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aainstitutetext: Department of Physics, Arizona State University, Tempe, AZ
85281, USAbbinstitutetext: Department of Electrical Engineering and Computer
Science, Syracuse University, NY 13210, USA

# Bounding Entanglement Entropy with Contracted Graphs

Cynthia Keeler a William Munizzi b and Jason Pollack<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Following on our previous work Keeler2022 ; Keeler:2023xcx  studying the
orbits of quantum states under Clifford circuits via ‘reachability graphs’, we
introduce ‘contracted graphs’ whose vertices represent classes of quantum
states with the same entropy vector. These contracted graphs represent the
double cosets of the Clifford group, where the left cosets are built from the
stabilizer subgroup of the starting state and the right cosets are built from
the entropy-preserving operators. We study contracted graphs for stabilizer
states, as well as W states and Dicke states, discussing how the diameter of a
state’s contracted graph constrains the ‘entropic diversity’ of its $2$-qubit
Clifford orbit. We derive an upper bound on the number of entropy vectors that
can be generated using any $n$-qubit Clifford circuit, for any quantum state.
We speculate on the holographic implications for the relative proximity of
gravitational duals of states within the same Clifford orbit. Although we
concentrate on how entropy evolves under the Clifford group, our double-coset
formalism, and thus the contracted graph picture, is extendable to generic
gate sets and generic state properties.

##  1 Introduction

One primary goal of quantum computation is to outperform classical computers:
that is, for certain tasks, to take a classical input and compute a classical
output more rapidly, or efficiently, than any known classical algorithm. (In
recent years, this goal has been achieved or brought within reach for certain
sets of problems arute2019quantum ; zhong2020quantum .) Intuitively, quantum
computers can only do better on these tasks because they’re doing something
intrinsically _quantum_ : if they weren’t, they couldn’t outperform the
classical method. Formalizing this intuitive result is an object of ongoing
research: precisely what feature of a particular quantum algorithm allows it
to gain an advantage?

→

G/Stab_G(—ψ⟩)

H \G/Stab_G(—ψ⟩)

Figure 1: A reachability graph and its reduction to a contracted graph. In
this example, discussed in more detail in Figure 11, $G$ is the subgroup of
the two-qubit Clifford group generated by Hadamard and $CNOT$ gates and $H$ is
the set of operations which leave entropy vectors unchanged.

Setting aside not-even-wrong explanations like “quantum computers act on each
term in a superposition simultaneously,” the folk wisdom is that the source of
quantum advantage has something to do with interference, superposition, and
entanglement. This appealing picture is challenged by the famous result that
Clifford circuits, which are generated by the one-qubit Hadamard and phase
gates and the two-qubit $CNOT$ gate, can be efficiently classically simulated
Gottesman1998 ; aaronson2004improved . That is, even though Clifford circuits
can, via $CNOT$ gate applications, produce entanglement, they _can’t_ give
quantum speedups. Evidently, if some kind of entanglement is the key to
quantum advantage, the type produced by Clifford gates doesn’t suffice.

In order to understand the evolution of entanglement as a state is evolved
through a quantum circuit, it’s useful to track the _entropy vector_ , which
characterizes the entanglement entropy of every subsystem of the state. In a
recent series of papers, we have investigated how the entropy vector changes
under the restricted action of Clifford gates acting on the first two qubits
of a state. We first obtained Keeler2022  the _reachability graphs_ , colored
by entropy vector, which show how stabilizer states evolve under the action of
the two-qubit Clifford group $\mathcal{C}_{2}$ and its subgroups. In our
second paper Keeler:2023xcx , having better understood the underlying group-
theoretic structures from which the reachability graphs are attained, we were
able to find a representation of $\mathcal{C}_{2}$ as generated by the
Clifford gates, as well as explore the reachability graphs produced from
initial non-stabilizer states.

Although reachability graphs are useful for directly showing the action of
explicit circuits and explicit states, they fail to fully illuminate the paths
by which the entropy vector can change. The problem, in short, is that some
circuits, even when they contain $CNOT$ gates, fail to change the entropy. For
example, one defining relation of $\mathcal{C}_{2}$ is Keeler:2023xcx

$\left(CNOT_{1,2}P_{2}\right)^{4}=P_{1}^{2}.$ (1)

Hence the structure of reachability graphs by themselves can only loosely
bound how the entropy vector might change.

In this paper, we accordingly pass to a more concise graphical representation,
the _contracted graphs_ , whose vertices represent not single states but
classes of states with the same entropy vector. We show how to construct these
graphs from the _double cosets_ of the Clifford group $\mathcal{C}_{2}$ and
its cosets. An example of this procedure is shown in Figure 1. Our protocol
for constructing contracted graphs is easily generalized to groups beyond the
Clifford group and state properties beyond the entropy vector, and might be of
use for other applications.

The remainder of this paper is organized as follows. In Section 2, we review
the Clifford group and stabilizer formalism, as well as the group-theoretic
concepts of cosets and double cosets. We also recall the objects used in our
previous papers: Cayley graphs, reachability graphs, and entropy vectors. In
Section 3, we give a general procedure for constructing the contracted graphs
which retain information about entropy-changing operations in a group. In
Section 4, we apply this procedure to $\mathcal{C}_{2}$ and its subgroup
$(HC)_{1,2}$. For each of the reachability graphs in our previous papers, we
obtain the resulting contracted graph, and show how these combine together
under the action of the full Clifford group. In Section 5 we consider the
diameter and entropic diversity of the reachability graphs, and discuss
implications for the available transformations on a dual geometry via
holography. In Section 6 we conclude and discuss future work. An appendix
collects additional details of our computations.

##  2 Review

###  2.1 Clifford Group and Stabilizer Formalism

The Pauli matrices are a set of unitary and Hermitian operators, defined in
the computational basis $\\{{|{0}\rangle},{|{1}\rangle}\\}$ as

$\mathbb{1}\equiv\begin{bmatrix}1&0\\\
0&1\end{bmatrix},\,\,\sigma_{X}\equiv\begin{bmatrix}0&1\\\
1&0\end{bmatrix},\,\,\sigma_{Y}\equiv\begin{bmatrix}0&-i\\\
i&0\end{bmatrix},\,\,\sigma_{Z}\equiv\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix}.$
(2)

The multiplicative matrix group generated by $\sigma_{X},\,\sigma_{Y},$ and
$\sigma_{Z}$ is known as the single-qubit Pauli group $\Pi_{1}$, which we
write

$\Pi_{1}\equiv\langle\sigma_{X},\,\sigma_{Y},\,\sigma_{Z}\rangle.$ (3)

When $\Pi_{1}$ acts on a Hilbert space $\mathcal{H}\equiv\mathbb{C}^{2}$, in
the fixed basis spanned by $\\{{|{0}\rangle},\,{|{1}\rangle}\\}$, it generates
the algebra of all linear operations on $\mathcal{H}$.

The Clifford group is likewise a multiplicative matrix group, generated by the
Hadamard, phase, and CNOT operations:

$H\equiv\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\\ 1&-1\end{bmatrix},\quad
P\equiv\begin{bmatrix}1&0\\\ 0&i\end{bmatrix},\quad
C_{i,j}\equiv\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{bmatrix}.$ (4)

The CNOT gate is a bi-local operation which, depending on the state of one
qubit, the control bit, may act with a $\sigma_{X}$ operation on a second
qubit, the target bit. For the gate $C_{i,j}$, the first subscript index
denotes the control bit and the second subscript the target bit. We define the
single qubit Clifford group $\mathcal{C}_{1}$ as the group $\langle
H,\,P\rangle$. Elements of $\mathcal{C}_{1}$ act as automorphisms on $\Pi_{1}$
under conjugation; hence $\mathcal{C}_{1}$ is the normalizer of $\Pi_{1}$ in
$L(\mathcal{H})$.

When considering the action of the Pauli and Clifford groups on multi-qubit
systems, we compose strings of operators which act collectively on an
$n$-qubit state. For an element of $\Pi_{1}$ which acts locally on the
$k^{th}$ qubit in an $n$-qubit system, for example, we write

$I^{1}\otimes\ldots\otimes I^{k-1}\otimes\sigma_{X}^{k}\otimes
I^{k+1}\otimes\ldots\otimes I^{n}.$ (5)

Eq. (5) is referred to as a Pauli string, where the weight of each string
counts the number non-identity insertions. The multiplicative group generated
by all Pauli strings of weight $1$ is the $n$-qubit Pauli group $\Pi_{n}$.

We similarly can extend the action of $\mathcal{C}_{1}$ to multiple qubits,
now incorporating $C_{i,j}$ into the generating set. Composing Clifford
strings analogously to Eq. (5), we define the $n$-qubit Clifford group
$\mathcal{C}_{n}$ as111This is not the minimal generating set for the Clifford
group, since some Clifford gates can be written in terms of the others, c.f.
Eqs. (4.5, 4.6) of Keeler:2023xcx . A more minimal definition of the Clifford
group is $C_{n}\equiv\langle\\{H_{i},P_{1},C_{i,j}\\}\rangle$ where
$i\in\\{1\ldots n\\}$, $j>i$.

$\mathcal{C}_{n}\equiv\langle
H_{1},...,\,H_{n},\,P_{1},...,P_{n},\,C_{1,2},\,C_{2,1},...,\,C_{n-1,n},\,C_{n,n-1}\rangle.$
(6)

When indicating the action of some local gate, Hadamard or phase, the gate
subscript denotes which qubit the gate acts on, e.g. $H_{1}$ for the action of
Hadamard on the first qubit of an $n$-qubit system.

Beginning with any $n$-qubit computational basis state, e.g.
${|{0}\rangle}^{\otimes n}$, the group $\mathcal{C}_{n}$ is sufficient to
generate the full set of $n$-qubit stabilizer states. As we noted in the
introduction, stabilizer states are notable in quantum computing as a set of
quantum systems which can be efficiently simulated with classical computing
aaronson2004improved ; Gottesman1997 . Additionally, stabilizer states
comprise the elements of $\mathcal{H}$ which are left invariant under a
$2^{n}$ element subgroup of $\Pi_{n}$. Since the group $\mathcal{C}_{n}$ is
finite, the set of $n$-qubit stabilizer states $S_{n}$ is also finite
doi:10.1063/1.4818950  and has order given by

$\left|S_{n}\right|=2^{n}\prod_{k=0}^{n-1}(2^{n-k}+1).$ (7)

###  2.2 Cosets and Double Cosets

Throughout this paper we support our graph models with parallel group-
theoretic arguments. Many of our explanations make substantial use of coset
and double coset constructions, which we review here. We also take this
opportunity to set notation and establish language that will be used
throughout the remainder of the paper.

Let $G$ be a group and $K\leq G$ an arbitrary subgroup. The set of all left
cosets of $K$ in $G$ are constructed as

$g\cdot K,\quad\forall g\in G.$ (8)

Each left coset built in Eq. (8) is an equivalence set of elements $[g_{i}]$,
which are equivalent under $K$ group action on the right:

$g_{i}\sim g_{j}\Longleftrightarrow\exists\,k\in K\mathrel{\mathop{\mathchar
58\relax}}g_{i}=g_{j}k.$ (9)

Any two cosets $[g_{i}]$ in $g\cdot K$ must be either equal or disjoint, and
every $g\in G$ must be found in, at most, one equivalence class. As a result,
the set of all $[g_{i}]$ gives a complete decomposition of $G$.

Eqs. (8) and (9), as well as the accompanying explanations, apply analogously
when generating all right cosets $H\cdot g$, for arbitrary $H\leq G$. We build
all right cosets by computing $H\cdot g$, for every $g\in G$, where each
equivalence class $[g_{i}]$ is now determined by left subgroup action

$g_{i}\sim g_{j}\Longleftrightarrow\exists\,h\in H\mathrel{\mathop{\mathchar
58\relax}}g_{i}=hg_{j}.$ (10)

When $H\leq G$ is normal in $G$, the left and right cosets are equal, and both
$H\cdot g$ and $g\cdot H$ form a group under the same binary operation which
defines $G$.

Two subgroups $H,K\leq G$ can be used to construct double cosets of $G$. We
build each $(H,K)$ double coset by acting on $g\in G$ on the right by subgroup
$K$, and on the left by $H$, explicitly

$H\cdot g\cdot K,\quad\forall g\in G,\,h\in H,\,k\in K.$ (11)

The double coset space built using Eq. (11) is denoted $H\backslash G/K$, and
is defined by the equivalence relation

$g_{i}\sim g_{j}\Longleftrightarrow\exists\,h\in H,\,k\in
K\mathrel{\mathop{\mathchar 58\relax}}g_{i}=hg_{j}k.$ (12)

In order to utilize the above coset constructions in this paper, we invoke
several foundational group theory concepts (see e.g. Alperin1995 ). First, for
a finite group $G$, the order of any subgroup $K\leq G$ partitions the order
of $G$ by Lagrange’s theorem

$\frac{|G|}{|K|}=[G\mathrel{\mathop{\mathchar 58\relax}}K],\quad\forall K\leq
G,$ (13)

where $[G\mathrel{\mathop{\mathchar 58\relax}}K]\in\mathbb{N}$ is the number
of left (or right) cosets of $K$ in $G$. When acting with $G$ on a set $X$,
the orbit-stabilizer theorem fixes the size of each orbit $[G\cdot x]$ to be

$[G\cdot x]=[G\mathrel{\mathop{\mathchar
58\relax}}K]=\frac{|G|}{|K|},\quad\forall x\leq X,$ (14)

where $K\leq G$ is the set of elements which map an $x\in X$ to itself.

We can likewise use Eq. (13) with Eq. (14) to compute the order of a double
coset space, i.e. the orbit of all left (or right) cosets under left (or
right) subgroup action. For finite $G$ and subgroups $H,K\leq G$, the
order222Note that a direct application of Lagrange’s theorem to the order of a
double coset space is false, i.e. the order of a double coset space of $G$
does not necessarily divide $|G|$. of $H\backslash G/K$ is computed as

$|H\backslash G/K|=\frac{1}{|H||K|}\sum_{(h,k)\in H\times K}|G^{(h,k)}|,$ (15)

where $G^{(h,k)}$ is the set of equivalence classes $[g_{i}]$ under Eq. (11).
The sum in Eq. (15) is taken over all ordered pairs $(h,k)$ of $h\in H$ and
$k\in K$.

###  2.3 Cayley Graphs and Reachability Graphs

A Cayley graph encodes in graphical form the structure of a group. For a group
$G$ and a chosen set of generators, we construct the Cayley graph of $G$ by
assigning a vertex for every $g\in G$, and an edge333Formally, each edge in a
Cayley graph is directed. However, for improved legibility, we will often
represent group generators which are their own inverse using undirected edges.
for every generator of $G$. When $G$ corresponds to a set of quantum operators
acting on a Hilbert space, paths in the Cayley graph represent quantum
circuits that can be composed using the generating gate set. Different paths
which start and end on the same pair of vertices indicate sequences of
operators whose action on any quantum state is identical. Loops in a Cayley
graph represent operations equivalent to the identity.

For a group $G\subset L(\mathcal{H})$, we define the stabilizer subgroup
$\textnormal{Stab}_{G}({|{\psi}\rangle})$ of some
${|{\psi}\rangle}\in\mathcal{H}$ as the subset of elements $g\in G$ which
leave ${|{\psi}\rangle}$ unchanged,

$\textnormal{Stab}_{G}({|{\psi}\rangle})\equiv\\{g\in
G\,|\,g{|{\psi}\rangle}=\psi\\}.$ (16)

In other words, the subgroup $\textnormal{Stab}_{G}({|{\psi}\rangle})$
consists of all $g\in G$ for which ${|{\psi}\rangle}$ is a $+1$ eigenvector.

Reachability graphs can be obtained more generally as quotients of Cayley
graphs Keeler:2023xcx ; githubStab ; githubCayley . To perform this procedure,
we first identify a group $G\in L(\mathcal{H})$ to act on a Hilbert space
$\mathcal{H}$, and a generating set for $G$. We then first quotient $G$ by any
subgroup of elements which act as an overall phase on the group. For
$\mathcal{C}_{n}$, this is the subgroup $\langle\omega\rangle$, where

$\omega\equiv\left(H_{i}P_{i}\right)^{3}=e^{i\pi/4}\mathbb{1}.$ (17)

Once we have removed overall phase and constructed the quotient group444Since
$\langle\omega\rangle<\mathcal{C}_{n}$ is normal, modding by
$\langle\omega\rangle$ builds a proper quotient. It therefore does not matter
whether we apply $\langle\omega\rangle$ on the left or right of $G$ when
building cosets, nor does it affect any subsequent double coset construction.
Accordingly, when it will not cause confusion we continue to use $G$ to refer
to the group modded by global phase, rather than using an alternative
notation. $\bar{G}=G/\langle\omega\rangle$, we identify a state
${|{\psi}\rangle}\in\mathcal{H}$. Selecting ${|{\psi}\rangle}$ immediately
defines the stabilizer subgroup
$\textnormal{Stab}_{\bar{G}}({|{\psi}\rangle})$. We then construct the left
coset space $\bar{G}/\textnormal{Stab}_{\bar{G}}({|{\psi}\rangle})$ whose
elements are

$g\cdot\textnormal{Stab}_{\bar{G}}({|{\psi}\rangle})\quad\forall g\in\bar{G}.$
(18)

To graphically represent this procedure, we begin with a graph
$\Gamma\equiv(V,E)$, which we quotient by first defining a partition on the
vertices $V$. This partition induces the equivalence relation $u\sim v$ iff
$u$ and $v$ lie in the same subset of the partition, defined for any $u,v\in
V$. In this way, each vertex in the quotient graph represents one subset of
the partition, and two vertices in the quotient graph are considered adjacent
if any two elements of their respective subsets are adjacent in $\Gamma$.

While the graphs in this paper often represent groups, constructing a graph
quotient is not equivalent to quotienting a group. Building a group quotient
requires modding by a normal subgroup, which ensures that the left and right
coset spaces of the chosen subgroup are equal, preserving the original group
action in quotient group. We do not impose such a requirement when building
graph quotients in this paper, even when our graphs illustrate the relation
between groups of operators. We distinguish graph quotients from group
quotients wherever potential confusion could occur.

###  2.4 Entropy Vectors and Entropy Cones

For a state ${|{\psi}\rangle}\in\mathcal{H}$, and some specified factorization
for $\mathcal{H}$, we can compute the von Neumann entropy of the associated
density matrix:

$S_{\psi}\equiv-\Tr\rho_{\psi}\ln\rho_{\psi},$ (19)

where $\rho_{\psi}\equiv{|{\psi}\rangle}\bra{\psi}$. For ${|{\psi}\rangle}$ a
pure state, the property $\rho_{\psi}^{2}=\rho_{\psi}$ implies $S_{\psi}=0$.
Throughout this paper, we measure information in bits, and entropies in Eq.
(19) are computed with $\log_{2}$.

For a multi-partite pure state ${|{\psi}\rangle}$, we can still observe non-
zero entanglement entropy among complementary subsystems of
${|{\psi}\rangle}$. Let ${|{\psi}\rangle}$ be some $n$-party pure state, and
let $I$ denote an $\ell$-party subsystem of ${|{\psi}\rangle}$. We can compute
the entanglement entropy between $I$ and its $(n-\ell)$-party complement,
$\bar{I}$, using

$S_{I}=-\Tr\rho_{I}\ln\rho_{I}.$ (20)

The object $\rho_{I}$ in Eq. (20) indicates the reduced density matrix of
subsystem $I$, which is computed by tracing out the complement subsystem
$\bar{I}$.

In general, there are $2^{n}-1$ possible subsystem entropies we can compute
for any $n$-qubit pure state ${|{\psi}\rangle}$. Computing each $S_{I}$, using
Eq. (20), and arranging all entropies into an ordered tuple defines the
entropy vector $\vec{S}\left({|{\psi}\rangle}\right)$. As an example, consider
the $4$-qubit pure state ${|{\psi}\rangle}$, where
$\vec{S}\left({|{\psi}\rangle}\right)$ is defined

$\vec{S}=(S_{A},S_{B},S_{C},S_{O};S_{AB},S_{AC},S_{AO},S_{BC},S_{BO},S_{CO};S_{ABC},S_{ABO},S_{ACO},S_{BCO};S_{ABCO}).$
(21)

In Eq. (21) we use a semicolon to separate entropy components for subregions
of distinct cardinality $|I|$. Additionally, for an $n$-qubit state it is
customary to denote the $n^{th}$ subsystem using $O$, as this region acts as a
purifier for the other $n-1$ parties.

For an $n$-party system, each entropy vector contains $2^{n}-1$ components,
with the first $n$ components representing single-qubit subsystems. We list
entropy vector components in lexicographic order: with the first region
denoted $A$, the second region denoted $B$, and so forth. Unlike what is
sometimes found in the literature, we use $O$ to represent a smaller
bipartition, instead of the one which does not contain the purifier. For
example, in Eq. (21) we declare $O$ a single-party subsystem which purifies
$ABC$, and write $S_{O}$ in place of $S_{ABC}$ among the single-party entries
of the entropy vector.

When ${|{\psi}\rangle}$ is a pure state, the condition $S_{\psi}=0$ implies an
additional equivalence between entropies of complement subsystems

$S_{I}=S_{\bar{I}}.$ (22)

Using Eq. (22) we can write $\vec{S}\left({|{\psi}\rangle}\right)$, for a pure
state ${|{\psi}\rangle}$, using only $2^{N-1}-1$ entropies. For example, the
entropy vector in Eq. (21) simplifies to the form

$\vec{S}=(S_{A},S_{B},S_{C},S_{O};S_{AB},S_{AC},S_{AO}).$ (23)

Since we are always considering pure states in this paper, all entropy vectors
are written using the reduced notation in Eq. (23).

##  3 Building Contracted Graphs

We now define a procedure to quotient reachability graphs by operations which
preserve some specified property of a quantum system. In this paper we focus
on the evolution of entanglement entropy under the action of the Clifford
group; however, this prescription is sufficiently general to study any state
property555In this work, the term state property refers to anything computable
from knowledge of the state, along with some additional information such as a
specified factorization of the Hilbert space. We do not restrict analysis to
properties which are observables; in fact, the main property discussed in this
paper, the entropy vector, is not itself an observable. under the action of
any finitely-generated group.

We build a contracted graph by identifying vertices in a reachability graph
which are connected by entropy-preserving circuits. In this way, a contracted
graph details the evolution of a state’s entropy vector under the chosen gate
set. The number of vertices in a contracted graph gives a strict upper bound
on the number of different entanglement vector values reachable via circuits
constructed using the chosen gate set. We will later use contracted graphs to
derive an upper bound on entropy vector variation in Clifford circuits.

We now give an algorithm for generating contracted graphs.

  1. 1.

We first select a group $G$, and a generating set for $G$, as well as a
property of our quantum system we wish to study under the action of $G$.

  2. 2.

We next build the Cayley graph for $G$ by assigning a vertex for every $g\in
G$, and a directed edge for each generator action on an element $g\in G$. We
quotient $G$, and its Cayley graph, by any subgroup which acts as a global
phase on the group, such as in Eq. (17).

  3. 3.

Next, we construct the reachability graph for some ${|{\psi}\rangle}$ under
the action of $G$, as detailed in Subsection 2.3, which we denote666A more
precise notation for such reachability graphs would be
$\mathcal{R}\left(\textnormal{Stab}_{G}\left({|{\psi}\rangle}\right)\right)$,
however we choose $\mathcal{R}_{G}\left({|{\psi}\rangle}\right)$ instead for
brevity. $\mathcal{R}_{G}\left({|{\psi}\rangle}\right)$. We determine the
stabilizer subgroup $\textnormal{Stab}_{G}\left({|{\psi}\rangle}\right)$ for
${|{\psi}\rangle}$, and generate the left coset space
$G/\textnormal{Stab}_{G}\left({|{\psi}\rangle}\right)$ using the equivalence
relation

$g_{i}\sim
g_{j}\Longleftrightarrow\exists\,s\in\textnormal{Stab}_{G}\left({|{\psi}\rangle}\right)\mathrel{\mathop{\mathchar
58\relax}}g_{i}=g_{j}s.$ (24)

We glue together vertices in the Cayley graph of $G$ that correspond to
elements which share an equivalence class $[g_{i}]$ in
$G/\textnormal{Stab}_{G}\left({|{\psi}\rangle}\right)$. This graph quotient
yields $\mathcal{R}_{G}\left({|{\psi}\rangle}\right)$.

  4. 4.

We now identify the subgroup $H\leq G$ of elements that leave the entropy
vector of any state invariant. The subgroup $H$ defines the equivalence
relation

$g_{i}\sim g_{j}\Longleftrightarrow\exists\,h\in H\mathrel{\mathop{\mathchar
58\relax}}g_{i}=hg_{j}.$ (25)

For any $G$, the group $H$ will at least contain all $g\in G$ which act as
local gates on a single qubit, since local action cannot modify entanglement.
However, $H$ may also contain additional circuits which do not change the
entropy vector.

  5. 5.

Finally, we build all double cosets $H\backslash
G/\textnormal{Stab}_{G}\left({|{\psi}\rangle}\right)$. We identify all
vertices in $\mathcal{R}_{G}\left({|{\psi}\rangle}\right)$ which share an
equivalence class in $H\backslash
G/\textnormal{Stab}_{G}\left({|{\psi}\rangle}\right)$, and subsequently
quotient $\mathcal{R}_{G}\left({|{\psi}\rangle}\right)$ to give the final
contracted graph.

We generate reachability graphs by building left cosets
$G/\textnormal{Stab}_{G}\left({|{\psi}\rangle}\right)$, defined by an
equivalence up to right subgroup action by
$\textnormal{Stab}_{G}\left({|{\psi}\rangle}\right)$ as in Eq. (24). Since
$\textnormal{Stab}_{G}\left({|{\psi}\rangle}\right)$ acts trivially on
${|{\psi}\rangle}$, appending any
$s\in\textnormal{Stab}_{G}\left({|{\psi}\rangle}\right)$ to the right of any
$g\in G$ does not change how $g$ transforms the state ${|{\psi}\rangle}$.
Conversely, we build a contracted graph by generating right cosets
$G\backslash H$, with equivalence defined up to left subgroup action as shown
in Eq. (25). Every element of $H$ preserves a state’s entropy vector,
therefore acting on the left of $g{|{\psi}\rangle}$ by any $h\in H$ does not
change the measurement of the full state entropy vector, for every $g\in G$.

Recall that there are two interpretations of a reachability graph. By
identifying a state ${|{\psi}\rangle}$ and group $G$ of operators acting on
that state, $\mathcal{R}_{G}\left({|{\psi}\rangle}\right)$ represents the
orbit of ${|{\psi}\rangle}$ under the action of $G$. In this state-orbit
interpretation, vertices of $\mathcal{R}_{G}\left({|{\psi}\rangle}\right)$
represent states reached in the orbit of ${|{\psi}\rangle}$. For simplicity,
we choose this state-orbit interpretation in this explanatory section. A more
general interpretation of reachability graphs exists which defines
$\mathcal{R}_{G}\left({|{\psi}\rangle}\right)$ as a quotient space of the
Cayley graph of the abstract group $G$. In this interpretation, vertices
represent equivalence classes of $g\in G$ defined by the left coset
$g\cdot\textnormal{Stab}_{G}\left({|{\psi}\rangle}\right)$.

##### Example:

For clarity, we now work through an explicit example. Consider the subgroup of
the two-qubit Clifford group777For additional detail on this Clifford subgroup
see Section 4.2 of Keeler:2023xcx , where all group elements are derived using
two-qubit Clifford group relations. generated by the $P_{2}$ and $CNOT_{1,2}$
gates,

$G\equiv\langle P_{2},\,CNOT_{1,2}\rangle.$ (26)

The group $\langle P_{2},\,CNOT_{1,2}\rangle$ consists of $32$ elements,
specifically

$\langle P_{2},\,CNOT_{1,2}\rangle=\\{p,\,pCp,\,C\bar{p}Cp\\},$ (27)

where we introduce the notations

$p\in\\{\mathbb{1},\,P_{2},\,P_{2}^{2},\,P_{2}^{3}\\},\qquad\bar{p}\in\\{\,P_{2},\,P_{2}^{2},\,P_{2}^{3}\\}.$
(28)

We select the state
${|{\psi}\rangle}=\left({|{00}\rangle}+2{|{01}\rangle}+4{|{10}\rangle}+3{|{11}\rangle}\right)/\sqrt{30}$,
which we choose for its particular entropic properties that we will discuss at
the end of the section. We construct the reachability graph
$\mathcal{R}_{G}\left({|{\psi}\rangle}\right)$ for ${|{\psi}\rangle}$, shown
in the left panel of Figure 2. The only element of $G$ which leaves
${|{\psi}\rangle}$ invariant is $\mathbb{1}$ in $G$, therefore

$\textnormal{Stab}_{G}({|{\psi}\rangle})=\\{\mathbb{1}\\}.$ (29)

Since the stabilizer group in Eq. (29) consists of just the identity, and is
therefore a normal subgroup, the group
$\textnormal{Stab}_{G}({|{\psi}\rangle})$ quotients $G$ and the reachability
graph $\mathcal{R}_{G}\left({|{\psi}\rangle}\right)$ is exactly the
$32$-vertex Cayley graph. In the more general case,
$\mathcal{R}_{G}\left({|{\psi}\rangle}\right)$ would not necessarily represent
a group quotient, but would represent a left coset space.

Figure 2: Reachability graph (left) of
${|{\psi}\rangle}=\left({|{00}\rangle}+2{|{01}\rangle}+4{|{10}\rangle}+3{|{11}\rangle}\right)/\sqrt{30}$,
highlighted in cyan, under action of $\langle P_{2},\ CNOT_{1,2}\rangle$, and
its associated contracted graph (right). The contracted graph has $4$ vertices
and $4$ edges connecting any two vertices, indicating the entropy vector can
maximally change $4$ times under any circuit built of $P_{2}$ and
$CNOT_{1,2}$. The $4$ entropy vector possibilities, defined by Eq. (21), are
given in the legend.

We construct the contracted graph of
$\mathcal{R}_{G}\left({|{\psi}\rangle}\right)$ by identifying the elements of
$G$ which cannot modify the entropy vector of ${|{\psi}\rangle}$. Since the
gate $P_{2}$ acts locally on a single qubit, it can never modify entanglement.
Accordingly, we initially contract
$\mathcal{R}_{G}\left({|{\psi}\rangle}\right)$ by gluing together all vertices
connected by a $P_{2}$ edge, represented by the orange dashed lines.
Additionally, as we recognized in Keeler:2023xcx ,

$\left(C_{1,2}P_{2}\right)^{4}=P_{1}^{2}.$ (30)

Hence all vertices connected by the circuit $\left(C_{1,2}P_{2}\right)^{4}$
must be identified together as well, since $P_{1}$ likewise does not change a
state’s entropy vector. The right panel of Figure 2 shows the final contracted
graph of $\mathcal{R}_{G}\left({|{\psi}\rangle}\right)$, which contains $4$
vertices. In this particular example, the contracted graph represents the
right coset space of the quotient group
$G/\textnormal{Stab}_{G}({|{\psi}\rangle})$. In general, however, the
contracted graph will represent the double coset space $H\backslash
G/\textnormal{Stab}_{G}({|{\psi}\rangle})$, where
$G/\textnormal{Stab}_{G}({|{\psi}\rangle})$ need not be a quotient group.

It is important to note that edges in a contracted graph do not represent any
one particular $C_{i,j}$ operation. Instead, every edge bearing a CNOT
coloration represents sequences of operations which, at least, include a
$C_{i,j}$ gate and are capable of modifying the entropy vector of a
sufficiently-general state. In this way, the edges of a contracted graph bound
the number of times the entropy vector of a system can change. Since the
process of building a contracted graph removes all group elements which leave
entanglement entropy unchanged, we are left with a graph structure that
represents the orbit of an entropy vector under the group action.

The number of vertices in a contracted graph give an upper bound on the number
of distinct entropy vectors which can be generated in particular reachability
graph. For example, the contracted graph in Figure 2 contains $4$ vertices,
indicating the maximum number of entropy vectors that can be achieved by
acting on ${|{\psi}\rangle}$ with $\langle P_{2},\,CNOT_{1,2}\rangle$. The
number of vertices in a contracted graph is fixed by the overall group
structure of $G$, as well as the group structure of $\textnormal{Stab}_{G}$;
however, the different ways in which those vertices can be colored according
to entanglement structure is set by the choice of state. The fact that this
contracted graph contains a unique entropy vector for each of its $4$
vertices, i.e. the contracted graph is maximally-colored, is the reason we
chose ${|{\psi}\rangle}$ as we did. While the number of vertices in a
contracted graph gives an upper bound on entropic diversity in reachability
graphs, there can be multiple entropic colorings of the same graph, depending
on factors such as qubit number or the specific state.

We have defined a procedure for building contracted graphs from the
reachability graph of arbitrary state ${|{\psi}\rangle}$. When considering a
group $G$ which acts on a Hilbert space, we build the reachability graph of
${|{\psi}\rangle}$ by decomposing $G$ into left cosets
$G/\textnormal{Stab}_{G}({|{\psi}\rangle})$, with elements equivalent up to
action by $\textnormal{Stab}_{G}({|{\psi}\rangle})$. We build the contracted
graph of the ${|{\psi}\rangle}$ reachability graph by building the double
coset space $H\backslash G/\textnormal{Stab}_{G}({|{\psi}\rangle})$, for a
subgroup $H\leq G$ of elements which preserve a state’s entropy vector.

We have demonstrated how contracted graphs illustrate the evolution of
entanglement entropy under the action of some quantum gate set. The number of
vertices in a contracted graph gives an upper bound on the maximal number of
times an entropy vector can change under the chosen set of gates. We have
chosen in this paper to construct contracted graphs from reachability graphs
in order to analyze the evolution of state entropy vectors; however, the
contraction procedure can be applied directly to Cayley graphs as well.

In the next section we use the techniques defined above to build contracted
graphs for all stabilizer state reachability graphs studied in Keeler2022 ;
Keeler:2023xcx , establishing upper bounds on the variation of entanglement
entropy in stabilizer state systems. We also extend our analysis beyond
stabilizer states, deriving upper bounds on the evolution of entanglement
entropy for any quantum state under the action of the Clifford group.

##  4 Contracted Clifford Reachability Graphs

In this section, we build contracted graphs to illustrate entropy vector
evolution in stabilizer and non-stabilizer state reachability graphs. We begin
by first considering stabilizer state reachability graphs under the action of
the $\mathcal{C}_{2}$ subgroup $(HC)_{1,2}\equiv\langle
H_{1},\,H_{2},\,C_{1,2},\,C_{2,1}\rangle$, as studied in Keeler2022 ;
Keeler:2023xcx . We demonstrate how the contracted version of each
$(HC)_{1,2}$ reachability graph explains the bounds on entanglement variation
observed in our earlier work Keeler2022 . We then extend our analysis to
consider the full action of $\mathcal{C}_{2}$ on stabilizer states, showing
how $\mathcal{C}_{2}$ contracted graphs constrain the evolution of
entanglement entropy in stabilizer systems under any $2$-qubit Clifford
circuit.

We extend our study beyond the stabilizer states to the set of $n$-qubit Dicke
states, a class of non-stabilizer quantum states possessing non-trivial
stabilizer group under Clifford action Munizzi:2023ihc . We construct
$(HC)_{1,2}$ and $\mathcal{C}_{2}$ reachability and contracted graphs for all
Dicke states, establishing constraints on entropy vector evolution for such
states. Finally we move toward complete generality, deriving an upper bound
for the number of entropy vectors that can be realized by any $n$-qubit
Clifford circuit, acting on an arbitrary quantum state.

###  4.1 Contracted Graphs of $g_{24}$ and $g_{36}$

The complete set of $n$-qubit stabilizer states can be generated by acting
with $\mathcal{C}_{n}$ on the state ${|{0}\rangle}^{\otimes n}$. However,
since we are motivated to better understand the evolution of entropy vectors
in stabilizer systems, we restrict analysis to $\mathcal{C}_{2}$ and its
subgroups, since all entanglement modification in Clifford circuits occurs
through bi-local operations. Acting with $\mathcal{C}_{2}$ on
${|{0}\rangle}^{\otimes n}$, for $n>1$, generates an orbit of $60$ states.

First, we consider the class of states with stabilizer subgroup888A
comprehensive derivation of all stabilizer subgroups, for stabilizer states
under the action of $(HC)_{1,2}$, is given in Section 5.3 of Keeler:2023xcx .
isomorphic to $\mathcal{S}_{HC}({|{0}\rangle}^{\otimes
n})\equiv\textnormal{Stab}_{(HC)_{1,2}}({|{0}\rangle}^{\otimes n})$, under the
action of $(HC)_{1,2}$. The state ${|{0}\rangle}^{\otimes n}$, and any other
state with stabilizer group isomorphic to
$\mathcal{S}_{HC}({|{0}\rangle}^{\otimes n})$, has an orbit of $24$ states
under $(HC)_{1,2}$.

####  4.1.1 $\left(HC\right)_{1,2}$ Contracted Graphs of $g_{24}$ and $g_{36}$

The stabilizer subgroup $\mathcal{S}_{HC}({|{0}\rangle}^{\otimes n})$ contains
$48$ elements. As a result, generating all left cosets of the $1152$-element
group $(HC)_{1,2}$ by $\mathcal{S}_{HC}({|{0}\rangle}^{\otimes n})$ builds a
coset space of $1152/48=24$ equivalence classes. The corresponding
reachability graph of ${|{0}\rangle}^{\otimes n}$ under $(HC)_{1,2}$ contains
$24$ vertices, which we appropriately term $g_{24}$. The left panel of Figure
3 shows the graph $g_{24}$, which is shared by all states with stabilizer
group isomorphic to $\mathcal{S}_{HC}({|{0}\rangle}^{\otimes n})$.

To build the associated contracted graph we quotient $g_{24}$ by all elements
of $(HC)_{1,2}$ which do not modify the entropy vector. One immediate
$(HC)_{1,2}$ subgroup which cannot modify entanglement entropy is $\langle
H_{1},\,H_{2}\rangle$, which describes all circuits composed of Hadamard gates
acting on two qubits. Additionally, as we recognized in Keeler:2023xcx , the
relation

$\left(C_{i,j}H_{j}\right)^{4}=P_{i}^{2},$ (31)

demonstrates that certain sequences of Hadamard and CNOT gates are actually
equivalent to phase operations. We therefore need to also identify all
vertices connected by the circuits in Eq. (31), since phase operations cannot
change entanglement. After identifying all vertices connected by entropy-
preserving edges, the reachability graph $g_{24}$ contracts to a graph with
$2$ vertices, shown on the right of Figure 3.

Figure 3: Reachability graph $g_{24}$ (left) and its contracted graph (right).
Any state with stabilizer group isomorphic to
$\mathcal{S}_{HC}({|{0}\rangle}^{\otimes n})$ will have reachability graph
$g_{24}$ under $(HC)_{1,2}$. The $g_{24}$ contracted graph has $2$ vertices,
indicating the maximum number of unique entropy vectors that can exist in any
$g_{24}$ graph. Each edge in the contracted graph represents a set of
entanglement-modifying circuits, each containing at least one CNOT gate.

The contracted graph of $g_{24}$ contains $2$ vertices, and is shown in the
right panel of Figure 3. These $2$ vertices represent the $2$ possible entropy
vectors that can be reached by all circuits in any $g_{24}$ graph, regardless
of qubit number. All states represented by blue vertices in $g_{24}$ are
connected by some circuit composed of $H_{1},\,H_{2},\,P_{1}^{2},$ and
$P_{2}^{2}$, and are therefore identified to a single blue vertex in the
contracted graph. Likewise, all red vertices in $g_{24}$ are identified to a
single red vertex in the contracted graph. For the specific case of
${|{0}\rangle}^{\otimes n}$, the two entropy vectors in $g_{24}$ correspond to
completely unentangled states, or states which share an EPR pair among two
qubits.

As a group-theoretic object, the vertices of a contracted graph represent the
equivalence classes of a double coset space, as defined in Eq. (11). For the
group $(HC)_{1,2}$ acting on $\mathcal{H}$, the subgroup

$(HP^{2})_{1,2}\equiv\langle H_{1},\,H_{2},\,P_{1}^{2},\,P_{2}^{2}\rangle$
(32)

can never modify the entropy vector of any state. Accordingly, the $2$
vertices of the contracted graph in Figure 3 indicate the $2$ distinct
equivalence classes in the double coset space
$(HP^{2})_{1,2}\backslash(HC)_{1,2}/\mathcal{S}_{HC}({|{0}\rangle}^{\otimes
n})$.

Acting with the gates $H_{1}$ followed by $P_{1}$ on the state
${|{0}\rangle}^{\otimes n}$, that is

${|{\phi}\rangle}=P_{1}H_{1}{|{0}\rangle}^{\otimes n},$ (33)

yields a state ${|{\phi}\rangle}$ with stabilizer group
$\mathcal{S}_{HC}({|{\phi}\rangle})$, consisting of $32$ elements, which is
not isomorphic to $\mathcal{S}_{HC}({|{0}\rangle}^{\otimes n})$. Consequently
the state ${|{\phi}\rangle}$, as well as any other state with stabilizer group
isomorphic to $\mathcal{S}_{HC}({|{\phi}\rangle})$, is not found on any
$g_{24}$ graph. Instead, each state stabilized by
$\mathcal{S}_{HC}({|{\phi}\rangle})$ resides on a reachability graph of $36$
vertices, which we term $g_{36}$, shown on the left of Figure 4. In general,
any state which is the product of a $2$-qubit stabilizer state and a generic
$(n-2)$-qubit state will either have reachability graph $g_{24}$ or $g_{36}$.

Figure 4: Reachability graph $g_{36}$ (left) and its contracted graph (right).
The $g_{36}$ contracted graph contains $4$ vertices, but only ever realizes
$2$ entropy vectors among those vertices. Different sets of blue vertices,
highlighted in cyan, yellow, and magenta, identify respectively to the three
blue vertices in the contracted graph. All red vertices in $g_{36}$ identify
to a single red vertex in the contracted graph. Non-trivial entropy-preserving
circuits, e.g. $(C_{i,j}H_{j})^{4}$ from Eq. (31), map vertices on opposite
sides of $g_{36}$ to each other.

The contracted graph of $g_{36}$, shown in the right panel of Figure 4,
contains $4$ vertices. All red vertices in $g_{36}$ identify to the same red
vertex in the contracted graph. There are three distinct sets of blue vertices
in $g_{36}$, highlighted with colors cyan, yellow, and magenta in Figure 4,
which identify to the three blue vertices in the contracted graph. All
vertices highlighted by the same color in $g_{36}$ are connected by circuits
which preserve the entropy vector.

The vertices of the $g_{36}$ contracted graph in Figure 4 represent the $4$
unique equivalence classes of the double coset space
$(HP^{2})_{1,2}\backslash(HC)_{1,2}/\mathcal{S}_{HC}({|{\phi}\rangle})$.
Examining the vertex identifications in Figure 4, we again observe that the
contraction map is not a quotient map on the original group. Vertex sets of
different cardinalities in $g_{36}$ are identified together under this graph
contraction, which cannot occur in a formal group quotient.

While the $g_{36}$ contracted graph contains four vertices, these vertices
only ever realize two different entropy vector possibilities. Specifically,
the two entropy vectors found on any $g_{36}$ graph are exactly the same as
those found on the $g_{24}$ graph in Figure 3. As we will show below, graph
$g_{24}$ attaches to $g_{36}$ when we add phase gates back to our generating
set. This connection of the $g_{24}$ and $g_{36}$ reachability graphs by local
operations constrains the number of distinct entropy vectors that can be found
on either graph.

####  4.1.2 $\mathcal{C}_{2}$ Contracted Graphs of $g_{24}$ and $g_{36}$

We now analyze the full action of $\mathcal{C}_{2}$ on states in a $g_{24}$ or
$g_{36}$ reachability graph under $(HC)_{1,2}$. Acting with $\mathcal{C}_{2}$
on any such state generates a reachability graph of $60$ vertices, which can
be seen in Figure 5. This $60$-vertex reachability graph consists of a single
copy of $g_{24}$ (top), attached to a single copy of $g_{36}$ (bottom) by sets
of $P_{1}$ and $P_{2}$ edges.

Figure 5: Reachability graph for all states with
$\mathcal{S}_{HC}({|{0}\rangle}^{\otimes n})$ under the action of
$\mathcal{C}_{2}$. This $60$-vertex reachability graph is the attachment of
$g_{24}$ (Figure 3) to $g_{36}$ (Figure 4) by $P_{1}$ and $P_{2}$ gates. This
reachability graph is likewise shared by all stabilizer product states.

Following the $P_{1}$ and $P_{2}$ edges in Figure 5, we can observe how
vertices of a certain color connect to other vertices of the same color. Blue
vertices in $g_{24}$ always connect to blue vertices in $g_{36}$, as is true
for red vertices. Red vertices in $g_{36}$ may connect to other red vertices
in $g_{36}$, or to red vertices in $g_{24}$. The three distinct batches of
blue vertices in $g_{36}$, highlighted in Figure 4, connect to each other via
sequences of $H_{1},\,H_{2},\,P_{1},$ and $P_{2}$, all of which leave the
entropy vector unchanged. We can also directly observe circuits such as
$\left(C_{1,2}H_{2}\right)^{4}$, as in Eq. (31), and verify that this sequence
is indeed equivalent to the entropy-preserving $P_{1}^{2}$ operation.

As before, we contract the $\mathcal{C}_{2}$ reachability graph in Figure 5 by
identifying vertices connected by entropy-preserving circuits. When performing
this contraction on the full $\mathcal{C}_{2}$ graph we do not rely on any
special operator relations, e.g. Eq. (31), since we are identifying vertices
connected by all $2$-qubit local operations, i.e. all operations built of
$H_{1},\,H_{2},\,P_{1},$ and $P_{2}$. The contracted graph of the
$\mathcal{C}_{2}$ reachability graph in Figure 5 is shown in the right panel
of Figure 6. The $2$ vertices in this contracted graph represent the $2$
equivalence classes in
$(HP^{2})_{1,2}\backslash\mathcal{C}_{2}/\mathcal{S}_{\mathcal{C}_{2}}({|{0}\rangle}^{\otimes
n})$.

Figure 6: Contracted graph (right) of $\mathcal{C}_{2}$ reachability graph in
Figure 5. The left panel shows the contracted graphs of $g_{24}$ (top) and
$g_{36}$ (bottom), connected by $P_{1}$ and $P_{2}$ circuits. Identifying
vertices connected by phase edges quotients the left graph to the $2$-vertex
contracted graph on the right. The $2$ vertices of this contracted graph
represent the $2$ unique entropy vectors that can be found in the reachability
graph in Figure 5.

Figure 5 depicts how sets of phase gates connect reachability graphs $g_{24}$
and $g_{36}$. Similarly, the left panel of Figure 6 shows how the respective
contracted graphs of $g_{24}$ and $g_{36}$ are connected by sets of phase
edges. The right panel of Figure 6 gives the final contracted graph after
quotienting the $\mathcal{C}_{2}$ reachability graph in Figure 5 by all
entropy-preserving edges. The contracted graph has $2$ vertices, corresponding
to the $2$ possible entropy vectors that can be found on any $\mathcal{C}_{2}$
reachability graph of the form shown in Figure 5. Furthermore, the $2$
vertices in the contracted explain why both graphs $g_{24}$ and $g_{36}$
individually only ever realize $2$ entropy vector colors among their vertices.

We examined the action of $(HC)_{1,2}$ and $\mathcal{C}_{2}$ on $n$-qubit
states with stabilizer group isomorphic to
$\mathcal{S}_{HC}({|{0}\rangle}^{\otimes n})$ and
$\mathcal{S}_{HC}({|{0}\rangle}^{P_{1}H_{1}\otimes n})$. We generated the
reachability graphs for all states with both stabilizer groups, and quotiented
each reachability graph by entropy-preserving operations to build the
associated contracted graphs. The number of vertices in each contracted graph
gave an upper bound on the number of different entropy vectors found in each
reachability graph. Similarly, the edges in each contracted graph indicated
the ways an entropy vector can change under all circuits comprising the
reachability graph. We will now consider the reachability graphs of $n>2$
qubit stabilizer states, where more-complicated entanglement structures can
arise.

###  4.2 Contracted Graphs of $g_{144}$ and $g_{288}$

When we consider the action of $(HC)_{1,2}$ and $\mathcal{C}_{2}$ on systems
of $n>2$ qubits, new reachability graph structures appear Keeler2022 .
Additionally at $n>2$ qubits, we observe new entanglement possibilities as
well as new entropy vector colorings for reachability graphs. In this
subsection, we define two new sets of stabilizer states which arise at $n=3$
qubits, defined by their stabilizer subgroup under $(HC)_{1,2}$ action. We
build all reachability graphs and contracted graphs for these two families of
states, and determine the bounds on entropy vector evolution in their
respective reachability graphs. We then consider the full action of
$\mathcal{C}_{2}$ on these classes of states, and again build all reachability
and contracted graphs.

At three qubits, acting with $(HC)_{1,2}$ on certain stabilizer states
produces an additional two reachability graphs beyond $g_{24}$ and $g_{36}$
discussed in the previous subsection. One new graph which arises at three
qubits contains $144$ vertices, shown on the left of Figure 7, and corresponds
to states which are stabilized by $8$ elements in $(HC)_{1,2}$. One example of
a state with $g_{144}$ reachability graph is the $3$-qubit GHZ state
${|{GHZ}\rangle}_{3}\equiv{|{000}\rangle}+{|{111}\rangle}$. The graph
$g_{144}$ is shared by all states with a stabilizer subgroup isomorphic to
$\mathcal{S}_{HC}({|{GHZ}\rangle}_{3})$. For reasons we will explain in a
moment, Figure 7 depicts the specific reachability graph for the $6$-qubit
state defined in Eq. (45).

Figure 7: Reachability graph $g_{144}$ (left), and its associated contracted
graph (right). The contracted graph contains $5$ vertices, corresponding to
the $5$ unique entropy vectors that can be found on $g_{144}$. We depict a
$g_{144}$ graph for the $6$-qubit state defined in Eq. (45), as it contains
the maximal number of $5$ entropy vectors among its vertices. Again we observe
that certain circuits, e.g. Eq. (31), do not modify entanglement and map
vertices of the same color together. The specific entropy vectors shown are
given in Table 6.

The contracted graph of $g_{144}$, shown on the right of Figure 7, contains
$5$ vertices. These $5$ vertices represent the $5$ unique entropy vectors that
can be found on any $g_{144}$ reachability graph. While the graph $g_{144}$ is
first observed among $3$-qubit systems, we do not find a maximal coloring of
$g_{144}$, i.e. a copy of $g_{144}$ with $5$ different entropy vectors, until
$6$ qubits. The specific graph shown in Figure 7 corresponds to the orbit of
the $6$-qubit state defined in Eq. (45), which we choose precisely because its
$g_{144}$ graph displays the maximum allowable entropic diversity. The
specific entropy vectors corresponding to the colors seen in Figure 7 can be
found in Table 6 of Appendix A.

Also beginning at three qubits, we witness a stabilizer state reachability
graph with $288$ vertices, which we denote $g_{288}$. States with reachability
graph $g_{288}$ are stabilized by $4$ elements of $(HC)_{1,2}$, specifically
by a subgroup isomorphic to

$\\{\mathbb{1},\,H_{2}(C_{1,2}H_{1})^{4},\,(C_{1,2}H_{1})^{4}H_{2},\,\left((C_{1,2}H_{1})^{3}C_{1,2}H_{2}\right)^{2}\\}.$
(34)

The left panel of Figure 8 depicts a $g_{288}$ reachability graph,
specifically for a $6$-qubit state stabilized by the group in Eq. (34).

Figure 8: Reachability graph $g_{288}$, and its contracted graph, for
$6$-qubit state stabilized by Eq. (34). While the $g_{288}$ contracted graph
has $12$ vertices, we only ever witness $5$ entropy vectors among those
vertices. The specific entropy vectors depicted are the same as those in
Figure 7, and can be found in Table 6.

The $g_{288}$ contracted graph shown in the right panel of Figure 8 contains
$12$ vertices, which provides a weak upper bound on the number of entropy
vectors that can be found on any $g_{288}$ graph. However, for reasons we will
soon explain, the $12$ vertices of this contracted graph are only ever colored
by $5$ different entropy vectors. The specific $5$ entropy vectors shown in
Figure 8 are exactly those seen in Figure 7, and are defined in Table 6.
Similar to the case of $g_{144}$ in Figure 7, the graph $g_{288}$ is first
observed among $3$-qubit systems, but only witnesses a maximal coloring
beginning at $n\geq 6$ qubits.

We now consider the full action of $\mathcal{C}_{2}$ on states with a
$g_{144}$ or $g_{288}$ reachability graph, returning $P_{1}$ and $P_{2}$ to
our generating set. Every state in a $g_{144}$ and $g_{288}$ reachability
graph under $(HC)_{1,2}$ is stabilized by $15$ elements of the full group
$\mathcal{C}_{2}$. The orbit of all such states under $\mathcal{C}_{2}$
therefore contains $768$ states, and the associated $768$-vertex reachability
graph is shown in Figure 9. The orange edges in the reachability graph, which
correspond to $P_{1}$ and $P_{2}$ gates, illustrate specifically how three
different copies of $g_{144}$ attach to a single copy of $g_{288}$ under phase
operations.

Figure 9: Reachability graph for states in $g_{144}$ and $g_{288}$ graphs,
under the full action of $\mathcal{C}_{2}$. This $768$-vertex graph is
composed of $3$ copies of $g_{144}$ and a single $g_{288}$. The graph
connectivity constrains the diversity of entropy vectors which can be found on
any single $g_{144}$ and $g_{288}$ graph. For clarity we choose not to color
vertices by their entropy vector here.

The contracted graphs for each $g_{144}$ and $g_{288}$ in Figure 9 are
compiled in the left panel of Figure 10. Each of the three copies of $g_{144}$
contracts to a $5$-vertex graph that is isomorphic to Figure 7, while the
single copy of $g_{288}$ contracts to the $12$-vertex graph seen in Figure 8.
These four contracted graphs attach to each other under phase operations,
adding connections which do not change a state’s entropy vector. The final
contracted graph of Figure 9 is shown on the right of Figure 10, and only has
$5$ vertices.

The full $\mathcal{C}_{2}$ contracted graph in Figure 10 is almost identical
to the $g_{144}$ contracted graph in Figure 7, but with an additional edge
connecting two of the vertices. Since every $g_{288}$ attaches to $3$ copies
of $g_{144}$ by phase gates, which do not modify entanglement, the maximum
number of entropy vectors on any $g_{288}$ is bounded by the entropic coloring
of each $g_{144}$ it connects to. This connectivity explains why we only
observe at most $5$ entropy vectors on any $g_{288}$ graph, as can be seen in
Figure 8.

Figure 10: Contracted graph of $\mathcal{C}_{2}$ reachability graph from
Figure 9. The left panel depicts the individual contracted graphs of the $3$
$g_{144}$ graphs attached to a single $g_{288}$ graph. The right panel shows
the final contracted graph, with $5$ vertices, and explains why we only ever
find $g_{288}$ and $g_{144}$ graphs with $5$ different entropy vectors (given
in Table 6).

Figure 10 depicts a symmetry between red and blue vertices which corresponds
to an equivalence of these two entropy vectors under an exchange of the first
two qubits. We likewise observe a symmetry between green, yellow, and magenta
vertices, reflecting the three ways to divide the $4$-qubit subsystem $CDEO$
into two groups of two qubits each. For each $g_{144}$ contracted graph in
Figure 10, the middle vertex corresponds to the entropy vector that occurs the
fewest number of times, specifically $16$ times, in each respective $g_{144}$
reachability graph. We again observe that the contraction procedure generates
a double coset space, rather than a group quotient, since the resulting
equivalence classes have different cardinalities.

In this subsection we built contracted graphs for the stabilizer state
reachability graphs $g_{144}$ and $g_{288}$, corresponding to states which are
stabilized by $4$ and $8$ elements of $(HC)_{1,2}$ respectively. We showed how
the contracted graph for $g_{144}$, with $5$ vertices, and the contracted
graph for $g_{288}$, with $12$ vertices, both witness a maximum of $5$
different entropy vectors. This constraint on the number of different entropy
vectors, perhaps surprising in the case of $g_{288}$, can be understood by
considering the full action of $\mathcal{C}_{2}$, which attaches three copies
of $g_{144}$ to $g_{288}$ by phase operations. The number of entropy vectors
found on any $g_{288}$ reachability graph is bounded by the number of entropy
vectors found on each of the $g_{144}$ graphs to which it attaches, since
$P_{1}$ and $P_{2}$ cannot modify entanglement. In the next subsection we
consider the action of $(HC)_{1,2}$ and $\mathcal{C}_{2}$ on generic quantum
states, which allows us to extend our analysis beyond the stabilizer states.

###  4.3 Contracted Graphs of $g_{1152}$ and Full $\mathcal{C}_{2}$

We now study the generic $(HC)_{1,2}$ reachability graph for any quantum state
stabilized by only the identity in $(HC)_{1,2}$. For stabilizer state systems,
this final $(HC)_{1,2}$ reachability graph structure arises at $n\geq 4$
qubits. The reachability graph, which we term $g_{1152}$, contains $1152$
vertices and is shown on the left of Figure 11.

Figure 11: Reachability graph $g_{1152}$ (left) and its contracted graph
(right). The graph $g_{1152}$ is shared by all stabilizer states stabilized by
only $\mathbb{1}\in(HC)_{1,2}$, as well as generic quantum states. In this
Figure, we illustrate an example $g_{1152}$ for the $8$-qubit state in Eq.
(46), where the contracted graph achieves a maximal coloring of $18$ different
entropy vectors (given in Figure 17).

The contracted graph of $g_{1152}$, shown in the right panel of Figure 11,
contains $18$ vertices. These $18$ vertices indicate the maximum number of
unique entropy vectors that can be generated for any quantum state using only
operations in $(HC)_{1,2}$. The $g_{1152}$ contracted graph is symmetric, and
achieves a maximal coloring at $8$ qubits. The specific instance of $g_{1152}$
in Figure 11 corresponds to the $8$-qubit state given in Eq. (46), for which
the entropy vectors are given in Table 17.

The full $2$-qubit Clifford group $\mathcal{C}_{2}$ is composed of $11520$
elements. A generic quantum state will only be stabilized by
$\mathbb{1}\in\mathcal{C}_{2}$, and therefore has an orbit of $11520$ states
under $\mathcal{C}_{2}$ action. Every state in an $11520$-vertex reachability
graph under $\mathcal{C}_{2}$ will trivially lie in a $g_{1152}$ graph under
$(HC)_{1,2}$, however, the converse999Since any state in an $11520$-vertex
graph under $\mathcal{C}_{2}$ is stabilized by only the identity, each state
will likewise be stabilized by only the identity in $(HC)_{1,2}$. There exist
states, however, which are stabilized by only the identity in $(HC)_{1,2}$,
but can be transformed under phase gates into states stabilized by more than
one element of $(HC)_{1,2}$. Subsection 4.4 discusses two classes of Dicke
states which demonstrate this counterexample.  is not always true. We display
the full $\mathcal{C}_{2}$ reachability graph, in a compressed format, to the
left of Figure 12. Each vertex in the left panel of Figure 12 represents a
distinct copy of $g_{1152}$ from Figure 11. Each of the $10$ copies of
$g_{1152}$ attaches to every other $g_{1152}$ via $P_{1}$ and $P_{2}$ gates.

Figure 12: The full $\mathcal{C}_{2}$ reachability graph (left) with $11520$
vertices. We present this reachability graph as a collection of attached
$g_{1152}$ graphs, illustrating how $(HC)_{1,2}$ reachability graphs connect
via $P_{1}$ and $P_{2}$ gates. We also remove all loops in the
$\mathcal{C}_{2}$ reachability graph, i.e. all phase edges which map a copy of
$g_{1152}$ to itself. The contracted graph of the $\mathcal{C}_{2}$
reachability graph is given to the right, and has $20$ vertices. These $20$
vertices give an upper bound on the number of distinct entropy vectors that
can be reached by applying any sequence of $2$-qubit operations on any quantum
state.

The contracted graph101010Since all states in the $11520$-vertex reachability
graph are stabilized by only $\mathbb{1}\in\mathcal{C}_{2}$, and since
$\langle\mathbb{1}\rangle$ is normal in $\mathcal{C}_{2}$, the object
$\mathcal{C}_{2}/\langle\mathbb{1}\rangle$ defines a formal group quotient on
$\mathcal{C}_{2}$. Consequently, the contracted graph to the right of Figure
12 actually represents the right coset space $\mathcal{C}_{2}\backslash\langle
H_{1},\,H_{2},\,P_{1},\,P_{2}\rangle$, as opposed to a double coset space. of
the $11520$-vertex $\mathcal{C}_{2}$ reachability graph contains $20$
vertices, and is shown on the right of Figure 12. This contracted graph is
complete and symmetric, and the $20$ entropy vectors shown in Figure 12 are
given in Table 17. Since we are considering the full action of
$\mathcal{C}_{2}$, the $20$ vertices in this contracted graph constrain the
number of entropy vectors that can be generated by any $2$-qubit Clifford
circuit. Otherwise stated, given a generic quantum state with arbitrary
entanglement structure, any unitary composed of $2$-qubit Clifford gates can
maximally achieve $20$ distinct entropy vectors.

In the remainder of the section we extend our discussion beyond stabilizer
states, examining contracted graphs for non-stabilizer Dicke states under
$(HC)_{1,2}$ and $\mathcal{C}_{2}$ action. We also derive a general upper
bound for the number of entropy vectors that can be achieved under any
$n$-qubit Clifford circuit, for arbitrary $n$.

###  4.4 Non-Stabilizer State Contracted Graphs

Keeler:2023xcx ; Munizzi:2023ihc  showed that certain non-stabilizer states
can have non-trivial stabilizer subgroups, i.e. they are stabilized by more
than just the identity, under the action of $\mathcal{C}_{n}$. One class of
states in particular, the set of $n$-qubit Dicke states nepomechie2023qudit ,
always admits a non-trivial $\mathcal{C}_{n}$ stabilizer group. In this
subsection, we discuss all $(HC)_{1,2}$ and $\mathcal{C}_{2}$ reachability
graphs for Dicke states and construct their associated contracted graphs. We
use the contracted graphs to bound the number of possible entropy vectors that
can be generated in Dicke state systems under Clifford group action
Schnitzer:2022exe ; Munizzi:2023ihc .

Each $n$-qubit Dicke state ${|{D^{n}_{k}}\rangle}$ is defined as an equal-
weight superposition over all $n$-qubit states of a fixed Hamming weight.
Using the $n$-qubit states $\\{{|{b}\rangle}\\}$, where $b$ denotes some
binary string of length $2^{n}$, we construct ${|{D^{n}_{k}}\rangle}$ as the
state

${|{D^{n}_{k}}\rangle}\equiv\binom{n}{k}^{-1/2}\sum_{b\in\\{0,1\\}^{n},\,h(b)=k}{|{b}\rangle},$
(35)

where $h(b)=k$ denotes the fixed Hamming weight of $b$. Some examples of Dicke
states include

$\begin{split}{|{D^{2}_{1}}\rangle}&=\frac{1}{\sqrt{2}}\left({|{01}\rangle}+{|{10}\rangle}\right),\\\
{|{D^{4}_{2}}\rangle}&=\frac{1}{\sqrt{6}}\left({|{1100}\rangle}+{|{1010}\rangle}+{|{1001}\rangle}+{|{0110}\rangle}+{|{0101}\rangle}+{|{0011}\rangle}\right).\end{split}$
(36)

Dicke states of the form ${|{D^{n}_{1}}\rangle}$ are exactly the non-
biseparable $n$-qubit $W$-states, while ${|{D^{n}_{n}}\rangle}$ are the
computational basis states ${|{1}\rangle}^{\otimes n}$.

For $n\geq 3$ qubits, the state ${|{D^{n}_{1}}\rangle}$ is not a stabilizer
state. Regardless, each ${|{D^{n}_{k}}\rangle}$ is stabilized by a subset of
$\mathcal{C}_{n}$ that contains more than just the identity. When considering
the action of $\mathcal{C}_{2}$ on ${|{D^{n}_{k}}\rangle}$, states of the form
${|{D^{n}_{1}}\rangle}$ and ${|{D^{n}_{n-1}}\rangle}$ share one particular set
of stabilizers, while those of the form ${|{D^{n}_{k}}\rangle}$ with $1<k<n-1$
share another. We discuss both cases below.

Dicke states of the form ${|{D^{n}_{1}}\rangle}$ and ${|{D^{n}_{n-1}}\rangle}$
are not stabilizer states for all $n\geq 3$. However, both
${|{D^{n}_{1}}\rangle}$ and ${|{D^{n}_{n-1}}\rangle}$ are stabilized by a
$4$-element subgroup111111There is a more compact representation of this
stabilizer group using $CZ$ gates (see also Latour:2022gsf ), which can be
written
$\mathcal{S}_{HC}({|{D^{n}_{1}}\rangle})=\\{\mathbb{1},\,CZ_{1,2},\,C_{1,2}C_{2,1}C_{1,2},\,CZ_{1,2}C_{1,2}C_{2,1}C_{1,2}\\}$.
of $\mathcal{C}_{2}$, specifically

$\begin{split}\mathcal{S}_{HC}({|{D^{n}_{1}}\rangle})&=\\{\mathbb{1},\,H_{2}C_{1,2}H_{2},\,C_{1,2}C_{2,1}C_{1,2},\,H_{2}C_{1,2}H_{2}C_{1,2}C_{2,1}C_{1,2}\\},\\\
&=\mathcal{S}_{HC}({|{D^{n}_{n-1}}\rangle}).\end{split}$ (37)

Furthermore, we note that the subgroup in Eq. (37) is contained in
$(HC)_{1,2}$. Therefore the left coset space
$(HC)_{1,2}/\mathcal{S}_{HC}({|{D^{n}_{1}}\rangle})$ contains $288$ elements.

The reachability graph for all ${|{D^{n}_{1}}\rangle}$ and
${|{D^{n}_{n-1}}\rangle}$, which we denote $g_{288^{*}}$, has $288$ vertices,
as dictated by the order of $\mathcal{S}_{HC}({|{D^{n}_{1}}\rangle})$ in Eq.
(37). While the graph $g_{288^{*}}$ has the same number of vertices as the
$g_{288}$ graph for stabilizer states, shown in Figure 8, its topology is
distinct from $g_{288}$ and the two graphs are not isomorphic. Graphs with the
topology of $g_{288^{*}}$ are never observed among stabilizer states, and
provide an example of non-stabilizer states that are stabilized by more than
just the identity in $\mathcal{C}_{2}$. The left panel of Figure 13 depicts an
example of $g_{288^{*}}$, specifically for the state ${|{D^{3}_{1}}\rangle}$.

Figure 13: Reachability graph $g_{288^{*}}$ (left) for ${|{D^{3}_{1}}\rangle}$
under the action of $(HC)_{1,2}$. The graph $g_{288^{*}}$ has different
topology than the $g_{288}$ graph for stabilizer states. The $g_{288^{*}}$
contracted graph (right) has $5$ vertices, and is isomorphic to the stabilizer
state contracted graph of $g_{144}$ from Figure 7. The exact, rather than
numerical, values of the $5$ entropy vectors given in the legend are shown in
Table 7.

The contracted graph of $g_{288^{*}}$ has $5$ vertices, and is shown on the
right of Figure 13. While the reachability graph $g_{288}$ for stabilizer
states has a contracted graph of $12$ vertices, the distinct connectivity of
$g_{288^{*}}$ yields a smaller contracted graph. Interestingly, the
$g_{288^{*}}$ contracted graph is isomorphic to the $g_{144}$ contracted graph
seen in Figure 7. There are $5$ possible entropy vectors found on any
$g_{288^{*}}$, and the graph achieves a maximal coloring beginning at $3$
qubits.

The orbit of ${|{D^{n}_{1}}\rangle}$ and ${|{D^{n}_{n-1}}\rangle}$ under the
full group $\mathcal{C}_{2}$ reaches $2880$ states, generating a reachability
graph of $2880$ vertices. The left panel of Figure 14 illustrates this
$2880$-vertex reachability graph for the state ${|{D^{3}_{1}}\rangle}$, which
is comprised of several attached copies of $(HC)_{1,2}$ reachability graphs.
For clarity, we allow each vertex of the $2880$-vertex reachability graph to
represent graphs $g_{288^{*}},\,g_{576}$ (introduced later in Figure 15), and
$g_{1152}$, focusing on the connectivity between different $(HC)_{1,2}$ orbits
under $P_{1}$ and $P_{2}$ operations.

Figure 14: Reachability graph (left) of ${|{D^{3}_{1}}\rangle}$ under the full
action of $\mathcal{C}_{2}$, containing $2880$ vertices. We illustrate this
reachability graph with vertices representing graphs $g_{288^{*}},g_{576},$
and $g_{1152}$ to illustrate the connectivity of certain $(HC)_{1,2}$
reachability graphs under phase gates. The right panel of the Figure depicts
the associated contracted for the $\mathcal{C}_{2}$ reachability graph, which
contains $6$ vertices.

The $\mathcal{C}_{2}$ reachability graph in Figure 14 is built of $2$ attached
copies of $g_{288^{*}}$, $2$ copies of $g_{576}$, and a single $g_{1152}$.
Every state in this $2880$-vertex reachability graph is stabilized by $4$
elements of $\mathcal{C}_{2}$. Certain states, such as ${|{D^{n}_{1}}\rangle}$
and ${|{D^{n}_{n-1}}\rangle}$, are stabilized by a $4$-element subgroup of
$\mathcal{C}_{2}$ which is also completely contained within $(HC)_{1,2}$, as
shown in Eq. (37). However, other states are stabilized by $4$ elements of
$\mathcal{C}_{2}$, but by only $2$ elements in $(HC)_{1,2}$ (see Footnote 9).
Accordingly, such states are found in one of the $g_{576}$ graphs in Figure
14. Still other states are stabilized by $4$ elements of $\mathcal{C}_{2}$,
but only by the identity in $(HC)_{1,2}$, and reside in the single copy of
$g_{1152}$ in Figure 14.

The $\mathcal{C}_{2}$ reachability graph of ${|{D^{3}_{1}}\rangle}$ contracts
to a $6$-vertex graph, seen to the right of Figure 14, after identifying
vertices connected by entropy-preserving circuits. While the contracted graph
in Figure 14 has $6$ vertices, we only ever observe $5$ different entropy
vectors among those vertices. We address this point further in the discussion.
The $5$ entropy vectors of the ${|{D^{3}_{1}}\rangle}$ contracted graph are
listed in Table 7.

All remaining Dicke states, those of the form ${|{D^{n}_{k}}\rangle}$ with
$1<k<n-1$, are stabilized by only $2$ elements in $\mathcal{C}_{2}$. For any
${|{D^{n}_{k}}\rangle}$ of this form, its stabilizer subgroup under
$\mathcal{C}_{2}$ action is given by

$\mathcal{S}_{\mathcal{C}_{2}}\left({|{D^{n}_{k}}\rangle}\right)=\\{\mathbb{1},\,C_{1,2}C_{2,1}C_{1,2}\\},\quad\forall\,1<k<n-1.$
(38)

We again note that the stabilizer group in Eq. (38) is also contained
completely within $(HC)_{1,2}$, and therefore the left coset space
$(HC)_{1,2}/\mathcal{S}_{\mathcal{C}_{2}}\left({|{D^{n}_{k}}\rangle}\right)$
consists of $576$ elements.

The reachability graph for ${|{D^{n}_{k}}\rangle}$ under $(HC)_{1,2}$, which
we denote $g_{576}$, has $576$ vertices. The left panel of Figure 15 depicts
$g_{576}$, specifically for the state ${|{D^{4}_{2}}\rangle}$. Reachability
graphs with $576$ vertices, under $(HC)_{1,2}$ action, are never observed for
stabilizer states. Again, as with $g_{288^{*}}$, the graph $g_{576}$
corresponds to non-stabilizer states which are non-trivially stabilized by
$\mathcal{C}_{n}$.

Figure 15: The $g_{576}$ reachability graph (left) for ${|{D^{4}_{2}}\rangle}$
under $(HC)_{1,2}$ action. Graphs of $576$ vertices are never observed among
stabilizer states under $(HC)_{1,2}$ action. The graph $g_{576}$ contracts to
a graph of $9$ vertices under entropy-preserving operations, with $6$
different entropy vectors among those vertices. The $6$ entropy vectors found
in this contracted graph are given in Table 8.

After identifying vertices in $g_{576}$ connected by entropy-preserving
operations, we are left with a contracted graph of $9$ vertices shown on the
right of Figure 15. These $9$ vertices are colored by $6$ different entropy
vectors, with maximal coloring beginning at $4$ qubits. Among the $6$ entropy
vectors in this contracted graph, there are symmetries shared among cyan,
magenta, and yellow vectors, and separately among red, blue, and green
vectors. The specific $6$ entropy vectors for the ${|{D^{4}_{2}}\rangle}$
contracted graph are given in Table 8.

Acting with the full group $\mathcal{C}_{2}$ on ${|{D^{n}_{k}}\rangle}$, for
$1<k<n-1$, generates an orbit of $5760$ states. The $\mathcal{C}_{2}$
reachability graph of ${|{D^{n}_{k}}\rangle}$ therefore has $5760$ vertices,
and is depicted in the left panel of Figure 16 for the case of
${|{D^{4}_{2}}\rangle}$. As before, we depict the full $5760$-vertex
reachability graph as $7$ attached copies of different $(HC)_{1,2}$
reachability graphs $g_{576}$ and $g_{1152}$.

Figure 16: Reachability graph of ${|{D^{4}_{2}}\rangle}$ under
$\mathcal{C}_{2}$ (left), and its associated contracted graph (right). We
display the $5760$-vertex reachability graph as a network of $(HC)_{1,2}$
graphs $g_{576}$ and $g_{1152}$, connected by $P_{1}$ and $P_{2}$ gates. The
contracted graph contains $10$ vertices, but we only ever observe $6$ entropy
vectors due to how the $g_{576}$ and $g_{1152}$ copies connect under phase
action. The $6$ different entropy vectors shown are given in Table 8.

The $5760$-vertex reachability graph in Figure 16 consists of $4$ copies of
$g_{576}$ and $3$ copies of $g_{1152}$, all connected via $P_{1}$ and $P_{2}$
operations. While every state in the full $5760$-vertex reachability graph is
stabilized by $2$ elements of $\mathcal{C}_{2}$, some states have a stabilizer
group completely contained within $(HC)_{1,2}$. States stabilized by $2$
elements of $(HC)_{1,2}$ are found in one of the $4$ copies of $g_{576}$ in
Figure 16. Alternatively, states which are stabilized by $2$ elements of
$\mathcal{C}_{2}$, but only the identity in $(HC)_{1,2}$, are found in one of
the $3$ copies of $g_{1152}$.

If we identify vertices connected by entropy-preserving operations in the
$\mathcal{C}_{2}$ reachability graph of ${|{D^{4}_{2}}\rangle}$, we are left
with a contracted graph containing $10$ vertices shown to the right of Figure
16. While this contracted graph has $10$ vertices, we only ever observe $6$
different entropy vectors among those $10$ vertices. We again return to this
point in the discussion. The contracted graph in Figure 16 also reflects the
symmetry among magenta, cyan, and yellow vertices observed in Figure 15. These
$6$ entropy vectors which can be generated from ${|{D^{4}_{2}}\rangle}$ under
$\mathcal{C}_{2}$ are given in Table 8.

In this subsection we extended our analysis beyond the stabilizer states,
building contracted graphs for non-stabilizer Dicke states under the action of
$(HC)_{1,2}$ and $\mathcal{C}_{2}$. States ${|{D^{n}_{k}}\rangle}$, for $k\neq
n$, are particularly interesting at $n\geq 3$ qubits as they comprise a class
of non-stabilizer states that are non-trivially stabilized by elements of
$\mathcal{C}_{n}$. We constructed the two possible reachability graphs for
${|{D^{n}_{k}}\rangle}$, one for states ${|{D^{n}_{1}}\rangle}$ and
${|{D^{n}_{n-1}}\rangle}$, and the other for all ${|{D^{n}_{k}}\rangle}$ with
$1<k<n-1$. We described how each Dicke state reachability graph under
$\mathcal{C}_{2}$ corresponds to a connection of $(HC)_{1,2}$ reachability
graphs $g_{288^{*}},\,g_{576},$ and $g_{1152}$ under $P_{1}$ and $P_{2}$
operations.

We built the contracted graphs for each ${|{D^{n}_{k}}\rangle}$ reachability
graph, both under $(HC)_{1,2}$ and $\mathcal{C}_{2}$ action. We illustrated
that states ${|{D^{n}_{1}}\rangle}$ and ${|{D^{n}_{n-1}}\rangle}$ can realize
$5$ different entropy vectors under $\mathcal{C}_{2}$. Alternatively, states
of the form ${|{D^{n}_{k}}\rangle}$ with $1<k<n-1$ can achieve $6$ different
entropy vectors under $\mathcal{C}_{2}$. In the next subsection we completely
generalize to an argument for $\mathcal{C}_{n}$ action on arbitrary quantum
states. We use our construction up to this point to bound the entropy vector
possibilities that can be achieved for any state under $n$-qubit Clifford
action.

###  4.5 Entanglement in $n$-Qubit Clifford Circuits

We now use our results to present an upper bound on entropy vector evolution
in Clifford circuits, for arbitrary qubit number. We begin by determining the
subset of $\mathcal{C}_{n}$ operations which cannot modify the entanglement
entropy of any state. We then build a contracted graph by identifying the
vertices in the $\mathcal{C}_{n}$ Cayley graph that are connected by entropy-
preserving circuits.

Local actions, i.e. all operations which act only on a single qubit in some
$n$-qubit system, will always preserve a state’s entropy vector. When
considering action by the Clifford group $\mathcal{C}_{n}$, the subgroup of
all local actions is exactly the group generated by $n$-qubit Hadamard and
phase gates, which we denote $(HP)_{n}$. We build $(HP)_{n}$ as the direct
product Keeler:2023xcx

$(HP)_{n}\equiv\prod_{i=1}^{n}\langle H_{i},\,P_{i}\rangle.$ (39)

Since $(HP)_{n}$ is a direct product, and $|\langle H_{i},\,P_{i}\rangle|=24$,
the order of $|(HP)_{n}|$ is just $24^{n}$. The order of the $n$-qubit
Clifford group is likewise known Walter2016 . We can compute
$|\mathcal{C}_{n}|$ as

$|\mathcal{C}_{n}|=2^{n^{2}+2n}\prod_{j=1}^{n}(4^{j}-1).$ (40)

Generating the right coset space $\mathcal{C}_{n}\backslash(HP)_{n}$
identifies all elements in $\mathcal{C}_{n}$ equivalent up to local gate
operations. Invoking Lagrange’s theorem (Eq. (13)) allows us to compute the
size of $\mathcal{C}_{n}\backslash(HP)_{n}$ as

$\frac{|\mathcal{C}_{n}|}{|(HP)_{n}|}=\frac{2^{n^{2}-n}}{3^{n}}\prod_{j=1}^{n}(4^{j}-1).$
(41)

It is important to note that $(HP)_{n}$ is not a normal subgroup of
$\mathcal{C}_{n}$, which we can immediately verify by considering any Hadamard
operation $H_{j}\in(HP)_{n}$. The element

$C_{i,j}H_{j}C_{i,j}^{-1}\notin\langle H_{i},\,P_{i},\,H_{j},\,P_{j}\rangle,$
(42)

which violates the necessity that any normal subgroup be invariant under group
conjugation. Accordingly, $(HP)_{n}$ does not generate a quotient of
$\mathcal{C}_{n}$.

The coset space $\mathcal{C}_{n}\backslash(HP)_{n}$ partitions
$\mathcal{C}_{n}$ into sets of Clifford circuits which are equivalent up to
local action. Consequently, Eq. (41) provides an upper bound on the number of
entropy vectors that can possibly be generated under any $n$-qubit Clifford
circuit, for any arbitrary quantum state. This upper bound is equivalently
captured by directly building a contracted graph from the $\mathcal{C}_{n}$
Cayley graph, and counting the number of vertices. The right panel of Figure
12 illustrates the $20$-vertex contracted graph of the $\mathcal{C}_{2}$
Cayley121212Formally, the left panel of Figure 12 depicts the reachability
graph for some set of states, rather than the Cayley graph of
$\mathcal{C}_{2}$. However, since the particular class of states is stabilized
by only the identity in $\mathcal{C}_{2}$, the reachability graph in the left
panel of Figure 12 is exactly the phase-modded $\mathcal{C}_{2}$ Cayley graph.
graph. Table 1 gives the explicit number of entropy vectors that can be
achieved using $n\leq 5$ qubit Clifford circuits.

$n$ | $|\mathcal{C}_{n}|/|(HP)_{n}|$
---|---
$1$ | $1$
$2$ | $20$
$3$ | $6720$
$4$ | $36556800$
$5$ | $3191262412800$
Table 1: Maximum number of entropy vectors that can be generated using
elements of the $n$-qubit Clifford group, for $n\leq 5$.

In Eq. (41) we count the right cosets of $\mathcal{C}_{n}$ by the subgroup of
entropy-preserving operations. This upper bound equivalently constrains the
number of entropy vectors which can be realized by a generic quantum state,
stabilized by only $\mathbb{1}\in\mathcal{C}_{n}$, under any Clifford circuit.
However, we can tighten this bound for states which are non-trivially
stabilized by some subset of $\mathcal{C}_{n}$. For a state ${|{\psi}\rangle}$
with stabilizer group $\mathcal{S}_{\mathcal{C}_{n}}({|{\psi}\rangle})$, the
number of achievable entropy vectors is bounded by the size of the double
coset space
$(HP)_{n}\backslash\mathcal{C}_{n}/\mathcal{S}_{\mathcal{C}_{n}}({|{\psi}\rangle})$.
As by Eq. (15), the size of
$(HP)_{n}\backslash\mathcal{C}_{n}/\mathcal{S}_{\mathcal{C}_{n}}({|{\psi}\rangle})$
is

$|(HP)_{n}\backslash\mathcal{C}_{n}/\mathcal{S}_{\mathcal{C}_{n}}({|{\psi}\rangle})|=\frac{1}{|(HP)_{n}||\mathcal{S}_{\mathcal{C}_{n}}({|{\psi}\rangle})|}\sum_{(h,s)\in(HP)_{n}\times\mathcal{S}_{\mathcal{C}_{n}}({|{\psi}\rangle})}|\mathcal{C}_{n}^{(h,s)}|,$
(43)

where $\mathcal{C}_{n}^{(h,s)}$ is defined by Eq. (11).

Applying Eq. (43) when ${|{\psi}\rangle}$ is a stabilizer state dramatically
reduces the number of possible entropy vectors that can be reached under
$\mathcal{C}_{n}$. Specifically, when restricting to group action by
$(HC)_{1,2}$, Eq. (43) computes the vertex count for each of the five
contracted graphs shown in Figures 3 – 12.

In this subsection we provided an upper bound on the number of entropy vectors
that can be generated by any Clifford circuit, at arbitrary qubit number. For
a generic quantum state, we showed that the number of possible entropy vectors
is bounded by the size of the right coset space
$\mathcal{C}_{n}\backslash(HP)_{n}$. Alternatively, for states stabilized by
additional elements in $\mathcal{C}_{n}$, the number of possible entropy
vectors is bounded by the size of the double coset space
$(HP)_{n}\backslash\mathcal{C}_{n}/\mathcal{S}_{\mathcal{C}_{n}}({|{\psi}\rangle})$.

##  5 From Entropic Diversity to Holographic Interpretation

The contracted graphs in Section 4 illustrate the diversity of entropy vectors
on $(HC)_{1,2}$ and $\mathcal{C}_{2}$ reachability graphs. We now analyze this
entropic diversity as we move towards a holographic interpretation of our
results. We begin by considering the maximum number of different entropy
vectors that can be found on each of the $(HC)_{1,2}$ and $\mathcal{C}_{2}$
graphs studied in the above section, as well as the minimum number of qubits
needed to realize that maximal diversity. We explore the implications of
entropic diversity and graph diameter as constraining the transformations of a
geometric gravitational dual in holography. We then present the number of
$(HC)_{1,2}$ subgraphs, including isomorphic subgraphs with different entropic
diversities, as we increase qubit number. We remark how our contracted graphs
encode information about entropy vector evolution through entropy space.

###  5.1 Clifford Gates in Holography

The AdS/CFT conjecture Maldacena:1997re  is a bulk/boundary duality which
relates gravitational objects in an asymptotically hyperbolic spacetime,
evaluated at some fixed timeslice $\Sigma$, with computable properties of a
quantum-mechanical system on the boundary of that spacetime $\partial\Sigma$.
For a special class of quantum states known as holographic states, the Ryu-
Takayanagi formula relates all components of the state’s entropy vector to
areas of extremal surfaces in the dual gravity theory Ryu:2006bv ;
Faulkner:2013ana . In this way, a description of the spacetime geometry in
$\Sigma$ is inherited from knowledge of the entanglement structure on
$\partial\Sigma$. For this relation to hold, holographic states are required
to have an entropy vector structure which satisfies a set of holographic
entropy inequalities Bao2015 ; HernandezCuenca2019 . One holographic
inequality, the monogamy of mutual information (MMI) Hayden2013 , reads

$S_{AB}+S_{AC}+S_{BC}\geq S_{A}+S_{B}+S_{C}+S_{ABC},$ (44)

and must be satisfied for all131313It is important to note that each
$A,B,C\subseteq\partial\Sigma$ may separately correspond to the disjoint union
of multiple qubits in the $n$-party boundary theory. Accordingly, the MMI
inequality in Eq. (44) must hold for disjoint triples $\\{A,B,C\\}$, as well
as those of the form $\\{AB,C,DE\\}$ or $\\{ABC,DE,F\\}$, and so on.
Furthermore, holographic states must saturate or satisfy MMI for all
permutations among any chosen $A,B,C\subseteq\partial\Sigma$.
$A,B,C\subseteq\partial\Sigma$. While MMI constitutes only one of many
holographic entropy inequalities, it arises at four qubits, while all other
holographic inequalities require more parties.

Understanding the entropy-vector dynamics of a state in $\partial\Sigma$ gives
insight into bulk geometric transformations in $\Sigma$. When a local operator
acts on ${|{\psi}\rangle}$ and modifies its entropy vector to another vector
within the holographic entropy cone, geodesics in the dual spacetime geometry
are likewise modified in accordance with the RT formula. Consequently,
analyzing how a group of operators transforms the entropy vector of a state
can reveal how gate action on $\partial\Sigma$ alters geometries in $\Sigma$.
When a sequence of Clifford gates causes the state to violate holographic
inequalities, the geometry may be only a semi-classical approximation.

The distance between vertices on reachability graphs encodes a natural notion
of circuit complexity. Entropy vectors which populate the same reachability
graph, e.g. under $(HC)_{1,2}$ or $\mathcal{C}_{2}$, may be considered close
in the sense that a limited number of gate applications is required to
transform a state with one entropy vector into some state with another. The
gravitational dual geometries of states with “nearby” entropy vectors may be
considered close in a similar sense, since a small number of manipulations are
needed to transform one dual geometry into each other.

Some $n$-qubit stabilizer states have entropy vectors which violate the
holographic entropy inequalities, beginning at $n=4$. Since stabilizer
entanglement is generated by bi-local gates, $2$-qubit Clifford operations are
sufficient to generate all stabilizer entropy vectors in an $n$-qubit system.
We can therefore explore the transition from holographic entropy vectors to
non-holographic stabilizer entropy vectors by observing entropy vector
evolution under $\mathcal{C}_{2}$. In the following subsections we discuss how
entropic diversity on $(HC)_{1,2}$ and $\mathcal{C}_{2}$ reachability graphs
can inform us about states which are geometrically close, and not so close, in
the dual gravitational theory.

###  5.2 Maximal Entropic Diversity for Stabilizer States

Each $(HC)_{1,2}$ and $\mathcal{C}_{2}$ reachability graph describes the full
orbit of some state ${|{\psi}\rangle}\in\mathcal{H}$ under the action of
$(HC)_{1,2}$ or $\mathcal{C}_{2}$ respectively. While we can construct
reachability graphs for an arbitrary $n$-qubit quantum state, including states
with arbitrary entanglement structure, the set of possible entropy vectors
that can be reached under $(HC)_{1,2}$ and $\mathcal{C}_{2}$ remains bounded
at the operator level. For a given reachability graph, we refer to the maximum
number of possible entropy vectors that can be generated in that graph as the
maximal entropic diversity of the graph.

Table 5.2 gives each stabilizer state $(HC)_{1,2}$ reachability graph, and the
maximal entropic diversity determined by its contracted graph. For certain
subgraphs, such as $g_{144},\,g_{288},$ and $g_{1152}$, the number of qubits
needed to realize the maximal entropic diversity is higher than the number of
qubits at which each graph first appears.

$(HC)_{1,2}$ Graph |  Max Entropic |  |
---|---|---|---
|  Stab. Qubit |  |
|  Stab. Qubit Num. |  |
| Max Diversity |  |
$g_{24}$ | $2$ | $2$ | $2$
$g_{36}$ | $2$ | $2$ | $2$
$g_{144}$ | $5$ | $3$ | $6$
$g_{288}$ | $5$ | $3$ | $6$
$g_{1152}$ | $18$ | $4$ |  $7$ or $8$

Table 2: Stabilizer state $(HC)_{1,2}$ graphs listed alongside their maximal
entropic diversities, set by contracted graphs. We give the qubit number when
each graph is first observed for stabilizer states, and the minimum qubit
number needed to realize the maximal entropic diversity for stabilizer states.
We have found $g_{1152}$ graphs with maximal diversity for $8$-qubit
stabilizer states, but have not completely ruled out a maximally diverse
$g_{1152}$ graph at $7$ qubits since an exhaustive search is computationally
difficult.

The entropy vectors on $g_{24}$ and $g_{36}$ correspond to maximal and minimal
$2$-qubit entanglement, and can therefore be achieved by entangling only $2$
qubits in an $n$-party system. These two entropy vectors are close in the
sense that they are connected by a single $C_{1,2}$ action. Since this single
gate acts on only $2$ out of the $n$ qubits, we expect states with these
entropy vectors to admit close dual (possibly semi-classical) geometries.
Analogously, altering only small segments of the boundary of a holographic
state will affect its geometry only inside the entanglement wedge of the union
of these segments.

For larger reachability graphs, the graph diameter upper bounds the
$(HC)_{1,2}$ gate distance, and thus the geometric closeness, of the included
entropy vectors. In particular, $g_{1152}$ is the $(HC)_{1,2}$ reachability
graph for generic quantum states, and its maximal entropic diversity gives an
upper bound on the number of distinct entropy vectors, and thus the number of
distinct semi-classical geometries, reachable under $(HC)_{1,2}$ action.

We additionally compile the entropic diversity data for all stabilizer state
$\mathcal{C}_{2}$ reachability graphs. As shown throughout Section 4, every
$\mathcal{C}_{2}$ graph is a complex of $(HC)_{1,2}$ subgraphs attached by
$P_{1}$ and $P_{2}$ edges. Table 5.2 lists the different $\mathcal{C}_{2}$
complexes, and the maximal entropic diversity of each.

$\mathcal{C}_{2}$ Graph |  Max Entropic |  |
---|---|---|---
|  Stab. Qubit |  |
|  Stab. Qubit Num. |  |
| Max Diversity |  |
$g_{24}+g_{36}$ | $2$ | $2$ | $2$
$3\cdot g_{144}+g_{288}$ | $5$ | $3$ | $6$
$10\cdot g_{1152}$ | $20$ | $4$ |  $7$ or $8$

Table 3: Each stabilizer state $\mathcal{C}_{2}$ graph, built of attached
$(HC)_{1,2}$ subgraphs. Each graph is listed alongside its maximal entropic
diversity, set by its contracted graph. We give the first time each graph
appears as a stabilizer state orbit, and the first time each graph achieves
maximal entropic diversity for stabilizer states.

The addition of $P_{1}$ and $P_{2}$ enables two more entropy vectors to be
reached by states in a $g_{1152}$ subgraph. Although this section has so far
concentrated on the stabilizer states, the $10\cdot g_{1152}$
$\mathcal{C}_{2}$ complex is actually the generic reachability graph for
arbitrary quantum states, which are not stabilized by any non-identity element
of a given two-qubit Clifford group. Accordingly, the $20$ entropy vectors in
this complex constrain the possible unique entropy vectors that can be
generated by starting with a generic quantum state and acting with $2$-qubit
Clifford operations.

In this subsection we provided Tables 5.2–5.2 which detailed the maximal
entropic diversity of each stabilizer state $(HC)_{1,2}$ and $\mathcal{C}_{2}$
reachability graph. Additionally, we provided the minimal system size needed
to realize each maximal entropic diversity in a stabilizer state orbit. Note
that for other quantum states with the same reachability graphs, maximal
entropic diversity could be achieved at lower qubit numbers. We speculated
that the maximal entropic diversity of reachability graphs constrains the
available transformations, and that the graph diameter constrains the
dissimilarity, of the dual geometries that can be generated from $(HC)_{1,2}$,
or $\mathcal{C}_{2}$, action on the boundary state. In the next subsection we
analyze the number, and diversity, of stabilizer state reachability graphs as
the number of qubits in the system increases.

###  5.3 $\mathcal{C}_{2}$ Subgraph Count by Qubit Number

The number of times each stabilizer state $\mathcal{C}_{2}$ reachability graph
in Section 4 occurs in the set of $n$-qubit stabilizer states increases with
every qubit added to the system. Furthermore, as we increase qubit number we
observe different entropic diversities which are possible on $\mathcal{C}_{2}$
reachability graphs. Table 4 gives a count for each variety of stabilizer
state $\mathcal{C}_{2}$ graph, with increasing qubit number, for $n\leq 5$
qubits.

|  $\mathcal{C}_{2}$ Graph
---|---
Qubit # | $g_{24}/g_{36}$ | $g_{144}/g_{288}$ | $g_{1152}$
2 | 1 (2) | 0 | 0
3 | 6 (2) | 1 (3) | 0
4 | 60 (2) | 12 (3), 18 (4) | 1 (2), 9 (4)
5 | 1080 (2) | 180 (3), 1080 (4) | 18 (2), 216 (4), 486 (6), 540 (7)
Table 4: Distribution of stabilizer state $\mathcal{C}_{2}$ reachability
graphs, and their different entropic diversities, for $n\leq 5$ qubits. The
first number in each cell gives the number of occurrences for each
$\mathcal{C}_{2}$ subgraph, while the number in parentheses gives the entropic
diversity of each subgraph variation.

The overall count of each $\mathcal{C}_{2}$ subgraph increases as the size of
the system grows. Graph $g_{1152}$ however, shown in the final column of Table
4, has an occurrence count which increases the fastest with qubit number. As
expected, when the system size grows large the percentage of states stabilized
by any non-identity $2$-qubit Clifford subgroup decreases.

Subgraphs $g_{144}/g_{288}$ can have an entropic diversity of $3,\,4,$ or $5$,
while states in a $g_{1152}$ $\mathcal{C}_{2}$ complex can reach up to $20$
different entropy vectors. As qubit number increases the number of
entanglement possibilities grows, yielding more complex entropy vectors.
Entropy vectors with sufficient complexity will change the maximal number of
allowed times under $\mathcal{C}_{2}$ action. We therefore expect the number
of $g_{144}/g_{288}$ graphs with $5$ entropy vectors, and $g_{1152}$
$\mathcal{C}_{2}$ graphs with $20$ entropy vectors, to dominate the subgraph
occurrence count in the large system limit. For larger subgraphs, e.g. those
composed of $g_{1152}$ subgraphs, understanding the precise distribution of
entropic diversity for arbitrary qubit number presents a challenging problem,
which we leave for future work. We now conclude this section with a discussion
of Dicke state entropic diversity in $(HC)_{1,2}$ and $\mathcal{C}_{2}$
reachability graphs.

###  5.4 Maximum Entropic Diversity for Dicke States

We now analyze the entropic diversity of the Dicke state
${|{D^{n}_{k}}\rangle}$ reachability graphs in Section 4.4. Subgraphs
$g_{288^{*}}$ and $g_{576}$ correspond to the two possible
${|{D^{n}_{k}}\rangle}$ orbits under $(HC)_{1,2}$ action, shown in Figures
13–15. Under the full action of $\mathcal{C}_{2}$, $P_{1}$ and $P_{2}$ edges
attach copies of $g_{288^{*}},g_{576},$ and $g_{1152}$ together, creating the
graph complexes seen in Figures 14–16. In Table 5.4 we present the maximal
entropic diversity of each ${|{D^{n}_{k}}\rangle}$ $(HC)_{1,2}$ and
$\mathcal{C}_{2}$ reachability graph, as determined by their contracted
graphs.

Graph |  Max Entropic |  |
---|---|---|---
|  First Appears |  |
|  Max Diversity |  |
| for ${|{D^{n}_{k}}\rangle}$ |  |
$g_{288^{*}}$ | $5$ | $3$ | $5$
$2\cdot g_{288^{*}}+2\cdot g_{576}+g_{1152}$ | $6$ | $3$ | $5$
$g_{576}$ | $9$ | $4$ | $6$
$4\cdot g_{576}+3\cdot g_{1152}$ | $10$ | $4$ | $6$

Table 5: All $(HC)_{1,2}$ reachability graphs (rows $1$ and $3$) and
$\mathcal{C}_{2}$ reachability graphs (rows $2$ and $4$) for Dicke states. We
give the maximal entropic diversity of each graph, as set by the contracted
graph, as well as the first time the graph appears for Dicke states and the
largest entropic diversity achieved among ${|{D^{n}_{k}}\rangle}$ states. For
$\mathcal{C}_{2}$ graphs in particular, we never observe a
${|{D^{n}_{k}}\rangle}$ orbit that achieves the maximum number of allowed
entropy vectors.

Both $\mathcal{C}_{2}$ reachability graphs in Table 5.4 do not achieve their
maximal entropic diversities as orbits of Dicke states. We expect that a state
with sufficiently general entanglement structure, which also shares one of
these reachability graphs141414Recall that the reachability graphs in Table
5.4 are shared by all states with stabilizer group given by Eqs. (37) or (38),
and are not restricted to ${|{D^{n}_{k}}\rangle}$ orbits. Since the entropy
vector is a state property, the state structure determines entropy vector
complexity and therefore how much an entropy vector can change under some
group action., would realize the maximum allowed number of distinct entropy
vectors, though we have not shown this explicitly. In Section 6 we speculate
on the highly symmetric structure of ${|{D^{n}_{k}}\rangle}$ entropy vectors
as a potential cause for the maximal diversity not being achieved in such
graphs.

In this section we analyzed the entropic diversity of reachability graphs
studied throughout Section 4. We detailed each reachability graph achieves its
maximal entropic diversity, and speculate implications for the geometric
interpretations of state entropy vectors in a dual gravity theory. We
demonstrated how certain $(HC)_{1,2}$ and $\mathcal{C}_{2}$ subgraphs appear
more frequently with increasing qubit number, as well as how different
entropic variations of each subgraph are distributed when the system size
grows large. We addressed the notable case of Dicke state reachability graphs,
which do not achieve their maximal entropic diversity as orbits of
${|{D^{n}_{k}}\rangle}$. We will now conclude this work with an overview of
our results and some ideas for future research.

##  6 Discussion and Future Work

In this work we presented a procedure for quotienting a reachability graph to
a contracted graph, which allowed us to analyze and bound entropy vector
evolution under group action on a Hilbert space. We first constructed a
reachability graph, built as a quotient of the group Cayley graph
Keeler:2023xcx , for a family of states defined by their stabilizer subgroup
under the chosen group action. As a group-theoretic object, the vertex set of
a reachability graph is the left coset space generated by the stabilizer
subgroup for the family of states. We then further quotiented this
reachability graph by identifying all vertices connected by edges that
preserve the entropy vector of a state. This second graph quotient corresponds
to the right coset space generated by the subgroup of elements which leave an
entropy vector invariant. The resultant object, after both graph quotients, is
a contracted graph. This contracted graph represents the double coset space
built of group elements which simultaneously stabilize a family of states, and
do not modify an entropy vector.

A contracted graph encodes the evolution of a state entropy vector under group
action. Specifically, the number of vertices in a contracted graph strictly
bounds the maximal number of distinct entropy vectors that can be found on a
reachability graph. The edges of a contracted graph detail the possible
changes an entropy vector can undergo through circuits composed of the group
generating set. We built contracted graphs for all stabilizer states under the
action of $(HC)_{1,2}$ and $\mathcal{C}_{2}$, and demonstrated how the vertex
count of each explains the reachability graph entropy distributions observed
in our previous work Keeler2022 ; Keeler:2023xcx .

Although we did derive a general upper bound on the number of different
entropy vectors that can be reached using any $n$-qubit Clifford circuit
starting from an arbitrary quantum state, much of our work focused on
$\mathcal{C}_{2}$ contracted graphs. However, we could use the same techniques
to extend our analysis to $\mathcal{C}_{n}$, for $n\geq 3$, increasing our
generating gate set for additional qubits. In fact, a presentation for
$\mathcal{C}_{n}$ is proposed in Selinger2013 , using Clifford relations up
through $3$ qubits. Understanding precisely how contracted graphs scale with
qubit number might offer tighter constraints on achievable entropy vectors in
$\mathcal{C}_{n}$ circuits, and enable us to study more general entropy vector
transformations. In AdS/CFT, we only expect systems with arbitrarily large
numbers of qubits to be dual to smooth classical qubits. Consequently, an
improved understanding of large-qubit-number contracted graph behavior would
strengthen the connection to previous holographic entropy cone work, and could
even yield insights for spacetime reconstruction efforts.

While our work in this paper has focused on Clifford circuits, the contracted
graph protocol can be applied equally to circuits composed of alternative gate
sets (for example, generators of crystal-like subgroups of $SU(N)$ such as
$\mathbb{BT}$ Gustafson:2022xdt ). When the chosen gate set generates a finite
group of operators, the associated Cayley graph will be finite, as will any
graph quotients. For all such cases, a contracted graph analysis follows
exactly as in Section 4, and can be used accordingly to bound entropy vector
evolution in different circuit architectures. By exploring different circuit
constructions, we can precisely tune our analysis to focus on operations which
may be preferred for specific experiments, e.g. arbitrary rotation gate
circuits, constructions which replace multiple CNOT gates with Toffoli gates,
and architectures that deliberately avoid gates which are noisy to implement.

Alternatively, if the chosen gate set is finite, but generates an infinite set
of operators, we can impose a cutoff at some arbitrary fixed circuit depth.
This cutoff truncates the associated Cayley graph, and enables an extension of
our methods toward the study of universal quantum circuits up to finite
circuit depth. Even without an imposed cutoff, we could use our graph analysis
to establish bounds on the rate of entanglement entropy per gate application.
This description is reminiscent of the notion of entanglement “velocity” in
universal quantum circuits Couch:2019zni ; Munizzi_2022 .

Although we were originally interested in entropy vector evolution under some
chosen gate set, our techniques are sufficiently general to study the
evolution of any state property (see footnote 5). Of immediate interest, for
example, is the amount of distillable quantum magic present in a state
Bravyi2004 ; Bao:2022mkc , and how this particular measure of non-
stabilizerness changes throughout a quantum circuit. Since magic is preserved
up to Clifford group action, one subgroup which leaves the amount of magic in
a state invariant is exactly the set $\mathcal{C}_{n}$.

In Section 5, we analyzed the maximal entropic diversity of reachability
graphs. A reachability graph has maximal entropic diversity when it realizes
the maximum number of possible entropy vectors permitted by its contracted
graph. We analyzed at which qubit number each $(HC)_{1,2}$ and
$\mathcal{C}_{2}$ reachability graph achieves maximal entropic diversity for
stabilizer states, and remarked on the growth of entropic diversity with
increasing qubit number.

Since contracted graphs are defined at the operator level, we are also able to
extend our analysis to non-stabilizer states. In this paper, we generated all
contracted graphs under $(HC)_{1,2}$ and $\mathcal{C}_{2}$ for $n$-qubit Dicke
states, a class of non-stabilizer states heavily utilized in optimization
algorithms Cerezo:2020jpv ; Niroula:2022wvn . For these states, we derived an
upper bound on the number of different entropy vectors that can exist in Dicke
state $(HC)_{1,2}$ and $\mathcal{C}_{2}$ reachability graphs. Interestingly,
we have not observed $\mathcal{C}_{2}$ graphs achieve a maximal entropic
diversity for Dicke states (see Figures 14–16). The contracted graphs of
$g_{288^{*}}$ and $g_{576}$ permit $6$ and $10$ unique entropy vectors
respectively, but we have only ever witnessed $5$ and $9$ entropy vectors for
Dicke states with these graphs. We suspect the reason no Dicke state orbit
attains its permitted maximal entropy diversity is due to additional
$\mathcal{C}_{2}$ elements which stabilize specifically the highly symmetric
entropy vectors of Dicke states Schnitzer:2022exe ; Munizzi:2023ihc .

In the body of this work, we connected our analysis of entropic diversity to
the holographic framework, where entropy vectors admit a description as
geometric objects in a dual gravity theory. We used our entropic diversity
results to speculate about constraints on geometric transformations in the
dual gravity theory, for states which are holographic or near-holographic. We
interpret a contracted graph as a coarse-grained map of an entropy vector’s
trajectory, through entropy space, under a set of quantum gates. Thus,
contracted graphs provide information about moving in entropy space, and
thereby moving between different entropy cones.

In future work, we plan to study precisely which Clifford operations move a
holographic entropy vector out of, and back into, the holographic entropy
cone. Furthermore, we will explore Clifford circuits that transition a
stabilizer entropy vector from satisfying, to saturating, to failing
holographic entropy conditions, particularly including the monogamy of mutual
information (MMI). We plan to concentrate on MMI since every explicit
stabilizer state we have checked either satisfies all holographic
inequalities, or violates at least one MMI condition. While _a priori_ we have
no reason to expect that all stabilizer states which are not holographic
necessarily violate MMI in particular, in practice we observe this to be the
case empirically for $n\leq 6$ qubits.

###### Acknowledgements.

The authors thank ChunJun Cao, Zohreh Davoudi, Temple He, Sergio Hernandez-
Cuenca, Bharath Sambasivam, Howard Schnitzer, Aaron Szasz, and Claire Zukowski
for helpful discussions. CAK and WM are supported by the U.S. Department of
Energy under grant number DE-SC0019470 and by the Heising-Simons Foundation
“Observational Signatures of Quantum Gravity” collaboration grant 2021-2818.
JP is supported by the Simons Foundation through _It from Qubit: Simons
Collaboration on Quantum Fields, Gravity, and Information_.

##  Appendix A Tables of Entropy Vectors

Below we include sets of entropy vectors referenced throughout the paper. The
states used to generate each entropy vector set are likewise given in bit-
address notation. A bit-address is the ordered set of coefficients multiplying
each basis ket of an n-qubit system, e.g. the bit-address $(1,0,0,1,0,0,i,i)$
indicates the state
${|{000}\rangle}+{|{011}\rangle}+i{|{110}\rangle}+i{|{111}\rangle}$. We order
index qubits within each ket from right to left, i.e. the rightmost digit
corresponds to the first qubit of the system, while the leftmost digit
represents the $n^{\text{th}}$ qubit of an $n$-qubit system.

###  A.1 Entropy Vectors for $6$-Qubit Stabilizer Graphs

Reachability graphs $g_{144}$ and $g_{288}$, shown in Figures 7–10, can be
generated by the action of $(HC)_{1,2}$ or $\mathcal{C}_{2}$ on the $6$-qubit
state in Eq. (45).

$\begin{split}\frac{1}{8}(1,&-1,1,1,-1,1,1,1,1,-1,1,1,1,-1,-1,-1,1,-1,-1,-1,-1,1,-1,-1,-1,1,1,\\\
&1,-1,1,-1,-1,-1,1,1,1,1,-1,1,1,-1,1,1,1,-1,1,1,-1,-1,-1,1,-1,1,1,1,\\\
&\qquad 1,-1,1,-1,-1,-1,1,1,1)\\\ \end{split}$ (45)

There are $5$ distinct entropy vectors that can be reached in the orbit of Eq.
(45) under $(HC)_{1,2}$ and $\mathcal{C}_{2}$, given in Table 6. The colors in
the table correspond to the vertex colors in Figures 7–10.

Label | Entropy Vector
---|---
| $\left(1,0,1,1,1,1,1,2,2,2,2,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2\right)$
| $\left(0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2\right)$
| $\left(1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,2\right)$
| $\left(1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,2,2,3,3\right)$
| $\left(1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,2,3,3,2,3\right)$
Table 6: Table of the $5$ entropy vectors found on $g_{144}$ and $g_{288}$
reachability graphs in Figures 7–10. Colors in the leftmost column correspond
to the vertex colors of these figures.

###  A.2 Entropy Vectors for $8$-Qubit $g_{1152}$

To construct the reachability graphs shown in Figure 11–12, we consider the
orbit of the $8$-qubit state in Eq. (46) under the action of $(HC)_{1,2}$ and
$\mathcal{C}_{2}$.

$\begin{split}\frac{1}{\sqrt{32}}(0,&0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,\\\
&\quad 0,1,0,-i,0,-1,0,-i,0,0,0,0,0,0,0,0,i,0,-1,0,-i,0,-1,0,0,0,0,0,0,0,\\\
&\quad\quad
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-i,0,\\\
&\quad\quad
0,0,0,0,0,0,0,0,-1,0,-i,0,0,i,0,-1,0,0,0,0,0,0,0,0,0,-i,0,-1,0,0,0,\\\
&\quad\quad\quad
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-i,0,\\\
&\quad\quad\quad\quad-1,0,0,0,0,0,0,0,0,0,i,0,-1,-1,0,-i,0,0,0,0,0,0,0,0,0,1,0,-i,0,\\\
&\quad\quad\quad\quad
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,\\\
&\quad\quad\quad\quad
0,0,0,0,i,0,1,0,-i,0,1,0,0,0,0,0,0,0,0,0,0,1,0,i,0,-1,0,i,0,0,0,0)\\\
\end{split}$ (46)

The entropy vectors generated along the $(HC)_{1,2}$ and $\mathcal{C}_{2}$
orbits of Eq. (46) are given in Figure 17. The color preceding each entropy
vector corresponds to the vertex coloring in Figures 11–12.

Figure 17: All $8$-qubit entropy vectors reached in the orbit of Eq. 46 under
the action of $\mathcal{C}_{2}$. Of these $20$ entropy vectors, $18$ can be
generated with $(HC)_{1,2}$ alone.

###  A.3 Entropy Vectors for W-State and Dicke States

The orbit of ${|{D^{3}_{1}}\rangle}$ under $(HC)_{1,2}$ and $\mathcal{C}_{2}$
reaches $5$ entropy vectors, built of $4$ different entangle entropy values.
We define these $4$ unique entropy values in Eq. (47).

$\begin{split}s_{0}&\equiv 1,\\\
s_{1}&\equiv\frac{2}{3}\log_{2}\left[\frac{3}{2}\right]+\frac{1}{3}\log_{2}\left[3\right],\\\
s_{2}&\equiv\frac{5}{6}\log_{2}\left[\frac{6}{5}\right]+\frac{1}{6}\log_{2}\left[6\right],\\\
s_{3}&\equiv\frac{3-\sqrt{5}}{6}\log_{2}\left[\frac{6}{3-\sqrt{5}}\right]+\frac{3+\sqrt{5}}{6}\log_{2}\left[\frac{6}{3+\sqrt{5}}\right],\end{split}$
(47)

The specific entropy vectors encountered in the $(HC)_{1,2}$ and
$\mathcal{C}_{2}$ orbit of ${|{D^{3}_{1}}\rangle}$ are given in Table 7. Each
entropy vector is built from the entanglement entropies given in Eq. 47.
Numerical approximations for each entropy vector were provided in Figure 13
when each first appeared.

Label | Entropy Vector
---|---
| $(s_{1},\,s_{1},\,s_{1})$
| $(s_{3},\,s_{1},\,s_{1})$
| $(s_{1},\,s_{3},\,s_{1})$
| $(s_{0},\,s_{0},\,s_{1})$
| $(s_{2},\,s_{2},\,s_{1})$
Table 7: Table showing the $5$ entropy vectors seen in Figures 13 and 14,
reached in the orbit of ${|{D^{3}_{1}}\rangle}$ under $(HC)_{1,2}$ and
$\mathcal{C}_{2}$. For clarity, we introduce variables in Eq. (47) to
succintly present each entropy vector.

Similarly for the orbit of ${|{D^{4}_{2}}\rangle}$ under $(HC)_{1,2}$ and
$\mathcal{C}_{2}$, we observe $6$ different entropy vectors. Following the
notation of Munizzi:2023ihc , we give these $6$ entropy vectors in terms of
their $5$ distinct entanglement entropy components, which we list in Eq. (48).

$\begin{split}s_{0}&\equiv\frac{5}{6}\log_{2}\left[\frac{12}{5}\right]+\frac{1}{6}\log_{2}\left[12\right],\\\
s_{1}&\equiv\frac{3-\sqrt{5}}{6}\log_{2}\left[\frac{12}{3-\sqrt{5}}\right]+\frac{3+\sqrt{5}}{6}\log_{2}\left[\frac{12}{3+\sqrt{5}}\right],\\\
s_{2}&\equiv\frac{2}{3}\log_{2}\left[\frac{3}{2}\right]+\frac{1}{3}\log_{2}\left[6\right],\\\
s_{3}&\equiv\frac{3-2\sqrt{2}}{6}\log_{2}\left[\frac{12}{3-2\sqrt{2}}\right]+\frac{3+2\sqrt{2}}{6}\log_{2}\left[\frac{12}{3+2\sqrt{2}}\right],\\\
s_{4}&\equiv 1,\\\
s_{5}&\equiv\frac{2}{3}\log_{2}\left[\frac{3}{2}\right]+\frac{1}{3}\log_{2}\left[3\right],\\\
s_{6}&\equiv\frac{5}{6}\log_{2}\left[\frac{6}{5}\right]+\frac{1}{6}\log_{2}\left[6\right].\end{split}$
(48)

The $5$ entropies in Eq. (48) build the $6$ entropy vectors in Table 8.

Label | Entropy Vector
---|---
| $\left(s_{4},\,s_{4},\,s_{4},\,s_{4},\,s_{2},\,s_{2},\,s_{2}\right)$
| $\left(s_{6},\,s_{5},\,s_{4},\,s_{4},\,s_{2},\,s_{1},\,s_{1}\right)$
| $\left(s_{5},\,s_{6},\,s_{4},\,s_{4},\,s_{2},\,s_{1},\,s_{1}\right)$
| $\left(s_{4},\,s_{4},\,s_{4},\,s_{4},\,s_{2},\,s_{0},\,s_{0}\right)$
| $\left(s_{4},\,s_{4},\,s_{4},\,s_{4},\,s_{2},\,s_{2},\,s_{2}\right)$
| $\left(s_{6},\,s_{6},\,s_{4},\,s_{4},\,s_{2},\,s_{3},\,s_{3}\right)$
Table 8: The $6$ entropy vectors in the orbit of ${|{D^{4}_{2}}\rangle}$ under
$(HC)_{1,2}$ and $\mathcal{C}_{2}$. The vectors appears in Figures 15 and 16,
and are built using the variables in Eq. (48).

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