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PAPER
Discontinuous Galerkin discretization for quantum simulation
of chemistry
Jarrod R McClean1,8, Fabian M Faulstich2,QinyiZhu
3, Bryan O'Gorman4,5, Yiheng
Qiu6, Steven R White6, Ryan Babbush1and Lin Lin3,7,8
1Google Research, 340 Main Street, Venice, CA 90291, United States of America
2Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, Oslo, Norway
3Department of Mathematics, University of California, Berkeley, CA 94720, United States of America
4Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720, United States of America
5Quantum Artificial Intelligence Laboratory, NASA Ames Research Center, Moffett Field, CA 94035, United States of America
6Department of Physics and Astronomy, University of California,Irvine, CA 92697, United States of America
7Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, United States of America
8Authors to whom any correspondence should be addressed.
E-mail: linlin@math.ber keley.edu and jmcclean@google.com
Keywords: quantum computing, quantum chemistry, density matrix renormalization group, discontinuous Galerkin methods,
electronic structure
Abstract
All-electron electronic structure methods based on the linear combination of atomic orbitals
method with Gaussian basis set discretization offer a well established, compact representation that
forms much of the foundation of modern correlated quantum chemistry calculations—on both
classical and quantum computers. Despite their ability to describe essential physics with relatively
few basis functions, these representations can suffer from a quartic growth of the number of
integrals. Recent results have shown that, for some quantum and classical algorithms, moving to
representations with diagonal two-body operators can result in dramatically lower asymptotic
costs, even if the number of functions required increases significantly. We introduce a way to
interpolate between the two regimes in a systematic and controllable manner, such that the
number of functions is minimized while maintaining a block-diagonal structure of the two-body
operator and desirable properties of an original, primitive basis. Techniques are analyzed for
leveraging the structure of this new representation on quantum computers. Empirical results for
hydrogen chains suggest a scaling improvement from O(N4.5) in molecular orbital representations
to O(N2.6) in our representation for quantum evolution in a fault-tolerant setting, and exhibit a
constant factor crossover at 15 to 20 atoms. Moreover, we test these methods using modern density
matrix renormalization group methods classically, and achieve excellent accuracy with respect to
the complete basis set limit with a speedup of 1–2 orders of magnitude with respect to using the
primitive or Gaussian basis sets alone. These results suggest our representation provides significant
cost reductions while maintaining accuracy relative to molecular orbital or strictly diagonal
approaches for modest-sized systems in both classical and quantum computation for correlated
systems.
1. Introduction
Predicting properties of both molecular and extended systems from first principles has long been the goal of
electronic structure in both correlated classical methods [1], including new approaches based on tensor
networks [2–5], and now many approaches based on quantum computing [6–15], some of which have even
been implemented on experimental devices [16–25]. A crucial aspect of such simulations is the
representation of the problem in a tractable discretization scheme such as a finite-difference method, or
(more commonly) a basis set, also known as a Galerkin discretization of the problem. The selection of the
basis influences not only the accuracy of the calculation but also the fundamental scaling of the cost of the
simulation as a function of the system size.
© 2020 The Author(s).Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche PhysikalischeGesellschaft
New J. Phy s. 22 (2020) 093015 JRMcCleanet al
Figure 1. A cartoon schematic of the general objective of this work depicted interms of the sparsity pattern of the two-electron
integrals. At the top, we depict a dense two-electron integral tensor with O(N4
a) nonzero two-electron integrals, but relatively few
basis functions. This is often the case with molecular orbital or diffuse Gaussianbasis sets. In the center we depict a diagonal
primitive basis set, such as Gausslets, with a relatively high number of basis functions but an overall scaling in O(N2
p). Finally, we
depict this for a discontinuous Galerkin (DG) basis set built from a diagonal primitive basis set that interpolates between the two
regimes by using fewer basis functions than strictly diagonal sets, while retaining an overall O(N2
d) scaling through its
block-diagonal structure. We note that the form of block-diagonal structure is a particular sparsity pattern which depends on the
arrangement of indices when consideringa matrix form; the text defines this sparsity pattern precisely.
The design of basis sets for correlated electronic structure has a long and rich history, packing much of
the essential physics and chemistry into very compact representations to exploit the power of existing
methods. While these basis sets have ranged from general purpose to those optimized for individual
computations, a common mainstay has been the use of Gaussian-based molecular orbitals [1,26–28].
Molecular orbitals tend to offer compact representations of correlated problems and also present an energy
ordering of orbitals that facilitates further reduction of the space through active-space methods. However, a
side effect of this reduction in the number of orbitals is often a Hamiltonian with a quartic number of terms
and orbitals that are delocalized in space (figure 1).
The utility of localizing orbitals in both the occupied and virtual space has been recognized in the
development of many linear-scaling methods [29], such as those based on pair natural orbitals [30–32]. In
these cases the localization helps not only in allowing the screening of some terms in the Hamiltonian, but
also reducing the correlations that need to be treated to reach a certain level of accuracy. However, even the
most localized orbitals that can be produced from transformations of a standard basis are often not strictly
local in the sense that they have heavy tails that can extend throughout the system to enforce orthogonality
between orbitals. While reasonably compatible with some methods, these tails negatively impact
tensor-network methods such as the density matrix renormalization group (DMRG), where an area-law
entanglement system that is solvable in modest polynomial time becomes a volume law system with
exploding bond dimension in a completely delocalized basis [33]. This effect, in combination with
consideration of the number of Hamiltonian terms, led to the recent development of Gausslet basis sets [34,
35], which use properties of wavelet transforms to maintain strict localization, orthogonality, and a
diagonal Coulomb operator.
In the case of quantum computing-based approaches, while basis localization may impact the
representational power of an ansatz-based variational approach, the structure and number of terms in the
Hamiltonian represent the dominant cost factor [36–39]. Moreover, some traditional density-based
truncations and localizations can be difficult to utilize in quantum computers, due to the need to measure
the density and rotate the basis to maximize the benefits. Recent advances in quantum-computing
approaches have shown that Hamiltonians with a diagonal structure with a quadratic number of terms
allow algorithms to execute numerically exact real-time dynamics and perform full configuration
interaction (assuming a reference state with nonvanishing overlap on the ground state can be prepared)
with a cost in terms of basis size that scales as roughly O(N1/3
p) in first quantization [40]orO(N2
p)insecond
quantization [41]. While the first work was done for periodic systems based on basis sets related to discrete
variable representations (DVR) [36], these results apply to all basis sets with these properties. This is in
contrast to the most advanced methods based on Gaussian molecular orbitals on quantum computers,
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New J. Phy s. 22 (2020) 093015 JRMcCleanet al
Figure 2. Compact description of the notation used throughout the paper in counting basis functions in different
representations for the electronic structure problem. Here discontinuous Galerkin (DG) is the block basis we construct from
primitive functions to represent the active-space orbitals with a block-diagonal Hamiltonian representation.
which have costs more like O(N3
a) in first quantization [42]orO(N4
a) in second quantization [38](see
figure 2for an overview of set size abbreviations used throughout the article).
While the scaling advantages of basis sets with diagonal representations appear at reasonable system
sizes, reduced representational flexibility can mean the cost of switching representations for equivalent
accuracy is still higher than current quantum computers with limited resources can afford. Thus, it is
desirabletobeabletosplitthedifferencebetweenstrictly diagonal basis sets and molecular orbitals. The
adaptive local basis set [43–47] was recently introduced to achieve this tradeoff in the setting of density
functional theory (DFT). The adaptive local basis is constructed on-the-fly to capture atomic and
environmental effects, by solving eigenvalue problems restricted to local domains called elements. Each
basis function is only supported on one element and is discontinuous on the level of the global domain.
These basis functions are ‘glued’ together to approximate the continuous electron density using the interior
penalty discontinuous Galerkin (DG) formalism [48,49]. Compared to typical numerical methods for
solving DFT, the DG formulation includes extra correction terms to handle the discontinuities of the basis
functions at the boundary of the elements. Numerics indicate that roughly 10 to 40 basis functions per
atom are often sufficient to achieve chemical accuracy for DFT calculations using pseudopotentials [50,51].
In this work, we introduce a variation of the DG approach for simulation on quantum computers,
which has two main differences compared to previous works [43,44]. First, our target is to model electronic
structure at the correlated level instead of the DFT level, and hence we can afford to obtain the best local
basis functions, e.g., by starting from molecular orbitals or approximate natural orbitals. This removes a
significant step of approximation in the original DG approach due to the solution of local eigenvalue
problems with certain artificial boundary conditions (such as the Dirichlet or periodic boundary
conditions). Second, our basis functions are represented as linear combination of primitive basis functions,
which means that the basis functions are no longer strictly discontinuous in real space. We demonstrate that
the two-body operator can still maintain a block-diagonal structure for efficient quantum simulation, and
for simplicity we will still refer to the basis set as the DG basis set. The use of the primitive basis functions
removes the need for correction terms that account for the discontinuity, which is similar in spirit to the
discrete discontinuous basis projection method [52] for DFT calculations. The DG basis set enables one to
interpolate between cheap diagonal representations and compact nondiagonal basis sets with blocks of
specified size.
We begin by introducing the standard discretization of the electronic structure problem in a basis, and
define precisely what is meant by a diagonal basis representation and strictly localized functions. We then
briefly review basis sets that exhibit spatial locality and diagonal interactions that have already been used in
the context of DMRG and quantum computing. This allows us to highlight the cost advantages of these
representations within each approach. Then, the DG approach is introduced as a general framework for
maintaining these properties by using those basis sets as a primitive building block. The block diagonal
structure of the resulting Hamiltonians leads us to new cost models based on swap networks for quantum
algorithms. We show that the new approach both maintains accuracy in correlated calculations, and
demonstrates a crossover to dramatically lower costs at modest system sizes between 15 and 20 atoms. We
finish with an outlook on how this approach will influence quantum and classical approaches to correlated
electronic structure alike.
2. Discretizing the electronic structure problem
A crucial aspect of essentially all algorithms for the simulation of electronic systems is the discretization of
the system into some tractable representation. This step takes the electronic Hamiltonian that acts in some
continuous space, and maps it to a discrete space. The continuous electronic structure Hamiltonian is in
atomic units given by
ˆ
H=−
i
∇2
ri
2−
I,j
ZI
|RI−rj|+
i<j
1
|ri−rj|+EII,(1)
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New J. Phy s. 22 (2020) 093015 JRMcCleanet al
where ridenotes the position of the ith electron, and we have assumed the Born–Oppenheimer
approximation such that the positions of nuclei, denoted by RIare constants (giving rise to a constant
energy correction EII for the nuclear–nuclear interaction). In the case of Galerkin discretizations, where one
chooses a basis set given by some set of orthonormal (single-particle) functions {χi(r)}(for simplicity we
assume a spin-restricted formulation), and enforces the antisymmetry of electrons in the operators. Then,
one may express the Hamiltonian in the standard second quantized form. It is possible to select basis
functions from a number of complete sets that allow the electron–electron interaction to be represented in
a way that is entirely diagonal under a Jordan–Wigner representation in the computational basis and also
exhibits a diagonal property under matricization. In this representation, the second quantized Hamiltonian
is given by
ˆ
H(p)=
Np
μ,ν=1
h(p)
μν ˆ
b†
μˆ
bν+1
2
Np
μ,ν=1
v(p)
μν ˆ
nμˆ
nν,(2)
where ˆ
b†
μis a creation operator in the primitive basis, and ˆ
nμ=ˆ
b†
μˆ
bμis a number operator. We refer to
Hamiltonians written in this form as ‘diagonal Hamiltonians’, and use such basis sets here as our ‘primitive’
basis (associated with the superscript (p)inˆ
H(p)) to efficiently construct compact bases that partially retain
this property.
Generally, the coefficients in equation (2) are given by the following integrals,
h(p)
μν =drχμ(r)−∇2
r
2−
I
ZI
|RI−r|χν(r),
vμσγν =drdrχμ(r)χσ(r)χγ(r)χν(r)
|r−r|,
(3)
and the defining property of our primitive basis sets may be written as vμσγν →v(p)
μν δμ,γδσ,ν, i.e., vbecomes
a diagonal matrix when we view (μ,σ)astherowindexand(γ,ν) as the column index, respectively. Note
that the relation between vμσγν and v(p)
μν δμ,γδσ,νis not necessarily an equality: what one requires is that
solutions to the Schrödinger equation using the two different forms of the interaction can systematically
approach each other as one approaches the complete basis set limit. A grid, defined via finite differences,
has the diagonal property, although there is no underlying basis. Here we consider only basis sets, but the
basis functions with the diagonal property are naturally associated with a uniform or nonuniform grid, with
typical spacings between grid points much smaller than the interatomic distances.
For a basis set, a sufficient condition for equality of these expressions is that one has functions with
strictly disjoint support, such that (formally) χμ(r)χσ(r)=0forallμ=σ. However, such a requirement
would be unrealistic because upon careful inspection it would imply a simultaneously diagonal kinetic and
potential operator if evaluated in the Galerkin formulation in equation (3) (in contrast to a finite-difference
or overlapping finite-element approach). Classical finite-element methods can produce near-diagonality by
allowing overlapping elements between only spatially neighboring sites, which retains the favorable scaling,
but generically introduces nonorthogonality in the basis [53–56]. Accordingly, methods requiring a return
to orthogonality, such as quantum-computing methods, suffer a transformation to O(N4
p)termson
orthogonalization, the introduction of unnecessary orthogonality tails, or both. More recent developments
in classical discretization have extended this idea of disjoint cells to allow for truly disjoint basis sets, but at
the cost of the introduction of additional surface terms and discontinuity penalties to reintroduce
physicality into the problem. These methods are known as discontinuous Galerkin (DG) methods [48,49]
and were first introduced to electronic structure in the context of density functional theory [43–46]. They
represent a general and rigorous framework for constructing problem representations that have this
property of strict (block) locality, and we develop a variation of these methods in this work to achieve this
goal for correlated electronic structure methods without the need for the introduction of surface terms.
While the condition of spatial disjointness is a sufficient condition to obtain a Hamiltonian with O(N2
p)
terms, it is not necessary. Two basis sets that have the diagonal property without the equality of the
individual matrix elements, and thus without the strict spatial disjointness property, are the plane-wave
dual basis [36] and Gausslet basis [34]. The plane-wave dual basis is related to periodic sinc functions and
discrete variable representations (DVR) [36] and is constructed from an aliased Fourier transform of plane
waves for a given box to yield a diagonal Hamiltonian. Here the diagonality originates from the plane wave
dual functions (also called the periodic sinc functions) that are Lagrange interpolation functions on a
uniform grid. It has the advantage that the basis is naturally periodic, and thus is well suited for the
treatment of materials and other condensed phase systems. Moreover, a modification using a truncated
Coulomb interaction can enable the treatment of isolated systems. Two key downsides of this approach are
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New J. Phy s. 22 (2020) 093015 JRMcCleanet al
that it inherently reflects a uniform discretization in space of the problem, lacking the ability to adaptively
refine sharp features such as the electron–nuclear cusp, and that it has long tails responsible for maintaining
the orthogonality of the basis. This first property is reflected in a large overhead for representing atomic
systems to a level of accuracy similar to Gaussians, and the second makes the method difficult to use with
geometric entanglement based approaches such as tensor networks.
When integrated with smooth functions, Gausslets behave like δfunctions, which results in the diagonal
property. The Gausslets are obtained as linear combinations of arrays of Gaussians using wavelet
transforms. Specific moment properties of the wavelet transforms make the Gausslets integrate polynomials
up to a certain order like a δfunction. The δ-function property survives under smooth coordinate
transformations, so unlike plane-wave dual bases, Gausslets can have variable resolutions, with more
degrees of freedom near the nuclei to represent the electron–nuclear cusp. Also in contrast to the
plane-wave dual basis functions, Gausslets have strong localization characteristics, avoiding long tails. The
lack of tails means these basis sets are naturally suited for tensor-network methods and, relatedly, a
variational quantum ansatz may have more expressive power at shorter circuit depths. The ability to more
naturally represent inhomogeneous features means the representational overhead is expected to be modest
relative to plane-wave representations.
We note, however, that both methods can benefit from the introduction of pseudopotentials, and that
the representational power of all single-particle basis sets are expected to be limited by the same asymptotic
scaling in the limit of a very large number of basis functions. This means that for very large basis set sizes
approaching the complete basis set limit, the representational overheads are expected to be negligible, and
the determining factors are the other properties of the basis sets. Despite this, however, there is considerable
interest in treating systems before reaching this limit, where the representational overhead for basis sets can
differ considerably.
In order to meet the demands of compactness one could start from a more generic basis set and use the
expressions in equation (3) to determine the Hamiltonian. However, an approach that will prove fruitful
here is to use the fact that it is equivalent to start from a complete primitive basis set, such as those above,
and project into a compact ‘active-space’ Hamiltonian. Gaussian, molecular orbitals, or other active-space
constructions can been seen as a specific case of the active space we refer to here.
Since the number of primitive basis functions is typically large compared to the number of electrons,
quantum chemistry calculations, especially at the correlated level, commonly utilize more physically
motivated basis sets that result in a more compact description (e.g. correlation-consistent basis sets in the
LCAO approach). Let {ϕp(r)}Na
p=1be a set of orthonormal single-particle functions; we refer to them as
active-space orbitals (for instance, canonical Hartree–Fock orbitals, or natural orbitals), which can be
expanded using a primitive basis set as
ϕp(r)=
μ
χμ(r)Φμp.(4)
Here Φ∈CNp×Nais a matrix with orthogonal columns. While this projection step represents an
approximation depending on the nature of the primitive and active bases, it is one that can be systematically
controlled and understood by increasing the size of the underlying primitive basis without significantly
increasing the size of the resulting block basis that we will construct subsequently. For flexibility and
accuracy, we will allow quite general definitions of the active-space basis set in this work. It will pertain to
traditional Hartree– Fock canonical orbitals as well as Gaussian basis sets such as the dunning cc-pVDZ
basis set [26], which allows us to use Gaussians with strictly block diagonal properties by going through the
primitive basis set using point sampling through the approximate delta function properties of the primitive
basis set. It has been shown previously that for the Gausslet primitive basis set we use, point sampling is
extremely accurate due to the delta function property of the basis [34]. We will also make use of this
construction to build hybrid active spaces, where we use a weighted density matrix from multiple basis sets
to define Φthrough its most important natural orbitals. In particular, we will combine the expressive power
of a large primitive basis, such as Gausslets, to capture static correlations through an unrestricted
Hartree–Fock defined active space, while including a Gaussian basis empirically refined to express dynamic
electronic correlation effects, e.g. cc-pVDZ through weighting. In this work we make use of a joined set of
density matrices built from a unrestricted Hartree–Fock (UHF) solution DUHF and Gaussian orbitals
αDGaussian,toformD=DUHF +αDGaussian in the primitive basis, and use a natural-orbital truncation of D
to define Φ. Empirically we use a value of α≈0.01 later in this work when we combine these basis sets,
which appears to give an excellent improvement in accuracy. A more detailed description of this procedure
and refinement of the value αis left to a future work. We term this the hybrid active-space approach. As we
only use this hybrid approach in conjunction with the DG-blocking procedure, we offload concerns about
orthogonality in the projected basis to the singular value decomposition (SVD) used in the DG procedure.
5
New J. Phy s. 22 (2020) 093015 JRMcCleanet al
Taking as granted the construction of the matrix Φ, we define a rotated set of creation and annihilation
operators in the active space as
ˆ
a†
p=
Np
μ=1
ˆ
b†
μΦμp,ˆ
ap=
Np
μ=1
ˆ
bμΦμp,(5)
where Φμpdenotes the complex conjugate and we may project the Hamiltonian as
ˆ
H(a)=
Na
p,q=1
h(a)
pq ˆ
a†
pˆ
aq+1
2
Na
p,q,r,s=1
v(a)
pqrsˆ
a†
pˆ
a†
qˆ
arˆ
as,(6)
which we refer to as the active-space Hamiltonian.
Generally, we see that our primitive basis sets have favorable scaling in number of terms in the
Hamiltonian (O(N2
p)vsO(N4
a)), which often corresponds to better scaling algorithms. While Npand Na
have the same asymptotic scaling, for modest sized calculations it is often observed that NpNain order
to achieve comparable accuracy. Here we will seek a way to split the difference between these two regimes by
forming a more compact basis that partially retains the diagonal properties, i.e., the resulting Hamiltonian
is block-diagonal.
3. Discontinuous Galerkin discretization
At a high level, we construct the block-diagonal basis by fitting spatially connected blocks of the primitive
basis set to the active basis set, while preserving the properties of the primitive basis set. We therewith
interpolate between the primitive basis set and the active basis set. We will refer to the general class of basis
sets that achieve both completeness in some limit and have the diagonal property as primitive basis sets.
Our goal is to systematically compress the active basis set {ϕp(r)}Na
p=1into a set of orthonormal basis
functions partitioned into elements (groups), so that basis functions associated with different elements have
mutually disjoint support. Assume that the index set Ω={1, ...,Np}can be partitioned into Nb
nonoverlapping index sets
K={κ1,κ2,...,κNb},(7)
so that ∪κ∈Kκ=Ω. Then the matrix Φcan be partitioned into Nbblocks Φκ:=[Φμp]μ∈κfor κ∈Kand
p∈{1, ...,Na}. Performing the singular value decomposition for Φκ,
Φκ≈UκSκV†
κ,(8)
where Uκis a matrix with orthonormal columns corresponding to the leading nκsingular values up to
some truncation tolerance τ, we obtain our compressed basis
φκ,j(r)=
μ∈κ
χμ(r)(Uκ)μ,j.(9)
The basis set is adaptively compressed with respect to the given set of basis functions, and are locally
supported (in a discrete sense) only on a single index set κ. In the absence of SVD truncation, we clearly
have span{ϕp}⊆span{φκ,j}. We refer to this basis set {φκ,j}as the DG basis set. Note that each DG basis
function φκ,jis a linear combination of primitive basis functions which are themselves continuous, so φκ,jis
also technically continuous in real space. In fact, φκ,jmight not be locally supported in real space if each
primitive basis function χpis delocalized. When the primitive basis functions are localized, φκ,jcan be very
close to a discontinuous function. (See figure 3for an example.) When computing the projected
Hamiltonian, we do not need to evaluate the surface terms in the DG formalism. If we form a
block-diagonal matrix
U=diag[U1,...,UNb], (10)
the total number of basis functions is thus Nd:=κ∈Knκ. We remark that the number of basis functions nκ
can be different across different elements.
To facilitate the complexity count below we may, without loss of generality, assume that nκis a constant
and that Nd=Nbnκ(for a schematic illustration of the subsequent procedure see figure 4). Then we have
defined a new set of creation and annihilation operators
ˆ
c†
κ,j=
μ
ˆ
b†
μ(Uκ)μj,ˆ
cκ,j=
μ
ˆ
bμ(Uκ)μj, (11)
with κ=1, ...,Nband j=1, ...,nκthat correspond to the DG basis set.
6
New J. Phy s. 22 (2020) 093015 JRMcCleanet al
Figure 3. Three DG functions in the X–Zplane, represented in real space. The axes are in units of Bohr and the color intensity
represent the amplitude |φκ,j|. Each of the functions is localized to its block, where the block divisions are shown in dotted gray
lines. The DG functions are represented by linear combination of plane wave dual basis functions. Each DG function is strictly
still a continuous function, but its nodal values (defined according to the center of each plane-wave dual function) are only
supported within only one block.
Figure 4. A schematic illustration of the compression process of delocalized active-space basis functions into DG basis functions.
Beginning with a matrix representing the projection of a primitive basis onto a chosen active basis (left), with the primitive basis
grouped into blocks represented by rows here. Those blocks are then reduced by a singular value decomposition (center), which
finally leads to the DG basis that has a block-diagonal two-electron integral representation (right).
Unlike equation (5), the basis set rotation in equation (11)isrestrictedtoeachelementκ.Wereadily
obtain the projected Hamiltonian in the DG basis as
ˆ
H(d)=
κ,κ;j,j
h(d)
κ,κ;j,jˆ
c†
κ,jˆ
cκ,j
+1
2
κ,κ;i,i,j,j
v(d)
κ,κ;i,i,j,jˆ
c†
κ,iˆ
c†
κ,iˆ
cκ,jˆ
cκ,j.(12)
The matrix elements are
h(d)
κ,κ;j,j=
μν
(Uκ)μjh(p)
μν (Uκ)νj, (13)
and
v(d)
κ,κ;i,i,j,j=
μν
(Uκ)μi(Uκ)νiv(p)
μν (Uκ)μj(Uκ)νj.(14)
In general, the one-body matrix h(d)can be a full dense matrix, but the two-body tensor v(d)always takes a
‘block diagonal’ form in the following sense (it has a specific sparsity pattern). In principle, the two-body
interaction in the DG basis set should take the form
1
2
κ,κ,λ,λ;i,i,j,j
vκ,i;κ,i;λ,j;λ,jˆ
c†
κ,iˆ
c†
κ,iˆ
cλ,jˆ
cλ,j.(15)
Compared to equation (12), we find that
vκ,i;κ,i;λ,j;λ,j=v(d)
κ,κ;i,i,j,jδκλδκλ.(16)
In other words, vcan be viewed as a block-diagonal matrix with respect to the grouped indices (κκ,λλ).
We remark that the convergence of the DG basis set is independent of the choice of the primitive basis
set so long as the primitive basis has sufficient degrees of freedom to form a good approximation to the
active-space functions of interest. At the end of this adaptive procedure, we expect the number of elements
in the Hamiltonian to scale as O(N2
bn4
κ). However, as we expect the number of basis functions required to
reach a fixed accuracy within a block (i.e., nκ) to be bounded by a constant as system size grows, and the
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New J. Phy s. 22 (2020) 093015 JRMcCleanet al
scaling, with system size becomes O(N2
d). We substantiate the rapid asymptotic convergence of nκfor real
systems later in this work, however, simple arguments from spatial locality and basis set completeness lead
to the same conclusion.
4. Quantum simulation with a DG basis
Here we introduce the methods used to exploit the properties of the DG basis on a quantum computer for
evolution under the Hamiltonian. We first describe a method that implements evolution using a
Suzuki–Trotter decomposition through swap networks, taking advantage of recent advances in quantum
swap networks. An alternative Trotter approach based on low-rank decompositions is detailed in appendix
A. We then describe a method for time evolution based on the linear combinations of unitaries approach,
which will provide the required background for determining the degree of advantage for using DG basis sets
for fault-tolerant quantum computations of chemistry.
4.1. Swap networks for block diagonal Hamiltonians
In the first work using a strictly diagonal basis in quantum computing for chemistry [36], the ability for
quantum computers to perform fast Fourier transforms on quantum wavefunctions was exploited to
capitalize on the representational advantages of being in either the plane-wave basis or its
Fourier-transformed dual. That method was originally restricted to Hamiltonians with that particular
structure in the coefficients, similar to split-operator Fourier transform methods used in classical
simulation of quantum systems. However, it was soon realized that the structure of any diagonal
Hamiltonian could be similarly exploited. This generalization used a linear, fermionic swap network to
achieve perfect parallelization of a Trotter step with depth that scales linearly in the number of orbitals [57],
even when gates are restricted to act on nearest neighbors of a line of qubits.
Fermionic swap networks are analogous to sorting networks from traditional computer science except
built upon the primitive of the fermionic-swap operation,
ˆ
fpq
swap =1+ˆ
a†
pˆ
aq+ˆ
a†
qˆ
ap−ˆ
a†
pˆ
ap−ˆ
a†
qaq, (17)
ˆ
fpq
swapˆ
a†
p(ˆ
fpq
swap)†=ˆ
a†
q, (18)
where ˆ
fpq
swap is the fermionic swap that swaps the labeling of modes. The fermionic-swap operation was
introduced in [58] and also studied in the context of tensor networks. The difference between such a swap
and a traditional swap is by swapping fermionic modes instead of assignments to qubits, nonlocal parity
strings used to enforce the fermionic anticommutation relations can be avoided.
The basic idea of the linear swap network is to fermionic swap all neighboring qubits, interact them with
their current neighbors, and repeat until all qubits have interacted with each other. Since the introduction
and use of these linear fermionic swap networks in quantum algorithms, they have been generalized for use
in nondiagonal Hamiltonians with some overhead. For example, the quantum chemistry Hamiltonian can
be decomposed into a sum of diagonal Hamiltonians (each in a rotated basis) using techniques similar to
Cholesky or density-fitting methods, where each diagonal Hamiltonian can then be implemented in
sequence [39]. We show how to use this method in the DG representation in appendix A.
For the general Hamiltonian in quantum chemistry, which is nondiagonal, a generalized swap network
that works directly with such Hamiltonians was developed [59]. This network implements time steps for
generic O(N4
a) Hamiltonians in a time that scales as O(N3
a), and we take advantage of it here with
specializations for the block diagonal structure. To implement a Trotter step of the Hamiltonian, the swap
network dynamically updates the mapping from qubits to orbitals so that for each term in the Hamiltonian,
the involved orbitals are mapped to adjacent qubits. We say that a swap network ‘acquaints’ a set of orbitals
when it brings them together at some point in this way, and represent that point by an empty box in the
circuit diagrams, which acts as a placeholder for the logical gate to be executed there. Prior work utilizing
swap networks has applied them to two extremal regimes with respect to the structure of the two-electron
terms in the Hamiltonian: the strictly diagonal case, which can be implemented with O(Np)depth[57]; and
the fully general case, which can be implemented in O(N3
a)[59]. Here we show how to interpolate between
these to achieve O(Nbn3
κ)=O(Ndn2
κ) depth for block-diagonal Hamiltonians. (For simplicity, in this section
we will assume that all blocks have the same size nκ, but the techniques generalize in a straightforward way
to nonuniform block sizes.)
We focus on how to implement the quartic terms in the Hamiltonian (i.e., two-electron terms involving
four distinct spin orbitals). The lower-order terms can be addressed with negligible additional resources by
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New J. Phy s. 22 (2020) 093015 JRMcCleanet al
Figure 5. Acquaintance strategy for block-diagonal Hamiltonian with Nb=4andnκ=10. ‘K4
n’ indicates a four-complete swap
network on nqubits, i.e., one that acquaints the n
4subsets of four qubits with each other. The other gates are double bipartite
swap networks, explained in figures D2 and D3.
incorporating them into the quartic terms. The quartic terms in the block-diagonal Hamiltonian satisfy the
following properties:
(a) Two orbitals are from one block κand two orbitals are from another block κ(or all four from the
same block when κ=κ).
(b) The orbital spins have even parity (i.e., all up, all down, or two and two).
We will exploit both of these properties in constructing our swap network, which uses primitives
originally designed for implementing unitary coupled cluster [59]. Figure 5shows the overall swap network.
Initially, the orbitals are arranged on the line in lexicographical ordering; only the block index κand spin
are indicated for concision. The logic of the strategy is as follows:
(a) The first layer acquaints all sets of four spin orbitals within each block in which all four orbitals have
the same spin. This is achieved by a ‘four-complete’ swap network on each half block of orbitals,
denoted by K4
nκ/2because the sets of orbitals it acquaints correspond to the edges of a complete
four-uniform hypergraph; it has depth O(n3
κ). Note that the edges of the complete k-uniform
hypergraph Kk
non nare the n
ksets of kvertices. The ‘uniform’ qualifier indicates that all of the
hyperedges have the same number of vertices. See [59]fordetails.
(b) Thesecondlayeracquaintsallsetsoffourspinorbitalswithineachblockinwhichtwoorbitalshave
spin up and the other two have spin down. This is achieved by a ‘double bipartite’ swap network on
each block in depth O(n3
κ); see figure D2.
(c) The third layer permutes, in O(nκ) depth, the orbitals within each block in preparation for the
inter-block acquaintances to follow.
(d) The rest of the strategy consists of Nbalternating layers that acquaint pairs of parts. In each layer, each
block of qubits is paired up with an adjacent one and a ‘balanced double bipartite’ swap network is
executed on the pair of blocks; see figure D3. Each balanced double bipartite swap network acquaints
thesetsoffourorbitalscontainingtwofromeachblock and with even (‘balanced’) spin parity. This
also has the effect of swapping the blocks, so overall a balanced double bipartite swap network is
applied to every pair of blocks. Each double bipartite swap network has depth O(n3
κ).
Overall, the depth is O(Nbn3
κ), dominated by the latter swap networks that effect the inter-block
interactions. The components of this approach are explained in more detail in appendix Dand an
alternative approach that may have asymptotic advantages in some regimes is discussed in appendix A.
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New J. Phy s. 22 (2020) 093015 JRMcCleanet al
4.2. LCU approaches for simulation
The quantum simulation algorithms discussed in the previous section and in appendix Aare useful for
implementing Trotter steps of the chemistry Hamiltonian. Such Trotter steps can be repeated to perform
time-evolution for modeling dynamics or for preparing eigenstates via the phase estimation algorithm [60,
61], but they can also serve as an ansatz for composing quantum variational algorithms [15,62]. This in
conjunction with these Trotter steps requiring only minimal (linear) connectivity makes them attractive
algorithms for near-term quantum computing. However, within cost models appropriate for a fault-tolerant
quantum computer, Trotter steps are not the most competitive technique for chemistry simulation. In such
a cost model, the key resource to minimize is the number of T gates required by the algorithm. This is
because in most practical error-correcting codes (e.g.,thesurfacecode),Tgatesrequiremanyphysical
qubits for their distillation and are orders of magnitude slower to implement than Clifford gates.
Currently, the lowest T complexity quantum algorithms for simulating chemistry are all based on linear
combinations of unitaries (LCU) methods [63]. This family includes Taylor series methods [64] (applied to
chemistry in [14,42]), qubitization [65] (applied to chemistry in [38,65,66]) and Hamiltonian simulation
in the interaction picture [41] (applied to chemistry in [40,41]). What all LCU methods have in common is
that they involve simulating the Hamiltonian from a representation where it can be accessed as a linear
combination of unitaries,
H=
L
=1
ωU,λ=
L
=1|ω|, (19)
where Uare unitary operators, ωare scalars, and λis a parameter that determines the complexity of these
methods. The second-quantized Hamiltonians discussed in this paper satisfy this requirement once mapped
to qubits (e.g., under the Jordan–Wigner transformation) since strings of Pauli operators are unitary.
LCU methods perform quantum simulation in terms of queries to two oracle circuits defined as
SELECT||ψ →|U|ψ, (20)
PREPARE|0⊗log L→
L
=1ω
λ|, (21)
where |ψis the system register and |is an ancilla register which usually indexes the terms in
equation (19) in binary and thus contains log Lancillae. Up to log factors in precision and other system
parameters, LCU methods can perform time-evolution with gate complexity scaling as
O((CS+CP)λt), (22)
where CSand CPare the gate complexities of SELECT and PREPARE respectively, and tis time.
To implement the LCU oracles one must be able to coherently translate the index into the associated
Uand ω. In the quantum chemistry context the Uare related to the second-quantized fermion operators
(e.g., ˆ
a†
pˆ
a†
qˆ
arˆ
as)andtheωare related to the molecular integrals. While the Uhave a structure that is
straightforward to unpack in a quantum circuit (see [38,66] for explicit implementations), the ωare
typically challenging to compute directly from this index (unless one pursues the highly impractical strategy
of computing the integrals on-the-fly, as in [14]). As a consequence, with state-of-the-art LCU methods for
simulating chemistry the primary bottleneck has been implementation of PREPARE rather than SELECT [38,
66].
In the most performant LCU approaches for chemistry (see [66] for plane waves and [38] for arbitrary
basis sets), PREPARE is implementing (and bottlenecked) by using a data-lookup routine referred to as
QROM (quantum read-only memory) to load the ωvalues into superposition. Using this approach, the
cost of PREPARE is a function of the number of unique coefficients ωin the Hamiltonian. Using the QROM
of [67] (which improves on the original concept from [66]), one can (up to log factors) implement PREPARE
with T complexity scaling as O(L/g+g)whereLis the number of molecular integrals and gis a free
parameter so long as at least gdirty ancilla are available during the implementation of PREPARE.AsPREPARE
acts only on the ancilla register, there are typically at least Nddirty ancilla available (from the |ψ
register).Thus (assuming L>N2
d)thescalingisO(L/Nd) without ancilla or O(√L)withO(√L)ancilla
(often a reasonable tradeoff within error-correction).
When simulating in a molecular orbital basis with L=O(N4
a), one can (up to log factors) evolve the
Hamiltonian for some time tand achieve T complexity O(N2
aλt)[38]. If a straightforward extension of the
approach in [38]isappliedtotheDGbasisHamiltonianwithL=n4
κN2
bthen this scaling would be reduced
to O(n2
κNbλt). However, if one can index the ωin a way that exploits symmetries in the coefficients, then
one can further reduce the effective value of ‘L’ in both of these expressions. For instance, in [66]itis
recognized that while there are N2
pcoefficients ωin the Hamiltonian of equation (2) (one for each value of
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New J. Phy s. 22 (2020) 093015 JRMcCleanet al
νand μ), there are only Npunique coefficients due to the translational invariance of the Coulomb operator
(essentially, one for each unique value of the index μ–ν), and the cost of the algorithm is reduced
accordingly. As a rough rule of thumb, the scaling becomes O(J/g+g)whereJcan be thought of as the
number of unique scalars needed to completely describe the Hamiltonian without recomputing the
molecular integrals. In [38], it is shown that because only J=O(N3
a) numbers are needed to describe the
low-rank factorized Coulomb operator in an arbitrary basis [39], one can improve the scaling to O(N3/2
aλt).
By tailoring such techniques to symmetries such as periodic crystalline symmetry or other redundancies in
the DG Hamiltonian (e.g., those exploited in the low-rank representation of appendix A)onecanalso
further improve the O(n2
κNbλt)scaling.
Finally, we note that the λvalue associated with the molecular orbital basis Hamiltonian is likely larger
than the λvalue associated with the DG Hamiltonian, and we quantify the extent to which that is the later
in this paper. This is yet another way in which the DG representation should lead to even more efficient
implementation of LCU methods for chemistry simulation. Note also, that the value of λis important for
the scaling of quantum simulation in a near-term cost model as well. In particular, for quantum variational
algorithms the number of measurements (corresponding to the number of circuit repetitions) required to
estimate the energy of a Hamiltonian to within error scales as O(λ2/2)[68].
5. Numerical results
To understand the performance of different discretization schemes, we examine the costs and associated
accuracy for each of the methods, using hydrogen chains of increasing lengths as test systems.
We show that the crossover point occurs around 15 to 20 atoms, above which the DG representation with
the block-diagonal Hamiltonian structure has significant lower costs. For two-dimensional and
three-dimensional hydrogen lattices, we expect that the number of basis functions will increase slightly from
our experience at the DFT level [43,51].
The subsequent calculations are twofold. We begin with numerical simulations designed to exhibit the
expected crossover behavior. We investigate hydrogen chains of increasing length (up to H30 ) in a Gaussian
cc-pVDZ basis [26](theactivespacetobefit),wherewechoosethe primitive diagonal basis set to be plane
wave dual functions with refinement built to match the accuracy of the Gaussian basis set to a specified
tolerance at the level of density functional theory with the Perdew–Burke–Ernzerhof (PBE)
exchange–correlation functionals [69]. We are not aware of any electronic structure software package using
a plane-wave basis set for practical all-electron DFT calculations. Hence we use the pseudopotential
formulation based on the ONCV pseudopotential [70]. As we fit to the span of the basis set itself, rather
than a density matrix generated with a fixed level of theory, these results should be representative of
performance across different levels of theory including, but not limited to, correlated methods. This is done
to observe the crossover without extrapolation, as by the size the crossover occurs, the more accurate
correlated calculations become prohibitively expensive using traditional classical methods.
In the second set of calculations, we instead fit to a natural-orbital active space using a Gausslet basis set
as the primitive diagonal basis set [34,35] to demonstrate that this representation maintains accuracy in
correlated calculations. This is possible as the properties of a Gausslet basis allow accurate DMRG
calculations for comparison on these systems. Similar DMRG calculations are expected to be more
expensive in basis sets with heavy tails, such as the plane-wave dual basis, which makes the one-particle
Hamiltonian a dense matrix, and result in excessive bond dimension requirements for accurate
descriptions. These calculations exhibit the versatility of this method and its ability to achieve accuracy,
even in a correlated active space built fit to a highly nonlocal active space. Furthermore, we demonstrate
that by fitting the DG basis simultaneously to two active basis sets (molecular orbitals at the UHF level and
the cc-pVDZ basis functions), the DG basis set can obtain accurate results when compared to quantum
Monte Carlo calculations at the complete basis set limit, while achieving a 1–2 orders of magnitude
reduction of computational cost over the primitive Gausslet basis or the Gaussian basis set.
5.1. Scaling crossover in a DG plane-wave dual basis
In order to demonstrate the performance of the DG approach for a long hydrogen chain, we use the
DFT-based MATLAB toolbox KSSOLV [71]. KSSOLV uses a plane-wave discretization in combination with
pseudopotentials to make larger systems computationally tractable. Figure 6shows that chemical accuracy is
achieved when the kinetic-energy cutoff (Ecut ) is set to around 20 hartree for H10 using the ONCV
pseudopotential. The kinetic-energy cutoff needs to be much larger for all-electron calculations using the
plane-wave basis set. With this kinetic-energy cutoff, we can perform calculations on hydrogen chains up to
H30.ForH
30 the box size is 10 ×10 ×118 a3
0,andthegridsizeis20×20 ×238, or 95 200 plane-wave dual
basis functions (around 3000 basis functions per atom).
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New J. Phy s. 22 (2020) 093015 JRMcCleanet al
Figure 6. Convergence of the total energy per atom for a H10 system with respect to the kinetic-energy cutoff using the
plane-wave basis set and the ONCV pseudopotential. Chemical accuracy (black dashed line) is achieved around Ecut =20
hartree, which is the value used for all examples in this section.
Figure 7. Potential energy surfaces for H20 in DG basis with different tolerances and full plane-wave basis. At equilibrium
position, the average number of DG basis per atom is given by nκfor each fit tolerance specified by DG ,whereis the SVD
cutoff. For comparison the number of primitive functions per atom here is approximately 3000. The primitive basis set is more
expressive by design than the active-space Gaussian cc-pVDZ basis against which the DG fit is performed. This allows even fairly
loose DG fits to match the accuracy of the active-space basis.
We start by constructing an active basis set using the cc-pVDZ basis with the given molecular geometry,
and sample it with an underlying grid of plane-wave dual functions. This yields a real-space discretization
of the cc-pVDZ basis, which is then transformed into a block-diagonal form by means of a DG blocking
procedure, also implemented in KSSOLV. As a technical note, the partitioning boundaries in the DG
approach are important. The hydrogen chains that are the subject of this numerical investigation are quasi
1D-problems. This and the fact that we use a real-space discretization suggests that the ideal partitioning of
the basis in terms of a DG-scheme (see section 3) corresponds to a nonuniform partitioning strategy, so
that each hydrogen atom is approximately located at the center of each element. We remark that by
construction the DG-blocking procedure is able to produce accurate results even if the partition is nonideal,
e.g. when an hydrogen atom is located near the boundary of an element. However, this can require a larger
number of basis functions per atom compared to the ideal partitioning strategy.
We first verify the accuracy of our DG-basis in figure 7by comparing potential energy surfaces (PES) in
DG discretizations of different tolerances with the corresponding PES in the primitive basis. The primitive
basis has been chosen to be much more expressive than the Gaussian active-space basis, showing a lower
absolute energy. This allows even a coarse DG fit to reliably match the Gaussian active-space accuracy, while
being far more compact than the original primitive basis set. In fact, the energies obtained from the DG
basis are slightly lower than those from the Gaussian basis set. This is because as nκincreases, the span of
the Gaussian basis set becomes approximately a subspace of the span of the DG basis set, and hence the DG
basis can possibly yield lower energies due to the variational principle. Also due to the variational principle,
the energies obtained from the DG basis are noticeably higher than those from the primitive plane-wave
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New J. Phy s. 22 (2020) 093015 JRMcCleanet al
Figure 8. Convergence of block size nκas a function of system size. The average number of DG-basisper atom at equilibrium
with SVD tolerances of 10−1,10
−2and 10−3. For a fixed accuracy, it is observed that the average number of DG functions per
block converges as a function of system size.
basis set, and the main limiting factor is the cc-pVDZ basis set to which DG is fitted. The average number of
functions per atom nκis shown for each tolerance and is approximately 20, as compared to the primitive
basis which is built from roughly 3000 functions per atom. The results suggest that the overall energetic
accuracy is relatively insensitive even to rather aggressive singular value truncations in the DG-blocking
procedure.
We then plot the average number of DG-basis functions per atom in figure 8for fixed SVD thresholds in
the DG-blocking procedure. The data confirms that, as the system size grows, nκapproaches a constant
value for a fixed SVD-truncation threshold. The combination of the energetic insensitivity beyond a
kinetic-energy cutoff of 20 hartree and the asymptotically constant nκstrongly supports the existence of a
crossover regime where a DG discretization is more cost effective than other representations. We confirm
the crossover directly by examining the quantities most relevant for fault-tolerant cost models of chemistry
on a quantum computer, i.e., the number of nonzero two-electron integrals in each representation and the
λfactor equation (B1)infigures9and 10, respectively (for λfactors for different bond lengths see appendix
B,figureB1). A threshold of 10−6is used to count an individual integral as 0 when calculating these
quantities empirically on the systems of interest. We choose to illustrate the cost crossover for the bond
length 1.7a0since we here detected the largest deviation with respect to the SVD-truncation threshold.
Considering the cost-model in the fault-tolerant setting outlined in section 4.2, one can already observe
a scaling advantage for the DG representation over simple Gaussian based active-space representations.
Recall from that section that the cost using an LCU method to evolve for some time tis roughly O(√Lλt)
where Lis the number of nonzero terms in the Hamiltonian. We consider here only the two-electron
integrals as they represent the dominant cost contribution in most cases. For molecular orbital
representations, this is generally O(N2
aλt), which was subsequently improved to O(N3/2
aλt) using low-rank
structure in the problem. For the hydrogen chains examined here in the Gaussian basis set, we see
empirically over the system sizes considered L∝N4
hand λ∝N2.5
h,whereNhis the number of hydrogen
atoms in the chain, leading to an expected cost scaling of O(N4.5
ht) when not exploiting further low rank
structure. In the DG representation for the same problem, we see that L∝2.25 and λ∝1.5, which suggests
an empirical scaling for this physical system of O(N2.6
ht). The same calculation for the primitive basis
suggests a scaling of O(N1.8
h), which is the lowest seen, however, the simulation cost and qubit counts
required are many orders of magnitude higher here as seen in the figures. In both quantities we observe a
constant factor crossover where the DG representation has strictly lower unique two-electron integrals and
λfactor when compared to the Gaussian active-space basis at around 15 to 20 hydrogen atoms.This suggests
that the DG representation is not only advantageous in the asymptotic scaling, but also for modest finite
system sizes for fault-tolerant implementations.
5.2. Correlated calculations in a DG Gausslet basis
Here we demonstrate that the performance of the DG basis set for a vastly different regime, which uses a
natural-orbital active space from an exact DMRG calculation fit to Gausslet primitive functions in a
correlated calculation. The Gausslet basis set is a recent approach to improve the discretization of quantum
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New J. Phy s. 22 (2020) 093015 JRMcCleanet al
Figure 9. The number of nonzero two-electron integrals in different representations, for equilibrium and dissociation bond
lengths with SVD-truncation tolerances of 10−1,10
−2and 10−3, plotted on a log –log scale. We fit a trendline plotted with black
dots from the second point onward to extract the scaling as a function of system size as Nα+cfor some constant c,andlistthe
exponent αbeside each representation in the legend. As predicted, for these system sizes the number of two-electron integrals lies
between the primitive and active-space representations, tending closer to the O(N2) scaling of the primitive representation,
requiring fewer functions. Note that for the Gaussian basis set, certain elements of the two-electron integrals can vanish due to
the symmetry of the atomic configuration of the hydrogen chain. This has a larger impact for small systemthan for large systems,
and therefore the scaling is observed to be slightly larger than O(N4
a).
Figure 10. λvalue for Gaussian and DG basis. A core quantity in determining the cost in quantum algorithms, λ,isplottedasa
function of system size for different representations. The notation DG indicates an SVD truncation threshold of in the
blocking procedure. We observe an advantageous crossover before or around H20 in all cases with respect to an actual value. We
fit a trendline plotted with black dots from the second point onward to extract the scaling as a function of system size as Nα+c
for some constant c,andlisttheexponentαbeside each representation in the figure legend. We see the scalingfor the DG basis is
significantly better in all cases than for the active-space basis as well.
chemical problems [34,35]. It has a special focus on sparsity, spatial locality, and orthonormality, to fulfill
the needs of strong-correlation methods like the DMRG, while keeping the number of basis functions lower
than other grid-based bases. Our calculations illustrate the accuracy of the coupled-cluster method with
single and double excitations (CCSD) method in a DG-basis with an underlying primitive Gausslet basis set
with respect to the original active space. The block-diagonal form of the Hamiltonian in a DG-basis yields
an asymptotic improvement in the number of nonzero two-electron integrals, which is directly related to
the circuit depth and size. The calculations presented here are limited to short hydrogen chains due to the
size of the Gausslet basis set and expense of correlated calculations. The goal of these calculations is to
demonstrate that the DG approach maintains accuracy for correlated calculations due to its construction as
a fit to the span of the active basis.
Specifically, we compare CCSD-energy results in an active-space basis set with calculations in a DG-basis
fit to the same active-space basis. This serves to quantify the overhead in restricting the basis to have block
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New J. Phy s. 22 (2020) 093015 JRMcCleanet al
Figure 11. Correlated potential-energy surfaces for H8. We plot the potential-energy surface computed using the Gausslet basis
as the primitive basis and a subset of the exact DMRG natural orbitals as an active space (AS). The CCSD energy is calculated in
the active space as well as the DG fit to that active space, showing excellent agreement with the active-space energetics in a
correlated calculation.The exact DMRG energy in the more expressive, full primitive basis (Gausslets) is shown for reference.
locality versus the totally delocalized natural-orbital active space. The calculations are performed for H2,
H4,H
6and H8with varying symmetrically stretched bond lengths. Figure 11 shows the calculation for H8
with optimal DG partitioning of the Gausslet basis set, the respective figures for H2,H
4,andH
6are
presented in appendix C(figures C1–C3). From these plots one may see that enforcing a block-diagonal
structure of the Hamiltonian does not reduce the accuracy further from the active-space approximation.
The Gausslet basis for computations resulting in figure 11 consists of 1336 Gausslets corresponding to
an average of 167 functions per atom. This is significantly lower than the number of basis functions per
atom in the plane-wave dual basis which reflects the variable resolution available with the Gausslet basis.
Averaging over all values along the PES, we find that the full two-electron integral tensor for this basis has
∼2 310 318 nonzero elements. Using a DG basis it is possible to reduce this number to ∼511 449 nonzero
two-electron integrals without losing accuracy compared to the respective active-space approximation. This
suggests cost reductions for correlated calculations that are similar to the observations in section 5.1.Note,
however, that H8is a small system compared to the systems considered in section 5.1, consequently, the
number of nonzero two-electron integrals in the DG-basis is still larger than in the active-space
approximation (∼52 346). This aligns with the results from the previous section, which showed that the
improvement of the nonzero two-electron integral count for the DG-basis becomes observable for 15 to 20
atoms, depending on the imposed truncation tolerance. A naive extrapolation of the nonzero two-electron
integral count in table C2 (see appendix Cfigure C4)suggestsacrossoveraround25atomsforthe
coupled-cluster calculations, which is again in agreement with the computations in section 5.1 with a low
truncation tolerance (see appendix Btable B1). We conclude that the trial calculations for small systems
performed here together with results from section 5.1 suggest that the DG approach maintains the accuracy
of active-space approximations for correlation methods but with a more efficient representation on
quantum devices once a certain system size is reached.
5.3. DMRG calculations in a DG basis
Here we examine the power of the DG technique to compactly and cheaply represent problems for classical,
correlated DMRG calculation by constructing an active space that takes static correlation from a UHF
calculation done in a flexible primitive basis set such as Gausslets, combined with contributions from a
basis set that has been empirically refined to capture dynamic correlations. These calculations have the dual
purpose of demonstrating the power of the DG approach even in highly-correlated calculations more
generally, including calculations on a quantum computer. Here we demonstrate a hybrid active-space
approach for cheaply finding a balanced active-space. Specifically, after performing a UHF calculation in a
Gausslet basis, we project both the cc-pVDZ basis set and the UHF orbitals onto the primitive Gausslet
basis. The UHF calculation can be performed cheaply. By including the UHF orbitals in our active-space,
we ensure that there are no HF-level errors in the basis. The only lack of completeness is associated with
correlation beyond HF. To (partially) capture correlation in a compact way, we include contributions from
the empirical Gaussian basis with a slightly smaller factor, α=0.01 here, then perform the DG blocking
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New J. Phy s. 22 (2020) 093015 JRMcCleanet al
Figure 12. Potential-energy surfaces for H10 constructed with the hybrid approach. We plot the potential-energy surface for
unrestricted Hartree– Fock (here denoted HF) calculations in both a cc-pVDZ (vDZ) Gaussian basis and a Gausslet basis, from
which the DG basis sets are derived to perform a DMRG calculation. We then contrast this with a calculation done in the DG
basis using the first 7 virtuals from the HF–DG basis (DG), and finally showcase the power of the hybrid approach to
cost-effectively reintroduce dynamic correlation without the cost of the original Gaussian basis by adding a point sampling of the
cc-pVDZ basis into the DG basis (DG+). This hybrid basis attains nearly the exact solution with respect to the complete basis set
limit with a fraction of the cost of using either the Gausslet or Gaussian basis set directly.
procedure to develop the basis as before. The resulting basis maintains the flexibility of the HF solution in
the Gausslet basis for the low-lying orbitals, while now including the refined features of the Gaussian
orbitals without losing the block-diagonal structure. Although it is difficult to compare the efficiency of
primitive and DG bases in a precise manner, we find that this approach can achieve a 1–2 orders of
magnitude reduction of computational cost over either the primitive Gausslet basis or Gaussian basis set,
while achieving excellent accuracy with respect to the complete basis set limit.
Numerical calculations for the H10 system are shown in figure 12. For comparison, we perform an
unrestricted Hartree– Fock calculation in both a traditional Gaussian basis set (cc-pVDZ), and a
multi-sliced Gausslet basis set. The Gaussian basis set contains 5 spatial orbitals per atom, totaling to 50
spatial orbitals with the associated nondiagonal two-electron integrals as one would expect. The Gausslet
basis is formed adaptively according to pre-determined cutoffs, and the number of functions ranges from
7000–10 000 spatial orbitals for these calculations, while retaining the diagonal property, making
calculations at the UHF level relatively straightforward, even with such a formidable number of basis
functions. This large primitive basis gives UHF results near the complete basis set limit, well beyond the
accuracy of this Gaussian basis. This high accuracy in the HF comes at little cost; the UHF is still fast
compared to the correlated calculations and the large number of primitive functions do not strongly affect
the size of the DG basis. For larger systems the accuracy could be reduced to keep the HF manageable.
To construct the DG basis, we construct 10 spatial blocks, 1 around each atom. We make use of the UHF
calculation density matrix and keep 7 total orbitals per block, yielding 70 total spatial orbitals, a number
almost identical to the number of Gaussians in the cc-pVDZ basis, but maintaining the block diagonal
property of the two-electron integrals. By construction, UHF in this basis can accurately match the UHF
results of the Gausslet basis set. The introduction of correlation through DMRG on this basis shows
improvement as expected but a relative offset from the exact answer due to UHF’s focus on static
correlation. By using the weigthing procedure, in order to include UHF orbitals in the DG construction, we
find excellent agreement with calculations done in the complete basis set limit. To that end, a weighting
factor of α=0.01 was imposed. The number of functions kept is 15 per block, yielding 150 functions in
total, which is about three times larger than the cc-pVDZ basis. However, the structure of the interactions
and spatially local construction of the DG functions allows the DMRG calculation to be done with a 1–2
order of magnitude reduction in computational cost as compared to using the DG or Gaussian basis sets
alone for the current implementation. This suggests that the hybrid active-space approach with the
DG-blocking procedure is a powerful technique for recovering both static and dynamic correlation in a
cost-effective manner.
To elaborate on the scaling of DMRG, mbe the bond dimension for the state required for the desired
accuracy. Then the computational cost of the Gaussian basis sets is expected to be O(N3
am3). In contrast, for
Gausslets and DG representations based on Gausslets, using matrix product operator (MPO) compression
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New J. Phy s. 22 (2020) 093015 JRMcCleanet al
and let Dbe the MPO dimension, one expects the asymptotic cost to be O(NpDm3)andO(NdDm3),
respectively. Due to the localization properties, Dis expected to depend weakly on length, and be
comparable for both the Gausslet and DG block representations. One can see this from considering the
MPO decomposition in the Gausslet representation then transformed to the DG representation. While
some expansion of the bond dimension could happen within a block, it is bounded by the Gausslet
dimension of the block from the properties of a Schmidt decomposition, and no inter-block mixing occurs.
As a result, the bond dimension will be comparable. Hence, it is the massive reduction in number of basis
functions, from 10 000 to 150 that reduces the cost by several orders of magnitude while maintaining
excellent accuracy. The Gaussian basis set, while advantageous in number of functions, suffers from lack of
spatial locality, but the cost of this nonlocality in terms of the required bond dimension for equivalent
accuracy has not been studied, and likely depends significantly on the completeness of the basis. It is easier
to compare the scaling of the Gaussian DMRG due to the lack of diagonality, resulting in a much larger set
of two-electron integrals. In this case, the block diagonality of the DG basis typically results in approximate
linear scaling in the number of atoms, versus cubic dependence with Gaussian DMRG. Hence in both cases,
the DG approach offers a significant reduction in computational cost for an accuracy that nearly approaches
the complete basis set limit, which was obtained through accurate quantum Monte-Carlo calculations
[35,72].
6. Conclusion
The discretization problem is a crucial aspect determining the cost and effectiveness of quantum chemistry
methods both for classical and quantum methods. The most popular basis sets for correlated calculations,
Gaussian based molecular orbitals, are notably more compact than alternatives and have a well developed
set of tools for their use. Unfortunately, in cost models for quantum computation, the overhead of a quartic
number of terms in the Hamiltonian leads to poor scaling with system size. On the other side of the
spectrum, representations that achieve a strictly quadratic number of terms and a diagonal representation,
such as Gausslets or plane-wave dual functions exhibit excellent scaling with system size, but have overheads
that make them undesirable for modest size implementations.
Here we introduced a systematic method for interpolating between the two regimes through the use of a
blocking procedure, motivated from the discontinuous Galerkin (DG) method. This method is able to use
any primitive basis with the diagonal property to represent a delocalized active-space basis, while
maintaining the diagonal property between blocks. By choosing a plane-wave dual primitive basis and
Gaussian active space, we were able to show how one can adaptively interpolate between these two regimes
to attain both a scaling and constant factor advantage over the target active space.
When these empirical results are put into the context of known costs for exact quantum algorithms for
chemistry, we observed a scaling improvement over Gaussian basis sets from O(N4.5
h)toO(N2.6
h)witha
constant factor crossover around 15 to 20 hydrogen atoms. This suggests that for modest sized systems, such
as those just beyond the classically tractable regime, this representation will be the optimal choice for
quantum algorithms. Moreover, we showed that for high accuracy DMRG calculations, one may take
advantage of this representation to achieve a high accuracy calculation with a cost reduction that is over an
order of magnitude with respect to traditional representations. In all cases, one may use this methodology
to scale between a compact representation and one with superior integral scaling depending on the
requirements of a particular method.
Acknowledgments
This work was partially supported by the Department of Energy under Grant No. DE-SC0017867, the
Quantum Algorithm Teams Program under Grant No. DE-AC02-05CH11231, the Google Quantum
Research Award (LL), the Research Council of Norway under CoE Grant No. 262695, the Peder Sather
Grant Program (FMF), the NASA Space Technology Research Fellowship (BO), and the Ning fellowship
(QZ).
Appendix A. Trotter step by low-rank factorization
As an alternative to the fixed swap networks used in the main text to evolve under the two-electron integral
term in the DG basis, one may use the low-rank factorization strategy in [39], but applied to the block
diagonal matricized tensor v(d)
κ,κ,λ,λ;i,i,j,j. For such a matrix, we know the maximum number of Cholesky
factors is given by the dimension of the matrix, N2
d, however, because of the block structure, each of the
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Figure B1. The number of non-zero two-electron integrals with bond lengths =1.0 to 3.6 in atomic units, tol =10−1to 10−3,
plotted on log– log scale. The crossover appears before or around H22 in all cases.
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Table B1. Crossover regions for different bond lengths and different truncation tolerances.
To l e r a n c e 1 0 −310−210−1
b=1.0H
16 –H18 H14–H16 H8–H10
b=1.2H
18 –H20 H14–H16 H6–H8
b=1.4H
20 –H22 H14–H16 H6–H8
b=1.6H
20 –H22 H12–H14 H6–H8
b=1.7H
20 –H22 H10–H12 H6–H8
b=1.8H
18 –H20 H8–H10 H8–H10
b=2.0H
16 –H18 H8–H10 H8–H10
b=2.4H
10 –H12 H8–H10 H8–H10
b=2.8H
8–H10 H8–H10 H8–H10
b=3.0H
8–H10 H8–H10 H8–H10
b=3.2H
8–H10 H8–H10 H6–H8
b=3.6H
8–H10 H8–H10 H4–H6
Figure C1. Potential energy surfaces for H2inaGaussletbasiscomputedbyrestrictedSCF,(DG-)CCSDandbenchmarkedwith
DMRG results. The averaged number of non-zero two-electron integrals per atom (DG-element κ) is 4144. The averaged
number of non-zero two-electron integrals of the CAS calculations is 205.
Figure C2. Potential energy surfaces for H4inaGaussletbasiscomputedbyrestrictedSCF,(DG-)CCSDandbenchmarkedwith
DMRG results. The averaged number of non-zero two-electron integrals per atom (DG-element κ) is 16 580. The averaged
number of non-zero two-electron integrals of the CAS calculations is 3269.
Cholesky factors need only have non-trivial support only within a κ,κblock. If nκbounds the larger of nκ,
nκ, then the dimension of one of these blocks is O(n2
κ). It is easy to see that the total matricized tensor has
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Figure C3. Potential energy surfaces for H6inaGaussletbasiscomputedbyrestrictedSCF,(DG-)CCSDandbenchmarkedwith
DMRG results. The averaged number of non-zero two-electron integrals per atom (DG-element κ) is 287 762. The averaged
number of non-zero two-electron integrals of the CAS calculations is 16 551.
Table C1. Averaged numbers Gausslets used to discretize the molecular system.
Atoms H2H4H6H8
Tot. no. of Gausslet 529 801 1066 1336
Gausslets in z-axis 11–13 15– 21 17–27 21 –33
Gausslets in x-resp. y-axis 6.33–7.40 6.92– 7.53 7.31–7.78 7.27 –8.01
Table C2. Averaged numerical values of non-zero two-electron integrals for H2,H
4,H
6and H8.
Atoms H2H4H6H8
No of Gausslets 529 801 1066 1336
nnz-tei Gausslet 351 084 816 232 1 459 901 2 310 318
DG per elem. 6 6 10 10
nnz-tei DG 4144 16 580 287 762 511 449
CAS 4 8 12 16
nnz-tei CAS 205 3269 16 551 52 346
Figure C4. Extrapolation of non-zero two-electron integrals with 10 basis functions per DG element.
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Figure D1. Notation and decomposition for a P-swap network with partition sizes (1, 2, 1, 2, 2). A P-swap network for a
partition (P1,P2,...,P|P|)ofthequbitsiPiacquaints every union of a pair of parts, i.e., {P∪P|P,P∈P}. At a high level,
the structure is similar to that of the simple linear swap network, except that instead of single qubits being swapped, groups are
(i.e., the parts of the partitionP). There are |P| layers of generalized swap gates, each of which swaps sets of qubits. For more
details, see [59].
Figure D2. Construction of the double bipartite swap network, with parts of size 4. The top half of the top circuit contains the
same swap gates as a linear swap network but with additional acquaintance opportunities. In the bottom half of the top circuit
are 4 linear swap networks in a row, one for each acquaintance layer of the linear swap network in the top half, which is copied
for each acquaintance layer of the bottom half. Overall, for every set of of four orbitals consisting of two from the top part and
two from the bottom part, there is a layer in the circuit in which both pairs are simultaneously acquainted. The bottom circuit,
depicting the double bipartite swap network, is formed by replacing each such acquaintance layer in the top circuit with a
P-swap network, where a pair of qubitsacquainted in the top circuit corresponds to a part of the partition P.TheP-swap
network acquaints the union of each pair of pairs; see figure D1. The final gate ensures that overall effect is to shift the parts.
dimension O(n2
κN2
b), and a number of non-zero entries scaling as O(n4
κN2
b). Hence it matches the primitive
limit as nκ→1andNb→Ndand the active space limit as nκ→Ndand Nb→1.
To execute a single Trotter step of the two-electron part of the Hamiltonian, one may start from a
factorization that is a product over κ,