# Custom fermionic codes for quantum simulation

Riley W. Chien<EMAIL_ADDRESS>Department of Physics and
Astronomy, Dartmouth College, Hanover, New Hampshire 03755  James D. Whitfield
<EMAIL_ADDRESS>Department of Physics and Astronomy, Dartmouth
College, Hanover, New Hampshire 03755

###### Abstract

Simulating a fermionic system on a quantum computer requires encoding the
anti-commuting fermionic variables into the operators acting on the qubit
Hilbert space. The most familiar of which, the Jordan-Wigner transformation,
encodes fermionic operators into non-local qubit operators. As non-local
operators lead to a slower quantum simulation, recent works have proposed ways
of encoding fermionic systems locally. In this work, we show that locality may
in fact be too strict of a condition and the size of operators can be reduced
by encoding the system quasi-locally. We give examples relevant to lattice
models of condensed matter and systems relevant to quantum gravity such as SYK
models. Further, we provide a general construction for designing codes to suit
the problem and resources at hand and show how one particular class of quasi-
local encodings can be thought of as arising from truncating the state
preparation circuit of a local encoding. We end with a discussion of designing
codes in the presence of device connectivity constraints.

##  I Introduction

A well-known duality between spins and fermions due to Jordan and Wigner
Wigner and Jordan (1928) has been famously employed in the solutions of spin
chains Lieb et al. (1961). Recent years have seen a new applications of spin-
fermion dualities as a way of encoding systems of indistinguishable fermions
into a system of distinguishable qubits. Such transformations are employed in
the simulation of fermionic systems on quantum computers. The idea of a
quantum simulator was conceived in Feynman (1999) and further expanded upon in
Lloyd (1996). It is now expected that quantum computers will become an
invaluable tool in the study of physical properties of strongly correlated
systems which are out of reach of classical computers such as for example
quantum chemistry McArdle et al. (2020). Target problems include chemical
reaction mechanisms Reiher et al. (2017) and the Hubbard model Abrams and
Lloyd (1997).

Working in second quantization, it is necessary to encode the anti-commuting
nature of the fermionic operators into the local qubit degrees of freedom. The
solution used in the Jordan-Wigner transformation (JW), is to map local
operators on one side of the duality to non-local operators on the other side.
When mapping from fermions to spins, the Pauli Z strings are non-local

$\displaystyle a_{j}$ $\displaystyle\to\prod_{k<j}Z_{k}(X_{j}+iY_{j})$ (1)
$\displaystyle a_{j}^{{\dagger}}$
$\displaystyle\to\prod_{k<j}Z_{k}(X_{j}-iY_{j}).$ (2)

In particular, the JW strings can lead to operators as large as the system
size.

Such non-local operators are known to lead to larger gate counts in the
quantum simulation experiment. In particular, the number of qubits an operator
acts on nontrivially, a number we refer to as the Pauli weight, determines the
required number of CNOT gates to perform a rotation generated by that
operator. Thus, lowering the Pauli weight is an important consideration in
particular for the Trotterization paradigm of time evolution which may be used
phase estimation,as a component of a variational state preparation step, or in
quantum imaginary time evolution Motta et al. (2020). It is important to
mention that in regard to phase estimation, recent work has sought to reduce
the T-gate complexity as this is likely to determine the dominant time cost of
a simulation given the magic state paradigm of universal fault tolerance
Bravyi and Kitaev (2005). This work does not seek to address this important
issue. Nevertheless, we expect this work to be of broad interest.

In addition to Jordan-Wigner, a number of other mappings that map $N$
fermionic modes to $N$ qubits are known. These include the parity mapping and
the Bravyi-Kitaev mapping Bravyi and Kitaev (2002). Recently, Jiang et al. put
forward an encoding based on ternary trees which they show to have an optimal
average Pauli weight across all Majorana operators defined on the system Jiang
et al. (2020).

Bravyi and Kitaev also proposed in Bravyi and Kitaev (2002) the superfast
encoding, an explicit method of encoding a system of fermions into qubit
degrees of freedom in such a way as to keep all operators local. This work was
closely followed by Ball (2005); Verstraete and Cirac (2005). The construction
by Bravyi and Kitaev work was explored in Whitfield et al. (2016); Setia and
Whitfield (2018); Chien et al. (2019) and generalized into the construction we
will employ here by Setia et al. in Setia et al. (2019). A number of other
encodings which have sought to reduce the necessary qubit resources and Pauli
weights of transformed operators have also been proposed Jiang et al. (2019);
Steudtner and Wehner (2019). We mention in particular the low weight encoding
of Derby and Klassen Derby and Klassen (2020) as their construction bears the
lowest qubit requirement and operator Pauli weight of known encodings on the
square lattice. An investigation of the resulting gauge theory was given by
Chen et al. along with a construction that gave more a more careful
consideration of spin structures in Chen et al. (2018); Chen (2019). Local
bosonization has also been implemented as a tensor network operator in Shukla
et al. (2020), mapping fermionic tensor networks to bosonic tensor networks.
Also, recent progress has been made in constructing spin duals of fermionic
models in translationally invariant settings using an algebraic formalism
Tantivasadakarn (2020) with focus towards topological and fracton models.

Understanding these local encodings is best done through the toric code model.
This famous model, proposed by Kitaev in Kitaev (2003), is a model for a
topological quantum memory. For our purposes here, it will be most important
to note that there are four basic anyon excitations, the vacuum 1, a bosonic
charge $e$, a bosonic flux $m$, and a fermionic composite of the charge and
flux $\varepsilon=e\times m$. These particles are all created, destroyed, and
moved at the endpoints of local string operators, made possible by the fact
that the system is topologically ordered Levin and Wen (2003, 2005). The
presence or absence of fermionic anyons at a site on the lattice corresponds
to the occupancy of the fermionic mode associated with that site. Finally, all
known local fermionic encodings are equivalent to the toric code defined on
some lattice up to a constant-depth local Clifford circuit. For example, the
Derby-Klassen low weight encoding Derby and Klassen (2020) uses as its
underlying topological state the Wen plaquette model Wen (2003) which is
equivalent to the toric code up to a local basis change on half of the qubits.
Thus, the initialization step of a quantum simulation utilizing a local
encoding is equivalent to preparing a toric code state. We will return to this
point at times throughout the paper when it becomes important.

In this paper, we first review a general construction for local fermionic
encodings before presenting our main result, a generalization of fermionic
encodings that is fully customizable to suit the available resources as well
as the problem geometry. The encoding presented here also encompasses a number
of existing methods as special cases which we mention along the way. We give a
number of examples including lattice models relevant to condensed matter
physics and highly connected systems relevant to the quantum gravity community
which frustrate existing encoding methods. Finally, given that a number of
quantum computing architectures are subject to limited connectivity, we
discuss designing codes under device connectivity constraints.

##  II Review of local fermionic encoding

In this section we review a certain construction introduced by Setia et al. in
Setia et al. (2019) for representing fermionic systems in terms of qubits in a
local fashion. This method is reminiscent of the construction of Nussinov et
al. (2012) for generating highly entangled states using Majorana modes at ends
of nanowires.

Typically, our problem setting will be that we are interested in some property
of a system of $N$ fermionic modes with dynamics governed by a Hamiltonian

$H=\sum_{jk}h_{jk}a^{{\dagger}}_{j}a_{k}+\sum_{jklm}h_{jklm}a^{{\dagger}}_{j}a^{{\dagger}}_{k}a_{l}a_{m}+\dots$
(3)

where the fermionic operators satisfy the usual anti-commutation relations
$\\{a_{j},a^{{\dagger}}_{k}\\}=a_{j}a^{{\dagger}}_{k}+a^{{\dagger}}_{k}a_{j}=\delta_{jk}$
and $\\{a_{j},a_{k}\\}=\\{a^{{\dagger}}_{j},a^{{\dagger}}_{k}\\}=0$. As
explored in Zohar and Cirac (2018), the Hamiltonian could also couple the
fermions to a gauge field, for example in the context of a high energy physics
simulation.

It will be useful to work in the Majorana basis of fermionic operators

$\displaystyle a_{j}$
$\displaystyle=\frac{1}{2}(\gamma_{2{j}-1}+i\gamma_{2{j}})$ (4) $\displaystyle
a_{j}^{{\dagger}}$ $\displaystyle=\frac{1}{2}(\gamma_{2{j}-1}-i\gamma_{2{j}})$
(5) $\displaystyle\\{\gamma_{j},\gamma_{k}\\}$ $\displaystyle=2\delta_{jk}.$
(6)

From these, we can form two types of operators quadratic in the Majoranas
which we will from here on refer to as edge and vertex operators,

$\displaystyle A_{jk}$
$\displaystyle=-i\gamma_{2{j}-1}\gamma_{2{k}-1}\quad\text{ (edge)}$ (7)
$\displaystyle B_{j}$ $\displaystyle=-i\gamma_{2{j}-1}\gamma_{2{j}}\quad\text{
(vertex)}.$ (8)

These operators suffice to generate the full algebra of parity preserving
fermionic operators. The vertex operator $B_{j}$ is the parity operator for
the mode $j$ and the edge operator $A_{jk}$ is involved in all hopping,
pairing, and scattering processes. Note that edge operators involve only odd-
indexed Majoranas. Appropriately multiplying an edge operator by vertex
operators allows for coupling two fermionic modes by their even indexed
Majoranas. For example, a hopping term can be expressed as

$a_{j}^{{\dagger}}a_{k}+a_{k}^{{\dagger}}a_{j}=-i(A_{jk}B_{k}+B_{j}A_{jk})/2.$
(9)

(a)

(b)

Figure 1: (a) Local Majorana modes for a degree $d=6$ vertex. (b) Two types of
plaquette stabilizers for the triangular lattice with the above encoded
Majoranas. (Left) Orange lines connecting Majoranas indicate the corresponding
edge operators. (Right) The corresponding qubit operators.

We here explicitly place the fermionic system on a graph which we refer to as
the interaction graph with fermionic modes associated to vertices $j\in V$.
When one writes the fermionic Hamiltonian in terms of edge and vertex
operators, if an edge operator, $A_{jk}$, coupling modes $j,k$ is required to
write the Hamiltonian in this form, then an edge is placed on the graph
connecting vertices $j$ and $k$, $(j,k)\in E$. The graph is the given as the
pair $\Gamma=(V,E)$. Further, we can identify the set of plaquettes on the
graph $P$ and the boundary of a given plaquette $p$ is the set of edges which
form it, $\partial p$.

For each vertex $j$ on the graph, we place a number of qubits equal to half
the degree of the vertex, $d(j)/2$ (actually $\lceil d(j)/2\rceil$ but we will
assume graphs of even degree for simplicity). We next define $d(j)$ Pauli
operators acting on the $d(j)/2$ qubits on site $j$ which we will refer to as
local Majoranas

$c_{j}^{1},\dots,c_{j}^{d(j)}\in\mathcal{P}_{d(j)/2}$ (10)

We use the letter $c$ for the local Majoranas to distinguish these Pauli
operators from the physical Majorana fermion operators $\gamma$. These local
Majoranas may be anything except that they must satisfy the following
properties: (1) they must satisfy Majorana anti-commutation relations with the
other operators defined on that vertex (they commute with operators defined on
other vertices) and (2) they must generate the full Pauli group on $d(j)/2$
qubits. Any choice of definition for the local Majoranas is related to any
other choice by a Clifford circuit acting only on qubits at that vertex. In
the figures throughout this paper, we will always use Jordan-Wigner to encode
the local Majoranas on a given vertex starting from the top clockwise,
$\\{c_{j}^{1},c_{j}^{2},c_{j}^{3},c_{j}^{4},\dots\\}\to\\{X_{j1},Y_{j1},Z_{j1}X_{j2},Z_{j1}Y_{j2}\dots\\}$
(See Fig. 1). A choice of definition for the Majoranas based on Fenwick trees
can give a decrease in the Pauli weight from $O(d)$ to $O(\log(d))$. As such,
the Fenwick tree choice will be preferable for graphs of large degree. We
refer readers to Setia et al. (2019); Whitfield et al. (2016) for a discussion
of Fenwick tree encodings. The recently proposed ternary tree construction of
Jiang et al. (2020) would provide a further reduction in Pauli weight.

Each local Majorana is associated to one of the edges incident to the vertex.
Each edge then has two local Majoranas associated to it, one from each vertex
at its endpoints and we will define our encoded edge operators to be

$\tilde{A}_{jk}=\epsilon_{jk}c_{j}^{p}c_{k}^{q}$ (11)

where $\epsilon_{jk}=+1$ if $j<k$ and $=-1$ if $j>k$ and $k$ is the $p$-th
neighbor of $j$ and $j$ is the $q$-th neighbor of $k$. Encoded vertex
operators are a product of all Majoranas at a given vertex,

$\tilde{B}_{j}=i^{d(j)/2}c_{j}^{1}c_{j}^{2}\dots c_{j}^{d(j)}.$ (12)

If Jordan-Wigner is used to encode the local Majoranas on a given site, then
the vertex operator will be Pauli Z acting on all the qubits on that vertex so
the occupancy of the mode is stored in the collective parity. Another choice
could be made such that the occupancy is stored in a single qubit. One could
achieve this an appropriate application of CNOTs.

Finally, for each plaquette $p$ on the graph, we have a stabilizer which is
given by a product of all the edge operators around the boundary of the
plaquette

$S(\partial p)=i^{|\partial p|}\prod_{(j,k)\in\partial p}\tilde{A}_{jk}.$ (13)

Whereas in the toric code, the plaquette stabilizers detected the presence of
a flux, here the plaquette stabilizers detect the presence of a flux without
an accompanying charge and vice-versa. In dimensions higher than 1, the
logical subspace is that in which charges have fluxes attached. Recall that
charge-flux pairs are fermionic in nature which provides the basis for this
construction. An example on a triangular lattice is shown in Fig. 1. For
geometries with non-contractible loops, e.g. a torus, we fix boundary
conditions to be periodic with a stabilizer consisting of a product of edge
operators around the non-contractible loop.

As a final consideration, we consider the odd fermionic parity sector. On a
graph with even degree it is not possible to directly encode an odd number of
particles. One must introduce an additional non-physical auxiliary mode and
create a pair of particles with one occupying the auxiliary mode. The total
parity of the physical modes is then odd and the odd-parity simulation can
proceed. If however there is a vertex on the graph with an odd degree, then
there will be a single unpaired Majorana operator on that vertex. Acting with
the unpaired Majorana operator can create or destroy a single particle at that
vertex without violating any of the stabilizers. As a simple example without
any stabilizers, we consider a 1D chain with open boundaries. At the two ends
of the chain, there are two unpaired local Majorana operators. If we for
example act with the unpaired Majorana on the first site in the chain, we
create a single particle. In this way, acting with unpaired Majorana operators
changes our parity sector. We can then proceed with our odd-parity sector
simulation. In the next section we show that if we pair the local Majorana
operators at the ends of the chain, we impose periodic boundary conditions and
lose the ability to enter the odd parity sector.

###  II.1 1D chain recovers Jordan-Wigner

To further illustrate the construction in a familiar setting that we hope will
give some intuition for the later sections, we will encode a 1D chain of
fermions with an onsite potential and periodic boundary conditions. The
Hamiltonian of this system is

$\displaystyle H$
$\displaystyle=t\sum_{j}(a^{{\dagger}}_{j+1}a_{j}+a^{{\dagger}}_{j}a_{j+1})+U\sum_{j}a^{{\dagger}}_{j}a_{j}$
(14)
$\displaystyle=\frac{-it}{2}\sum_{j}(A_{j,j+1}B_{j+1}+B_{j}A_{j,j+1})+U\sum_{j}B_{j}.$
(15)

Each vertex in this 1D chain obviously has $d=2$ so a single qubit is placed
at each vertex. We choose $c_{j}^{1}=Y_{j}$, $c_{j}^{2}=X_{j}$ for the
Majoranas at each site. All the edge operators are then
$\tilde{A}_{j,j+1}=X_{j}Y_{j+1}$ and the vertex operators are
$\tilde{B}_{j}=Z_{j}$. The transformed Hamiltonian then takes the familiar
form of the XY chain

$\tilde{H}=\frac{t}{2}\sum_{j}(X_{j}X_{j+1}+Y_{j}Y_{j+1})+U\sum_{j}Z_{j}.$
(16)

Notice that given the basis chosen for the local Majoranas at each vertex, the
Jordan-Wigner transformation is recovered. Indeed, an edge operator between
two modes not nearest-neighbor connected is necessarily a product of edge
operators in a path connecting the two targeted vertices

$\displaystyle A_{j,j+n}$ $\displaystyle=A_{j,j+1}\dots A_{j+n-1,j+n}$ (17)
$\displaystyle\tilde{A}_{j,j+n}$ $\displaystyle=(-i)^{n-1}X_{j}Z_{j+1}\dots
Z_{j+n-1}Y_{j+n}$ (18)

giving back the Jordan-Wigner strings of Pauli Zs that all local fermionic
encodings are attempting the alleviate.

Finally, as we have periodic boundary conditions and thus a (large) loop in
our interaction graph, we have a stabilizer which is the product of all edge
operators around the loop. This operator, which is given by
$S=\prod_{j}Z_{j}$, corresponds to the fact that global fermionic parity is
preserved and constrained to be even. We could as well choose not to restrict
to the subspace stabilized by the loop as previous described in our discussion
of the odd parity sector. In that case, edge operators coupling sites $1,N$
would have a Pauli weight extensive in the system size. The total fermionic
parity will always be a symmetry of the Hamiltonian by virtue of only even
parity operators being physical. The above shows a consequence of imposing the
total parity to be a symmetry of the states as well. This clear physical
interpretation does not however generalize to plaquette stabilizers in higher
dimensions.

##  III Main result: Custom fermionic codes

(a)

(b)

Figure 2: (a) Two nearest-diagonal-neighbor couplings (left) fermion picture
(right) qubit picture (b) The same two nearest-diagonal-neighbor couplings
with diagonal edges omitted from the system graph.

Our main result is centered on the idea that regardless of the interaction
graph determined by the Hamiltonian terms, we may choose to encode the system
into whatever geometry we wish. As such, we will begin discussing two separate
graphs for the remainder of the paper, the interaction graph as determined by
the Hamiltonian and the system graph that we will encode with our qubit
system. The system graph must have at least as many vertices as the
interaction graph in order to accommodate the fermionic modes. Also, if an
edge operator coupling two modes is present in the Hamiltonian is expressed
using the operators of (7 \- 8), then a path must connect those two vertices
on the system graph. Otherwise the system graph may be arbitrary.

We will discuss a number of useful modifications of the interaction graph. We
will briefly discuss sparsifications of the interaction graph - simply
omitting edges in the case of lattice models. We then discuss using a virtual
geometry including virtual modes that provide paths across which interactions
may take place. We there give an example of a system featuring all-to-all
interactions.

Next, we give a construction that allows for balancing qubit resource
requirements and code locality in the case of lattice models through a
blocking construction. We will find it convenient at that time to discuss
state preparation. Finally, we will discuss encodings in the presence of
constraints on connectivity between qubits using the recently proposed heavy-
hexagon lattice as an example.

###  III.1 Omitting edges

As shown above in (18), if two modes are meant to be coupled with an edge
operator but the two modes do not share an edge on the system graph, the two
modes are still able to be coupled together, but the interaction will be not
strictly local. This will lead to a generalization of the JW Z string where
the intermediary vertices will all be acted on by a product of two local
Majoranas. These generalized Jordan-Wigner strings are similar to string
operators in quantum error correcting codes.

For concreteness, consider a $L\times L$ square lattice of fermionic modes
interacting with nearest-neighbors and nearest-diagonal-neighbors, those
across the diagonal of a square. An example of such a problem is the nearest-
diagonal-neighbor Hubbard model. The vertices of the interaction graph are
then of degree $d(j)=8$. If we choose the system graph to match the
interaction graph, then we require $4$ qubits at every vertex giving $4L^{2}$
qubits in total. As shown in Fig. 2b, we omit the diagonal edges from our
system graph such that each vertex has only degree $d(j)=4$. The diagonal edge
operators are then a product of two nearest-neighbor edge operators. The path
taken around the square plaquette does not matter as the upper and lower path
in each case are equal up to multiplication by a stabilizer. We see that in
this case, the Pauli Weight of the qubit operators is smaller for the two
paths without the diagonal edges in the system graph. This can be seen in Fig.
2. Also the presence of the additional qubits also increases the Pauli weight
of the nearest-neighbor edge operators and the vertex operators. So, we have
shown that strict locality is not always optimal and relaxing to quasi-
locality is in some cases beneficial.

The system graph can be sparsified arbitrarily relative to the interaction
graph to save qubits as long as paths connect modes that must be coupled with
edge operators. In this construction, the qubit requirement is determined by
the vertex degrees and not the number of edges as in the local mappings of
Bravyi and Kitaev (2002); Chen et al. (2018). Thus, if a reduction of qubit
requirement is sought, one should delete edges with the aim of reducing the
degrees of vertices with target degrees being even numbers.

We would like to mention here that if our system has a square lattice
interaction graph and we sparsify the graph in certain ways, we recover the
auxiliary qubit mappings of Steudtner and Wehner Steudtner and Wehner (2019).
Thus, this construction contains the auxiliary qubit mappings as special case.

###  III.2 Adding virtual modes

Figure 3: (left) Schematic of an operator coupling two modes at the boundary
of the MERA geometry. The orange arc connecting the two endpoints indicates
the path of the generalized Jordan-Wigner string. The path through the virtual
space depicted in gray is shorter than the path along the boundary. (right)
The qubit operator corresponding to a coupling with the generalized Jordan-
Wigner string following the discrete geodesic.

Additional virtual modes can be added to the system graph which can in some
cases lead to a reduction in the Pauli weight of the transformed Hamiltonians.
As always we require closed loops in the system graph to be stabilized by the
corresponding plaquette operators. Virtual modes are stabilized by their
vertex operators as they will always be unoccupied and so have parity $+1$.

Using virtual modes to reduce Pauli weights could be especially useful in
cases where one has a complete or nearly complete interaction graph. Nearly
complete interaction graphs (within spin sectors) are known to arise in small
molecular Hamiltonians using atomic orbital basis sets leading to large
simulation costs with strictly local encodings Chien et al. (2019). A number
of all-to-all interacting fermionic models have also become popular in recent
years in the study of scrambling of quantum information and of AdS/CFT, the
most notable of which is the SYK model Sachdev and Ye (1993); Sachdev (2015).

####  III.2.1 All-to-all coupled fermions

We now give an example of a system for which the quantum simulation cost is
decreased by using virtual modes. The SYK model consists of $2N$ Majorana
fermions with random strength $q$-body interactions coupling all Majoranas.
Proposals regarding quantum simulation of the $q=4$ SYK model have previously
been put forward García-Álvarez et al. (2017); Babbush et al. (2019); Cao et
al. (2020).

We will consider the $q=2$ case,

$H=-i\sum_{j<k}^{2N}J_{jk}\gamma_{j}\gamma_{k}.$ (19)

We will pair the $2N$ Majorana fermions into $N$ complex fermions and let the
indices now run over complex fermions. It will also be convenient for us to
break the terms up by whether the Majoranas are odd or even indexed

$H=\\\
-i\sum_{j<k}^{N}J_{2j-1,2k-1}\gamma_{2{j}-1}\gamma_{2{k}-1}-i\sum_{j\leq
k}^{N}J_{2j-1,2k}\gamma_{2{j}-1}\gamma_{2{k}}\\\
-i\sum_{j<k}^{N}J_{2j,2k-1}\gamma_{2{j}}\gamma_{2{k}-1}-i\sum_{j<k}^{N}J_{2j,2k}\gamma_{2{j}}\gamma_{2{k}}$
(20)

We can then write the Hamiltonian in terms of edge and vertex operators

$H=\sum_{j<k}^{N}J_{2j-1,2k-1}A_{jk}+\sum_{j}^{N}J_{2j-1,2j}B_{j}\\\
+i\sum_{j<k}^{N}J_{2j-1,2k}A_{jk}B_{k}+i\sum_{j<k}^{N}J_{2j,2k-1}B_{j}A_{jk}\\\
-\sum_{j<k}^{N}J_{2j,2k}A_{jk}B_{j}B_{k}$ (21)

The interaction graph is a complete graph so each vertex has degree
$d(j)=N-1$. We calculate the qubit requirement and the Pauli Weight for a
number of different system geometries):

  1. 1.

A complete graph with $N$ vertices (Complete Graph)

  2. 2.

A 1D chain with periodic boundary conditions (Linear)

  3. 3.

A geometry with a single virtual mode of degree $N$ connected to all physical
modes which only connect to the central virtual mode (N Branches)

  4. 4.

Ternary tree with physical modes at the leaves and virtual vertices of degree
$4$ (Ternary Tree)

  5. 5.

Ternary MERA-like geometry with virtual vertices of degree $4$ (Ternary MERA)
(a cutout is shown in Fig. 3)

  6. 6.

Hyperbolic geometry where each vertex is degree $6$ and faces are $4$-sided.
Physical modes are identified with legs at the boundary of the disk. ($d=6$
Hyperbolic Tiling)

Figure 4: (Top) Qubit requirements to encode the system of $N$ complex
fermions ($2N$ Majorana fermions) given the specified geometry (Bottom) The
worst-case Pauli weight of the transformed Hamiltonians.

Some of these geometries have taken their inspiration from tensor networks
such as MERA Vidal (2004); Swingle (2012) as well as existing literature
regarding hyperbolic codes Pastawski et al. (2015). For the geometries
containing vertices with vertex degrees growing with system size, we use the
Fenwick tree as proposed in Setia et al. (2019) which reduces the worst-case
weight of operators to logarithmic in the degree. Note also that the ternary
tree geometry used here is unrelated to the ternary tree geometry construction
of Jiang et al. (2020).

We present the qubit requirements and the worst-case Pauli weight for each
geometry vs the number of complex fermions $N$ in Fig. 4. For the tree, MERA,
and hyperbolic geometries, the Pauli weight data presented should be
interpreted as approximate given that only certain numbers of modes completely
fill levels in the hierarchical constructions.

We now summarize the results shown in Fig. 4. All geometries except the
complete graph required a number of qubits scaling linearly in the number of
modes. The complete graph qubit requirements scaled quadratically in the
number of modes. While all geometries except the linear chain provided a Pauli
weight that scaled as $O(N^{2}\log{N})$. The linear geometry had a Pauli
weight that scaled as $O(N^{3})$.

(a)

(b)

(c)

Figure 5: (a) A section of a square lattice of fermionic modes (white circles)
with nearest neighbor interaction graph (gray lines). Schematics of two edge
operators are shown in orange ( The operators are products of Majoranas from
the two fermionic modes). (b) The lattice is partitioned into $2\times 2$
blocks of fermionic modes (white). The system graph (gray) is modified from
nearest-neighbor square lattice in (a) to a coarser lattice of blocks. The top
left mode in each block remains in the lattice while the remaining modes
within the block are connected as a 1D chain. The same two edge operators from
(a) are shown here. Operators coupling modes within a block run along the
chain. Operators coupling sites in adjacent blocks have a generalized Jordan-
Wigner string that traverses the lattice from one block to the next. (c) The
qubit-encoded version of the system graph is shown. Each fermionic vertex on
the lattice, corresponding to fermionic modes, contain either one or three
qubits. The qubit operator versions of the edge operators from (a) and (b) are
shown. All operators remain local with respect to the coarse-grained lattice.

In this case, we see that the matching the system graph to the interaction
graph exactly is not the most economical encoding strategy. It requires the
greatest qubit requirements of all geometries shown, $N(N-1)/2$, while giving
a Pauli weight scaling that was comparable to other geometries. The opposite
limit, the 1D geometry, requires the least number of qubits, $N$, but due to
the non-locality of the string operators, the Pauli weight scales as
$O(N^{3})$.

Geometries where we include virtual modes which provide shorter paths between
vertices perform favorably at the cost of more qubits. For example, we can
include a single central mode connected to all others which serves as a
midpoint for interactions. Using this geometry, all couplings are products of
two edge operators from the physical modes to the virtual one and we require
$1.5N$ qubits. The Pauli weight still scales as $O(N^{2}\log{N})$ but with the
Fenwick tree encoding of local Majorana operators was the lowest of all
geometries tested.

The other three geometries presented in Fig. 4 also all require a number of
qubits that is linear in $N$. The hierarchical structure of the geometries
means that points separated by $l$ on the boundary have paths through the
virtual space that are length $\sim\log{l}$. As a result, they offer Pauli
weights that scale as $O(N^{2}\log{N})$. In the case of the ternary MERA and
$d=6$ hyperbolic geometries we can give a nice interpretation, the couplings
feature generalized JW string operators traversing discrete versions of
geodesics in the virtual hyperbolic geometry, Fig 3.

We note that in a number of the geometries investigated above, we placed
physical modes at the boundaries of a hyperbolic disk or at the leaves of a
tree while the other modes were considered virtual i.e. not corresponding to
physical modes. We could also have chosen to associate the vertices in the
bulk of graph with physical modes. This would provide a savings in the
required number of qubits.

Previous proposals for simulation of the SYK model have used Jordan-Wigner
which in the $q=2$ case leads to a Pauli weight of $O(N^{3})$. As we have
shown this can be reduced to $O(N^{2}\log{N})$. Thus we can reduce the
complexity of the simulation with a more careful consideration of encoding
geometry.

Given that simulations of all-to-all interacting systems can benefit from a
virtual geometry, it would be interesting to extend our investigation to
sparsified graphs for example the sparsified SYK model proposed in Xu et al.
(2020). In addition to contributing towards progress in studies quantum chaos
and holography, we also believe that investigations of quantum simulations of
such highly connected systems will also benefit simulations of quantum
chemistry as chemical Hamiltonians can also feature regions of high
connectivity as we have shown in Chien et al. (2019).

###  III.3 Qubit vs. locality trade-off with blocking

We will now consider an approach to finding a balance between locality and
qubit requirement for a system on a $L\times L$ square lattice. A linear (JW)
geometry will require $L^{2}$ qubits but will suffer from long JW strings
whereas a strictly local encoding will require roughly $2L^{2}$ qubits (minus
a few on the boundaries). Again, we can reduce the required number of qubits
by relaxing the necessity for strictly local interactions. We will divide the
system into a number of blocks. Interactions within blocks will be non-local
incurring Jordan-Wigner strings that increase in length with the size of the
blocks. Interactions between blocks, however, remain local in that the Pauli
weight of operators are independent of the size of the full lattice.

The blocking goes as follows. Partition the lattice into the desired number of
blocks, $b$, determined by the available resources. Treat the modes within the
block as though they were on a 1D chain. The lattice is then a coarser lattice
with each vertex connected to a 1D chain. The first mode in the chain remains
connected to the lattice such that those vertices are of degree $d=5$ in the
bulk of the system and so require $3$ qubits. The total number of qubits
required is then $L^{2}+2b$ (up to boundaries). With this construction, we are
free to interpolate between a strictly local encoding with $L^{2}$ blocks of
size $1$ and $1$ block of size $L^{2}$ by choosing blocks to be of the desired
size.

The idea of using a segmented encoding was explored in Whitfield et al. (2016)
where segmented versions of the Bravyi-Kitaev and Fenwick tree encodings were
explored for the 2D Hubbard model.

####  III.3.1 As truncated state preparation

We now hope to provide an intuitive picture for the blocking construction.
This will also be a convenient time to address state preparation. For
simplicity, we will again consider a square lattice of dimension $L\times L$.
As previously mentioned, this encoding manages to encode the fermions in a
local way by representing them as excitations of the toric code which are odd
under particle exchange. Thus, to use this encoding, one must prepare a toric
code state which is well known to be topologically ordered and therefore long-
range entangled Chen et al. (2010).

Utilizing a MERA quantum circuit, one can prepare a toric code state by
introducing entanglement scale-by-scale beginning with long-range entanglement
and ending with entanglement at the final lattice scale Aguado and Vidal
(2008). The circuit is comprised of a number of levels $U=U_{1}\dots
U_{\log{L}}$. Each level $k$ takes as input a state on a lattice of linear
size $l$ and a number of ancilla qubits and outputs a state of linear size
$2l$,

$U_{k}\ket{\psi_{l}}\ket{0\dots 0}=\ket{\psi_{2l}}.$ (22)

The state at each level has four times as many plaquettes as the previous
level and so, with corrections at the boundaries, has about four times as many
qubits in addition to the $L^{2}$ qubits associated to the fermionic modes.
Again, at the final level, the total number of qubits is $2L^{2}$ with
correction at the boundaries. Each layer consists of acting with Hadamards and
CNOTs locally with respect to the lattice at the given level. The exact form
of the circuit can be found in Aguado and Vidal (2008). Upon application of
the $\log{L}$ layers of the circuit, the toric state is prepared. At this
point, a constant depth unitary is performed to satisfy the stabilizers which
differ slightly from the true toric code model. The exact form of this circuit
depends on the basis chosen for the local Majoranas. If a Jordan-Wigner basis
is chosen, the circuit consists merely of a single layer of Pauli X and Z
gates.

Figure 6: Here we show the trade-off between qubit requirement and operator
locality. The circles represent the fermionic modes while the lines represent
the lattice of the topologically ordered state underpinning the local
encoding. Each level represents a case in which one additional level of the
state preparation circuit is applied, creating a finer lattice for the
underlying toric code. Operators are local only with respect to the lattice
spacing of the toric code state. At the top level with no topological order,
the fermionic operators may be fully non-local e.g. long JW strings. At the
bottom level, the operators are fully local with respect to the lattice
spacing of the fermions but twice as many qubits are required.

The blocking construction can be thought of as a truncation of the state
preparation circuit. Truncating the circuit results in a coarse-grained toric
code state relative to the lattice of fermionic modes. As such, the fermionic
operators which are local due to the topological order of the toric code
state, are now only local with respect to the toric code lattice. Operators
may be non-local up to the scale of the toric code lattice spacing. The
benefit, however, is a savings in qubit resources as each MERA layer requires
additional qubits. Thus by utilizing topological order on a coarse-grained
lattice, one can realize the trade-off between operator locality and qubit
requirement as depicted in Fig. 6.

Although each level of the MERA unitary is local with respect to the lattice
at each level, it is not local with respect to the final lattice of qubits.
With strictly local operations, preparing the topologically ordered toric code
state takes a time proportional to the linear size of the lattice Bravyi et
al. (2006).

####  III.3.2 Further generalizations

We now discuss a number of ways this above construction can be generalized.
Going beyond a square lattice of blocks, one could perform the partitioning of
the system of $N$ modes into a set of general sites
$S=\\{s_{1},\dots,s_{|S|}\\}$ each containing a number of modes $n(s_{i})$ and
where each site is connected to $d(s_{i})$ others on the lattice of sites.
Then given the construction above, the total number of qubits required would
be

$\\#\text{ of qubits
}=N+\sum_{i}^{|S|}\left\lceil\frac{d(s_{i})}{2}\right\rceil.$ (23)

Further, one is free to choose any encoding for the modes within each site.
For example, one could choose to encode the modes on a 1D chain as described
or choose to use a Fenwick tree Whitfield et al. (2016) or Jiang et al.’s
ternary tree encoding Jiang et al. (2020).

To emphasize the generality of the construction we are proposing here and to
illustrate how the geometry of the interactions should inform the geometry of
the qubit system, we would like to sketch how one might encode a system of
interest lately, that being a lattice of SYK islands Gu et al. (2017); Song et
al. (2017). The system is a lattice of islands $S=\\{s_{1},\dots,s_{|S|}\\}$
each containing $n(s_{i})$ Majorana fermions with two types of interactions,
quartic interactions between Majoranas within each island with random strength
and quadratic interactions between Majoranas on adjacent islands on the
lattice. We propose that such a system would be best simulated as a lattice of
sites, where the modes on each site, are placed at vertices on the “$N$
branches” geometry or a hierarchical geometry as described above. The chosen
geometry is then attached to the lattice at the central vertex. We highlight
such a system as it frustrates many of the existing encoding schemes,
featuring both highly connected regions as well as a notion of locality.

Finally, we reiterate that one is free to use any basis for the local Majorana
operators at each vertex. A Jordan-Wigner encoding was chosen for simplicity
but an improvement in Pauli weight can be achieved by using a different local
basis. For example, a Fenwick tree basis for local Majorana operators as
proposed in Setia et al. (2019) or Jiang et al.’s ternary tree basis Jiang et
al. (2020) would give a Pauli weight for Majorana operators logarithmic in the
number of qubits.

###  III.4 Device connectivity constraints

Figure 7: (top left) The heavy-hexagon geometry as presented in Chamberland et
al. (2020). 65 qubits are shown. (top right) We show a choice of grouping
pairs of qubits together to encode the degree $3$ vertices. (bottom) We show
the system graph where each larger circle corresponds to one of the 49 encoded
fermionic modes. Smaller circles represent the qubits associated to each mode.

Many of the quantum computing platforms under development are subject to
connectivity constraints. Notably, these include superconducting qubits which
have recently been used to achieve a “quantum supremacy” result Arute et al.
(2019). The processor used in the supremacy experiment features qubits laid
out on a square grid with nearest-neighbor connectivity. Recent work on
increasing the capabilities of quantum computers as measured in so-called
quantum volume Cross et al. (2019) has led to progress on devices with qubits
laid out on lower-degree graphs. In particular, the heavy-hexagon lattice has
been identified as a candidate system geometry for realizing quantum error
correction while mitigating hardware challenges presented by cross-talk and
frequency collisions Chamberland et al. (2020). This lattice features qubits
placed on vertices of a hexagonal lattice as well as on edges.

On such devices subject to connectivity constraints, it may be preferable to
encode a fermionic system into a graph informed by the connectivity of the
device. To that end, we present here a candidate geometry for encoding a
fermionic system into a heavy-hexagon lattice. In Fig. 7, we show a $65$ qubit
heavy-hexagon lattice. To each degree $3$ vertex, we associate an additional
qubit. Thus, with $16$ degree $3$ vertices, we are able to encode 49 fermionic
modes into the 65 qubit heavy-hexagon lattice. As above, coupling modes which
do not share an edge in the system geometry will require edge operators
containing generalized Jordan-Wigner strings.

Designing device-specific encodings for other platforms would proceed
similarly, identifying qubits to group together to encode vertices of
appropriate degree and embedding a problem withing the device-informed
geometry. We leave an investigation of optimal device-specific geometries to
future work.

##  IV Conclusion

In this paper, we have presented a very general construction for designing
quantum codes to simulate fermions on quantum computers. The construction
realizes the trade-off between qubit resources and operator locality in such a
way that one can tailor the encoding to best fit the resources at hand. We
have also shown that in some cases, locality is too strict of a constraint and
one is better off seeking a quasi-local encoding. We showed this occurs in
systems such as square lattice models with nearest- and nearest-diagonal-
neighbor coupling where it is best to simply encode the square lattice and
realize the diagonal couplings quasi-locally. We also presented the case of a
fermionic system with all-to-all connectivity and demonstrated that one should
encode this system with a virtual geometry so that generalized Jordan-Wigner
strings traverse paths through the virtual geometry. We discussed how quasi-
local codes can be interpreted as local codes with truncated state preparation
circuits. Finally, we considered the design of custom codes to suit device
connectivity constraints. We expect that the encoding construction presented
here will find use in quantum simulation experiments ranging from quantum
chemistry, to quantum gravity, to condensed matter physics.

## Acknowledgements

RWC and JDW were funded by the NSF (PHYS-1820747) and the Department of Energy
(Grant DE-SC0019374). JDW is also supported by NSF (EPSCoR-1921199) and by the
U.S. Department of Energy, Office of Science, Office of Advanced Scientific
Computing Research under programs Quantum Computing Application Teams and
Accelerated Research for Quantum Computing program.

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# Irreducibility of a universal Prym-Brill-Noether locus

Andrei Bud

###### Abstract

For genus $g=\frac{r(r+1)}{2}+1$, we prove that via the forgetful map, the
universal Prym-Brill-Noether locus $\mathcal{R}^{r}_{g}$ has a unique
irreducible component dominating the moduli space $\mathcal{R}_{g}$ of Prym
curves.

##  1 Introduction

The moduli space $\mathcal{R}_{g}$ of Prym curves was brought to the attention
of algebraic geometers by Mumford in his influential paper [Mum74], as a way
of understanding principally polarized Abelian varieties. For an element
$[C,\eta]$ of $\mathcal{R}_{g}$ we let $\pi\colon\widetilde{C}\rightarrow C$
be the associated double cover and let
$\mathrm{Nm}_{\pi}\colon\mathrm{Pic}^{2g-2}(\widetilde{C})\rightarrow\mathrm{Pic}^{2g-2}(C)$
be the norm map of this morphism of curves. In this situation, the preimage of
$\omega_{C}$ consists of two disjoint Abelian varieties, namely

$P^{+}=\left\\{L\in\mathrm{Pic}^{2g-2}(\widetilde{C})\ |\
\mathrm{Nm}(L)=\omega_{C}\ \mathrm{and}\ h^{0}(\widetilde{C},L)\equiv 0\
(\mathrm{mod}\ 2)\right\\}$

and

$P^{-}=\left\\{L\in\mathrm{Pic}^{2g-2}(\widetilde{C})\ |\
\mathrm{Nm}(L)=\omega_{C}\ \mathrm{and}\ h^{0}(\widetilde{C},L)\equiv 1\
(\mathrm{mod}\ 2)\right\\}.$

The intersection of $W^{0}_{2g-2}(\widetilde{C})$ with $P^{+}$ is twice a
theta divisor, and this allows us to associate to $[C,\eta]$ a principally
polarized Abelian variety.

Following this development, Welters pointed out in [Wel85] that Prym-Brill-
Noether theory can be employed in order to understand the geometry of
subvarieties of Prym varieties. More precisely, he considered the loci

$V^{r}(C,\eta)\coloneqq\left\\{L\in\mathrm{Pic}^{2g-2}(\widetilde{C})\ |\
\mathrm{Nm}(L)\cong\omega_{C},\ h^{0}(\widetilde{C},L)\geq r+1,\ \mathrm{and}\
h^{0}(\widetilde{C},L)\equiv r+1\ (\mathrm{mod}\ 2)\right\\}$

in order to study the singularities of the theta divisor of the associated
Prym variety. The relation between Prym-Brill-Noether theory and the study of
singularities of theta divisors attracted other mathematicians to this topic.
The two papers [Wel85] and [Ber87] showed that when $g\geq\frac{r(r+1)}{2}+1$,
the locus $V^{r}(C,\eta)$ is non-empty of dimension at least
$g-1-\frac{r(r+1)}{2}$. Moreover, for a generic $[C,\eta]\in\mathcal{R}_{g}$,
the locus $V^{r}(C,\eta)$ has exactly this dimension when
$g\geq\frac{r(r+1)}{2}+1$ and is empty when $g<\frac{r(r+1)}{2}+1$, see
[Sch17]. Subsequently in [DCP95], De Concini and Pragacz viewed
$V^{r}(C,\eta)$ as a Lagrangian degeneracy locus (cf. [Mum71]) and computed
the class of $V^{r}(C,\eta)$ in the Prym variety when it has the expected
dimension $g-1-\frac{r(r+1)}{2}$.

In recent years, two new perspective for the study of Prym-Brill-Noether
theory emerged. On one hand, it has been studied from the point of view of
tropical geometry, see [CLRW20] and [LU21], thus providing another proof for
the dimension estimate of $V^{r}(C,\eta)$ for a generic $[C,\eta]$ and, on the
other hand, from the perspective of moduli theory, in order to understand the
birational geometry of $\mathcal{R}_{g}$ for small values of $g$. It is
natural to ask when $g\geq\frac{r(r+1)}{2}+1$ whether the universal Prym-
Brill-Noether locus

$\mathcal{V}^{r}_{g}\coloneqq\left\\{[C,\eta,L]\ |\
[C,\eta]\in\mathcal{R}_{g}\ \mathrm{and}\ L\in V^{r}(C,\eta)\right\\}$

has a unique irreducible component dominating the moduli space
$\mathcal{R}_{g}$. This is true for $g>\frac{r(r+1)}{2}+1$ because the fibre
above a general $[C,\eta]\in\mathcal{R}_{g}$ is irreducible, see [Deb00,
Exemples 6.2]. However, as pointed out in [JP21], this was not known for
$g=\frac{r(r+1)}{2}+1$. The goal of this paper is to show that when
$g=\frac{r(r+1)}{2}+1$, the moduli space $\mathcal{V}^{r}_{g}$ has a unique
irreducible component dominating $\mathcal{R}_{g}$. To prove this result, we
will consider the compactification $\overline{\mathcal{R}}_{g}$ of the moduli
space of Prym curves $\mathcal{R}_{g}$, see [BCF04] and [FL10]. Finally, we
degenerate to the boundary locus of $\overline{\mathcal{R}}_{g}$ and employ
the theory of limit linear series, adapted to our situation.

Acknowledgements: I would like to thank my advisor Gavril Farkas for
suggesting this problem and for all his help along the way.

##  2 Prym linear series

Let $[C,\eta]\in\mathcal{R}_{g}$ be a generic Prym curve. Then, we know from
[Wel85, Lemma 3.2] that a generic element $L\in V^{r}(C,\eta)$ satisfies
$h^{0}(\widetilde{C},L)=r+1$. Moreover, when $g=\frac{r(r+1)}{2}+1$ we know
from [Sch17, Theorem 1.1] that all $L\in V^{r}(C,\eta)$ satisfy
$h^{0}(\widetilde{C},L)=r+1$. In particular, the line bundle $L$ can be viewed
as a $g^{r}_{2g-2}$ on the curve $\widetilde{C}$. Furthermore, up to
restricting to an open subset, we can view all irreducible components of
$\mathcal{V}^{r}_{g}$ dominating $\mathcal{R}_{g}$ as contained in the moduli
space $\mathcal{G}^{r}_{2g-2}(\mathcal{R}_{g})$ parametrizing limit
$g^{r}_{2g-2}$ over double covers $[\pi\colon\widetilde{C}\rightarrow C]$
where $\widetilde{C}$ is of compact type. We ask what points can appear in the
compactification of $\mathcal{V}^{r}_{g}$ inside this space.

Let $[\pi\colon\widetilde{C}\rightarrow C]\in\overline{\mathcal{R}}_{g}$ such
that $C$ is of compact type and admits a unique irreducible component $X$
satisfying $\eta_{X}\ncong\mathcal{O}_{X}$. For this component $X$, we denote
by $p^{X}_{1},\ldots,p^{X}_{s_{X}}$ its nodes and by
$g^{X}_{1},\ldots,g^{X}_{s_{X}}$ the genera of the connected components of
$C\setminus X$ glued to $X$ at these points. For an irreducible component $Y$
of $C$, different from $X$, we denote by $q^{Y}$ the node glueing $Y$ to the
connected component of $C\setminus Y$ containing $X$, and by
$p^{Y}_{1},\ldots,p_{s_{Y}}^{Y}$ the other nodes of $Y$. We denote by
$g^{Y}_{0},g^{Y}_{1},\ldots,g^{Y}_{s_{Y}}$ the genera of the connected
components of $C\setminus Y$ glued to $Y$ at these points. Using the above
notations, we can define the concept of a Prym limit $g^{r}_{2g-2}$:

######  Definition 2.1.

A Prym limit $g^{r}_{2g-2}$, denoted $L$, is a crude limit $g^{r}_{2g-2}$ on
$\widetilde{C}$ satisfying the following two conditions:

  1. 1.

For the unique component $\widetilde{X}$ of $\widetilde{C}$ above $X$, the
$\widetilde{X}$-aspect $L_{\widetilde{X}}$ of $L$ satisfies

$Nm_{\pi_{|\widetilde{X}}}L_{\widetilde{X}}\cong\omega_{X}(\sum_{i=1}^{s}2g^{X}_{i}p_{i})$

  2. 2.

For a component $Y$ of $C$ different from $X$, we denote by $Y_{1}$ and
$Y_{2}$ the two irreducible components of $\widetilde{C}$ above it. We
identify these two components with $Y$ via the map $\pi$. With this
identification the $Y_{1}$ and $Y_{2}$ aspects of $L$ satisfy:

$L_{Y_{1}}\otimes
L_{Y_{2}}\cong\omega_{Y}\text{\large(}(2g-2+2g^{Y}_{0})q^{Y}+\sum_{i=1}^{s}g_{i}^{Y}p_{i}^{Y}\text{\large)}$

Because the points in the boundary need to respect the norm condition, we
immediately obtain that:

######  Lemma 2.2.

Let $[\pi\colon\widetilde{C}\rightarrow C]\in\overline{\mathcal{R}}_{g}$ with
$\widetilde{C}$ of compact type and let $\overline{\mathcal{V}}^{r}_{g}$ the
closure of $\mathcal{V}^{r}_{g}$ inside
$\mathcal{G}^{r}_{2g-2}(\mathcal{R}_{g})$. Then the fibre of the map
$\overline{\mathcal{V}}^{r}_{g}\rightarrow\overline{\mathcal{R}}_{g}$ over the
point $[\pi\colon\widetilde{C}\rightarrow C]$ is contained in the locus of
Prym limit $g^{r}_{2g-2}$ on $[\pi\colon\widetilde{C}\rightarrow C]$.

We are now ready to use a degeneration argument in order to prove our main
result.

######  Theorem 2.3.

When $g=\frac{r(r+1)}{2}+1$, the space $\mathcal{V}^{r}_{g}$ has a unique
irreducible component dominating $\mathcal{R}_{g}$.

###### Proof.

Let $[Y_{1}\cup_{x_{1}}\widetilde{E}\cup_{x_{2}}Y_{2}\rightarrow Y\cup_{x}E]$
be the double cover associated to a generic element of $\Delta_{1}$. We want
to describe the locus of Prym limit $g^{r}_{2g-2}$’s on such a double cover.

Let $L$ be a Prym limit $g^{r}_{2g-2}$ on
$[Y_{1}\cup_{x_{1}}\widetilde{E}\cup_{x_{2}}Y_{2}\rightarrow Y\cup_{x}E]$. The
additivity of the Brill-Noether numbers implies:

$\rho(2g-1,r,2g-2)=-r\geq\rho(L_{Y_{1}},x_{1})+\rho(\widetilde{E},x_{1},x_{2})+\rho(L_{Y_{2}},x_{2})$

But we know from [EH87, Theorem 1.1] and [Far00, Proposition 1.4.1] that
$\rho(L_{Y_{1}},x_{1})\geq 0,\ \rho(L_{Y_{2}},x_{2})\geq 0$ and
$\rho(\widetilde{E},x_{1},x_{2})\geq-r$. It is clear that these are in fact
equalities and $L$ is a refined limit $g^{r}_{2g-2}$.

We denote by $0\leq a_{0}<a_{1}<\cdots<a_{r}\leq 2g-2$ and $0\leq
b_{0}<b_{1}<\cdots<b_{r}\leq 2g-2$ the vanishing orders for the $Y_{1}$ and
$Y_{2}$ aspects respectively. The equality
$\rho(\widetilde{E},x_{1},x_{2})=-r$ implies that $a_{i}+b_{r-i}=2g-2$ for all
$0\leq i\leq r$.

The genericity of $[Y_{2},x_{2}]\in\mathcal{M}_{g-1,1}$ together with
$\rho(L_{Y_{2}},x_{2})=0$ imply that
$h^{0}(Y_{2},L_{Y_{2}}(-b_{i}x_{2}))=r+1-i$ for all $0\leq i\leq r$. Using
that $L_{Y_{1}}\otimes L_{Y_{2}}\cong\omega_{Y}(2g\cdot x)$ and the Riemann-
Roch theorem we obtain

$h^{0}\text{\large(}Y_{1},L_{Y_{1}}(-(2+a_{r-i})q)\text{\large)}=g+r-1-a_{r-i}-i$

Choosing $i=0$ we get $a_{r}=g+r-1$. Inverting the roles of the $a_{i}$’s and
$b_{i}$’s we obtain that $a_{0}=g-r-1$. Because we have the divisorial
equivalences

$a_{i}x_{1}+b_{r-i}x_{2}\equiv a_{j}x_{1}+b_{r-j}x_{2}$

on the elliptic curve $E$ for every $0\leq i,j\leq r$, we obtain that
$a_{i}-a_{i-1}\geq 2$ for every $1\leq i\leq r$. This implies that
$a_{i}=g-r+2i-1$ for every $0\leq i\leq r$.

We now view the moduli space $\mathcal{M}_{g-1,1}$ as embedded in
$\overline{\mathcal{R}}_{g}$ via the map
$\pi\colon\mathcal{M}_{g-1,1}\rightarrow\overline{\mathcal{R}}_{g}$ sending a
pointed curve $[Y,x]\in\mathcal{M}_{g-1,1}$ to
$[Y\cup_{x}E,\mathcal{O}_{Y},\eta_{E}]$ where $[E,x]$ is a generic elliptic
curve and $\eta_{E}$ is a 2-torsion line bundle on $E$. For the ramification
sequence $\alpha=(g-r-1,g-r,\ldots,g-1)$ associated to the vanishing orders
$a=(a_{0},\ldots,a_{r})=(g-r-1,\ldots,g+r-1)$, we consider the locus
$\mathcal{G}^{r}_{2g-2}(\alpha)$ parametrizing pairs $[C,p,L]$ where
$[C,p]\in\mathcal{M}_{g-1,1}$ and $L$ is a $g^{r}_{2g-2}$ having vanishing
orders greater or equal to $a$ at the point $p$. Then the locus of Prym limit
$g^{r}_{2g-2}$ over $\mathrm{Im}(\pi)$ is birationally isomorphic to
$\mathcal{G}^{r}_{2g-2}(\alpha)$.

We know from [EH89, Lemma 3.6] that $\mathcal{G}^{r}_{2g-2}(\alpha)$ has a
unique irreducible component dominating $\mathcal{M}_{g-1,1}$. Moreover

$\mathrm{deg}\text{\large(}\mathcal{G}^{r}_{2g-2}(\alpha)\rightarrow\mathcal{M}_{g-1,1}\text{\large)}=2^{\frac{r(r-1)}{2}}\cdot(g-1)!\cdot\prod_{i=1}^{r}\frac{(i-1)!}{(2i-1)!}$

as stated on the second page of [FT16]. On the other hand we have from [DCP95,
Theorem 9] that

$\mathrm{deg}\text{\large(}\mathcal{V}^{r}_{g}\rightarrow\mathcal{R}_{g}\text{\large)}=2^{\frac{r(r-1)}{2}}\cdot(g-1)!\cdot\prod_{i=1}^{r}\frac{(i-1)!}{(2i-1)!}$

.

We conclude that all dominant irreducible components of $\mathcal{V}^{r}_{g}$
contain $\mathcal{G}^{r}_{2g-2}(\alpha)$ in their closure. From this we get
that each such component map to $\mathcal{R}_{g}$ with degree at least
$2^{\frac{r(r-1)}{2}}\cdot(g-1)!\cdot\prod_{i=1}^{r}\frac{(i-1)!}{(2i-1)!}$,
implying unicity. ∎

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Humboldt-Universität zu Berlin, Institut für Mathematik, Rudower Chausee 25
12489 Berlin, Germany

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