# Block Encodings of Discrete Subgroups on Quantum Computer

Henry Lamm<EMAIL_ADDRESS>Fermi National Accelerator Laboratory, Batavia,
Illinois, 60510, USA  Ying-Ying Li<EMAIL_ADDRESS>Interdisciplinary
Center for Theoretical Study, University of Science and Technology of China,
Hefei, Anhui 230026, China  Peng Huanwu Center for Fundamental Theory, Hefei,
Anhui 230026, China  Jing Shu<EMAIL_ADDRESS>School of Physics and State Key
Laboratory of Nuclear Physics and Technology, Peking University, Beijing
100871, China  Center for High Energy Physics, Peking University, Beijing
100871, China  Beijing Laser Acceleration Innovation Center, Huairou, Beijing,
101400, China  Yi-Lin Wang<EMAIL_ADDRESS>Interdisciplinary
Center for Theoretical Study, University of Science and Technology of China,
Hefei, Anhui 230026, China  Peng Huanwu Center for Fundamental Theory, Hefei,
Anhui 230026, China  Bin Xu<EMAIL_ADDRESS>School of Physics and State Key
Laboratory of Nuclear Physics and Technology, Peking University, Beijing
100871, China

###### Abstract

We introduce a block encoding method for mapping discrete subgroups to qubits
on a quantum computer. This method is applicable to general discrete groups,
including crystal-like subgroups such as $\mathbb{BI}$ of $SU(2)$ and
$\mathbb{V}$ of $SU(3)$. We detail the construction of primitive gates – the
inversion gate, the group multiplication gate, the trace gate, and the group
Fourier gate – utilizing this encoding method for $\mathbb{BT}$ and for the
first time $\mathbb{BI}$ group. We also provide resource estimations to
extract the gluon viscosity. The inversion gates for $\mathbb{BT}$ and
$\mathbb{BI}$ are benchmarked on the Baiwang quantum computer with estimated
fidelities of $40^{+5}_{-4}\%$ and $4^{+5}_{-3}\%$ respectively.

††preprint: USTC-ICTS/PCFT-24-15, FERMILAB-PUB-24-0242-T

##  I Introduction

Gauge symmetries play important roles in quantum field theories, with the
$SU(3)\times SU(2)\times U(1)$ gauge symmetry being particularly important as
it encapsulates the interactions in the well-established Standard Model for
particle physics. Accurate predictions in the strongly-coupled regime of these
interactions require substantial computational resources. Over the past few
decades, Monte Carlo methods in lattice gauge theory (LGT) have flourished,
benefiting from advances in supercomputing capabilities and algorithmic
innovations. Nevertheless, challenges persist, particularly in scenarios
involving dynamic processes such as transport coefficients of the quark-gluon
plasma [1, 2, 3], parton physics in hadron collisions [4, 5, 6, 7, 8, 9], and
out-of-equilibrium evolution in the early universe [10, 11, 12, 13] due to
sign problems from the complex-valued nature of the Boltzmann sampling weight.
Quantum computers offer a promising avenue to circumvent this challenge by
enabling real-time simulations within the Hamiltonian formalism [14, 15, 16,
17, 18].

The Hilbert space of LGT is infinite-dimensional, requiring digitization
methods to facilitate its mapping onto a finite quantum memory (see Sec VI.b
of [17] for different digitization methods). These include the loop-string-
hadron (LSH) formulations [19, 20, 21] where the Hilbert space is built from
gauge-invariant operators, digitizations of independent Wilson loops [22, 23,
24, 25, 26, 27], qubitization formulations of gauge theories [28, 29, 30], and
the focus of this paper -the discrete subgroup approximation[31, 32].
Understanding and reducing the theoretical errors in the digitization is an
area of active research [33, 34, 35, 36, 37, 38, 39].

LGT calculations are performed at lattice spacing $a=a(\beta)$ which
approaches zero for asymptomatic free theories as the coupling parameter
$\beta\to\infty$. To perform extrapolation to the continuous spacetime limit,
calculations need to be done in the scaling regime with $\beta>\beta_{s}$. For
the discrete subgroup approximation, gauge links will become “frozen” to the
identity when $\beta>\beta_{f}$, leading to different behaviors from the
continuous groups, which makes the discrete subgroup approximations to be only
valid in the regime $\beta<\beta_{f}$. Thus to extrapolate to the correct
continuous spacetime limit as the continuous group, the discrete subgroup
approximations need to satisfy the condition $\beta_{f}>\beta_{s}$. The
discrete subgroup approximation has been explored for the Abelian group $U(1)$
[40, 41] and $SU(N)$ gauge theories [42, 43, 44, 45, 36, 46, 47, 35, 48, 49],
including fermions [50, 51]. For the case of $SU(2)$ which we consider in this
work, there are three crystal-like subgroups: $\mathbb{BT}$, $\mathbb{BO}$,
and $\mathbb{BI}$ with $\beta_{f}^{d}$ shown in Tab. 1. The smallest group,
$\mathbb{BT}$ has $\beta_{f}^{3+1}<\beta_{s}^{3+1}\sim 2.2$, while the other
two subgroup $\mathbb{BI}$, $\beta^{d}_{f}\gg\beta^{d}_{s}$ as shown in Fig. 1
for both $2+1d$ and $3+1d$ theories. The larger $\beta_{f}$ of $\mathbb{BI}$
implies that digitization errors should be smaller in comparison to
$\mathbb{BT}$ and $\mathbb{BO}$. Other attempts to increase $\beta_{f}$ with
modified actions are also being discussed [52].

Table 1:  Freezing couplings as a function of spatial dimension $d$, $\beta_{f}^{d+1}$, for crystal-like subgroups of $SU(2)$. For comparison, $\beta_{s}^{2+1d}=3$ and $\beta_{s}^{2+1d}=2.2$. $G$ | $\beta^{2+1d}_{f}$ | $\beta^{3+1d}_{f}$
---|---|---
$\mathbb{BT}$111Numerical results from [45] | 3.45(5) | 2.25(5)
$\mathbb{BO}$222Numerical results from [53] | 5.45(5) | 3.25(5)
$\mathbb{BI}$ | 9.65(5) | 5.85(5)

Figure 1:  Average plaquette or lattice energy density $\langle E_{0}\rangle$
of $\mathbb{BI}$ as a function of Wilson coupling parameter $\beta$ on $8^{d}$
lattices for (top) $2+1d$ with $\beta^{2+1}_{f}=9.65(5)$ and (bottom) $3+1d$
with $\beta^{3+1}_{f}=5.85(5)$. The shaded region corresponds to the phase
where $\beta\leq\beta_{s}$.

Though reducing the uncertainties in the extrapolation to the continuum with
larger $\beta_{f}$, quantum simulations of larger discrete group will be more
complicated. To simulate the discrete subgroup theory on a quantum computer,
one need to obtain the mapping between the elements of the discrete group and
the qubits. The usual way is to write the elements of certain discrete
subgroup as an ordered product of group generators with the exponents mapped
to qubits or qudits on quantum computer [45, 53, 54, 55]. However, for the
largest crystal-like $\mathbb{BI}$ and $\mathbb{V}$ subgroups of $SU(2)$ and
$SU(3)$ theories, the ordered product expression is unknown yet, and may not
be possible. This potential obstacle motivates the search for other encodings
of discrete groups.

In this work, we consider a new encoding – _block encoding_ – expressing
groups as $d$-dimensional matrices over a finite field $\mathbf{F}$. According
to Cayley’s theorem, any finite group of order $n$ can be mapped by an
injective homomorphism to the general linear group $GL(d,\mathbf{F})$ over
certain finite field $\mathbf{F}$, with dimension $d\leq n$. Each matrix
element will be sorted in its own register, and then the register representing
the group element is built from them. This encoding methods can be applied to
generic discrete subgroups, including the largest discrete crystal-like
$\mathbb{BI}$ subgroup of $SU(2)$ and $\mathbb{V}$ of $SU(3)$. After reviewing
the basics of group representation in Sec. II, we introduce the block encoding
methods. In Sec. III, we review the primary gates to implement group element
operations and basic logic quantum gates set in constructing the primary
gates. We then present the construction of primary gates based on the block
encoding for the $\mathbb{BT}$, $\mathbb{BI}$ in Sec. IV. Resource
requirements are estimated and compared to other digitization methods in Sec.
V. Fidelities for the block encoding with quantum errors are analyzed in Sec.
VI, followed by the benchmarking of the inverse gates in Sec. VII. In Sec.
VIII, we present the conclusion and outlook. Appendices include an alternative
constructions based on two’s complement for block encoding.

##  II Discrete subgroup encodings

For any encoding of discrete groups, the group elements $g$ are mapped to at
least as many integers as $p=[0,|G|-1]$ where $|G|$ is the group dimension.
Previous work focused on the _ordered product_ encoding which maps onto the
integers the integer exponents $\\{o_{k}\\}$ of group generators
$\\{\lambda_{k}\\}$ in a particular ordering:

$\displaystyle g_{\\{o_{k}\\}}=$ $\displaystyle\prod_{k}\lambda^{o_{k}}_{k}$
(1)

where $0\leq o_{k}<O_{k}$. $O_{k}$ can be as large as the generator’s order
$\lambda_{k}^{O_{k}}=\mathbb{1}$ but often lower when redundancies occur e.g.
$\lambda_{1}^{O_{1}}=\lambda_{2}$. An integer mapping is then defined as

$\displaystyle
p=o_{k}+O_{k}(o_{k-1}+O_{k-1}(\ldots+O_{2}(o_{1}+O_{1}o_{0}))).$ (2)

With this, one may consider encoding $p$ onto quantum memory by decomposing
$p$ via Eq. (2) where $\\{O_{k}\\}$ are replaced by the dimensionality of the
qudits $\\{D_{k}\\}$ available. Both $\mathbb{BT}$ and $\mathbb{BO}$ have been
formulated in this way [45, 53]. As an example, the group elements of
$\mathbb{BT}$ can be represented as

$g=(-1)^{m}\mathbf{i}^{n}\mathbf{j}^{o}\mathbf{l}^{p}.$ (3)

where $\mathbf{i,j,k}$ are the unit quaternions and
$\mathbf{l}=-(\mathbf{1+i+j+k})/2$. $m,n,o$ are 3 binary variables, while $p$
is tertiary, and thus can be encoded for example in a ($p=8$) quoctit and
($p=3$) qutrit or in 5 qubits with some forbidden states. A similar
construction for $\mathbb{BO}$ could be realized with 6 qubits, or a qutrit
with either 2 quoctits ($p=4$) or a qudecasexit ($p=16$) [54]. The
corresponding qubit primitive gates to implement group inverse, product, trace
and Fourier transformation has been worked out recently [45, 53]. While this
encoding can be efficient in quantum memory, it is not a given that all finite
groups can be encoding with ordered products.

Due to the potential limitations of the ordered product encoding, we consider
a different encoding, block encoding, which instead maps the finite fields in
the matrices representing $g$ to a set of qubits. This encoding relies on the
ability to represent the matrix elements themselves as valued only over a
finite field. Here, we will consider how to encode $\mathbb{BT}$ and
$\mathbb{BI}$ as examples.

The $\mathbb{BT}$ group is isomorphic to the special linear group $SL(2,3)$ —
the group of all $2\times 2$ matrices with unit determinant over the three-
element finite field $\mathbf{F}_{3}$ (which can be defined as the ring of
integers modulo 3). The character table for $\mathbb{BT}$ is found at the
bottom of Tab. 2. Using the isomorphism, $g$ can be represented by

$\displaystyle g\in\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\bigg{|}\,a,b,c,d\in\mathbf{F}_{3},ad-
bc\equiv\,1\,\text{mod}(3)\right\\}$ (4)

This leads to another way of encoding $\mathbb{BT}$ group elements using four
ternary variables $a,b,c,d=\\{0,1,2\\}$ as $\ket{g}=\ket{abcd}$ with a Hilbert
space of $3^{4}-57=24$ states where 57 states are removed by the determinant
constraints.

In this work, we encode each matrix element in Eq. (4) as a binary integer
into two qubits, e.g. the non-negtive encoding with $\ket{0}=\ket{00}$,
$\ket{1}=\ket{01}$, $\ket{2}=\ket{10}$ and a forbidden state $\ket{11}$. Given
the redundancy in this encoding, we can also use the two’s complement encoding
as $\ket{0}=\ket{00},\ket{1}=\ket{01}$ and $\ket{2}=\ket{11}$. In the
$\mathbb{BT}$ case, the two’s component encoding is the same as the Gray code
[56, 57, 58, 59]. Alternatively, the matrix elements could each be encoding in
a ($p=3$) qutrit without forbidden states. In this paper, we adopt the non-
negative encoding, resulting in a 8-qubit group register while leaving
optimizations of quantum resources among different encoding to the future as
considerations to noise is crucial [54].

Table 2: Character Tables of (left) $\mathbb{BT}$ and (right) $\mathbb{BI}$
including a representative element in the given conjugacy class, where
$\omega=e^{2\pi i/3}$, $\tau=(1+\sqrt{5})/2$, and $\sigma=(1-\sqrt{5})/2$.

$\mathbb{BT}$ | $C_{1}$ | $C^{\prime}_{1}$ | $C_{4}$ | $C^{\prime}_{4}$ | $C^{\prime\prime}_{4}$ | $C^{\prime\prime\prime}_{4}$ | $C_{6}$
---|---|---|---|---|---|---|---
Ord. | 1 | 2 | 3 | 3 | 6 | 6 | 4
$\chi_{1}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1
$\chi_{1^{\prime}}$ | 1 | 1 | $\omega$ | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | 1
$\chi_{1^{\prime\prime}}$ | 1 | 1 | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | $\omega$ | 1
$\chi_{2}$ | 2 | -2 | -1 | -1 | 1 | 1 | 0
$\chi_{2^{\prime}}$ | 2 | -2 | $-\omega$ | $-\omega^{2}$ | $\omega$ | $\omega^{2}$ | 0
$\chi_{2^{\prime\prime}}$ | 2 | -2 | $-\omega^{2}$ | $-\omega$ | $\omega^{2}$ | $\omega$ | 0
$\chi_{3}$ | 3 | 3 | 0 | 0 | 0 | 0 | -1
$|g\rangle$ | $\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$ | $\begin{pmatrix}2&0\\\ 0&2\end{pmatrix}$ | $\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$ | $\begin{pmatrix}1&2\\\ 0&1\end{pmatrix}$ | $\begin{pmatrix}2&2\\\ 0&2\end{pmatrix}$ | $\begin{pmatrix}2&1\\\ 0&2\end{pmatrix}$ | $\begin{pmatrix}0&1\\\ 2&0\end{pmatrix}$

$\mathbb{BI}$ | $C_{1}$ | $C^{\prime}_{1}$ | $C_{30}$ | $C_{20}$ | $C^{\prime}_{20}$ | $C_{12}$ | $C^{\prime}_{12}$ | $C^{\prime\prime}_{12}$ | $C^{\prime\prime\prime}_{12}$
---|---|---|---|---|---|---|---|---|---
Ord. | 1 | 2 | 4 | 3 | 6 | 5 | 10 | 5 | 10
$\chi_{1}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
$\chi_{2}$ | 2 | -2 | 0 | -1 | 1 | $-\tau$ | $\tau$ | $-\sigma$ | $\sigma$
$\chi_{2^{\prime}}$ | 2 | -2 | 0 | -1 | 1 | $-\sigma$ | $\sigma$ | $-\tau$ | $\tau$
$\chi_{3}$ | 3 | 3 | -1 | 0 | 0 | $\tau$ | $\tau$ | $\sigma$ | $\sigma$
$\chi_{3^{\prime}}$ | 3 | 3 | -1 | 0 | 0 | $\sigma$ | $\sigma$ | $\tau$ | $\tau$
$\chi_{4}$ | 4 | 4 | 0 | 1 | 1 | -1 | -1 | -1 | -1
$\chi_{5}$ | 5 | 5 | 1 | -1 | -1 | 0 | 0 | 0 | 0
$\chi_{4^{\prime}}$ | 4 | -4 | 0 | 1 | -1 | -1 | 1 | -1 | 1
$\chi_{6}$ | 6 | -6 | 0 | 0 | 0 | 1 | -1 | 1 | -1
$|g\rangle$ | $\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$ | $\begin{pmatrix}4&0\\\ 0&4\end{pmatrix}$ | $\begin{pmatrix}0&1\\\ 4&0\end{pmatrix}$ | $\begin{pmatrix}0&1\\\ 4&4\end{pmatrix}$ | $\begin{pmatrix}0&4\\\ 1&1\end{pmatrix}$ | $\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$ | $\begin{pmatrix}4&4\\\ 0&4\end{pmatrix}$ | $\begin{pmatrix}1&2\\\ 0&1\end{pmatrix}$ | $\begin{pmatrix}4&3\\\ 0&4\end{pmatrix}$

The block encoding can be extended to other groups. The $\mathbb{BI}$ group is
isomorphic to the special linear group $SL(2,5)$ — the group of all $2\times
2$ matrices over the finite field $\mathbf{F}_{5}$ with unit determinant. The
character table for $\mathbb{BI}$ is found at the bottom of Tab. 2 and the
group element can be represented by

$\displaystyle g\in\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\bigg{|}\,a,b,c,d\in\mathbf{F}_{5},ad-bc\equiv 1\text{
mod}(5)\right\\}$ (5)

We can encode the $\mathbb{BI}$ group elements with four quinary variables
$a,b,c,d=\\{0,1,2,3,4\\}$, and thus the matrix elements can be represented
with three qubits with forbidden states $\ket{101},\ket{110},\ket{111}$, and
the group element with 12 qubits. Alternatively, the matrix elements could be
encoding into ($p=5$) ququints without forbidden states. In this work, we
derive the primitive gates of $\mathbb{BI}$ with a 12-qubit group register.

It is worth noting that the largest crystal-like subgroup of $SU(3)$ –
$\mathbb{V}$ – is isomorphic to a subgroup of $GL(3,4)$ and can be encoded
with nine 9 ququarts or 18 qubits, though further investigations are needed to
implement the quantum gates that can project the Hilbert space to its
$\mathbb{V}$ subspace.

##  III Primitive gate overviews

Quantum circuits for pure gauge theories can be constructed out of a set of
primitive gates [60] acting on one or more group element registers. The
necessary gates for simulation and extraction of observables are:

  * •

the inverse gate: $\mathfrak{U}_{-1}|g\rangle=\left|g^{-1}\right\rangle$,
which computes in-place the inverse of a group element,

  * •

the trace gate:
$\mathfrak{U}_{\operatorname{Tr}}(\theta)|g\rangle=e^{i\theta\operatorname{Re}\operatorname{Tr}g}|g\rangle$,
which introduces a phase based on the trace of a group element in a particular
representation and weighted by a coupling $\theta$ which can depend on the
Hamiltonian and approximation used in time evolution,

  * •

the left multiplication gate:
$\mathfrak{U}_{\times}|g_{i}\rangle|g_{j}\rangle|{\rm
anc}\rangle=|g_{i}\rangle|g_{j}\rangle|g_{i}g_{j}\rangle$. In this work, we
consider a novel definition of $\mathfrak{U}_{\times}$ which stores the result
in an ancilla group register. The right multiplication gate, if desired, can
be defined via the $\mathfrak{U}_{-1}$, and $\mathfrak{U}_{\times}$,

  * •

the group Fourier gate:
$\mathfrak{U}_{F}\sum_{g}f(g)\left|g\right>=\sum_{\rho}\hat{f}(\rho)_{ij}\left|\rho,i,j\right>$
with $\hat{f}$ denoting the Fourier transform of $f$.

For qubit-based computers, we need 8 qubits to encode a $\mathbb{BT}$ register
and 12 qubits for $\mathbb{BI}$. We will present the primitive gates
construction using five entangling gates: the two qubit CNOT and SWAP, the
three qubit CSWAP (known as the Fredkin gate), and the multi-controlled phase
${\rm C}^{n}{\rm P(\phi)}$ and CnNOT quantum gates. The special case of C2NOT
is commonly called the Toffoli gate. The first three implement the operations:

$\displaystyle\text{CNOT}\ket{q_{1}}\ket{q_{2}}$
$\displaystyle=\ket{q_{1}}\left|q_{1}\oplus q_{2}\right>$ (6)
$\displaystyle\text{SWAP}\ket{q_{1}}\ket{q_{2}}$
$\displaystyle=\ket{q_{2}}\ket{q_{1}}$ (7) $\displaystyle{\rm
CSWAP}\ket{q_{1}}\ket{q_{2}}\ket{q_{3}}$
$\displaystyle=\ket{q_{1}}\ket{q_{3}}\ket{q_{2}}.$ (8)

For the multi-controlled gates, which apply and operation to one qubit based
on the state of $n-1$ others we have

$\displaystyle{\rm C}^{n}{\rm P(\phi)}\ket{q_{1}}\dots\left|q_{n}\right>$
$\displaystyle=\ket{q_{1}}\dots e^{i\phi q_{1}\dots
q_{n-1}}\left|q_{n}\right>$ (9) $\displaystyle{\rm C}^{n}{\rm
NOT}\ket{q_{1}}\dots\left|q_{n}\right>$
$\displaystyle=\ket{q_{1}}\dots\left|q_{n}\oplus q_{1}\dots q_{n-1}\right>.$
(10)

##  IV Primitive Gates for $\mathbb{BT}$ and $\mathbb{BI}$

To construct the primitive gates based on block encodings, we adopt the
following conventions. We use parentheses to denote operations that return
values with mod $n$ applied for $SL(2,n)$ group. The values of matrix element
are the remainders mod $n$. A feature of the block encoding is that the
primitive gates for $\mathbb{BT}$ and $\mathbb{BI}$ are quite similar, with
the difference mainly arising from the modular arithmetic. Therefore we will
consider the two groups in parallel as we construct their gates.

###  IV.1 Inverse Gate

For the construction of $\mathfrak{U}_{-1}$, we note that the inverse $g^{-1}$
is given in terms of the matrix elements of $g$ by

$\displaystyle g^{-1}=\begin{pmatrix}d&-b\\\ -c&a\end{pmatrix},$ (11)

From this, we see that the inverse operation corresponds to swapping the
values of $a,d$ and flipping the sign of $b,c$. Using this, the
$\mathfrak{U}_{-1}$ circuits can be derived and are presented in the left of
Fig. 2 and Fig. 3 for $\mathbb{BT}$ and $\mathbb{BI}$ respectively.

Figure 2: Inverse gate $\mathfrak{U}_{-1}$ (left) and Trace gate
$\mathfrak{U}_{\rm Tr}(\theta)$ (right) for $\mathbb{BT}$ group.

Figure 3: Inverse gate $\mathfrak{U}_{-1}$ (left) and trace gate
$\mathfrak{U}_{\rm Tr}$ (right) for $\mathbb{BI}$ group.

###  IV.2 Trace Gate

Our interest here is in the traces of each group elements in the faithful
representation, which are denoted by $\chi_{2}$ in Tab. 2. With one
representative element in the given conjugacy class in Tab. 2, one can
determine the other elements in the class. This enables a derivation of the
rules for obtaining $\operatorname{Tr}(g)$ in the blocking encoding. For
$\mathbb{BT}$, these rules are

$\displaystyle\operatorname{Tr}(g)=\begin{dcases}2&{\rm
if~{}}(a+d)=2\,\&\,(c-b)=0\\\ 1&{\rm if~{}}(a+d)=1\,\&\,(c-b)\neq 0\\\ 0&{\rm
if~{}}(a+d)=0\\\ -1&{\rm if~{}}(a+d)=2\,\&\,(c-b)\neq 0\\\ -2&{\rm
if~{}}(a+d)=1\,\&\,(c-b)=0\end{dcases}$ (12)

While for $\mathbb{BI}$, they are found to be

$\displaystyle\operatorname{Tr}(g)=\begin{dcases}0&{\rm if~{}}(a+d)=0\\\
1&{\rm if~{}}(a+d)=1\\\ -1&{\rm if~{}}(a+d)=4\\\ 2&{\rm
if~{}}(a+d)=2\,\&\,(c^{2}+b^{2})=0\\\ -\tau&{\rm
if~{}}(a+d)=2\,\&\,(c^{2}+b^{2})=1{\rm~{}or~{}}2\\\ -\sigma&{\rm
if~{}}(a+d)=2\,\&\,(c^{2}+b^{2})=3{\rm~{}or~{}}4\\\ -2&{\rm
if~{}}(a+d)=3\,\&\,(c^{2}+b^{2})=0\\\ \tau&{\rm
if~{}}(a+d)=3\,\&\,(c^{2}+b^{2})=1{\rm~{}or~{}}2\\\ \sigma&{\rm
if~{}}(a+d)=3\,\&\,(c^{2}+b^{2})=3{\rm~{}or~{}}4\\\ \end{dcases}.$ (13)

By inspecting these rules, one notices that to realize $\mathfrak{U}_{\rm
Tr}(\theta)$, we first implement an addition operation $\mathfrak{U}_{\rm
a}\ket{a}\ket{b}=\ket{a}\ket{b\oplus a}$. These are constructed for
$\mathbb{BT}$ and $\mathbb{BI}$ in Fig. 4. In the case of $\mathbb{BI}$, it is
useful to further define subroutines $\mathfrak{U}_{\rm+n}$ for $n=1,2$ which
increment the matrix element $\ket{a}$ to $\ket{a+n}$ as shown in Fig. 5. The
operation of $(b-a)$ can be implemented using $\mathfrak{U}^{\dagger}_{\rm
a}$. With $\mathfrak{U}_{a}$, we can construct $\mathfrak{U}_{\rm
Tr}(\theta)$, as shown in the right of Fig. 2 and Fig. 3 where $\theta$
depends on couplings and approximations in the time evolution.

=  =

Figure 4: Implementation of $\mathfrak{U}_{\rm a}$ for $\mathbb{BT}$ (left)
and $\mathbb{BI}$ group(right).

Figure 5: Elementary operations $\mathfrak{U}_{\rm+1}$ and
$\mathfrak{U}_{\rm+2}$ that add one and two to a single matrix element,
respectively for the $\mathbb{BI}$ group.

=  =

Figure 6: Implementation of $\mathfrak{U}_{\rm p}$ for $\mathbb{BT}$ (left)
and $\mathbb{BI}$ group(right).

###  IV.3 Multiplication Gate

Moving on to $\mathfrak{U}_{\times}$, the multiplication of two group elements
$g_{i}$ and $g_{j}$ in terms of the matrix elements is

$\begin{pmatrix}a_{i}&b_{i}\\\
c_{i}&d_{i}\end{pmatrix}\begin{pmatrix}a_{j}&b_{j}\\\
c_{j}&d_{j}\end{pmatrix}=\begin{pmatrix}a_{i}a_{j}+b_{i}c_{j}&a_{i}b_{j}+b_{i}d_{j}\\\
c_{i}a_{j}+d_{i}c_{j}&c_{i}b_{j}+d_{i}d_{j}\end{pmatrix}$ (14)

The multiplication of group elements can thus be built from blocks of
$\mathfrak{U}_{\rm a}$ and a new subroutine $\mathfrak{U}_{\rm p}$ which
calculate the product of two elements (See Fig. 6). In order to leave the
matrix elements intact, ancillary qubits store the product. One ancillary
register will be required to store each matrix element in Eq. (14).
Nevertheless, to optimise the parallelization, we actually introduce one
ancillary register for each product in Eq. (14), which doubles the ancillary
qubits required. The group multiplication circuit $\mathfrak{U}_{\times}$ is
subsequently constructed as in Fig. 7.

Figure 7: Multiplication gate $\mathfrak{U}_{\times}$ for $\mathbb{BT}$ group
and $\mathbb{BI}$ group.

###  IV.4 Fourier Gate

To reduce the quantum gates required for implmenting Fourier gate, we can
project both $\mathbb{BT}$ and $\mathbb{BI}$ group to the subspace satisfying
the determinate constraints using circuit $\mathfrak{U}_{proj}$ and ancillary
qubits, which will reduce the encoding from 8 and 12 qubits to 6 and 9 qubits,
respectively. $\mathfrak{U}_{proj}$ is implemented with 4 CNOT gates, 3 CSWAP
gates and 8 Tofolli gates for $\mathbb{BT}$ group. For the $\mathbb{BI}$
group, $\mathfrak{U}_{proj}$ requires 8 CNOT gates, 3 CSWAP gates, 8 Toffoli
gates and 2 $\mathfrak{U}_{\rm p}$ gates. With such projection, transforming
to the Fourier basis can be decomposed using the Qiskit transpiler. The
resource required for Fourier gate implemented this way for both groups are
shown in Tab. 3.

| $\mathfrak{U}_{F}$
---|---
| $R_{X}$ | $R_{Y}$ | $R_{Z}$ | CNOT
$\mathbb{BT}$ | 186 | 2052 | 3491 | 1941
$\mathbb{BI}$ | 10743 | 131879 | 223044 | 125919
Table 3: $\mathfrak{U}_{F}$ decompositions for $\mathbb{BT}$ and $\mathbb{BI}$
group.

$\mathfrak{U}_{F}$ is usually the most costly primitive gate for quantum
simulations. Future direction should consider the fast Fourier transformation
in [61] where sub-exponential growth $\mathcal{\exp(\sqrt{\log|G|})}$ of the
circuit depth in the group size is possible. In particular, it should be
noticed that the Fourier transformation of $\mathbb{BI}$ group can be built
upon that of $\mathbb{BT}$ group using the following natural tower:

$\mathbb{BI}>\mathbb{BT}>\mathbb{Z}_{6}>\mathbb{Z}_{3}>\mathbb{Z}_{1}=\\{1\\}.$
(15)

which could simplify the constructions for the $\mathbb{BI}$ group once the
construction for $\mathbb{BT}$ is realized. It will be valuable to realize
such fast Fourier transformation and compare the resources required with our
brutal direct decomposition, which we will leave for the future work.

With these four primitive gates, it is possible to perform a resource estimate
and compare to other implementations of $SU(2)$. This will be done in Sec. V.
Additionally, the primitive gates for the two’s component encoding are
discussed in App. A. There we find that total gate costs are similar, but
given the different distribution of qubit states, they are anticipated to have
different noise robustness.

##  V Resource Requirements

As T gate counts are known to require costly encodings for error correction,
it is usually used in fault-tolerent resource analysis [62, 63]. We estimate
the T gates required for simulating both $\mathbb{BT}$ or $\mathbb{BI}$ group
with the block encoding method. The Toffoli gate requires 7 T gates [62];
${\rm C}^{n}{\rm NOT}$ can be constructed with $(2[\log_{2}(n+1)]-1)$ toffoli
gates and $n-2$ dirty ancilla qubits for $n>2$ [62]. ${\rm C}^{n}{\rm SWAP}$
require the same number of T gates as ${\rm C}^{n+1}{\rm NOT}$, as it can be
decomposed to ${\rm C}^{n+1}{\rm NOT}$ and CNOTs using the symmetric
decomposition [64]. Approximating $R_{Z}$ gates to the precision of $\epsilon$
will require $1.15\log_{2}(1/\epsilon)$ T gates [53], while $R_{X}$ and
$R_{Y}$ can be implemented with at most three $R_{Z}$ gates. ${\rm C}{\rm
P(\phi)}$ requires 8 T gate and one $R_{Z}$ gate with one clean ancillary
qubit [65], and ${\rm C}^{n}{\rm P(\theta)}$ can be decomposed to $2(n-1)$
Toffoli gates and one ${\rm C}{\rm P(\theta)}$ with $n-1$ clean ancilla [66].
The estimated number of T gates to realize each primary gates are listed in
Tab. 4.

gate | T ($\mathbb{BT}$) | T ($\mathbb{BI}$)
---|---|---
$\mathfrak{U}_{-1}$ | 0 | 14
$\mathfrak{U}_{\rm Tr}$ | 172 + 4.6 $\log_{2}(1/\epsilon)$ | 666 + 9.2 $\log_{2}(1/\epsilon)$
$\mathfrak{U}_{\times}$ | 336 | 3640
$\mathfrak{U}_{F}$ | $11735.8\log_{2}(1/\epsilon)$ | $748547\log_{2}(1/\epsilon)$
Table 4: Number of T gates required to realize the primary gates.

We compare the resources required for the calculation of the viscosity as that
in [67]. The total T gate count for certain Hamiltonian $H$ is given by
$N^{H}_{T}=C^{H}_{T}\times dL^{d}N_{t}$ for a $d$ spatial lattice simulated
for a time $t=N_{t}\delta t$ [53], where $C^{H}_{T}$ is the average number of
T gates require per link per $\delta t$, and $dL^{d}$ is the total number of
links. We first consider the simulation of the $\mathbb{BT}$ group. With the
primitive gates per link per $\delta t$ listed in [53], we get

$\displaystyle C^{H_{\rm KS}}_{T}$
$\displaystyle=2102(d-1)+(23469.2+2.3d)\log_{2}\frac{1}{\epsilon}$ (16)
$\displaystyle C^{H_{I}}_{T}$
$\displaystyle=8994d-7650+(46936.1+6.9d)\log_{2}\frac{1}{\epsilon},$ (17)

using the Kogut-Susskind Hamiltonian $H_{\rm KS}$ and the improved Hamiltonian
$H_{I}$ studied in [68], respectively. With this, the total synthesis error
$\epsilon_{T}$ can be estimated as the sum of $\epsilon$ from each $R_{Z}$:

$\displaystyle\epsilon^{H_{KS}}_{T}$
$\displaystyle=2(10204+d)dL^{d}N_{t}\times\epsilon$ (18)
$\displaystyle\epsilon^{H_{I}}_{T}$
$\displaystyle=2(20407+3d)dL^{d}N_{t}\times\epsilon.$ (19)

For simulating the $\mathbb{BI}$, the cost increase moderately:

$\displaystyle C^{H_{\rm KS}}_{T}$
$\displaystyle=22215(d-1)+(1497090+4.6d)\log_{2}\frac{1}{\epsilon}$ (20)
$\displaystyle C^{H_{I}}_{T}$
$\displaystyle=95793d-81205+(2994170+13.8d)\log_{2}\frac{1}{\epsilon},$ (21)

and total synthesis errors:

$\displaystyle\epsilon^{H_{KS}}_{T}$
$\displaystyle=4(325454+d)dL^{d}N_{t}\times\epsilon$ (22)
$\displaystyle\epsilon^{H_{I}}_{T}$
$\displaystyle=12(216969+d)dL^{d}N_{t}\times\epsilon.$ (23)

To calculate the shear viscosity with the total synthesis error
$\epsilon_{T}=10^{-8}$ on a $d=3$ lattice with $L^{3}=10^{3}$ for $N_{t}=50$,
finding $N^{H_{\rm KS}}_{T}=2.1\times 10^{11}$ and $N^{H_{I}}_{T}=4.2\times
10^{11}$. For the ordered product method in simulating $\mathbb{BT}$ group
[45], the total number of T gates are estimated as 1.1$\times 10^{11}$ for
$H_{\rm KS}$ and 4.1$\times 10^{11}$ for $H_{I}$. We note that $H_{\rm KS}$ is
inadequate to reach the scaling regime with $\mathbb{BT}$, but for the case of
$H_{I}$, the T gate cost is only $\sim 3\%$ higher – and in both cases
dominated by $\mathfrak{U}_{F}$. With only 60% higher qubit costs, we conclude
that this encoding is comparable to ordered product on these simple metrics,
and further analysis of other metrics like noise robustness should be
undertaken. For the larger $\mathbb{BI}$ group – where no ordered product
encoding currently exists – the costs increase to $N^{H_{\rm
KS}}_{T}=1.4\times 10^{13}$ and $N^{H_{I}}_{T}=2.9\times 10^{13}$ T gates
using $H_{KS}$ and $H_{I}$, respectively. But given $\beta_{f}\gg\beta_{s}$
for $\mathbb{BI}$, the increase in gate costs may be an acceptable trade-off
due to the reduced systematic error from digitization.

We can also compare the the block encoding of discrete subgroups to other
digitization methods, such as LSH formalism, in which the T gate counts has
been estimated for $d=1$ in [69]. For comparison, we take the finite
arithmetic precision errors to be smaller than the synthesis error
$\epsilon_{T}=10^{-8}$ by taking minimally $n=7$ steps and $m=42$ bits to
evaluate the inverse-square-root functions using Newton’s method. We choose 5
and 7 qubits to encode one group register in LSH to match the minimal number
of qubits that can encode $\mathbb{BT}$ and $\mathbb{BI}$ group, respectively.
With $L=10$, LSH will require in trotter step using singular-value-
decomposition roughly $3.9\times 10^{7}$ and $4.2\times 10^{7}$ T gates for
the 5 qubits and 7 qubits case, respectively, while for the blocking encoding,
$\mathbb{BT}$ and $\mathbb{BI}$ requires $1.3\times 10^{7}$ and $8.0\times
10^{8}$ T gates. Given this similarity in resource costs, quantifying the
systematic and statistical errors in digitization methods becomes an important
concern. In the next section, we consider the relative resilience of the block
encoding versus the ordered producted encoding. Additionally, a comprehensive
comparision to other forms of approximating the time evolution in LGT [33, 70]
are desirable, but understanding the theoretical LGT errors must be understood
for a complete comparison.

##  VI Resilience to Quantum Errors

In this section, we investigate the resilience to errors for a single register
using the block encoding and the ordered product encoding, considering
$\mathbb{BT}$. Given that redundant degrees of freedom are introduced in both
encodings, one can utilize these redundancies in QEC [71, 72, 73, 54, 74, 75,
76, 77, 78, 79, 80, 81, 82]. In the following, we derive the error rate
threshold [82] below which the block encoding would provide higher fidelity
than the ordered product one for a single register as more redundancies are
introduced in the block encoding. We will take the bit-flip error as an
example and consider the fidelity as the one averaging over all group
elements. Error mitigation using post-selection remove trials affected by
detectable errors. For the error channel with only one bit-flip error
$\mathcal{N}_{i}$ affecting qubit $i$ at an error rate of $\epsilon$, we count
the number of $\mathcal{N}_{i}$ that transform the group element into
forbidden state, which are detectable errors. Using the ordered product
encoding, there are in total 16 detectable $\mathcal{N}_{i}$ for all 24 group
elements, while for the block encoding, 160 $\mathcal{N}_{i}$. The lower bound
for the fidelity is the probability of no errors after removing trials
affected by these detectable one bit-flip errors:

$\displaystyle\mathcal{F}^{{\rm ps}}_{\rm BE}$ $\displaystyle\geq$
$\displaystyle\frac{(1-\epsilon)^{8}}{1-160/24(1-\epsilon)^{7}\epsilon},$
$\displaystyle\mathcal{F}^{{\rm ps}}_{\rm OPE}$ $\displaystyle\geq$
$\displaystyle\frac{(1-\epsilon)^{5}}{1-16/24(1-\epsilon)^{4}\epsilon}.$ (24)

For logical error rates $\epsilon\lesssim 0.1$, we found the lower bound of
the fidelity for the current encoding methods can be higher. As post-
selections require resources that are exponential [83] in system size, we also
consider correcting one bit-flip error to reduce the effects of quantum
errors. The correctable one bit-flip errors transform the group element to a
forbidden state that cannot be transformed from other group element via any
one bit-flip error (Knill-Laflamme condition [84]). Under this condition, when
a forbidden state is seen, the error channel can be inferred and corrected.
Using the ordered product encoding, there are in total 0 correctable
$\mathcal{N}_{i}$ for all 24 group elements, while for the block encoding, 80
$\mathcal{N}_{i}$. Correcting trials affected by correctable
$\mathcal{N}_{i}$, we obtain the lower bound of the averaged fidelity as the
probability with no errors or only correctable one bit-flip errors:

$\displaystyle\mathcal{F}^{\rm cor}_{\rm BE}$ $\displaystyle\geq$
$\displaystyle(1-\epsilon)^{8}+\frac{80}{24}(1-\epsilon)^{7}\epsilon,$
$\displaystyle\mathcal{F}^{\rm cor}_{\rm OPE}$ $\displaystyle\geq$
$\displaystyle(1-\epsilon)^{5}.$ (25)

We observe that in this situation for $\epsilon\lesssim 0.1$, the lower bound
of the fidelity for the current encoding methods can be mildly higher.

##  VII Experimental Results

Figure 8: Fidelity of $\mathfrak{U}^{\rm trans}_{-1}$ gate of Baiwang for each
group element $|g\rangle=|abcd\rangle$ of $\mathbb{BT}$ and $\mathbb{BI}$,
which is labeled by the lexicographic order in the range $\\{0,|G|-1\\}$. The
averaged fidelity $\bar{\mathcal{F}}$ over all group elements are also shown.
On the top right, the qubit graph on Baiwang used to represent $\ket{g}$ for
(left) $\mathbb{BT}$ and (right) $\mathbb{BI}$ respectively.

Using the block encoding method, we benchmark the fidelity of the inverse gate
for $\mathbb{BT}$ and $\mathbb{BI}$ group on the Quafu platform [85]. Given
the availability, we used the Baiwang quantum real machine on Quafu which has
144 qubits and 136 available. The qubits are arranged in a $12\times 12$
lattice, where qubits in each row are connected adjacently and adjacent rows
are connected by only 2-4 qubit connections [85]. 8 qubits and 12 qubits are
selected for the simulation of $\mathbb{BT}$ and $\mathbb{BI}$ group,
respectively, with their positions and connectivity shown in Fig. 8. Starting
from state $|\psi_{0}\rangle=|0\rangle^{\otimes n_{q}}$ with $n_{q}=8(12)$ for
$\mathbb{BT}(\mathbb{BI})$ group, we can prepare the initial state $|g\rangle$
with circuit $\mathfrak{U}_{g}$: $|g\rangle=\mathfrak{U}_{g}|\psi_{0}\rangle$.
For single states, $\mathfrak{U}_{g}$ is simply a tensor product of X gates
used to initialize some qubits to $|1\rangle$. Adapting to the connectivity of
the qubits chosen on Baiwang quantum chip, $\mathfrak{U}_{-1}$ requiring 12
(29) CNOT gates333We have decomposed the Fredkin gate to 7 CNOTs [64]) is
transpiled to $\mathfrak{U}^{\rm trans}_{-1}$ with 18 (31-43) CNOTs for
$\mathbb{BT}(\mathbb{BI})$ where the number of CNOTs is counted from the
transpiled circuits. We subsequently apply $\mathfrak{U}^{\rm trans}_{-1}$ to
$|g\rangle$, resulting in the final state $|g^{\prime}\rangle$. The fidelity
is defined as:

$\mathcal{F}=|\langle g^{-1}|g^{\prime}\rangle|^{2}=|\langle
g^{-1}|\mathfrak{U}^{\rm trans}_{-1}\mathfrak{U}_{g}|\psi_{0}\rangle|^{2},$
(26)

which is the probability of measuring the correct $|g^{-1}\rangle$.

We show the fidelity for each $g$ in Fig. 8 for both $\mathbb{BT}$ and
$\mathbb{BI}$ group calculated from 5 runs, with $N=5000$ shots for each run.
The group element are labeled by the lexicographic order in the range
$\\{0,|G|-1\\}$ on the x-axis. We also present the average fidelity
$\bar{\mathcal{F}}$ over all group elements in Fig. 8. The fidelity of an X
gate is found to be $\sim$97% on Baiwang, and CNOT gate to be around
$\sim$95%. For the $\mathbb{BT}$ case, different X gates are needed to prepare
state, which are denoted with different shapes in the upper plot of Fig. 8.
This variance in state preparation causes noticeable variations in
$\mathcal{F}$ observed in Fig. 8.

We found that for $|g\rangle$ in $\mathbb{BT}$, the error rate grows slightly
with number of X gates in $\mathfrak{U}^{\rm trans}_{-1}$. We can achieve
$\bar{\mathcal{F}}=40^{+5}_{-4}\%$ for $\mathbb{BT}$. Previously, the fidelity
of a 5-qubit ordered product encoding $\mathfrak{U}_{-1}$ was benchmarked on
ibm_nairobi with higher connectivity and Pauli twirling used for error
mitigation [45]. Despite these striking differences, a value of
$\bar{\mathcal{F}}=37.0^{+8}_{-8}\%$ was found, suggesting that effective
digitizations depend on connectivity and noise channels in nontrivial ways,
and therefore should not be neglecting in the discussion of digitization.

For $\mathbb{BI}$, $\mathfrak{U}^{\rm trans}_{-1}$ can require varying numbers
of CNOTs depending on the transpilation, including the orders of mapping
matrix elements to the qubits. We show the fidelities for each group element
of $\mathbb{BI}$ in Fig. 8 (bottom) with shapes indicating the number of CNOTs
in the transpiled circuits. As more than 30 CNOT gates are involved, noises
from CNOT gates contribute mostly to the quantum errors which reduce the
average fidelity to only $\bar{\mathcal{F}}=4\%$. This low fidelity could be
improved by improving transpilation that reduces CNOT counts and implementing
error mitigation strategies such as Pauli twirling which have proven effective
in the past for LGT [86, 68, 87, 45, 88, 89, 90].

##  VIII Conclusions and outlook

In this work, we introduced the block encoding method – a general method for
digitizing discrete groups on quantum computers and developed the primitive
gate set for the two important discrete subgroup of $SU(2)$, including the
first implementation ever of $\mathbb{BI}$. The realization of quantum
circuits for this largest crystal-like subgroup of $SU(2)$ allows simulating
physics of $SU(2)$ deep into the scaling regime. We have shown that the qubit
and T costs as well as robustness to noise are comparable to other
digitizations of $SU(2)$. In particular both theoretical analysis and
experimental results support the idea that the connection between number of
qubits/gates and the true performance in a noisy environment is nontrivial,
and such metrics should be considered in evaluating digitization schemes.

A number of directions for research exist following these results. Primary is
that given its predominance in the gate costs, determination of a quantum
Fourier transformation gate within the block encoding method would be
invaluable, as it could radically improve the resources costs. Further, given
the block encoding method separates group registers into registers of finite
fields, it may greatly benefit from formulations on qudit-based devices
similar to [91, 92, 93, 94, 95, 96, 97, 9, 55, 98]. Finally, the outstanding
goal of the discrete group approximation is the largest crystal-like subgroup
of $SU(3)$ – $\mathbb{V}$ – which similar to $\mathbb{BI}$ faces obstacles in
the order product encoding. Given it is isomorphic to a subgroup of $GL(3,4)$,
the block encoding method provides an avenue for encoding it onto nine 9
ququarts or 18 qubits.

###### Acknowledgements.

This work is supported by the Department of Energy through the Fermilab
QuantiSED program in the area of “Intersections of QIS and Theoretical
Particle Physics”. Fermilab is operated by Fermi Research Alliance, LLC under
contract number DE-AC02-07CH11359 with the United States Department of Energy.
Y.-Y. L is supported by the NSF of China through Grant No. 12305107, 12247103.
J.S. is supported by Peking University under startup Grant No. 7101302974 and
the National Natural Science Foundation of China under Grants No. 12025507,
No.12150015; and is supported by the Key Research Program of Frontier Science
of the Chinese Academy of Sciences (CAS) under Grants No. ZDBS-LY-7003. This
work was supported by the Munich Institute for Astro-, Particle and BioPhysics
(MIAPbP), which is funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy–EXC-2094–390783311.

## References

  * Cohen  _et al._ [2021] T. D. Cohen, H. Lamm, S. Lawrence, and Y. Yamauchi (NuQS), Quantum algorithms for transport coefficients in gauge theories, Phys. Rev. D 104, 094514 (2021), arXiv:2104.02024 [hep-lat] .
  * Turro  _et al._ [2024] F. Turro, A. Ciavarella, and X. Yao, Classical and Quantum Computing of Shear Viscosity for $2+1D$ SU(2) Gauge Theory (2024), arXiv:2402.04221 [hep-lat] .
  * Farrell  _et al._ [2024a] R. C. Farrell, M. Illa, and M. J. Savage, Steps Toward Quantum Simulations of Hadronization and Energy-Loss in Dense Matter (2024a), arXiv:2405.06620 [quant-ph] .
  * Lamm  _et al._ [2020a] H. Lamm, S. Lawrence, and Y. Yamauchi (NuQS), Parton physics on a quantum computer, Phys. Rev. Res. 2, 013272 (2020a), arXiv:1908.10439 [hep-lat] .
  * Kreshchuk  _et al._ [2020] M. Kreshchuk, W. M. Kirby, G. Goldstein, H. Beauchemin, and P. J. Love, Quantum Simulation of Quantum Field Theory in the Light-Front Formulation (2020), arXiv:2002.04016 [quant-ph] .
  * Echevarria  _et al._ [2020] M. Echevarria, I. Egusquiza, E. Rico, and G. Schnell, Quantum Simulation of Light-Front Parton Correlators (2020), arXiv:2011.01275 [quant-ph] .
  * Bauer  _et al._ [2021] C. W. Bauer, M. Freytsis, and B. Nachman, Simulating Collider Physics on Quantum Computers Using Effective Field Theories, Phys. Rev. Lett. 127, 212001 (2021), arXiv:2102.05044 [hep-ph] .
  * Li  _et al._ [2022] T. Li, X. Guo, W. K. Lai, X. Liu, E. Wang, H. Xing, D.-B. Zhang, and S.-L. Zhu (QuNu), Partonic collinear structure by quantum computing, Phys. Rev. D 105, L111502 (2022), arXiv:2106.03865 [hep-ph] .
  * Zache  _et al._ [2023] T. V. Zache, D. González-Cuadra, and P. Zoller, Fermion-qudit quantum processors for simulating lattice gauge theories with matter (2023), arXiv:2303.08683 [quant-ph] .
  * Polkovnikov and Gritsev [2008] A. Polkovnikov and V. Gritsev, Breakdown of the adiabatic limit in low-dimensional gapless systems, Nature Physics 4, 477 (2008).
  * Baum  _et al._ [2021] S. Baum, M. Carena, N. R. Shah, C. E. M. Wagner, and Y. Wang, Nucleation is more than critical: A case study of the electroweak phase transition in the NMSSM, JHEP 03, 055, arXiv:2009.10743 [hep-ph] .
  * Riera  _et al._ [2012] A. Riera, C. Gogolin, and J. Eisert, Thermalization in nature and on a quantum computer, Phys. Rev. Lett. 108, 080402 (2012).
  * Czajka  _et al._ [2022] A. M. Czajka, Z.-B. Kang, H. Ma, and F. Zhao, Quantum simulation of chiral phase transitions, JHEP 08, 209, arXiv:2112.03944 [hep-ph] .
  * Feynman [1982] R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys. 21, 467 (1982).
  * Jordan  _et al._ [2018] S. P. Jordan, H. Krovi, K. S. Lee, and J. Preskill, BQP-completeness of Scattering in Scalar Quantum Field Theory, Quantum 2, 44 (2018), arXiv:1703.00454 [quant-ph] .
  * Bañuls  _et al._ [2020] M. C. Bañuls _et al._ , Simulating Lattice Gauge Theories within Quantum Technologies, Eur. Phys. J. D 74, 165 (2020), arXiv:1911.00003 [quant-ph] .
  * Bauer  _et al._ [2023a] C. W. Bauer _et al._ , Quantum Simulation for High-Energy Physics, PRX Quantum 4, 027001 (2023a), arXiv:2204.03381 [quant-ph] .
  * Di Meglio  _et al._ [2023] A. Di Meglio _et al._ , Quantum Computing for High-Energy Physics: State of the Art and Challenges. Summary of the QC4HEP Working Group (2023), arXiv:2307.03236 [quant-ph] .
  * Anishetty  _et al._ [2009] R. Anishetty, M. Mathur, and I. Raychowdhury, Irreducible su(3) schwinger bosons, Journal of Mathematical Physics 50, 10.1063/1.3122666 (2009).
  * Anishetty and Raychowdhury [2014] R. Anishetty and I. Raychowdhury, Su(2) lattice gauge theory: Local dynamics on nonintersecting electric flux loops, Phys. Rev. D 90, 114503 (2014).
  * Raychowdhury and Stryker [2020] I. Raychowdhury and J. R. Stryker, Loop, string, and hadron dynamics in su(2) hamiltonian lattice gauge theories, Phys. Rev. D 101, 114502 (2020).
  * Zohar  _et al._ [2013] E. Zohar, J. I. Cirac, and B. Reznik, Quantum simulations of gauge theories with ultracold atoms: local gauge invariance from angular momentum conservation, Phys. Rev. A88, 023617 (2013), arXiv:1303.5040 [quant-ph] .
  * Kaplan and Stryker [2020] D. B. Kaplan and J. R. Stryker, Gauss’s law, duality, and the hamiltonian formulation of u(1) lattice gauge theory, Phys. Rev. D 102, 094515 (2020).
  * Bender and Zohar [2020] J. Bender and E. Zohar, Gauge redundancy-free formulation of compact qed with dynamical matter for quantum and classical computations, Phys. Rev. D 102, 114517 (2020).
  * Yamamoto [2021] A. Yamamoto, Real-time simulation of (2+1)-dimensional lattice gauge theory on qubits, PTEP 2021, 013B06 (2021), arXiv:2008.11395 [hep-lat] .
  * Bauer and Grabowska [2021] C. W. Bauer and D. M. Grabowska, Efficient representation for simulating u(1) gauge theories on digital quantum computers at all values of the coupling (2021), arXiv:2111.08015 [hep-ph] .
  * Ciavarella and Bauer [2024] A. N. Ciavarella and C. W. Bauer, Quantum simulation of su(3) lattice yang mills theory at leading order in large n (2024), arXiv:2402.10265 [hep-ph] .
  * Buser  _et al._ [2020] A. J. Buser, T. Bhattacharya, L. Cincio, and R. Gupta, State preparation and measurement in a quantum simulation of the $o(3)$ sigma model, Phys. Rev. D 102, 114514 (2020).
  * Bhattacharya  _et al._ [2021] T. Bhattacharya, A. J. Buser, S. Chandrasekharan, R. Gupta, and H. Singh, Qubit regularization of asymptotic freedom, Phys. Rev. Lett. 126, 172001 (2021).
  * Alexandru  _et al._ [2022a] A. Alexandru, P. F. Bedaque, A. Carosso, and A. Sheng, Universality of a truncated sigma-model, Physics Letters B 832, 137230 (2022a).
  * Zohar and Burrello [2015] E. Zohar and M. Burrello, Formulation of lattice gauge theories for quantum simulations, Phys. Rev. D 91, 054506 (2015).
  * Lamm  _et al._ [2019a] H. Lamm, S. Lawrence, and Y. Yamauchi (NuQS Collaboration), General methods for digital quantum simulation of gauge theories, Phys. Rev. D 100, 034518 (2019a).
  * Shaw  _et al._ [2020] A. F. Shaw, P. Lougovski, J. R. Stryker, and N. Wiebe, Quantum Algorithms for Simulating the Lattice Schwinger Model, Quantum 4, 306 (2020), arXiv:2002.11146 [quant-ph] .
  * Tong  _et al._ [2022] Y. Tong, V. V. Albert, J. R. McClean, J. Preskill, and Y. Su, Provably accurate simulation of gauge theories and bosonic systems, Quantum 6, 816 (2022), arXiv:2110.06942 [quant-ph] .
  * Ji  _et al._ [2020] Y. Ji, H. Lamm, and S. Zhu (NuQS), Gluon Field Digitization via Group Space Decimation for Quantum Computers, Phys. Rev. D 102, 114513 (2020), arXiv:2005.14221 [hep-lat] .
  * Carena  _et al._ [2022a] M. Carena, E. J. Gustafson, H. Lamm, Y.-Y. Li, and W. Liu, Gauge theory couplings on anisotropic lattices, Phys. Rev. D 106, 114504 (2022a).
  * Bauer  _et al._ [2023b] C. W. Bauer, I. D’Andrea, M. Freytsis, and D. M. Grabowska, A new basis for Hamiltonian SU(2) simulations (2023b), arXiv:2307.11829 [hep-ph] .
  * Ciavarella [2023] A. N. Ciavarella, Quantum Simulation of Lattice QCD with Improved Hamiltonians (2023), arXiv:2307.05593 [hep-lat] .
  * Hanada  _et al._ [2022] M. Hanada, J. Liu, E. Rinaldi, and M. Tezuka, Estimating truncation effects of quantum bosonic systems using sampling algorithms (2022), arXiv:2212.08546 [quant-ph] .
  * Creutz  _et al._ [1979] M. Creutz, L. Jacobs, and C. Rebbi, Monte Carlo Study of Abelian Lattice Gauge Theories, Phys. Rev. D20, 1915 (1979).
  * Creutz and Okawa [1983] M. Creutz and M. Okawa, Generalized Actions in $Z(p$) Lattice Gauge Theory, Nucl. Phys. B220, 149 (1983).
  * Bhanot and Rebbi [1981] G. Bhanot and C. Rebbi, Monte Carlo Simulations of Lattice Models With Finite Subgroups of SU(3) as Gauge Groups, Phys. Rev. D24, 3319 (1981).
  * Petcher and Weingarten [1980] D. Petcher and D. H. Weingarten, Monte Carlo Calculations and a Model of the Phase Structure for Gauge Theories on Discrete Subgroups of SU(2), Phys. Rev. D22, 2465 (1980).
  * Bhanot [1982] G. Bhanot, SU(3) Lattice Gauge Theory in Four-dimensions With a Modified Wilson Action, Phys. Lett. 108B, 337 (1982).
  * Gustafson  _et al._ [2022] E. J. Gustafson, H. Lamm, F. Lovelace, and D. Musk, Primitive quantum gates for an SU(2) discrete subgroup: Binary tetrahedral, Phys. Rev. D 106, 114501 (2022), arXiv:2208.12309 [quant-ph] .
  * Hackett  _et al._ [2019] D. C. Hackett, K. Howe, C. Hughes, W. Jay, E. T. Neil, and J. N. Simone, Digitizing Gauge Fields: Lattice Monte Carlo Results for Future Quantum Computers, Phys. Rev. A 99, 062341 (2019), arXiv:1811.03629 [quant-ph] .
  * Alam  _et al._ [2022] M. S. Alam, S. Hadfield, H. Lamm, and A. C. Y. Li (SQMS), Primitive quantum gates for dihedral gauge theories, Phys. Rev. D 105, 114501 (2022), arXiv:2108.13305 [quant-ph] .
  * Ji  _et al._ [2023] Y. Ji, H. Lamm, and S. Zhu (NuQS), Gluon digitization via character expansion for quantum computers, Phys. Rev. D 107, 114503 (2023), arXiv:2203.02330 [hep-lat] .
  * Alexandru  _et al._ [2022b] A. Alexandru, P. F. Bedaque, R. Brett, and H. Lamm, Spectrum of digitized QCD: Glueballs in a S(1080) gauge theory, Phys. Rev. D 105, 114508 (2022b), arXiv:2112.08482 [hep-lat] .
  * Weingarten and Petcher [1981] D. H. Weingarten and D. N. Petcher, Monte Carlo Integration for Lattice Gauge Theories with Fermions, Phys. Lett. 99B, 333 (1981).
  * Weingarten [1982] D. Weingarten, Monte Carlo Evaluation of Hadron Masses in Lattice Gauge Theories with Fermions, Phys. Lett. 109B, 57 (1982), [,631(1981)].
  * Alexandru  _et al._ [2019] A. Alexandru, P. F. Bedaque, S. Harmalkar, H. Lamm, S. Lawrence, and N. C. Warrington (NuQS Collaboration), Gluon field digitization for quantum computers, Phys. Rev. D 100, 114501 (2019).
  * Gustafson  _et al._ [2024a] E. J. Gustafson, H. Lamm, and F. Lovelace, Primitive quantum gates for an SU(2) discrete subgroup: Binary octahedral, Phys. Rev. D 109, 054503 (2024a), arXiv:2312.10285 [hep-lat] .
  * Gustafson and Lamm [2023] E. J. Gustafson and H. Lamm, Robustness of Gauge Digitization to Quantum Noise (2023), arXiv:2301.10207 [hep-lat] .
  * Gustafson  _et al._ [2024b] E. J. Gustafson, Y. Ji, H. Lamm, E. M. Murairi, and S. Zhu, Primitive Quantum Gates for an SU(3) Discrete Subgroup: $\Sigma(36\times 3)$ (2024b), arXiv:2405.05973 [hep-lat] .
  * Vartiainen  _et al._ [2004] J. J. Vartiainen, M. Möttönen, and M. M. Salomaa, Efficient Decomposition of Quantum Gates, Phys. Rev. Lett. 92, 177902 (2004).
  * Sawaya  _et al._ [2020] N. P. D. Sawaya, T. Menke, T. H. Kyaw, S. Johri, A. Aspuru-Guzik, and G. G. Guerreschi, Resource-efficient digital quantum simulation of d-level systems for photonic, vibrational, and spin-s Hamiltonians, npj Quantum Information 6, 49 (2020), arXiv:1909.12847 [quant-ph] .
  * Di Matteo  _et al._ [2021] O. Di Matteo, A. McCoy, P. Gysbers, T. Miyagi, R. M. Woloshyn, and P. Navrátil, Improving Hamiltonian encodings with the Gray code, Phys. Rev. A 103, 042405 (2021).
  * Chang  _et al._ [2021] C. C. Chang, K. S. McElvain, E. Rrapaj, and Y. Wu, Improving Schrödinger Equation Implementations with Gray Code for Adiabatic Quantum Computers (2021), arXiv:2103.08056 [quant-ph] .
  * Lamm  _et al._ [2019b] H. Lamm, S. Lawrence, and Y. Yamauchi (NuQS), General Methods for Digital Quantum Simulation of Gauge Theories, Phys. Rev. D100, 034518 (2019b), arXiv:1903.08807 [hep-lat] .
  * Moore  _et al._ [2003] C. Moore, D. Rockmore, and A. Russell, Generic quantum fourier transforms (2003), arXiv:quant-ph/0304064 [quant-ph] .
  * Chuang and Nielsen [1997] I. L. Chuang and M. A. Nielsen, Prescription for experimental determination of the dynamics of a quantum black box, Journal of Modern Optics 44, 2455 (1997).
  * Kitaev [1997] A. Y. Kitaev, Quantum computations: algorithms and error correction, Russian Mathematical Surveys 52, 1191 (1997).
  * Cruz and Murta [2024] P. M. Q. Cruz and B. Murta, Shallow unitary decompositions of quantum Fredkin and Toffoli gates for connectivity-aware equivalent circuit averaging, APL Quantum 1, 016105 (2024), arXiv:2305.18128 [quant-ph] .
  * Kim and Choi [2018] T. Kim and B.-S. Choi, Efficient decomposition methods for controlled-rn using a single ancillary qubit, Scientific Reports 8 (2018).
  * Benenti  _et al._ [2004] G. Benenti, G. Casati, and G. Strini,  _Principles of Quantum Computation and Information Volume I: Basic Concepts_ (World Scientific, 2004).
  * Kan and Nam [2022] A. Kan and Y. Nam, Lattice quantum chromodynamics and electrodynamics on a universal quantum computer (2022), arXiv:2107.12769 [quant-ph] .
  * Carena  _et al._ [2022b] M. Carena, H. Lamm, Y.-Y. Li, and W. Liu, Improved Hamiltonians for Quantum Simulations of Gauge Theories, Phys. Rev. Lett. 129, 051601 (2022b), arXiv:2203.02823 [hep-lat] .
  * Davoudi  _et al._ [2023] Z. Davoudi, A. F. Shaw, and J. R. Stryker, General quantum algorithms for hamiltonian simulation with applications to a non-abelian lattice gauge theory, Quantum 7, 1213 (2023).
  * Rhodes  _et al._ [2024] M. Rhodes, M. Kreshchuk, and S. Pathak, Exponential improvements in the simulation of lattice gauge theories using near-optimal techniques (2024), arXiv:2405.10416 [quant-ph] .
  * Lieu  _et al._ [2020] S. Lieu, R. Belyansky, J. T. Young, R. Lundgren, V. V. Albert, and A. V. Gorshkov, Symmetry breaking and error correction in open quantum systems, Phys. Rev. Lett. 125, 240405 (2020).
  * Bonati  _et al._ [2021a] C. Bonati, A. Pelissetto, and E. Vicari, Breaking of Gauge Symmetry in Lattice Gauge Theories, Phys. Rev. Lett. 127, 091601 (2021a).
  * Bonati  _et al._ [2021b] C. Bonati, A. Pelissetto, and E. Vicari, Lattice gauge theories in the presence of a linear gauge-symmetry breaking, Phys. Rev. E 104, 014140 (2021b).
  * Stannigel  _et al._ [2014] K. Stannigel, P. Hauke, D. Marcos, M. Hafezi, S. Diehl, M. Dalmonte, and P. Zoller, Constrained dynamics via the Zeno effect in quantum simulation: Implementing non-Abelian lattice gauge theories with cold atoms, Phys. Rev. Lett. 112, 120406 (2014).
  * Tran  _et al._ [2021] M. C. Tran, Y. Su, D. Carney, and J. M. Taylor, Faster digital quantum simulation by symmetry protection, PRX Quantum 2, 010323 (2021).
  * Lamm  _et al._ [2020b] H. Lamm, S. Lawrence, and Y. Yamauchi (NuQS), Suppressing Coherent Gauge Drift in Quantum Simulations (2020b), arXiv:2005.12688 [quant-ph] .
  * Stryker [2019] J. R. Stryker, Oracles for Gauss’s law on digital quantum computers, Phys. Rev. A99, 042301 (2019), arXiv:1812.01617 [quant-ph] .
  * Rajput  _et al._ [2023] A. Rajput, A. Roggero, and N. Wiebe, Quantum error correction with gauge symmetries, npj Quantum Information 9, 41 (2023).
  * Mathew and Raychowdhury [2022] E. Mathew and I. Raychowdhury, Protecting local and global symmetries in simulating (1+1)D non-Abelian gauge theories, Phys. Rev. D 106, 054510 (2022).
  * Van Damme  _et al._ [2021] M. Van Damme, J. Mildenberger, F. Grusdt, P. Hauke, and J. C. Halimeh, Suppressing nonperturbative gauge errors in the thermodynamic limit using local pseudogenerators (2021), arXiv:2110.08041 [quant-ph] .
  * Halimeh and Hauke [2022] J. C. Halimeh and P. Hauke, Stabilizing Gauge Theories in Quantum Simulators: A Brief Review (2022) arXiv:2204.13709 [cond-mat.quant-gas] .
  * Carena  _et al._ [2024] M. Carena, H. Lamm, Y.-Y. Li, and W. Liu, Quantum error thresholds for gauge-redundant digitizations of lattice field theories (2024), arXiv:2402.16780 [hep-lat] .
  * Cai  _et al._ [2023] Z. Cai, R. Babbush, S. C. Benjamin, S. Endo, W. J. Huggins, Y. Li, J. R. McClean, and T. E. O’Brien, Quantum error mitigation (2023), arXiv:2210.00921 [quant-ph] .
  * Knill and Laflamme [1997] E. Knill and R. Laflamme, Theory of quantum error-correcting codes, Phys. Rev. A 55, 900 (1997).
  * bib [2024] Quafu quantum computing cloud platform (2024).
  * Yeter-Aydeniz  _et al._ [2022] K. Yeter-Aydeniz, Z. Parks, A. Nair, E. Gustafson, A. F. Kemper, R. C. Pooser, Y. Meurice, and P. Dreher, Measuring NISQ Gate-Based Qubit Stability Using a 1+1 Field Theory and Cycle Benchmarking (2022), arXiv:2201.02899 [quant-ph] .
  * Farrell  _et al._ [2022] R. C. Farrell, I. A. Chernyshev, S. J. M. Powell, N. A. Zemlevskiy, M. Illa, and M. J. Savage, Preparations for Quantum Simulations of Quantum Chromodynamics in 1+1 Dimensions: (I) Axial Gauge (2022), arXiv:2207.01731 [quant-ph] .
  * Atas  _et al._ [2022] Y. Y. Atas, J. F. Haase, J. Zhang, V. Wei, S. M. L. Pfaendler, R. Lewis, and C. A. Muschik, Simulating one-dimensional quantum chromodynamics on a quantum computer: Real-time evolutions of tetra- and pentaquarks (2022), arXiv:2207.03473 [quant-ph] .
  * Charles  _et al._ [2023] C. Charles, E. J. Gustafson, E. Hardt, F. Herren, N. Hogan, H. Lamm, S. Starecheski, R. S. Van de Water, and M. L. Wagman, Simulating $\mathbb{Z}_{2}$ lattice gauge theory on a quantum computer (2023), arXiv:2305.02361 [hep-lat] .
  * Farrell  _et al._ [2024b] R. C. Farrell, M. Illa, A. N. Ciavarella, and M. J. Savage, Quantum Simulations of Hadron Dynamics in the Schwinger Model using 112 Qubits (2024b), arXiv:2401.08044 [quant-ph] .
  * Gustafson  _et al._ [2021] E. Gustafson _et al._ , Large scale multi-node simulations of $\mathbb{Z}_{2}$ gauge theory quantum circuits using Google Cloud Platform, in  _IEEE/ACM Second International Workshop on Quantum Computing Software_ (2021) arXiv:2110.07482 [quant-ph] .
  * Gustafson [2021] E. Gustafson, Prospects for Simulating a Qudit Based Model of (1+1)d Scalar QED, Phys. Rev. D 103, 114505 (2021), arXiv:2104.10136 [quant-ph] .
  * Popov  _et al._ [2024] P. P. Popov, M. Meth, M. Lewenstein, P. Hauke, M. Ringbauer, E. Zohar, and V. Kasper, Variational quantum simulation of U(1) lattice gauge theories with qudit systems, Phys. Rev. Res. 6, 013202 (2024).
  * Kurkcuoglu  _et al._ [2022] D. M. Kurkcuoglu, M. S. Alam, J. A. Job, A. C. Y. Li, A. Macridin, G. N. Perdue, and S. Providence, Quantum simulation of $\phi^{4}$ theories in qudit systems (2022), arXiv:2108.13357 [quant-ph] .
  * Calajò  _et al._ [2024] G. Calajò, G. Magnifico, C. Edmunds, M. Ringbauer, S. Montangero, and P. Silvi, Digital quantum simulation of a (1+1)D SU(2) lattice gauge theory with ion qudits (2024), arXiv:2402.07987 [quant-ph] .
  * González-Cuadra  _et al._ [2022] D. González-Cuadra, T. V. Zache, J. Carrasco, B. Kraus, and P. Zoller, Hardware efficient quantum simulation of non-abelian gauge theories with qudits on Rydberg platforms (2022), arXiv:2203.15541 [quant-ph] .
  * Illa  _et al._ [2024] M. Illa, C. E. P. Robin, and M. J. Savage, Qu8its for quantum simulations of lattice quantum chromodynamics (2024), arXiv:2403.14537 [quant-ph] .
  * Roy  _et al._ [2024] T. Roy, T. Kim, A. Romanenko, and A. Grassellino, Qudit-based quantum computing with SRF cavities at Fermilab, PoS LATTICE2023, 127 (2024).

##  Appendix A Primitive Gates for $\mathbb{BT}$ Group with Two’s Component
Encoding

In this section, we present the primitive gates for $\mathbb{BT}$ group with
two’s component encoding, e.g.
$\left|a=0\right>=\left|00\right>,\left|a=1\right>=\left|01\right>$ and
$\left|a=-1\right>=\left|11\right>$. Given the addition and multiplication
table below, the quantum circuit to implement matrix element additions
$\mathfrak{U}_{\rm a}$ and productions $\mathfrak{U}_{\rm p}$ are shown in
Fig. 9. The construction of the group multiplication circuit
$\mathfrak{U}_{\times}$ is similar as Fig. 7 in the main text. The inverse and
trace operations follow the same rules as Eq. (11) and Eq. (12) in the main
text, but with different quantum circuits given in Fig. 10.

Table 5: Addition. | 00 | 01 | 11
---|---|---|---
00 | 00 | 01 | 11
01 | 01 | 11 | 00
11 | 11 | 00 | 01
Table 6: Multiplication. | 00 | 01 | 11
---|---|---|---
00 | 00 | 00 | 00
01 | 00 | 01 | 11
11 | 00 | 11 | 10

=  =

Figure 9: Quantum circuit implementing matrix element additions and
productions in $\mathbf{F}_{5}$.

Figure 10: Inverse gate $\mathfrak{U}_{-1}$ and trace gate $\mathfrak{U}_{\rm
Tr}$ for $\mathbb{BT}$ group with the two’s component encoding.


# On the necessity of the assumptions used to prove Hardy-Littlewood and Riesz
Rearrangement Inequalities

H. Hajaiej

###### Abstract

We prove that supermodularity is a necessary condition for the generalized
Hardy-Littlewood and Riesz rearrangement inequalities. We also show the
necessity of the monotonicity of the kernels involved in the Riesz–type
integral.

Keywords and phrases: rearrangement, inequality, supermodular integrand .

AMS Subject Classification: 26D15 .

##  1 Introduction

Let $F:\mathbb{R}_{+}^{m}\to\mathbb{R}$ be a Borel measurable function. We say
that $F$ is supermodular if:

$\displaystyle F(y+he_{i}+ke_{j})+F(y)\geq F(y+he_{i})+F(y+ke_{j})$ (1.1) for
every $i,j\in\\{1,\ldots,m\\}$, $i\not=j$, $k,h>0$,

where $y=(y_{1},\ldots,y_{m})$, and $e_{i}$ denotes the standard basis vector
in $\mathbb{R}^{m}$, $i=1,\ldots,m$.
This condition has been first analysed in the context of integral inequalities
by G. Lorentz [19].
Note that if $F$ is $C^{2}$, (1.1) is equivalent to

$\frac{\partial^{2}F}{\partial x_{i}\partial x_{j}}(x)\geq 0,\quad\mbox{for
every $i,j\in\\{1,\ldots,m\\}$, $i\not=j$.}$ (1.2)

The supermodularity condition has been widely used in functional analysis. It
was the main assumption to prove the generalized Hardy-Littlewood
rearrangement inequality,

$\int_{X}F(u_{1}(x),\ldots,u_{m}(x))\,d\mu(x)\leq\int_{X}F(u_{1}^{*}(x),\ldots,u_{m}^{*}(x))\,d\mu(x),$
(1.3)

Here $u_{i}$ are nonnegative measurable functions that vanish at infinity and
$u_{i}^{*}$ are their symmetric decreasing rearrangements. The integral is
taken over $X=\mathbb{R}^{n}$, $\mathbb{H}^{n}$ or ${\cal S}^{n}$. (1.3) has
been studied by many authors [17, 19, 11, 6, 7]. Its main applications are
economics [11], chemistry [24], and nonlinear optics where the profile of
stable electromagnetic waves traveling along a planar waveguide are given by
the ground states of the energy functional

$E(u):=\frac{1}{2}\int_{\mathbb{R}}\left(u^{{}^{\prime}2}-G(|x|,u)\right)\,dx,$
(1.4)

under the constraint $\|u\|_{2}=c>0$. A crucial step to prove that (1.4)
admits an even ground state $u=u(x)$ which is decreasing for $x>0$, is (1.3),
[15].
In [16], we showed that the ground state of the $m\times m$ elliptic system

${\bf(S)}\left\\{\begin{array}[]{l}\Delta
u_{1}+\lambda_{1}u_{1}+g_{1}(u_{1},\ldots,u_{m})=0\\\ \cdots\\\ \Delta
u_{m}+\lambda_{m}u_{m}+g_{m}(u_{1},\ldots,u_{m})=0\end{array}\right.,$

where $g_{i}=(\partial G/\partial u_{i})$, ($i=1,\ldots,m$), exists when $G$
is supermodular, and satisfies some further conditions.
Notice that supermodularity condition is equivalent to the cooperativity of
${\bf(S)}$. As proved by W.C. Troy [24], the latter condition is also
necessary for the existence of solutions of ${\bf(S)}$. In [10], G. Carlier
viewed the generalized Hardy–Littlewood rearrangement inequality as an optimal
transportation problem. He showed that the left-hand side of (1.3) achieves
its maximum (i.e. the cost is minimized) under the supermodularity assumption.
In this paper, we prove, among other things, that (1.1) is necessary for the
inequality (1.3) to hold.
The generalized Riesz rearrangement inequality is

$\displaystyle\int\cdots\int
F(u_{1}(x_{1}),\ldots,u_{m}(x_{m}))\prod_{i<j}K_{ij}(d(x_{i},x_{j}))\,dx_{1}\cdots
dx_{m}$ (1.5) $\displaystyle\leq$ $\displaystyle\int\cdots\int
F(u_{1}^{*}(x_{1}),\ldots,u_{m}^{*}(x_{m}))\prod_{i<j}K_{ij}(d(x_{i},x_{j}))\,dx_{1}\cdots
dx_{m},$

where $d$ denotes distance and the functions $K_{ij}$ are decreasing.
It is closely related to the Brunn-Minkowski inequality of convex geometry.
Notice that the integral in (1.5) can represent a physical interaction
potential.
The most relevant case is $K_{ij}(d(x_{i},x_{j}))=\omega_{ij}(|x_{i}-x_{j}|)$
where $\omega_{ij}$ are nonnegative nonincreasing functions. (1.5) is then
equivalent to

$\displaystyle\int\cdots\int
F(u_{1}(x_{1}),\ldots,u_{m}(x_{m}))\omega_{ij}(|x_{i}-x_{j}|)\,dx_{1}\cdots
dx_{m}$ (1.6) $\displaystyle\leq$ $\displaystyle\int\cdots\int
F(u_{1}^{*}(x_{1}),\ldots,u_{m}^{*}(x_{m}))\omega_{ij}(|x_{i}-x_{j}|)\,dx_{1}\cdots
dx_{m}.$

(1.5) and (1.6) have been studied in [22, 23, 3, 4, 5, 1, 20, 13, 21, 12, 25,
7]. All these results use the supermodularity of $F$. Among the numerous
applications of (1.6), let us mention that it was extremely useful to prove
the existence and uniqueness of the minimizing solution of Choquard’s equation
[18]. In the special case that $F$ is a product and $m=2$, (1.6) hold on the
standard spheres and hyperbolic spaces [2, 3] and it still contains the
isoperimetric inequality as a limit. In this paper we will show (see
Proposition 3.2 below) that (1.1) is necessary for (1.6) to hold. Notice that
, in [7], cases of equality of (1.3) and (1.6) were established under the
strict supermodularity condition, that is to say (1.1) with the strict
inequality sign. We will also prove that such a condition is inescapable to
establish cases of equality in (1.3) and (1.6) (see Remark 1 below).

### 2\. Notation and preliminaries

We fix $n\in\mathbb{N}$. Let $\mu$ the Lebesgue measure on $\mathbb{R}^{n}$.
If $x\in\mathbb{R}^{n}$ and $r>0$, let
$B_{r}(x)=\\{y\in\mathbb{R}^{n}:\,|y-x|<r\\}$. For any set
$M\subset\mathbb{R}^{n}$, let ${\bf 1}_{M}$ denote its characteristic
function. By ${\cal M}$ we denote the set of all measurable functions on
$\mathbb{R}^{n}$. For a Borel measurable function
$F:\mathbb{R}^{m}_{+}\to\mathbb{R}$, nonincreasing functions
$w_{ij}:\mathbb{R}_{+}\to\mathbb{R}_{+}$ and nonnegative $u_{i}\in{\cal M}$,
we study the following generalized Hardy–Littlewood type functional,

$I(u_{1},\ldots,u_{m})=\int_{\mathbb{R}^{n}}F(u_{1}(x),\ldots,u_{m}(x))\,dx,$
(2.1)

and the generalized Riesz type functional.

$J(u_{1},\ldots,u_{m})=\int_{\mathbb{R}^{n}}\cdots\int_{\mathbb{R}^{n}}F(u_{1}(x_{1}),\ldots,u_{m}(x_{m}))\omega_{ij}(|x_{i}-x_{j}|)\,dx_{1}\cdots
dx_{m}.$ (2.2)

If $f\in{\cal F}$, let $f^{*}$ denote its Schwarz symmetrization which is the
unique lower continuous function which is radially symmetric, radially
nonincreasing and such that $\mu\\{a<f\leq b\\}=\mu\\{a<f^{*}\leq b\\}$ for
all numbers $\inf f<a<b$ (see [1]). Notice that if $f={\bf 1}_{M}$, where $M$
is Lebesgue measurable with finite measure, then $f^{*}={\bf 1}_{B_{R}(0)}$
where $R$ is chosen such that $\mu(B_{R}(0))=\mu(M)$ . Accordingly, a function
$u\in{\cal F}$ is called Schwarz symmetric  if it is radial and radially
decreasing. We say that it is strictly Schwarz symmetric if it is radial and
strictly radially decreasing.

##  3\. Results

Proposition 3.1 ( Necessity of the supermodularity condition in the
generalized Hardy–Littlewood inequality )
Let $F:(\mathbb{R}_{+})^{m}\to\mathbb{R}$ be a Borel measurable function which
vanishes on hyperplanes, that is, $F(y_{1},\ldots,y_{m})=0$ if $y_{k}=0$ for
some $k\in\\{1,\ldots,m\\}$. If

$\int_{\mathbb{R}^{n}}F(u_{1}(x),\ldots,u_{m}(x))\,dx\leq\int_{\mathbb{R}^{n}}F(u_{1}^{*}(x),\ldots,u_{m}^{*}(x))\,dx,$

for any $(u_{1},\ldots,u_{m})\in{\cal F}^{m}$ then $F$ satisfies (1.1).
Proof: Suppose that (1.1) is not true. Then there exist
$i,j\in\\{1,\ldots,m\\}$, $i\not=j$, $0\leq a<b$, $0\leq c<d$ and $m-2$
nonnegative numbers $\alpha_{l}$, ($l\in\\{1,\ldots,m\\}\setminus\\{i,j\\}$),
such that:

$\displaystyle
F(\alpha_{1},\ldots,a,\ldots,d,\ldots,\alpha_{m-2})+F(\alpha_{1},\ldots,b,\ldots,c,\ldots,\alpha_{m-2})$
$\displaystyle>$ $\displaystyle
F(\alpha_{1},\ldots,b,\ldots,d,\ldots,\alpha_{m-2})+F(\alpha_{1},\ldots,a,\ldots,c,\ldots,\alpha_{m-2}).$

Now let $E$ and $F$ be two measurable sets such that $E\cap F=\emptyset$,
$\mu(F)=\mu(E)<\infty$, and $u_{i}=a{\bf 1}_{E}+b{\bf 1}_{F}$, $u_{j}=c{\bf
1}_{F}+d{\bf 1}_{E}$, and $u_{l}=\alpha_{l}({\bf 1}_{F}+{\bf 1}_{E})$ for
$l\in\\{1,\ldots,m\\}\setminus\\{i,j\\}$. Then $u_{i}^{*}=b{\bf
1}_{B_{r}(0)}+a{\bf 1}_{A}$, where $r$ is chosen such that
$\mu(B_{r}(0))=\mu(F)=\mu(E)$, and $A$ is the annulus
$B_{r^{\prime}}\setminus\overline{B_{r}}$ and $\mu(A)=\mu(F)=\mu(E)$. It
follows that $u_{j}^{*}=d{\bf 1}_{B_{r}(0)}+c{\bf 1}_{A}$,
$u_{l}^{*}=\alpha_{l}{\bf 1}_{B_{r^{\prime}}(0)}$, and

$\displaystyle\int F(u_{1}(x),\ldots,u_{n}(x))\,dx$ $\displaystyle=$
$\displaystyle[F(\alpha_{1},\ldots,a,\ldots,d,\ldots,\alpha_{m-2})+F(\alpha_{1},\ldots,b,\ldots,c,\ldots,\alpha_{m-2})]\mu(E),$
$\displaystyle\int F(u_{1}^{*}(x),\ldots,u_{n}^{*}(x))\,dx$ $\displaystyle=$
$\displaystyle[F(\alpha_{1},\ldots,b,\ldots,d,\ldots,\alpha_{m-2})+F(\alpha_{1},\ldots,a,\ldots,c,\ldots,\alpha_{m-2})]\mu(E).$

Hence we have in view of (3\. Results),

$\int F(u_{1}(x),\ldots,u_{n}(x))\,dx>\int
F(u_{1}^{*}(x),\ldots,u_{n}^{*}(x))\,dx,$

a contradiction.
Remark 1:  Following the same approach we can easily conclude that the strict
supermodularity assumption, that is, (1.1) with strict inequality sign, is
necessary to establish cases of equality. In [7], (1.3) was proven under the
supermodularity assumption (1.1) and an integrability condition, and cases of
equality were established assuming (1.1) with strict inequality sign.
Proposition 3.2: ( Necessity of supermodularity assumption in Riesz–type
integral )
Let $j:(0,+\infty)\to\mathbb{R}$ be a function which is nonincreasing and not
identically equal to zero, and such that

$\lim_{r\to+\infty}r^{n-1}j(r)=0.$ (3.2)

Suppose that $\Psi:[0,+\infty)\times[0,+\infty)\to\mathbb{R}$ is a Borel
measurable function satisfying

$\Psi(u,0)=\Psi(0,v)=0\quad\forall u\geq 0,\ v\geq 0.$ (3.3)

Finally assume that for all nonnegative functions $f,g\in
L^{\infty}(\mathbb{R}^{n})$ with compact support there holds

$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\Psi(f(x),g(y))j(|x-y|)\,dxdy\leq\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\Psi(f^{*}(x),g^{*}(y))j(|x-y|)\,dxdy.$
(3.4)

Then $\Psi$ is supermodular, that is,

$\Psi(a+a^{\prime},b+b^{\prime})-\Psi(a+a^{\prime},b)-\Psi(a,b+b^{\prime})+\Psi(a,b)\geq
0\quad\forall a,a^{\prime},b,b^{\prime}\in[0,+\infty).$ (3.5)

Proof:  In view of (3.2), and since $j$ is non-increasing and nontrivial, $j$
is non-negative, too, and there exist positive numbers $\varepsilon$ and
$t_{0}$ with $t_{0}>2\varepsilon$ and such that

$j(t)<j(s)\quad\mbox{ whenever $0\leq s\leq\varepsilon$ and
$|t-t_{0}|\leq\varepsilon$.}$ (3.6)

We fix $z\in\mathbb{R}^{n}$ with $|z|=t_{0}$, and $\rho>t_{0}+\varepsilon$.
Now let $a,a^{\prime},b,b^{\prime}\in[0,+\infty)$. Then, if $f=a{\bf
1}_{B_{R}(0)}+a^{\prime}{\bf 1}_{B_{\varepsilon}(0)}$, and $g=b{\bf
1}_{B_{\rho}(0)}+b^{\prime}{\bf 1}_{B_{\varepsilon}(z)}$, where
$R>\varepsilon$, we have that $f=f^{*}$, and $g^{*}=b{\bf
1}_{B_{\rho}(0)}+b^{\prime}{\bf 1}_{B_{\varepsilon}(0)}$. Using (3.3), a short
computation shows that (3.4) is equivalent to

$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\left\\{\Psi(a+a^{\prime},b+b^{\prime})-\Psi(a+a^{\prime},b)-\Psi(a,b+b^{\prime})+\Psi(a,b)\right\\}\cdot$
$\displaystyle\cdot\int_{B_{\varepsilon}(0)}\left[\int_{B_{\varepsilon}(0)}j(|x-y|)\,dy-\int_{B_{\varepsilon}(z)}j(|x-y|)\,dy\right]dx$
$\displaystyle+$
$\displaystyle\left(\Psi(a,b+b^{\prime})-\Psi(a,b)\right)\int_{B_{R}(0)}\left[\int_{B_{\varepsilon}(0)}j(|x-y|)\,dy-\int_{B_{\varepsilon}(z)}j(|x-y|)\,dy\right]dx.$

Set

$H(x):=\int_{B_{\varepsilon}(0)}j(|x-y|)\,dy,\quad\,x\in\mathbb{R}^{n}.$

Since $j$ is nonnegative and nonincreasing, $H$ is nonnegative, radial and
radially nonincreasing, that is we can write $H(x)=:h(|x|)$, with $h$
nonincreasing. Since $j$ satisfies (3.2), we also have
$\lim_{t\to\infty}t^{n-1}h(t)=0$. Setting

$I(t):=\int_{B_{t}(0)}H(x)\,dx-\int_{B_{t}(z)}H(x)\,dx,\quad(t>0),$

it follows that $I$ is nonnegative, and moreover, (3.6) implies that

$I(\varepsilon)>0.$ (3.8)

Finally, if $t>t_{0}$, then

$I(t)\leq\int_{B_{t+t_{0}}(0)\setminus
B_{t-t_{0}}(0)}H(x)\,dx=n\omega_{n}\int_{t-t_{0}}^{t+t_{0}}s^{n-1}h(s)\,ds,$

where $\omega_{n}$ denotes the volume of the unit ball in $\mathbb{R}^{n}$.
Hence

$\lim_{t\to\infty}I(t)=0.$ (3.9)

Now inequality (3\. Results) becomes

$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\left\\{\Psi(a+a^{\prime},b+b^{\prime})-\Psi(a+a^{\prime},b)-\Psi(a,b+b^{\prime})+\Psi(a,b)\right\\}I(\varepsilon)$
$\displaystyle+$
$\displaystyle\left(\Psi(a,b+b^{\prime})-\Psi(a,b)\right)I(R).$

Sending $R\to+\infty$, we obtain

$0\leq\left\\{\Psi(a+a^{\prime},b+b^{\prime})-\Psi(a+a^{\prime},b)-\Psi(a,b+b^{\prime})+\Psi(a,b)\right\\}I(\varepsilon),$
(3.10)

and (3.5) follows from (3.8).
Remark 2:  Proposition 3.2 tells us in particular that $\Psi$ is supermodular
if (3.4) holds for $j={\bf 1}_{B_{R}(0)}$, with some $R>0$.
Proposition 3.3. Let $\Psi:[0,+\infty)\times[0,+\infty)$ be nontrivial, and
satisfies (3.5) and (3.3), and let $h\in C(\mathbb{R}^{n})$. Assume that there
holds

$\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\Psi(f(x),g(y))h(x-y)\,dxdy\leq\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\Psi(f^{*}(x),g^{*}(y))h(x-y)\,dxdy$
(3.11)

for all bounded nonnegative functions $f$ and $g$ with compact support . Then
$h$ is radial and radially nonincreasing, that is, it can be written as
$j(|x|)=h(x)$, ($x\in\mathbb{R}^{n}\setminus\\{0\\}$), where
$j:(0,+\infty)\to\mathbb{R}$ is a nonincreasing function.
Proof : By choosing $a=b=0$ in (3.5) and taking into account (3.3), we have
that $\Psi$ is nonnegative, and since $\Psi$ is nontrivial, there are numbers
$a>0$, $b>0$ such that $\Psi(a,b)>0$. Let $z_{1},z_{2}\in\mathbb{R}^{n}$ with
$0<|z_{1}|<|z_{2}|$, and set $R:=(|z_{1}|+|z_{2}|)/2$. Let $f=a({\bf
1}_{B_{R}(0)}+{\bf 1}_{B_{\varepsilon}(z_{2})}-{\bf
1}_{B_{\varepsilon}(z_{1})})$, and $g=b{\bf 1}_{B_{\varepsilon}(0)}$, where
$0<\varepsilon<(|z_{2}|-|z_{1}|)/2$. Then $g=g^{*}$ and $f^{*}=a{\bf
1}_{B_{R}(0)}$. Hence (3.11) gives

$0\leq\Psi(a,b)\left\\{\int_{B_{\varepsilon}(z_{1})}\int_{B_{\varepsilon}(0)}h(x-y)\,dxdy-\int_{B_{\varepsilon}(z_{2})}\int_{B_{\varepsilon}(0)}h(x-y)\,dxdy\right\\}.$

Sending $\varepsilon\to 0$ and taking into account $\Psi(a,b)>0$ we find that
$h(z_{1})\geq h(z_{2})$. Since $h$ is continuous, this implies $h(x)\geq h(y)$
iff $|x|\leq|y|$. Hence $h$ is radial and radially nonincreasing as claimed.
Acknowledgement: The author is very grateful to F. Brock for his kind
invitation to visit AUB in June 2008 and June 2009, and for his precious help.

## References

  * [1] F. Almgren, E.H. Lieb, Symmetric decreasing rearrangement is sometimes continuous. Journal of the A.M.S. 2 (1989), 683–773.
  * [2] A. Baernstein II, B.A. Taylor, Spherical rearrangements, subharmonic functions and $*$-functions in $n$-space. Duke Math. J. 43 (1976), 245–268.
  * [3] W. Beckner, Sobolev inequalities, the Poisson semigroup and the analysis on the sphere ${\cal S}^{n}$. Proc. A.M.S. 89 (1992), 4816–4819.
  * [4] W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger Inequality. Ann. Math. (2) 138 (1993), 213–242.
  * [5] H.J. Brascamp, E.H. Lieb, M.J. Luttinger, A general rearrangement inequality for multiple integrals. J. Funct. Anal. 17 (1974), 227–237.
  * [6] F. Brock, A general rearrangement inequality a la Hardy–Littlewood. J. Inequ. Appl. 5 (2000), 309–320.
  * [7] A. Burchard, H. Hajaiej, Rearrangement inequalities for functionals with monotone integrands. Journal of Functional Analysis 233 (2006), 561–582.
  * [8] A. Burchard, M. Schmuckenschläger, Comparison theorems for exit times. Geom. Funct. Anal. 11 (2001), 651–692.
  * [9] G.R. Burton, Vortex–rings of prescribed impulse. Proc. Cambridge Philos. Soc. (3) 134 (2003), 515–528.
  * [10] G. Carlier, On a class of multidimensional optimal transportation problems. J. Convex Anal. 10 (2003), 517–529.
  * [11] J.A. Crowe, J.A. Zweibel, P.C. Rosenbloom, Rearrangement of functions. J. Funct. Anal. 66 (1986), 432–438.
  * [12] C. Draghici, A general rearrangement inequality. Proc. Amer. math. Soc. 133 (2005), 735–743.
  * [13] R. Friedberg, J.M. Luttinger, (a) Rearrangement inequalities for periodic functions, (b) A new rearrangement inequality for multiple integrals. Arch. Rat. Mech. Anal. 61 (1976), 35–44, and 45–64.
  * [14] H. Hajaiej, Balls are the maximizers of the Riesz–type functionals with supermodular integrands. to appear in: Annali de Matematica Pura ed Applicata.
  * [15] H. Hajaiej, C.A. Stuart, Existence and nonexistence of Schwarz symmetric ground states for eigenvalue problems. Ann. Mat. Pura Appl. (4) 184 (2005), no. 3, 297–314.
  * [16] H. Hajaiej, Symmetric ground states solutions of m-coupled nonlinear Schr dinger equations. Nonlinear Analysis: Methods, Theory and Applications 71, no.2 (2009).
  * [17] G.E. Hardy, J.E. Littlewood, G. Polya, Inequalities. Cambridge Univ. Press, London and N.Y, 1st edn. 1934, 2nd edn. 1952.
  * [18] E.H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Studies in Applied Mathematics 57 (1977), 93–105.
  * [19] G.G. Lorentz, An inequality for rearrangements. Amer. Math. Monthly 60 (1953), 176–179.
  * [20] J.M. Luttinger, Generalized isoperimetric inequalities I,II,III. J. Math. Physics 14 (1973), 586–593, 1444–1447, 1448–1450.
  * [21] C. Morpurgo, Sharp inequalities for functional integrals and traces of conformally invariant operators. Duke Math. J. 114 (2002), 447–553.
  * [22] F. Riesz, Sur une inegalite integrale. J. London Math. Soc. 5 (1930), 162–168.
  * [23] S.L. Sobolev, On a theorem in functional analysis. Math. Sb. (N.S.) 4 (1938), 471–497; AMS Transl.(2) 34 (1963), 39–68.
  * [24] W. Troy, Symmetry properties in systems of semilinear elliptic equations. J. Differential Equations 42 (1981), no. 3, 400–413.
  * [25] A. Zygmund, On an integral inequality. J. London Math. Soc. 8 (1933), 175–178.