# Quantum Simulation of Lattice QCD with Improved Hamiltonians

Anthony N. Ciavarella  0000-0003-3918-4110<EMAIL_ADDRESS>InQubator for
Quantum Simulation (IQuS), Department of Physics, University of Washington,
Seattle, Washington 98195-1550, USA

###### Abstract

Quantum simulations of lattice gauge theories are anticipated to directly
probe the real time dynamics of QCD, but scale unfavorably with the required
truncation of the gauge fields. Improved Hamiltonians are derived to correct
for the effects of gauge field truncations on the SU(3) Kogut-Susskind
Hamiltonian. It is shown in $1+1D$ that this enables low chromo-electric field
truncations to quantitatively reproduce features of the untruncated theory
over a range of couplings and quark masses. In $3+1D$, an improved Hamiltonian
is derived for lattice QCD with staggered massless fermions. It is shown in
the strong coupling limit that the spectrum qualitatively reproduces aspects
of two flavor QCD and simulations of a small system are performed on IBM’s
Perth quantum processor.

††preprint: IQuS@UW-21-056

##  I Introduction

The real time dynamics of quantum chromodynamics (QCD) are of relevance to a
number of phenomena in particle and nuclear physics. These range from
collisions of hadrons at high energies to the behavior of quark-gluon plasma
in the early universe. The simulation of QCD discretized onto a lattice has
enabled non-perturbative calculations of static observables in QCD such as
hadron masses and form factors [1, 2, 3, 4, 5, 6].

Quantum computers are expected to be able to directly probe the real time
dynamics of quantum field theories. The recent developments in quantum
hardware have inspired studies into how to implement simulations of lattice
gauge gauge theories on quantum computers. The first quantum simulations of
pure non-Abelian lattice gauge theories have been performed in low dimensions
on quantum hardware [7, 8, 9, 10, 11, 12, 13, 14]. There have also been
quantum simulations of non-Abelian gauge theories coupled to matter in one
spatial dimension [15, 16, 17, 18]. Theoretical studies have been performed
into how to scale up these calculations to larger systems [19, 20, 21, 22, 23,
24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39] and large
scale simulations have been performed of Abelian gauge theories [40, 41, 42,
43, 44]. However, all these approaches to simulating gauge theories require
the gauge field to be truncated and scale poorly with the gauge field
truncation. Similar problems were found in the classical simulation of lattice
gauge theories with the scaling of errors with lattice spacing. These problems
were mitigated through the development of improved Symanzik actions with more
favorable scaling of errors with lattice spacing [45]. Progress has been made
towards deriving improved Hamiltonians that reduce errors from finite lattice
spacing [46]. It is expected that improved Hamiltonians can be found that
mitigate the effects of truncating the gauge field as well.

In this work, improved Hamiltonians are derived for lattice gauge theories
through the application of the similarity renormalization group (SRG). SU(3)
gauge fields coupled to fermions in 1+1D are used as a case study for the
improved Hamiltonians. Tensor network simulations are used to demonstrate that
the improved Hamiltonians derived in 1+1D correctly reproduce observables on
large lattices. An improved Hamiltonian for lattice QCD with two flavors is
derived for 3+1D and a small simulation is performed on IBM’s quantum
processors.

##  II 1+1D

###  II.1 1+1D Hamiltonian

Gauge theories in one spatial dimension have been used as toy models to study
the quantum simulation of gauge theories in higher dimensions as they share
many qualitative features and their reduced complexity makes simulation more
tractable. Previous simulations on quantum hardware have studied the dynamics
of hadrons in one spatial dimension [15, 16, 18] and $\beta$ decay [17].

In this work, the SU(3) Kogut Susskind Hamiltonian [47] with a single flavor
of staggered fermions in $1+1D$ will be used as a toy model to study the
effects of gauge field truncation and the performance of improved
Hamiltonians. The Hamiltonian describing this theory is

$\displaystyle\hat{H}=\hat{H}_{Kin}+\hat{H}_{m}+\hat{H}_{E}$
$\displaystyle\hat{H}_{Kin}=\sum_{x,a,b}\frac{1}{2}\hat{\psi}_{x,a}^{\dagger}\hat{U}^{a,b}_{x,x+1}\hat{\psi}_{x+1,b}+\text{h.c.}$
$\displaystyle\hat{H}_{m}=m\sum_{x,a}(-1)^{x}\hat{\psi}_{x,a}^{\dagger}\hat{\psi}_{x,a}$
$\displaystyle\hat{H}_{E}=\sum_{x,c}\frac{g^{2}}{2}\hat{E}_{x,x+1}^{c}\hat{E}_{x,x+1}^{c}\
\ \ ,$ (1)

where $g$ is the gauge coupling, $m$ is the fermion mass, $\hat{\psi}_{x,a}$
is the fermion field at site $x$ with color $a$, $\hat{U}^{a,b}_{x,x+1}$ is
the parallel transporter on the link between the sites $x,x+1$ and
$\hat{E}_{x,x+1}^{c}$ is the chromo-electric field operator. By working with
open boundary conditions in the axial gauge, and enforcing Gauss’s law, the
gauge fields in this theory can be completely integrated out yielding the
Hamiltonian

$\displaystyle\hat{H}=\hat{H}_{Kin}+\hat{H}_{m}+\hat{H}_{E}$
$\displaystyle\hat{H}_{Kin}=\sum_{x,a}\frac{1}{2}\hat{\psi}_{x,a}^{\dagger}\hat{\psi}_{x+1,a}+\text{h.c.}$
$\displaystyle\hat{H}_{m}=m\sum_{x,a}(-1)^{x}\hat{\psi}_{x,a}^{\dagger}\hat{\psi}_{x,a}$
$\displaystyle\hat{H}_{E}=\sum_{x,c}\frac{g^{2}}{2}\left(\sum_{y<x}\hat{Q}_{y}^{c}\right)\left(\sum_{y<x}\hat{Q}_{y}^{c}\right)\
\ \ ,$ (2)

where $\hat{Q}_{y}^{c}$ is the chromo-electric charge at site $x$ defined by

$\hat{Q}_{y}^{c}=\sum_{a,b}\hat{\psi}^{\dagger}_{y,a}T^{c}_{a,b}\hat{\psi}_{y,b}\
\ \ ,$ (3)

where $T^{c}_{a,b}$ are the Gell-Mann matrices. By working with this
Hamiltonian, we can directly study the untruncated theory and the performance
of improved Hamiltonians that correct for the gauge field truncation.

###  II.2 Strong Coupling Expansion $m=0$

Before the Hamiltonian in Eq. (1) can be mapped onto a quantum computer, it
must first be truncated to a finite Hilbert space. Typically, this is done by
working in the basis of the chromo-electric field and truncating the field
below some cutoff. It has been shown numerically for some small systems [48,
10, 25, 49, 36] and rigorously proven in general [50] that the error induced
by this truncation falls off exponentially with the truncation. The error due
to gauge field truncation can be reduced even further by first performing a
unitary rotation on the Hamiltonian to reduce the coupling to the higher
electric field states and then truncating. In other words, there is a low-
energy subspace coupled to a high-energy subspace and one would like to derive
an effective field theory description of the low-energy subspace with the
high-energy subspace decoupled. Previous work has explored how to perform this
decoupling variationally [51, 52]. One alternative method to construct such an
effective Hamiltonian is Schrieffer-Wolff perturbation theory which
systematically constructs approximate unitary transformations that decouple
the high-energy subspace [53, 54].

As an example, we will consider the Hamiltonian in Eq. (1) on two staggered
sites (one physical site) with massless fermions, truncated at zero electric
field. This is the harshest possible truncation that can be applied, and the
only physical states left in the Hilbert space are those where sites are
unoccupied or have three fermions present forming a color singlet, i.e., a
baryon. At this truncation, the Hamiltonian in Eq. (1) is trivial, and there
are no dynamics. The states kept in this truncation span the zero electric
energy subspace while all states with higher electric energy are being
discarded. Using the Schrieffer-Wolff perturbation theory, an effective
Hamiltonian for the zero electric energy subspace at leading order is given by

$\displaystyle\hat{H}_{eff}$
$\displaystyle=\sum_{x}\frac{9}{16g^{2}}\hat{Z}_{x}\hat{Z}_{x+1}+\frac{27}{32g^{4}}\left(\hat{X}_{x}\hat{X}_{x+1}+\hat{Y}_{x}\hat{Y}_{x+1}\right)$
$\displaystyle+\mathcal{O}(g^{-6})\ \ \ ,$ (4)

where $\hat{X}_{x}$, $\hat{Y}_{x}$, $\hat{Z}_{x}$ are the corresponding Pauli
matrices at site $x$ on the lattice. In this basis, spin up states correspond
to a site being unoccupied and spin down states correspond to a baryon being
present on the site. The details of this derivation and how to systematically
derive higher order terms are in Appendix A. In this context, the Schrieffer-
Wolff expansion corresponds to performing a strong coupling expansion around
the zero electric energy subspace. Note that similar results have been derived
for SU(2) lattice gauge theories and the Schwinger model with multiple
flavors, showing that they are equivalent to spin systems in the strong
coupling limit [55, 56, 57].

The effective Hamiltonian in Eq. (4) requires only a single qubit per site to
be mapped onto a quantum computer. The Hamiltonian in Eq. (2) with gauge
fields integrated out requires three qubits per site to represent the state of
the system. By using this effective Hamiltonian to describe a subspace of the
system, the computational resources required are reduced. However, the
Schrieffer-Wolff expansion is known to have a finite radius of convergence
[54], so this effective Hamiltonian should only be valid over a limited range
of couplings. The energy gap for the effective Hamiltonians obtained at
different orders in the Schrieffer-Wolff expansion over a range of couplings
are shown in Fig. 1. Note that both the ground state and first excited state
are in the baryon number zero sector. As this figure shows, the effective
Hamiltonians obtained through the Schrieffer-Wolff expansion are only valid
for strong couplings, and the expansion fails to converge at weak couplings.

Figure 1: Energy gaps as a function of coupling $g$ for the improved
Hamiltonian derived with Schrieffer-Wolff perturbation theory. The black
dashed curve is the energy gap of the exact Hamiltonian in Eq. (2) and the
blue curve is the energy gap of the Hamiltonian in Eq. (4). The other curves
correspond to including higher order terms in the Schrieffer-Wolff expansion
of the improved Hamiltonian.

###  II.3 Similarity Renormalization Group $m=0$

The strong coupling expansion in the previous section was able to yield an
improved Hamiltonian to correct for the chromo-electric field truncation for a
small system. However, the performance of the improved Hamiltonian was limited
by the convergence of the strong coupling expansion. An alternative approach
to derive an improved Hamiltonian is the similarity renormalization group
(SRG). This method works by choosing a generator of unitary rotations that
should decouple the high energy subspace and then continuously flowing to
decouple the high energy subspace. The similarity renormalization group was
originally introduced to help with the renormalization of light-front quantum
field theories [58, 59, 60]. Similar to the calculations in the previous
section, it was found that perturbative calculations in light front field
theories suffered from convergence issues and it was found that the SRG could
mitigate these issues. It has also been used in low energy nuclear physics to
derive low energy nuclear potentials with improved convergence properties
[61]. Explicitly the Hamiltonian being flowed is parametrized as

$\hat{H}_{s}=\hat{H}_{\Lambda}+\hat{V}_{s}\ \ \ ,$ (5)

where $\hat{H}_{\Lambda}$ determines the energy scales that should be
decoupled, $\hat{V}_{s}$ is the remaining terms in the Hamiltonian and $s$ is
the flow parameter. The generator of the SRG flow is traditionally taken to be

$\hat{\eta}_{s}=[\hat{H}_{\Lambda},\hat{H}_{s}]\ \ \ .$ (6)

Note that because $\hat{\eta}_{s}$ is anti-Hermitian, the evolution under the
SRG is unitary and prior to truncation $\hat{H}_{s}$ is the same as the
original Hamiltonian just expressed in a different basis. The evolution of the
Hamiltonian under SRG is given by

$\displaystyle\frac{d\hat{H}_{s}}{ds}$
$\displaystyle=\frac{d\hat{V}_{s}}{ds}=[[\hat{H}_{\Lambda},\hat{V}_{s}],\hat{H_{s}}]$
$\displaystyle=[[\hat{H}_{\Lambda},\hat{V}_{s}],\hat{H}_{\Lambda}]+[[\hat{H}_{\Lambda},\hat{V}_{s}],\hat{V}_{s}]\
\ \ .$ (7)

By flowing to $s\rightarrow\infty$, the low and high energy sectors will be
decoupled. It can be seen that these sectors decouple because the commutator
$[\hat{H}_{\Lambda},\hat{V}_{s}]$ must vanish as $s\rightarrow\infty$ for the
SRG to converge and when this commutator vanishes the two sectors have
decoupled. In the following sections, it will be shown how the SRG can be used
to derive improved Hamiltonians that correct for the effects of gauge field
truncation.

####  II.3.1 Two Staggered Sites

Once again, the Hamiltonian in Eq. (1) on two staggered sites (one physical
site), truncated at zero electric field will be used as an example to
construct an improved Hamiltonian. The generator of the SRG flow will be
chosen to decouple states with different electric energies, i.e.
$\hat{H}_{\Lambda}=\hat{H}_{E}$. The SRG equations can then be solved
numerically to recover an improved Hamiltonian of the form

$\displaystyle\hat{H}_{SRG}$
$\displaystyle=A(g)\left(\hat{X}_{1}\hat{X}_{2}+\hat{Y}_{1}\hat{Y}_{2}\right)+B(g)\hat{Z}_{1}\hat{Z}_{2}\
\ \ ,$ (8)

where $A(g)$ and $B(g)$ are constants computed numerically. It can be seen
that the improved Hamiltonian must take this form due to the symmetries of the
full Hamiltonian. The Hamiltonian in Eq. 1 conserves quark number and this
symmetry commutes with the truncation of the Hamiltonian. Combined with the
chiral symmetry of this Hamiltonian at $m=0$, the only allowed operators in
the improved Hamiltonian are $\hat{X}_{1}\hat{X}_{2}+\hat{Y}_{1}\hat{Y}_{2}$
and $\hat{Z}_{1}\hat{Z}_{2}$. Note that this Hamiltonian takes the same form
as that derived in the strong coupling expansion in Eq. (4) except now the
coefficients multiplying the operators have been determined through SRG
instead of a perturbative expansion. The energy gap for this Hamiltonian as a
function of the coupling is shown in Fig. 2. Unlike the improved Hamiltonian
obtained through the strong coupling expansion, the improved Hamiltonian
obtained through the SRG suffers from no convergence issues and is able to
correctly reproduce the energy gap at all values of the coupling.

Figure 2: Energy gaps as a function of coupling $g$ for the improved
Hamiltonian derived with the SRG. The black dashed curve is the energy gap of
the exact Hamiltonian in Eq. (2) and the blue points are the energy gap of the
Hamiltonian in Eq. (8).

####  II.3.2 Larger Systems

As shown in the previous section, the SRG was capable of producing an improved
Hamiltonian that correctly describes the physics of a small system. In
practice, improved Hamiltonians will be needed for larger systems. The setup
of the SRG used in the previous section does not scale efficiently to larger
lattices. This is because as the SRG evolves, the number of operators
generated can be exponential in the system size. This can be mitigated through
the use of the in-medium similarity renormalization group (IMSRG) which
truncates operators in the SRG flow above a certain weight [62]. In practice
IMSRG can be implemented by performing SRG on a lattice of the same size as
the operator size truncation and matching the obtained improved Hamiltonian to
a translationally invariant Hamiltonian defined on a larger lattice. The cost
of performing the IMSRG scales exponentially with the size truncation. However
the convergence with operator size is also exponential due to the exponential
decay of correlations in low energy states.

As an explicit example, improved Hamiltonians for the zero electric field
truncation will be derived with IMSRG. The smallest nontrivial operator size
truncation is at two staggered sites. The improved Hamiltonian derived with
IMSRG at this truncation with coupling $g$ on $L$ staggered sites is

$\displaystyle\hat{H}_{SRG}=$
$\displaystyle\sum_{x<L}A(g)\left(\hat{X}_{x}\hat{X}_{x+1}+\hat{Y}_{x}\hat{Y}_{x+1}\right)$
$\displaystyle+B(g)\hat{Z}_{x}\hat{Z}_{x+1}\ \ \ .$ (9)

This improved Hamiltonian comes from doing the same calculation as the
previous section and matching the two staggered site improved Hamiltonian to a
translationally invariant one defined on a larger lattice. The accuracy of the
improved Hamiltonians derived through IMSRG at this electric field truncation
can be improved by computing the IMSRG flow for larger operator size
truncations. In general, one would expect this method to work well when the
operator size truncation used is comparable to the correlation length of the
system in question. Explicitly, the form of the improved Hamiltonians obtained
by truncating at operators defined on three staggered sites takes the form

$\displaystyle\hat{H}_{3,SRG}=$
$\displaystyle\sum_{x}A_{1}(g)\left(\hat{X}_{x}\hat{X}_{x+1}+\hat{Y}_{x}\hat{Y}_{x+1}\right)+B_{1}(g)\hat{Z}_{x}\hat{Z}_{x+1}$
$\displaystyle+A_{2}(g)\left(\hat{X}_{x}\hat{X}_{x+2}+\hat{Y}_{x}\hat{Y}_{x+2}\right)+B_{2}(g)\hat{Z}_{x}\hat{Z}_{x+2}$
(10)

where $A_{i}(g)$, and $B_{i}(g)$ are constants determined from solving the SRG
equations numerically. Note that this takes the same form as Eq. (8) just with
the inclusion of next to nearest neighbor hopping. The performance of the
improved Hamiltonians can be improved further by truncating the operator size
at four staggered sites. The improved Hamiltonian obtained at this truncation
takes the form

$\displaystyle\hat{H}_{4,SRG}=$
$\displaystyle\sum_{x}A_{1}(g)\left(\hat{b}_{x}\hat{b}^{\dagger}_{x+1}+\hat{b}^{\dagger}_{x}\hat{b}_{x+1}\right)+B_{1}(g)\hat{Z}_{x}\hat{Z}_{x+1}$
$\displaystyle+A_{2}(g)\left(\hat{b}_{x}\hat{b}^{\dagger}_{x+2}+\hat{b}^{\dagger}_{x}\hat{b}_{x+2}\right)+B_{2}(g)\hat{Z}_{x}\hat{Z}_{x+2}$
$\displaystyle+A_{3}(g)\left(\hat{b}_{x}\hat{b}^{\dagger}_{x+3}+\hat{b}^{\dagger}_{x}\hat{b}_{x+3}\right)+B_{3}(g)\hat{Z}_{x}\hat{Z}_{x+3}$
$\displaystyle+C_{1}(g)\left(\hat{b}_{x}\hat{b}^{\dagger}_{x+1}+\hat{b}^{\dagger}_{x}\hat{b}_{x+1}\right)\hat{Z}_{x+2}\hat{Z}_{x+3}$
$\displaystyle+C_{2}(g)\left(\hat{b}_{x}\hat{b}^{\dagger}_{x+2}+\hat{b}^{\dagger}_{x}\hat{b}_{x+1}\right)\hat{Z}_{x+1}\hat{Z}_{x+3}$
$\displaystyle+C_{2}(g)\left(\hat{b}_{x+1}\hat{b}^{\dagger}_{x+3}+\hat{b}^{\dagger}_{x+1}\hat{b}_{x+3}\right)\hat{Z}_{x}\hat{Z}_{x+2}$
$\displaystyle+C_{3}(g)\left(\hat{b}_{x}\hat{b}^{\dagger}_{x+3}+\hat{b}^{\dagger}_{x}\hat{b}_{x+1}\right)\hat{Z}_{x+1}\hat{Z}_{x+2}$
$\displaystyle+C_{4}(g)\left(\hat{b}_{x+1}\hat{b}^{\dagger}_{x+2}+\hat{b}^{\dagger}_{x+1}\hat{b}_{x+2}\right)\hat{Z}_{x}\hat{Z}_{x+3}$
$\displaystyle+C_{5}(g)\hat{Z}_{x}\hat{Z}_{x+1}\hat{Z}_{x+2}\hat{Z}_{x+3}$
$\displaystyle+D_{1}(g)\left(b^{\dagger}_{x}b^{\dagger}_{x+1}b_{x+2}b_{x+3}+b_{x}b_{x+1}b^{\dagger}_{x+2}b^{\dagger}_{x+3}\right)$
$\displaystyle+D_{2}(g)\left(b^{\dagger}_{x}b_{x+1}b_{x+2}^{\dagger}b_{x+3}+b_{x}b^{\dagger}_{x+1}b_{x+2}b^{\dagger}_{x+3}\right)$
$\displaystyle+D_{3}(g)\left(b^{\dagger}_{x}b_{x+1}b_{x+2}b_{x+3}^{\dagger}+b_{x}b^{\dagger}_{x+1}b^{\dagger}_{x+2}b_{x+3}\right)$
(11)

where $\hat{b}_{x}=\frac{1}{2}\left(\hat{X}_{x}+i\hat{Y}_{x}\right)$ is a
qubit annihilation operator at site $x$ and $A_{i}(g)$, $B_{i}(g)$,
$C_{i}(g)$, and $D_{i}(g)$ are constants determined from solving the SRG
equations numerically. Note that it can be seen that the improved Hamiltonians
must take the forms in Eq. 10 and Eq 11 at these size truncations by
considering the symmetries of the full Hamiltonian as discussed in the
previous section.

Figure 3: Baryon mass as a function of lattice size for $g=2$. The black
points show the baryon mass for the Hamiltonian in Eq. (2). The different
solid curves correspond to the baryon mass in the various improved
Hamiltonians derived through the use of IMSRG with different operator size
truncations.

Figure 4: Baryon mass as a function of lattice size for $g=1$. The black
points show the baryon mass for the Hamiltonian in Eq. (2). The different
solid curves correspond to the baryon mass in the various improved
Hamiltonians derived through the use of IMSRG with different operator size
truncations.

To test the performance of the improved Hamiltonians derived through SRG,
density matrix renormalization group (DMRG) calculations were performed using
the C++ iTensor library [63, 64, 65, 66, 67] to obtain the vacuum state and
the single baryon ground state of the Hamiltonian in Eq. (2) and the improved
Hamiltonians described above for lattices with up to fifteen physical sites
with open boundary conditions. Fig. 3 shows the mass of the baryon (difference
of the energy of the single baryon state and vacuum state) for the full
Hamiltonian and the improved Hamiltonians for the zero electric field
truncation for $g=2$. As this figure shows, the relative error in the baryon
mass computed with the improved Hamiltonians grows with system size and then
saturates. By using improved Hamiltonians with a larger operator size
truncation in the IMSRG, the relative error in the baryon mass can be reduced
down to the percent level. The baryon mass for $g=1$ was also computed and is
shown in Fig. 4. At this weaker coupling, the correlation length is longer and
the relative error in the baryon mass grows uncontrollably with the lattice
size for the improved Hamiltonian obtained by the two staggered site
truncation IMSRG. However, increasing the size of the operator truncation used
in the IMSRG decreases the error in the baryon mass to controllable levels.
This rapid convergence with the operator size is due to the IMSRG correctly
reproducing all correlations smaller than the operator size truncation. It is
expected that once the operator size truncation is larger than the inverse
baryon mass, the improved Hamiltonian derived through IMSRG will accurately
describe single baryon states. In general, it is expected that the improved
Hamiltonians derived with IMSRG will accurately describe states containing
correlations that are smaller than the operator size truncation.

In addition to studying the energy of different states on the lattice, the
IMSRG flows of operators can be computed and their expectation values can be
computed using improved Hamiltonians. As an explicit example, the SRG flow of
the chromo-electric energy density was computed. The operators corresponding
to the chromo-electric operators in the improved basis are the same as those
that show up in the improved Hamiltonians, just with different coefficients.
The vacuum expectation of the chromo-electric energy density is shown in Fig.
5 for $g=1$ and $g=2$. As before, increasing the size of the operator
truncation in the IMSRG improves the accuracy of the improved Hamiltonians.
Remarkably, even though the improved Hamiltonians are being truncated at zero
electric field, their ground states still reproduce the electric energy
density of the full untruncated theory.

Figure 5: The expectation of the electric energy on each link for a lattice
with 15 physical sites. The black points were computed using the Hamiltonian
in Eq. (2). The other points were computed using the improved Hamiltonians for
the zero electric field truncation computed using SRG.

###  II.4 Similarity Renormalization Group $m\neq 0$

In the previous section, IMSRG was used to derive an improved Hamiltonian that
describes the dynamics of baryons in QCD in one dimension with massless
quarks. The same technique can be used to setup improved Hamiltonians in the
case of massive quarks as well.

In a theory with massive quarks, the piece of the Hamiltonian that should be
used to generate the SRG flow is the combination of the mass and electric
energy terms. At the zero electric energy truncation, the only state left
after truncation is the one with matter sites empty and anti-matter sites
filled. Therefore with massive quarks, there are no dynamics at this level of
truncation. The next lowest truncation in the SRG flow depends on the relative
size of the fermion mass $m$ and the coupling $g$. If $\frac{2}{3}g^{2}>m$,
then the next lowest lying state in the spectrum consists of a baryon at a
site. The improved Hamiltonian derived by truncating at this level takes the
same form as in the previous section except with the addition of a mass term
for the baryons. If instead $\frac{2}{3}g^{2}<m$, then the next lowest lying
state in the spectrum corresponds to a quark anti-quark pair connected by a
link of electric flux. In the strong coupling limit, this corresponds to a
meson at the excited link.

Denoting the trivial vacuum state by $\ket{Vac}$, and the state with a
$q\bar{q}$ pair on link $l$ by $\ket{l}$, the Hamiltonian obtained under IMSRG
flow truncating the energy at single link excitations and the operator size at
two link operators takes the form

$\displaystyle\hat{H}_{SRG}=E_{0}(g,m)\ket{Vac}\bra{Vac}$
$\displaystyle+\sum_{l}h(g,m)\left(\ket{l+1}\bra{l}+\ket{l}\bra{l+1}\right)+E_{1}(g,m)\ket{l}\bra{l}\
\ \ ,$ (12)

where $E_{0}(g,m)$, $E_{1}(g,m)$, and $h(g,m)$ are constants determined
through numerically solving the SRG flow. Note that this Hamiltonian has the
same form as that of a single non-relativistic particle. The Hamiltonian in
Eq. (12) can be viewed as a Hamiltonian for a single link excitation (or
meson) and can be mapped onto a second quantized Hamiltonian to describe a
system with more excited links. Explicitly, the single excitation sector of

$\displaystyle\hat{H}_{SRG}$
$\displaystyle=\sum_{l}\frac{h(g,m)}{2}\left(\hat{X}_{l}\hat{X}_{l+1}+\hat{Y}_{l}\hat{Y}_{l+1}\right)$
$\displaystyle+\frac{E_{0}(g,m)-E_{1}(g,m)}{2}\hat{Z}_{l}\ \ \ ,$ (13)

will be identical to the Hamiltonian in Eq. (12). This improved Hamiltonian
will also be capable of describing states with multiple links excited as well.
The description of these states with multiple links excited can be improved by
raising the truncation of states kept after SRG flow to include states where
two links are excited. By keeping these states after the SRG flow and keeping
the other truncations as before, the improved Hamiltonian given by

$\displaystyle\hat{H}_{SRG2}$
$\displaystyle=\sum_{l}\frac{h(g,m)}{2}\left(\hat{X}_{l}\hat{X}_{l+1}+\hat{Y}_{l}\hat{Y}_{l+1}\right)$
$\displaystyle+s(g,m)\hat{Z}_{l}\hat{Z}_{l+1}+\frac{E_{0}(g,m)-E_{1}(g,m)}{2}\hat{Z}_{l}\
\ \ ,$ (14)

will have single and two excitation sectors that match the improved
Hamiltonians derived through SRG. As a test of the performance of this
improved Hamiltonian, the mass of the meson was computed on a lattice with two
physical sites for $g=1$ and various values of $m$ in Fig. 6. Similar to the
massless case, the improved Hamiltonian derived with the SRG performs well
when there is a large separation in energy scales between the states being
decoupled. Note that in principle, the same comparison can be done with larger
lattices, however the meson is in the same baryon number sector as the vacuum
which complicates the calculation of the meson mass. It is expected that this
improved Hamiltonian scales to larger lattices as in the massless case.

Figure 6: Meson mass as a function of quark mass $m$ for $g=1$ on a lattice of
2 physical sites (4 staggered). The black dashed curves shows the meson mass
for the Hamiltonian in Eq. (2). The blue curve shows the meson mass for the
improved Hamiltonian in Eq. (14).

####  II.4.1 Quantum Simulation

As an example of how these improved Hamiltonians can be used for quantum
simulation, a simulation will be performed of a meson’s time evolution on
three physical sites with open boundary conditions. Using the Hamiltonian in
Eq. (2) would require a quantum computer with 18 qubits to encode the state,
and non-local interactions between the qubits to implement the electric energy
piece of the Hamiltonian. Using the improved Hamiltonian in Eq. (14) requires
only 5 qubits to represent the state and only requires nearest neighbor
interactions on the quantum computer to perform time evolution.

Figure 7: Time evolution of a single meson on 3 physical sites performed on
the IBM Perth quantum processor. Each color corresponds to the probability of
a different link being excited. The solid lines show the exact time evolution.
The dashed lines show a classical simulation of the Trotterized time evolution
that was implemented on the quantum processor. The data points were obtained
using self-mitigating circuits on IBM Perth.

Fig. 7 shows the real time evolution of a single meson on three physical sites
with $g=1,m=1$ simulated on IBM’s Perth quantum processor [68, 69]. A meson
state was prepared on the quantum processor by applying an $\hat{X}$ gate to
the qubit assigned to the leftmost link. Time evolution was performed using a
first order Trotter formula. Explicitly, the Hamiltonian was decomposed as
$\hat{H}=\sum_{l=1}^{4}\hat{H}_{l}$ where

$\hat{H}_{l}=\frac{h(g,m)}{2}\left(\hat{X}_{l}\hat{X}_{l+1}+\hat{Y}_{l}\hat{Y}_{l+1}\right)+s(g,m)\hat{Z}_{l}\hat{Z}_{l+1}\
\ \ ,$ (15)

and the Trotterized time evolution operator was given by

$\hat{U}(\Delta t)=e^{-i\hat{H}_{2}\Delta t}e^{-i\hat{H}_{4}\Delta
t}e^{-i\hat{H}_{3}\Delta t}e^{-i\hat{H}_{1}\Delta t}\ \ \ .$ (16)

Each individual $e^{-i\hat{H}_{l}\Delta t}$ was decomposed into a circuit with
3 CNOT gates using standard techniques [70, 71]. The sum over Pauli $\hat{Z}$
operators can be ignored when performing time evolution because it commutes
with the full Hamiltonian and the operators being measured. The noise in the
quantum simulation was mitigated using self-mitigation combined with Pauli
twirling [72, 9, 8]. For each Trotter step, $50$ circuits describing the time
evolution were used along with $50$ circuits with $\Delta t=0$ used to
determine the strength of the depolarizing noise channel. Each circuit was
sampled $10,000$ times. As Fig. 7 shows, the quantum hardware is able to
describe the time evolution well at short times, but at long times the
hardware noise begins to dominate. However, despite the presence of hardware
noise at late times, the location of the peak of the wavepacket of the meson
can still be located at late times.

##  III 3+1D

###  III.1 3+1D Hamiltonian

Performing a quantum simulation of lattice QCD requires a choice of
Hamiltonian to be used. This choice is complicated by the phenomena of fermion
doubling, where the naive discretization of the Dirac field on the lattice in
$d$ dimensions actually describes $2^{d}$ fermions. Furthermore, the Nielson-
Ninomiya theorem forbids the presence of chiral symmetry on the lattice when
all doublers are removed [73, 74]. In this work, staggered fermions will be
used. Staggered fermions work by distributing the components of the Dirac
field across different sites of the lattice. This preserves some chiral
symmetry at the cost of still having some fermion doublers remain. In lattice
QCD calculations on classical computers, space and time are both discretized
leading to staggered fermions describing 4 types of fermions, referred to as
tastes in the literature. For practical calculations, these can be reduced to
a single flavor through the process of rooting [75, 76, 77, 78, 79]. In
quantum simulation, time is left continuous and only space is discretized.
This changes the counting of the number of tastes present. Explicitly, with
three dimensions of space discretized and time left continuous, staggered
fermions describe two tastes. This is a feature, not a bug for using lattice
QCD to study nuclear physics as one taste can be identified as an up quark and
the other can be identified as a down quark. Therefore, we would expect
lattice QCD with a single staggered fermion on a quantum computer to describe
two flavor QCD where both quarks have the same mass. With massless quarks,
this lattice regularization should reproduce the predictions of chiral
perturbation theory as the continuum limit is approached. Explicitly, the
Hamiltonian that should be used for $3+1$ dimensional two flavor massless
lattice QCD on a quantum computer is

$\displaystyle\hat{H}$ $\displaystyle=\hat{H}_{K}+\hat{H}_{E}+\hat{H}_{B}$
$\displaystyle\hat{H}_{K}$
$\displaystyle=\sum_{\vec{r},\hat{\mu},a,b}\eta_{\vec{r},\hat{\mu}}\frac{1}{2}\hat{\psi}_{\vec{r},a}^{\dagger}\hat{U}^{a,b}_{\vec{r},\vec{r}+\hat{\mu}}\hat{\psi}_{\vec{r}+\hat{\mu},b}+\text{h.c.}$
$\displaystyle\hat{H}_{E}$
$\displaystyle=\frac{g^{2}}{2}\sum_{l\in\text{links},c}\hat{E}_{l}^{c}\hat{E}_{l}^{c}$
$\displaystyle\hat{H}_{B}$
$\displaystyle=-\frac{1}{2g^{2}}\sum_{p\in\text{plaquettes}}\Box_{p}\ \ \ ,$
(17)

where $\psi_{\vec{r},a}$ is a fermion field at site $\vec{r}$ with color $a$,
$\hat{\mu}$ is a unit vector in the $\hat{x}$, $\hat{y}$, or $\hat{z}$
directions, $\eta_{\vec{r},\hat{\mu}}$ are the spin diagonalization phases,
$\hat{U}^{a,b}_{\vec{r},\vec{r}+\hat{\mu}}$ is an SU(3) parallel transporter
between sites $\vec{r}$ and $\vec{r}+\hat{\mu}$, $\hat{E}_{l}^{c}$ is the
SU(3) chromo-electric field on link $l$ and $\Box_{p}$ is the Hermitian
component of the trace over color indices of the product of parallel
transporters on plaquette $p$. Previous work has shown that this Hamiltonian
has a discrete chiral symmetry corresponding to translation by one lattice
site that is spontaneously broken and an isospin symmetry that corresponds to
diagonal translations [80, 81, 82].

###  III.2 Improved Hamiltonian

As is the case for 1D QCD, mapping the Hamiltonian in Eq. (17) onto qubits is
challenging, especially if one wishes to perform a quantum simulation with
existing hardware. Improved Hamiltonians can also be derived for performing
quantum simulations of this theory. Following the discussions of the previous
sections, IMSRG can be applied to this theory with a truncation in operator
size. The smallest non-trivial operator size IMSRG can be applied to is a
single link and the lowest electric field truncation that can be used is zero
electric field. The resulting improved Hamiltonian on the 3 dimensional
lattice will take the same form as in the 1D case except now the hopping terms
will have phases that result from the spin diagonalization.

Explicitly, the improved Hamiltonian obtained through SRG at this truncation
in operator size and electric field is

$\displaystyle\hat{H}_{SRG}$
$\displaystyle=\sum_{\vec{r}}A(g)\left(\hat{\psi}_{\vec{r}}^{\dagger}\hat{\psi}_{\vec{r}+\hat{x}}+\hat{\psi}_{\vec{r}+\hat{x}}^{\dagger}\hat{\psi}_{\vec{r}}\right)$
$\displaystyle+A(g)(-1)^{r_{1}}\left(\psi_{\vec{r}}^{\dagger}\hat{\psi}_{\vec{r}+\hat{y}}+\hat{\psi}_{\vec{r}+\hat{y}}^{\dagger}\hat{\psi}_{\vec{r}}\right)$
$\displaystyle+A(g)(-1)^{r_{1}+r_{2}}\left(\hat{\psi}_{\vec{r}}^{\dagger}\hat{\psi}_{\vec{r}+\hat{z}}+\hat{\psi}_{\vec{r}+\hat{z}}^{\dagger}\hat{\psi}_{\vec{r}}\right)$
$\displaystyle+B(g)\sum_{\hat{\mu}}\left(2\hat{\psi}^{\dagger}_{\vec{r}}\hat{\psi}_{\vec{r}}-1\right)\left(2\hat{\psi}^{\dagger}_{\vec{r}+\hat{\mu}}\hat{\psi}_{\vec{r}+\hat{\mu}}-1\right)\
\ \ ,$ (18)

where $\psi_{\vec{r}}$ is a colorless fermion field at site $\vec{r}$ and
$A(g)$ and $B(g)$ are numerical constants determined through solving the SRG
equations.

Note that this improved Hamiltonian only describes the QCD Hamiltonian
accurately for large coupling $g$ as the lattice size is increased. This is
because as $g$ is decreased, the correlation length increases. To accurately
describe the QCD Hamiltonian at these couplings without increasing the
electric field truncation, higher weight operators must be included as in the
$1$D case. Alternatively, the electric field truncation could be increased. At
large coupling, the $\pi$ meson is massive and is integrated out of this
improved Hamiltonian. By increasing the chromo-electric field truncation of
states kept after the SRG flow, states with quark-antiquark pairs separated by
a link will be included in the low energy Hilbert space kept after truncation
and will yield an improved Hamiltonian that describes meson degrees of freedom
as well.

####  III.2.1 Spectrum

The improved Hamiltonian in Eq. (18) will describe the untruncated theory
accurately in the limit of large $g$. While the continuum limit of lattice QCD
is in the limit of $g\rightarrow 0$, large couplings can be used to study the
theory at finite lattice spacing. In the limit $g\rightarrow\infty$,
$A(g)\rightarrow 0$ and some qualitative features of low energy QCD are
recovered. In particular, it has been shown that in the strong coupling limit
this theory has an isospin symmetry and a spontaneously broken chiral symmetry
[80, 81, 82]. In addition to the previously studied features of this
regularization, the strong coupling limit of this Hamiltonian also reproduces
the approximate Wigner SU(4) spin flavor symmetry of nuclear physics.

As an example, we will study the improved Hamiltonian in Eq. (18) on a single
cube. The fermionic fields will be mapped onto qubits using a Jordan-Wigner
encoding. When $A(g)=0$, the Hamiltonian in Eq. (18) can be rewritten in terms
of Pauli matrices as

$\hat{H}_{SRG}=\frac{9}{16g^{2}}\sum_{\hat{\mu}}\hat{Z}_{\vec{r}}\hat{Z}_{\vec{r}+\hat{\mu}}\
\ \ .$ (19)

The ground state is in the baryon number $B=0$ sector and is a degenerate Néel
state. For the rest of this discussion, we will only consider the sector that
is even under reflection across the $\hat{z}$ axis. The lowest lying excited
states in the $B=0$ sector correspond to performing a SWAP operation on one of
the links. Denoting the energy cost of flipping one link as
$\Delta=\frac{9}{8g^{2}}$, this set of excited states has energy $4\Delta$ and
there are 12 of them. These 12 states should correspond to spin one and spin
zero baryon anti-baryon pairs, i.e., $p\bar{p}$, $n\bar{n}$, $n\bar{p}$ and
$p\bar{n}$ states.

The lowest lying energy states in the $B=1$ sector correspond to flipping one
site from the Néel state on the cube. There are four corners that can be
flipped in the Néel state to end up in the $B=1$ sector so there are four
degenerate states with energy $3\Delta$. These correspond to the two spin
modes of the proton and neutron. Note that the proton and neutron mass are
degenerate which should be expected from isospin symmetry.

In the $B=2$ sector, the lowest lying states correspond to flipping two spins
in the Néel state. This results in six degenerate states with energy
$6\Delta$. These states correspond to spin 1 $pn$ states and spin $0$ $pp$,
$pn$ and $nn$ states. The fact that these states are degenerate is reflective
of spin-flavor symmetry which is approximately present in low energy nuclear
physics. The spin-flavor symmetry has been shown to emerge in the large
$N_{c}$ limit of QCD [83] and is related to the minimization of entanglement
in low energy nucleon scattering [84, 85, 86, 87, 88, 89]. We also see that in
the strong coupling limit, the deuteron has binding energy zero. Similar
calculations can be done in the higher baryon number sectors which also show
that these sectors also demonstrate spin-flavor symmetry and nuclei with
binding energy = 0. It is also interesting to note that the nucleon-nucleon
scattering lengths are large. As a result, the pionless EFT describing nucleon
scattering is an expansion around a non-trivial fixed point where the binding
energy of nuclei vanishes as is the case in this lattice regularization [90,
91, 92, 93].

####  III.2.2 Quantum Simulation

The Hilbert space describing the Hamiltonian in Eq. (18) consists of a single
fermion mode for each site. Using the Jordan-Wigner encoding, the state of
each site can be represented with a single qubit. In this encoding, a list of
fermion operators $\psi_{1},\hat{\psi}_{2},...,\hat{\psi}_{N}$ are mapped onto
qubit operators as

$\hat{\psi}_{n}=\bigotimes_{k<n}\frac{1}{2}\hat{Z}_{k}\left(\hat{X}_{n}+i\hat{Y}_{n}\right)\
\ \ .$ (20)

For a local one dimensional fermionic theory, this fermion encoding leads to a
Hamiltonian that is local in qubits. However, in higher dimensions, the
operators in the Hamiltonian will include strings of Pauli $\hat{Z}$ operators
that wrap around the lattice. These long range operators are necessary to
enforce the anti-commutation relations of the fermionic operators and may make
it difficult to practically scale to calculations on a large lattice.

Figure 8: Connectivity of the system described by the improved Hamiltonian in
Eq. (21).

As a demonstration of how this improved Hamiltonian works in practice, time
evolution on six vertices connected to a single vertex at the center as shown
in Fig. 8 will be simulated. This is the smallest non-trivial subsystem of a
full three dimensional lattice that will be repeated periodically and will be
useful for understanding how simulations on a larger lattice will work. Each
of the seven vertices can be mapped onto a single qubit. The Hamiltonian
describing their time evolution is given by

$\displaystyle\hat{H}_{SRG}$
$\displaystyle=\sum_{v}A(g)\left(\hat{\psi}_{0}^{\dagger}\hat{\psi}_{v}+\hat{\psi}_{v}^{\dagger}\hat{\psi}_{0}\right)$
$\displaystyle+B(g)\sum_{v}\left(2\hat{\psi}^{\dagger}_{v}\hat{\psi}_{v}-1\right)\left(2\hat{\psi}^{\dagger}_{0}\hat{\psi}_{0}-1\right)\
\ \ ,$ (21)

where the $0$ subscript denotes the vertex at the center and the sum is over
the other vertices. The quantum processor is initialized with the center qubit
in the $1$ state and the remaining qubits are in the $0$ state. In the
staggered fermion lattice regularization, sites are alternatively identified
with matter and anti-matter degrees of freedom so this state should correspond
to the trivial vacuum. By evolving with the Hamiltonian in Eq. (21), it should
be possible to observe matter anti-matter fluctuations. Note that with this
initial state, a single Trotter step can be performed without having to
implement CNOT gates from the Jordan-Wigner strings. A single Trotter step was
implemented on IBM Perth with the size of the time step being varied to sample
different times. Due to the connectivity of the hardware, this circuit
required 28 CNOT gates. Fig. 9 shows the results of performing a single
Trotter step for $g=2$ on IBM Perth. For small times, the quantum simulation
is able to describe the evolution of the system accurately, however beyond
$t=1$, the error in the single Trotter step used is large and limits the
accuracy of the quantum simulation.

Figure 9: Probability of staying in the trivial vacuum state computed on the
IBM Perth quantum processor. The solid black line shows the exact time
evolution. The blue dashed line shows the probability computed using a first
order Trotter step computed on a classical computer. The blue data points were
obtained using self-mitigating circuits on IBM Perth.

While the Jordan-Wigner encoding is efficient in the number of qubits used,
the Hamiltonian generated has long range interactions which are necessary to
preserve the anti-commutation relation of the fermions. Scaling these
calculations to a larger lattice will require making use of a more efficient
fermion encoding. For example, the Bravyi-Kitaev superfast encoding can be
used to map fermions onto qubits [94]. In this encoding, a qubit is associated
with each link on the lattice and represents the parity of the number of
fermions on the link. The length of the strings of Pauli $\hat{Z}$ operators
for an operator on a link extends only to neighboring links. For a large
lattice, this will limit the circuit depth necessary to perform time evolution
and potentially allow for larger calculations to be performed.

##  IV Discussion

In this work, the SRG has been used to derive improved Hamiltonians that
mitigate the effects of gauge field truncation. It was demonstrated in 1+1D
that the improved Hamiltonians derived this way outperform those derived
through the strong coupling expansion for small systems. Tensor network
calculations were performed to demonstrate that these improved Hamiltonians
perform well as the system size is increased. These techniques were also
applied to 3+1D giving an improved Hamiltonian capable of describing two
flavour QCD on the lattice. Real time dynamics on small systems were simulated
on IBM’s Perth quantum processor.

Previous strategies for quantum simulation of lattice gauge theories improved
accuracy by increasing the truncation of the gauge field. This comes at the
cost of needing more qubits to represent the system and a more complicated
circuit to implement the time evolution. The improved Hamiltonians introduced
in this work are capable of improving accuracy only at the cost of requiring
more complicated circuits to simulate.

Improved Hamiltonians have been derived for a single flavor of staggered
fermions coupled to SU(3) gauge fields truncated at low electric field. This
has enabled quantum simulation of systems that would otherwise be out of reach
of current quantum hardware. The same approach introduced here can be used to
derive improved Hamiltonians for larger electric field truncations and with
more flavors of fermions. Future work will extend these methods to higher
spatial dimensions with larger operator truncations where the plaquette terms
will modify the SRG flow. This will enable quantum simulations of lattice
gauge theories in multiple dimensions to be performed in the near term.

###### Acknowledgements.

The authors would like to acknowledge useful conversations about SRG with
Zhiyao Li on a related project. We would also like to thank Marc Illa and
Roland Farrell for feedback in preparing this manuscript. The authors would
also like to acknowledge many useful conversations with Martin Savage,
Francesco Turro, Xiaojun Yao and Niklas Mueller. The material presented here
was funded by U.S. Department of Energy, Office of Science, Office of Nuclear
Physics, Inqubator for Quantum Simulation (IQuS)111https://iqus.uw.edu under
Award Number DOE (NP) Award DE-SC0020970 via the program on Quantum Horizons:
QIS Research and Innovation for Nuclear Science. This work was enabled, in
part, by the use of advanced computational, storage and networking
infrastructure provided by the Hyak supercomputer system at the University of
Washington222https://itconnect.uw.edu/research/hpc. We acknowledge the use of
IBM Quantum services for this work. The views expressed are those of the
authors, and do not reflect the official policy or position of IBM or the IBM
Quantum team.

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##  Appendix A Schrieffer-Wolff Perturbation Theory

The improved Hamiltonians derived in this work are based on performing a
unitary transformation before truncating the electric field to reduce the
coupling to the states being removed by the truncation. This can be done
perturbatively through the use of Schrieffer-Wolf perturbation theory (SWPT).
In this section, the application of SWPT to the Hamiltonian in Eq. (2) with
$m=0$ will be demonstrated. The Hamiltonian for lattice gauge theories in 1D
we wish to simulate takes the form

$\hat{H}=\hat{H}_{E}+\hat{H}_{D}+\hat{V}\ \ \ ,$ (22)

where $\hat{H}_{E}$ is the electric Hamiltonian, $\hat{V}$ couples the low
energy subspace to the high energy subspace and $\hat{H}_{D}$ describes
dynamics in the high energy Hilbert space. Note that the kinetic term of Eq.
(2) is equal to $\hat{H}_{D}+\hat{V}$. For the zero electric field truncation,
$\hat{V}$ is the piece of the kinetic term that corresponds to a baryon on a
site ejecting a quark to a neighboring site and $\hat{H}_{D}$ is the piece of
the kinetic term that describes a quark propagating freely between sites. SWPT
systematically generates a unitary, $e^{\hat{S}}$ that decouples the selected
low energy subspace. For lattice gauge theories, we will be decoupling the
electric vacuum and states with low energy relative to the electric
Hamiltonian. To leading order we have

$\displaystyle e^{\hat{S}_{1}}\hat{H}e^{-\hat{S}_{1}}$
$\displaystyle=\hat{H}_{E}+[\hat{S}_{1},\hat{H}_{E}+\hat{H}_{D}]+\hat{H}_{D}+\hat{V}+[\hat{S}_{1},\hat{V}]$
$\displaystyle+\frac{1}{2}[\hat{S}_{1},[\hat{S}_{1},\hat{H}_{E}+\hat{H}_{D}]+\mathcal{O}(\hat{V}^{3})\
\ \ .$ (23)

The leading order coupling between the low and high energy subspace comes from
$\hat{V}$ and be cancelled at leading order by choosing $\hat{S}_{1}$ such
that $[\hat{S}_{1},\hat{H}_{E}+\hat{H}_{D}]=-\hat{V}$. Explicitly, the matrix
elements of $\hat{S}_{1}$ are

$(S_{1})_{ab}=\frac{1}{E_{a}-E_{b}}V_{ab}\ \ \ ,$ (24)

where the indices label eigenstates of $\hat{H}_{E}+\hat{H}_{D}$ with
eigenvalues $E_{a}$. To leading order, the effective Hamiltonian is

$\hat{H}_{eff}^{1}=\hat{H}_{E}+\frac{1}{2}[\hat{S}_{1},\hat{V}]\ \ \ ,$ (25)

and provided that the low energy subspace has an electric energy of 0, the
commutator is equal to

$\frac{1}{2}[\hat{S}_{1},\hat{V}]=-\hat{V}\frac{1}{\hat{H}_{E}+\hat{H}_{D}}\hat{V}=-\sum_{n}\hat{V}\left(-\hat{H}_{E}^{-1}\hat{H}_{D}\right)^{n}\frac{1}{\hat{H}_{E}}\hat{V}\
\ \ .$ (26)

Therefore to $\mathcal{O}(H_{E}^{-2})$, the effective Hamiltonian is given by

$\hat{H}_{eff}^{1}=\hat{H}_{E}-\hat{V}\frac{1}{\hat{H}_{E}}\hat{V}+\hat{V}\frac{1}{\hat{H}_{E}}\hat{H}_{D}\frac{1}{\hat{H}_{E}}\hat{V}+\mathcal{O}(\hat{H}_{E}^{-3})\
\ \ .$ (27)

To derive Eq. (4) from Eq. 27, it will be helpful to work in the gauge
invariant subspace of the theory. The gauge invariant states of the two
staggered site Hamiltonian are

$\displaystyle\ket{\Omega}$
$\displaystyle=\hat{\psi}^{\dagger}_{1,b}\hat{\psi}^{\dagger}_{1,g}\hat{\psi}^{\dagger}_{1,r}\ket{0}$
$\displaystyle\ket{M}$
$\displaystyle=\frac{1}{\sqrt{3}}\sum_{c}\hat{\psi}^{\dagger}_{2,c}\hat{\psi}_{1,c}\ket{\Omega}$
$\displaystyle\ket{T}$
$\displaystyle=\frac{1}{2\sqrt{3}}\sum_{c^{\prime}}\hat{\psi}^{\dagger}_{2,c^{\prime}}\hat{\psi}_{1,c^{\prime}}\sum_{c}\hat{\psi}^{\dagger}_{2,c}\hat{\psi}_{1,c}\ket{\Omega}$
$\displaystyle\ket{B}$
$\displaystyle=\hat{\psi}^{\dagger}_{2,b}\hat{\psi}^{\dagger}_{2,g}\hat{\psi}^{\dagger}_{2,r}\ket{\Omega}$
$\displaystyle\ket{\bar{B}}$
$\displaystyle=\hat{\psi}_{1,b}\hat{\psi}_{1,g}\hat{\psi}_{1,r}\ket{\Omega}$
$\displaystyle\ket{B\bar{B}}$
$\displaystyle=\hat{\psi}_{1,b}\hat{\psi}_{1,g}\hat{\psi}_{1,r}\hat{\psi}^{\dagger}_{2,b}\hat{\psi}^{\dagger}_{2,g}\hat{\psi}^{\dagger}_{2,r}\ket{\Omega}\
\ \ ,$ (28)

where $\ket{0}$ is the state with all fermion modes unoccupied,
$\hat{\psi}_{x,c}$ is the quark field operator on site $x$ with color $c$,
$\ket{\Omega}$ is the electric vacuum, $\ket{M}$ is a meson state in the
strong coupling limit with massive quarks, $\ket{T}$ is a tetra-quark state,
$\ket{B}$ is a state with a single baryon, $\ket{\bar{B}}$ is a state with a
single anti-baryon and $\ket{B\bar{B}}$ is a state with a baryon anti-baryon
pair. When these basis states are enumerated in the order
$\\{\ket{\bar{B}},\ket{\Omega},\ket{B\bar{B}},\ket{B},\ket{M},\ket{T}\\}$, the
electric energy operator written as a matrix is

$\hat{H}_{E}=\begin{pmatrix}0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
0&0&0&0&0&0\\\ 0&0&0&0&\frac{2}{3}g^{2}&0\\\
0&0&0&0&0&\frac{2}{3}g^{2}\end{pmatrix}$ (29)

and the kinetic term is

$\hat{H}_{K}=\begin{pmatrix}0&0&0&0&0&0\\\ 0&0&0&0&\frac{\sqrt{3}}{2}&0\\\
0&0&0&0&0&\frac{\sqrt{3}}{2}\\\ 0&0&0&0&0&0\\\ 0&\frac{\sqrt{3}}{2}&0&0&0&1\\\
0&0&\frac{\sqrt{3}}{2}&0&1&0\end{pmatrix}\ \ \ .$ (30)

Note that this ordering of basis states has been chosen so that the encoding
onto qubits corresponds to the upper left $4\times 4$ corner of this matrix.
The decomposition of $\hat{H}_{K}$ into $\hat{V}+\hat{H}_{D}$ is given by

$\hat{V}=\begin{pmatrix}0&0&0&0&0&0\\\ 0&0&0&0&\frac{\sqrt{3}}{2}&0\\\
0&0&0&0&0&\frac{\sqrt{3}}{2}\\\ 0&0&0&0&0&0\\\ 0&\frac{\sqrt{3}}{2}&0&0&0&0\\\
0&0&\frac{\sqrt{3}}{2}&0&0&0\end{pmatrix}\ \ \ ,$ (31)

and

$\hat{H}_{D}=\begin{pmatrix}0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
0&0&0&0&0&0\\\ 0&0&0&0&0&1\\\ 0&0&0&0&1&0\end{pmatrix}\ \ \ .$ (32)

Truncating to the basis states
$\\{\ket{\bar{B}},\ket{\Omega},\ket{B\bar{B}},\ket{B}\\}$, the leading term in
the effective Hamiltonian is given by

$-\hat{V}\frac{1}{\hat{H}_{E}}\hat{V}=\begin{pmatrix}0&0&0&0\\\
0&-\frac{9}{8g^{2}}&0&0\\\ 0&0&-\frac{9}{8g^{2}}&0\\\ 0&0&0&0\end{pmatrix}\ \
\ ,$ (33)

and after dropping a term proportional to the identity matrix this is
equivalent to $\frac{9}{16g^{2}}\hat{Z}_{1}\hat{Z}_{2}$. Similarly, the
subleading term is

$\hat{V}\frac{1}{\hat{H}_{E}}\hat{H}_{D}\frac{1}{\hat{H}_{E}}\hat{V}=\begin{pmatrix}0&0&0&0\\\
0&0&\frac{27}{16g^{4}}&0\\\ 0&\frac{27}{16g^{4}}&0&0\\\ 0&0&0&0\end{pmatrix}\
\ \ ,$ (34)

and written in terms of Pauli matrices this is
$\frac{27}{32g^{4}}\left(\hat{X}_{1}\hat{X}_{2}+\hat{Y}_{1}\hat{Y}_{2}\right)$.
Adding together the expressions in Eq. 33 and Eq. 34 yields the improved
Hamiltonian in Eq 4. Techniques for performing this expansion to higher orders
can be found in Ref. [54].


# On the Chow groups of Plücker hypersurfaces in Grassmannians

Robert Laterveer  Institut de Recherche Mathématique Avancée, CNRS –
Université de Strasbourg, 7 Rue René Descartes, 67084 Strasbourg CEDEX,
FRANCE<EMAIL_ADDRESS>
###### Abstract.

Motivated by the generalized Bloch conjecture, we formulate a conjecture about
the Chow groups of Plücker hypersurfaces in Grassmannians. We prove weak
versions of this conjecture.

###### Key words and phrases:

Algebraic cycles, Chow groups, motive, Bloch-Beilinson conjectures

###### 2010 Mathematics Subject Classification:

Primary 14C15, 14C25, 14C30.

##  1\. Introduction

Given a smooth projective variety $Y$ over $\mathbb{C}$, let
$A_{i}(Y):=CH_{i}(Y)_{\mathbb{Q}}$ denote the Chow groups of $Y$ (i.e. the
groups of $i$-dimensional algebraic cycles on $Y$ with
$\mathbb{Q}$-coefficients, modulo rational equivalence). Let
$A_{i}^{hom}(Y)\subset A_{i}(Y)$ denote the subgroup of homologically trivial
cycles.

The “generalized Bloch conjecture” [27, Conjecture 1.10] predicts that the
Hodge level of the cohomology of $Y$ should have an influence on the size of
the Chow groups of $Y$. In case $Y$ is a surface, this is the notorious Bloch
conjecture, which is still an open problem. In the case of hypersurfaces in
projective space, the precise prediction is as follows:

######  Conjecture 1.1.

Let $Y\subset\mathbb{P}^{n}$ be a smooth hypersurface of degree $d$. Then

$A_{i}^{hom}(Y)=0\ \ \ \forall\ i\leq{n\over d}-1\ .$

Conjecture 1.1 is still open; partial results have been obtained in [20],
[24], [22], [8], [9].

In the case of Plücker hypersurfaces in Grassmannians, we hazard the following
prediction (cf. subsection 2.1 below for motivation):

######  Conjecture 1.2.

Let $\operatorname{Gr}(k,n)$ denote the Grassmannian of $k$-dimensional
subspaces of an $n$-dimensional vector space, and let

$Y=\operatorname{Gr}(k,n)\cap H\ \ \ \subset\ \mathbb{P}^{{n\choose k}-1}$

be a smooth hyperplane section (with respect to the Plücker embedding). Then

$A_{i}^{hom}(Y)=0\ \ \ \forall\ i\leq n-2\ .$

The main result of this note is that a weak version of Conjecture 1.2 is true:

######  Theorem  (=Theorem 3.1).

Let

$Y=\operatorname{Gr}(k,n)\cap H\ \ \ \subset\ \mathbb{P}^{{n\choose k}-1}$

be a smooth hyperplane section (with respect to the Plücker embedding). Then

$A_{i}^{hom}(Y)=0\ \ \ \forall i\leq n-k\ .$

The argument proving Theorem 3.1 is very easy and straightforward; it combines
the recent construction of jumps among subvarieties of Grassmannians [2] and a
motivic version of the Cayley trick [12].

In some cases, we can do better and the conjecture is completely satisfied:

######  Theorem  (=Theorem 3.2).

Let

$Y=\operatorname{Gr}(3,n)\cap H\ \ \ \subset\ \mathbb{P}^{{n\choose 3}-1}$

be a smooth hyperplane section (with respect to the Plücker embedding). Assume
$n\leq 13$, $n\not=12$. Then

$A_{i}^{hom}(Y)=0\ \ \ \forall i\leq n-2\ .$

The $n=10$ case of Theorem 3.2 was already proven by Voisin as an application
of her technique of spread of algebraic cycles [26, Theorem 2.4]. Hyperplane
sections $Y\subset\operatorname{Gr}(3,10)$ are also known as Debarre–Voisin
hypersurfaces, because they give rise to the Debarre–Voisin hyperkähler
fourfolds [7]. The new proof of [26, Theorem 2.4] provided by Theorem 3.2 does
not rely on Voisin’s spread technique, nor on the relation with hyperkähler
fourfolds.

As a consequence of Theorem 3.2, some instances of the generalized Hodge
conjecture are verified:

######  Corollary  (=Corollary 4.1).

Let $Y$ be as in Theorem 3.2. Then $H^{\dim Y}(Y,\mathbb{Q})$ is supported on
a subvariety of codimension $n-1$.

As another consequence, we find some new examples of varieties with finite-
dimensional motive:

######  Corollary  (=Corollary 4.2).

Let

$Y=\operatorname{Gr}(3,9)\cap H\ \ \ \subset\ \mathbb{P}^{83}$

be a smooth hyperplane section (with respect to the Plücker embedding). Then
$Y$ has finite-dimensional motive (in the sense of [14]).

Varieties $Y$ as in Corollary 4.2 are studied in [2, Section 5.1], where they
are related to Coble cubics and abelian surfaces.

######  Conventions .

In this note, the word variety will refer to a reduced irreducible scheme of
finite type over $\mathbb{C}$. A subvariety is a (possibly reducible) reduced
subscheme which is equidimensional.

All Chow groups will be with rational coefficients: we denote by
$A_{j}(Y):=CH_{j}(Y)_{\mathbb{Q}}$ the Chow group of $j$-dimensional cycles on
$Y$ with $\mathbb{Q}$-coefficients; for $Y$ smooth of dimension $n$ the
notations $A_{j}(Y)$ and $A^{n-j}(Y)$ are used interchangeably. The notations
$A^{j}_{hom}(Y)$ and $A^{j}_{AJ}(X)$ will be used to indicate the subgroup of
homologically trivial (resp. Abel–Jacobi trivial) cycles.

For a vector bundle $E$, we write $\mathbb{P}(E)$ for
$\hbox{Proj}(\oplus_{m>0}\hbox{Sym}^{m}E)$.

##  2\. Preliminaries

###  2.1. Motivating the conjecture

######  Theorem 2.1 (Bernardara–Fatighenti–Manivel [2], Kuznetsov [17]).

Let

$Y=\operatorname{Gr}(k,n)\cap H\ \ \ \subset\ \mathbb{P}^{{n\choose k}-1}$

be a smooth hyperplane section (with respect to the Plücker embedding). Assume
either $n>3k>6$, or $n$ and $k$ are coprime. Then $Y$ has Hodge coniveau
$n-1$. More precisely, the Hodge numbers verify

$h^{p,\dim Y-p}(Y)=\begin{cases}1&\hbox{if}\ p=n-1\ ,\\\ 0&\hbox{if}\ p<n-1\
.\\\ \end{cases}$

###### Proof.

The case $n>3k>6$ is [2, Theorem 3]. In case $n$ and $k$ are coprime,
Kuznetsov [17, Corollary 4.4] has constructed an exceptional collection for
the derived category of $Y$ whose right orthogonal is a Calabi–Yau category of
dimension $k(n-k)+1-2n$. Taking Hochschild homology, one obtains the assertion
about the Hodge numbers. ∎

As mentioned in [2], the assumptions on $n$ and $k$ are probably not optimal.
(And in view of the examples given in loc. cit., it seems likely that for any
$n,k$, the Hodge coniveau of $Y$ is $\geq n-1$, while one needs some condition
on $n,k$ to get an equality.)

The generalized Bloch conjecture [27, Conjecture 1.10] predicts that any
variety $Y$ with Hodge coniveau $\geq c$ has

$A_{i}^{hom}(Y)=0\ \ \ \forall\ i<c\ .$

This motivates Conjecture 1.2. Note that at least for $n>3k>6$ (or $n$ and $k$
coprime), the bound of Conjecture 1.2 is optimal: assuming $A_{i}^{hom}(Y)=0$
for $j\leq n-1$ and applying the Bloch–Srinivas argument [5], one would get
the vanishing $h^{n-1,k(n-k)-n}(Y)=0$, contradicting Theorem 2.1.

###  2.2. Jumps

######  Proposition 2.2 (Bernardara–Fatighenti–Manivel [2]).

Let

$Y=\operatorname{Gr}(k,n)\cap H\ \ \ \subset\ \mathbb{P}^{{n\choose k}-1}$

be a general hyperplane section (with respect to the Plücker embedding). There
is a Cartesian diagram

$\begin{array}[c]{ccccc}F&\hookrightarrow&q^{\ast}Y&\xrightarrow{q}&Y\\\
&&&&\\\ \downarrow{}&&\ \ \downarrow{\scriptstyle p}&&\\\ &&&&\\\
T&\hookrightarrow&\operatorname{Gr}(k-1,n)&&\\\ \end{array}$

Here the morphism $q$ is a $\mathbb{P}^{k-1}$-bundle, and the morphism $p$ is
a $\mathbb{P}^{n-k-1}$-bundle over $\operatorname{Gr}(k-1,n)\setminus T$ and a
$\mathbb{P}^{n-k}$-bundle over $T$. The subvariety
$T\subset\operatorname{Gr}(k-1,n)$ is smooth of codimension $n-k+1$, given by
a section of $Q^{\ast}(1)$ (where $Q$ denotes the universal quotient bundle on
$\operatorname{Gr}(k-1,n)$).

###### Proof.

This is a special case of the construction of a jump in [2, Section 3.3]. The
idea is to consider the flag variety $Fl(k-1,k,n)$ as a correspondence

$\begin{array}[c]{ccc}Fl(k-1,k,n)&\xrightarrow{q}&\operatorname{Gr}(k,n)\\\
&&\\\ \ \ \downarrow{\scriptstyle p}&&\\\ &&\\\ \operatorname{Gr}(k-1,n)\\\
\end{array}$

and look at what happens over $Y$. The hyperplane
$Y\subset\operatorname{Gr}(k,n)$ corresponds to a $k$-form $\Omega$ on an
$n$-dimensional vector space. The variety $q^{\ast}Y\subset Fl(k-1,k,n)$ is
defined by $q^{\ast}\Omega$; this is the inverse image of $Y$ under $q$. The
flag variety $Fl(k-1,k,n)$ can be identified with the bundle of hyperplanes
$\mathbb{P}(Q^{\ast}(1))$ on $\operatorname{Gr}(k-1,n)$, and the locus
$T\subset\operatorname{Gr}(k-1,n)$ where the fiber dimension of $p\colon
q^{\ast}Y\to\operatorname{Gr}(k-1,n)$ jumps is the zero locus of the section
of $Q^{\ast}(1)$ defined by $\Omega$. For general $Y$, the locus $T$ will be
smooth of codimension equal to $\operatorname{rank}Q^{\ast}(1)=n-k+1$. (For
the smoothness of $T$, we note that $Q^{\ast}(1)$ is globally generated and so
the universal family $\mathcal{T}$ of zero loci of sections of $Q^{\ast}(1)$
is a projective bundle over $Gr(k,n)$ hence $\mathcal{T}$ is smooth; the
smoothness of general $T$ then follows from generic smoothness applied to
$\mathcal{T}\to\mathbb{P}H^{0}(\operatorname{Gr}(k,n),Q^{\ast}(1))$.) ∎

###  2.3. Cayley’s trick and Chow groups

######  Theorem 2.3 (Jiang [12]).

Let $E\to X$ be a vector bundle of rank $r\geq 2$ over a smooth projective
variety $X$, and let $T:=s^{-1}(0)\subset X$ be the zero locus of a section
$s\in H^{0}(X,E)$ such that $T$ is smooth of dimension $\dim
X-\operatorname{rank}E$. Let $Y:=w^{-1}(0)\subset\mathbb{P}(E)$ be the zero
locus of the section $w\in
H^{0}(\mathbb{P}(E),\mathcal{O}_{\mathbb{P}(E)}(1))$ that corresponds to $s$
under the natural isomorphism $H^{0}(X,E)\cong
H^{0}(\mathbb{P}(E),\mathcal{O}_{\mathbb{P}(E)}(1))$. There are
(correspondence-induced) isomorphisms of Chow groups

$A_{i}(Y)\ \cong A_{i+1-r}(T)\oplus\bigoplus_{j=0}^{r-2}A_{i-j}(X)\ \ \
\forall\ i\ .$

In particular, there are isomorphisms

$A^{hom}_{i}(Y)\ \cong\
A_{i+1-r}^{hom}(T)\oplus\bigoplus_{j=0}^{r-2}A_{i-j}^{hom}(X)\ \ \ \forall\ i\
.$

###### Proof.

The first statement is a special case of [12, Theorem 3.1] (the statement is
actually true with integer coefficients). Both the isomorphism and its inverse
are explicitly described. The crucial point is that the projection $Y\to X$ is
a $\mathbb{P}^{r-2}$-fibration over $X\setminus T$, and a
$\mathbb{P}^{r-1}$-fibration over $T$.

As for the second statement, one observes that the first statement also holds
on the level of Chow motives (this is [12, Corollary 3.8]). This implies that
there is a commutative diagram (where vertical arrows are cycle class maps)

$\begin{array}[c]{ccc}A_{i}(Y)&\cong&A_{i+1-r}(T)\ \oplus\
{\displaystyle\bigoplus_{j=0}^{r-2}}A_{i-j}(X)\\\ &&\\\
\downarrow&&\downarrow\\\ &&\\\ H_{2i}(Y,\mathbb{Q})&\cong&\ \ \ \
H_{2i+2-2r}(T,\mathbb{Q})\ \oplus\
{\displaystyle\bigoplus_{j=0}^{r-2}}H_{2i-2j}(X,\mathbb{Q})\ .\\\ \end{array}$

This proves the second statement. ∎

######  Remark 2.4.

In the set-up of Theorem 2.3, a cohomological relation between $Y$, $X$ and
$T$ was established in [16, Prop. 4.3] (cf. also [10, section 3.7], as well as
[2, Proposition 46] for a generalization). A relation on the level of derived
categories was established in [21, Theorem 2.10] (cf. also [13, Theorem 2.4]
and [2, Proposition 47]).

###  2.4. A variant of the Cayley trick

######  Proposition 2.5.

Let

$\begin{array}[c]{ccc}Y_{T}&\hookrightarrow&Y\\\ &&\\\ \ \ \downarrow{}&&\ \
\downarrow{\scriptstyle p}\\\ &&\\\ T&\hookrightarrow&X\\\ \end{array}$

be a Cartesian diagram of projective varieties, with $T\subset X$ of
codimension $c$. Assume that $p$ is a proper morphism which is a
$\mathbb{P}^{n}$-bundle over $X\setminus T$, and a $\mathbb{P}^{m}$-bundle
over $T$. Assume also that there exists $h\in A^{1}(Y)$ such that
$h|_{Y\setminus Y_{T}}$ is relatively ample for the $\mathbb{P}^{n}$-bundle
and $h|_{Y_{T}}$ is relatively ample for the $\mathbb{P}^{m}$-bundle. Then
there is an exact sequence

$\bigoplus_{j=n+1}^{m}A_{i-j}(T)\ \to\ A_{i}(Y)\ \to\
\bigoplus_{j=0}^{n}A_{i-j}(X)\ \to\ 0\ .$

###### Proof.

We use Bloch’s higher Chow groups $A_{i}(-,j)$ [3], [4]. There is a
commutative diagram with long exact rows

$\begin{array}[c]{ccccccc}A_{i}(Y\setminus Y_{T},1)&\xrightarrow{}&\
A_{i}(Y_{T})&\xrightarrow{}&A_{i}(Y)&\to&A_{i}(Y\setminus Y_{T})\ \ \ \to\ \ \
0\\\ &&&&&&\\\ \downarrow{\cong}&&\ \ \ \ \ \ \ \ \ \
\downarrow{\scriptstyle\Phi_{T}}&&\downarrow{\scriptstyle\Phi}&&\downarrow{\cong}\\\
&&&&&&\\\ {\displaystyle\bigoplus_{j=0}^{n}}A_{i-j}(X\setminus
T,1)&\to&{\displaystyle\bigoplus_{j=0}^{n}}A_{i-j}(T)&\xrightarrow{}&{\displaystyle\bigoplus_{j=0}^{n}}A_{i-j}(X)&\to&{\displaystyle\bigoplus_{j=0}^{n}}A_{i-j}(X\setminus
T)\ \to\ 0\ .\\\ \end{array}$

The vertical arrows in this diagram are defined as follows: the map $\Phi$ is
$\sum_{j=0}^{n}p_{\ast}(h^{j}\cap-)$, and the map $\Phi_{T}$ is
$\sum_{j=0}^{n}p_{\ast}((h|_{Y_{T}})^{j}\cap-)$. Similarly, the left and right
vertical maps are defined by restricting $h$ to $Y\setminus Y_{T}$.
Commutativity of the diagram is proven as in [19, diagram (10)], by looking at
the level of the underlying complexes. The left and right vertical arrows are
isomorphisms because of the projective bundle formula for higher Chow groups
[3].

The projective bundle formula says that $\Phi_{T}$ is surjective with kernel

$\ker\Phi_{T}\ \cong\ {\displaystyle\bigoplus_{j=n+1}^{m}}A_{i-j}(T)\ .$

A diagram chase now yields the desired exact sequence. ∎

######  Remark 2.6.

The case $(m,n)=(r-2,r-1)$ of Proposition 2.5 gives back a weak version of
Jiang’s result (Theorem 2.3). Versions of Proposition 2.5 on the level of
cohomology and on the level of derived categories are given in [2, Appendix
A], resp. [2, Appendix B].

At least when all varieties are smooth, it seems likely that a stronger
version of Proposition 2.5 is true: we guess that in this case there is an
isomorphism of Chow groups

$A_{i}(Y)\ \cong\bigoplus_{j=n+1}^{m}A_{i-j}(T)\ \oplus\
\bigoplus_{j=0}^{n}A_{i-j}(X)\ .$

Since this is not needed below, we have not pursued this guess.

###  2.5. Spreading out

######  Proposition 2.7.

Let $\mathcal{Y}\to B$ be a family of smooth projective varieties. Assume
there is some $c\in\mathbb{N}$ that

(1) $A_{i}^{hom}(Y_{b})=0\ \ \ \forall i\leq c\ $

for the very general fiber $Y_{b}$. Then

$A_{i}^{hom}(Y_{b})=0\ \ \ \forall i\leq c\ $

for every fiber $Y_{b}$.

###### Proof.

Let $B^{\circ}\subset B$ denote the intersection of countably many Zariski
open subsets such that the vanishing (1) holds for all $b\in B^{\circ}$.

Doing the Bloch–Srinivas argument [5] (cf. also [18]), this implies that for
each $b\in B^{\circ}$ one has a decomposition of the diagonal

(2) $\Delta_{Y_{b}}=\gamma_{b}+\delta_{b}\ \ \ \hbox{in}\ A^{\dim
Y_{b}}(Y_{b}\times Y_{b})\ $

where $\gamma_{b}$ is completely decomposed (i.e. $\gamma_{b}\in
A^{\ast}(Y_{b})\otimes A^{\ast}(Y_{b})$) and $\delta_{b}$ is supported on
$Y_{b}\times W_{b}$ with $\hbox{codim}\,W_{b}=c+1$.

Using Hilbert schemes as in the proofs of [28, Theorem 2.1(i)], [25,
Proposition 3.7], the fiberwise data

$(\gamma_{b},\,\delta_{b},\,W_{b})$

can be encoded by a countably infinite number of varieties, each carrying a
universal object. By a Baire category argument, one of these varieties must
dominate $B$. Taking a linear section, this means that after a generically
finite base change the $\gamma_{b},\delta_{b},W_{b}$ exist relatively. That
is, there exist a generically finite morphism $B^{\prime}\to B$, a cycle
$\gamma\in
A^{\ast}(\mathcal{Y}^{\prime}\times_{B^{\prime}}\mathcal{Y}^{\prime})$ that is
completely decomposed (i.e.
$\gamma\in(p_{1})^{\ast}A^{\ast}(\mathcal{Y}^{\prime})\cdot(p_{2})^{\ast}A^{\ast}(\mathcal{Y}^{\prime})$),
a subvariety $\mathcal{W}\subset\mathcal{Y}^{\prime}$ of codimension $c+1$,
and a cycle $\delta$ supported on
$\mathcal{Y}^{\prime}\times_{B^{\prime}}\mathcal{W}$ such that

$\Delta_{\mathcal{Y}^{\prime}}|_{b}=\gamma|_{b}+\delta|_{b}\ \ \ \hbox{in}\
A^{\dim Y_{b}}(Y_{b}\times Y_{b})\ \ \ \forall\ b\in B^{\circ}\ .\ $

(Here $\mathcal{Y}^{\prime}:=\mathcal{Y}\times_{B}B^{\prime}$.)

Let $\bar{\gamma},\bar{\delta}\in A^{\dim
Y_{b}}(\mathcal{Y}^{\prime}\times_{B^{\prime}}\mathcal{Y}^{\prime})$ be cycles
that restrict to $\gamma$ resp. $\delta$. The spread lemma [28, Proposition
2.4], [27, Lemma 3.2] then implies that

$\Delta_{\mathcal{Y}^{\prime}}|_{b}=\bar{\gamma}|_{b}+\bar{\delta}|_{b}\ \ \
\hbox{in}\ A^{\dim Y_{b}}(Y_{b}\times Y_{b})\ \ \ \forall\ b\in B\ .\ $

Given any $b_{1}\in B\setminus B^{\circ}$, one can find representatives for
$\bar{\gamma}$ and $\bar{\delta}$ in general position with respect to the
fiber $Y_{b_{1}}\times Y_{b_{1}}$. Restricting to the fiber, this implies that
the diagonal of $Y_{b_{1}}$ has a decomposition as in (2), and so (2) holds
for all $b\in B$. Letting the decomposition (2) act on Chow groups, this shows
that

$A_{i}^{hom}(Y_{b})=0\ \ \ \forall\ i\leq c\ ,\ \ \ \forall\ b\in B\ .$

∎

##  3\. Main results

######  Theorem 3.1.

Let

$Y=\operatorname{Gr}(k,n)\cap H\ \ \ \subset\ \mathbb{P}^{{n\choose k}-1}$

be a smooth hyperplane section (with respect to the Plücker embedding). Then

$A_{i}^{hom}(Y)=0\ \ \ \forall i\leq n-k\ .$

###### Proof.

In view of Proposition 2.7, it suffices to prove this for generic hyperplane
sections, and so we may assume $Y$ is as in Proposition 2.2. The jump of
Proposition 2.2 gives rise to a commutative diagram

$\begin{array}[c]{ccccc}F&\hookrightarrow&q^{\ast}Y&\xrightarrow{q}&Y\\\
&&&&\\\ \downarrow{}&&\ \ \downarrow{\scriptstyle p}&&\\\ &&&&\\\
T&\hookrightarrow&\ \operatorname{Gr}(k-1,n)\ .&&\\\ \end{array}$

The morphism $q^{\ast}Y\to Y$ is a $\mathbb{P}^{k-1}$-bundle, and so the
projective bundle formula implies there are injections

(3) $A_{i}^{hom}(Y)\ \hookrightarrow\ A_{i}^{hom}(q^{\ast}Y)\ \ \ \forall i\
.$

For $Y$ sufficiently general, the locus $T$ will be smooth of codimension
$n-k+1$ (Proposition 2.2). The set-up is thus that of Cayley’s trick, with
$X=\operatorname{Gr}(k-1,n)$ and $E=Q^{\ast}(1)$. Applying Theorem 2.3 (with
$r=\operatorname{rank}Q^{\ast}(1)=n-k+1$), we find that there are isomorphisms

(4) $A_{i}^{hom}(q^{\ast}Y)\ \cong\ A_{i-n+k}^{hom}(T)\ .$

But $T$ is a smooth Fano variety (indeed, using adjunction one can compute
that the canonical bundle of $T$ is $\mathcal{O}_{T}(1-k)$), hence $T$ is
rationally connected [6], [15] and so in particular $A_{0}(T)\cong\mathbb{Q}$.
It follows that

(5) $A_{i-n+k}^{hom}(T)=0\ \ \ \forall\ i\leq n-k\ .$

Combining (3), (4) and (5), the theorem is proven. ∎

######  Theorem 3.2.

Let

$Y=\operatorname{Gr}(3,n)\cap H\ \ \ \subset\ \mathbb{P}^{{n\choose 3}-1}$

be a smooth hyperplane section (with respect to the Plücker embedding). Assume
$n\leq 13$, $n\not=12$. Then

$A_{i}^{hom}(Y)=0\ \ \ \forall i\leq n-2\ .$

###### Proof.

In view of Theorem 3.1, it only remains to treat the case $i=n-2$.

Applying Proposition 2.2, we may assume $Y$ is sufficiently generic. Doing the
jump as in the proof of Theorem 3.1 above, one finds a smooth variety
$T\subset\operatorname{Gr}(2,n)$ (of codimension $n-2$) and an injection of
Chow groups

(6) $A_{n-2}^{hom}(Y)\ \hookrightarrow\ A_{1}^{hom}(T)\ .$

In order to understand $A_{1}^{hom}(T)$, we perform a second jump. This second
jump (cf. [2, Section 3.4]) induces a diagram

$\begin{array}[c]{ccccc}F&\hookrightarrow&q^{\ast}T&\xrightarrow{q}&T\\\
&&&&\\\ \ \ \ \ \ \ \downarrow{\scriptstyle p|_{F}}&&\ \
\downarrow{\scriptstyle p}&&\\\ &&&&\\\ P^{\prime}&\hookrightarrow&\ P\ .&&\\\
\end{array}$

Here $q^{\ast}T\subset Fl(1,2,n)$, and the morphism $q^{\ast}T\to T$ is a
$\mathbb{P}^{1}$-bundle. The projective bundle formula gives an injection

(7) $A_{1}^{hom}(T)\ \hookrightarrow\ A_{1}^{hom}(q^{\ast}T)\ .$

The varieties $P$ and $P^{\prime}$ depend on the parity of $n$:

  * •

If $n$ is even, $P=\mathbb{P}^{n-1}$ and $P^{\prime}$ is the
$(n-4)$-dimensional Pfaffian variety called $P(1,n)$ in [2]. For $n\leq 10$,
the generic $P(1,n)$ is smooth and in this case $p$ is the blow-up of
$P=\mathbb{P}^{n-1}$ with center $P^{\prime}=P(1,n)$.

  * •

If $n$ is odd, $P$ is the Pfaffian hypersurface
$P(1,n)\subset\mathbb{P}^{n-1}$ (in the notation of [2]). For $n\leq 15$, the
generic $P(1,n)$ has singular locus $P^{\prime}\subset P(1,n)$ of codimension
$5$ and $P^{\prime}$ is smooth. In this case, the morphism $p$ is a
$\mathbb{P}^{1}$-bundle over $P\setminus P^{\prime}$, and a
$\mathbb{P}^{3}$-bundle over $P^{\prime}$.

Let us first treat the case $n$ even, $n\leq 10$. The blow-up formula gives an
isomorphism

$A_{1}^{hom}(q^{\ast}T)=A_{0}^{hom}(P^{\prime})\ .$

Since $P^{\prime}=P(1,n)$ is a smooth Fano variety, we have

$A_{0}^{hom}(P^{\prime})=0\ ,$

and so (combining with (6) and (7)) the theorem follows for $n$ even and
generic $Y$.

Next, let us treat the case $n$ odd, $n\leq 13$. In this case, we apply
Proposition 2.5 (with $h\in A^{1}(q^{\ast}T)$ the restriction of the
relatively ample class for $Fl(1,2,n)\to\mathbb{P}^{n-1}$). This gives a
(correspondence-induced) isomorphism

$A_{1}(q^{\ast}T)=A_{0}(P)\oplus A_{1}(P)\ ,$

and in particular an injection

$A_{1}^{hom}(q^{\ast}T)\ \hookrightarrow\ A^{hom}_{0}(P)\oplus A^{hom}_{1}(P)\
.$

But $P=P(1,n)\subset\mathbb{P}^{n-1}$ is a (singular) hypersurface of degree
$(n-3)/2$. For $n\leq 13$, it is known that any hypersurface
$P\subset\mathbb{P}^{n-1}$ of degree $\leq(n-3)/2$ has

$A_{0}^{hom}(P)=A_{1}^{hom}(P)=0\ .$

(For smooth $P$ this is proven in [22], the extension to singular $P$ is done
in [9]. Note that, at least for smooth $P$, Conjecture 1.1 states that the
restriction to $n\leq 13$ is not necessary.) Combined with (6) and (7), the
theorem follows for $n$ odd and $Y$ generic. ∎

##  4\. Some consequences

######  Corollary 4.1.

Let $Y$ be as in Theorem 3.2. Then $H^{\dim Y}(Y,\mathbb{Q})$ is supported on
a subvariety of codimension $n-1$.

###### Proof.

This follows in standard fashion from the Bloch–Srinivas argument. The
vanishing of Theorem 3.2 is equivalent to the decomposition

$\Delta_{Y}=\gamma+\delta\ \ \ \hbox{in}\ A^{\dim Y}(Y\times Y)\ ,$

where $\gamma$ is a completely decomposed cycle (i.e. $\gamma\in
A^{\ast}(Y)\otimes A^{\ast}(Y)$), and $\delta$ has support on $Y\times W$ with
$W\subset Y$ of codimension $n-1$. Let $H^{\dim Y}_{tr}(Y,\mathbb{Q})$ denote
the transcendental cohomology (i.e. the complement of the algebraic part under
the cup product pairing). The cycle $\gamma$ does not act on $H^{\dim
Y}_{tr}(Y,\mathbb{Q})$. The action of $\delta$ on $H^{\dim
Y}_{tr}(Y,\mathbb{Q})$ factors over $W$, and so

$H^{\dim Y}_{tr}(Y,\mathbb{Q})\ \ \subset\ H^{\dim Y}_{W}(Y,\mathbb{Q})\ .$

Since the algebraic part of $H^{\dim Y}(Y,\mathbb{Q})$ is (by definition)
supported in codimension $\dim Y/2$, this settles the corollary. ∎

######  Corollary 4.2.

Let

$Y=\operatorname{Gr}(3,9)\cap H\ \ \ \subset\ \mathbb{P}^{83}$

be a smooth hyperplane section (with respect to the Plücker embedding). Then

$A^{\ast}_{AJ}(Y)=0\ .$

In particular, $Y$ has finite-dimensional motive (in the sense of [14], [1]).

###### Proof.

Theorem 3.2 implies that

$A_{i}^{hom}(Y)=0\ \ \ \forall\ i\leq 7\ .$

The Bloch–Srinivas argument [5], [18] then implies that

$A^{j}_{AJ}(Y)=0\ \ \ \forall\ j\leq 9\ .$

Since $Y$ is $17$-dimensional, these two facts taken together mean that

$A^{\ast}_{AJ}(Y)=0\ ,$

as claimed. The fact that any variety $Y$ with $A^{\ast}_{AJ}(Y)=0$ is Kimura
finite-dimensional is [23, Theorem 4]. ∎

######  Acknowledgments .

Thanks to Kai and Len for making me listen to Sébastien Patoche. Thanks to the
referee for many constructive comments that significantly improved the paper.

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