Quantum Computation for Periodic Solids in Second Quantization Aleksei V. Ivanov1Christoph S underhauf1Nicole Holzmann1 Tom Ellaby2Rachel N. Kerber2Glenn Jones2and Joan Camps1

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Quantum Computation for Periodic Solids in Second Quantization
Aleksei V. Ivanov,1, ∗Christoph S¨underhauf,1Nicole Holzmann,1
Tom Ellaby,2Rachel N. Kerber,2Glenn Jones,2and Joan Camps1
1Riverlane, St Andrews House, 59 St Andrews Street, Cambridge, CB2 3BZ, United Kingdom
2Johnson Matthey Technology Centre, Blounts Court, Sonning Common, RG4 9NH, United Kingdom
In this work, we present a quantum algorithm for ground-state energy calculations of periodic
solids on error-corrected quantum computers. The algorithm is based on the sparse qubitization
approach in second quantization and developed for Bloch and Wannier basis sets. We show that
Wannier functions require less computational resources with respect to Bloch functions because:
(i) the L1norm of the Hamiltonian is considerably lower and (ii) the translational symmetry of
Wannier functions can be exploited in order to reduce the amount of classical data that must
be loaded into the quantum computer. The resource requirements of the quantum algorithm are
estimated for periodic solids such as NiO and PdO. These transition metal oxides are industrially
relevant for their catalytic properties. We find that ground-state energy estimation of Hamiltonians
approximated using 200–900 spin orbitals requires ca. 1010–1012 T gates and up to 3 ·108physical
qubits for a physical error rate of 0.1%.
I. INTRODUCTION
Quantum mechanical simulation of molecules and materials is a promising application area of quantum comput-
ers [1–3] that will enable the calculation of key properties of chemical systems with controllable errors using physically
accurate models. Following Feynman’s original idea of modelling quantum systems on quantum computers [4] and the
first formalized procedures for carrying out such simulations [5–7], a plethora of quantum algorithms for calculating
energies of molecular systems have been developed in recent years [8–34]. Similarly, but to a lesser degree, quantum
algorithms taking into account the specifics of condensed matter applications have also been conceived. These include
the development of different flavors of variational quantum eigensolvers (VQE) [35–38], the quantum imaginary time
evolution algorithm [39], and fault-tolerant algorithms [24,34,40–43] for simulation of model Hamiltonians, such as
the Hubbard model, as well as first-principles Hamiltonians.
Quantum computers can provide a computational advantage over classical computers only for hard classical prob-
lems. These include the simulation of so-called strongly correlated systems and, more practically, problems that are
not solved with sufficient accuracy using classical methods with low computation cost — such as Kohn-Sham density
functional theory (KS-DFT)) [44–46] or coupled-cluster theory [47,48]. Notwithstanding varying definitions and
interpretations of “strong correlation”, and an ongoing debate regarding the extent to which KS-DFT can describe
such systems [49,50], the general consensus is that molecular and solid state systems with a large number of localised
dor felectrons present a significant challenge for classical simulations. Examples of such systems include transition
metal oxides such as NiO and PdO used in heterogeneous catalysis applications. The number of localised sites in such
systems is formally infinite as the solids should be simulated at the thermodynamic limit. In practice, one restricts
calculations to a periodic finite-sized cell (also referred to as supercell) with ca. 30–100 unique transition metal atoms;
all other atoms in the solid are replicas of those in this computational cell.
The ability to accurately model the electronic structure of materials such as NiO and PdO would no doubt prove
extremely useful in the study of heterogeneous catalysis, a field with no shortage of materials that are poorly described
by DFT. It is often the case that the interpretation of calculated results (e.g. regarding trends in activity) must be
presented with significant caveats regarding the underlying nature of the models used.
In this work, we focus on the calculation of the ground state energy of electrons in materials within the Born-
Oppenheimer approximation [51]. This corresponds to finding the lowest eigenvalue of the electronic Hamiltonian
for a fixed position of the nuclei. Two main families of quantum algorithms can perform such calculation: VQE [13]
and quantum phase estimation (QPE) [52–54]. While VQE might have its merits in certain use cases, it appears the
emerging consensus is that QPE has a superior scaling with the system size [1,55]. In order to estimate the eigenvalues
of the Hamiltonian with QPE, one has to implement a unitary operator encoding the spectrum of the Hamiltonian.
QPE requires deep quantum circuits, and as such it will need to run on error-corrected quantum computers. In such
error-corrected implementations one must strive to minimize the number of T gates needed to encode the Hamiltonian,
∗Corresponding author: aleksei.ivanov@riverlane.com
arXiv:2210.02403v2 [quant-ph] 15 May 2023
2
as these gates are the costliest to implement (see e.g. [56]). To this date, the most cost-efficient approaches for such
encodings are based on the so-called qubitization technique [25–28,31,32,57]. Previous work on second quantized
Hamiltonians for realistic solid state systems has mainly focused on the Trotterization approach [24,40]. In this work,
we adapt the sparse qubitization approach to the simulation of crystalline solids with QPE and estimate the resources
required for calculating the ground state energy of crystals in error-corrected quantum computers.
The quantum resources required to simulate a Hamiltonian strongly depend on the single-electron basis sets used
to represent electron interactions. For crystalline solids, plane waves (PW) currently appear to be one of the most
efficient basis sets both in first and second quantization [24,34]. An advantage of using PW basis sets is the sparse
representation of the electronic Hamiltonian. This advantage is always exploited in classical computations such as
KS-DFT [58,59]. The number of two-body terms in PW representation scales cubically with the size of the basis set.
The main disadvantage of such basis sets, however, is that they require a large number of basis functions, especially
in all-electron calculations. For crystalline solids one can exploit Bloch functions instead, which are plane waves times
a periodic function with the periodicity of the unit cell. In the Bloch representation, the number of terms also scales
cubically with the system size, and at the same time such a representation allows using localised atomic orbitals
as the periodic constituent of the orbitals. The other commonly used representation in computational condensed
matter physics is the Wannier representation, in which orbitals are localized in space [60–62]. Wannier orbitals can be
related to Bloch functions through Fourier transformation, and can be localized using unitary optimization in order
to produce maximally localized Wannier functions [60]. When the periodic function in the Bloch representation is a
constant, the Wannier representation coincides with the PW dual representation introduced in the context of quantum
computing in Ref. [24]. At the same time, Wannier orbitals can be spanned in the localised atomic orbital basis which
in turn can significantly reduce the size of the basis set for an accurate description of finite band-gap solids. In this
work, we investigate Bloch and Wannier representations in the context of qubitized QPE. We note that such basis
sets have recently been investigated in the context of the VQE algorithm [36].
Quantum computation with qubitization-based QPE requires a large number of gates in a circuit. In order to
perform large quantum computations, one has to encode a logical qubit using several physical qubits with a tech-
nique known as quantum error correction [63,64]. In order to estimate the total number of physical and logical
qubits required for the implementation of quantum algorithms as well as their runtime, we have followed Litinski’s
approach [65]. This scheme operates the surface code [66] with lattice surgery [56,67], and compiles logical quantum
circuits down to just multi-qubit T gates and multi-qubit measurements—all Clifford gates are commuted past the
end of the circuit. In this way, runtime is directly related to T-gate count.
The article is organized as follows. In Sec. II, we first describe the relevance of modelling bulk materials such as
NiO and PdO for applications to heterogeneous catalysis—an area where quantum computation can provide high
accuracy results when error-corrected quantum computers become available. In Sec. III, the Hamiltonian, basis sets,
and quantum algorithms for modelling of crystalline solids are introduced. In Sec. IV, we discuss the performance
of quantum algorithms and provide quantum resource estimations for several solid state systems. Finally, discussion
and conclusions are presented in Sec. V. Detailed logical qubit and Toffoli gate counts of the sparse qubitization are
provided in Appendix A.
II. MATERIALS AND HETEROGENEOUS CATALYSTS
Catalysts are used in practically every industrial chemical process, with applications in agriculture, transportation
and energy production, among many others. The function of a catalyst to ultimately reduce the energy requirements
of a process to make it viable or more efficient means that catalytic processes are a key component for ensuring a
sustainable future and reducing human impact on the environment. Transition metal oxide catalysts are essential
components for many important industrial processes (such as refining and petrochemistry, fuel cells, hydrogen pro-
duction, biomass conversion, photocatalysis) where they are used both directly, as the active material (providing
the active site), and indirectly, as a support material (commonly as a reducible oxide taking a secondary role in the
catalysis). The overall performance of the solid catalyst depends on many factors, including the particle size, particle
shape, crystallinity, chemical composition, and all preparation and activation procedures. High catalytic efficiencies
are achieved as the number of active surface sites grows, while the structural flexibility of supported metal catalysts
(dynamic structural changes) is key for the catalytic reactivity when we consider that the surface sites repeatedly
participate in adsorption/desorption cycles.
The systems considered in this work, nickel oxide (NiO) and palladium oxide (PdO), both form the basis of
industrially relevant catalyst materials. In the field of energy and environment, natural gas reforming is the most
common process used in industry to produce H2from fossil fuels, known as methane steam reforming (MSR). Here
NiO is reduced to Ni which functions as a high-temperature catalyst. Despite its age and ubiquity, the MSR process
still has many technical challenges, for instance around deactivation from carbon whisker formation and stability at
3
high-temperature. Under certain operating conditions, a local oxidizing environment can form within the reactor,
leading to the deactivation of Ni due to NiO being present. Thermodynamics can predict the conditions at which this
can occur [68]. However, in general, it is still a challenge to obtain reliable or accurate thermodynamic parameters
for strongly correlated oxide materials, especially when they deviate form the bulk limit such as in nanoparticles.
Methane has an estimated greenhouse warming potential (GWP 100) of 27.9 [69], meaning its emissions contribute
significantly to global warming and climate change; it is therefore necessary to reduce them wherever possible. Among
the many different technologies for methane abatement, methane combustion catalysts based on palladium can be
found. Such technologies include after-treatment for combustion of natural gas engines (CNG) and diesel oxidation
catalysts (DOC) as well as in mine ventilation systems.
In the above applications, methane is efficiently combusted over palladium (or alloyed) oxide catalyst to produce
H2O and CO2, with activity in this process influenced by a number of factors. A technical target in practical catalysis
is to reduce the temperature at which this occurs, allowing for a lower operating temperature and more efficient
handling of emissions. Partial oxidation can sometimes occur, and may indeed be desirable in the development of
processes to produce precursors for more complex chemicals. The ability to simulate accurately not only the activity
but also the selectivity, which is a measure of a catalyst’s ability to promote the formation of the desired product(s)
over other possibilities, is crucial to the prediction of new catalysts.
Figure 1(a) shows a schematic of a typical catalyzed reaction. The presence of a catalyst provides additional
reaction coordinates, or reaction intermediates, with their own activation energies (ET S1,ET S2). For an effective
catalyst, these energies are necessarily lower than the uncatalyzed activation energy Ea. In the case of heterogeneous
catalysts, reaction intermediates are typically adsorption steps, where one or more of the reactants binds to a surface
site of the catalyst. Depending on the complexity of the reaction mechanisms, there may be a large number of these
intermediates as well as branches and side reactions that must be considered when studying a reaction in order to
determine the key step(s). It is often necessary to find these steps, which govern the activity and/or selectivity of
a catalyst, as in doing so, the problem is reduced to fewer dimensions and descriptors that facilitate a more rapid
study. For example, in a kinetic analysis the largest activation energy is usually of most interest, as this will be the
rate determining step. Whilst this knowledge may be well established in well known reactions, it can be necessary
to perform many calculations in more novel applications. Furthermore, whilst the accuracy of current computational
approaches may be good enough to predict trends in similar systems, obtaining chemical accuracy and absolute values
for detailed kinetic studies remains a challenge. Figure 1(b) shows a typical set of model systems that would be used
to estimate the energetics of a heterogeneous catalytic process.
When running simulations of a catalyst, consideration needs to be made of the question at hand and the level of
accuracy that is needed. Broadly speaking, we are interested in activity, selectivity and stability. When simulating
activity, we often need a kinetic model which can provide rates or turn-over frequencies. If we are interested in
screening for materials, it is often sufficient to correlate these rates with descriptors [71].
For example, following the Sabatier principle [72], which is employed primarily for materials screening, calculating
the (heterogeneous) catalytic activity of a material is performed by determining the binding strengths of the reactants,
products and any important intermediates of a given reaction with the surface of that material. These binding
strengths can be determined from energy calculations using a wide variety of models, each with their own trade-offs
between accuracy, transferability and computational cost.
However, if we are interested in predicting reactor performance or process conditions then we need significantly
greater precision in the simulated parameters. Likewise, simulating the often subtle differences in competing reactions
(which result in different products) typically requires greater accuracy in calculations to predict selectivity.
Whilst the questions of activity and selectivity are crucial for a material’s function as a catalyst, when looking
for a technical solution, the question of stability becomes critical. Catalysts often need to operate over many years
under harsh conditions (high temperature, pressure, contaminated conditions and, in the case of electrocatalysis,
high potentials and corrosive environments). The simulation of stability introduces a whole range of other problems;
for instance, predicting morphological changes and thermal degradation of a catalyst requires a large number of
calculations, often of large model systems, to allow sintering of nanoparticles or ceramic supports to be conducted.
Material complexity (e.g. simulation of realistic metal/ceramic interfaces), bridging time and length scales where
accurate atomic-scale materials properties can be fed into multi-scale models, are all open challenges in this area.
DFT is one of the most successful and widely used models for calculating the energies of molecular and solid state
systems relevant to industrial processes. It is an ab initio method that uses functionals of the electronic density to
calculate energy rather than attempt to deal directly with the many-body wavefunction. In KS-DFT, the electronic
density is constructed using a fictitious set of non-interacting single electron wavefunctions and approximating an
unknown correction term. This term, known as the exchange-correlation (XC) functional, includes exchange and
correlation effects as well as discrepancy between the real and non-interacting kinetic energy. There are many choices,
though all of them approximated, for its form.
Ultimately, it is the use of single-particle wavefunctions in DFT that leads to some of its most prominent short-
4
Reaction coordinate
Energy
Ea
reactants
products
activated
complex
intermediate
ETS1 ETS2
ΔE
C
H
H
H
H
C
H
H
H
H
Bulk Slab Adsorption structure Reactants and products
(a)
(b)
FIG. 1. (a) Schematic of a reaction pathway with and without a catalyst, and (b) diagram of the necessary calculation steps
required to determine these properties. Relative energies between the reactants, products and intermediates (if they are known)
can be calculated via density functional theory or other computational approaches. Reaction barriers for a catalyzed reaction
can be approximated via adsorption energy calculations in many cases [70], or by transition state searches, which require the
calculation of forces as well as the energy.
comings. In the case of NiO, and indeed most transition metal oxides, the strong electron-electron interactions of
the d-electrons in these materials is poorly described by approximate KS-DFT, leading to over-delocalisation of these
bands (and to the prediction of more metallic electronic structures than the reality). A Hubbard U [73] correction can
be used alongside local density approximation (LDA) and generalised gradient approximation (GGA) XC functionals
to mitigate this issue in some cases, although it is overly empirical in nature. While the use of hybrid XC functionals
such as PBE0 [74] can sometimes perform better [75], due to the inclusion of Hartree-Fock exact exchange, the fraction
of exact exchange to use can be varied (depending on the XC functional used), which again leads to empirical fitting.
Hybrid functionals are also incomplete (and incorrect) in their description of the electronic structure, and are by no
means a guaranteed improvement over GGA functionals in their prediction of transition metal oxide properties [76].
To model the bulk properties of materials effectively, the use of periodic boundary conditions (PBCs) is required,
allowing for a simulation box to include only the primitive unit cell in highly ordered systems. Even in disordered
systems, periodicity is still imposed (on a larger unit cell), as the approximation still provides more representative
models than any non-periodic alternative, without extending the system far beyond practical limits.
The study of heterogeneous catalysis primarily concerns the properties of surfaces, so slab models are often used.
These are also periodic, albeit in 2 dimensions rather than 3. Bulk calculations are also required in order to determine
the surface energies of the facets of a material, which, for example, allow for the prediction of the expected shape of
nanoparticles, as well as which facets are most predominant and relevant for catalysis. The stability of a material is
another important aspect that can be predicted by energy calculations on bulk systems.
III. METHODOLOGY
A. Hamiltonian for Periodic Systems
The Hamiltonian of interacting electrons in the Born-Oppenheimer approximation can be written as follows:
ˆ
H=ˆ
H(0) +ˆ
H(1) +ˆ
H(2),(1)
5
where ˆ
H(0) is a constant term describing nuclear repulsion, ˆ
H(1) and ˆ
H(2) are one-body and two-body terms, respec-
tively [77, p. 32]:
ˆ
H(1) =ZZ dxdx0ˆ
ψ†(x)h(x,x0)ˆ
ψ(x0) (2)
ˆ
H(2) =1
2ZZ dxdx0ˆ
ψ†(x)ˆ
ψ†(x0)g(x,x0)ˆ
ψ(x0)ˆ
ψ(x),(3)
xdenotes position and spin, (r, σ), of an electron and the integration domain is over the volume of the macroscopic
crystal, V. In this work, we do not consider the external magnetic field or spin-orbit coupling and therefore, the one-
and two-body kernels are diagonal w.r.t. spin degrees of freedom. The spatial part of one-body kernel is:
h(r) = −1
2∇2+U(r) (4)
where U(r) is the nuclei potential
U(r) = X
a∈V
Za
|r−Pa|,(5)
Zaand Paare the nuclear charge and position of nucleus a. The spatial part of two-body kernel is
g(r,r0) = g(|r−r0|) = 1
|r−r0|(6)
We assume Born-von-K´arm´an periodic boundary conditions at the boundaries of the macroscopic crystal which is
defined by the vectors L1,L2,L3:
A(r+Lα) = A(r), A =ˆ
ψ, h, U, g α = 1,2,3 (7)
In this case, the external potential and two-body kernel are defined in terms of their Fourier series:
U(r) = X
a∈VX
K
4πZae−iKPa
VK2eiKr (8)
g(r) = X
K
4π
VK2eiKr (9)
where Ksatisfies:
KLα= 2πmα, mα∈Z, α = 1,2,3 (10)
Crystalline solids consist of unit cells and each unit cell Vuc is defined by translation lattice vectors, a1,a2,a3. Each
unit cell can be labeled with Rindicating a node of the Bravais lattice:
R=ATn;n= (n1, n2, n3)T, nα∈Z;A= [a1,a2,a3]T(11)
Let Nα≥1 be the number of unit cells along aα,nα= 0,1, .., Nα−1 and thus, the total number of unit cells which
spans the whole finite macroscopic crystal is N=N1N2N3. We also introduce the reciprocal lattice which is defined
as:
G=Bn;n= (n1, n2, n3)T, nα∈Z;B= [b1,b2,b3]=2πA−1(12)
The vectors b1,b2,b3and a1,a2,a3satisfy the following relations:
aαbβ= 2πδαβ (13)
In the case of crystalline solids, the external potential can also be rewritten in terms of reciprocal lattice vectors,
because it is has periodicity of the lattice:
U(r) = X
a∈Vuc X
G
4πZae−iGPa
VucG2eiGr (14)
摘要：

QuantumComputationforPeriodicSolidsinSecondQuantizationAlekseiV.Ivanov,1,ChristophSunderhauf,1NicoleHolzmann,1TomEllaby,2RachelN.Kerber,2GlennJones,2andJoanCamps11Riverlane,StAndrewsHouse,59StAndrewsStreet,Cambridge,CB23BZ,UnitedKingdom2JohnsonMattheyTechnologyCentre,BlountsCourt,SonningCommon,RG4...

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