# Symmetry-informed transferability of optimal parameters in the Quantum
Approximate Optimization Algorithm

Isak Brundin<EMAIL_ADDRESS>Laura García-Álvarez  Department of
Microtechnology and Nanoscience (MC2), Chalmers University of Technology,
SE-412 96 Göteborg, Sweden

###### Abstract

One of the main limitations of variational quantum algorithms is the classical
optimization of the highly dimensional non-convex variational parameter
landscape. To simplify this optimization, we can reduce the search space using
problem symmetries and typical optimal parameters as initial points if they
concentrate. In this article, we consider typical values of optimal parameters
of the Quantum Approximate Optimization Algorithm for the MaxCut problem with
$d$-regular tree subgraphs and reuse them in different graph instances. We
prove symmetries in the optimization landscape of several kinds of weighted
and unweighted graphs, which explains the existence of multiple sets of
optimal parameters. However, we observe that not all optimal sets can be
successfully transferred between problem instances. We find specific
transferable domains in the search space and show how to translate an
arbitrary set of optimal parameters into the adequate domain using the studied
symmetries.

##  I Introduction

Variational quantum algorithms (VQAs) are among the most popular and
accessible quantum algorithms for solving classically hard problems, with a
notable example being the Quantum Approximate Optimization Algorithm (QAOA)
[1], used to tackle combinatorial optimization problems (COPs). The QAOA is a
hybrid algorithm consisting of a quantum circuit with several layers of
parameter-dependent operations applied iteratively, followed by a classical
optimization of the parameters. The quality of the solutions output by this
algorithm depends on these classically optimized variational parameters and
the depth of the quantum circuit. In theory, the solution improves as the
circuit depth increases but, in practice, deep circuits introduce noise that
affects the quantum state. However, shallow depth QAOA is often inferior to
state-of-the art classical algorithms [2, 3, 4, 5, 6]. Therefore, we have to
choose an algorithm depth that achieves an acceptable solution while
maintaining quantum coherence. Alternatively, one can modify the standard QAOA
to mitigate the effects of decoherence. Multiple alternative QAOA inspired
algorithms adapted to specific COPs show better theoretical performance at
lower depths. Yet their success also depends on finding good variational
parameters.

The number of variational parameters scales linearly with the depth of the
circuit, and the number of random initialization points needed to find the
global optima is exponential in the number of parameters [7]. All QAOA
variants suffer from this curse of dimensionality and, in general, the
classical optimization routine of QAOA is known to be an NP-hard problem [8].
Barren plateaus in the optimization landscape further increase the difficulty
in finding the optimal parameters [9, 10, 8, 11], and it has been suggested
that problems that are classically hard to simulate will always have these
plateaus [12]. As a result, efficient strategies for finding optimal
parameters are vital in the viability of QAOA [13].

Optimizing the variational parameters is an active field of research, and a
multitude of techniques have been studied, including standard classical
optimizers [14], gradient-free optimizers [14, 15], classical genetic
algorithms [16], machine-learning assisted optimization [17, 18, 19], and
extrapolation techniques [7, 20]. Furthermore, we can benefit from prior
knowledge of the problem, either by initializing a quantum state close to the
solution [21, 22, 23] or bypassing the optimization by transferring pre-
trained optimal parameters [24, 25, 26, 27, 28, 29, 30, 31, 32]. The
optimization can also be simplified by employing fixed schemes such as
Fourier-series extrapolation [7], discrete adiabatic paths [7, 20], or linear
ramp schedules [33]. While the main purpose of QAOA is finding approximate
solutions to COPs, the sampling from the output quantum state is related to
pseudo-Boltzmann distributions that are classically intractable [34, 35]. In
these cases, the quality of the algorithm parameters affects the effective
temperature of the probability distribution, with the optima leading to the
lowest temperature.

In some cases, the optimal parameters concentrate on typical values in
parameter space. This behavior was initially observed for MaxCut on
$3$-regular graphs [36], and later for MaxCut on other kinds of graphs [7, 34,
37, 28]. This phenomenon has been linked with the locality of QAOA at shallow
depths, where the cost Hamiltonian of a graph can be reformulated as a sum of
local Hamiltonians of subgraphs [1, 7]. The typical optimal parameters are
those optimal to the local Hamiltonian that dominate the expectation value of
the total cost Hamiltonian [24, 26, 38, 25]. At algorithm depth $p=11$,
numerical studies show that MaxCut on $3$-regular graphs with typical
parameters has performance guarantees superior to the best classical
algorithms [39]. Furthermore, it is possible to achieve near-optimal QAOA
performances by transferring the typical values from a donor problem instance
to a receiver problem instance drawn from a different ensemble. These
successful transfers have been numerically confirmed between regular graphs
with either even or odd degrees [25, 40], between the Sherrington-Kirkpatrick
model and MaxCut on high-girth graphs [37, 36], between unweighted and
weighted graphs [27, 28], and between the MaxCut problem and other COPs [29].

The classical optimization can also be simplified by identifying symmetries in
the parameter space and limiting the search space accordingly [7, 41, 32]. In
principle, any optimal parameters related by symmetry operations output the
same quantum state and, thus, perform equally when solving the classical
problem. However, when transferring these parameters to other instances, not
all sets of optimal parameters are suitable [25, 40]. In particular, one
should consider domains in the search space containing optimal parameters for
both the donor and the receiver instances. To this end, it is essential to
study the symmetries in both the donor and receiver problem instances.

In this article, we identify symmetries in the optimization landscape of QAOA
for MaxCut that include certain kinds of weighted graphs, previously not
addressed in the literature. In addition, we study the performance of QAOA
using transferred parameters between problem instances for circuit depths
$p=1$ and $p=2$. We define a function characterizing the transferability of
parameters and provide an analytical closed-form expression for triangle-free
regular graphs. We observe that, in general, these transferred parameters
reach a high performance if they belong to certain domains in the search
space. Our analysis extends to $d$-regular graphs with integer weights and
unweighted random graphs from different models.

The article is structured as follows. In Sec. II, we review the formulation of
QAOA and the COP MaxCut, as well as the known symmetries of the variational
parameters. We prove a generalization of these symmetries to include a wider
family of graphs. Then, in Sec. III, we analyze the transferability of optimal
variational parameters and identify which optimal sets are suitable for reuse.
Lastly, we summarize our results in Sec. IV.

##  II Symmetries of QAOA for MaxCut

We begin by reviewing the formulation of QAOA, an algorithm that can be easily
adapted to tackle the optimization of classical functions, and introducing the
notation we use throughout the manuscript. In this work, we consider a
fundamental NP-complete problem in graph theory, namely MaxCut, whose solution
is also hard to approximate. This problem seeks to divide the vertices of an
undirected graph into two sets to maximize the total weight of edges—or the
number of edges in the case of unweighted graphs—crossing between these sets.
That is, given an undirected graph $G=(V,E)$ with weights $w_{ij}$ on the
edges $(i,j)\in E$ joining the vertices $V$, we seek to maximize

$C(x)=\frac{1}{2}\sum_{(i,j)\in E}w_{ij}(1-x_{i}x_{j}),$ (1)

with $x\in\\{-1,1\\}^{n}$ the length-$n$ string to be optimized, and $n$ the
order of the graph $|V|$. For a given problem instance, the quality of a
feasible solution $x$ can be quantified by the approximation ratio,

$r(x)=\frac{C(x)-C_{\rm{min}}}{C_{\rm{max}}-C_{\rm{min}}},$ (2)

which is equal to one for optimal solutions and approaches zero as the quality
decreases. For MaxCut, the best classical algorithms guarantee an
approximation ratio of at least 0.8785 [42].

The MaxCut problem can be encoded on a quantum computer by rewriting the cost
function of Eq. (1) in terms of quantum operators. We replace each variable
$x_{i}$ by the Pauli operator $\hat{\sigma}_{z}^{i}$ to obtain the cost
Hamiltonian

$\hat{H}_{C}=\frac{1}{2}\sum_{(i,j)\in
E}w_{ij}\left(\mathbb{1}-\hat{\sigma}_{z}^{i}\hat{\sigma}_{z}^{j}\right),$ (3)

where the problem’s solutions are related to the measurement outcomes $\pm 1$
of each qubit in the computational basis. The QAOA leverages the problem
structure by encoding it in the cost unitary
$\hat{U}_{C}(\gamma)=e^{-i\gamma\hat{H}_{C}}$. The algorithm iteratively
applies this cost unitary and a mixer unitary
$\hat{U}_{M}(\beta)=e^{-i\beta\hat{H}_{M}}$—with the mixer Hamiltonian
$\hat{H}_{M}=\sum_{i}\hat{\sigma}_{x}^{i}$—to the symmetric superposition of
computational basis states $\ket{+}^{\otimes n}$ to prepare the $n$-qubit
variational state $\ket{\bm{\gamma},\bm{\beta}}$. The state depends on the
real-valued parameters $\bm{\gamma}=(\gamma_{1},\dots,\gamma_{p})$ and
$\bm{\beta}=(\beta_{1},\dots,\beta_{p})$ assigned to the cost and mixing
operations for the total $p$ iterations—depth $p$—and aims to optimize the
expectation value of the cost Hamiltonian $\hat{H}_{C}$,

$C(\bm{\gamma},\bm{\beta})=\bra{\bm{\gamma},\bm{\beta}}\hat{H}_{C}\ket{\bm{\gamma},\bm{\beta}}.$
(4)

We can thus consider the approximation ratio over the distribution of QAOA
solution outputs as a quality measure,

$r(\bm{\gamma},\bm{\beta})=\frac{C(\bm{\gamma},\bm{\beta})-C_{\rm{min}}}{C_{\rm{max}}-C_{\rm{min}}}.$
(5)

The cost landscape represents the relationship between the objective function
of Eq. (4) and the parameters $\bm{\gamma}$ and $\bm{\beta}$. The classical
optimization of $\bm{\gamma}$ and $\bm{\beta}$ is generally challenging due to
highly non-convex landscapes containing several local minima and maxima. The
landscape symmetries help simplify the optimization process by restricting the
search space. In other words, these symmetries help divide the search space
into domains containing optimal parameters that perform equally when finding
good approximate solutions to a COP or creating final states from which one
can perform “pseudo-thermal” sampling [35, 34].

First, we notice that the cost and mixer Hamiltonians are real symmetric
matrices in the computational basis and, therefore, QAOA exhibits a “time-
reversal” symmetry $(\bm{\gamma},\bm{\beta})\to(-\bm{\gamma},-\bm{\beta})$ [7,
32]. This symmetry inverts the sign of all elements of the vectors
simultaneously and is valid for any graph with real weights. Second, we
consider the standard mixer Hamiltonian
$\hat{H}_{M}=\sum_{i}\hat{\sigma}_{x}^{i}$ and MaxCut problem instances with
integer weights so that the cost Hamiltonian has integer eigenvalues and only
two-qubit interactions. Therefore, we begin by restricting the parameter
optimization to the domain $\mathcal{A}^{p}$, such that
$\bm{\gamma}\in[-\pi,\pi)^{p}$ and $\bm{\beta}\in[-\pi/4,\pi/4)^{p}$—as shown
in Fig. 1 for $p=1$—since the QAOA cost function for MaxCut remains unchanged
when any $\gamma_{i}$ and $\beta_{i}$ parameters shift by integer multiples of
$2\pi$ and $\pi/2$, respectively.

In the following, we review additional symmetries of the optimal parameters
within $\mathcal{A}=[-\pi,\pi)\times[-\pi/4,\pi/4)$, summarized in Table 1,
and extend the known results for MaxCut unweighted graph instances [7, 32] to
specific families of weighted graphs.

Table 1: Transformations of $(\bm{\gamma},\bm{\beta})\to(\bm{\gamma}^{\prime},\bm{\beta}^{\prime})$ such that $C(\bm{\gamma},\bm{\beta})=C(\bm{\gamma}^{\prime},\bm{\beta}^{\prime})$. The EWS symmetry refers to integer-weighted graphs where the sum of weights connected to each vertex is even, i.e., $\Sigma_{j}w_{jk}=2\mathbb{Z},\,\forall k\in V$. Analogously, OWS symmetry refers to integer-weighted graphs where the sum of the weights connected to each vertex is odd, i.e., $\Sigma_{j}w_{jk}=\mathbb{Z}\setminus 2\mathbb{Z},\,\forall k\in V$. Graph | Symmetry
---|---
EWS | $(\gamma_{1},...,\gamma_{i},...,\gamma_{p})\to(\gamma_{1},...,\gamma_{i}\pm\pi,...,\gamma_{p})$
OWS | $\left\\{\begin{tabular}[]{l}$(\gamma_{1},...,\gamma_{i},...,\gamma_{p})\to(\gamma_{1},...,\gamma_{i}\pm\pi,...,\gamma_{p})$\\\ $(\beta_{1},...,\beta_{i},...,\beta_{p})\to(\beta_{1},...,-\beta_{i},...,-\beta_{p})$\end{tabular}\right.$
Integer weights | $(\gamma_{1},...,\gamma_{i},...,\gamma_{p})\to(\gamma_{1},...,\gamma_{i}\pm 2\pi,...,\gamma_{p})$
$(\beta_{1},...,\beta_{i},...,\beta_{p})\to(\beta_{1},...,\beta_{i}\pm\pi/2,...,\beta_{p})$
Real weights | $\left\\{\begin{tabular}[]{l}$(\gamma_{1},...,\gamma_{i},...,\gamma_{p})\to-(\gamma_{1},...,\gamma_{i},...,\gamma_{p})$\\\ $(\beta_{1},...,\beta_{i},...,\beta_{p})\to-(\beta_{1},...,\beta_{i},...,\beta_{p})$\end{tabular}\right.$

#### Graphs where the sum of weights connected to each vertex is even.

The cost in Eq. (4) is invariant under parameter changes
$\gamma_{i}\to\gamma_{i}\pm\pi$ for any layer $i$ in QAOA, for graphs where
the sum of the weights connected to each vertex is even, i.e.,
$\Sigma_{j}w_{jk}=2\mathbb{Z},\forall k\in V$ (see proof in Appendix A). This
even weight sum (EWS) symmetry captures the previously known result for
unweighted graphs where all vertices have even degree [32]. More examples that
exhibit this parameter pattern include graphs with only even weights and
graphs where all vertices have even degrees with only odd weights, for
instance, $w_{ij}\in\\{-1,1\\}$.

#### Graphs where the sum of weights connected to each vertex is odd.

The cost in QAOA is invariant under the simultaneous parameter change
$\gamma_{i}\to\gamma_{i}\pm\pi$ and $\beta_{j}\to-\beta_{j}$ in all layers
$j\geq i$, for graphs where the sum of the weights connected to each vertex is
odd, i.e., $\Sigma_{j}w_{jk}=\mathbb{Z}\setminus 2\mathbb{Z},\forall k\in V$
(see proof in Appendix A). This odd weight sum (OWS) symmetry contains the
case of unweighted graphs when all vertices have odd degree [32]. Moreover, we
also include graphs where each vertex has an odd degree and all weights are
odd, for example, $w_{ij}\in\\{-1,1\\}$.

Figure 1: Symmetries for optimal QAOA parameters $(\gamma,\beta)$ at depth
$p=1$ for unweighted regular graphs with odd (top) and even (bottom) degrees,
which follow OWS and EWS symmetries, respectively. Each solid green area
contains one set of optimal parameters, while they do not appear in red-
crossed areas. The domains $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ contain
optimal parameters in both cases.

As previously mentioned, these symmetries relate to patterns in the
optimization landscape and help identify sets of optimal parameters that lead
to the same QAOA performance for a given MaxCut problem instance. The optimal
parameters are known analytically for unweighted $d$-regular triangle-free
graphs at depth $p=1$ [43], and empirically for the unweighted 3-regular tree
subgraph up to $p=11$ [39]. Similar empirical results are known for unweighted
regular graphs of degree $d\leq 11$ at depth $p=2$ [39], and exhaustive sets
of small graphs with $n\leq 9$ vertices at depths $p\leq 3$ [32]. In Fig. 1,
we show the parameter space domains that contain the optimal sets for
unweighted regular graphs with even and odd degrees. We observe that, in both
cases, the domains $\mathcal{U}_{1}=[0,\pi/2)\times[0,\pi/4)$ and
$\mathcal{U}_{2}=[-\pi/2,0)\times[-\pi/4,0)$ contain a set of optimal
parameters.

#### Symmetry patterns at higher depths.

At higher algorithm depths $p$, graphs that obey either the EWS symmetry or
OWS symmetry will have $2^{p+1}$ areas containing an optimal set of parameters
within $\mathcal{A}^{p}$. As shown in Table 1, the cost function will be
invariant under the $m$-th algorithm layer parameter changes
$\gamma_{m}\to\gamma_{m}\pm\pi$ for instances with EWS symmetry, and
$\gamma_{m}\to\gamma_{m}\pm\pi$ and $\beta_{i}\to-\beta_{i}$ for $i\geq m$ for
instances with OWS symmetry. These symmetries provide two optimal parameter
choices at each layer, either the initial set or the symmetric one, which
results in $2^{p}$ possible combinations of optimal parameters. Lastly, the
cost also remains invariant under the “time-reversal symmetry”, which changes
the sign of all parameters in the algorithm, thus doubling the number of
optimal choices. In total, there are $2^{p+1}$ sets of optimal parameters
within $\mathcal{A}^{p}$ providing the same QAOA performance. We could
visualize them as different domains in the energy landscape, as shown in Fig.
1 for $p=1$.

In summary, we have identified parameter symmetries in QAOA for weighted
MaxCut instances that explain optimal parameter patterns from numerical
simulations. Given the proven symmetries, for a given instance, all sets of
optimal parameters can be retrieved knowing a single set, provided there is a
single optimum in the domain $[0,\pi/2)\times[-\pi/4,\pi/4)$. The choice of
parameters in the algorithm determines the quantum circuit. Therefore, a
higher number of optimal parameter sets results in more choices of quantum
circuits and gates, potentially improving the hardware implementation of the
algorithm.

##  III Transferability of optimal parameters

In this section, we analyze the performance of QAOA when transferring the
different sets of optimal parameters identified in Sec. II from one problem
instance to another. To that end, we define a figure of merit to quantify how
transferable the parameters are. We study this transferability property both
analytically and numerically for several families of graphs, with different
degree and weight distributions.

Previous research showed the viability of transferring optimal parameters of
QAOA for MaxCut between unweighted $3$-regular graphs of different sizes [24,
39], from unweighted to weighted graphs [28, 27], between graphs with
different degrees [25, 40] and between different COPs [29, 33]. The
transferability of optimal parameters can be explained by the decomposition of
a given graph instance into local subgraphs at shallow QAOA depths. Following
a light cone argument, one can show that the algorithm sees a sum of local
subgraphs instead of the whole graph. Thus, the QAOA cost function only
depends on these subgraphs, and the algorithm will perform similarly for two
different problem instances with similar local subgraphs. For example, the
cost function of Eq. (4) for all unweighted $3$-regular graphs for QAOA with
depth $p=1$ is a sum of the cost of three subgraphs, times their respective
multiplicity [1]. Such locality arguments for shallow algorithm depths also
explain concentration properties of QAOA, and have been linked to performance
limitations for certain problems [2, 5, 12, 44].

The number of possible subgraphs for a $d$-regular graph grows exponentially
with the algorithm depth $p$, if QAOA does not see the whole graph, i.e., if
$p\sim\mathcal{O}(\log n)$ with $n$ the number of vertices. However, despite
this exponential growth, in a sufficiently large random $d$-regular graph,
almost all subgraphs will be the cycle-free tree subgraph [38, 2, 22].
Consequently, the optimal QAOA parameters for a $d$-regular graph will be
close to the optimal parameters of the $d$-regular tree subgraph, which
explains the successful parameter transferability between them at shallow
depths. However, this argument fails to explain whether we can reuse optimal
parameters between graphs with different degrees (or average degrees).
Particularly, prior research suggests that, on average, we can successfully
transfer any optimal parameters between regular graphs with the same parity
(both have either odd or even degrees). However, for graphs with opposite
parity, we can only reuse a subset of the optimal parameters [25, 40].

#### Transferability error.

To measure the success of reusing parameters of QAOA, we define the
transferability error $\eta_{{R},{D}}$ as

$\eta_{{R},{D}}=r_{R}(\bm{\gamma}_{R}^{*},\bm{\beta}_{R}^{*})-r_{R}(\bm{\gamma}_{D}^{*},\bm{\beta}_{D}^{*}).$
(6)

The error $\eta_{{R},{D}}$ is the difference between the approximation ratios
$r_{R}$ for a receiver problem instance $R$—defined in Eq. (5)—when using its
optimal parameters $(\bm{\gamma}^{*}_{R},\bm{\beta}_{R}^{*})$, and when using
parameters that are optimal to a donor instance $D$,
$(\bm{\gamma}^{*}_{D},\bm{\beta}_{D}^{*})$. A low error indicates a successful
transfer.

#### Analytic expression of the transferability error for simple cases at
$p=1$.

We begin our study of the parameters’ transferability with the simplest
example of unweighted $d$-regular graphs, for which we can rely on previously
known analytical results. Given the locality of QAOA at shallow depths, we
choose the $d^{\prime}$-regular tree subgraph—the most common subgraph—as the
donor instance. Indeed, using the optimal parameters for the $3$-regular tree
subgraph is known to give performance guarantees for arbitrary $3$-regular
graphs at $p=1$ [1] and $p=2$ [39]. Furthermore, the cost value for MaxCut
with triangle-free graphs at depth $p=1$ has a known closed form [43]. Using
the closed form, we can derive an analytical expression for the
transferability error between a $d^{\prime}$-regular tree subgraph and a
$d$-regular graph with girth $g>3$, i.e., triangle-free regular graphs. Thus,
following our definition of Eq. (6), and identifying the receiver ($R$) and
donor ($D$) graphs by their degree $d$ and $d^{\prime}$, we have for depth
$p=1$

$\eta_{{d},{d^{\prime}}}^{g>3}=\frac{1}{2k\sqrt{d}}\biggl{[}\left(\frac{d-1}{d}\right)^{\tfrac{d-1}{2}}-\sqrt{\frac{d}{d^{\prime}}}\left(\frac{d^{\prime}-1}{d^{\prime}}\right)^{\tfrac{d-1}{2}}\biggr{]},$
(7)

with $k=C_{\text{max}}/|E|$, the fraction of edges cut in the maximum cut. In
general, $k\in(1/2,1]$ and $k=1$ for bipartite graphs, and Eq. (7) is valid
for sufficiently large graphs [1], that is, graphs of order larger than
$n\sim(d-1)^{2p}$ (see Appendix B for a detailed derivation).

The function in Eq. (7) allows us to calculate the transferability error for
regular graphs up to an arbitrary degree. We visualize this expression in Fig.
2 for bipartite receiver graphs with degrees up to $d=70$ and tree donor
subgraphs of degree $d^{\prime}\in\\{3,4,5\\}$. The transferability error
$\eta_{{d},{d^{\prime}}}^{g>3}$ is minimized for donor and receiver graphs
with the same degree $d=d^{\prime}$, in agreement with previous results [24,
25]. We observe that as the difference between the receiver and donor degrees
$d$ and $d^{\prime}$ grows, QAOA with the parameters transferred from the tree
subgraphs performs as random guessing with an approximation ratio of $r=1/2$
[45]. Therefore, while the error tends to $0$ as $d\to\infty$, transferring
parameters becomes pointless in the large $d$ limit. Moreover, when the error
eventually reaches $0$ for large $d$, it also indicates that the approximation
ratio of QAOA with depth $p=1$ for any optimal parameters is $r\approx 1/2$
[2].

Figure 2: Transferability error $\eta_{{d},{d^{\prime}}}^{g>3}$ from Eq. (7)
for $d$-regular bipartite graphs ($k=1$) with a donor tree subgraph with
degrees $d^{\prime}=3$ (red solid line), $d^{\prime}=4$ (blue dash-dotted
line) and $d^{\prime}=5$ (green dotted line). The black dashed line represents
the transferability error when QAOA with donated parameters is equivalent to
random guessing.

###  III.1 Numerical analysis

Here, we simulate and study the transferability of optimal QAOA parameters,
characterized by Eq. (6), for different families of graphs.

#### Transferability from the $d^{\prime}$-regular tree subgraph to
$d$-regular unweighted graphs at $p=1$.

This case is approximated by the analytic expression in Eq. (7), visualized in
Fig. 2. Here, we consider ten instances of order $n=120$ of receiver regular
graphs for each degree ranging from $d=3$ to $d=9$. We use the optimal
parameters for the triangle-free graphs, equivalent to the optimal parameters
of the tree subgraphs,
$\gamma_{d^{\prime}}^{*}=\arctan(1/\sqrt{d^{\prime}-1})$ and
$\beta_{d^{\prime}}^{*}=\pi/8$ [43]. In Fig. 3, we observe that Eq. (7)
reproduces the numerical simulations of Eq. (6), and that the transferability
error increases when the difference between the degrees of the donor and
receiver graphs grows.

We note that, in this case, the success of the transfer does not depend on the
relative parity between the degrees $d$ and $d^{\prime}$. Indeed, the set of
optimal parameters from the donor graph lies in the domain $\mathcal{U}_{1}$
(see Fig. 1) and, as discussed in Sec. II, $\mathcal{U}_{1}$ contains optimal
parameters for regular graphs with both odd and even degree. Previous works
also suggest that only parameters from such shared domains, $\mathcal{U}_{1}$
and $\mathcal{U}_{2}$, are the ones transferable between unweighted regular
graphs with degrees of different parity [25]. However, considering the problem
symmetries, if we instead have access to an optimal set of parameters from a
different domain, it can be easily translated into either $\mathcal{U}_{1}$ or
$\mathcal{U}_{2}$ following the rules of Table 1. In view of this, knowing the
problem symmetries and the shared domains between instances, we can transfer
any optimal parameter between any $d$-regular graph independently of the
graphs’ degrees.

Figure 3: Average transferability error $\eta_{{d},{d^{\prime}}}$ of Eq. (6)
at depth $p=1$ for $d$-regular receiver graphs with $n=120$ vertices and donor
tree subgraphs with degrees $d^{\prime}=3$ (red circle) and $d^{\prime}=4$
(blue cross). The error bars overlap with the symbols and denote the 25th and
75th percentiles. Comparison with the analytic results $\eta_{{d},{3}}^{g>3}$
(red solid line) and $\eta_{{d},{4}}^{g>3}$ (blue dashed-dotted line) of Eq.
(7). The black dashed line indicates the average transferability error when
QAOA with donated parameters performs as random guessing.

#### Transferability from the $d^{\prime}$-regular tree subgraph to random
graphs at $p=1$.

In this case, we use again the unweighted $d^{\prime}$-regular tree subgraph
as a donor graph, for general random receiver graphs. Unlike regular graphs,
random graphs do not have a fixed degree and, at low depths, QAOA sees a local
subgraph with a varying degree around each vertex. The random graph models
will have different distributions of degrees. Here, we study three models of
random graphs: the Erdős-Rényi (ER) model [46], the Barabási-Albert (BA) model
[47], and the Watts-Strogatz (WS) model [48].

We use ten instances of each type of random receiver graph with order $n=200$,
and the same average degree $\bar{d}\approx 6$. The degree of the donor tree
subgraph takes the values $d^{\prime}=\\{3,...,9\\}$. In contrast to the
previous case, these random unweighted graphs have neither the OWS nor the EWS
symmetry described in Sec. II, and we find that a transfer is only successful
if the donor parameters are from the domains $\mathcal{U}_{1}$ or
$\mathcal{U}_{2}$. As we observe in Fig. 4, the transferability error is low
for parameters from $\mathcal{U}_{1}$ and increases for parameters from
neither $\mathcal{U}_{1}$ nor $\mathcal{U}_{2}$. Given the degree distribution
of the models, the random graphs have both odd and even degree subgraphs. As a
result, even the parameters optimal to the $6$-regular tree subgraph perform
poorly if they are not included in $\mathcal{U}_{1}$ or $\mathcal{U}_{2}$. We
can compare this result to Ref. [25], where they fix the percentage of even-
regular subgraphs in $20$-node graphs and find different QAOA performances for
parameters transferred from each domain. As in the previous case, the
transferability error is minimized when the donor and receiver graphs have
similar degrees $d^{\prime}=\bar{d}$, that is, $\eta_{{\bar{d}},{6}}\approx 0$
when the donor is a $6$-regular tree in our analysis. Furthermore, for the
random graph models we studied, the impact of the variance of the degree
distribution compared to the average degree is negligible. Additionally, the
relative parity between the degree of the donor subgraph and the average
degree only affects the transferability fo r the BA graphs, when using optimal
parameters outside $\mathcal{U}_{1}$ or $\mathcal{U}_{2}$.

Figure 4: Average transferability error $\eta_{{\bar{d}},{d^{\prime}}}$ of Eq.
(6) at depth $p=1$ for random receiver graphs with $n=200$ vertices and
average degree $\bar{d}\approx 6$, sampled from the ER model (red solid line),
the WS model (green dashed line), and the BA model (blue dash-dotted line). We
consider donor regular tree subgraphs with degrees $3\leq d^{\prime}\leq 9$
and their optimal parameters from a) $\mathcal{U}_{1}$, and b) neither
$\mathcal{U}_{1}$ nor $\mathcal{U}_{2}$. The error bars overlap with the
symbols and denote the 25th and 75th percentiles. The black lines in a)
represent the average transferability error when QAOA with donated parameters
is equivalent to random guessing for ER (solid line), WS (dashed line), and BA
(dash-dotted line).

#### Transferability from the $d^{\prime}$-regular tree subgraph to
$d$-regular graphs with integer weights at p=1.

We study the transferability of parameters from unweighted tree donor
subgraphs with degrees $d^{\prime}=3$ and $d^{\prime}=4$ to $d$-regular graphs
with uniformly distributed integer weights $w_{ij}\in\\{-1,1\\}$, and degrees
$d$ varying from $3$ to $9$. To this end, we use ten instances of $20$-node
receiver graphs for each degree $d$. We choose the optimal parameter set of
the donor graphs from the domain $\mathcal{U}_{1}$. Following the discussion
in Sec. II, we know that the receiver regular graphs with this weight
distribution follow the EWS or the OWS symmetry for even or odd degrees,
respectively.

In Fig. 5, we observe a low transferability error, especially when $d\approx
d^{\prime}$, confirming that the donor parameters are transferable to the
integer weighted graphs. The small size of the simulated graphs ($n=20$)
increases the numerical results’ variability and moves us out of the large
graph regime with triangle-free graphs [2], especially for the higher degrees.
Without this large graph approximation, the tree subgraph is a less suitable
donor. As in all previous examples, the error increases as the difference
between $d$ and $d^{\prime}$ increases and, in this case, it grows faster than
for the unweighted $d$-regular graphs. The smaller sizes of the receiver
graphs can contribute to this faster deterioration of the transferability.
Additionally, we confirm with an initial numerical exploration that the
transferability error increases when using parameters outside
$\mathcal{U}_{1}$ and $\mathcal{U}_{2}$. Since the symmetries that apply are
the same as for the unweighted $d$-regular graphs, we can follow a similar
argumentation as in the unweighted graphs case. Namely, the universally
transferable domains for the weighted graphs are those universal to the
unweighted $d$-regular graphs, i.e., $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$.
However, we know $d$-regular graphs with these integer weights satisfy either
the OWS or the EWS symmetry, and using the rules in Table 1 we can shift a
given set of parameters so that the transferability becomes independent of the
relative parity between $d$ and $d^{\prime}$.

Figure 5: Average transferability error $\eta_{{d},{d^{\prime}}}$ of Eq. (6)
at $p=1$ for $d$-regular receiver graphs with $n=20$ vertices, and uniformly
distributed weights $w_{ij}\in\\{-1,1\\}$. The donor graphs are unweighted
tree subgraphs with degrees $d^{\prime}=3$ (red circle), and $d^{\prime}=4$
(blue cross). The error bars overlap with the symbols and denote the 25th and
75th percentiles. The black dashed line shows the average transferability
error when QAOA with transferred parameters behaves as random guessing.

#### Transferability at higher algorithm depths.

As we show in Sec. II, at depth $p=2$, QAOA for MaxCut on $d$-regular graphs
has eight sets of optimal parameters in $\mathcal{A}^{2}$, instead of the four
sets at $p=1$. We present these parameters in Appendix C (see Table 2) for the
$3$-regular tree subgraph. Here, we study the performance of QAOA for
unweighted regular graphs of order $n=120$ and degrees $d\in\\{3,4,5\\}$,
considering ten graphs per degree, and using—without shifting to a common
domain—the eight sets of optimal parameters of the $3$-regular tree donor
subgraph.

Figure 6:  Average transferability error $\eta_{{d},{d^{\prime}}}$ of Eq. (6)
at depth $p=2$ for unweighted regular receiver graphs of degrees $d=3$ (red
circles), $d=4$ (blue crosses), and $d=5$ (green stars), with an unweighted
tree donor subgraph of degree $d^{\prime}=3$. The error bars overlap with the
symbols and denote the 25th and 75th percentiles. We use the eight optimal
sets of parameters of the donor graph (see Table 2), numbered from one to
eight, as shown on the $x$-axis. The sets $1$ and $8$ lie in $\mathcal{U}_{1}$
and $\mathcal{U}_{2}$, respectively. The black dashed line shows the average
transferability error when QAOA with transferred parameters performs as random
guessing.

In Fig. 6, we observe an analogous behavior to the case of $p=1$. That is,
only parameters from the higher dimensional domains
$\mathcal{U}_{1}^{\,2}=[0,\pi/2)^{2}\times[0,\pi/4)^{2}$ and
$\mathcal{U}_{2}^{\,2}=[-\pi/2,0)^{2}\times[-\pi/4,0)^{2}$ can be transferred
(without using symmetry rules) independently of the relative parity between
the donor and receiver. When transferring these parameters to a receiver graph
of degree $d=3$ or $d=5$, i.e., with the same parity, the error remains low
for all eight sets of parameters. Conversely, when transferring to $4$-regular
graphs, the error is significantly higher for parameters from neither
$\mathcal{U}_{1}^{\,2}$ nor $\mathcal{U}_{2}^{\,2}$. We notice that these six
sets of parameters could be translated into $\mathcal{U}_{1}^{\,2}$ or
$\mathcal{U}_{2}^{\,2}$ using the symmetry relations of the cost function.
Unlike the case of tree subgraphs at $p=1$, there is no analytically known set
of optimal parameters in $\mathcal{U}_{1}$ for a general $d$-regular tree at
$p=2$. However, as mentioned in Sec. II, previous studies show numerical
evidence for optimal parameters in $\mathcal{U}_{1}^{\,2}$ for $d\leq 11$ and
$p=2$ [39]. When restricting to the $3$-regular tree subgraph, the same
authors provide numerically optimized values in $\mathcal{U}_{1}^{\,p}$ at
even higher depths, $p\leq 11$ [39]. Moreover, other works that consider
interpolation and pseudo-adiabatic paths for finding parameters at higher
depths use the initial optimal parameters in $\mathcal{U}_{1}$ and find that
$(\bm{\gamma},\bm{\beta})\in\mathcal{U}_{1}^{\,p}$ [7, 20]. These adiabatic
interpolation techniques fail when considering initial optimal parameters to
$3$-regular graphs outside $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$,
$(\gamma^{*},\beta^{*})\in[\pi/2,\pi)\times[0,\pi/4)$ [20].

Summarizing, prior work confirms the existence of optimal parameters in
$\mathcal{U}_{1}^{\,p}$ for the tree subgraph up to $p=11$. However, more work
is necessary to conclude whether these parameters are universally transferable
at depths $p>2$ for regular graphs of any degree, either odd or even. For
depths $p>11$, it remains an open question whether optimal parameters
(transferrable or not) exist in $\mathcal{U}_{1}^{\,p}$. For instance, if we
consider a linear ramp interpolating the optimal values of the parameters, we
can expect that their values will eventually lie outside
$\mathcal{U}_{1}^{\,p}$. Nevertheless, any optimal parameter at those depths
will follow the symmetry rules discussed in Sec. II.

##  IV Conclusions and outlook

Optimizing the variational circuits remains a big obstacle for realizable and
successful VQAs. One useful simplification in the optimization task is
limiting the search space using symmetries in the energy landscape.
Additionally, the optimal parameters concentrate on typical values for
different instances of certain problems. Thus, we can use these typical values
as pretrained parameters and transfer them between instances, further
simplifying the optimization process.

In this article, we extend the known energy landscape symmetries of MaxCut and
show how to leverage them to transfer optimal QAOA parameters. While previous
studies focus on unweighted graphs, we find that for all graphs where the sum
of the weights connected to each vertex is even, the landscape follows the EWS
symmetry (see Table 1). Analogously, the landscape obeys the OWS symmetry for
all graphs where the sum of the weights connected to each vertex is odd. Among
the examples following these symmetries, we find regular graphs with weights
$w_{ij}\in\\{-1,1\\}$.

Regarding the optimal parameters for $p=1$, it is known that unweighted
triangle-free regular graphs have an optimal set in the domain
$\mathcal{U}_{1}=[0,\pi/2)\times[0,\pi/4)$ [43], and from the symmetries,
other optimal sets in the green solid areas of Fig. 1, depending on the parity
of the degree. Here, we analyze the transferability of these optimal
parameters, i.e. those of a $d^{\prime}$-regular tree subgraph, to triangle-
free $d$-regular graphs, $d$-regular graphs, different kinds of random graphs,
and $d$-regular graphs with integer weights. Our work shows that, in all
cases, the graphs are viable recipients for the transferred parameters.

To measure the success of transferring parameters, we define a transferability
error in Eq. (6), $\eta_{{R},{D}}$, as a function of the approximation ratios
of QAOA with the optimal parameters of the receiver ($R$) and donor ($D$)
graphs. Using an analytically known set of optimal parameters to the
$d^{\prime}$-regular tree subgraphs, we derive a closed-form expression for
the error when transferring to triangle-free $d$-regular graphs at $p=1$. The
expression allows us to analyze the performance of transferability for large
graphs with high $d$ without needing to simulate large QAOA circuits.
Furthermore, this approximated expression reproduces the numerical simulations
of low-girth graphs. The transferability error is minimized and close to zero
if $d\approx d^{\prime}$ for $d$-regular graphs with a $d^{\prime}$-regular
donor tree subgraph. We also find that the error goes to zero in the
$d\to\infty$ limit. However, we can explain the latter result by QAOA
performing equally poorly for all parameters.

We observe that the success of a transfer is independent of the relative
parity between the degree of the donor and receiver when the transferred set
of parameters comes from $\mathcal{U}_{1}$ or $\mathcal{U}_{2}$ (see Fig. 1),
as already reported in previous works [25]. However, we provide the rules to
translate any optimal set of parameters into these domains using the donor
graph symmetry, thus making all sets of optimal parameters transferable in
practice.

Similarly, our results with random receiver graphs show that a transfer is
only successful if the donor parameters come from $\mathcal{U}_{1}$ or
$\mathcal{U}_{2}$. In all the cases with random graphs—generated from the ER
model, the BA model, and the WS model—the success depends on the closeness
between the degree of the donor and the average degree of the receiver graph,
rather than on the degree distribution of the model. In this case, we find
that a transfer is only successful if the donor parameters come from
$\mathcal{U}_{1}$ or $\mathcal{U}_{2}$, independently of the relative parity
between donor and receiver graphs. When considering weighted graphs obeying
either the EWS or the OWS symmetry, the transferability error behaves
similarly to the case of unweighted graphs. That is, the minimum error occurs
when $d=d^{\prime}$, and the domains $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$
contain optimal parameters that work regardless of the relative parity between
the degrees of the donor and receiver graphs.

At the higher algorithm depth $p=2$, we also find successful transfers between
any $d$-regular and $d^{\prime}$-regular graph, if the parameters come from
the higher dimensional domains
$\mathcal{U}_{1}^{\,2}=[0,\pi/4)^{2}\times[0,\pi/2)^{2}$ or
$\mathcal{U}_{2}^{\,2}=[-\pi/4,0)^{2}\times[-\pi/2,0)^{2}$. Similarly, in this
case, we can translate any optimal set of parameters to these domains using
symmetries. We notice that the number of domains that contain an optimal set
of parameters within $\mathcal{A}^{p}=[-\pi,\pi)^{p}\times[-\pi/4,\pi/4)^{p}$
scale as $2^{p+1}$, and our numerical simulations suggest that the number of
shared domains suitable for transferring parameters is two. However, more
studies are needed to conclude this.

In future work, our study on symmetries and transferability can be extended to
higher algorithm depths and other kinds of problems and versions of VQAs, such
as the multiangle QAOA, with different circuit parameterizations. These
efforts may lead to further simplifications of the classical optimization task
in VQAs, one of the most challenging aspects of their successful
implementation.

###### Acknowledgements.

We use the Python-based packages QTensor [49] and Qiskit [50] for the
numerical simulations of QAOA and Matplotlib [51] for data visualization. We
acknowledge useful discussions with Zeidan Zeidan and Göran Johansson. We
acknowledge support from the Knut and Alice Wallenberg Foundation through the
Wallenberg Centre for Quantum Technology (WACQT). L.G.-Á. further acknowledges
funding under Horizon Europe programme HORIZON-CL4-2022-QUANTUM-01-SGA via the
project 101113946 OpenSuperQPlus100.

##  Appendix A Symmetries of optimal QAOA parameters on MaxCut

Here, we provide detailed proofs for the QAOA parameter symmetries presented
in Sec. II, extending the results presented in Ref. [32] for unweighted graphs
where all vertices have either odd or even degree. We identify two symmetries
for regular graphs with integer weights that can be classified according to
the parity of the weights’ sum on each vertex. In the following subsections,
we present the results for graphs in which the weights of the edges connected
to each vertex sum to an even and odd quantity, respectively.

###  A.1 Symmetries for graphs where the sum of weights connected to each
vertex is even

In this section, we prove the symmetry called EWS in Section II. We consider a
variational state constructed by a QAOA circuit of depth $p$. We focus on the
$m$-th iteration of the algorithm, in which the operation
$\hat{U}_{M}(\beta_{m})\hat{U}_{C}(\gamma_{m})$ acts on the intermediate
variational state

$\ket{\Psi_{m-1}}=\prod_{\mathclap{j=m-1}}^{1}e^{-i\beta_{j}\hat{H}_{M}}e^{-i\gamma_{j}\hat{H}_{C}}\ket{+}.$
(8)

Let us now assume that all eigenvalues of the diagonal cost Hamiltonian
$\hat{H}_{C}$ are even, that is, for a computational basis $n$-qubit state
$\ket{y}$—with $y=y_{1}y_{2}\dots y_{n}$ using the binary
representation—$\hat{H}_{C}\ket{y}=2\ell_{y}\ket{y}$, for
$\ell_{y}\in\mathbb{Z}$. In this case, the operations
$\hat{U}_{C}(\gamma_{m})$ and $\hat{U}_{C}(\gamma_{m}\pm\pi)$ are equivalent,
as we can observe by their action on
$\ket{\Psi_{m-1}}=\sum_{y=0}^{2^{n}-1}c_{y}\ket{y}$. Explicitly,

$\displaystyle e^{-i\gamma_{m}\hat{H}_{C}}{\textstyle\sum_{y}}c_{y}\ket{y}$
$\displaystyle={\textstyle\sum_{y}}c_{y}e^{-i\gamma_{m}2\ell_{y}}\ket{y}$
$\displaystyle={\textstyle\sum_{y}}c_{y}e^{-i\gamma_{m}2\ell_{y}}e^{\pm
i2\ell_{y}\pi}\ket{y}$
$\displaystyle=e^{-i(\gamma_{m}\pm\pi)\hat{H}_{C}}{\textstyle\sum_{y}}c_{y}\ket{y},$
(9)

where we have introduced the global phases $e^{\pm i2\ell_{y}\pi}=1$. That is,
the output of the QAOA is invariant under changes
$\gamma_{m}\to\gamma_{m}\pm\pi$ if the costs associated with each
computational state $\ket{y}$ are even. For a graph instance of order $n$, the
cost associated with the MaxCut problem introduced in Eq. (1) evaluates to
zero for the string of $n$ ones $x^{1}=1^{n}$, $C(x^{1})=0$. Consequently, in
the quantum formulation, $\hat{H}_{C}$ will have the even eigenvalue zero for
the product state $\ket{1}^{\otimes n}$, $\hat{H}_{C}\ket{1}^{\otimes n}=0$.
In the following, we consider an arbitrary string $x\in\\{-1,1\\}^{n}$ and
show that changing the value of one bit changes its cost by an even amount.
Therefore, since the cost of $x^{1}$ is even, all the costs are even in both
the classical and quantum formulations, which in turn proves the symmetry.

The costs of Eq. (1) for two strings $x$ and $x^{\prime}$ that differ only by
the $k$-th bit, such that $x^{\prime}_{k}=-x_{k}$, are

$C(x)=\frac{1}{2}\sum_{i\neq
k}\sum_{j>i}w_{ij}(1-x_{i}x_{j})+\frac{1}{2}\sum_{j}w_{kj}(1-x_{k}x_{j}),$
(10)

and

$C(x^{\prime})=\frac{1}{2}\sum_{i\neq
k}\sum_{j>i}w_{ij}(1-x_{i}x_{j})+\frac{1}{2}\sum_{j}w_{kj}(1-x^{\prime}_{k}x_{j}),$
(11)

respectively. Thus, the cost difference is

$C(x)-C(x^{\prime})=\sum_{j}w_{kj}x^{\prime}_{k}x_{j}.$ (12)

Since $x^{\prime}_{k}x_{j}=\pm 1$, this difference will be an even quantity if
the sum of all weights connected to a vertex $k$ is even. This requirement is
equivalent to saying that the number of edges with odd integer weights
connected to a vertex $k$ is even. For the symmetry to hold for all bit
changes, we require this criterion to hold for each vertex.

Particularly, any graph with only even integer weights will show the parameter
symmetry $\gamma_{m}\to\gamma_{m}\pm\pi$, for any layer $m$, regardless of
each vertex degree. Moreover, graphs with only odd weights where all vertices
have even degree (thus including regular graphs of even degree) will share the
same symmetry, with the specific examples of all odd weights
$w_{ij}\in{\\{-1,1\\}}$, and unweighted graphs when all vertices have even
degree. This last example is already proven in Ref. [32]. In Sec. III.1, we
study numerically the transferability of the parameters for regular graphs of
even degree with weights $w_{ij}\in{\\{-1,1\\}}$.

###  A.2 Symmetries for graphs where the sum of weights connected to each
vertex is odd

In this section, we prove the symmetry called OWS in Section II. As in the
previous case, we consider the general case of a QAOA circuit of depth $p$ and
how the change $\gamma_{m}\rightarrow\gamma_{m}\pm\pi$ in the $m$-th layer
changes the final output state. Given the cost Hamiltonian $\hat{H}_{C}$ of
Eq. (3) for MaxCut on a weighted graph, the algorithm operation of layer $m$
is

$\displaystyle\hat{U}_{C}(\gamma_{m}\pm\pi)=\prod_{\mathclap{(i,j)\in
E}}\exp[-i\frac{w_{ij}}{2}(\gamma_{m}\pm\pi)(\mathbb{1}-\hat{\sigma}_{z}^{i}\hat{\sigma}_{z}^{j})]$
$\displaystyle=\hat{U}_{C}(\gamma_{m})\prod_{\mathclap{(i,j)\in E}}\exp(\mp
i\frac{w_{ij}\pi}{2})\exp(\mp
i\frac{w_{ij}\pi}{2}\hat{\sigma}_{z}^{i}\hat{\sigma}_{z}^{j}).$ (13)

Here, we can ignore the global phase factors $\exp(\mp iw_{ij}\pi/2)$ and
focus on the operational terms differing from $\hat{U}_{C}(\gamma_{m})$, which
can be rewritten as

$\exp(\mp
i\frac{w_{ij}\pi}{2}\hat{\sigma}_{z}^{i}\hat{\sigma}_{z}^{j})=\cos(\frac{w_{ij}\pi}{2})\mp
i\sin(\frac{w_{ij}\pi}{2})\hat{\sigma}_{z}^{i}\hat{\sigma}_{z}^{j}.$

We only consider graphs with integer weights $w_{ij}$, which can be either
even or odd. If a given weight $w_{ij}$ is even, only the cosine differs from
zero, which just adds a global phase. Otherwise, if a weight $w_{ij}$ is odd,
only the sine function is different from zero, and the resulting operation of
the $m$-th layer in QAOA changes. For convenience, we rewrite Eq. (A.2) as

$\displaystyle\hat{U}_{C}(\gamma_{m}\pm\pi)=\hat{U}_{C}(\gamma_{m})e^{i\kappa}e^{i\delta}\prod_{\mathclap{\begin{subarray}{c}(i,j)\in
E\\\ w_{ij}\textrm{
odd}\end{subarray}}}\hat{\sigma}_{z}^{i}\hat{\sigma}_{z}^{j},$ (14)

where the global phases encompass all the phase factors that do not change the
final variational state. These phases include the contribution of all edges
with even weights and the phase factors from the sine functions,

$\displaystyle e^{i\kappa}$ $\displaystyle=\prod_{\mathclap{(i,j)\in
E}}\exp(\mp
i\frac{w_{ij}\pi}{2})\prod_{\mathclap{\begin{subarray}{c}(k,\ell)\in E\\\
w_{k\ell}\textrm{ even}\end{subarray}}}\cos(\frac{w_{k\ell}\pi}{2}),$
$\displaystyle e^{i\delta}$ $\displaystyle=\mp
i\prod_{\mathclap{\begin{subarray}{c}(i,j)\in E\\\ w_{ij}\textrm{
odd}\end{subarray}}}\sin(\frac{w_{ij}\pi}{2}).$

As we can observe from Eq. (14), we recover the symmetry
$\gamma_{m}\rightarrow\gamma_{m}\pm\pi$ for any graph with even integer
weights, already proven in the previous section.

In the following, we address the case in which a graph can have both even and
odd integer weights. We denote the set of edges connected to a vertex $k$ as
$E_{k}$ and focus on that vertex and the weights of $E_{k}$. The contribution
of these edges to the product in Eq. (14) is

$\prod_{\mathclap{\begin{subarray}{c}(i,k)\in E_{k}\\\ w_{ik}\textrm{
odd}\end{subarray}}}\hat{\sigma}_{z}^{i}\hat{\sigma}_{z}^{k}=\left(\hat{\sigma}_{z}^{k}\right)^{\mu_{k}}\prod_{\mathclap{\begin{subarray}{c}i|(i,k)\in
E_{k}\\\ w_{ik}\textrm{ odd}\end{subarray}}}\hat{\sigma}_{z}^{i},$ (15)

where $\mu_{k}$ is the number of weights of the set $E_{k}$ with an odd
integer value. Now, we require that $\mu_{k}$ is either even or odd for all
vertices in the graph, $k\in V$, that is, that every vertex has the same
parity for the number of odd weighted edges connected to it. We can simplify
Eq. (14) as

$\displaystyle\hat{U}_{C}(\gamma_{m}\pm\pi)=\hat{U}_{C}(\gamma_{m})e^{i\kappa}e^{i\delta}\prod_{\mathclap{k\in
V}}\left(\hat{\sigma}_{z}^{k}\right)^{\mu_{k}},$ (16)

and we have two cases: if $\mu_{k}$ is even, then
$(\hat{\sigma}_{z}^{k})^{\mu_{k}}=\mathbb{1}$; otherwise, if $\mu_{k}$ is odd,
then $(\hat{\sigma}_{z}^{k})^{\mu_{k}}=\hat{\sigma}_{z}^{k}$. In the first
case, with $\mu_{k}$ even for all $k\in V$, we recover again the symmetry rule
$\gamma_{m}\rightarrow\gamma_{m}\pm\pi$ described in the previous section,
since the sum of all weights connected to each vertex is even.

We explore now the case in which $\mu_{k}$ is odd for all vertices $k\in V$
and include the action of the entire $m$-th QAOA layer, also with the mixer
unitary $\hat{U}_{M}(\beta_{m})$. In particular, we consider changing its
parameter $\beta_{m}\rightarrow\beta^{\prime}_{m}$ such that we find algorithm
symmetries. The complete algorithm step is then given by

$\displaystyle\hat{U}_{M}(\beta^{\prime}_{m})\hat{U}_{C}(\gamma_{m}\pm\pi)=$
$\displaystyle\prod_{\mathclap{k\in
V}}\left[\cos(\beta^{\prime}_{m})-i\sin(\beta^{\prime}_{m})\hat{\sigma}_{x}^{k}\right]$
$\displaystyle\times
e^{i\kappa}e^{i\delta}\hat{\sigma}_{z}^{k}\hat{U}_{C}(\gamma_{m}).$ (17)

Using the relation
$\hat{\sigma}_{z}^{k}\hat{\sigma}_{x}^{k}=-\hat{\sigma}_{x}^{k}\hat{\sigma}_{z}^{k}$,
and $e^{i\theta}=e^{i\kappa}e^{i\delta}$, we can rewrite Eq. (A.2) as

$\displaystyle\hat{U}_{M}(\beta^{\prime}_{m})\hat{U}_{C}(\gamma_{m}\pm\pi)=e^{i\theta}\prod_{\mathclap{k\in
V}}\hat{\sigma}_{z}^{k}\hat{U}_{M}(-\beta^{\prime}_{m})\hat{U}_{C}(\gamma_{m}),$
(18)

which allows us now to study the action of these parameter changes in the
final QAOA expectation value of the objective Hamiltonian,
$C(\bm{\gamma},\bm{\beta})$ in Eq. (4). In particular, we recall the
intermediate variational state $\ket{\Psi_{m-1}}$ of Eq. (8), and construct
the final state for the modified parameters as

$\displaystyle\ket{\bm{\gamma}^{\prime},\bm{\beta}^{\prime}}=$ $\displaystyle\
e^{i\theta}\prod_{\mathclap{j=p}}^{\mathclap{m+1}}\hat{U}_{M}(\beta^{\prime}_{j})\hat{U}_{C}(\gamma_{j})\prod_{\mathclap{k\in
V}}\hat{\sigma}_{z}^{k}$
$\displaystyle\times\hat{U}_{M}(-\beta^{\prime}_{m})\hat{U}_{C}(\gamma_{m})\ket{\Psi_{m-1}},$
(19)

with
$\bm{\gamma}^{\prime}=(\gamma_{1},\dots,\gamma_{m-1},\gamma_{m}\pm\pi,\gamma_{m+1},\dots,\gamma_{p})$
and
$\bm{\beta}^{\prime}=(\beta_{1},\dots,\beta_{m-1},\beta^{\prime}_{m},\beta^{\prime}_{m+1},\dots,\beta^{\prime}_{p})$.
Using again the commutation relations of the Pauli operators, we can rewrite
Eq. (A.2) as

$\displaystyle\ket{\bm{\gamma}^{\prime},\bm{\beta}^{\prime}}=\
e^{i\theta}\prod_{\mathclap{k\in
V}}\hat{\sigma}_{z}^{k}\prod_{\mathclap{j=p}}^{\mathclap{m}}\hat{U}_{M}(-\beta^{\prime}_{j})\hat{U}_{C}(\gamma_{j})\ket{\Psi_{m-1}}.$
(20)

The expectation value of the objective Hamiltonian for the new parameters,
$C(\bm{\gamma}^{\prime},\bm{\beta}^{\prime})$, can be constructed from the
probabilities of measuring each computational state $\ket{y}$,
$P^{\prime}(y)=\bra{\bm{\gamma}^{\prime},\bm{\beta}^{\prime}}\ket{y}\bra{y}\ket{\bm{\gamma}^{\prime},\bm{\beta}^{\prime}}$,
with

$\displaystyle P^{\prime}(y)=$
$\displaystyle\bra{\Psi_{m-1}}\prod_{\mathclap{j=m}}^{\mathclap{p}}\hat{U}^{\dagger}_{C}(\gamma_{j})\hat{U}^{\dagger}_{M}(-\beta^{\prime}_{j})\prod_{\mathclap{k\in
V}}\hat{\sigma}_{z}^{k}\ket{y}$
$\displaystyle\times\bra{y}\prod_{\mathclap{k\in
V}}\hat{\sigma}_{z}^{k}\prod_{\mathclap{j=p}}^{\mathclap{m}}\hat{U}_{M}(-\beta^{\prime}_{j})\hat{U}_{C}(\gamma_{j})\ket{\Psi_{m-1}}.$
(21)

Given that the $\hat{\sigma}_{z}^{k}$ operators commute with $\ket{y}\bra{y}$,
and $\prod_{k\in V}(\hat{\sigma}_{z}^{k})^{2}=\mathbb{1}$, the probabilities
become

$\displaystyle
P^{\prime}(y)=\bra{\bm{\gamma},\bm{\beta}^{\prime}}\ket{y}\bra{y}\ket{\bm{\gamma},\bm{\beta}^{\prime}},$
(22)

with

$\displaystyle\bm{\gamma}$ $\displaystyle=(\gamma_{1},\dots,\gamma_{p}),$
$\displaystyle\bm{\beta}^{\prime}$
$\displaystyle=(\beta_{1},\dots,\beta_{m-1},\beta^{\prime}_{m},\beta^{\prime}_{m+1},\dots,\beta^{\prime}_{p}).$

We observe that the QAOA symmetry, for which $P^{\prime}(y)=P(y)$, with
$P(y)=|\innerproduct{y}{\bm{\gamma},\bm{\beta}}|^{2}$, occurs then for the
simultaneous parameter changes

$\displaystyle\bm{\gamma}$
$\displaystyle\rightarrow(\gamma_{1},\dots,\gamma_{m}\pm\pi,\dots,\gamma_{p}),$
$\displaystyle\bm{\beta}$
$\displaystyle\rightarrow(\beta_{1},\dots,\beta_{m-1},-\beta_{m},-\beta_{m+1},\dots,-\beta_{p}).$
(23)

Thus, graphs where every vertex has an odd number of odd weighted edges
connected to it are symmetric under the parameter changes of Eq. (A.2). In
other words, graphs where the sum of weights connected to each vertex is odd
will exhibit these parameter symmetries.

In particular, graphs with only odd weights where all the vertices have odd
degree (thus also regular graphs of odd degree) will exhibit this symmetry,
which includes the unweighted graphs studied in Ref. [32]. We study
numerically the transferability of the parameters for regular graphs of odd
degree with weights $w_{ij}\in{\\{-1,1\\}}$ in Sec. III.1.

##  Appendix B Derivation of the analytic expression of the transferability
error

In this section, we show a detailed derivation of the closed form expression
for $\eta_{{d},{d^{\prime}}}^{g>3}$ from Eq. (7). The expression is exact for
unweighted triangle-free $d$-regular graphs, shown in Fig. 2, and approximate
for unweighted low-girth $d$-regular graphs, shown in Fig. 3.

From Ref. [43], the objective function of an unweighted triangle-free graph is
given by

$C_{d}=\frac{|E|}{2}\left[1+\sin 4\beta\sin\gamma\cos^{d-1}\gamma\right],$
(24)

evaluated at
$(\gamma_{d}^{*},\beta_{d}^{*})=(\arctan\frac{1}{\sqrt{d-1}},\frac{\pi}{8})$
yields

$C_{d}^{*}=\frac{|E|}{2}\left[1+\frac{1}{\sqrt{d}}\left(\frac{d-1}{d}\right)^{\frac{d-1}{2}}\right].$
(25)

The maximum number of edges that can be cut is $|E|$, and the minimum number
of edges that can be cut is
$\frac{|E|}{2}\left(1+\frac{1}{d}\right)-\frac{1}{4}$. For our purpose, it is
sufficient to lower bound the cut to half the edges, so
$\frac{C_{\text{max}}}{|E|}=k\in(0.5,1]$. The minimum cut value is
$C_{\text{min}}=0$, so $r_{d}=\frac{C_{d}}{C_{\text{max}}}$. From the
definition of $\eta_{{d},{d^{\prime}}}$ from Eq. (6) we get

$\eta_{{d},{d^{\prime}}}=\frac{1}{k|E|}[C_{d}(\gamma_{d}^{*},\beta_{d}^{*})-C_{d}(\gamma_{d^{\prime}}^{*},\beta_{d^{\prime}}^{*})].$
(26)

Since we consider triangle-free receiver graphs, we replace
$C_{d}(\gamma_{d}^{*},\beta_{d}^{*})$ with Eq. (25). The donor graph is the
$d^{\prime}$-regular tree subgraph so
$C_{d}(\gamma_{d^{\prime}}^{*},\beta_{d^{\prime}}^{*})$ is Eq. (24) evaluated
at
$(\gamma_{d^{\prime}}^{*},\beta_{d^{\prime}}^{*})=(\arctan\frac{1}{\sqrt{d^{\prime}-1}},\frac{\pi}{8})$.
Then, we get

$\displaystyle\eta_{{d},{d^{\prime}}}^{g>3}=$
$\displaystyle\frac{1}{2k}\Biggr{[}\frac{1}{\sqrt{d}}\left(\frac{d-1}{d}\right)^{(d-1)/2}$
$\displaystyle-\cos^{d-d^{\prime}}\gamma_{d^{\prime}}^{*}\frac{1}{\sqrt{d^{\prime}}}\left(\frac{d^{\prime}-1}{d^{\prime}}\right)^{(d^{\prime}-1)/2}\Biggr{]}.$
(27)

Using the trigonometry relation

$\cos^{n}(\arctan x)=\left(\frac{1}{1+x^{2}}\right)^{\frac{n}{2}},$ (28)

we get the final expression

$\displaystyle\eta_{{d},{d^{\prime}}}^{g>3}=$
$\displaystyle\frac{1}{2k\sqrt{d}}\Biggr{[}\left(\frac{d-1}{d}\right)^{(d-1)/2}$
$\displaystyle-\sqrt{\frac{d}{d^{\prime}}}\left(\frac{d^{\prime}-1}{d^{\prime}}\right)^{(d-1)/2}\Biggr{]}.$
(29)

Note that the exponent is $\frac{d-1}{2}$ for both terms.

##  Appendix C Optimal parameters for the 3-regular tree subgraph at $p=2$

Table 2:  Eight sets of optimal parameters to the $3$-regular tree subgraph at depth $p=2$ within $\mathcal{A}^{2}=[-\pi,\pi)^{2}\times[-\pi/4,\pi/4)^{2}$. Set 1 and 8 are in the domains $\mathcal{U}_{1}^{\,2}$ and $\mathcal{U}_{2}^{\,2}$, respectively.  | $\gamma_{1}$ | $\beta_{1}$ | $\gamma_{2}$ | $\beta_{2}$
---|---|---|---|---
Set 1 | $0.156\pi$ | $0.177\pi$ | $0.286\pi$ | $0.0933\pi$
Set 2 | $0.156\pi$ | $0.177\pi$ | $-0.714\pi$ | $-0.0933\pi$
Set 3 | $0.844\pi$ | $0.177\pi$ | $0.714\pi$ | $-0.0933\pi$
Set 4 | $0.844\pi$ | $0.177\pi$ | $-0.286\pi$ | $0.0933\pi$
Set 5 | $-0.844\pi$ | $-0.177\pi$ | $0.286\pi$ | $-0.0933\pi$
Set 6 | $-0.844\pi$ | $-0.177\pi$ | $-0.714\pi$ | $0.0933\pi$
Set 7 | $-0.156\pi$ | $-0.177\pi$ | $0.714\pi$ | $0.0933\pi$
Set 8 | $-0.156\pi$ | $-0.177\pi$ | $-0.286\pi$ | $-0.0933\pi$

Here, we present the eight sets of optimal parameters to the $3$-regular tree
subgraph at $p=2$ that we used as donors in Fig. 6 of Sec. III. The sets are
taken from Ref. [39], where the authors define the parameters
$(\bm{\gamma}^{*},\bm{\beta}^{*})\in[0,2\pi)^{2}\times[0,\pi/2)^{2}$. However,
to follow the convention used in this article, we have translated the
parameters into $\mathcal{A}^{p}$ using the OWS symmetry. We provide the
optimal parameter values in Table 2.

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# Coexistence of Canted Antiferromagnetism and Bond-order in $\nu=0$ Graphene

Ankur Das  Department of Condensed Matter Physics, Weizmann Institute of
Science, Rehovot, 76100 Israel  Department of Physics & Astronomy, University
of Kentucky, Lexington, KY 40506, USA  Ribhu K. Kaul  Ganpathy Murthy
Department of Physics & Astronomy, University of Kentucky, Lexington, KY
40506, USA

###### Abstract

Motivated by experimental studies of graphene in the quantum Hall regime, we
revisit the phase diagram of a single sheet of graphene at charge neutrality.
Because of spin and valley degeneracies, interactions play a crucial role in
determining the nature of ground state. We show that, generically, in the
regime of interest there is a region of coexistence between magnetic and bond
orders in the phase diagram. We demonstrate this result both in continuum and
lattice models, and argue that the coexistence phase naturally provides an
explanation for unreconciled experimental observations on the quantum Hall
effect in graphene.

Introduction: The quantum Hall effect is a fundamental manifestation of
topology, quantum mechanics and many-particle physics in two dimensions Prange
and Girvin (1990); Das Sarma and Pinczuk (1996). Discovered originally in
semiconductor heterostructures, it found a new realization in graphene two
decades later Berger  _et al._ (2004); Novoselov (2004); Zhang  _et al._
(2005a); Castro Neto  _et al._ (2009). Graphene brings several tantalizing
twists to the original quantum Hall problem that arise due to its honeycomb
lattice Zhang  _et al._ (2005b); Geim and Novoselov (2007); Castro Neto  _et
al._ (2009); Das Sarma  _et al._ (2011). At low energies graphene has a
relativistic linear dispersion leading to an effective Dirac equation near
charge neutrality, which leads to a distinct Landau spectrum Zheng and Ando
(2002); Gusynin and Sharapov (2005). Additionally, there are two copies of the
Landau levels due to valley degeneracy, causing electron-electron interactions
to play a crucial role in selecting the ground state even for integer fillings
Zhang  _et al._ (2006); Checkelsky  _et al._ (2009); Zhang  _et al._ (2009).

Here we focus on the ground state at charge neutrality ($\nu=0$), which
corresponds to an electron count that fills precisely two of the four (almost)
degenerate $n=0$ Landau levels (LLs). We will call this manifold of states the
zero-energy LLs (ZLLs). At the noninteracting level, the Zeeman energy splits
the four degenerate $n=0$ LLs into pairs of two-fold degenerate ones, picking
a fully polarized ground state Abanin  _et al._ (2006). Since the ZLLs have
equal contributions from particle-like and hole-like states, at the edge one
linear combination of the valleys has a particle-like dispersion, while the
orthogonal linear combination has a hole-like dispersion. The edge of a fully
polarized bulk state develops a pair of counter-propagating charged chiral
modes protected by spin-rotation symmetry, manifesting the quantum spin Hall
effect Abanin  _et al._ (2006). The addition of Coulomb interactions gaps the
single-particle electron spectrum everywhere, but preserves the two gapless
counter-propagating charge modes (protected by $S_{z}$ conservation),
promoting them into a helical Luttinger liquid Brey and Fertig (2006).

From pioneering experiments Zhang  _et al._ (2006); Jiang  _et al._ (2007);
Young  _et al._ (2012); Maher  _et al._ (2013), we now know that the ground
state depends on the balance between the orbital magnetic field, $B_{\perp}$
(perpendicular to the graphene sheet) and the total field, $B_{\rm tot}$
(which enters via the Zeeman energy $E_{Z}$ and can be tuned by applying an
in-plane field). For $E_{Z}$ less than a critical value $E_{Z}^{*}$ all charge
excitations in the bulk and the edges are completely gapped. However, for
$E_{Z}>E_{Z}^{*}$, one obtains a gapped bulk with a two-terminal edge
conductance of (almost) $2e^{2}/h$ Young  _et al._ (2014), which is expected
of the helical Luttinger liquid. While the nature of the phase for
$E_{Z}<E_{Z}^{*}$ has not been conclusively identified in experiment, a
continuous phase transition to it from the fully polarized state is observed
Young  _et al._ (2014). Based on a Hartree-Fock (HF) treatment of a continuum
model which keeps only the ZLLs with ultra-short range interactions, it is
believed that the $E_{Z}<E_{Z}^{*}$ phase is a canted antiferromagnet
Kharitonov (2012). While this proposal is consistent with recent magnon
transmission experiments Takei  _et al._ (2016); Wei  _et al._ (2018);
Stepanov  _et al._ (2018) that imply that the state is magnetic, it is in
tension with STM studies Li  _et al._ (2019) which find evidence for bond
order in the $E_{Z}<E_{Z}^{*}$ insulating phase at $\nu=0$.

In this Letter we offer a resolution to this paradox. We propose that the
seemingly contradictory observations arise from the coexistence of magnetism
and bond order at charge neutrality, which was absent in previous theoretical
phase diagrams. We show by HF methods (known to be reliable in the integer QHE
Prange and Girvin (1990)), both in the continuum and on the lattice, that
coexistence is a generic feature in the regime of interest. In the continuum
model, justified at weak $B_{\perp}$ relevant to experiment, we first show
that a general HF analysis in the ZLLs depends only on six couplings constants
that parametrize the electron-electron interactions. We then show that generic
choices of these couplings lead to coexistence. In a complementary, more
microscopic, HF analysis on the lattice in a magnetic field with $1/q$ quanta
of flux penetrating each unit cell, we find ubiquitous evidence for
coexistence for small and moderate values of $q$ up to $36$. Careful
extrapolation to large $q$ of our numerical data demonstrates that the
coexistence survives in the $B_{\perp}$ regime relevant to experiments (for
reference, $B_{\perp}=10$T gives $q\simeq 10000$). Since coexistence is
generically present in both limiting cases, we argue that it can explain the
experimental observations Young  _et al._ (2014); Li  _et al._ (2019),
especially since disorder, which pins the bond-order, will only enhance its
presence in the physical system.

A microscopic model for graphene in a magnetic field that is expected to
harbor all the phenomena discussed takes the general form,

$H_{\rm latt}=-\sum_{\langle
ij\rangle}t_{ij}c^{\dagger}_{is}c_{js}-E_{Z}\sum_{is}sc^{\dagger}_{is}c_{is}+H^{(4)}_{\rm
int}$ (1)

where $c_{is}$ destroys an electron on the $i^{\rm th}$ site of the honeycomb
lattice with spin $s=\pm 1$. The Zeeman term, $E_{Z}=g\mu_{B}B_{\rm tot}/2$
and the hopping $t_{ij}=te^{i\int_{i}^{j}\vec{A}.d\vec{l}}$ with $\vec{A}$
chosen so $\nabla\times\vec{A}=\hat{z}B_{\perp}$, together describe the free
part of the Hamiltonian. The magnetic field introduces the length scale
$\ell=\sqrt{\frac{h}{eB}}$, such that an area of $2\pi\ell^{2}$ is pierced by
one flux quantum. Since for $B_{\perp}=1\ T$ $\ell=25\ nm$, it is clear that
$\ell\gg a$, where $a$ is the lattice spacing. $H^{(4)}_{\rm int}$ is a four-
fermi electron-electron interaction whose precise form is unknown – we shall
discuss specific forms for it below.

Continuum: In this limit justified for $\ell\gg a$, one restricts attention to
low-energy states near the $K,K^{\prime}$ points, linearizing the band
structure to Dirac equations at each valley. Momentum conservation, when
applied to two-body interactions, forces the conservation of particle number
in the two valleys independently, leading to a $U(1)$ symmetry in the valley
space Alicea and Fisher (2006). An orbital $B$ field is introduced by minimal
coupling into the Dirac equation, leading to four copies (spin and valley) of
a relativistic Landau level spectrum. The interacting Hamiltonian projected
into the ZLLs is,

$\displaystyle H_{\rm cont}=-E_{Z}\sum\limits_{\alpha,k,s}sc^{\dagger}_{\alpha
ks}c_{\alpha ks}+\sum\limits_{{\bf q}\mu}\frac{v_{\mu}({\bf
q})\mathrel{\mathop{\mathchar 58\relax}}\rho_{\mu}({\bf q})\rho_{\mu}(-{\bf
q})\mathrel{\mathop{\mathchar 58\relax}}}{2L_{x}L_{y}}$ (2)
$\displaystyle\rho_{\mu}({\bf
q})=\sum\limits_{k,s,\alpha,\beta}e^{-(\frac{q^{2}}{4}+iq_{x}(k-\frac{q_{y}}{2}))\ell^{2}}c^{\dagger}_{\alpha
k-q_{y}s}\tau_{\mu}^{\alpha\beta}c_{\beta ks}$

where $c_{\alpha ks}$ destroys an electron with spin $s$ in valley $\alpha$
and $y$-momentum $k$, and $\tau_{\mu}$ are Pauli matrices in the valley space.
We work in the Landau gauge $\vec{A}=(0,B_{\perp}x)$ on an $L_{x}\times L_{y}$
sample with periodic boundary conditions in $y$. Since the valley and
sublattice indices are tied in the ZLLs, no sublattice index appears. The
functions $v_{\mu}({\bf q})$ are the Fourier transforms of the effective
interactions (in the ZLLs) in the $\mu=0,x,y,z$ valley channels ($\tau_{0}$ is
the unit matrix). The $U(1)$ valley symmetry forces $v_{x}({\bf q})=v_{y}({\bf
q})$. The phase diagram of Eq. (2) can be calculated in the HF approximation
with the averages $\langle c^{\dagger}_{\alpha
ks}c_{\alpha^{\prime}k^{\prime}s^{\prime}}\rangle=\delta_{kk^{\prime}}\Delta^{ss^{\prime}}_{\alpha\alpha^{\prime}}$
preserves translation invariance up to an inter-valley coherence. Inter-valley
coherence signifies incipient bond-order, though to realize a bond-ordered
state breaking lattice translation symmetries requires physics beyond the
continuum model. Kharitonov Kharitonov (2012) assumed ultra-short-ranged
interactions in real-space ($v_{\mu}({\bf q})\equiv v_{\mu}$ constant), and
found a HF phase diagram with four phases: canted antiferromagnetic (CAF,
characterized by the order parameter $Tr(\tau_{z}\sigma_{x}\Delta)\neq 0$ and
$Tr(\sigma_{z}\Delta)<2$), fully polarized (F, characterized by
$Tr(\sigma_{z}\Delta)=2$), charge-density-wave (CDW, characterized by
$Tr(\tau_{z}\Delta)\neq 0$), and bond-ordered (BO, characterized by
$Tr(\tau_{x}\Delta)\neq 0$). There is no coexistence of order parameters in
this model, and all transitions except for CAF to F are first-order.
Experimental graphene samples are believed to be in the CAF regime for purely
perpendicular fields, which needs $v_{x}=v_{y}<0$, and $v_{z}>|v_{x}|$.
Kharitonov found in his model that $E_{Z}^{*}=|v_{x}|/\pi\ell^{2}$, leading to
the conclusion that increasing $E_{Z}$ while keeping $B_{\perp}$ fixed will
eventually lead to a fully polarized bulk state for $E_{Z}>E_{Z}^{*}$ via a
second-order phase transition, consistent with experiment Young  _et al._
(2014).

We now show that considering a more general form of the interaction by
relaxing the ultra-short-range assumption leads generically to coexistence
between the canted antiferromagnet and bond-ordered states near their phase
boundary in the ultra-short-range model. While the functions $v_{\mu}({\bf
q})$ have an infinite number of degrees of freedom, the translation-invariant
HF ground state energy depends only on two specific numbers for each
$v_{\mu}$: The Hartree coupling $g_{\mu,H}=\frac{v_{\mu}({\bf 0})}{2\pi
l^{2}}$ and the Fock coupling $g_{\mu,F}=\int\frac{d{\bf
q}}{(2\pi)^{2}}v_{\mu}({\bf q})e^{-q^{2}l^{2}/2}$. These six coupling
constants completely characterize the HF energies of all translation-invariant
ground states in the ZLLs. The assumption in previous work Kharitonov (2012)
that the interactions remain short-range on the lattice scale $a\ll\ell$ even
in the effective theory in the ZLLs forces $g_{\mu,H}=g_{\mu,F}$. As we now
show, it is this restrictive assumption that leads to the lack of coexistence
in the phase diagram in previous work Kharitonov (2012).

In the regime of coupling constants of interest in real graphene samples,
where the ground states are CAF and/or BO, we find that three of the couplings
$g_{0,H},g_{0,F},g_{z,H}$ play no role in selecting the ground state. We are
left with just three independent couplings $g_{z,F},g_{xy,H},g_{xy,F}$. We
assume an ansatz for the two occupied orbitals that interpolates between the
CAF and the BO states Murthy  _et al._ (2014).

$\displaystyle|a\rangle=$
$\displaystyle\frac{1}{\sqrt{2}}\big{(}c_{a}|{K\uparrow}\rangle-
s_{a}|{K\downarrow}\rangle+c_{a}|{K^{\prime}\uparrow}\rangle+s_{a}|{K^{\prime}\downarrow}\rangle\big{)}$
(3) $\displaystyle|b\rangle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(-c_{b}|{K\uparrow}\rangle+s_{b}|{K\downarrow}\rangle+c_{b}|{K^{\prime}\uparrow}\rangle+s_{b}|{K^{\prime}\downarrow}\rangle\right)$
(4)

where $c_{\alpha}=\cos{\frac{\psi_{\alpha}}{2}}$ and
$s_{\alpha}=\sin{\frac{\psi_{\alpha}}{2}}$. The CAF state corresponds to
$\psi_{a}=\psi_{b}=\theta$, the canting angle, and the BO state corresponds to
$\psi_{a}=0,\ \psi_{b}=\pi$. In a generic state, these two angles are
independently minimized. We have verified that this ansatz correctly describes
the states of interest by numerically carrying out iterative HF starting from
random “seed” $\Delta$-matrices. We find two necessary conditions for
coexistence: $|g_{xy,F}|>|g_{xy,H}|$ and $E_{Z}>0$. Fig. 1 shows the order
parameters for the BO, CAF and F states as a function of $E_{Z}$ for a
particular choice of our parameters. With this choice, the system starts in
the BO phase at zero $E_{Z}$, undergoes a phase transition to a phase with
coexistence between BO and CAF for intermediate $E_{Z}$, goes through another
transition to a pure CAF phase, and finally to the F phase. All transitions
are second-order. Fig. 2 is a section of the phase diagram at constant
$g_{xy,F}=-1,E_{Z}=1$, clearly showing that coexistence is absent with the
usual ultra-short range assumption $g_{xy,H}=g_{xy,F}$, but appears when
$g_{xy,H}-g_{xy,F}>0$. Evidently, $g_{xy,H}-g_{xy,F}$ determines the sign of
the energy-energy coupling between the two order parameters Bruce and Aharony
(1975) in a Landau theory of the phase transition.

Figure 1:  Order parameters obtained from our generalized HF study of the
continuum theory, Eq. (2) plotted as a function of the Zeeman energy, $E_{Z}$.
We have chosen the interaction parameters $g_{z,F}=0.1$, $g_{xy,H}=-0.75$,
$g_{xy,F}=-1$. The bar at the bottom shows the phase the system is in based on
which orders have condensed. For $E_{z}=0$ the system is in the BO (bond-
ordered) phase. For $E_{z}$ very large the system is in (F) ferromagnetic
phase. Varying $E_{Z}$ between these limits, the system goes through two
intermediate phases, a canted anti-ferromagnet (CAF) without and with bond
order coexistent (CAF+BO) (all three order parameters are non-zero). All the
transitions are continuous in our HF theory.

In order for $g_{\mu,H}$ to be significantly different from $g_{\mu,F}$ one
needs the relevant function $v_{\mu}({\bf q})$ to vary on the scale of the
magnetic length $\ell$ in real-space and be non-monotonic. While the
renormalization of the effective interactions in the ZLLs from very high-
energy states should be independent of $\ell$, the Dirac-Landau quantization
of energy levels, in combination with LL-mixing induced by the Coulomb
interaction Feshami and Fertig (2016), naturally introduces this scale into
the effective interactions while integrating out lower energies. We show an
explicit model calculation of this effect in the supplemental material (SM) SM
.

Lattice: A canonical way to proceed is to carry out a renormalization group
calculation to determine the effective interactions in the ZLL due to LL-
mixing. Unfortunately, because of the large LL-mixing induced by the Coulomb
interaction Feshami and Fertig (2016), there is no natural small parameter
that would justify such a calculation. We will proceed in another direction by
carrying out a lattice HF calculation in the presence of a small orbital flux
(for the noninteracting limit, see, for example Rhim and Park (2012); Das  _et
al._ (2020)) per unit cell Jung and MacDonald (2009); Lado and Fernández-
Rossier (2014); Lukose and Shankar (2016); Mishra  _et al._ (2016, 2017);
Mishra and Lee (2018); Mastropietro (2019); Giuliani  _et al._ (2020). Since
no projection to the low-energy manifold is performed, all LL-mixing effects
are automatically included. Furthermore, lattice scale physics (C3 symmetry,
reciprocal lattice vectors, etc) that plays an important role in the bond
order is kept fully, while it is absent in the continuum. This approach also
allows us to answer the interesting question of whether short-range
interactions on the lattice can lead to effective interactions in the ZLL with
structure on the scale of $\ell$, at least at the level of HF.

Figure 2: A section of the HF phase diagram obtained from our continuum
theory, Eq. 2. Coexistence between CAF and BO at fixed $E_{Z}$ can be seen in
a robust region. The plots are made for $g_{xy,F}=-1,E_{z}=1.0$. Two necessary
conditions for coexistence are $0>g_{xy,H}>g_{xy,H}$ and $E_{Z}>0$. The ultra-
short-range result is the dotted vertical line at $g_{xy,H}=-1$. Figure 3:
Illustrative HF results for the lattice model defined by Eqs. (1) and (5). The
numerical results are obtained on lattices with a flux of $1/q$ quanta per
unit cell and then extrapolated to the weak field regime
($q\rightarrow\infty$). (a) The extrapolated order parameters for
$g=0.3,~{}U=3.5$ as a function of $E_{Z}$. Note that there are two phase
transitions from zero to large Zeeman coupling, consistent with our continuum
result. The phases are labeled in the bottom bar. (b,c) Examples of the
extrapolations of the AFxy and BO order parameters to the $q\rightarrow\infty$
limit used to produce (a). For finite-$q$ coexistence between magnetism and BO
is ubiquitous, but the BO vanishes at intermediate Zeeman as
$q\rightarrow\infty$, resulting in a pure CAF phase for $E_{Z}>0.6$.

We use,

$H_{int}^{(4)}=\frac{U}{2}\sum\limits_{i}\big{(}{n}_{i}\big{)}^{2}-2g\sum\limits_{\langle
ij\rangle}{\vec{S}_{i}}\cdot{\vec{S}}_{j},$ (5)

where $n_{i}=\sum_{s}c^{\dagger}_{is}c_{is}$ and
${\vec{S}}_{i}=\frac{1}{2}\sum_{s,s^{\prime}}c^{\dagger}_{is}{\vec{\sigma}}_{ss^{\prime}}c_{is^{\prime}}$.
The first term is the Hubbard interaction, and the second is a nearest-
neighbor Heisenberg spin exchange. We treat this model in HF approximation
allowing for translation symmetry breaking SM .

As expected, the phase diagram we find is much richer than that found in the
continuum, with several different types of magnetic order and bond order
making their appearance in different ranges of parameters. We focus on the
issue of interest, setting aside the full phase diagram for a future
publication. Computational resources limit us to a maximum $q$ of 36, which
corresponds to $B_{\perp}$ much larger than experimentally accessible fields.
We circumvent this shortcoming by extrapolating our data to the large-$q$
limit, which corresponds to experimentally realizable fields. The extrapolated
order parameters are shown in Fig. 3(a) for a particular choice of couplings
$U,g$. There are two distinct phase transitions at $E_{Z1}$ and $E_{Z2}$ as
$E_{Z}$ is increased. The other two panels show how the extrapolation is done
for representative points in the CAF/BO coexistent phase (b), and for the pure
CAF phase (c). For $E_{Z}<E_{Z1}$, as shown in Fig. 3(b), the order parameters
of the CAF and BO both saturate to nonzero values in the limit $q\to\infty$.
However, for $E_{Z1}<E_{Z}<E_{Z2}$ (Fig. 3(c)) the bond-order vanishes in the
continuum limit, while the CAF saturates to a nonzero value.

Strikingly, the lattice results, when extrapolated to the weak-field limit,
show the same sequence of phases with increasing Zeeman coupling as in the
continuum. Even thought the two calculations approach the problem from
opposite limits they converge on the same generic nature of the coexistence
between CAF and BO. As anticipated, even though $U$ and $J$ are ultra-short-
range interactions, the LL-mixing inherent in the full lattice calculation has
succeeded in generating structure in the effective $v_{\mu}({\bf q})$ on the
scale of $\ell$.

Discussion: Our theoretical results show that there is no contradiction
between magnon transmission experiments Wei  _et al._ (2018) that provide
direct evidence of low-energy spin excitations and STM experiments Li  _et
al._ (2019) which show evidence for bond-order. Furthermore they imply that
the interactions in charge-neutral graphene lie in a region of parameter space
supporting the coexistence of CAF and bond-order at low Zeeman coupling, and
that the effective interactions $v_{\mu}({\bf q})$ in the ZLLs necessarily
have structure on the scale of the magnetic length. This raises the
possibility that other theoretical results (for bilayer graphene, for example)
based on ultra-short-range interactions may need to be revisited. For future
experiments, we predict a new phase transition. Indeed, as we have shown bond-
order will decrease as $E_{Z}$ increases, vanishing continuously at a “lower”
critical Zeeman coupling $E_{Z1}$. This is distinct from the “upper” critical
Zeeman field $E_{Z2}$ at which there is a continuous transition to the fully
spin-polarized state. STM experiments carried out over a range of in-plane
fields can test this prediction.

An important aspect of the experiment not taken into account in our studies is
the effect of disorder. We generically expect disorder to enhance bond order,
though of course it will have other effects as well Hong  _et al._ (2021).
While bond-order breaks translational invariance spontaneously, disorder
breaks this symmetry explicitly, favoring the bond-ordered state over the
translation-invariant CAF state. Thus, we can expect STM experiments to see
bond-order over a wider range of $E_{Z}$ than we found theoretically. While
technically, based on the mapping to a random-field Ising model Imry and Ma
(1975); Binder (1983); Aizenman and Wehr (1989), one may conclude that long-
range bond-order is destroyed by disorder, this clearly does not have
implications for STM experiments, which measure the local strength of bond-
order.

In summary, we have resolved a seeming contradiction in the nature of the low-
Zeeman charge-neutral state of graphene in the quantum Hall regime. By two
complementary methods we find that coexistence between CAF and BO orders is
generic. From the theoretical side, the neighborhood of the phase transition
between the CAF and the BO phases in $\nu=0$ graphene is interesting, because
it may host an approximate $SO(5)$ symmetry Wu  _et al._ (2014); Wang  _et
al._ (2021) and field theories for this transition contain topological terms
Lee and Sachdev (2015) which allow certain excitations in either phase to
carry the quantum numbers of the other. These intriguing ideas provide further
motivation for future experimental and theoretical work on bond order in
$\nu=0$ graphene.

###### Acknowledgements.

The authors acknowledge supported from NSF DMR-1611161, German-Israeli
Foundation (GIF) Grant No. I-1505-303.10/2019, Minerva Foundation, Dean of
Faculty and Israel planning and budgeting committee for financial support
(AD), NSF DMR-2026947 (RKK) and US-Israel Binational Science Foundation Grant
no. 2016130 (GM). The authors thank Benjamin Sacépé, Alexis Coissard, Adolfo
Grushin, and Cécile Repellin for stimulating conversations, and the Aspen
Center for Physics (NSF Grant PHY-1607611) (RKK, GM) where this work was
completed. We would also like to thank the University of Kentucky Center for
Computational Sciences and Information Technology Services Research Computing
for their support and use of the Lipscomb Compute Cluster and associated
research computing resources.

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