequilibrium dynamics. 

Scientific Challenges

Quantum simulations of chemical reactions and catalytic pathways 

Mastering the guiding principles that underlie catalytic chemical reactions is crucial to our transition to a 

more sustainable economy. Indeed, 

catalysis represents an essential component 

in the production of most chemicals and 

fuels, with different catalysts functioning 

to enable diverse chemical tasks ranging 

from the splitting of water and nitrogen 

fixation to the selective transformation of 

complex molecules for pharmaceutical 

synthesis.

5 Despite its importance and 

ubiquity, predicting catalytic reactions 

using computer simulations remains an 

open challenge, encompassing outstanding 

problems such as the simulation of 

chemical reactions in condensed phases, 

the importance of nonadiabatic 

phenomena, and the characterization of 

pathways often involving (multiple) 

electronic excitations.

6

Figure 3. Pictorial representation of a water-splitting reaction 

occurring on a catalytic surface, which starts with harvesting 

light energy to form charge carriers and can involve protoncoupled electron transfer (PCET) processes. The full quantum 

chemical simulation of such a reaction (by solving the timedependent Schrödinger equation) is expected to be 

unfeasible with classical computers. | Image courtesy of Giulia 

Galli, University of Chicago
6

Although these problems are being pursued with classical computers, with steady progress, quantum 

computations could provide the necessary leap to solve the time-dependent Schrödinger equation for 

complex systems, inclusive of surfaces in contact with fluids and eventually in the presence of electric 

fields (Figure 3). However, a simpler yet ambitious near-term goal is the full quantum simulation of a 

small-scale chemical reaction, such as OH + CO →H + CO2, going beyond the Born-Oppenheimer 

approximation (i.e., including the coupled quantum dynamics of electrons and nuclei). This simulation 

could proceed in two steps, with the first focusing on the reaction in the gas phase and the second 

focusing on the reaction in the presence of a solid surface.

7 The sidebar “Mapping a Chemistry Problem 

onto a Quantum Computer” explains how a chemical reaction is mapped into a set of operations for a 

quantum computer.

Mapping a Chemistry Problem onto a Quantum Computer

TYPICAL CHEMISTRY PROBLEM WORKFLOW

Chemistry simulations have been identified as a leading application for near-term quantum computers and 

quantum simulators. However, mapping a chemistry problem from the molecular domain to the qubits of a 

quantum computer is a task requiring a number of important steps8,9:

1. Specify a molecule geometry, spin, charge, and basis set.

2. Calculate the integrals within the chosen basis set that define the problem.

3. Perform a Hartree-Fock or correlated calculation to set the reference vacuum state

4. Transform the Hamiltonian into qubits from the second-quantized Hamiltonian.

5. Select the property and algorithm of interest and generate the circuits.

6. Map the quantum circuit to the constraints of the target hardware.

To maximize contributions by scientists in this area, it is key that software tools be built to bridge gaps of 

knowledge, enabling domain scientists from both chemistry and quantum information to tackle open 

questions. An open-source example, OpenFermion (http://www.openfermion.org), has been developed to

assist users in taking the problem through all of the aforementioned steps.

Image at top courtesy of Jarrod McClean, Google. Quantum circuit at bottom left is based on work reported in and is 

presented with permission from Nature Publishing Group. Nature, State preservation by repetitive error detection in a 

superconducting quantum circuit, J. Kelly, et al., copyright 2015.
7

Among the many challenges associated with computing the nonequilibrium dynamics of a chemical 

reaction, one of the most essential is to establish a rigorous metric assessing the reliability of results. In 

the case of catalysis, a particularly appealing comparison would be to benchmark these quantum 

simulation results with the direct measurement of basic pathways. Such measurements have recently been 

achieved by exploiting the sensitive, broadband, and high-resolution capabilities of time-resolved cavityenhanced direct frequency comb spectroscopy.

10

To capture the full reaction dynamics present in catalytic chemistry, it is expected that between ~102 and 

103 qubits would be required, with a gate depth (the number of gate operations carried out by each qubit) 

of ~103 gates per real-time evolution step.

11 To this end, while near-term quantum computers will help 

provide insight into the short-time behavior of reactions, to access the long-time dynamics may require 

tailored algorithms that implement simple forms of error correction. 

Quantum computing for the discovery of new topological matter

Spurred by recent progress in the melting, enhancement, and induction of ordered states of electrons using 

out-of-equilibrium techniques, a tantalizing prospect is to use a near-term quantum computer to discover 

new dynamical phases of matter.

12 In particular, in the presence of strong, broad optical pump pulses, the 

electronic properties of quantum materials can be dramatically altered, leading to a novel landscape of 

transient Floquet steady states.

13,14 By manipulating the underlying symmetries governing the material, 

including for example, inversion and time-reversal, nonequilibrium driving offers the opportunity to 

create strongly interacting states of matter with no static analogs.

This strategy can be applied to materials to alter 

their topological properties. For example, 

frustrated quantum magnets host a delicate 

interplay of competing orders, as well as 

putative spin liquid phases, so that weak 

symmetry-breaking optical perturbations could 

be expected to have an outsized effect in 

determining the transient state. Recently, it has 

been shown that shining circularly polarized 

light on a Kagome Mott antiferromagnet can 

transiently induce a Floquet chiral spin liquid 

(Figure 4).

15 Circularly polarized optical 

pumping below the charge gap dynamically 

breaks time reversal and promotes a scalar-spinchirality term to the spin dynamics. The 

resulting chiral spin liquid has emergent 

excitations that obey the “semionic” statistics 

characteristic of a ν=1/2 Kalmeyer-Laughlin 

fractional quantum Hall state. Such emergent, 

fractionalized excitations are also believed to form the basis for quantum computation, which is immune 

to local noise and decoherence.

16 Therefore, simulating and understanding the dynamics of optically 

“switchable” interacting topological systems is highly desirable both from a fundamental physics 

perspective and as a possible new arena for noise-resistant quantum algorithms. 

However, accessing and predicting the quantum properties of strongly driven, nonequilibrium quantum 

materials is a tremendous numerical challenge. To date, numerical simulation (even on large-scale 

computational clusters) has been limited to small 1D and 2D amenable simulations to exact 

diagonalization and DMRG techniques. To this end, quantum computing has the potential to dramatically 

Figure 4. Phase diagram of the driven 2D Kagome 

antiferromagnet showing a transition into a chiral spin 

liquid with energy gap ∆CSL in circularly polarized light 

fields | Image reprinted from M. Claassen, et al., Nat. 

Commun. 8, 1192, 2017, under the terms and conditions of the 

Creative Commons CC BY license
8

increase the understanding of how novel topological states of matter can be realized and stabilized in 

realistic materials away from equilibrium. Moreover, once such materials are identified, a quantum 

processor will be able to simulate the dynamics of fractionalized excitations, suggesting novel strategies 

for the control and manipulation of these emergent degrees of freedom.

17 Finally, by scaling up the 

quantum simulation of such driven systems, it may become possible to discern certain blueprints for the 

engineering of topological matter from the bottom up using a combination of existing materials and 

optical pulses.

Understanding entangled states in materials for metrology with a quantum computer

At a fundamental level, a quantum computer is powerful because it can prepare and maintain complex 

superpositions across many quantum degrees of freedom. This allows the design of new sensing materials

with improved ability to interrogate chemical systems and quantum materials in order to precisely 

measure their properties, which is crucial for both fundamental and applied sciences. Current quantum 

coherent sensors use an ensemble of independent qubits (e.g., nitrogen vacancy centers in diamond), with 

which one can achieve a significant enhancement in measurement sensitivity, with errors scaling as 

~1/√N, where N is the number of sensors.

18 While the aforementioned scaling suggests that increasing the 

number of particles always improves the signal-to-noise ratio, crucially, this argument does not capture 

the effect of inter-qubit interactions. Indeed, above a certain density, interactions among the nitrogen 

vacancy centers will fundamentally limit the precision of any measurement and thus the maximum 

achievable sensitivity. More intuitively, once the sensing qubits become too close to one another, their 

accumulated spectroscopic signal will be dominated by the field of nearby particles rather than the 

external field being measured. 

To overcome this fundamental limit, one must be able to turn the interactions among the sensing particles 

into a resource and not a hindrance. This can be achieved if the interactions between the particles generate 

specially entangled quantum states, such as the Greenberger–Horne–Zeilinger (GHZ) state;

19 in this case, 

the nonclassical properties of the GHZ state enable precision measurements that reduce noise more 

rapidly than 1/√N (the standard statistical reduction from multiple independent sensors). However, an 

outstanding challenge for materials theory is to understand which strongly entangled states are actually 

generated in real materials as the density of sensing defects increases, while avoiding enhanced 

decoherence effects. To this end, a near-term quantum information processor capable of deterministically 

creating a variety of entangled states would significantly benefit quantum metrology, with enhancements 

in terms of both measurement sensitivity and bandwidth.

20

In addition to questions about the equilibrium states of 

sensing materials, quantum computing could test how using 

nonequilibrium pulsed controls might energetically 

stabilize these entangled states beyond their normal 

lifetimes (Figure 5).

Simulating frustrated quantum dynamics 

In strongly frustrated systems, competing interactions can 

conspire with fluctuations to prevent simple classical order 

from emerging.

21 When interactions are short-range, 

frustration typically relies upon geometry. This is most 

pertinent in the solid state, in which frustrated exchange 

interactions have led to the discovery of a number of 

exciting topological materials in layered 2D Mott 

insulators. An alternate route to frustration is provided by 

longer-range interactions, commonplace in chemical 

systems. For example, understanding the role of long-range 

Figure 5. Using a three-ion GHZ state 

enables a ~1.42× enhancement in 

spectroscopy over the projection | Image 

courtesy of Wineland Group, National Institute of 

Standards and Technology
9

ion-dipole forces (Figure 6) in generating dynamics in aqueous solutions remains a particularly 

challenging task.

22 By virtue of their intrinsic frustration, the energetic landscape of such systems can be 

highly complex with multiple nearly degenerate minima. To this end, methods to simulate the dynamics 

of frustrated quantum systems remain scarce.

By dynamically tuning the energy landscape, a quantum 

computer may be able to harness tunneling effects to more 

efficiently sample the landscape of minima in a frustrated 

system. Moreover, if quantum control techniques can generate 

finite overlap with a desired state, purification methods can 

systematically enhance this initial overlap. Finally, 

nonequilibrium frustrated dynamics can also emerge in 

disordered systems; in these cases, the disorder can cause 

localization, which prevents a system from becoming ergodic 

and sampling all of its allowed phase space.

23 Such systems 

present a possible alternative to stabilizing quantum coherent 

behavior in a many-body system without requiring complete

isolation from the environment.

Computational Challenges

The primary computational challenge for near-term quantum 

simulations of nonequilibrium quantum dynamics is the gate 

depth required to achieve interesting simulations.

24,25 At present 

it is believed that devices will be able to achieve a gate depth of 

~103 before error correction is required. It is not yet known if this is sufficient to reach time scales of 

physical interest for problems in chemical and materials systems. To this end, the further development of 

novel, testbed-specific algorithms will be essential. While a gate depth of ~103 may be physically 

feasible, the combination of gate fidelity limitations (currently at 99.9% or less) and qubit lifetimes will 

put limits on the accuracy of the results of quantum computing–driven simulations.

26 In a post-error 

correction setting, the difficulties are more likely to be related to initial state preparation, runtime, and 

analysis. 

These considerations lead naturally to the quantification of the impact of different quantum errors on the 

algorithms of interest. The most common error metric for quantum computations today is the fidelity of 

the quantum state; however, many alternatives may be preferable.

27 For the simulation of physical 

systems, it is clear that high fidelity is generally too stringent a requirement and moreover offers little 

insight into the degree of error expected in practice. This can be seen from a simple example of a 

collection of quantum spins or particles, where in a computational state of all 0s and 1s, the flip of a 

single spin will reduce the overlap fidelity with the original state to zero. If the physical system has local

interactions, the error in a quantity of interest may scale only as a small polynomial, e.g., linear or 

quadratic function of the number of spins flipped. It is also clear in such a model that as more spins are 

flipped, the error in the energy increases, while the fidelity remains zero. Therefore, it is important to 

identify improved metrics that accurately capture the error of a nonequilibrium quantum dynamics 

simulation. It is likely that the best choices will depend on the specific algorithm, the physical system 

being studied, and the quantum hardware in use.

The need to quantify errors in quantum simulation naturally leads to the idea of error model–specific 

algorithms. That is, if there is a reasonable idea of the errors that are expected to occur on a particular 

quantum hardware setup, are there steps that can be taken to mitigate the impact of these errors on the 

simulation? For example, if a setup incurs significantly more errors of one type than another, e.g., more 

Figure 6. Charge-dipole interactions, 

where four water molecules are 

interacting favorably with a 

magnesium dication. | Image 

reproduced from Molecular Interactions

courtesy of Loren Dean Williams, Georgia 

Institute of Technology
10

dephasing errors than bit flips, it may be possible to alter the algorithm or encoding to account for this. A 

number of near-term error mitigation strategies have been suggested along these lines in the context of 

quantum simulations.

28,29 In these cases, a reasonable error model allows a marked reduction of the 

impact on the calculation.

30,31

The additional consideration of the initial state is an important one in the context of dynamics simulations 

for both practical and theoretical reasons. On the practical side, some of the most efficient schemes for 

simulating chemical systems allow for both quantum and classical nuclei. However, they introduce the 

challenge of defining and preparing initial quantum states of both the electrons and the nuclei, which are 

no longer simple point charges in the system. Although preparing the exact initial state of interest may be 

difficult, recent work suggests that sampling over an ensemble of initial states can provide 

approximations to the thermal distributions of interest. Moreover, for ergodic systems with sufficiently 

long simulation times, this may not be an issue for error-corrected quantum devices. On the theoretical 

side, the simulation of a quantum system is formally promised to be efficient only if the initial state is 

efficient to prepare. While we expect this to be true for most natural physical systems, it remains an 

outstanding question in the field.

Quantum Computing Requirements

The quantum hardware requirements for simulating nonequilibrium dynamics differ greatly, depending on 

the approach used and the physical system of interest. In fact, in many cases, the exact quantification of 

the resources required remains an open question. In the case of hybrid analog and digital approaches, it 

may be possible, using recent sublinear depth algorithms32,33 for the simulation of a driven lattice or 

Hubbard model (relevant to quantum materials), to reach dynamics of interest on a system with as few as 

~50 qubits and a gate depth of ~103

. 

For early quantum computers, one restricted form of simulation is a hybrid-quantum classical exploration 

of nuclear dynamics. That is, the potential energy surface is discretized under the Born-Oppenheimer 

approximation, and the energies of the ground and excited states are calculated using a quantum computer 

at each nuclear configuration, yielding a more accurate energy surface than by existing means. Dynamics 

or alternative analyses could then be run on these surfaces using a classical computer. However, as the 

exact storage of these surfaces grows exponentially with the number of variables, the approach is most 

suitable for relatively low-dimensional reaction surfaces, such as simple dissociations. For higherdimensional systems, a treatment with more coupling between electron and nuclear motion is needed, 

using either surface hopping or eventually full quantum dynamics on a quantum processor.

Finally, the exploration and operation of a quantum information processor by scientific users presents 

many challenges driven by the experimental nature of quantum hardware and the absence of the essential 

software needed to program the hardware. Software for quantum computing is in its infancy, and the 

development of executable code for quantum hardware using current strategies is an arduous and 

unsustainable enterprise, locking many potential users out of exploring and using this technology. Key for 

an end-user software platform is the development of a top-to-bottom open-source software platform for 

quantum computing.

Potential Impact

The control and simulation of nonequilibrium dynamics has the potential to solve a number of 

outstanding open questions at the interface of the chemical and materials sciences. In the case of catalysis, 

one such question is the identification of the active site; namely, where the reaction occurs and how many 

atoms are involved.

5 The ability to simulate chemical reactions in the condensed phase will pave the way 

to identifying active sites and to mapping the pathways that lead to catalysis, in turn opening the door to 

controlling and optimizing such pathways. Once pathways are identified and controlled, they may be used 
11

to design catalysts for specific purposes.

34 Through careful engineering, pathway control may eventually 

enable designer catalysts for important reactions such as nitrogen fixation or carbon capture.

11 On the 

other hand, the quantum simulation of strongly driven systems also unlocks the ability to characterize 

entirely new materials and to stabilize nonstandard electronic states, which may not exist in traditional, 

equilibrium settings. These could include Floquet topological states as well as frustrated long-range 

systems.

12 Finally, the future use of a near-term quantum computer to prepare and analyze strongly 

entangled states can enable enhanced metrology, allowing for the direct imaging of nanoscale magnetic, 

electrical, and thermal properties (of both chemical reactions and quantum materials) while also 

optimizing the trade-off between sensitivity, spatial resolution, and detection bandwidth.

13

PRO 2: Unraveling the Physics and Chemistry of Strongly Correlated Electron 

Systems

Contributors: Thomas Maier, Bela Bauer, Gus Scuseria, Joel Moore, and Thomas Devereaux

Overview

Strong electronic correlations in molecules and solids give rise to complex many-body dynamics wherein 

electrons are entangled and multiple electronic states are nearly degenerate. This situation presents both 

opportunities and challenges. The opportunities range from new molecular magnets and nitrogen fixation 

technologies to novel superconducting, magnetic, and improved quantum information storage materials. 

The challenge is to understand and provide guidance in the design of this important class of systems. 

To address this problem, quantum scientists have designed a myriad of methods to deal with strong 

correlations. Although much progress has been made in the understanding of correlated systems, classical 

approaches to the exact solution are fundamentally hampered by the exponential growth of the storage 

required to represent and manipulate the quantum many-body wavefunction on a classical computer. In 

many of these systems, the key orbitals and interactions are well understood and there is a simple unit cell 

or other repeated structure; but computational difficulty grows so rapidly with size that predictions are not 

possible for the materials or molecules of greatest interest.

By storing the state in entangled qubits, a quantum computer can avoid this problem. Quantum computing 

therefore promises to reduce the simulation complexity from exponential to polynomial in system size 

and thus presents a major opportunity in the simulation of strongly correlated molecules and solids. 

Although the large gate counts and low noise levels required for quantum simulations will likely restrict 

near-term applications to problem sizes hardly competitive with classical computation, current research 

on quantum hardware and algorithms promises to make important progress on these issues.

Simulations on quantum hardware thus provide a unique opportunity to advance our understanding of 

strongly correlated molecules and solids and address a number of questions that are out of the reach of 

present classical approaches. These include the origin of high-temperature superconductivity, the 

robustness of spin liquids, the magnetization behavior of molecular magnets, and the catalytic activity of 

nanoparticles. Progress in our understanding of these issues can lead to revolutionary advances in several 

important energy-related technologies. 

Scientific Challenges

Theoretical modeling and simulation are essential elements in understanding the behavior of molecules 

and solids and thus guiding the search for systems with optimized properties. The progress made over the 

past decades allows for quantitative and predictive modeling of weakly correlated molecules and solids 

from first principles. These systems can be described starting from the standard electronic structure 

paradigm based on Hartree-Fock or DFT approaches, with many extensions made over the past decades. 

But systems in which electronic correlations are strong have successfully eluded such mainstream 

theoretical modeling. 

The difficulty of accurately modeling strongly correlated molecules and solids arises from the fact that, in 

these systems, the residual correlation potential originating from the Coulomb interaction between the 

electrons is comparable in size to the terms retained in standard approximations. This competition 

produces complex many-body dynamics, in which particles behave collectively and regularly produce 

nearly degenerate states. 
14

The family of copper oxide–based high-temperature superconductors provides a clear example of these 

principles in materials science.

35 The pseudogap phase at weak hole doping36 from which 

superconductivity emerges at lower temperatures hosts a number of different states with charge, 

magnetic, and nematic order, possibly intertwined with a Cooper pair-density wave,

37,38 and their relation 

to superconductivity remains puzzling. 

The families of magnetically frustrated spin systems also have rich phase diagrams in which the 

frustration of the magnetic interactions prevents the formation of long-range magnetic order.

39,40 This 

failure leads to many different magnetic states that lie close in energy. As a result, these systems are 

sensitive to small changes in pressure, strain, or magnetic field, which can switch the system between 

states. 

In molecular systems, the same type of situation arises when correlations are strong. In this case, several 

states are nearly degenerate, and their correlation effects are critical for accurately describing the 

formation and breaking of chemical bonds. Conical intersections (regions of potential energy surfaces 

where electronic states are nearly degenerate) are another example of how even relatively small molecules 

present considerable challenges for current methods when dynamics are considered.

To address this complexity, researchers need to use methods that treat the relevant degrees of freedom on 

an equal footing. Techniques based on approximations of a single electron moving in a mean-field of all 

the other electrons, such as Hartree-Fock or DFT, cannot describe the strong entanglement of the 

electrons that underlies this behavior. For these cases, quantum chemists and condensed matter physicists 

have designed a myriad of methods to deal with the strong correlation problem. However, the exponential 

growth of the many-body Hilbert space with the number of degrees of freedom remains a limiting factor 

for these approaches; this complexity cannot be overcome with today’s petascale and future exascale 

computing hardware. 

Quantum algorithms executed on quantum computing hardware do not face this problem, in principle. 

They promise to reduce the simulation complexity for quantum many-body systems from exponential to 

just polynomial,

41 thus presenting a fundamental advantage over classical approaches and creating an 

exceptional opportunity in the simulation of strongly correlated molecules and solids. This speedup from 

exponential to polynomial is similar in spirit to the remarkable speedup of factoring by Shor’s algorithm, 

a key theoretical development that helped drive quantum computing research.

Near-term opportunities are in providing high-quality benchmark data for molecules as small as a few 

atoms. For example, chemical accuracy (4 kJ/mol) has not yet been achieved for a relatively small 

benchmark system, Be2,

26 and in simulating simple effective models of solids, such as the Fermi Hubbard 

model, on small lattice sizes. Such proof-of-principle applications will provide important information 

regarding the feasibility and utility of quantum computing in addressing grand-challenge scientific 

problems in the area of correlated molecules and materials. Another opportunity lies in the quantum 

computation of dynamical quantities of correlated systems, such as the single-particle spectral function or 

the inelastic magnetic neutron scattering spectrum. Computing these observables remains highly difficult 

for classical approaches.

Computational Challenges

The main computational challenge of classical approaches to the quantum many-body problem arises 

from the exponential growth of the Hilbert space. That is, the memory required to store and manipulate 

the quantum many-body wave-function on a classical computer increases exponentially with the number 

of degrees of freedom. 
15

The coupled cluster method, the gold standard in quantum chemistry for weakly correlated systems, 

reduces this complexity to polynomial in system size.

42 But its predictions become very poor for strongly 

correlated molecules because of the lack of a dominating mean field. Quantum chemists have designed a 

myriad of methods to deal with this situation. Nevertheless, despite decades of efforts, techniques to deal 

with strong correlations either lack the polynomial computational cost as a function of size that coupled 

cluster theory enjoys, thus precluding their use on larger systems, or they are simply not accurate enough 

to be deemed paradigmatic. The quest for an accurate polynomial cost wave function ansatz for 

describing molecular strong correlations remains unabated.

In condensed matter physics, the full 

Hamiltonian of a solid is first reduced to 

a manageable model Hamiltonian that 

describes the low-energy degrees of 

freedom. We focus here on cases in 

which the model Hamiltonian is already 

known; developing improved ways to 

extract model Hamiltonians from initial 

chemical/structural information is 

discussed later in this document. The 

Fermi Hubbard model43 illustrated in 

Figure 7 is the most prominent example 

and is particularly relevant to the 

physics of the high-temperature 

superconducting cuprates.

44 Over the 

past decade, there has been significant 

progress in the development of 

numerical techniques for studying such 

models. Techniques ranging from 

determinantal quantum Monte Carlo 

(QMC)45 and exact diagonalization46 to DMRG,

47 tensor network methods,48 and dynamical mean-field 

theory (DMFT)49,50 provide complementary information on ground states and phase diagrams.

Similar to coupled cluster theory, QMC methods reduce the exponential complexity to polynomial. But 

this improvement is generally limited by the fermion sign problem, which arises from mapping the ddimensional quantum problem to an equivalent d + 1-dimensional classical problem.

51 This problem gives 

rise to an exponential growth of the statistical error and hence the simulation time, effectively restoring 

the original exponential scaling for all but a few limiting cases. 

The DMRG method is very efficient for 1D systems, but it suffers from a computational effort that scales 

exponentially with the system width owing to the increasing entanglement content of the wavefunction 

and the bias of this method toward states with low entanglement. More general tensor network algorithms

such as projected entangled pair states or multiscale entanglement renormalization ansatz—which were in 

fact originally inspired by quantum information science—work directly in 2D; but they scale with a very 

high power according to the number of quantum correlations in the wavefunction.

48

In addition, calculating the dynamic properties of a system, even in thermodynamic equilibrium—such as 

the single-particle spectral function measured in photoemission experiments, or the dynamic magnetic 

structure factor measured in neutron scattering—remains a challenge for classical approaches. For 

example, QMC methods do not work directly in real time, but rather in imaginary time; and the analytic 

continuation required to extract the real-time dynamics remains a key challenge in connecting theory with 

experiments,

52 a situation that provides additional motivation for PRO 1. 

Figure 7. The Fermi Hubbard model provides a low-energy 

effective description of the copper-oxygen planes in cuprate 

high-temperature superconductors. It models electrons moving 

on a 2D square lattice with a nearest neighbor hopping 

amplitude t and a local (on-site) Coulomb repulsion for two 

electrons with opposite spins on the same site. This model is not 

integrable, and classical simulations are limited by the 

exponential growth of the Hilbert space. | Image courtesy of 

Thomas Maier, Oak Ridge National Laboratory
16

Nevertheless, considerable progress in the development of classical approaches over the last few decades 

has led to advanced simulation capabilities that will be challenging to surpass with quantum computing in 

the immediate future. Simulations of bigger or more complex systems on quantum hardware will require 

large numbers of qubits and gate operations. A quantum simulation of the ground state of the Fermi 

Hubbard model, for example, for lattice sizes competitive with those of classical computations, will 

require at least on the order of 100 logical qubits and millions of gates.

53 For materials science problems, 

quantum algorithms that simulate the system directly in the thermodynamic limit instead of on a finitesize real-space lattice could potentially circumvent this problem. 

Another challenge for quantum computations of materials is the efficient preparation of an initial quantum 

state (the “ansatz”) from which a variational quantum eigensolver (VQE)54 tries to find the ground state 

of a given Hamiltonian by successively minimizing the expectation value of the Hamiltonian, i.e., the 

energy. This task is difficult, if not impossible, if the ansatz state does not have sufficient overlap with the 

true ground state of the system and if there are a number of nearly degenerate states. For example, if the 

true ground state wave function has superconducting order, but the initial state does not, this procedure 

will converge to a state that is not the true ground state of the system. 

Quantum Computing Requirements

A number of quantum approaches are known for tackling the low-energy behavior of strongly correlated 

electron systems. For a recent nontechnical overview of the developments in the field of quantum 

chemistry, see an article by Mueck.

55 Most prominent is a direct simulation approach, akin to full 

configuration interaction in the chemistry context, or exact diagonalization in condensed matter. This 

approach was first described for chemistry applications in Aspuru-Guzik3 and discussed in the condensedmatter context in Wecker et al.53 Although the minimal number of error-corrected qubits required is 

relatively small, achieving high accuracy in this approach often comes with the challenge of relatively 

large gate counts of at least millions of error-corrected gates. Especially for chemistry applications, a 

large body of literature has attacked this problem, and researchers have reduced the gate counts drastically 

in many relevant cases compared with early estimates. However, the challenge still remains considerable 

for near-term quantum computing applications. 

In the condensed matter context, a crucial challenge is that the problems exhibit more locality and are thus 

often more amenable to classical simulation, thus raising the number of qubits at which a quantum 

computer would exhibit a clear advantage. However, relevant cases become feasible at on the order of 

100–200 qubits and 106 gates. An additional complication is that in many interesting cases, preparing the 

correct low-energy states may be difficult, since several competitive low-energy states may be very close 

to each other in energy. However, quantum computers promise a significant advantage for computing 

dynamical properties (such as spectral functions or dynamical structure factors), which are largely 

inaccessible to classical methods.

To go beyond the direct, full simulation of the problem, several alternative approaches have been 

discussed. A promising direction is the use of variational quantum algorithms.

54 These may considerably 

reduce the requirements for the number of gates that must be coherently executed in a single run and are 

naturally resilient to certain types of errors in the gate implementation. However, as these are heuristic 

approaches, they are chronically difficult to analyze without access to a quantum computer, making it 

hard to assess the number of gates that need to be coherently executed for a given problem to reach a 

desired accuracy. 

An alternative approach is to embed a quantum computer as a solver in a classical embedding method, 

such as DMFT or the density matrix embedding theory. This approach has been proposed for complex 

materials, which defy a direct approach because of the large number of qubits that would be required.

56
17

These approaches have the advantage that they work directly in the thermodynamic limit and can solve 

complex problems with a relatively small number of qubits; however, they require additional 

approximations that are difficult to control. Furthermore, the gate depth of the relevant circuits is 

relatively deep, requiring the coherent execution of about 108 gate operations.

56

A central question is whether any relevant problems can be solved with noisy qubits, i.e., with highquality physical qubits, but without error correction. Current experimental evidence suggests a skeptical 

outlook: for circuit depths beyond classical simulation capability, noise destroys all quantum advantages. 

However, as both the hardware and the understanding of the algorithms are likely to see drastic 

improvements, this conclusion may change. Some recent theoretical work has investigated how to solve 

these issues,

57,28 but it remains to be seen how these approaches can be brought to bear on practical 

calculations.

Potential Impact

Prototypical examples of molecular systems in 

which strong correlations are ubiquitous include the 

single-molecule magnet Mn12 acetate (see Figure 8) 

and related compounds, a case that exhibits an 

extremely slow relaxation of its magnetization 

below a blocking temperature. Another example is 

the iron-molybdenum cofactor (FeMoco) active site 

of nitrogenase, the enzyme that breaks the N2 bond 

at room temperature. The current process for 

making ammonia from nitrogen uses high pressure 

and high temperature and consumes 1% of the 

world’s energy. Accurate quantum calculations of 

strongly correlated enzyme active sites could 

unravel their secrets and lead to understanding that 

would provide the basis for important technological 

breakthroughs in catalysis and many other relevant 

areas. Design of novel molecular magnets could 

have a large impact on low-cost, environmentally 

friendly materials for storage and a myriad of other 

technological applications.

A successful quantum computation of the Fermi Hubbard model would have significant impact in 

condensed matter physics; this model is commonly believed to have the right ingredients to enable an 

understanding of high-temperature superconductivity in the cuprates. Although it is too simple for a 

quantitatively accurate description of these systems, it allows researchers to study the essential behavior 

arising from strong correlations, such as magnetism and superconductivity. Calculating the ground state 

and finite temperature properties of this model and understanding the mechanism(s) that give rise to these 

behaviors could lead to a detailed understanding of high-temperature superconductivity in the cuprates. It 

would allow definition of the minimal set of ingredients necessary to explain these phenomena. 

Furthermore, the knowledge gained would allow scientists to define a set of principles for designing 

systems with improved properties, and thus help guide the search for superconductors with higher 

transition temperatures that could potentially revolutionize US energy technologies.

Moreover, progress in the simulation of frustrated spin systems—such as the Heisenberg model on a 

Kagome lattice (Figure 9)—could significantly advance our understanding of spin liquids by providing 

Figure 8. Mn12-acetate single-molecule magnet. 

Manganese atoms with oxidation states of III and 

IV are in orange and green, respectively; oxygen is 

in red and carbon in black. | Image reproduced with 

permission from Jens Kortus from Molecular Magnetism 
18

new insight into the competition between nearly 

degenerate low-energy states, including exotic spin 

liquids and more conventional magnetically 

ordered states. Elastic and inelastic neutron 

scattering can measure many properties of the 

magnetism that are currently beyond what can be 

accessed by numerical simulation. Such progress is 

important to understand the robustness of desired 

spin liquid ground states and thus could open new 

avenues in the experimental realization of quantum 

spin liquids. The spin liquid phase has been linked 

to unconventional superconductivity and may be 

useful for further advancing quantum computing 

research. 

Figure 9. The mineral herbertsmithite (left) is 

composed of a lattice of copper ions sitting on a 

kagome lattice (right) that consists of cornersharing triangles. This lattice structure frustrates the 

magnetism despite a strong exchange coupling 

and thus can host a quantum spin liquid state, in 

which spins do not order down to the lowest 

temperatures. The exact nature of this state and its 

competition with other more conventional 

magnetically ordered states is an active area of 

current research. | Image by Rob Lavinsky; reproduced 

from https://commons.wikimedia.org/wiki/File:

Herbertsmithite-163165.jpg#filelinks under the terms and 

conditions of the Creative Commons CC BY license.
19

PRO 3: Embedding Quantum Hardware in Classical Frameworks 

Contributors: Peter Love, Sue Coppersmith, John Preskill, Birgitta Whaley, and Marivi Fernandez-Serra

Overview

This section discusses computational approaches that merge quantum and classical pieces by passing 

information back and forth in real time, with applications to complex molecules and materials and to 

“open” quantum systems in which quantum orbitals interact with an effectively classical environment, 

such as a liquid. Applications of interest to BES include the development of complex, bio-inspired 

molecules with the potential for creating chemical energy from light (artificial photosynthesis) and the 

creation of new quantum materials, ranging from improving the understanding of defects that limit the 

scaling of current computers to ever smaller sizes, to new families of magnets and high-temperature 

superconductors.

Many quantum chemistry methods are efficient because they separate a problem into a relatively easy part 

that can be treated with computationally inexpensive approaches, and a “hard” or strongly correlated part 

that can be treated only with more expensive techniques. For example, organic molecules and transition 

complexes containing only one or two transition metal atoms are well modeled by the current generation 

of ab initio quantum chemistry methods. These methods—such as complete active space perturbation 

theory second order (CASPT2)58 or n-electron valence perturbation theory second order (NEVPT2)59—

provide an ab initio, systematically improvable framework that describes weakly correlated, delocalized, 

s- and p- electrons by an approximate and computationally inexpensive method, while the strongly 

correlated, spatially confined, d-electrons are modeled at a higher level of theory.

Although these methods remain valuable, once the number of transition metal atoms exceeds two, the 

difficulty of simulating these transition metal–rich compounds quickly becomes insurmountable because 

of the exponential growth of the Hilbert space necessary to model the strongly correlated electrons.

46 The 

preceding PRO concentrated on cases in which the strongly correlated problem is known from experiment 

or an initial one-shot classical calculation, often because there is a simple unit cell or repeated structure. 

However, in most compounds, there is a need to embed algorithms that could pass information back and 

forth between quantum and classical pieces in real time and achieve self-consistency.

60

Such hybrid algorithms would allow major progress on complex transition metal–containing molecules 

that are of great fundamental and applied interest. In the past decade, the adaptation of the DMRG61,62,63

to chemical systems allowed accurate calculation of systems with up to three and four transition metal 

atoms. However, ab initio evaluation of systems with larger numbers of transition metal atoms currently 

remains completely inaccessible. Thus, active centers of biologically important enzymes, such as 

FeMoco‡

, which is the primary cofactor of nitrogenase in nitrogen fixation, or the Mn4O5Ca core of the 

oxygen evolving complex§ (see Figure 10), remain far beyond the current capabilities of ab initio 

quantum chemical methods.

 ‡ See https://en.wikipedia.org/wiki/FeMoco. § See https://en.wikipedia.org/wiki/Oxygen-evolving_complex.
20

Scientific Challenges

For these strongly correlated large molecules, and for 

solids for which a simple model Hamiltonian such as the 

Hubbard model is unavailable, finding a systematically 

improvable and accurate description of the physical 

ground state remains far beyond the capabilities of 

current quantum chemistry methods. Spintronic and 

battery applications of correlated solids also involve not 

only the description of the ground state but additionally 

the description of multiple excited states, particularly to 

model function at nonzero temperatures. Currently, the 

description of temperature effects beyond a few lowestlying excited states cannot be done at the ab initio level. 

The lack of such a description, beyond the smallest 

possible cases, inhibits finding new families of hightemperature superconductors and creates problems for 

above-room-temperature materials applications such as 

those appearing in nuclear energy applications and warm 

dense matter. 

Similarly, simulations of quantum dynamics 

and open dissipative systems are frequently 

beyond the reach of traditional methods in 

quantum chemistry and condensed matter. 

The quantum simulation of sequential 

absorption of single photons, subsequent 

energy transfer, and charge separation that 

is required to probe these quantum 

dynamics within a realistic model of 

photosystem II (see Figure 11) is well 

beyond the capability of current classical 

computers and represents a substantial 

challenge for quantum computers. 

In all the aforementioned examples, the 

common feature that renders the current 

generation of ab initio methods unsuitable is 

the growth of the Hilbert space necessary to 

describe the strongly correlated part of the 

problem. To overcome this difficulty in both 

the quantum chemical and condensed matter 

communities, multiple embedding methods 

were designed. In an embedding 

method,

49,64,65 all electrons present in the 

system are separated into “easy to describe” 

weakly correlated electrons and “hard to 

describe” strongly correlated electrons. The 

weakly correlated electrons are treated by a 

low-level, computationally inexpensive 

method with sufficient accuracy. The 

Figure 10. Crystal structure of the oxygen 

evolving complex of Cytochrome P450 at 1.9 

Å resolution. Multiple manganese and oxygen 

atoms make the description of such an 

enzymatic center very challenging for 

traditional methods. | Image reproduced from 

commons.wikimedia under the terms of the GNU 

Free Documentation License.

Figure 11. The core complex contains the reaction center 

(not shown), and the peripheral light-harvesting complexes 

(LHCII, CP29, and CP26) supply excitation energy to the 

reaction center. PsbW is a core subunit typical of plants 

that mediates the association of LHCII with the core |

Nature Publishing Group. Nature, Structure of spinach 

photosystem II: LHCII supercomplex at 3.2 Å resolution, P. Wei et 

al., copyright 2016
21

description of the strongly correlated electrons requires solving a difficult problem for which the size of 

the Hilbert space grows exponentially. Furthermore, the behavior of the strongly correlated electrons 

influences the weakly correlated part of the problem, e.g., by modifying the screening of interactions; thus 

a self-consistent solution typically requires iteration and the passage of information between weakly 

correlated and strongly correlated pieces. Consequently, embedding methods that initially were developed 

for classical computers can be adapted for use in quantum computations, because the part of the 

calculation involving strongly correlated electrons can be successfully run on a quantum computer. For 

details, see Figure 12. It is reasonable to envision that in the near future, when machines with around 50–

100 qubits become available, embedding calculations could become a dominant technique in the quantum 

chemistry and condensed matter communities. 

Computational Challenges

For the foreseeable future, quantum 

computers will need to interface 

intimately with classical computers 

and software that control the 

computation and the flow of input 

data and output results. This is 

especially true in the short term 

while quantum computers lack the 

number of qubits, robustness, and 

functionality required to run 

complete simulations. Although 

rigorous theoretical results underpin 

our expectations of the benefit of 

quantum simulation, we anticipate 

that many practical computations 

will have to step beyond these proofs 

and employ algorithms that initially 

have limited formal proofs. 

At present, the main computational 

challenge is thus to design and optimize classical algorithms that are suitable for possible future hybrid 

quantum classical computations. Currently, there are several classes of classical algorithms that would 

benefit from the interface with quantum computation/algorithms. Examples include

• Optimizing (e.g., stochastic) of parameters in the quantum simulation, including its initial state

• Searching for (e.g., by combinatorial chemistry) dynamics of molecules and materials driven by 

results of quantum simulations

• Interfacing with big data analytics, e.g., feeding the results of quantum simulations into classical 

machine learning, or taking real-world data (e.g., from BES light sources and microscopes) and 

feeding them into quantum machine learning algorithms, or otherwise leveraging the vast databases 

of scientific data of modern science

• Physical embedding (e.g., using DMFT,

49 self-energy embedding theory [SEET64], and density matrix 

embedding theory65) of quantum simulations of finite systems into models capable of treating 

extended complex systems at finite temperatures.

• Studying open quantum systems (a quantum system interacting with a classical environment)

• Simulating time evolution in complex molecules, building on PRO 1

Figure 12. An example workflow between classical and quantum 

machines. Embedding methods developed for classical 

computers can be easily adapted to this mixed quantum classical 

computation scheme. | Image courtesy of Dominika Zgid, University of 

Michigan
22

Another challenge for current quantum chemical methods on classical computers is creating a series of 

basis sets that are suitable for quantum computers. Most of the current quantum computations are 

performed within very small basis sets. It is conceivable, but far from certain, that in the near future, good 

basis sets will be formed in which the number of orbitals can be kept small while good accuracy is 

maintained. In the long term, basis sets that can approach completeness in a systematic manner while 

being sufficiently well conditioned are highly desirable. Such bases are very important for extended 

systems and in systems with many length scales.

In the current form, the quantum computer is more efficient when Hamiltonians are sparse and the 

number of required two-body Coulomb integrals is small. Learning how to prepare and optimize 

Hamiltonians and initial states that are suitable for efficient quantum simulations will certainly constitute 

a short-term challenge for the current generation of classical quantum chemistry methods. 

Finally, although the possibility of interfacing classical and quantum simulation is exciting, further 

developments are necessary to create a unifying formalism that can first be tested on classical computers 

and that is compatible for molecules, solids with periodic structure, temperature-dependent problems, and 

nonequilibrium problems. For example, formalisms based on Green’s functions not only are general 

enough to access a broad class of systems but also could predict both equilibrium and nonequilibrium 

processes.

Quantum Computing Requirements

Since, in the short term, results from quantum simulation will be inherently noisy, robust schemes must 

be developed that will couple the classical and quantum computations and give accurate answers without 

an accumulation of errors from the noise. Existing computational classical embedding methods such as 

DMFT, density matrix embedding theory, and SEET are already capable in principle of using QMC

solvers that yield noisy data.

66 However, there should be additional development of algorithms and a 

careful evaluation of their noise robustness.

Moreover, current calculations on quantum computers mostly focus on evaluating electronic energy. 

When a classical–quantum hybrid approach is used, quantum computers should be able to evaluate first-, 

second-, and higher-order density matrices or Green’s functions. A major short-term advantage would be 

to use quantum computers of a few hundred qubits to evaluate high-level density matrices in quantum 

chemistry calculations such as CASPT2 or NEVPT2. Similarly, quantum computing algorithms should be 

optimized to evaluate Green’s functions in quantum impurity problems, since the evaluation of these 

problems is a bottleneck for current classical methods such as DMFT or SEET. The outputs of the 

quantum simulation, such as density matrices and Green’s functions, will then be fed into classical 

algorithms that might leverage existing implementations of these approaches. The main challenge is to 

evaluate these quantities on a quantum computer with sufficiently small noise.

57,28

A separate set of quantum computing requirements is present for quantum dynamics and quantum 

simulation of open systems. The major computational challenge is to develop efficient propagators for 

open quantum systems such as LHCII (see Figure 11). Doing so requires simulation of both unitary and 

dissipative evolution, for both localized “site” excitons and separated charge transfer states. Several 

components are involved. First these states, which are of variable spatial locality, need to be mapped to a 

qubit representation of well-defined connectivity.

67 Then either digital (stroboscopic) or continuous 

Liouvillian propagators need to be developed. For Markovian systems, a number of such schemes exist, 

some of which have been implemented experimentally. For example, Bell and GHZ states have been 

prepared as steady states of digital evolution in an ion trap experiment using stroboscopic cycling of 

sequences of unitary and dissipative terms, where the latter were realized by the addition of ancilla states 
23

and optical pumping.

68 Fewer options are known for propagation in quantum simulation of nonMarkovian systems,

69,70 and further development of efficient Liouvillian propagators is desirable.

Another challenge will be handling noise encountered as a result of decoherence of the quantum system, 

as well as at the classical–quantum interface. Instead of treating quantum decoherence as counterproductive, there is an opportunity to engineer the coupling between the classical and quantum systems. 

Doing so would allow for controlled dissipation to reach computationally relevant states, as well as for 

preparing targeted thermal and mixed states.

Finally, the last challenge for classical–quantum hybrid machines will lie in reproducibility, verification, 

and validation. To advance classical–quantum hybrid machines, a community agreed-upon set of 

approaches and a consistent set of results/benchmarks is necessary to verify and validate the results of 

hybrid simulations. Such simulations introduce new types of errors (e.g., model, systematic, statistical, 

experimental, technological) that are not commonly found in classical simulations. It took a concerted 

community much effort over decades to develop effective approaches for classical simulation, and there 

are still many lingering questions about reproducibility. We must address this problem head-on and early 

in the life of classical–quantum simulations.

Potential Impact

In the long term, quantum computation promises fully predictive simulations of many highly important 

chemical and material systems, with the same or greater ease with which researchers now use 

approximate density functional methods that have no systematic basis for controlling error or improving 

accuracy. The revolutionary advances that quantum computation promises will translate into new 

materials and technologies that will form the basis of new industries and have a profound impact on our 

economy and nation. For instance, computational insights into spin-crossover complexes and magnetic

ad-atoms on surfaces can be used to optimize future nano-devices. Predictive optimization of strongly 

correlated active centers of enzymes can be used to advance medical research, in addition to the catalysis 

and photosynthesis advances mentioned earlier.

It is also expected that entirely ab initio prediction of new classes of high-temperature superconductors 

will be possible. New superconductors could have a tremendous impact on energy losses in electricity 

transmission and other applications. Since hybrid classical–quantum simulation will include all orbitals 

present in the realistic system (not just a model system), we can expect that chemical and physical 

changes controlling superconductivity will be understood and used in the design of future hightemperature superconductors. The predictive nature of hybrid classical–quantum simulations should also 

be helpful in simulation of phase diagrams of actinides, a complex materials class of DOE interest.

Finally, the possibility of simulating open dissipative systems as well as time evolution (see PRO 1) 

should provide an essential theoretical component that can guide many experiments dealing with 

nonequilibrium processes that are notoriously hard to interpret. In summary, the major impact of 

classical–quantum computations will lie in delivering a series of modern, systematically improvable tools 

that can be used in both quantum chemistry and quantum materials. We think that these tools will become 

essential in understanding the physics phenomena present in new materials, as well as enabling the 

controlled design of materials that are industrially relevant.

25

PRO 4: Bridging the Classical–Quantum Computing Divide

Contributors: Dominika Zgid, Robert Harrison, Antonio Mezzacapo, and James Whitfield

Overview

Quantum computation faces the daunting challenge of outperforming 70 years of exponential growth of 

classical computational power reflected in Moore’s Law. In the near term, by which we mean 50 to 100 

physical qubits, quantum computers may pass the threshold of “quantum supremacy”—meaning they 

themselves cannot be simulated by any existing, or conceivable, classical computer. A more stringent 

condition is for a quantum computer to solve a useful and interesting computational problem that classical 

computing cannot solve, and eventually to access grand-challenge science.

However, classical and quantum computing do not exist only in competition. PRO 3 discussed the realtime exchange of information between classical and quantum hardware in hybrid algorithms. This section 

considers bridging between classical and quantum computing—for example, cases in which the strengths 

of quantum algorithms can be used to improve the key inputs of classical algorithms or to capture key 

quantum processes in “open systems” in which decoherence by coupling to the environment is present. In 

other words, PRO 3 discusses hybrid or co-processor computing, while PRO 4 discusses integrating 

results of separate quantum and classical computations to solve scientific challenges. Open systems 

appear across chemistry and materials science, including technologically important examples such as 

solid-liquid interfaces and optical absorption in liquid backgrounds. Such approaches also have the 

potential to scale up as the ratio of quantum to classical resources used grows over time.

Scientific Challenges

Our PROs are in the areas of quantum chemical and materials sciences. Specifically, this PRO considers 

the improvement of exact solutions for small instances of the electronic structure problem and the 

improvement of DFT for larger problems such as clusters of water molecules.

The first example is the determination of exact electronic energies for particularly difficult cases of the 

electronic structure problem. We call these electronic structure challenge problems that are defined by 

the failure of classical approximate methods. This means that (1) they must exhibit large correlation 

energy so that self-consistent field approaches fail; (2) they must exhibit a severe fermion sign problem so 

that QMC fails; and (3) coupled-cluster or similar gold standard correlation approaches must not yield the 

desired accuracy. In addition, they must present a sufficiently severe multi-reference character to 

confound full configuration interaction QMC approaches. Clearly, these conditions mean that small 

example problems will be few in number, and the best classical approaches may require high-accuracy 

solutions. A bridging approach uses the best classical results as the starting point for a quantum 

variational method. The quantum resources are used to broaden the space of possible ansatz states 

available to the variational method and can therefore improve upon the best classical approach.

The second example is the quantum simulation of small clusters of water by VQE. High-accuracy 

simulations of these systems can inform density functionals that enable DFT to better treat the structure of 

water in a wide variety of problems. The accuracy requirements are not expected to be as stringent, but 

the necessity to treat a cluster will require more orbitals for representation of the problem. Such a 

challenge problem for smaller (50–100 qubit) quantum computers will help investigate whether the best 

quantum solution on a smaller basis is better than the classical solution on a larger basis. An improvement

is possible owing to the higher quality of solution available by quantum means, but is by no means 

guaranteed. Such problems therefore represent an interesting short-term test case for these methods.
26

The search for new density functionals that are capable of accurately describing different phases of 

matter, and simultaneously do not compromise on simulation performance with respect to system size and 

length scale simulations (in ab initio molecular dynamics approaches), is currently on the rise. The search 

is motivated by the availability of very accurate experimental data and by the push of machine learning 

and data science–motivated approaches to fitting complex functions. The parameterization and 

optimization of complex models is a pervasive problem in many areas, particularly in the development of 

new density functionals optimized to treat specific systems. It has been shown that—using highly 

accurate quantum chemical calculations—a standard, generalized gradient approximation and a van der 

Waals-type of density functional can be optimized to describe liquid water, or general systems with 

accuracy close to that of the data to which it was fitted (Figure 13).

71,72 This approach would certainly 

benefit from the availability of even more “exact” data, not only total energies but also density matrices 

and possibly other observables that can be directly computed using DFT. Such sets of data are directly 

accessible using quantum computers but are not obtainable with direct experiments; hence they represent 

the exact numerical experimental results. Combining machine learning with density functional 

development (including functionals of the density matrix, or other nonstandard approaches) or advancing 

on the systematic development or optimization of more accurate density functionals71–73 will enormously 

benefit from quantum simulation results. The data availability would accelerate and possibly give birth to 

new ideas for optimization and data fitting, in addition to providing new physical and chemical insight 

from which new theories or models will emerge.

Figure 13. An accurate and fully quantum mechanical description of liquid water will have an impact in 

many BES priority areas. In the picture, the catalytic properties of SrTiO3 surfaces are strongly coupled to 

their interactions with ambient water, and a full quantum mechanical atomistic simulation (left) is needed 

to interpret experimental results (right, atomic force microscope image of SrTiO3 under water after 

irradiation by ultraviolet light). | Images courtesy of M. V. Fernandez-Serra (left) and M. Dawber (right), both of Stony 

Brook University.

The techniques to be employed on quantum hardware are VQE with two possible types of states for the 

initial ansatz that is improved by the quantum computation. Both the coupled-cluster ansatz and tensor 

network state ansatz will be employed. Both of these can be regarded as gold standard methods for 

electronic structure methods. Both, fortunately, admit a straightforward route to marry the best classical 

solutions to quantum algorithms that can improve those solutions. 

The use of coupled cluster in VQE has already been investigated theoretically, and it is quite natural to 

take a coupled-cluster ansatz obtained by classical means and use the parameters of the classical ansatz to 

create a short quantum circuit producing the corresponding ansatz for the quantum computer. One can 

therefore use this as a starting point for the VQE solver, and any enhancement of the space of variational 

wavefunctions will cause the variational estimate of the energy to improve (decrease). In an arms race 

between classical coupled-cluster and quantum VQE methods, a user is therefore able to transfer any 

classical improvements immediately to the quantum case. The only possibility of failure is that the 
27

classical method may be able to solve larger problems (with more orbitals) than there are qubits available. 

As qubit resources grow, the quantum methods must therefore win out.

Similarly, the use of DMRG, matrix-product, and tensor network methods is widespread in condensed 

matter physics, materials science, and chemistry. These methods suffer from exponential scaling with 

bond dimension on classical computers, but not on quantum computers. Again, a best-in-class classical 

tensor network approximation could be used as a starting point for a quantum variational calculation. 

Computational Challenges

In the case of variational approaches to electronic structure problems, quantum hardware can access a 

larger space of variational states than is possible classically. As discussed earlier, the best-case classical 

variational states can be used as a starting point for such quantum variational calculations. In principle, 

this means that the use of quantum hardware can only improve upon (or in the worst case, match) 

classical variational algorithms. In practice, the efficacy of such approaches will be determined by the 

ability to optimize an ansatz on quantum hardware with more parameters than in the classical case. 

Therefore, it is worth considering the possibility that quantum approaches to optimization may enable 

taking a further step forward using VQE-based algorithms. 

Optimization is a vast field of numerical analysis that is hugely important in many areas of science and 

technology. One important application relevant to BES is optimization of time for processing chemical 

and material transformations. Time-optimal control of processes such as chemical reactions; materials 

processing; engineering of control of quantum systems; and larger-scale manufacturing processes that rely 

on exterior features such as workflow, logistics, and transportation requires the optimization of a function 

that is generally highly nonlinear in the parameters. Minimization of the time to target can mean 

significant savings in resources and can often increase energy efficiency.

74

Another key application for BES is the variational optimization of energies for electronic structure 

calculations. Hybrid classical–quantum computation schemes for electronic structure calculations, such as 

VQEs, which have been proposed for near-term quantum computers, rely on the ability to perform 

efficient classical optimization—for example, quantum gate parameters for state initialization, which are 

potentially off-line from the quantum aspects of the calculation (state preparation and Hamiltonian 

tomography). However, while the Hamiltonian is 2-local (i.e. has pairwise interactions, but these may be 

long-range), the complexity of the classical optimization step rises rapidly with the number of qubits on 

account of the growth in number of gate parameters. Thus, any demonstration of quantum enhancement in 

optimization algorithms would be greatly impactful. Optimization tasks are also intimately related to 

questions of coherent control, and control theory provides an additional perspective for physics-inspired 

approaches to optimization. For example, hybrid schemes such as VQE can be regarded as examples of 

open loop learning algorithms.

Recent developments in quantum algorithms have provided key components that might enable such an 

enhancement. First, a quantum algorithm recently proposed for semidefinite programming shows 

exponential speedup in certain instances, in particular, when the input matrices are mapped to quantum 

states of low-rank Hamiltonians.

75 This finding has significant implications for all convex optimization 

problems, which constitute a very large subgroup of optimization instances, as well as for any problem 

that can be mapped onto sequential convex optimization steps in an iterative procedure. Second, a 

recently proposed quantum algorithm for gradient calculations76 shows a significant speedup in efficiency 

and an improvement in complexity, implying a significant quantum advantage for the performance of 

optimization calculations in large-scale systems. Additional developments in coherent control theory for 

quantum systems are showing progress in techniques to search for the global optimal solution via analysis 
28

of the typically high-dimensional control “landscape,” i.e., the topographic map of the quantum metrics 

for successful control as a function of the control parameters.

Together with the developing capability to perform quantum Hamiltonian simulation, these developments 

suggest that a proven quantum advantage in simulation—combined with quantum speedups in convex 

optimization and/or quantum gradient calculations—could enable a significant speedup for many-variable 

optimization problems, particularly those that can be cast in convex form. The time optimal control 

problem mentioned earlier is one such problem, in which a nonlinear time evolution might be either 

linearized or made quadratic over small time scales and the local optimization then implemented by a 

quantum semi-definite programing algorithm.

77 From a fundamental perspective, this problem is highly 

important because one might thereby find a route to the global optimum of the control landscape.

78 From 

a practical perspective, an advance of this type would be highly valuable for two application directions of 

interest to BES: time optimal control of processing and variational hybrid classical–quantum approaches 

to electronic structure of molecules and materials.

Quantum Computing Requirements

In the case of VQEs, the requirements in terms of qubits can be quite modest—simulations in the 

quantum supremacy regime of 50–100 qubits are expected to be comparable to the best classical 

approaches. An example application is given by the proposed simulation of the uniform electron gas—

Jellium—by the Google group.

79 In general, for VQEs, the constraint will not be the number of qubits 

(once the threshold of direct classical simulatability is crossed at around 45–50 qubits). In these 

algorithms, the number of gate operations or measurements required can easily grow to astronomical 

numbers.

80 Rapid repetition of measurements or large parallelism of many small independent quantum 

devices may therefore be necessary.

Potential Impact

In the next 5 years, quantum computers will cross a threshold beyond which classical simulation of the 

devices will become impossible. Although such quantum supremacy demonstrations are of intrinsic 

scientific interest, to be truly impactful, this next generation of quantum hardware needs to be applied to 

addressing problems of grand-challenge science. The bridging approach proposed in this report is one 

way to reach that goal. Using quantum techniques such as VQE that can improve upon best-in-class 

classical results should enable researchers to address interesting scientific questions. The electronic 

structure challenge problems and improvement of water functionals would have significant impact and 

help establish quantum simulation of materials properties as a reality in the 21st century.
29

3. Crosscutting Computational Themes and Challenges

Contributor: James Whitfield

The BES Roundtable was successful in identifying four PROs, which are discussed in the preceding 

chapters. During the roundtable, it became clear that several computational aspects appear in multiple 

PROs. The crosscutting aspects identified by the roundtable participants are error correction or robustness 

in the presence of quantum noise, software stack developments, and the validation of quantum simulation 

results.

Quantum decoherence is responsible for the emergence of classical physics from an underlying quantum 

system. Since all of the PROs attempt to exploit quantum effects for enhanced computations, it is 

important to understand and mitigate these decoherence effects. There are two broad ways to fight against 

the effects of decoherence: fault-tolerant quantum error correction81 and error mitigation techniques. The 

first of these two methods has seen continuous development since the 1990s; however, full fault tolerance 

requires significant qubit overheads that will require significant technological development before such 

methods become standard. Given the status of current and near-future quantum hardware, requiring fullfledged quantum error correction would greatly hinder applications in the foreseeable future. 

Consequently, the second option of developing error mitigation will be an important scientific driver in 

the near future. Error suppression techniques and intermediate levels of quantum error correction in 

various experimental setups can optimize quantum hardware utilization. In addition, the design of 

pre-/post-processing filtering is expected to enhance quantum computing noise robustness.

The second crosscutting theme, software stack development, refers to computational tools and software 

packages needed to assist scientists with minimal backgrounds in quantum computing to set up and 

execute quantum simulations. Since the technology is still being developed, the concept of the stack is not 

completely defined. Ideally, the software tools should maintain a user-friendly abstraction level while 

compiling inputs to experimental pulses/gates/controls of quantum devices. Doing so will require 

maintaining hardware backend interfaces, useful encapsulations going up and down stack, and the 

flexibility to integrate new quantum algorithms into software stacks. Good integration of optimal classical 

and quantum methods will build the quantum computing user base and create the needed infrastructure 

for the applications outlined in the PROs. 

The third theme identified that overlaps with each of the PROs is the scientific challenge of validating the 

output of quantum simulations. It will be important to establish the correct metrics for quantum 

simulations/measurements. Examples include local observables, correlation functions, and the fidelity of 

the computed wave function. The specific metric will ultimately be determined by the specific quantum 

computing application and the available hardware resources. Even for small quantum systems, full 

tomography of the wave function is expensive. Hence, near-term quantum validation will likely focus on 

test cases from solvable models (e.g., the 1D Hubbard model), well characterized molecular test sets, and 

high-quality benchmarks (for a small representative collection see Lynch and Truhlar82). Based on the 

current pace of development, quantum computation technology will soon possess computational power 

that exceeds that of direct classical computational simulation. Hence, alternative validation schemes will 

be needed. Theoretically, a number of works explore the validation of quantum algorithms without

requiring a classical numerical reference. Blind quantum computation and interactive proofs83,84 have 

been studied in recent years but have not yet been translated to realistic quantum simulations.

Significant progress in these three crosscutting computational themes will be a key enabler of the research 

identified in the PROs. Another important enabler mentioned by several participants is increased 

collaboration among quantum computing experts and theoretical chemists and materials scientists. At the 

moment, these groups tend to attend separate scientific meetings and to supervise distinct groups of junior 
30

researchers. Accomplishing grand-challenge science with quantum computers will require a concerted 

effort from both computing and applications communities.
31

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37

Appendix A: Workshop Participants

BES Roundtable on Opportunities for Basic Research for 

Quantum Computing in Chemical and Materials Sciences

Chair: Joel Moore, Lawrence Berkeley National Lab/University of California–Berkeley, 

and Alán Aspuru-Guzik, Harvard University

PRO 1: Enabling and Solving Non-Equilibrium Quantum Dynamics

Giulia Galli, University of Chicago, Argonne National Laboratory: Lead

Norman Yao, Lawrence Berkeley National Laboratory/University of California–Berkeley

Wibe (Bert) de Jong, Lawrence Berkeley National Laboratory

Thomas Devereaux, SLAC National Accelerator Laboratory

Jarrod McClean, Google

PRO 2: Unraveling the Physics and Chemistry of Strongly Correlated Electron Systems

Thomas Maier, Oak Ridge National Laboratory: Lead

Bela Bauer, Microsoft

Gus Scuseria, Rice University

Joel Moore, Lawrence Berkeley National Laboratory/University of California–Berkeley

Thomas Devereaux, SLAC National Accelerator Laboratory

PRO 3: Inercalated Quantum and Classical Approaches

Peter Love, Tufts University: Lead

Sue Coppersmith, University of Wisconsin

John Preskill, CalTech

Birgitta Whaley, University of California–Berkeley 

Marivi Fernandez-Serra, Stony Brook University

PRO 4: Hybrid Quantum–Classical Methods

Dominika Zgid, University Michigan: Lead

Robert Harrison, Brookhaven National Laboratory/Stony Brook University

Antonio Mezzacapo, IBM

James Whitfield, Dartmouth University

Crosscutting Themes

James Whitfield: Lead

Invited Participants

Alán Aspuru-Guzik, Harvard University

Bela Bauer, Microsoft

Sue Coppersmith, University of Wisconsin

Wibe (Bert) de Jong, Lawrence Berkeley National Laboratory

Thomas Devereaux, SLAC National Accelerator Laboratory

Marivi Fernandez-Serra, Stony Brook University
38

Giulia Galli, University of Chicago, Argonne National Laboratory

Robert Harrison, Brookhaven National Laboratory/Stony Brook University

Peter Love, Tufts University

Thomas Maier, Oak Ridge National Laboratory

Antonio Mezzacapo, IBM

Jarrod McClean, Google

Chris Monroe, University of Maryland, IonQ

Joel Moore, Lawrence Berkeley National Laboratory/University of California–Berkeley

John Preskill, CalTech

Gus Scuseria, Rice University

Birgitta Whaley, University of California–Berkeley 

James Whitfield, Dartmouth University

Norman Yao, Lawrence Berkeley National Laboratory/University of California–Berkeley

Dominika Zgid, University of Michigan

Invited Observers

Kramer Akli, DOE Office of Science, Fusion Energy Sciences

Dimitri Argyriou, Ames Laboratory

David Awschalom, University of Chicago/Argonne National Laboratory

Paul Bayer, DOE Office of Science, Biological and Environmental Research

Jeff Blackburn, National Renewable Energy Laboratory

Steve Binkley, DOE Office of Science

Tof Carim, DOE Office of Science, High Energy Physics

Lali Chatterjee, DOE Office of Science, High Energy Physics

Hans Christen, Oak Ridge National Laboratory

Claire Cramer, DOE Office of Science, Advanced Scientific Computing Research

David Dean, Oak Ridge National Laboratory 

Jim Davenport, DOE Office of Science, Basic Energy Sciences 

Jonathan DuBois, Lawrence Livermore National Laboratory

Chris Fecko, DOE Office of Science, Basic Energy Sciences

Bruce Garrett, DOE Office of Science, Basic Energy Sciences

Matthias Graf, DOE Office of Science, Basic Energy Sciences

Barbara Helland, DOE Office of Science, Advanced Scientific Computing Research

Craig Henderson, DOE Office of Science, Basic Energy Sciences

Linda Horton, DOE Office of Science, Basic Energy Sciences

Jim Horwitz, DOE Office of Science, Basic Energy Sciences

Helen Kerch, DOE Office of Science, Basic Energy Sciences

Robert Konik, Brookhaven National Laboratory

Refik Kortan, DOE Office of Science, Basic Energy Sciences

Jeff Krause, DOE Office of Science, Basic Energy Sciences

Harriet Kung, DOE Office of Science, Basic Energy Sciences

Peter Lee, DOE Office of Science, Basic Energy Sciences

Eliane Lessner, DOE Office of Science, Basic Energy Sciences

Ramana Madupu, DOE Office of Science, Biological and Environmental Research

John Mandrekas, DOE Office of Science, Fusion Energy Sciences

George Maracas, DOE Office of Science, Basic Energy Sciences

Gail McLean, DOE Office of Science, Basic Energy Sciences

Raul Miranda, DOE Office of Science, Basic Energy Sciences 

Rick Muller, Sandia National Laboratories

Jim Murphy, DOE Office of Science, Basic Energy Sciences

Mick Pechan, DOE Office of Science, Basic Energy Sciences

Mark Pederson, DOE Office of Science, Basic Energy Sciences
39

Jim Rhyne, DOE Office of Science, Basic Energy Sciences

Tom Russell, DOE Office of Science, Basic Energy Sciences

Jim Rustad, DOE Office of Science, Basic Energy Sciences

John Sarrao, Los Alamos National Laboratory

Andy Schwartz, DOE Office of Science, Basic Energy Sciences

Tom Settersten, DOE Office of Science, Basic Energy Sciences

Jim Siegrist, DOE Office of Science, High Energy Physics

Panagiotis Spentzouris, Fermi National Accelerator Laboratory

Peter Sushko, Pacific Northwest National Laboratory

Ceren Susut-Bennett, DOE Office of Science, Advanced Scientific Computing Research

41

Appendix B: Workshop Agenda

BES Roundtable on Opportunities for Basic Research for 

Quantum Computing in Chemical and Materials Sciences

Gaithersburg Marriott Washingtonian Center ● October 31 – November 1, 2017

Tuesday, October 31, 2017

1:00 – 1:30 p.m. Registration — Lakeside 1 and 2 

1:30 – 2:00 p.m. Welcome from DOE and Roundtable Charge

Harriet Kung, Associate Director of Science for Basic Energy Sciences

2:00 – 2:15 p.m. Introduction of Participants

2:15 – 2:30 p.m. Opening Remarks

Joel Moore, Lawrence Berkeley National Lab/University of California–Berkeley

2:30 – 3:10 p.m. Keynote Presentation

John Preskill, California Institute of Technology

(Introduction by Alán Aspuru-Guzik)

3:10 – 3:20 p.m. Break — Refreshments Available

3:20 – 4:00 p.m. Opportunities for Quantum Computation in Chemistry: Perspective by Alán AspuruGuzik (includes overview of homework and time for discussion)

4:00 – 4:40 p.m. Opportunities for Quantum Computation for Materials: Perspective by Joel Moore 

(includes overview of homework and time for discussion)

4:40 – 4:50 p.m. Break

4:50 – 6:00 p.m. Discussion — Scientific Challenges

1. What are the long-term chemical and materials sciences challenges that 

would be opportunities for quantum computing? 

2. What are the short-term opportunities to begin demonstrating the value of 

quantum computing for BES chemical and materials sciences?

3. Why can’t these challenges be addressed with today’s petascale or 

tomorrow’s exascale classical computers?

4. What advantages will quantum computers provide for these problems?

5. What computational approaches and algorithms are needed to solve 

these problems on a quantum computer?

6:00 – 7:30 p.m. Break for dinner (on your own)

7:30 – 8:00 p.m. Synthesize High-priority Scientific Challenges

8:00 – 9:00 p.m. Discussion: Computational/Algorithmic Challenges

Focus discussion on the priority scientific challenges

1. What computational approaches and algorithms are needed to solve 

these problems on a quantum computer?

2. How do we evaluate these algorithms for correctness and scaling?

3. How do we test algorithms on testbeds or in classical simulations to know 

how they are working?
42

Wednesday, November 1, 2017

7:00 – 8:00 a.m. Continental Breakfast — Lakeside 1 and 2

8:00 – 9:00 a.m. Quantum Computing Industry Panel Discussion

Alán Aspuru-Guzik, moderator

10 minutes per industry participant + discussion

1. What applications are you targeting in the near term, particularly in 

chemistry and materials? 

2. What are the current challenges in developing quantum computers? How 

will those challenges impact addressing BES science challenges?

3. What types of architectures (e.g., type and number of qubits) are being 

pursued by your company?

4. What progress has been made on the software stack?

9:00 – 10:00 a.m. Discussion: QC Hardware and Software Tools

Focus discussion on priority scientific directions

1. What software tools, compilers, etc. would accelerate progress in solving 

chemical/material problems?

2. What are the characteristics of QC hardware (e.g., coherence times, 

fidelity) needed for these algorithms? 

10:00 – 10:15 a.m. Break

10:15 – 11:30 a.m. Discussions of Priority Science Challenges

Develop outline from science challenge to algorithms to software and hardware 

characteristics (can break into groups). Begin with presentation of the report 

template

11:30 – 1:00 p.m. Working lunch (brief presentations from groups)

1:00 – 4:00 p.m. Breakouts for writing assignments

4:00 – 5:00 p.m. Presentation of Major Conclusions and Closing Remarks
Disclaimer

This report was prepared as an account of work sponsored by an agency of the 

United States Government. Neither the United States Government nor any 

agency thereof, nor any of their employees, makes any warranty, express or 

implied, or assumes any legal liability or responsibility for the accuracy, 

completeness, or usefulness of any information, apparatus, product, or process 

disclosed, or represents that its use would not infringe privately owned rights. 

Reference herein to any specific commercial product, process, or service by 

trade name, trademark, manufacturer, or otherwise, does not necessarily 

constitute or imply its endorsement, recommendation, or favoring by the United 

States Government or any agency thereof. The views and opinions of authors 

expressed herein do not necessarily state or reflect those of the United States 

Government or any agency thereof.