:

# Accelerating Quantum Computations of Chemistry Through Regularized
Compressed Double Factorization

Oumarou Oumarou  Covestro Deutschland AG, Leverkusen 51373, Germany
Maximilian Scheurer  Covestro Deutschland AG, Leverkusen 51373, Germany
Robert M. Parrish<EMAIL_ADDRESS>QC Ware Corporation, Palo Alto, CA
94301, USA  Edward G. Hohenstein  QC Ware Corporation, Palo Alto, CA 94301,
USA  Christian Gogolin<EMAIL_ADDRESS>Covestro Deutschland
AG, Leverkusen 51373, Germany

(05/04/2023)

###### Abstract

We propose the regularized compressed double factorization (RC-DF) method to
classically compute compressed representations of molecular Hamiltonians that
enable efficient simulation with noisy intermediate scale (NISQ) and error
corrected quantum algorithms. We find that already for small systems with 12
to 20 qubits, the resulting NISQ measurement scheme reduces the number of
measurement bases by roughly a factor of three and the shot count to reach
chemical accuracy by a factor of three to six compared to truncated double
factorization (DF) and we see order of magnitude improvements over Pauli
grouping schemes. We demonstrate the scalability of our approach by performing
RC-DF on the Cpd I species of cytochrome P450 with 58 orbitals and find that
using the resulting compressed Hamiltonian cuts the run time of qubitization
and truncated DF based error corrected algorithms almost in half and even
outperforms the lambda parameters achievable with tensor hypercontraction
(THC) while at the same time reducing the CCSD(T) energy error heuristic by an
order of magnitude.

The run time of algorithms on most of today’s noisy intermediate scale (NISQ)
hardware platforms is largely independent of the anyway shallow circuit depth.
Instead, it is mostly a function of the number of distinct circuits that need
to be evaluated and the number of repeated circuit executions because re-
programming the quantum device to perform a different circuit and the time for
measurement and qubit reset dominate the overall circuit execution time [1].

Many NISQ algorithms such as the variational quantum eigensolver (VQE) [2] are
essentially methods to reduce circuit depth at the expense of requiring many
repetitions, also called shots. In a similar fashion, error mitigation
techniques [3, 4] create a tolerance for the noise of NISQ devices by further
increasing the number of shots needed to obtain a final result. The total
number of shots is thus often the limiting factor on the path towards quantum
advantage.

This is a particularly pressing issue in in quantum chemistry simulation.
Here, molecular Hamiltonians, which in their second-quantized form have
$O(n^{4})$ terms, where $n$ is the number of spatial orbitals, need to be
measured with very high accuracy. Naive measurement schemes require an
extremely fast growing number of distinct observables and total number of
shots [5] for reaching the required accuracy.

Methods to cope with this problem fall in two broad classes. First, methods
[6, 7, 8] which, starting from a decomposition of the observable into Pauli
operators, group or otherwise combine these Pauli operators into sets that are
jointly measurable with no or only minimal increase in circuit depth. Second,
methods which yield a compressed and possibly approximate representation of
the original Hamiltonian in the form of a tensor contraction. For fermionic
second-quantized Hamiltonians, these are mainly density fitting [9], tensor
hypercontraction (THC) [10, 11, 12], and double factorization (DF) [13, 14]
(see [14] for a comparison). While the Pauli grouping methods are applicable
to general qubit Hamiltonians, the second class of methods typically yields
better performance when applicable [5].

These compressed representations also enable drastic resource reductions in
leading fault tolerant algorithms for the simulation of chemistry based on
linear combinations of unitaries (LCU) and qubitization [15, 12, 16, 17, 11].
Here run time is mainly a function of the so-called lambda parameter. Its
precise definition depends on the algorithm and will be discussed later, but
it can be thought of as a norm-like quantity that depends on the magnitude of
the coefficients of the representation of the Hamiltonian. THC typically
yields lower lambda parameters than existing DF schemes. The fact that some
tensors in the THC decomposition are non-square and non-unitary causes other
overheads and complications [17] which does not make THC a viable option for
typical NISQ quantum algorithms.

In contrast, explicit double factorization (X-DF) and compressed double
factorization (C-DF) [18, 19, 20, 21, 22] naturally yield a NISQ-friendly
measurement scheme that only requires a linear depth orbital/Givens rotation
circuit before the final measurements, is compatible with particle number post
selection, and also an LCU representation of the Hamiltonian suitable for
error corrected algorithms based on qubitization [23].

The X-DF measurement scheme reduces the number of distinct measurement bases
to at most $n(n+1)/2$ and drastically decreases the number of shots to reach a
target accuracy, when compared to Pauli-based schemes. The number of bases can
be further reduced by truncating the X-DF representation of the Hamiltonian,
thereby making the representation approximate. This can reduce the required
number of shots, but the error resulting from the now approximate
representation of the Hamiltonian quickly outweighs this. C-DF is designed to
overcome this issue by performing a tighter least-squares numerical tensor
fitting of the molecular Hamiltonian to truncated double-factorized form. By
lifting a rank constraint in the equation defining the X-DF Hamiltonian and
using the resulting additional freedom to improve the representation of the
molecular Hamiltonian by means of parameter optimization starting from a
truncated X-DF guess, it achieves lower approximation errors than truncated
X-DF. However, when attempting practical deployment of C-DF in the context of
quantum algorithms, one encounters an additional major barrier: the
optimization of the C-DF tensor fitting to minimize least squares error does
not consider the variance properties of the resulting representation. In
practice, this means that the variance of the resulting energy estimator can
erratically fluctuate and can be orders of magnitude higher than the variance
of the X-DF energy estimator and the approximation error of both X-DF and
C-DF.

In this work we propose the regularized compressed double factorization method
(RC-DF) to fix this. RC-DF uses the same functional form of the compressed
Hamiltonian as C-DF but it adds a regularization term to the C-DF cost
function that is used when optimizing the parameters of the compressed
representation 111When working out the implications of RC-DF for fault
tolerant quantum algorithms, we became aware that a similar erratic behavior
of the $\lambda$ parameter of THC had been observed in [11] and an L1
regularization has been proposed as a cure there.. The regularization term
stabilizes the optimization and reduces the variance of the resulting NISQ
energy estimator as well as the $\lambda$ parameter determining the resources
of fault tolerant quantum algorithms. We find that RC-DF consistently
outperforms both previous double factorization schemes in terms of variance,
approximation error, and lambda parameter and even yields lambda parameters
lower than THC.

##  Appendix A Comparison of factorization methods

We start from the well known form of the second-quantized electronic structure
Hamiltonian

$\displaystyle\hat{H}=E_{\rm c}$ $\displaystyle+\sum_{pq}(p|\hat{h}_{\rm
c}|q)\hat{E}_{pq}$
$\displaystyle+\frac{1}{2}\sum_{pqrs}(pq|rs)\left(\hat{E}_{pq}\hat{E}_{rs}-\delta_{qr}\hat{E}_{ps}\right),$
(1)

where $(p|\hat{h}_{\rm c}|q)$, $(pq|rs)$ are the symmetric one-electron
integrals, the real and 8-fold symmetric two-electron integrals, and
$\hat{E}_{pq}\coloneqq\hat{p}^{\dagger}\hat{q}+\hat{\bar{p}}^{\dagger}\hat{\bar{q}}$
is the singlet excitation operator. The exact X-DF representation of the
Hamiltonian is determined by diagonalizing the modified one-electron integrals
tensor $\mathcal{F}_{pq}$ and doubly diagonalizing the two-electron integrals
tensor as in

$\displaystyle\mathcal{F}_{pq}$
$\displaystyle\coloneqq(p|\hat{h}_{c}|q)-\frac{1}{2}\sum_{r}(pr|qr)+\sum_{r}(pq|rr)$
(2)
$\displaystyle=\sum_{k}U^{\varnothing}_{pk}\,f^{\varnothing}_{k}\,U^{\varnothing}_{qk}$
(3) $\displaystyle(pq|rs)$
$\displaystyle=\sum_{t=1}^{n_{t}}V^{t}_{pq}\,g_{t}\,V^{t}_{rs}$ (4)
$\displaystyle=\sum_{t=1}^{n_{t}}\sum_{kl=1}^{n}U^{t}_{pk}\,U^{t}_{qk}\,Z^{t}_{kl}\,U^{t}_{rl}\,U^{t}_{sl},$
(5)

where the $U^{t}_{pk}$ result from diagonalizing the
$V^{t}_{pq}=\sum_{k}^{n}U^{t}_{pk}\Lambda_{k}U^{t}_{qk}$ and consequently
$Z^{t}_{kl}=\Lambda_{k}g^{t}\Lambda_{l}$ is, for every $t$, a symmetric rank
one outer product and the $U^{t}_{pk}$ are unitary (in fact without loss of
generality special orthogonal). The second factorization is possible whenever
$(pq|rs)$ is real and 8-fold symmetric (as is always the case for non-
relativistic Coulomb repulsion integrals), as this is enough to ensure that
the $V^{t}_{pq}$ are not only orthogonal but also real and symmetric for every
$t$ (see Lemma 1 in Appendix D). With $n_{t}$ equal to the maximum number
$n(n+1)/2$ of non-zero eigenvalues of $(pq|rs)$ the Hamiltonian can then be
written exactly as

$\displaystyle\begin{aligned}
\hat{H}=&~{}\mathcal{E}-\frac{1}{2}\sum_{k}\mathcal{F}^{\varnothing}_{k}U^{\varnothing}\left(\hat{Z}_{k}+\hat{Z}_{\bar{k}}\right)(U^{\varnothing})^{\dagger}\\\
&+\frac{1}{8}\sum_{t=1}^{n_{t}}\sum_{kl=1}^{n}Z^{t}_{kl}U^{t}\left(\hat{Z}_{k}\hat{Z}_{l}-\delta_{kl}+\hat{Z}_{k}\hat{Z}_{\bar{l}}+\hat{Z}_{\bar{k}}\hat{Z}_{l}+\hat{Z}_{\bar{k}}\hat{Z}_{\bar{l}}-\delta_{\bar{k}\bar{l}}\right)(U^{t})^{\dagger},\end{aligned}$
(6)

where

$\mathcal{E}=E_{\rm c}+\sum_{p}(p|\hat{h}_{\rm
c}|p)+\frac{1}{2}\sum_{pq}(pp|qq)-\frac{1}{4}\sum_{pq}(pq|pq),$ (7)

is independent of the state, and $U^{\varnothing}$ and $U^{t}$ rotate the
orbitals for each $t$ according to $U^{t}_{pk}$.

If the sum over $t$ is ordered according to $|g_{t}|$, the Hamiltonian can be
approximated with a truncated X-DF representation with fewer terms (also
called leafs).

In (R)C-DF the rank-one constraint on the $Z^{t}_{kl}$ is lifted and they are
allowed to be arbitrary symmetric matrices. The orbital rotations $U^{t}_{pk}$
and coefficients $Z^{t}_{kl}$ are then obtained using a two-step gradient
based optimization procedure by first exponentially parametrizing the orbital
rotations $U^{t}_{pq}\coloneqq\exp(X^{t})_{pq}$ via anti-symmetric generators
$X^{t}_{pq}$ and then minimizing the squared Frobenius norm (in [16] this is
called the incoherent error)

$\frac{1}{2}\Bigg{|}\Bigg{|}\underbrace{\sum_{t=1}^{n_{t}}\sum_{kl=1}^{n}U^{t}_{pk}\,U^{t}_{qk}\,Z^{t}_{kl}\,U^{t}_{rl}\,U^{t}_{sl}-(pq|rs)}_{\Delta_{pqrs}}\Bigg{|}\Bigg{|}_{\mathcal{F}}^{2}$
(8)

of the difference between the left and right hand side of (5) for some pre-set
$n_{t}\leq n\,(n+1)/2$ starting from a truncated X-DF initial guess (for
details see [18]).

The most favorable resource estimates of quantum algorithms for error
corrected simulation of chemistry with qubitization have been obtained with
THC [17, 11]. In this context THC approximates the two-body part of the
Hamiltonian according to

$(pq|rs)\approx\sum_{kl=1}^{M}\chi^{k}_{p}\,\chi^{k}_{q}\,\zeta_{kl}\,\chi^{l}_{r}\,\chi^{l}_{s},$
(9)

with $\sum_{k}^{M}\chi^{k}_{p}\chi^{k}_{q}\leq 1$ and $M\leq n^{2}$ the THC
rank. Also here the symmetric $\zeta_{kl}$ and rectangular $\chi^{k}_{p}$ are
found by means of minimizing the Frobenius norm error. When DF is used with
qubitization [12, 16] it is usually presented and performed according to

$\displaystyle(pq|rs)$ $\displaystyle=\sum_{t=1}^{n^{2}}L^{t}_{pq}L^{t}_{rs}$
(10)
$\displaystyle=\frac{1}{2}\sum_{t=1}^{n^{2}}U^{t}\Big{(}\sum_{k}^{n}\lambda^{t}_{k}\,(\hat{Z}_{k}+\hat{Z}_{\bar{k}})\Big{)}^{2}(U^{t})^{\dagger},$
(11)

where the $L^{t}_{pq}$ can be found with Cholesky decomposition or eigen
decomposition as $L^{t}_{pq}=\sqrt{g_{t}}V^{t}_{pq}$ and the scalar
$\lambda^{t}_{k}=\sqrt{g_{t}}\Lambda_{k}$ are the eigenvectors and the $U^{t}$
the diagonalizing unitaries of the $L^{t}_{pq}$ (here $g_{t}$, $V^{t}_{pq}$,
and $\Lambda_{k}$ refer to the quantities introduced in the context of X-DF
above). For NISQ measurement schemes it makes no sense to partially discard a
leaf, but in qubitization truncation can be done on the level of setting
individual $\lambda^{t}_{k}$ equal to zero.

The (R)C-DF form of the Hamiltonian, in which $Z^{t}_{kl}$ is no longer rank
one, can be re-cast as a sum of operator squares similar to (10). By taking
the positive matrix square root $W^{t}_{kl}\coloneqq(\sqrt{Z^{t}})_{kl}$ so
that $Z^{t}_{kl}=\sum_{i}^{n}W^{t}_{ki}W^{t}_{li}$ one can write

$\displaystyle\begin{aligned}
(pq|rs)=\sum_{t}^{n_{t}}\sum_{i}^{n}&\left(\sum_{k}U^{t}_{pk}\,U^{t}_{qk}\,W^{t}_{ki}\right)\\\
&\times\left(\sum_{l}W^{t}_{il}\,U^{t}_{rl}\,U^{t}_{sl}\right).\end{aligned}$
(12)

The block encoding of the qubitization method needs the Hamiltonian in the
form of an LCU. The number of ancillary qubits and $T$ gates is then
determined by the number of terms of the LCU and a sort of normalization
factor, called the lambda parameter. The result of (truncated) X-DF, C-DF, and
RC-DF is itself an LCU with a lambda factor

$\displaystyle\lambda^{\mathrm{LCU}}_{\mathrm{DF}}$
$\displaystyle\coloneqq\sum_{k}^{n}|\mathcal{F}^{\varnothing}_{k}|+\sum_{t=1}^{n_{t}}\left(\sum_{k<l}^{n}|Z^{t}_{kl}|+\frac{1}{4}\sum_{k}^{n}|Z^{t}_{kk}|\right).$
(13)

Alternatively one can use the algorithm from [16], which achieves a
contribution from the two-body part of the Hamiltonian to lambda of
$1/4\,\sum_{t}\|(L^{t}_{pq})_{pq}\|_{1}^{2}$ for Hamiltonians of the form (10)
(where $\|\cdot\|_{1}$ is the Schatten $1$-norm) and using (12) one can thus
obtain

$\displaystyle\lambda^{\mathrm{Burg}}_{\mathrm{DF}}$
$\displaystyle\coloneqq\sum_{k}^{n}|\mathcal{F}^{\varnothing}_{k}|+\frac{1}{4}\,\sum_{t}^{n_{t}}\sum_{i}^{n}\Big{\|}(\sum_{k}^{n}U^{t}_{pk}U^{t}_{qk}W^{t}_{ki})_{pq}\Big{\|}_{1}^{2}$
(14)
$\displaystyle=\sum_{k}^{n}|\mathcal{F}^{\varnothing}_{k}|+\frac{1}{4}\,\sum_{t}^{n_{t}}\sum_{i}^{n}\left(\sum_{k}^{n}|W^{t}_{ki}|\right)^{2}.$
(15)

For THC, Lee et al. [17] have obtained a lambda of

$\displaystyle\lambda^{\mathrm{Lee}}_{\mathrm{THC}}$
$\displaystyle\coloneqq\sum_{k}^{n}|\mathcal{F}^{\varnothing}_{k}|+\frac{1}{2}\,\sum_{kl}^{M}|\chi_{kl}|.$
(16)

The precise run times of the algorithms corresponding to the different lambda
values differ and depend on factorization-specific quantities such as the THC
rank $M$ but they all scale like their respective lambda divided by the
allowable phase estimation energy error times the sum of run times of certain
circuit primitives plus logarithmic overheads. The differences between the
lambda values have turned out to outweigh the influence of other factors when
comparing algorithm run times for similar overall target accuracies [17].

##  Appendix B Regularized compressed double factorization

While C-DF allows to reduce the number of leafs needed for good accuracy from
close to $n^{2}$ for X-DF to roughly linear in $n$ while maintaining an
approximate but sufficiently accurate representation of the Hamiltonian, it
turns out that the optimization of C-DF often converges to $Z^{t}_{kl}$
tensors with very large entries. This is problematic since the variance of the
NISQ energy estimator and both lambda parameters grow with the number and
magnitude of $|Z^{t}_{kl}|$ values. To solve this issue, we propose to add to
the C-DF cost function from (8) a regularization term penalizing large
$|Z^{t}_{kl}|$. We have tested both weighted $\mathrm{L1}$ and $\mathrm{L2}$
regularizations of the form
$(\sum_{tkl}\rho_{tkl}|Z^{t}_{kl}|^{\gamma})^{2/\gamma}$ with $\gamma\in{1,2}$
but concentrate on $\mathrm{L2}$ regularization in the rest of the main text.

As in C-DF, a joint optimization of the $U^{t}_{pq}$ and $Z^{t}_{pq}$ has
unfavorable performance also with regularization, but the the two-step
optimization of C-DF proposed in [18] can be adopted to the regularized case.
Further, for large $n$, a very expensive 6-index matrix inversion can be
circumvented by carrying it out in a matrix-free manner with, e.g., a
conjugate gradient algorithm (for details see Appendix A). We have found that
this step benefits from the regularization ($\mathrm{L2}$), as it improves the
conditioning of the matrix. In RC-DF, initialization can be done either from
X-DF truncated to the target number of C-DF leafs or one can start from the
full X-DF factorization and put a high penalty on the leafs that are to be
truncated in the end. In practice, contrary to the difficulties reported in
[11, 17] on converging THC, RC-DF seems to be rather well behaved. Convergence
may take thousands of iterations, but we had no difficulty converging RC-DF in
large active spaces to much tighter residual Frobenius norm errors (8) and
coupled cluster with singles, doubles, and perturbative triples (CCSD(T))
energy errors than those reported for THC [11] (See Appendix A for further
details of the optimization procedure).

##  Appendix C Numerical results

Unless otherwise explicitly stated all results shown in the following were
obtained with L2 regularization starting from a truncated X-DF guess and with
a uniform regularization factor for all $n_{t}$ leafs of
$\rho_{tkl}\eqqcolon\rho$.

We first investigate the advantages of RC-DF over other NISQ measurement
schemes. The performance of any such scheme is determined by both the
systematic error introduced in case the Hamiltonian is approximated and the
variance, $\mathrm{Var}$, of the estimator. We quantify the overall
performance by the mean squared error and take the square root to obtain a
quantity that has units of energy
$\sqrt{\mathrm{MSE}}\coloneqq\sqrt{\langle\hat{H}-\hat{H}^{\prime}\rangle^{2}+\mathrm{Var}}$,
where $\hat{H}^{\prime}$ is the compressed Hamiltonian obtained with the
respective flavor of double factorization and for the Pauli grouping based
schemes $\hat{H}^{\prime}=\hat{H}$.

The variance further depends on the state, the overall shot budget, and the
shot distribution. In the main text we show data for the state being the
complete active space configuration interaction (CASCI) ground state, but the
plots look very similar for representative states along a VQE optimization
trajectory. We consider two shot distribution schemes: A “uniform”
distribution which divides the total number of shots uniformly among all bases
in which measurements need to be preformed, and an “according to weights”
distribution which distributes the shots according to the L2 norm of the
coefficients of each group of jointly measurable Pauli operators. We chose the
overall shot budget so that the best method is able to achieve chemical
accuracy of $10^{-3}$ Hartree. Lastly, we refer the reader to appendix E for a
detailed comparison with the fluid fermionic method ($F^{3}$) [21]. In half of
the considered systems, RC-DF yields a better shot budget, even though,
contrary to $F^{3}$ it does not requires a guess state. To combine the
advantages of both approaches, the RC-DF Hamiltonian can be used as input for
the $F^{3}$ procedure, however we did not explore this possibility in this
study.

###  C.1 NISQ measurement of the para-benzyne ground state

As a first test case we consider the CASCI ground state in a (6e, 6o) active
space of FON-HF/cc-pVDZ (with fractional occupations determined with
temperature $0.1[\mathrm{1/Eh}]$ and Gaussian broadening within the active
space) [24, 25, 26] orbitals of para-benzyne (see Figure 1). The weighted shot
distribution yields lower variance than uniform shot distribution in all cases
by directing more shots to the more important first leafs. RC-DF beats the
second best method (X-DF) by approximately a factor of five and while the
maximum number $21=6\,(6+1)/2$ of leafs yields the lowest $\mathrm{MSE}$,
chemical accuracy is consistently achievable with RC-DF from $n_{t}=7$ on. The
$\mathrm{MSE}$ of C-DF fluctuates widely and while it by chance achieves a
good variance and approximation error for $8$ leafs, this is of limited use
for practical applications. The results are virtually unchanged over a broad
range of $\rho$ values and good values for $\rho$ can be found in a systematic
way even when running on quantum hardware (see Appendix B).

Figure 1: Performance of measurement schemes based on C-DF, X-DF, and RC-DF
(with L2 regularization $\rho=10^{-6}$) in comparison with the naive termwise
Pauli scheme as well as the tensor product basis [27] and a minimum clique
cover [6] based Pauli grouping methods as implemented in [28] for $3\times
10^{5}$ shots each in computing a single point energy in the (6e, 6o) CASCI
ground state of para-benzyne on 12 qubits. To reach a $\mathrm{MSE}$ on par
with that of RC-DF at $n_{t}=7$ would require over $1.1\times 10^{6}$ shots
with X-DF and $n_{t}=17$ and about $3.8\times 10^{6}$ shots with the best
Pauli grouping scheme. We also compare shot distribution schemes (see main
text) and find that “according to weights” improves performance.

###  C.2 NISQ measurement of the singlet-triplet gap of naphthalene

As a second test case, we investigate the singlet-triplet energy gap in a
(10e, 10o) active space consisting of the $\pi$ system of naphthalene
constructed with AVAS [29] as implemented in PySCF [30, 31]. The reference
state was computed at the HF/def2-SVP[32] level of theory. We only show data
for double factorization based measurement schemes since the Pauli grouping
based schemes become increasingly less competitive for larger systems. Since
the “according to weights” method is significantly better than uniform shot
distribution, we use it exclusively in this case. We compute the factorization
once per $n_{t}$ and then evaluate the energetically lowest singlet and
triplet energies from the same decomposition. The $\mathrm{MSE}$ of the
singlet-triplet energy gap is given by the square of the difference between
the noiseless energy gaps $\Delta$ computed with the exact and the gap
$\Delta^{\prime}$ from the compressed Hamiltonian plus the sum of the
variances of the singlet $\mathrm{Var}_{\mathrm{S}}$ and triplet
$\mathrm{Var}_{\mathrm{T}}$ energies
$\mathrm{MSE}=(\Delta-\Delta^{\prime})^{2}+\mathrm{Var}_{\mathrm{S}}+\mathrm{Var}_{\mathrm{T}}$.
We find the variances $\mathrm{Var}_{\mathrm{S}}$ and
$\mathrm{Var}_{\mathrm{T}}$ of both states to be very similar, and for
(R)C-DF, the $\mathrm{MSE}$ is dominated by the variance contributions for
$n_{t}\geq 8$ whereas for X-DF, the systematic energy error remains high until
$n_{t}\approx 25$. Surprisingly, RC-DF reaches chemical accuracy already at
$n_{t}=6$ with the assigned shot budget and is consistently more accurate than
the other two factorization methods for all $n_{t}>2$. By contrast, the
variance of C-DF singlet-triplet gap estimations is quite erratic, as
explained previously. While the accuracy in predicting the energy gap with
X-DF is well controllable, an approximately 12 times larger shot budget and
more leafs would be needed to reach chemical accuracy. This test case shows
the reliable performance of RC-DF for medium-sized active spaces and indicates
that the method could be employed for other chemical properties such as
activation energies.

Figure 2:  Square-root of the MSE for the singlet-triplet energy gap of
naphthalene on 20 qubits (i.e., (10e, 10o) active space, shown in the inset).
For each factorization method, $2\times 10^{6}$ shots were used in total for
the singlet and triplet state, distributed according to weights. A
regularization factor of $\rho=10^{-3}$ was used for RC-DF. RC-DF reaches
chemical accuracy with only $n_{t}=6$ leafs. X-DF would require at least
$n_{t}=25$ leafs and at least $12\times 10^{6}$ shots to achieve the same
accuracy as RC-DF with only $n_{t}=6$ leafs.

###  C.3 RC-DF for error corrected quantum computing

$\mathsf{n_{t}=100}$ Figure 3: Comparison of the achievable CCSD(T) error
heuristic and lambda values $\lambda^{\mathrm{Burg}}_{\mathrm{DF}}$ for the
truncated DF method based on (10), X-DF and RC-DF, as well as
$\lambda^{\mathrm{Lee}}_{\mathrm{THC}}$ for THC. The color scheme represents
the number of leafs $n_{t}$ for double factorization schemes, or the THC rank
$M$. The active space Hamiltonian of the Cpd I model of cytochrome P450 and
the data for THC and truncated DF were taken from [11]. The encircled THC data
point was used for the resource estimates there. To compare different levels
of convergence we vary the squared Frobenius norm error (8) at which we abort
the RC-DF optimization (Conv. Tol.) and use $\rho=10^{-3}$. The data is
tabulated in Appendix F.

We now turn to exploring the usefulness of RC-DF for error corrected
algorithms based on qubitization. As example we take the
(34$\alpha$+29$\beta$e, 58o) active space of the Cpd I species of cytochrome
P450 proposed in [11]. As this system is beyond the regime accessible with
CASCI, we use, as in [17, 11], the CCSD(T) energy error as a heuristic to
assess the quality of the compressed representation. Computational details can
be found in Appendix F.

The aim is then to find compressed representation of the Hamiltonian that
achieves both a low CCSD(T) error and a low lambda value. We find
$\lambda^{\mathrm{Burg}}_{\mathrm{DF}}<\lambda^{\mathrm{LCU}}_{\mathrm{DF}}$
in all cases and thus only display and compare the former. As can be seen in
Figure 3, RC-DF outperforms both previous DF methods and THC by a substantial
amount. Compared to truncated DF or X-DF we can almost cut
$\lambda^{\mathrm{Burg}}_{\mathrm{DF}}$ in half, and thereby also the run time
of the quantum computer, while at the same time achieving a CCSD(T) error
smaller than $5\times 10^{-5}$, which was not reached with THC [11]. Further
comparing with THC, we can reduce the CCSD(T) error by an order of magnitude
and the lambda value by roughly one third or, alternatively, RC-DF can achieve
a $\lambda^{\mathrm{Burg}}_{\mathrm{DF}}$ that is only 60% of
$\lambda_{\mathrm{THC}}$ at comparable CCSD(T) error. This improvement becomes
even more noteworthy when recognizing that when qubitizing the RC-DF
Hamiltonian one need not worry about the complications caused by the non-
orthogonal nature of THC that had to be worked around in [17]. Converging the
best RC-DF data point with $n_{t}=100$ took $2944$ L-BFGS iterations and
approximately $11$ hours on a single GPU and our not highly optimized JAX [33]
based implementation.

To investigate the scaling of the achievable lambda values with system size we
considered the hydrogen chain benchmark in the STO-6G basis previously
proposed in [12, 17]. Also there we find that at $n=100$ orbitals
(corresponding to $100$ hydrogen atoms) RC-DF achieves values of
$\lambda^{\mathrm{Burg}}_{\mathrm{DF}}$ that are lower than previously
reported values for $\lambda_{\mathrm{THC}}$ and we find, as for THC, an
approximately linear scaling of $\lambda^{\mathrm{Burg}}_{\mathrm{DF}}$ with
$n$ if $n_{t}$ is chosen such that the CCSD(T) error per particle is constant
(see Appendix C for details).

##  Appendix D Conclusions

We proposed the regularized compressed double factorization (RC-DF) method
which, from a unified framework yields both a NISQ compatible measurement
scheme with only linear circuit overhead and can be used in conjunction with
qubitization in error corrected quantum algorithms for the simulation of
chemistry. We found that in both of these scenarios RC-DF leads to lower
quantum run times when compared to previous double factorization (DF) and
tensor hypercontraction (THC) schemes.

In the NISQ setting, this advantage is a consequence of the fact that the
regularization guides the optimization towards compressed representations of
the Hamiltonian with smaller coefficients, which reduces the variance of the
resulting energy estimator. In qubitization schemes, the smaller coefficients
reduce the norm-like lambda parameter of the Hamiltonian on which the $T$ gate
count depends in a multiplicative fashion.

Avoiding a six-index intermediate quantity during the RC-DF optimization and
adopting a two step gradient based scheme previously developed by some of us
for non-regularized compressed DF, we were able to make RC-DF scale well into
the regime where quantum computers may provide an advantage over classical
methods. More work is needed to understand the precise scaling of RC-DF with
active space and basis set size and to explore other options for
regularization.

## Data Availability

The data supporting the findings of this manuscript have been uploaded to
Zenodo with the DOI 10.5281/zenodo.7866658 [34], including instructions on how
to load the data using Python.

## Acknowledgements

We thank Gian-Luca Anselmetti, Fotios Gkritsis, and Pauline Ollitrault, Artur
Izmaylov and Seonghoon Choi for valuable discussions. RMP thanks Mario Motta
and Jeff Cohen for many discussions regarding C-DF. QC Ware Corp. acknowledges
generous funding from Covestro for the undertaking of this project. Covestro
acknowledges funding from the German Ministry for Education and Research
(BMBF) under the funding program quantum technologies as part of project HFAK
(13N15630).

Conflict of Interest: EGH and RMP own stock/options in QC Ware Corp.

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##  Appendix A RC-DF optimization procedure

The RC-DF cost function is

$C(X,Z)\coloneqq\frac{1}{2}\|\Delta_{pqrs}\|_{\mathcal{F}}^{2}+\sum_{tkl}\rho_{tkl}|Z^{t}_{k}l|^{\gamma},$
(17)

with regularization tensor $\rho_{tkl}$ and $\gamma=1$ in the $\mathrm{L1}$
case and $\gamma=2$ in the $\mathrm{L2}$ case. Just like C-DF as outlined in
[18], also for RC-DF it is advisable to use a nested two-step optimization
process to minimize $C$ with respect to $X$ and $Z$ following these steps:

  1. 1.

Update the $X^{t}_{pq}$ for fixed $Z^{t}_{kl}$. This can be done with a
gradient based optimizer using the gradient

$\frac{\partial C}{\partial
X^{t}_{pq}}=\frac{\partial\|\Delta_{pqrs}\|^{2}_{\mathcal{F}}}{\partial
X^{t}_{pq}}=\sum_{mn}\frac{\partial C}{\partial U^{t}_{mn}}\frac{\partial
U^{t}_{mn}}{\partial X^{t}_{pq}}.$ (18)

  2. 2.

Determine the optimal $Z^{t}_{kl}$ given the updated $X^{t}_{pq}$ by solving

$\frac{\partial C}{\partial Z^{t}_{kl}}=0.$ (19)

The gradient with respect to the orbital rotations’ generators $X^{t}_{pq}$
for a given $Z^{t}_{pq}$ is independent of the regularization and stays
unchanged compared to the original C-DF as the $U^{t}_{pq}$ do not appear in
the regularization contribution to the cost function. Therefore

$\frac{\partial C}{\partial
U^{t}_{mn}}=-4\sum_{qrsl}\Delta_{mqrs}U^{t}_{qn}Z^{t}_{nl}U^{t}_{rl}U^{t}_{sl}.$
(20)

Let us now look at the second step. In the $\mathrm{L2}$ case (19) yields

$\frac{\partial C}{\partial
Z^{t}_{kl}}=-\sum_{pqrs}\Delta_{pqrs}U^{t}_{pk}U^{t}_{qk}U^{t}_{rl}U^{t}_{sl}+\lambda^{t}_{kl}Z^{t}_{kl}=0,$
(21)

which implies

$\displaystyle\begin{aligned}
&\sum_{pqrs}(pq|rs)U^{t}_{pk}U^{t}_{qk}U^{t}_{rl}U^{t}_{sl}=\lambda_{kl}^{t}Z^{t}_{kl}+\sum_{omn}\left[\sum_{p}U^{t}_{pk}U^{o}_{pm}\right]\left[\sum_{q}U^{t}_{qk}U^{o}_{qm}\right]Z^{o}_{mn}\left[\sum_{r}U^{t}_{rl}U^{o}_{rn}\right]\left[\sum_{s}U^{t}_{sl}U^{o}_{sn}\right],\end{aligned}$
(22)

which is of the form

$\displaystyle b^{t}_{kl}((pq|rs),U^{t}_{pq})$
$\displaystyle=\sum_{omn}A^{tkl}_{omn}(U^{t}_{pq},\lambda^{t}_{kl})Z^{o}_{mn}$
(23)

for

$\displaystyle
b((pq|rs),U^{t}_{pq})_{tkl}=\sum_{pqrs}(pq|rs)U^{t}_{pk}U^{t}_{qk}U^{t}_{rl}U^{t}_{sl}$
(24) $\displaystyle\begin{aligned}
A(U^{t}_{pq},&\lambda^{t}_{kl})_{tkl,omn}=\sum_{omn}\left(\left[\sum_{p}U^{t}_{pk}U^{o}_{pm}\right]\left[\sum_{q}U^{t}_{qk}U^{o}_{qm}\right]\left[\sum_{r}U^{t}_{rl}U^{o}_{rn}\right]\left[\sum_{s}U^{t}_{sl}U^{o}_{sn}\right]+\delta_{omn,tkl}\lambda_{tkl}\right).\end{aligned}$
(25)

In the $\mathrm{L1}$ case, (19) yields

$\frac{\partial C}{\partial
Z^{t}_{kl}}=-\sum_{pqrs}\Delta_{pqrs}U^{t}_{pk}U^{t}_{qk}U^{t}_{rl}U^{t}_{sl}+\lambda^{t}_{kl}\mathrm{sign}(Z^{t}_{kl})=0,$
(26)

which implies

$\displaystyle\begin{aligned}
&\sum_{pqrs}(pq|rs)U^{t}_{pk}U^{t}_{qk}U^{t}_{rl}U^{t}_{sl}-\lambda^{t}_{kl}\mathrm{sign}(Z^{t}_{kl})=\sum_{omn}\left[\sum_{p}U^{t}_{pk}U^{o}_{pm}\right]\left[\sum_{q}U^{t}_{qk}U^{o}_{qm}\right]Z^{o}_{mn}\left[\sum_{r}U^{t}_{rl}U^{o}_{rn}\right]\left[\sum_{s}U^{t}_{sl}U^{o}_{sn}\right]\end{aligned}$
(27)

which is of the form

$\displaystyle
b^{t}_{kl}\left((pq|rs),U^{t}_{pq},\lambda^{t}_{kl},Z^{t}_{kl}\right)$
$\displaystyle=\sum_{omn}A^{tkl}_{omn}(U^{t}_{pq})Z^{o}_{mn}.$ (28)

for

$\displaystyle\begin{aligned}
&b^{t}_{kl}((pq|rs),U^{t}_{pq},\lambda^{t}_{kl},Z^{t}_{kl})=\sum_{pqrs}(pq|rs)U^{t}_{pk}U^{t}_{qk}U^{t}_{rl}U^{t}_{sl}-\lambda^{t}_{kl}\mathrm{sign}(Z^{t}_{kl})\end{aligned}$
(29) $\displaystyle\begin{aligned}
&A(U^{t}_{pq})_{tkl,omn}=\sum_{omn}\left[\sum_{p}U^{t}_{pk}U^{o}_{pm}\right]\left[\sum_{q}U^{t}_{qk}U^{o}_{qm}\right]\left[\sum_{r}U^{t}_{rl}U^{o}_{rn}\right]\left[\sum_{s}U^{t}_{sl}U^{o}_{sn}\right].\end{aligned}$
(30)

So in both cases one can find the optimal $Z^{t}_{kl}$ by pseudo-inverting
$A_{tkl,omn}$. While pseudo-inverting the six-index tensor $A_{tkl,omn}$ to
determine the $Z^{t}_{kl}$ can be done for medium size systems, it is
intractable for large systems. To circumvent this problem, the inversion can
be carried out in a matrix-free manner with, e.g., a conjugate gradient
algorithm. This procedure only requires the matrix-vector product and the
matrix diagonal rather than the dense matrix. Lastly, any gradient-descent
based optimization algorithm can be used. For the result presented in this
work, we have used the L-BFGS [35] algorithm as implemented in SciPy [36]. All
time critical routines for the steps above were just-in-time compiled and ran
on an NVIDIA Tesla V100 with the help of JAX [33].

##  Appendix B Tuning of the regularization

The regularization tensor $\rho_{tkl}$ is a hyper-parameter that controls how
much effort is put on achieving small $|Z^{t}_{pq}|$ versus reducing the
Frobenius norm error. A good balance must be found to obtain both a small
systematic energy error and a small variance and lambda value. We concentrate
only the case that after truncating some number of $n_{t}$ leafs the
regularization is chosen uniformly $\rho_{tkl}\eqqcolon\rho$.

In the main text we have quantified the overall performance of the measurement
scheme by means of the $\mathrm{MSE}$, which combines the systematic error in
the energy because of the approximation of the Hamiltonian and the statistical
error due to variance and shot noise. Here let us look at those two
contributions separately. We consider the same data underlying Figure 1 from
the main text.

In Figure 4 we show only the systematic error that results from RC-DF
approximating the two body part of the Hamiltonian and represents what is
achievable in the limit of infinitely many shots. The energy approximation
error grows roughly linearly with $\rho/10$ and when the number of leafs is
increased it seems to become easier for the optimizer to find $Z^{t}_{kl}$
that are not too much distorted by the regularization and thus represent the
$(pq|rs)$ tensor well leading to a smaller energy error.

In Figure 5 we show the standard deviation $\sqrt{\mathrm{Var}}$ of the ground
state energy estimator, quantifying how far from the infinite shot budget
limit a measured energy value is likely to lie. Larger regularization factors
$\rho$ manifestly reduce the standard deviation. If the shot distribution
takes into account the weight of the leafs this effect is greatly enhanced.
Very strong regularization does not seem to help. Increasing or decreasing the
total shot budget would simply move the data point up and down according to
the well known one over square root raw. As can be nicely seen there is a
rather large window between $\rho\in[10^{-6},10^{-2}]$ where both errors (and
thus their sum) is well below $10^{-}3$. In practice one find a good $\rho$ by
estimating the standard deviation on a quantum device starting from a
comparably large regularization and then reduce the regularization to reduce
the systematically error until the variance becomes too large to be
compensated for by the affordable shot budget.

Figure 4: RC-DF systematic ground state energy approximation error of para-
benzyne as a function of the regularization factor $\rho$ with different
numbers of leafs $n_{t}$. Figure 5: Standard deviation of the ground state
energy estimator of para-benzyne for a budget of 300.000 shots as a function
of $\rho$ for different numbers of leafs $n_{t}$.

##  Appendix C Lambda scaling with system size

To investigate the scaling of the lambda parameters achievable with double
factorization in the large system size limit we use the hydrogen chain
benchmark previously employed in the context of qubitization combined with low
rank factorization in [12] and with THC in [17]. In this benchmark system
neutral hydrogen atoms are placed with 1.4 Bohr distance on a line and
represented in the STO-6G basis, which has one spatial orbital per hydrogen so
that the size of the complete active space $n$ is equal to the number of
hydrogen atoms.

The lambda factor of the qubitization scheme from [12] is the sum of a one-
body contribution $\lambda_{T}$ and a two-body contribution $\lambda_{W}$ and
is strongly dominated by the latter. Between $n=10$ and $n=100$ the authors
find that $\lambda_{W}$ grows roughly proportional to $n^{2.5}$ to values up
to about $2\times 10^{5}$ (see Fig 11(a) in [12]).

In [17] a THC based qubitization scheme is proposed that improves over what
the authors call the “naïve” approach. In Fig 20 of that work the two-body
contribution $\lambda_{2}$ of these two schemes is compared, again for $10\leq
n\leq 100$. The “naïve” $\lambda_{2}$ is found to grow approximately
proportional to $n^{3.16}$ to values well beyond $5\times 10^{6}$. The
$\lambda_{2}$ achievable with the non-orthogonal basis THC based qubitization
proposed in [17] (which corresponds to the two-body part of
$\lambda^{\mathrm{Lee}}_{\mathrm{THC}}$) is found to only grow approximately
proportionally to $n^{1.16}$ and reaches approximately $4\times 10^{2}$ at
$n=100$. Fig 9 of the same work compares the full lambda values (including the
one-body contribution) for different factorization methods including THC (the
main focus of that work and the most competitve of the factorization methods
in this plot) and the Cholesky based DF described around (10). The parameters
of these methods are chosen to yield a CCSD(T) correlation energy error per
atom of at most 50 micro Hartree, amounting to $0.5\times 10^{-3}$ Hartree at
$n=100$. The authors find a scaling of roughly $n^{1.88}$ for DF and of
$n^{1.11}$ for THC and at $n=100$ values of $\lambda>2\times 10^{3}$ for DF
and $\lambda\approx 5\times 10^{2}$ for THC.

With a regularization of $\rho=5\times 10^{-5}$ and even just a linear number
of leafs $n_{t}=\lfloor(n+1)/2\rfloor$ (red line in Fig. 6) we find that RC-DF
(with two norm regularization $\gamma=2$) is able to yield a constant
Frobenius norm error and an absolute CCSD(T) error per atom
$|\Delta_{\mathrm{CCSD(T)}}|/n$ in line with the 50 micro Hartree per atom
used in Fig 9 of [17] (see the green line in Fig. 6(b)). At these parameters
$\lambda^{\mathrm{Burg}}_{\mathrm{RC-DF}}$ is found to scale approximately
like $n^{1.08\pm 0.10}$ (fit through the values for $70\leq n\leq 100$) and
reaches approximately $2.5\times 10^{2}$ at $n=100$, roughly a factor of two
better than the THC results from [17]. Compared to X-DF (which requires
$n_{t}$ to grow faster than linearly to obtain acceptable accuracy (see the
blue and yellow lines in Fig. 6(b)), RC-DF yields roughly one order of
magnitude lower lambda values at $n=100$, mostly owing to the fact that we
find that the X-DF lambda values scale roughly quadratic with $n$.
$\lambda^{\mathrm{Burg}}$ and $\lambda^{\mathrm{LCU}}$ seem to have a similar
scaling for all double factorization schemes considered here.

(a)

(b)

(c)

(d)

Figure 6: Comparison of lambda values achievable with X-DF and two-norm
($\gamma=2$) RC-DF with regularization $\rho=5\times 10^{-5}$ and convergence
tolerance $||\Delta_{pqrs}||_{\mathcal{F}}\leq 10^{-5}$ for the hydrogen chain
benchmark in the STO-6G basis previously use in [12, 17] (panel 6(a)). The
black triangles are the lambda values eyeballed from Fig. 9 of [17]. The thin
lines are power law fits to the date for $70\leq n\leq 100$ used to estimate
the exponents stated in the text and displayed in the figure. Two different
scalings of $n_{t}$ with $n$ were chosen such that the CCSD(T) error per
particle $|\Delta_{\mathrm{CCSD(T)}}|/n$ (panel 6(b)) is approximately below
50 micro Hartree per atom for all cases execpt X-DF with linear number of
leafs. The Frobenius norm error of the two-body tensor
$\|\Delta_{pqrs}\|_{\mathcal{F}}$ (panel 6(c)) seems to be well correlated
with $|\Delta_{\mathrm{CCSD(T)}}|$ for X-DF (panel 6(d)) but their magnitues
can differ by several orders for other methods, so care needs to be taken when
drawing conlusions base on $\|\Delta_{pqrs}\|_{\mathcal{F}}$ (The RC-DF data
points are on a vertical line because $\|\Delta_{pqrs}\|_{\mathcal{F}}\leq
10^{-5}$ was used as an abort criterion for the RC-DF optimization).

##  Appendix D Necessary and sufficient condition for the symmetry of the
$V^{t}_{pq}$

In this section we present a proof that shows that 8-fold symmetry of the
$(pq|rs)$ tensor is a sufficient condition for the $V^{t}_{pq}$ to be
symmetric for every $t$ where the eigenvalue $g_{t}\neq 0$. Hence, the
$Z^{t}_{pq}$ are real, which is required for the X-DF procedure to work, and
the $U^{t}_{pq}$ are orthogonal (and thud can be chosen to be special
orthogonal without loss of generality), which is essential for their
implementation on a quantum computer by means of a fabric of givens rotations.
Slightly abusing notation, in the following lemma, we use $(pq|rs)$ for a four
index tensor that does not necessarily arise from electron overlap integrals,
but has the stated properties.

######  Lemma 1.

Let $(pq|rs)$ be any real, symmetric, i.e., $(pq|rs)=(rs|pq)$ tensor of shape
$n\times n\times n\times n$. By grouping the indices $pq$ and $rs$, let
$(g_{t})_{t}$ be its $n^{2}$ eigenvalues, ${V^{t}_{pq}}$ its diagonalizing
unitary and let $T_{+}=\left\\{t:g_{t}>0\right\\}$,
$T_{-}=\left\\{t:g_{t}<0\right\\}$, and $T\coloneqq T_{+}\cup T_{-}$. Then
$(pq|rs)$ can be written in the form

$(pq|rs)=\sum_{t\in T}V^{t}_{pq}\,g^{t}\,V_{rs}^{t}.$ (31)

Further, if and only if $(pq|rs)$ is in addition 8-fold symmetric, i.e.,
$(qp|rs)=(qp|sr)=(pq|sr)=(pq|rs)=(rs|pq)=(sr|pq)=(sr|qp)=(rs|qp)$ the matrices
$(V^{t}_{pq})_{pq}$ are symmetric, i.e., $V^{t}_{pq}=V^{t}_{qp}\ \forall t\in
T$, and $|T|\leq n(n+1)/2$.

###### Proof.

That symmetry of the $V^{t}_{pq}$ implies 8-fold symmetry of $(qp|rs)$ can be
verified directly from (31) and whenever the $V^{t}_{pq}$ are symmetric, since
they are also orthogonal for different values of $t$, there can be at most
$n\,(n+1)/2$ of them and thus the bond on the size $|T|$ of $T$ holds.

To show that 8-fold symmetry implies symmetry of all $V^{t}_{pq}$ with $t\in
T$ let us first define

$\displaystyle(pq|rs)_{\pm}$ $\displaystyle\coloneqq\sum_{t\in
T_{\pm}}V^{t}_{pq}\,g^{t}\,V_{rs}^{t}$ (32) $\displaystyle|(pq|rs)|$
$\displaystyle\coloneqq\sum_{t\in T}V^{t}_{pq}\,|g^{t}|\,V_{rs}^{t}.$ (33)

so that we have $(pq|rs)\pm|(pq|rs)|=2\,(pq|rs)_{\pm}$. Since we also have
$|(pq|rs)|=(pq|rs)\,R$ with

$\displaystyle R$ $\displaystyle\coloneqq\sum_{t\in
T}V^{t}_{pq}\,\mathrm{sign}(g^{t})\,V_{rs}^{t}.$ (34)

symmetric and invertible, $8$-fold symmetry of $(pq|rs)$ implies $8$-fold
symmetry of $|(pq|rs)|$ and thereby $8$-fold symmetry of both $(pq|rs)_{+}$
and $(pq|rs)_{-}$ individually. The symmetry $(pq|rs)_{\pm}=(qp|rs)_{\pm}$
then implies that

$\sum_{t\in
T_{\pm}}(V^{t}_{pq}V^{t}_{rs}-V^{t}_{qp}V^{t}_{rs})\,g^{t}=0\quad\forall
p,q,r,s.$ (35)

For $p=r$ and $q=s$ we can specialized this to

$\sum_{t\in T_{\pm}}g^{t}V^{t}_{pq}V^{t}_{qp}=\sum_{t\in
T_{\pm}}g^{t}(V^{t}_{pq})^{2}\quad\forall p,q.$ (36)

The symmetry $(pq|pq)_{\pm}=(qp|qp)_{\pm}$ together with (31) implies

$\sum_{t\in T_{\pm}}g_{t}(V^{t}_{pq})^{2}=\sum_{t\in
T_{\pm}}g_{t}(V^{t}_{qp})^{2}$ (37)

and thus the left and right hand side of (36) can be identified to be the left
and right hand side of the Cauchy-Schwarz inequalities of the inner product of
the vectors $(\sqrt{g_{t}}^{*}V^{t}_{pq})_{t\in T_{\pm}}$ and
$(\sqrt{g_{t}}V^{t}_{qp})_{t\in T_{\pm}}$:

$\displaystyle\Big{(}\sum_{t\in T_{\pm}}(\sqrt{g_{t}}^{*}V^{t}_{pq})^{*}$
$\displaystyle\sqrt{g_{t}}V^{t}_{qp}\Big{)}^{2}=$ (38)
$\displaystyle\Big{(}\sum_{t\in
T_{\pm}}g_{t}\,V^{t}_{pq}V^{t}_{qp}\Big{)}^{2}$ $\displaystyle\leq\sum_{t\in
T_{\pm}}|g_{t}|(V^{t}_{pq})^{2}\sum_{t\in T_{\pm}}|g_{t}|(V^{t}_{qp})^{2}$
(39) $\displaystyle=\sum_{t\in T_{\pm}}\pm|g_{t}|(V^{t}_{pq})^{2}\sum_{t\in
T_{\pm}}\pm|g_{t}|(V^{t}_{qp})^{2}$ (40) $\displaystyle=\sum_{t\in
T_{\pm}}g_{t}\,(V^{t}_{pq})^{2}\sum_{t\in T_{\pm}}g_{t}\,(V^{t}_{qp})^{2}$
(41) $\displaystyle=\Big{(}\sum_{t\in
T_{\pm}}g_{t}\,(V^{t}_{qp})^{2}\Big{)}^{2}\quad\forall p,q,$ (42)

where in the last step we have used (37). Thus (36) implies that these Cauchy-
Schwarz inequalities are indeed fulfilled with equality, which is the case if
and only if the two vectors in the Cauchy-Schwarz inequality are identical up
to a prefactor, i.e., for each $p,q$ there must exists a scalar $\alpha$ such
that

$\displaystyle\sqrt{g_{t}}^{*}V^{t}_{pq}=\alpha\,\sqrt{g_{t}}V^{t}_{qp}\quad\forall
t\in T_{\pm}$ (43) $\displaystyle\Leftrightarrow
V^{t}_{pq}=\pm\alpha\,V^{t}_{qp}\quad\forall t\in T_{\pm}~{}.$ (44)

Since $(pq|rs)=(qp|rs)$, we further have

$\displaystyle(pq|rs)$ $\displaystyle=\sum_{t\in
T_{\pm}}V^{t}_{pq}g^{t}V^{t}_{rs}=\pm\alpha\sum_{t\in
T_{p}m}V^{t}_{qp}g^{t}V^{t}_{rs}$ (45) $\displaystyle=\sum_{t\in
T\pm}V^{t}_{qp}g^{t}V^{t}_{rs}=(qp|rs),$ (46)

which is possible only if $\pm\alpha=1$. Thus we have as claimed
$V^{t}_{pq}=V^{t}_{qp}\ \forall t\in T$.

∎

##  Appendix E Comparison of RC-DF and $F^{3}$

In this section, We compare the shot budget required by RC-DF to that of the
Fluid Fermionic Fragments ($F^{3}$) method reported in [21]. We see in Table 1
that in half of the cases RC-DF performs better whereas in the other the
$F^{3}$ produce better shot budgets. Note that in all the studied cases, the
RC-DF systematic energy error is brought down below the $10^{-5}$ a.u.
threshold and the regularization factor is fixed to $10^{-5}$. Lastly, the
shot distribution strategy used with RC-DF is a weighted distribution
according the variances of the terms.

Table 1: Comparion of the required shot budget (in millions) to achieve an accuracy of $10^{-3}$ a.u. in the ground state energy between RC-DF and $F^{3}$ [21]. $n_{s}$ refers to the number of spatial orbitals, $F^{3}$ the best score reported in [21]. RC-DF refers to the shot budget of RC-DF and $n_{t}$ is the corresponding number of leafs. The shot budget, with RC-DF, is distributed according to the variances. molecule |  $n_{s}$ |  $F^{3}$ |  RC-DF |  $n_{t}$ |  $n_{s}/n_{electrons}$
---|---|---|---|---|---
NH3 |  8 |  0.990 |  5.550 |  28 |  8/10
CH2 |  7 |  0.689 |  1.500 |  22 |  7/8
BeH2 |  7 |  0.341 |  0.249 |  22 |  7/6
H2O |  7 |  0.709 |  6.427 |  24 |  7/10
LiH |  6 |  0.122 |  0.067 |  16 |  6/4
HF |  6 |  0.278 |  5.300 |  16 |  6/10
H6 |  6 |  0.519 |  0.252 |  16 |  6/6
H4 |  4 |  0.316 |  0.121 |  10 |  4/4

##  Appendix F Computational methodology for Cpd I test case

The active space integrals for Cpd I were obtained from the deposited data
[37] of Ref. [11]. The system under consideration here is labelled “X” in Ref.
[11], and consists of a (34$\alpha$+29$\beta$, 58o) active space. We
factorized the two-body integrals using RC-DF with
$n_{t}\in\\{30,40,50,60,70,80,90,100,150,200\\}$ and additional
$n_{t}=400,~{}800$ with X-DF. For RC-DF, we employed $\rho=10^{-3}$ and varied
the convergence tolerance from $10^{-1}$ to $10^{-4}$. From the factorized
two-body integrals, a CCSD(T) energy was computed as described in Ref. [11]
using the chemftr Python library (https://github.com/ncrubin/chemftr)
interfaced with PySCF [30, 31]. The CCSD(T) energy error is the energy
difference of the CCSD(T) energies with exact two-body integrals and the two-
body integrals reconstructed from the factorization. For both factorization
schemes, we evaluated $\lambda_{\mathrm{DF}}^{\mathrm{Burg}}$ and
$\lambda_{\mathrm{DF}}^{\mathrm{LCU}}$. Since the data for the truncated DF
scheme in Ref. [11] are not shown in the paper, we recomputed the
factorization using chemftr and checked that for exact factorizations, the
lambda parameters agree exactly with our X-DF implementation. The RC-DF and
X-DF results are shown in Tables 2 and 3, respectively. In addition, Table 4
contains the recomputed results for truncated DF from Ref. [11]. The
factorized Hamiltonians, together with the resulting energy errors and lambda
factors were deposited on Zenodo [34].

Table 2: RC-DF performance summary for Cpd I Conv. Tol. | $n_{t}$ |  $\|\Delta_{pqrs}\|_{\mathcal{F}}$ | CCSD(T) error [mEh] |  $\lambda_{\mathrm{DF}}^{\mathrm{Burg}}$ |  $\lambda_{\mathrm{DF}}^{\mathrm{LCU}}$
---|---|---|---|---|---
$10^{-1}$ | 30 |  0.4472 | $-23.5907$ |  300.6 |  456.5
| 40 |  0.4412 | $-22.6134$ |  311.0 |  468.6
| 50 |  0.4382 | $-22.4729$ |  320.6 |  478.0
| 60 |  0.4415 | $-22.7342$ |  327.5 |  478.1
| 70 |  0.4452 | $-21.4039$ |  336.3 |  483.7
| 80 |  0.4352 | $-18.2827$ |  343.0 |  490.5
| 90 |  0.4310 | $-17.3526$ |  349.9 |  496.1
| 100 |  0.4278 | $-17.2627$ |  355.8 |  498.1
| 150 |  0.4232 | $-14.8979$ |  385.9 |  521.0
| 200 |  0.4258 | $-15.0266$ |  417.9 |  545.2
$10^{-2}$ | 30 |  0.1411 | $-3.0769$ |  278.2 |  430.4
| 40 |  0.1411 | $-3.6537$ |  290.8 |  448.5
| 50 |  0.1408 | $-2.9621$ |  299.5 |  462.7
| 60 |  0.1414 | $-3.2143$ |  304.6 |  470.0
| 70 |  0.1406 | $-2.8366$ |  308.6 |  475.4
| 80 |  0.1407 | $-3.6169$ |  313.8 |  482.2
| 90 |  0.1407 | $-2.6580$ |  318.7 |  489.2
| 100 |  0.1409 | $-2.1238$ |  321.7 |  492.1
| 150 |  0.1410 | $-2.6875$ |  337.2 |  508.8
| 200 |  0.1413 | $-2.5034$ |  351.2 |  524.2
$10^{-3}$ | 30 |  0.0447 | $-0.2292$ |  244.4 |  386.6
| 40 |  0.0447 | $-0.3581$ |  257.1 |  394.1
| 50 |  0.0447 | $-0.2405$ |  268.0 |  409.6
| 60 |  0.0447 | $-0.2515$ |  276.5 |  422.8
| 70 |  0.0447 | $-0.5585$ |  284.0 |  434.3
| 80 |  0.0447 | $-0.1995$ |  288.6 |  442.6
| 90 |  0.0447 | $-0.3395$ |  292.2 |  450.4
| 100 |  0.0447 | $-0.0137$ |  295.8 |  456.6
| 150 |  0.0446 | $-0.4667$ |  310.3 |  481.2
| 200 |  0.0446 | $-0.2690$ |  321.1 |  499.3
$10^{-4}$ | 100 |  0.0141 | $0.0086$ |  258.0 |  390.8
| 150 |  0.0141 | $-0.0423$ |  275.7 |  414.4
| 200 |  0.0141 | $-0.0340$ |  287.9 |  435.3
Table 3: X-DF performance summary for Cpd I $n_{t}$ |  $\|\Delta_{pqrs}\|_{\mathcal{F}}$ | CCSD(T) error [mEh] |  $\lambda_{\mathrm{DF}}^{\mathrm{Burg}}$ |  $\lambda_{\mathrm{DF}}^{\mathrm{LCU}}$
---|---|---|---|---
30 |  1.01455 | $15.5688$ |  418.5 |  759.2
40 |  0.72606 | $-7.3444$ |  431.6 |  784.9
50 |  0.58478 | $-8.5971$ |  440.3 |  802.0
60 |  0.47298 | $-14.4322$ |  447.9 |  816.8
70 |  0.38068 | $-3.1890$ |  453.7 |  828.2
80 |  0.29495 | $-9.6247$ |  458.8 |  838.3
90 |  0.21062 | $0.5473$ |  462.2 |  844.9
100 |  0.15487 | $-3.9773$ |  464.3 |  848.9
150 |  0.04288 | $-0.0570$ |  469.8 |  859.8
200 |  0.01609 | $0.1208$ |  471.7 |  863.4
400 |  0.00126 | $0.0092$ |  473.0 |  866.0
800 |  0.00001 | $0.0003$ |  473.2 |  866.3
Table 4: Truncated DF performance summary for Cpd Ia)

$n_{t}$b) |  thresholdc) |  $\|\Delta_{pqrs}\|_{\mathcal{F}}$ | CCSD(T) error [mEh] |  $\lambda_{\mathrm{DF}}^{\mathrm{Burg}}$
---|---|---|---|---
99 |  0.07500 |  0.4909 | $-73.8021$ |  428.8
114 |  0.05000 |  0.3631 | $-17.4890$ |  439.9
131 |  0.02500 |  0.1892 | $1.1578$ |  455.1
178 |  0.01000 |  0.0836 | $2.6993$ |  464.3
194 |  0.00750 |  0.0644 | $4.1638$ |  466.1
207 |  0.00500 |  0.0457 | $1.8040$ |  467.9
241 |  0.00250 |  0.0245 | $-0.1019$ |  470.1
309 |  0.00100 |  0.0106 | $0.0266$ |  471.7
364 |  0.00050 |  0.0054 | $0.0155$ |  472.3
505 |  0.00010 |  0.0012 | $-0.0086$ |  473.0
568 |  0.00005 |  0.0006 | $0.0037$ |  473.1
702 |  0.00001 |  0.0001 | $0.0007$ |  473.1

a) Recomputed with chemftr.
b) Referred to as $L$ in Refs. [12, 17].
c) Eigenvector screening threshold with which the accuracy of the
factorization is tuned, see Ref. [12].