# Unitary k-designs from random number-conserving quantum circuits

Sumner N. Hearth  Michael O. Flynn  Anushya Chandran  Chris R. Laumann
Department of Physics, Boston University, 590 Commonwealth Avenue, Boston,
Massachusetts 02215, USA

###### Abstract

Local random circuits scramble efficiently and accordingly have a range of
applications in quantum information and quantum dynamics. With a global $U(1)$
charge however, the scrambling ability is reduced; for example, such random
circuits do not generate the entire group of number-conserving unitaries. We
establish two results using the statistical mechanics of $k$-fold replicated
circuits. First, we show that finite moments cannot distinguish the ensemble
that local random circuits generate from the Haar ensemble on the entire group
of number-conserving unitaries. Specifically, the circuits form a
$k_{c}$-design with $k_{c}=O(L^{d})$ for a system in $d$ spatial dimensions
with linear dimension $L$. Second, for $k<k_{c}$, the depth $\tau$ to converge
to a $k$-design scales as $\tau\gtrsim kL^{d+2}$. In contrast, without number
conservation $\tau\gtrsim kL^{d}$. The convergence of the circuit ensemble is
controlled by the low-energy properties of a frustration-free quantum
statistical model which spontaneously breaks $k$ $U(1)$ symmetries. The
associated Goldstone modes are gapless and lead to the predicted scaling of
$\tau$. Our variational bounds hold for arbitrary spatial and qudit
dimensions; we conjecture they are tight.

##  I Introduction

(a)

[width=]tikz/bricklayer-single

(b)

[width=]tikz/bricklayer

Figure 1:  (a) A number-conserving brick-layer circuit in $d=1$ spatial
dimensions with $L=5$ qudits, $t=3$ layers and $b=2$ body gates. The conserved
number (red circles) spreads under the action of the circuit. Each gate
$U_{g}$ is independently and uniformly sampled from the set of local number-
conserving unitaries. A layer unitary $U_{l}$ is the product of $U_{g}$ for
each gate $g$ in a layer, and $U_{t}$ is the product of $t$ layers. (b)
Replica model for the $k=3$ moment operator with $k=3$ replicas and
$*$-replicas. Each replica and $*$-replica carries a separately conserved
number current.

Consider a system of $L^{d}$ qudits arranged on a $d$-dimensional lattice with
a conserved $U(1)$ charge $\hat{N}$. Let the qudits evolve according to a
local unitary number-conserving circuit (see Fig. 1(a)) . When the gate
unitaries are drawn uniformly at random, such circuits have a range of
applications in quantum information and scrambling theory Knill  _et al._
(2008); Harrow and Low (2009); Hayden and Preskill (2007); Brown and Fawzi
(2015); Fisher  _et al._ (2023); McCulloch  _et al._ (2023); Agrawal  _et al._
(2022); Scott (2008); Brown and Fawzi (2013). For example, they provide
minimal models for local unitary dynamics with conservation laws, permitting
the study of operator hydrodynamics and scrambling Rakovszky  _et al._ (2018);
Khemani  _et al._ (2018); Gharibyan  _et al._ (2018); Chan  _et al._ (2018);
Žnidarič (2020); Rakovszky  _et al._ (2021); Fisher  _et al._ (2023). They
also enable randomized benchmarking protocols for measuring the purity of
systems with number conservation such as ultracold atoms van Enk and Beenakker
(2012); Elben  _et al._ (2018); Brydges  _et al._ (2019), and produce output
distributions from which it is difficult to sample Boixo  _et al._ (2018);
Bouland  _et al._ (2018); Aaronson and Gunn (2020).

As the depth $t$ of the random circuit grows, the circuit unitary $U_{t}$
executes a random walk on the group of number-conserving unitaries
$U(\mathcal{H}|\hat{N})$ acting on Hilbert space $\mathcal{H}$. The key
question in the above applications boils down to how ‘scrambling’ the circuit
is – for our purposes, how close the distribution of $U_{t}$ is to a steady-
state Haar distribution on $U(\mathcal{H}|\hat{N})$. This can be quantified in
several ways as these are very high dimensional distributions. Luckily,
physical applications typically only depend on low order moments of the
circuit unitary. These are naturally organized by the $k$’th _moment operator_
,

$\displaystyle\hat{T}_{k}^{U_{t}}$ $\displaystyle\equiv\int
dU_{t}\,p(U_{t})\left(U_{t}\otimes U_{t}^{*}\right)^{\otimes k}$ (1)

which encodes all of the $(k,k)$’th order moments 111Unbalanced moments in the
matrix elements of $U_{t}$ and $U_{t}^{*}$ are encoded by
$\hat{T}_{k,\bar{k}}^{U_{t}}$. These can be non-zero for sufficiently large
$k$, or $\bar{k}$ with number conservation, see Sec. V. in the matrix elements
of $U_{t}$ and $U_{t}^{*}$. Here, $p(U_{t})$ is the probability distribution
of $U_{t}$; for the Haar measure on a unitary group $G$, we also write
$\hat{T}_{k}^{G}$ in a small abuse of notation.

If $\hat{T}_{k}^{U_{t}}=\hat{T}_{k}^{G}$ then the ensemble of random unitaries
$U_{t}$ is said to form an _exact $k$-design_ on $G$ – the distribution of
$U_{t}$ is statistically indistinguishable from the Haar measure up to $k$’th
order moments Roberts and Yoshida (2017); Gross  _et al._ (2007). Typically,
finite depth local circuits do not form exact designs; rather, they form
_approximate $k$-designs_ which converge in the large depth limit

$\displaystyle\hat{T}_{k}^{U_{t}}$
$\displaystyle\xrightarrow{t\to\infty}\hat{T}_{k}^{G}.$ (2)

In the absence of number conservation, for $G=U(\mathcal{H})$, this limit is
well-studied Harrow and Low (2009); Diniz and Jonathan (2011); Brandão  _et
al._ (2016); Haferkamp (2022); Nakata  _et al._ (2017); Harrow and Mehraban
(2023); Hunter-Jones (2019); Brandão  _et al._ (2021); Haferkamp  _et al._
(2022); Low (2010); Jian  _et al._ (2022); Emerson  _et al._ (2005); Deshpande
_et al._ (2022).

Number conservation affects the limit, Eq. (2), in two ways. First, the group
generated by number-conserving $b$-body circuits of arbitrary depth,
$U_{b}(\mathcal{H}|\hat{N})$, is a smooth proper subgroup of
$U(\mathcal{H}|\hat{N})$ with codimension $O(L^{d})$ Marvian (2022). Thus, the
moment operators for these two groups must differ for sufficiently large $k$.
In particular, for arbitrary $k$, Eq. (2) holds for the Haar measure on
$U_{b}(\mathcal{H}|\hat{N})$, not $U(\mathcal{H}|\hat{N})$. Second, charge
transport is diffusive in a local number-conserving circuit. The depth
required for convergence should thus grow at least as quickly as $L^{2}$. In
contrast, $b$-body circuits without conservation laws converge to the full
unitary group $U(\mathcal{H})$ with depth scaling with $L$ in $d=1$ Harrow and
Low (2009); Diniz and Jonathan (2011); Brandão  _et al._ (2016); Nakata  _et
al._ (2017); Harrow and Mehraban (2023); Hunter-Jones (2019).

We present two main results in this article. The first, in Sec. V, is that all
finite moments are identical for the Haar measures on
$U_{b}(\mathcal{H}|\hat{N})$ and $U(\mathcal{H}|\hat{N})$ as $L\to\infty$.
More precisely, we show that for $b$-body circuits with $b\geq 2$,

$\displaystyle\hat{T}_{k}^{U_{b}(\mathcal{H}|\hat{N})}$
$\displaystyle=\hat{T}_{k}^{U(\mathcal{H}|\hat{N})},\quad k<k_{c}$ (3)

with $k_{c}\geq L^{d}$. Finite moments _cannot_ detect that $b$-body circuits
diffuse on $U_{b}(\mathcal{H}|\hat{N})$ rather than $U(\mathcal{H}|\hat{N})$
in the thermodynamic limit.

Second, in Sec. VI, we show that $\hat{T}_{k}^{U_{t}}$ converges to
$\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}$ exponentially with a rate $\Delta$
limited by diffusion,

$\displaystyle\hat{T}_{k}^{U_{t}}$
$\displaystyle\approx\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}+\hat{C}e^{-\Delta
t}+\cdots,\qquad\Delta\lesssim\frac{1}{L^{2}}$ (4)

This follows from interpreting $\hat{T}_{k}^{U_{t}}$ as a replicated quantum
statistical mechanical model (see Fig. 1(b)) whose low energy properties
govern the late time convergence. The frustration-free ground state of the
replicated model spontaneously breaks $k$ $U(1)$ symmetries. The rate $\Delta$
in Eq. (4) is thus bounded by the gap to the lowest wavenumber Goldstone mode
(Fig. 2(b)). Finally, putting these results together in Sec. VII, we find that
random number-conserving circuits form $\varepsilon$-approximate $k$-designs
on $U(\mathcal{H}|\hat{N})$ at a depth $\tau$ (See Fig. 2(c)), with

$\displaystyle\tau\gtrsim
kL^{d+2}\ln(Q)+\frac{1}{2}L^{2}\ln(\frac{k}{\varepsilon^{2}})$ (5)

where the $\sim$ hides geometry-dependent constants and $Q$ is the total
Hilbert space dimension of each qudit.

The paper proceeds as follows. Sec. II presents the high-level spontaneous
symmetry breaking analysis that leads to Eq. (4). Sec. III collects
definitions and symbols used throughout the paper. Sec. IV reviews general
properties of the Haar measure and the moment operators for number-conserving
unitary groups. Sec. V presents the first of our main results: all finite
moments of $U_{b}(\mathcal{H}|\hat{N})$ and $U(\mathcal{H}|\hat{N})$ are
identical in the thermodynamic limit. Sec. VI analyzes the low energy
properties of the replicated circuit model and an associated Hamiltonian
model, both of which are frustration-free. There we prove our second main
result: $\hat{T}_{k}^{U_{t}}$ converges exponentially to
$\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}$ with a rate $\Delta\sim 1/L^{2}$. Sec.
VII translates the convergence rate $\Delta$ into a lower bound on the depth,
$\tau$, beyond which the circuit is an approximate unitary $k$-design. We
conclude with a discussion in Sec. VIII.

(a)

[width=]tikz/bands-noncon

(b)

[width=]tikz/bands-con

(c)

[width=]tikz/simple-slope

Figure 2:  The low energy spectrum of the replica model (a) without and (b)
with number conservation. Dashes on the x-axis mark the discrete set of
momenta at finite size, and the red dots mark the corresponding excitation
energies. Without number conservation, the ground state spontaneously breaks
the $S_{k}\times S_{k}$ symmetry down to a diagonal subgroup $S_{k}$. The
domain wall excitations are gapped. With number conservation, the ground state
additionally spontaneously breaks the $U(1)^{k}\times U(1)^{k}$ symmetry down
to a diagonal $U(1)^{k}$ subgroup. The resulting $k$ gapless Goldstone modes
having dynamical exponent $z=2$, owing to the frustration-freeness of the
model. This leads to a finite-size gap $\Delta\sim L^{-2}$. (c) The spectral
gap $\Delta$ sets the timescale for convergence to an approximate $k$-design.
The convergence time $\tau$ is the depth required to reach an error, or
diamond norm difference, of $\varepsilon$ (Sec. VII)

##  II Overview of Replica Model

In what follows, we present the arguments that lead to the second main result,
Eq. (4), in more detail. The moment operator for $U_{t}$ is a product of
moment operators on layers, each of which is product of moment operators on
gates,

$\displaystyle\hat{T}_{k}^{U_{t}}$
$\displaystyle=\prod_{\mathrm{Layers~{}}l}\underbrace{\left(\prod_{\text{Gates
}g\in
l}\hat{T}_{k}^{U_{g}}\right)}_{\hat{T}_{k}^{U_{l}}}=\left(\hat{T}_{k}^{U_{l}}\right)^{t}$
(6)

Clearly, $\hat{T}_{k}^{U_{l}}$ plays the role of a transfer matrix in the
treatment of the replicated system as a quantum statistical mechanical model
in $d+1$ dimensions Nahum  _et al._ (2017); Zhou and Nahum (2019). The
convergence of $\hat{T}_{k}^{U_{t}}$ is thus governed by the maximum
eigenvalue of $\hat{T}_{k}^{U_{l}}$ and the spectral gap below it.

To make progress, we make several observations. First, each gate operator,
$\hat{T}_{k}^{U_{g}}$, is a Hermitian projector, with eigenvalues $+1$ and
$0$. This elementary result holds for the moments of any Haar distributed
unitary (see Sec. IV for a review of general properties of the moment operator
for Haar ensembles). As a product of projectors, the maximum possible
eigenvalue of the transfer matrix $\hat{T}_{k}^{U_{l}}$ is $+1$, carried by
steady states which are simultaneous $+1$ eigenstates of _all_ of the gate
projectors $\hat{T}_{k}^{U_{g}}$. So long as this joint steady state space is
non-empty, $\hat{T}_{k}^{U_{t}}$ converges to a finite projector, as expected
in Eq. (2).

While it is possible to further analyze $\hat{T}_{k}^{U_{l}}$ directly (see
Sec. VI.2), it is instructive to consider the closely related layer
Hamiltonian of the replicated system,

$\displaystyle\hat{H}_{k}$ $\displaystyle=\sum_{\text{Gates~{}}g\in
l}(1-\hat{T}_{k}^{U_{g}})$ (7)

This Hamiltonian is _frustration-free_ : its zero energy ground state space is
the joint zero energy ground state space of each term – that is, the ground
state space of $\hat{H}_{k}$ is the steady state space of
$\hat{T}_{k}^{U_{l}}$. Technically, there are two advantages to shifting our
attention to the Hamiltonian $\hat{H}_{k}$ rather than $\hat{T}_{k}^{U_{l}}$.
First, while $\hat{H}_{k}$ has all of the spatial structure of the gates in a
circuit layer, it does not keep track of their ordering between sublayers as
in Fig. 1(b), which simplifies its analysis. We do not expect this to
qualitatively matter to physical properties of the low temperature limit,
where the layers are iterated many times; indeed, we find it does not. Second,
it leads us to interpret the steady states of the replicated system in terms
of spontaneous symmetry breaking in the ground states of the frustration-free
Hamiltonian $\hat{H}_{k}$.

Let us then turn to the symmetries of the replicated system. As in all replica
treatments, the replicated system has $S_{k}\times S_{k}$ replica (and
$*$-replica) permutation symmetry. In the case of unitary circuits without
symmetries, it is well known that the replicated system spontaneously breaks
the $S_{k}\times S_{k}$ symmetry down to a diagonal subgroup $S_{k}$ Zhou and
Nahum (2019). More precisely, the ground state space of $\hat{H}_{k}$ is
spanned by states labeled by permutations $\sigma\in S_{k}$ which bind the
state on replica $\alpha$ to that on $*$-replica $\sigma(\alpha)$,

$\displaystyle\ket{\sigma}$
$\displaystyle=\sum_{\boldsymbol{i},\bar{\boldsymbol{i}}}\underbrace{\left(\prod_{\alpha=1}^{k}\delta[\bar{i}_{\alpha}=i_{\sigma(\alpha)}]\right)}_{\delta[\bar{\boldsymbol{i}}=\sigma(\boldsymbol{i})]}\ket{\boldsymbol{i},\bar{\boldsymbol{i}}}=\sum_{\boldsymbol{i}}\ket{\boldsymbol{i},\sigma(\boldsymbol{i})}$
(8)

where $i_{\alpha}$ ($\bar{i}_{\alpha}$) runs over basis states for the
$\alpha$ replica ($*$-replica). In the simplest case, when the dimension of
the Hilbert space that each gate acts on is larger than $k$, these states are
linearly independent and we obtain a ground state space with dimension $k!$.
As a discrete symmetry breaking phase in a lattice system, the Hamiltonian
$\hat{H}_{k}$ has a spectral gap $\Delta\sim O(L^{0})$ for large $L$;
accordingly, $\hat{T}_{k}^{U_{t}}$ converges exponentially with an order one
timescale. The lowest lying states above the gap are well-described by single
domain wall excitations, see Fig. 2(a).

Returning to the case of a number-conserving circuit, each replica (and
$*$-replica) additionally carries its own conserved charge associated to the
replicated $U(1)$ symmetry – the system has a global $U(1)^{k}\times U(1)^{k}$
symmetry. This breaks spontaneously down to a diagonal subgroup $U(1)^{k}$,
which is intertwined with the choice of replica permutation symmetry breaking.
Explicitly, the ground states of $\hat{H}_{k}$ are spanned by the symmetry-
breaking vacua,

$\displaystyle{\ket{\boldsymbol{\Phi};\sigma}}$
$\displaystyle=e^{i\sum_{\alpha}\Phi_{\alpha}\hat{N}_{\alpha}}\ket{\sigma}$
(9)

where $\boldsymbol{\Phi}$ is a collection of $k$ phases, one to each replica.
Although the parameters $\boldsymbol{\Phi}$ vary continuously, they span a
ground state space of dimension $k!M^{k}$ where $M$ is the number of global
charge sectors in the system (again in the simplest case for small enough
$k$); indeed, a basis for these states may be obtained by projecting the
parent ${\ket{\boldsymbol{\Phi};\sigma}}$ states into the global charge
sectors (Sec. IV.3).

As $\hat{H}_{k}$ spontaneously breaks a continuous symmetry, it harbors $k$
Goldstone modes associated with long-wavelength fluctuations of the phases
$\boldsymbol{\Phi}$, see Fig. 2(b). Such excitations can be created with
momentum $q$ above any symmetry breaking vacuum using the modulated number
operator,

$\displaystyle\hat{n}^{\alpha}_{q}$ $\displaystyle=\sum_{x}e^{-iq\cdot
x}\hat{n}^{\alpha}_{x}$ (10)

where $\hat{n}^{\alpha}_{x}$ is the local number operator at qudit $x$ in
replica $\alpha$. The excitation energy of these modes provides a rigorous
upper bound on the spectral gap $\Delta$ of $\hat{H}_{k}$. Indeed,
$\Delta\lesssim\frac{1}{L^{2}}$, as expected for Goldstone modes in
frustration-free systems, independent of $k$, spatial dimension $d$, and local
Hilbert space structure.

The frustration free Hamiltonian $\hat{H}_{k}$ has been previously studied in
two special cases in $d=1$ Gharibyan  _et al._ (2018); Barratt  _et al._
(2022). In Ref. Gharibyan  _et al._ (2018), the qudit on each site is composed
of a charged qubit and a neutral qudit with an infinitely large Hilbert space
dimension. Ref. Barratt  _et al._ (2022) considers the case of qubits on each
site. In both cases, $\hat{H}_{k}$ is a nearest neighbor Heisenberg model in
each replica. The nearest neighbor Heisenberg model is special because it has
an enlarged $SU(2)$ symmetry and is integrable in one dimension. Indeed, Ref.
Gharibyan  _et al._ (2018) obtained the spectral gap from the integrability
literature. Our work shows that $\hat{H}_{k}$ can be derived for _any_ local
qudit structure in any dimension, and in general, need not be integrable or
have an enlarged symmetry. It is however frustration-free and spontaneously
breaks the replicated $U(1)$-symmetry, which guarantees $\Delta\lesssim
1/L^{2}$.

Although we have sketched the arguments here for the associated Hamiltonian
system, Eq. (7), the physics is the same in the discrete time system defined
by Eq. (6). Sec. VI.2 deals with the mathematical tweaks required to analyze
the discrete time case.

##  III Definitions and Notation

We consider a system of $V=L^{d}$ qudits in $d$ spatial dimensions with linear
dimension $L$. Under the $U(1)$ symmetry, each qudit’s local Hilbert space
$\mathcal{H}^{(1)}$ can be decomposed into $M^{(1)}$ number sectors,

$\displaystyle\mathcal{H}^{(1)}$
$\displaystyle=\bigoplus_{n=0}^{M^{(1)}-1}\mathcal{H}^{(1)}_{n}.$ (11)

We denote the local Hilbert space dimension of number sector $n$ as
${d^{(1)}_{n}}=|\mathcal{H}^{(1)}_{n}|$. For example, in a lattice model of
single-orbital spin-$1/2$ fermions, the local Hilbert space may have $0,1$ or
$2$ particles and we write $d^{(1)}=(1,2,1)$. In a spin-$1/2$ system with
$S^{z}$ conservation, $d^{(1)}=(1,1)$.

The local dimensions $d^{(1)}$ fully specify the decomposition of the multi-
qudit Hilbert space. For example, a pair of qudits has joint Hilbert space
$\mathcal{H}^{(2)}$, which can be similarly decomposed into number sectors

$\displaystyle\mathcal{H}^{(2)}=\mathcal{H}^{(1)}\otimes\mathcal{H}^{(1)}$
$\displaystyle=\sum_{n=0}^{M^{(2)}-1}\mathcal{H}^{(2)}_{n}$ (12)
$\displaystyle{d^{(2)}_{n}}\equiv|\mathcal{H}^{(2)}_{n}|=\sum_{n_{1}+n_{2}=n}{d^{(1)}_{n_{1}}}{d^{(1)}_{n_{2}}}.$
(13)

The Hilbert space of the $V$ qudits is denoted by
$\mathcal{H}=\mathcal{H}^{(V)}$, with ${\mathcal{D}}=|\mathcal{H}|$ (we omit
sub/superscripts to denote properties of a complete set of qudits).
$\mathcal{H}$ and ${\mathcal{D}}$ may in turn be broken into global number
sectors as $\mathcal{H}=\sum_{n}\mathcal{H}_{n}$ and
${\mathcal{D}}=\sum_{n}d_{n}$ with $d_{n}=|\mathcal{H}_{n}|$.

We take the standard basis $\ket{i}$ to be compatible with the decomposition
of $\mathcal{H}$ into number sectors and qudits. That is, we write the global
number operator

$\displaystyle\hat{N}=\sum_{x}\hat{n}_{x}$ (14)

where $x$ runs over spatial sites. Then, the site-resolved number operators
are diagonal

$\displaystyle\hat{n}_{x}\ket{i}=n_{x}(i)\ket{i}.$ (15)

We consider circuits with brick-layer architectures as in Fig. 1(b). Each of
the $t$ layers is obtained by stacking a finite number of sub-layers of gates
acting on all even bonds along each spatial dimension, and odd bonds along
each spatial dimension. The ordering of the sub-layers is not important. Thus,
for example, in $d=1$, each layer is composed of two sub-layers (Fig. 1(b)),
while in $d=2$, there are four sub-layers and the resulting interaction graph
is that of a square lattice. The total number of layers is denoted by $t$.
Although the geometry of each layer is repeated, the particular gates $U_{g}$
are not – they are each independently sampled from the uniform measure on the
group of number-conserving 2-body gates.

Table 1 provides a comprehensive symbol glossary.

Name | Symbol | Notes
---|---|---
Number of qudits | $V=L^{d}$ |
Number of global number sectors | $M$ |
Spatial dimension | $d$ |
Dimension of $n$th global number sector | $d_{n}$ |
Number of qudits acted on by a gate | $b$ |
Replica index | $\alpha$ | $\alpha\in 1\cdots k$
Qudit index | $x$ | $x\in 1\cdots V$
Number sector index | $n$ | $n\in 0\cdots M-1$
Number-conserving unitary group |  $U(\mathcal{H}|\hat{N})$ | Eq. (28)
Unitaries realizable from arbitrary depth $b$-body circuits |  $U_{b}(\mathcal{H}|\hat{N})$ |  Sec. V
Moment operator for random unitary $U$ | $\hat{T}_{k}^{U}$ | Eq. (1)
Moment operator for Haar measure on group $G$ | $\hat{T}_{k}^{G}$ | Eq. (17)
Moment operator for layer/transfer matrix | $\hat{T}_{k}^{U_{l}}$ | Eq. (6)
Hamiltonian of associated replica model | $\hat{H}_{k}$ | Eq. (53)
Gap of replica model | $\Delta$ | Eq. (4)
Time to converge to approximate design | $\tau$ | Sec. VII
Standard basis of Hilbert space $\mathcal{H}$ | $\ket{i}$ | Sec. III
Standard basis of replica space $\left(\mathcal{H}\otimes\mathcal{H}^{*}\right)^{\otimes k}$ | $\ket{\boldsymbol{i},\boldsymbol{\bar{i}}}$ | Sec. III
Permutation state | $\ket{\sigma}$ | Eq. (23)
Number-permutation state | ${\ket{\boldsymbol{n};\sigma}}$ | Eq. (32)
Phase-permutation state | ${\ket{\boldsymbol{\Phi};\sigma}}$ | Eq. (9)
Table 1: Symbol glossary. Vectors (eg. $\boldsymbol{i}$) have components
across $k$ replicas ($i_{\alpha}$). Bar indicates $*$-replicas.

##  IV Moments of Global Unitary Groups

We briefly review key properties of the Haar ensemble on compact unitary
(matrix) groups $G$ acting on a Hilbert space $\mathcal{H}$. We begin with
features shared by all compact unitary groups and then turn to the cases of
the general unitary group $U(\mathcal{H})$ and the number-conserving unitary
group $U(\mathcal{H}|\hat{N})$. The results on $U(\mathcal{H}|\hat{N})$ apply
directly to the individual gates in the local circuit model.

###  IV.1 General Properties

Suppose $G$ is a compact group of unitaries acting on the Hilbert space
$\mathcal{H}$ with dimension ${\mathcal{D}}$. This could be the general
unitary group $U(\mathcal{H})$ or some compact subgroup of it, such as
$U(\mathcal{H}|\hat{N})$. The Haar measure on $G$ is the unique normalized
measure which is invariant under both left- and right- shifts, as well as
inversion Diestel and Spalsbury (2014). That is, for arbitrary functions $f$
on $G$ and elements $U,V\in G$,

$\displaystyle{\int dUf(U)}={\int dUf(VU)}={\int dUf(UV)}={\int dUf(U^{-1})}$
(16)

These invariance properties completely characterize the measure.

#### The Moment Operator—

Moments of $U$ are encoded in the moment operator

$\displaystyle\hat{T}_{k,\bar{k}}^{G}$ $\displaystyle\equiv{\int
dU\left(U^{\otimes k}\otimes(U^{*})^{\otimes\bar{k}}\right)}$ (17)

which acts on the $(k+\bar{k})$-fold replicated Hilbert space
$\mathcal{H}^{\otimes k}\otimes\left(\mathcal{H}^{*}\right)^{\otimes\bar{k}}$.

When the number of starred and unstarred replicas are unequal, $k\neq\bar{k}$,
the moment operator is called unbalanced. In many cases, the unbalanced moment
operators are identically zero. Thus, we largely focus on the balanced moment
operators, $\hat{T}_{k}^{G}\equiv\hat{T}_{k,k}^{G}$ and only discuss
unbalanced moments when we need to show that they vanish (see Sec. IV.2).

The matrix elements of the moment operator encode the ${\mathcal{D}}^{4k}$
moments of order $(k,k)$ of the matrix elements of $U$ and $U^{*}$. For
arbitrary compact unitary groups $G$, $\hat{T}_{k}^{G}$ is a _Hermitian
projector_ – a fact that follows directly from the unitarity of $G$ and the
invariance properties in Eq. (16). Explicitly, shift-invariance implies

$\displaystyle\left(\hat{T}_{k}^{G}\right)^{2}$ $\displaystyle=\int
dUdV\left(UV\otimes U^{*}V^{*}\right)^{\otimes k}$ (18)
$\displaystyle=\left(\int dV\right)\hat{T}_{k}^{G}$
$\displaystyle=\hat{T}_{k}^{G}.$

while inversion-invariance gives

$\displaystyle\left(\hat{T}_{k}^{G}\right)^{\dagger}$
$\displaystyle=\left(\int dU(U\otimes U^{*})^{\otimes k}\right)^{\dagger}$
(19) $\displaystyle=\int dU(U^{-1}\otimes U^{-1*})^{\otimes k}$
$\displaystyle=\hat{T}_{k}^{G}.$

Eq. (18) and (19) together imply $\hat{T}_{k}^{G}$ is a Hermitian projector,
with eigenvalues of $+1$ and $0$.

As a projector, $\hat{T}_{k}^{G}$ is completely characterized by its $+1$
eigenspace. Below, we construct these spaces explicitly for the Haar measure
on several unitary groups. First, we review a few useful general
relationships.

#### Restricted Measures and Subgroups—

Suppose unitary $V$ is sampled according to an arbitrary non-Haar measure on
$G$. The moment operator for $V$, $\hat{T}_{k}^{V}$, need not be a projector.
Nonetheless, shift-invariance of the Haar measure implies that
$\hat{T}_{k}^{V}\hat{T}_{k}^{G}=\hat{T}_{k}^{G}\hat{T}_{k}^{V}=\hat{T}_{k}^{G}$.
Less abstractly, the $+1$ eigenstates of the Haar moment projector are also
$+1$ eigenstates of $\hat{T}_{k}^{V}$.

In particular, if $V$ is sampled from the Haar measure on a compact subgroup
$H\subset G$, $\hat{T}_{k}^{V}$ projects onto a superspace of
$\hat{T}_{k}^{G}$. This is useful for constructing the $+1$ eigenspace of
restricted unitary groups like $U(\mathcal{H}|\hat{N})$, as we will see in
Sec. IV.3.

#### Replica Symmetry—

From its definition, the moment operator $\hat{T}_{k}^{G}$ is clearly
symmetric under the $S_{k}\crossproduct S_{k}$ separate permutations of
replicas and $*$-replicas. Indeed, replica permutation symmetry holds for any
ensemble on $G$. To make this explicit and set some notation, let us introduce
an orthonormal basis for the $k$ replicas labeled by
$\boldsymbol{i}=(i_{1}\cdots i_{k})$, and similarly for the $k$ $*$-replicas
$\bar{\boldsymbol{i}}=(\bar{i}_{1}\cdots\bar{i}_{k})$, where each
$i_{\alpha},\bar{i}_{\alpha}=1,\dots\mathcal{D}$. $\hat{T}_{k}^{G}$ acts on
states
$\ket{\boldsymbol{i},\bar{\boldsymbol{i}}}\in\left(\mathcal{H}\otimes\mathcal{H}^{*}\right)^{\otimes
k}$. Consider permutations $\sigma,\tau\in S_{k}$ and associated operator
$\hat{R}_{(\sigma,\tau)}=\sum_{\boldsymbol{i},\bar{\boldsymbol{i}}}\ket{\sigma(\boldsymbol{i}),\tau(\bar{\boldsymbol{i}})}\bra{\boldsymbol{i},\bar{\boldsymbol{i}}}$.
The action on $\hat{T}_{k}^{G}$ can be understood,

$\displaystyle\bra{\boldsymbol{j},\bar{\boldsymbol{j}}}\hat{R}_{(\sigma,\tau)}^{-1}\hat{T}_{k}^{G}$
$\displaystyle\hat{R}_{(\sigma,\tau)}\ket{\boldsymbol{i},\bar{\boldsymbol{i}}}$
$\displaystyle=\int
dU\prod_{\alpha}U_{i_{\sigma(\alpha)}j_{\sigma(\alpha)}}\prod_{\beta}U^{*}_{\bar{i}_{\tau(\beta)}\bar{j}_{\tau(\beta)}}$
$\displaystyle=\bra{\boldsymbol{j},\bar{\boldsymbol{j}}}\hat{T}_{k}^{G}\ket{\boldsymbol{i},\bar{\boldsymbol{i}}}.$
(20)

Rearranging the matrix elements in the integrand has no effect on the ensemble
average, and the moment operator is invariant under replica and $*$-replica
permutations.

#### Associated Hamiltonian—

Replica symmetry implies that the $+1$ eigenspace must be invariant under
$S_{k}\times S_{k}$ permutations, though the action inside that space may be
non-trivial. Indeed, a useful perspective is provided by considering the
replica-symmetric Hamiltonian whose ground state space is the $+1$ eigenspace
of $\hat{T}_{k}^{G}$,

$\displaystyle\hat{H}_{k}=1-\hat{T}_{k}^{G}$ (21)

If the subgroup $\mathcal{S}\subseteq S_{k}\times S_{k}$ leaves a particular
ground state of $\hat{H}_{k}$ invariant, then we can say that the $S_{k}\times
S_{k}$ symmetry is spontaneously broken down to $\mathcal{S}$.

#### Spatial permutation symmetry—

Suppose the Hilbert space $\mathcal{H}$ is a tensor product of single qudit
Hilbert spaces, so that the moment operator has $2kV$ incoming and outgoing
qudit ‘legs’, $V$ per replica and $*$-replica. Let the operator
$\hat{L}_{\pi}=\hat{l}_{\pi}^{\otimes 2k}$ apply the same permutation $\pi\in
S_{V}$ to the spatial indices in each replica. Left- and right- shift
invariance of the Haar measure implies that the moment operator has left- and
right- leg permutation symmetry,

$\displaystyle\hat{T}_{k}^{G}\hat{L}_{\pi}=\hat{L}_{\pi}\hat{T}_{k}^{G}=\hat{T}_{k}^{G}.$
(22)

###  IV.2 General Unitary Group $U(\mathcal{H})$

Let us now focus on the moments $\hat{T}_{k}^{U(\mathcal{H})}$ for the general
unitary group $U(\mathcal{H})$ without number conservation. As a consequence
of Schur-Weyl duality Roberts and Yoshida (2017) (see App. A),
$\hat{T}_{k}^{U(\mathcal{H})}$ projects onto the space spanned by the $k!$
permutation states,

$\displaystyle\ket{\sigma}$
$\displaystyle=\sum_{\boldsymbol{i},\bar{\boldsymbol{i}}}\underbrace{\left(\prod_{\alpha=1}^{k}\delta[\bar{i}_{\alpha}=i_{\sigma(\alpha)}]\right)}_{\delta[\bar{\boldsymbol{i}}=\sigma(\boldsymbol{i})]}\ket{\boldsymbol{i},\bar{\boldsymbol{i}}}=\sum_{\boldsymbol{i}}\ket{\boldsymbol{i},\sigma(\boldsymbol{i})}$
(23)

which bind replicas to $*$-replicas. Accordingly,
$\hat{T}_{k}^{U(\mathcal{H})}$ has an operator expansion of the form

$\displaystyle\hat{T}_{k}^{U(\mathcal{H})}$
$\displaystyle=\sum_{\sigma,\tau\in
S_{k}}V^{\sigma\tau}\ket{\sigma}\bra{\tau}.$ (24)

Pictorially (for $k=3$),

$\hat{T}_{k}^{U(\mathcal{H})}$
$\displaystyle=\sum_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tau}\in
S_{k}}V^{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tau}}\left(\leavevmode\hbox
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The coefficients $V^{\sigma\tau}$, better known as the Weingarten functions
Weingarten (1978), are determined by the overlaps of the states $\ket{\sigma}$
onto which $\hat{T}_{k}^{U(\mathcal{H})}$ projects. These are neither
orthogonal nor normalized – indeed, they are not linearly independent for
$k>{\mathcal{D}}$. The formulae for $V^{\sigma\tau}$ are presented in Appendix
B for any $k$ for completeness, but will not be needed in the main text.
Furthermore, the states $\ket{\sigma}$ are linearly independent only if
$k\leq{\mathcal{D}}$. If $k$ is too large we must instead draw from a linearly
independent set of states labeled by $\sigma$ in the set
$S_{k}^{\leq{\mathcal{D}}}\subseteq S_{k}$. The details of this set are left
to Appendix B. Rather than clutter the already burdened notation, we leave the
set from which permutations are drawn implicit.

The states $\ket{\sigma}$ transform non-trivially under the
$S_{k}\crossproduct S_{k}$ replica symmetry:

$\displaystyle\hat{R}_{(\sigma,\tau)}\ket{\gamma}=\ket{\sigma^{-1}\tau\gamma}$
(26)

That is, the space spanned by $\\{\ket{\sigma}\\}$ is invariant under replica
symmetry $S_{k}\times S_{k}$ (as it had to be), but any given state
$\ket{\sigma}$ is invariant only under a subgroup isomorphic to $S_{k}$. In
the language of the ground state space of the Hamiltonian
$\hat{H}_{k}=1-\hat{T}_{k}^{U(\mathcal{H})}$, we say that the $S_{k}\times
S_{k}$ replica symmetry is spontaneously broken down to the diagonal subgroup
$S_{k}$.

The unbalanced moment operators are identically zero, a simple consequence of
shift invariance. Consider left-multiplying by the unitary $V=e^{i\theta}$.

$\displaystyle V^{\otimes
k}(V^{*})^{\otimes\bar{k}}\hat{T}_{k,\bar{k}}^{U(\mathcal{H})}$
$\displaystyle=e^{i\theta{\mathcal{D}}(k-\bar{k})}\hat{T}_{k,\bar{k}}^{U(\mathcal{H})}\equiv\hat{T}_{k,\bar{k}}^{U(\mathcal{H})}$
(27)

As the equality should hold for any $\theta$,
$\hat{T}_{k,\bar{k}}^{U(\mathcal{H})}=0$ if $k\neq\bar{k}$.

###  IV.3 Number-Conserving Unitary Group $U(\mathcal{H}|\hat{N})$

With number conservation, the unitary group on the Hilbert space $\mathcal{H}$
is restricted to those unitaries which commute with the global number
operator,

$\displaystyle U(\mathcal{H}|\hat{N})$ $\displaystyle=\\{U\in
U(\mathcal{H})\,|\,[U,\hat{N}]=0\\}$ (28)

Any unitary in $U(\mathcal{H}|\hat{N})$ may be block decomposed

$\displaystyle U=\bigoplus_{n=0}^{M-1}U_{n}$ (29)

where $U_{n}$ acts unitarily on $\mathcal{H}_{n}$ and is zero for all other
sectors. The group $U(\mathcal{H}|\hat{N})$ is a compact subgroup of
$U(\mathcal{H})$, isomorphic to the direct product of unitary groups on the
$n$-subspaces.

The $U(\mathcal{H}|\hat{N})$ Haar ensemble is the unique shift-invariant
ensemble on the number-conserving unitary group. In particular, if
$U=\bigoplus_{n}U_{n}$ is sampled from the $U(\mathcal{H}|\hat{N})$ Haar
ensemble, then each $U_{n}$ is independently sampled from the Haar ensemble on
$U(\mathcal{H}_{n})$, with $\mathcal{H}_{n}$ the Hilbert space of states with
total number $n$. This allows us to bootstrap results in Sec. IV.2 to the
number-conserving case.

Using Eq. (29), the $k$-th moment operator is:

$\displaystyle\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}={\int
dU\bigoplus_{\boldsymbol{n},\bar{\boldsymbol{n}}}\left(\bigotimes^{k}_{\alpha=1}U_{n_{\alpha}}\otimes
U^{*}_{\bar{n}_{\alpha}}\right)}$ (30)

Here $\alpha$ labels the replicas and $\boldsymbol{n}$, $\bar{\boldsymbol{n}}$
labels the number sector for the replicas and $*$-replicas. Once again, shift-
and inversion-invariance of the Haar measure imply that
$\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}$ acts as a Hermitian projector on
$\left(\mathcal{H}\otimes\mathcal{H}^{*}\right)^{\otimes k}$. In addition to
$S_{k}\times S_{k}$ replica symmetry, $\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}$
is invariant under $2k$ replicated number operators,

$\displaystyle[\hat{T}_{k}^{U(\mathcal{H}|\hat{N})},\hat{N}_{\alpha}]$
$\displaystyle=0$
$\displaystyle[\hat{T}_{k}^{U(\mathcal{H}|\hat{N})},\hat{\bar{N}}_{\alpha}]$
$\displaystyle=0,$ (31)

which generate a $U(1)^{k}\times U(1)^{k}$ symmetry group.

Let us explicitly construct a set of states which span the $k$’th moment space
for $U(\mathcal{H}|\hat{N})$. We begin with the parent permutation states
$\ket{\sigma}$ of Eq. (23); as $U(\mathcal{H}|\hat{N})$ is a subgroup of
$U(\mathcal{H})$, these states are automatically $+1$ eigenstates of the
number-conserving moment operator, Eq. (30). The number symmetry, Eq. (31),
implies that we can act on $\ket{\sigma}$ with any operator constructed from
$\hat{N}_{\alpha}$ and $\hat{\bar{N}}_{\alpha}$ and obtain another $+1$
eigenstate. In particular, the $U(1)^{k}$ _phase states_ ,
${\ket{\boldsymbol{\Phi};\sigma}}=e^{i\sum_{\alpha}\Phi_{\alpha}\hat{N}_{\alpha}}\ket{\sigma}$,
are all $+1$ eigenstates of $\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}$. We note
that the permutation states are invariant under the $k$ generators
$\hat{N}_{\alpha}-\hat{\bar{N}}_{\sigma(\alpha)}$, so that there are only $k$
phases available to parameterize the moment space.

Since they are continuously parameterized, the phase states are clearly
linearly dependent. An alternative, discrete basis may be obtained by
projecting $\ket{\sigma}$ onto the number sectors
$\boldsymbol{n}=(n_{1},\cdots,n_{k})$ of each replica,

$\displaystyle{\ket{\boldsymbol{n};\sigma}}$
$\displaystyle=\Pi_{\boldsymbol{n}}\ket{\sigma}$ (32)
$\displaystyle=\sum_{\boldsymbol{i}}\underbrace{\left(\prod_{\alpha}\delta[N_{\alpha}(i_{\alpha})=n_{\alpha}]\right)}_{\delta[n(\boldsymbol{i})=\boldsymbol{n}]}\ket{\boldsymbol{i},\sigma(\boldsymbol{i})}$

It is straightforward to check that, (i) states in Eq. (32) with different
number labels are orthogonal to one another, (ii) there are $M^{k}k!$ such
states, (iii) the states are linearly independent if $k\leq d_{n}$ for all
$n$, and span the $+1$ eigenspace of $\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}$,
and (iv) a linearly independent and spanning subset can be chosen if $k>d_{n}$
for some $n$. See Appendix C. We thus obtain a Weingarten like representation
of the $k$-th moment operator for $U(\mathcal{H}|\hat{N})$

$\displaystyle\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}$
$\displaystyle=\sum_{\boldsymbol{n}}\sum_{\sigma,\tau}{\ket{\boldsymbol{n};\sigma}}V_{\boldsymbol{n}}^{\sigma\tau}{\bra{\boldsymbol{n};\tau}}.$
(33)

As in the non-conserving case in Sec. IV.2, the coefficients
$V_{\boldsymbol{n}}^{\sigma\tau}$ follow from the inverse of a suitably
restricted overlap matrix for any value of $k$. Explicit formulae follow from
a straightforward generalization of Appendix B.

We note that the states ${\ket{\boldsymbol{\Phi};\sigma}}$ transform non-
trivially under the $U(1)^{k}\times U(1)^{k}$ number symmetry and the
$S_{k}\times S_{k}$ replica symmetry. In the language of the ground state
space of the Hamiltonian $\hat{H}_{k}=1-\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}$,
we say that the $U(1)^{k}\times U(1)^{k}$ number symmetry is spontaneously
broken down to the diagonal subgroup $U(1)^{k}$ which is intertwined with the
diagonal $S_{k}$ chosen by the $S_{k}\times S_{k}$ replica symmetry breaking.

The unbalanced moment operators, number sector by number sector, are
identically zero following the same argument as in Sec. IV.2.

##  V Moments of $b$-body Circuit Ensembles

We now turn our attention to the group of unitaries generated by $b$-body
circuits – that is, circuits of arbitrary depth composed of unitary gates
which act on at most $b$ qudits. Without number conservation, this group is
simply the full unitary group, $U_{b}(\mathcal{H})=U(\mathcal{H})$ Nielsen and
Chuang (2010). As noted in Eq. (2), this implies that the moment operators of
brick-layer random circuits converge to the moment operators of the full
unitary group.

It came as a surprise when Marvian Marvian (2022) showed that number-
conserving $b$-body circuits fail to generate the full group of number-
conserving unitaries, $U(\mathcal{H}|\hat{N})$. Rather, $b$-body circuits
generate a proper subgroup $U_{b}(\mathcal{H}|\hat{N})\subset
U(\mathcal{H}|\hat{N})$. However, since the codimension of
$U_{b}(\mathcal{H}|\hat{N})$ is very small (linear in $L^{d}$) compared to the
full dimension of $U(\mathcal{H}|\hat{N})$ (exponential in $L^{d}$), one might
expect that the moment operators are distinguishable only for rather high-
order moments. Indeed, we argue here that this is the case: in the
thermodynamic limit $L\to\infty$, _all_ finite moments of the $b$-body circuit
ensemble converge to those of the Haar measure on $U(\mathcal{H}|\hat{N})$.
More precisely, the moment index $k$ must exceed a critical value $k_{c}\geq
L^{d}$ to distinguish between the Haar measure on $U(\mathcal{H}|\hat{N})$ and
the Haar measure on $U_{b}(\mathcal{H}|\hat{N})$.

Our argument proceeds in several steps. First, in Sec. V.1, we show that the
moment operators, $\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}$ and
$\hat{T}_{k}^{U_{b}(\mathcal{H}|\hat{N})}$ can be factored into the product of
a projector on the space of $U(1)$ charges and the moment operator,
$\hat{T}_{k}^{\text{SU}(\mathcal{H}|\hat{N})}$, for a related group of special
unitary matrices
$SU(\mathcal{H}|\hat{N})\equiv\prod_{n}^{M}SU(\mathcal{H}_{n})$. This
factorization reveals that $\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}$ and
$\hat{T}_{k}^{U_{b}(\mathcal{H}|\hat{N})}$ differ only in the first factor,
see Eq. (40). In Sec. V.2, we analyze this factor and obtain linear
constraints, Eq. (45), on the number sectors in which the difference can be
nonzero. In Sec. V.3, we turn to the second factor,
$\hat{T}_{k}^{\text{SU}(\mathcal{H}|\hat{N})}$, which provides further
constraints on the number sectors in which the difference is non-zero, Eq.
(48). Finally, in Sec. V.4, we combine these constraints to obtain $k_{c}\geq
L^{d}$.

###  V.1 Decomposition of Moment Operators

Consider a number-conserving unitary $U\in U(\mathcal{H}|\hat{N})$. It may be
block diagonalized,

$\displaystyle U$ $\displaystyle=\bigoplus_{n}e^{i\theta_{n}}\tilde{U}_{n},$
(34)

where $\det\tilde{U}_{n}=1$. This factorization is not unique as one may shift
$\theta_{n}\to\theta_{n}+2\pi/d_{n}$ and simultaneously multiply
$\tilde{U}_{n}$ by a compensating phase. Nonetheless, the Haar measure on
$U(\mathcal{H}|\hat{N})$ factorizes uniformly,

$\int
dU=\int\left(\prod_{n}^{M}\frac{d\theta_{n}}{2\pi}d\tilde{U}_{n}\right)=1$
(35)

From these factorizations,

$\displaystyle\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}$ $\displaystyle=\int
dU\left(\bigoplus_{n}e^{i\theta_{n}}\tilde{U}_{n}\otimes\bigoplus_{\bar{n}}e^{-i\theta_{\bar{n}}}\tilde{U}^{*}_{\bar{n}}\right)^{\otimes
k}$
$\displaystyle=\bigoplus_{\\{n_{\alpha},\bar{n}_{\alpha}\\}}\underbrace{\int\left(\prod_{n}^{M}\frac{d\theta_{n}}{2\pi}\right)\exp{i\sum_{\alpha}\left(\theta_{n_{\alpha}}-\theta_{\bar{n}_{\alpha}}\right)}}_{\hat{P}_{\boldsymbol{n}\bar{\boldsymbol{n}}}}\times\underbrace{\int\left(\prod_{n}^{M}d\tilde{U}_{n}\right)\bigotimes_{\alpha}^{k}\left(\tilde{U}_{n_{\alpha}}\otimes\tilde{U}_{\bar{n}_{\alpha}}^{*}\right)}_{(\hat{T}_{k}^{\text{SU}(\mathcal{H}|\hat{N})})_{{\boldsymbol{n}\bar{\boldsymbol{n}}}}}$
$\displaystyle=\hat{P}~{}\hat{T}_{k}^{\text{SU}(\mathcal{H}|\hat{N})}$ (36)

In the final step, we have exploited that both $\hat{P}$ and
$\hat{T}_{k}^{\text{SU}(\mathcal{H}|\hat{N})}$ are block diagonal in the
$M^{2k}$ number blocks labeled by $\boldsymbol{n},\bar{\boldsymbol{n}}$.
Indeed, both $\hat{P}$ and $\hat{T}_{k}^{\text{SU}(\mathcal{H}|\hat{N})}$ are
moment operators in the sense of Sec. IV.1 for Haar measures on $U(1)^{M}$ and
$SU(\mathcal{H}|\hat{N})$, respectively.

Let us now derive an analogous factorization to Eq. (V.1) for the $b$-body
unitary circuits, $U_{b}(\mathcal{H}|\hat{N})$. As demonstrated in Ref.
Marvian, 2022, the structure of $b$-body unitary circuits imposes a set of
linear constraints among the phases $\\{\theta_{n}\\}$ in (34). We denote by
$\\{\phi_{\ell}\\}$ the set of unconstrained phases which parameterize the
$(b+1)$-dimensional subtorus of $U(1)^{M}$ accessible by $b$-body circuits.
The moment operator $\hat{T}_{k}^{U_{b}(\mathcal{H}|\hat{N})}$ can then be
written as

$\hat{T}_{k}^{U_{b}(\mathcal{H}|\hat{N})}=\hat{P}_{b}\hat{T}_{k}^{\text{SU}(\mathcal{H}|\hat{N})}$
(37)

where

$\displaystyle\hat{P}_{b}$
$\displaystyle=\bigoplus_{\\{n_{\alpha},\bar{n}_{\alpha}\\}}\int\left(\prod_{\ell}^{b+1}\frac{d\phi_{\ell}}{2\pi}\right)\exp{i\sum_{\alpha}\left(\theta_{n_{\alpha}}-\theta_{\bar{n}_{\alpha}}\right)}$
(38)

In this representation, the $\theta_{n}$ are given by,

$\displaystyle\theta_{n}=\sum_{\ell}G_{n\ell}\phi_{\ell}$ (39)

where the integer-valued matrix $G$ encodes the linear embedding of the
$\phi$-torus into the $\theta$-torus. Explicit forms for $G$ are available in
Ref. Marvian (2022), but are unnecessary for our purposes: what is essential
is that $\hat{P}_{b}$ is a projector. This follows either by explicit
evaluation of the Fourier representation Eq. (38) or from regarding
$\hat{P}_{b}$ as a moment operator of the Haar measure on the $\phi$-torus
$U(1)^{b+1}$.

Thus, the difference of the moment operators can be written,

$\hat{T}_{k}^{U_{b}(\mathcal{H}|\hat{N})}-\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}=\left(\hat{P}_{b}-\hat{P}\right)\hat{T}_{k}^{\text{SU}(\mathcal{H}|\hat{N})}$
(40)

As is clear from the definitions in Eqs. (V.1), (38), the factor,
$\hat{P}_{b}-\hat{P}$, only depends on the number sector
$\\{n_{\alpha},\bar{n}_{\alpha}\\}$ and, as we show in Sec. V.2, vanishes
except in special circumstances. In those circumstances, we turn to the second
factor, $\hat{T}_{k}^{\text{SU}(\mathcal{H}|\hat{N})}$, to determine the
minimal $k$ at which the moment operators are distinguishable, Sec. V.3.

###  V.2 First Factor: Number Sector Projectors

The projectors, $\hat{P}_{b}$ and $\hat{P}$, select number blocks in the
replicated space. Without the $b$-body restriction, the projector $\hat{P}$
simply projects onto number-balanced blocks. To make this more precise, define
the counting operators,

$\displaystyle\hat{\\#}_{n}$
$\displaystyle=\sum_{\alpha}\delta[\hat{N}_{\alpha}=n]$
$\displaystyle\hat{\bar{\\#}}_{n}$
$\displaystyle=\sum_{\alpha}\delta[\hat{\bar{N}}_{\alpha}=n]$ (41)

which determine the number of replicas ($*$-replicas) in the number sector
$n$. As these operators are diagonal in number and in our choice of standard
basis, Eq. (15), we will also abuse notation slightly by writing
$\\#_{n}(\boldsymbol{n})$ and $\\#_{n}(\boldsymbol{j})$ for the appropriate
diagonal matrix elements of the operator $\hat{\\#}_{n}$ (and similarly for
$\hat{\bar{\\#}}_{n}$). The Fourier integral defining $\hat{P}$, Eq. (V.1),
imposes that these counts cancel,

$\displaystyle\hat{P}=\prod_{n}^{M}\delta[\hat{\\#}_{n}=\hat{\bar{\\#}}_{n}]$
(42)

as claimed.

To understand the projector $\hat{P}_{b}$, we make use of the shift invariance
of the Haar measure on $U(1)^{b+1}$. Consider the effect of a global 0-body
phase shift $\gamma$, which acts on operators $U\in U(\mathcal{H}|\hat{N})$ as
$U\to e^{i\gamma}U$. This transformation acts entirely on the phase part of
the factorization, Eq. (34), which leads matrix elements of $\hat{P}_{b}$ to
acquire non-trivial phases,

$\displaystyle\bra{\boldsymbol{i},\bar{\boldsymbol{i}}}\hat{P}_{b}\ket{\boldsymbol{i},\bar{\boldsymbol{i}}}\to\exp\left[i\gamma\sum_{n}\left(\\#_{n}(\boldsymbol{i})-\bar{\\#}_{n}(\bar{\boldsymbol{i}})\right)\right]$
$\displaystyle\hfill\times\bra{\boldsymbol{i},\bar{\boldsymbol{i}}}\hat{P}_{b}\ket{\boldsymbol{i},\bar{\boldsymbol{i}}}$
(43)

Similarly, we can implement 1-body phase shifts
$U=\prod_{x,\alpha}e^{i\beta\hat{n}_{x,\alpha}}$, under which matrix elements
transform as

$\displaystyle\bra{\boldsymbol{i},\bar{\boldsymbol{i}}}\hat{P}_{b}\ket{\boldsymbol{i},\bar{\boldsymbol{i}}}\to\exp\left[i\beta\sum_{n}n\left(\\#_{n}(\boldsymbol{i})-\bar{\\#}_{n}(\bar{\boldsymbol{i}})\right)\right]$
$\displaystyle\hfill\times\bra{\boldsymbol{i},\bar{\boldsymbol{i}}}\hat{P}_{b}\ket{\boldsymbol{i},\bar{\boldsymbol{i}}}$
(44)

However, shift invariance of the Haar measure guarantees that both global
phase shifts and 1-body phase shifts leave $\hat{P}_{b}$ unchanged (for $b\geq
1$); equations (V.2) and (V.2) are therefore satisfied for arbitrary
$\gamma,\beta$ only if

$\displaystyle\sum_{n}\left(\\#_{n}-\bar{\\#}_{n}\right)$ $\displaystyle=0$
(45) $\displaystyle\sum_{n}n\left(\\#_{n}-\bar{\\#}_{n}\right)$
$\displaystyle=0.$

The number balanced sectors, or the $+1$ eigenspace of $\hat{P}$, provide a
trivial solution of these constraints, thus $\hat{P}_{b}$ projects onto a
superspace of the image of $\hat{P}$, as expected.

It follows that the moment operators,
$\hat{T}_{k}^{U_{b}(\mathcal{H}|\hat{N})}$ and
$\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}$, are indistinguishable within number-
balanced blocks of the replicated space. Thus, in order to find differences,
we need to consider nontrivial solutions to the constraints Eq. (45). We
return to these constraints to construct a bound on the critical moment index,
$k_{c}$, in Sec. V.4.

###  V.3 Second Factor: Moments of the Special Unitary Group

Consider the moment operator $\hat{T}_{k}^{\text{SU}(\mathcal{H}|\hat{N})}$
for the Haar measure on the special unitary group
$SU(\mathcal{H}|\hat{N})\equiv\prod_{n}^{M}SU(\mathcal{H}_{n})$. The operator
is block-diagonal with $M^{2k}$ blocks, labeled by the replicated numbers,
$\boldsymbol{n}$, and $\bar{\boldsymbol{n}}$. Here, we show that if the block
$(\hat{T}_{k}^{\text{SU}(\mathcal{H}|\hat{N})})_{\boldsymbol{n}\bar{\boldsymbol{n}}}$
is non-zero, then
$\\#_{n}(\boldsymbol{n})-\bar{\\#}_{n}(\boldsymbol{\bar{n}})$ is an integer
multiple of the dimension $d_{n}$ for every $n=0,\cdots,M-1$.

Consider the unitary $W\in SU(\mathcal{H}|\hat{N})$

$\displaystyle W$
$\displaystyle=\exp\left({i\frac{2\pi}{d_{n}}\delta[\hat{N}=n]}\right)$ (46)

which acts as the identity in all number sectors except $n$, in which it
applies a global phase $e^{i2\pi/d_{n}}$. Right shift invariance of the Haar
measure implies,

$\displaystyle\exp\left[i\frac{2\pi}{d_{n}}\left(\\#_{n}(\boldsymbol{j})-\bar{\\#}_{n}(\boldsymbol{\bar{j}})\right)\right]$
$\displaystyle\hat{T}_{k}^{\text{SU}(\mathcal{H}|\hat{N})}\ket{\boldsymbol{j},\bar{\boldsymbol{j}}}$
(47) $\displaystyle\hskip
10.0pt=\hat{T}_{k}^{\text{SU}(\mathcal{H}|\hat{N})}\ket{\boldsymbol{j},\bar{\boldsymbol{j}}}.$

If
$\hat{T}_{k}^{\text{SU}(\mathcal{H}|\hat{N})}\ket{\boldsymbol{j},\bar{\boldsymbol{j}}}$
is nonzero with $\boldsymbol{j}$, $\bar{\boldsymbol{j}}$ in the
$\boldsymbol{n}$, $\bar{\boldsymbol{n}}$ number sector, then we must have that
the phase in Eq. (47) is 1. Therefore,

$\displaystyle\\#_{n}(\boldsymbol{n})-\bar{\\#}_{n}(\boldsymbol{\bar{n}})=c_{n}d_{n}$
(48)

for some integer $c_{n}$.

###  V.4 A Bound on $k_{c}$

According to Sec. V.2, in order to distinguish
$\hat{T}_{k}^{U_{b}(\mathcal{H}|\hat{N})}$ from
$\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}$, we must look in an unbalanced number
block where $\\#_{n}\neq\bar{\\#}_{n}$ for some $n$; this implies that the
constraint Eq. (45) holds non-trivially. Such blocks vanish in the common
factor $\hat{T}_{k}^{\text{SU}(\mathcal{H}|\hat{N})}$ unless they further
satisfy Eq. (48). Combining these two conditions, we find the linear system

$\displaystyle\sum_{n}d_{n}c_{n}$ $\displaystyle=0$ (49)
$\displaystyle\sum_{n}nd_{n}c_{n}$ $\displaystyle=0$

where $c_{n}$ are the integers in Eq. (48).

Any nontrivial vector of integers $\boldsymbol{c}=(c_{0}\cdots c_{M-1})$ which
satisfy the conditions (49) must contain a minimum of three nonzero entries.
With only two, the restricted coefficient matrix always has a full rank and
only admits zero solutions. Translating back to
$\boldsymbol{n},\boldsymbol{\bar{n}}$, there must be at least three unbalanced
number sectors.

Due to Eq. (48), any unbalanced number sector $n$ has
$|\\#_{n}(\boldsymbol{n})-\bar{\\#}_{n}(\boldsymbol{\bar{n}})|=|c_{n}d_{n}|\geq
d_{n}$. This requires $k\geq d_{n}$. Accordingly, the minimum value of $k$ to
distinguish $\hat{T}_{k}^{U_{b}(\mathcal{H}|\hat{N})}$ from
$\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}$ grows at least as quickly as the
dimension of the third-smallest number sector. A worst-case estimate for the
scaling of different number sectors is provided by qubit systems with $S_{z}$
conservation, defined by single-particle dimensions $d^{(1)}=(1,1)$. The two
smallest number sectors correspond to polarized states, $d_{0}=d_{V}=1$. The
third smallest sector, which is degenerate with $d_{1}=d_{V-1}=V$, grows
extensively, corresponding to the number of choices of qubit flips in the
fully polarized states. We therefore conclude that

$\displaystyle k_{c}\geq V$ (50)

in qubit systems.

Any other qudit structure, $d^{(1)}$, leads to at least as large a lower bound
on $k_{c}$, typically exponentially large in $V$.

##  VI Gap Analysis of Circuit Moment Operators

Now we turn to an analysis of the gap of the quantum statistical mechanical
system defined by replica averaging the random circuits of depth $t$, see Fig.
1(b). This gap controls the convergence of the finite depth moments,
$\hat{T}_{k}^{U_{t}}$, to the global Haar moments,
$\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}$. Throughout this section, we assume
$k<k_{c}$ and neglect the distinction between $U(\mathcal{H}|\hat{N})$ and
$U_{b}(\mathcal{H}|\hat{N})$.

We analyze two closely related statistical models in $d+1$ dimensions. The
primary model of interest gives the imaginary time representation of the
moment operator, with partition function,

$\displaystyle Z^{(k)}_{t}$ $\displaystyle=\int dU_{t}~{}p(U_{t})|\Tr
U_{t}|^{2k}=\Tr[\hat{T}_{k}^{U_{t}}].$ (51)

The associated transfer matrix,

$\displaystyle\hat{T}_{k}^{U_{l}}=\prod_{\text{Gates }g\in
l}\hat{T}_{k}^{U_{g}}$ (52)

propagates the system between layers in imaginary time. This model
spontaneously breaks the $U(1)^{k}\times U(1)^{k}$ symmetry down to $U(1)^{k}$
in a frustration-free manner. In Sec. VI.2, we bound the gap of the transfer
matrix due to the associated Goldstone modes.

Before turning to the circuit model, in Sec. VI.1 we analyze the closely
related, but technically simpler, Hamiltonian system

$\displaystyle\hat{H}_{k}$ $\displaystyle=\sum_{\text{Gates~{}}g\in
l}(1-\hat{T}_{k}^{U_{g}})$ (53)

where the sum runs over all gates present in a layer of the replicated
circuit. The ground state space of $\hat{H}_{k}$ is constructed to be
identical to the $+1$ eigenspace of the layer transfer matrix
$\hat{T}_{k}^{U_{l}}$. Again, the Goldstone modes control the Hamiltonian gap,
which in turn controls the large imaginary time properties.

Both models have gaps scaling as $1/L^{2}$. This is the result we use in Sec.
VII to bound the convergence of the circuit to an $\varepsilon$-approximate
$k$-design.

###  VI.1 Variational Bound on Gap of $\hat{H}_{k}$

As discussed in Sec. I, the ground states of $\hat{H}_{k}$ are spanned by the
phase states,

$\displaystyle{\ket{\boldsymbol{\Phi};\sigma}}$
$\displaystyle={\ket{\Phi_{1},\ldots,\Phi_{k};\sigma}}=e^{i\sum_{\alpha}\Phi_{\alpha}\hat{N}_{\alpha}}\ket{\sigma}$
(54)

These states transform non-trivially under $U(1)$ rotations associated with
the conserved charges $\hat{N}_{\alpha}$, and $\hat{\bar{N}}_{\alpha}$,

$\displaystyle
e^{i\left(\theta\hat{N}_{\alpha}+\bar{\theta}\hat{\bar{N}}_{\sigma(\alpha)}\right)}{\ket{\boldsymbol{\Phi};\sigma}}={\ket{\Phi_{1},\ldots,\Phi_{\alpha}+\theta+\bar{\theta},\ldots,\Phi_{k};\sigma}}$
(55)

Thus, of the original $2k$ symmetries, $k$ are unbroken.

Each broken symmetry yields a Goldstone mode. To bound the gap, $\Delta$, we
consider long-wavelength modulations of the phase,

$\displaystyle e^{i\theta\sum_{x}e^{-iqx}\hat{n}^{\alpha}_{x}}\ket{\Psi}$
$\displaystyle=(1+i\theta\hat{n}^{\alpha}_{q}+...)\ket{\Psi}$
$\displaystyle=\ket{\Psi}+i\theta\ket{q_{\alpha}}+...$ (56)

where $\hat{n}_{q}^{\alpha}=\sum_{x}e^{-iqx}\hat{n}_{x}^{\alpha}$ and
$\ket{\Psi}$ is an arbitrary ground state of $\hat{H}_{k}$ and $\theta$ is a
perturbative parameter. The finite momentum states $\ket{q_{\alpha}}$ are
necessarily orthogonal to the ground states of $\hat{H}_{k}$, which have leg-
permutation symmetry (see Eq. (22)) and therefore carry zero momentum. The
variational energy

$\displaystyle\Delta_{q}\equiv\frac{\bra{q_{\alpha}}\hat{H}_{k}\ket{q_{\alpha}}}{\innerproduct{q_{\alpha}}{q_{\alpha}}}.$
(57)

provides an upper bound on the true gap $\Delta$. Note that the replica index,
$\alpha$, is just a spectator and is omitted going forward.

We now evaluate the numerator of (57) using symmetries of $\hat{H}_{k}$,

$\displaystyle\bra{q}\hat{H}_{k}\ket{q}$
$\displaystyle=\sum_{x,y,g}e^{iq(y-x)}\langle\hat{n}_{x}(1-\hat{T}_{k}^{U_{g}})\hat{n}_{y}\rangle.$
(58)

Here and henceforth, $\langle\cdot\rangle$ implicitly denotes an average with
respect to the ground state $\ket{\Psi}$. We further specialize to the case of
$2$-body gates in a nearest neighbor square lattice for simplicity, although
the final result can be readily generalized. Each gate operator
$\hat{T}_{k}^{U_{g}}$ commutes with the total number operator for sites in the
support of that gate. For example, if gate $g$ acts on $x$ and $y$,

$\displaystyle[\hat{T}_{k}^{U_{xy}},\hat{n}_{x}+\hat{n}_{y}]=0.$ (59)

Further, leg permutation symmetry, Eq. (22), and
$\hat{T}_{k}^{U_{xy}}\ket{\Psi}=\ket{\Psi}$ implies an identity for number
operators connected by a gate,

$\displaystyle\hat{T}_{k}^{U_{xy}}\hat{n}_{x}\ket{\Psi}$
$\displaystyle=\frac{1}{2}(\hat{n}_{x}+\hat{n}_{y})\ket{\Psi}.$ (60)

Applying (60) to each term in (58), we remove all dependence on the moment
operator and replica count $k$,

$\displaystyle\langle\hat{n}_{x}\hat{T}_{k}^{U_{xy}}\hat{n}_{x}\rangle=\langle\hat{n}_{x}\hat{T}_{k}^{U_{xy}}\hat{n}_{y}\rangle$
$\displaystyle=\frac{\langle\hat{n}_{x}^{2}\rangle+\langle\hat{n}_{x}\hat{n}_{y}\rangle}{2}.$
(61)

Correlation functions of number operators are strongly constrained by
invariance under translations and permutations:

$\langle\hat{n}_{x}\hat{n}_{y}\rangle=\langle\hat{n}_{0}\hat{n}_{y-x}\rangle=\delta_{x,y}\langle\hat{n}_{0}^{2}\rangle+\left(1-\delta_{x,y}\right)\langle\hat{n}_{0}\hat{n}_{1}\rangle$
(62)

where in the first equality we have applied translation invariance, and in the
second, permutation invariance. Applying this result to Eq. (61), we find

$\displaystyle\bra{q}\hat{H}_{k}\ket{q}$
$\displaystyle=L^{d}\left(\langle\hat{n}_{0}^{2}\rangle-\langle\hat{n}_{0}\hat{n}_{1}\rangle\right)\left(d-\sum^{d}_{\mu}\cos
q_{\mu}\right).$ (63)

We now turn our attention to the denominator of (57). Using the structure of
correlation fuctions in Eq. (62), we find

$\displaystyle\innerproduct{q}{q}$
$\displaystyle=L^{d}\langle\hat{n}_{0}^{2}\rangle+L^{d}\langle\hat{n}_{0}\hat{n}_{1}\rangle(\delta_{q,0}L^{d}-1).$
(64)

Putting (63) and (64) together we find a variational gap,

$\displaystyle\Delta_{q}$ $\displaystyle=d-\sum_{\mu}\cos q_{\mu}.$ (65)

Note that $\Delta_{q}$ is independent of our initial choice of $\ket{\Psi}$ as
well as the qudit structure. One exception, however, are fully polarized
states, which are eigenstates of the number operator on each site and have no
number fluctuations; in these sectors, one cannot construct a state with
nonzero momentum and the variational analysis breaks down. However, it is
clear that in such cases that conservation laws are irrelevant, and the
results of previous studiesHunter-Jones (2019); Jian  _et al._ (2022) predict
an $O(1)$ gap.

All finite momentum states are orthogonal to the ground state manifold and the
variational gap $\Delta_{q}$ provides an upper bound for $\Delta$.
Minimization of $\Delta_{q}$ with respect to $q$ in a finite system provides
the bound on the gap in the square lattice system

$\displaystyle\Delta\leq\frac{2\pi^{2}}{L^{2}}.$ (66)

Careful readers may have recognized that Eq. (65) is the spatial Fourier
transform of the Laplace operator on the square lattice. This is no accident;
on general lattices with higher body interactions, the charge undergoes a
continuous time random walk on the spatial lattice. Consider the imaginary
time correlation function (for $t>0$),

$\displaystyle C_{xy}(t)$
$\displaystyle=\langle\hat{n}_{x}(t)\hat{n}_{y}(0)\rangle$ (67)

where

$\displaystyle\hat{n}_{x}(t)$ $\displaystyle=e^{+tH}\hat{n}_{x}e^{-tH}$ (68)

The correlator satisfies the equation of motion,

$\displaystyle\frac{d}{dt}C_{xy}(t)$
$\displaystyle=\langle[H,n_{x}(t)]n_{y}\rangle=\langle\hat{n}_{x}(t)H\hat{n}_{y}\rangle$
(69)

where in the second equality we have used the frustration free condition. That
is, the right hand side is precisely the numerator of Eq. (57). Using number
conservation and leg permutation symmetries of the gates it is straightforward
to show that $[H,\hat{n}_{x}]$ is a linear combination of
$\hat{n}_{x^{\prime}}$ operators when evaluated within the ground space. This
leads Eq. (69) to close,

$\displaystyle\frac{d}{dt}C_{xy}(t)$
$\displaystyle=-\sum_{x^{\prime}}\Gamma_{x^{\prime},x}C_{x^{\prime}y}(t)$ (70)

where $\Gamma_{x^{\prime},x}$ is a transition rate matrix which satisfies
$\sum_{x^{\prime}}\Gamma_{x^{\prime},x}=0$ in order to conserve number. That
is, we have obtained a continuous time random walk. For any particular choice
of gate geometry, it is straightforward to compute $\Gamma$ explicitly.

For a general short-ranged, translation-invariant Hamiltonian, we can Fourier
transform $\Gamma$ and find a long-wavelength gap which closes as $q^{2}$.
Thus, we obtain

$\displaystyle\Delta\leq\frac{C}{L^{2}}.$ (71)

where $C$ is a geometry dependent factor.

###  VI.2 Variational Gap on $\hat{T}_{k}^{U_{t}}$

We now consider the layer transfer matrix, $\hat{T}_{k}^{U_{l}}$, governing
the primary circuit model. There are two technical complications compared to
the Hamiltonian case. First, as $\hat{T}_{k}^{U_{l}}$ is not Hermitian, it is
not guaranteed to have real eigenvalues. However, it is frustration-free and
has a degenerate $+1$ eigenspace identical to the ground state space of the
Hamiltonian. The relevant gap is thus between the $+1$ eigenvalues and the
real part of the next-largest eigenvalue. Second, the circuit
$\hat{T}_{k}^{U_{l}}$ has structure in time corresponding to the sequence in
which gates are applied. This complicates the mathematical treatment slightly,
but does not change the scaling of the Goldstone gap with $L$.

Recall that the moment operator for depth $t$ circuits can be broken into $t$
layers composed of individual gates obtained by Haar-averaging,

$\displaystyle\hat{T}_{k}^{U_{t}}$
$\displaystyle=\left(\hat{T}_{k}^{U_{l}}\right)^{t}.$ (72)

We write the second-largest-magnitude eigenvalue of $\hat{T}_{k}^{U_{l}}$ as
$\lambda\equiv e^{-\Delta}$, with $\text{Re}(\Delta)>0$. To bound $\Delta$, we
again consider the Goldstone excitations, $\ket{q}=\hat{n}_{q}\ket{\Psi}$. The
reduction in norm of $\ket{q}$ under the action of the transfer matrix sets a
bound,

$\displaystyle\frac{||\hat{T}_{k}^{U_{l}}\ket{q}||^{2}}{\innerproduct{q}{q}}\leq|e^{-2\Delta}|.$
(73)

The denominator is familiar – see Eq. (64).

The numerator of Eq. (73) can be mapped onto a discrete time random walk
reminiscent of the continuous time walk obtained in the Hamiltonian case.
Every step of the walk spreads the operators $\hat{n}_{x}$ equally to the
outgoing legs of each gate, so that the moves of the conserved charge are
determined by the geometry of the circuit. This process is encoded in a
doubly-stochastic matrix $R$,

$\displaystyle\hat{T}_{k}^{U_{l}}\hat{n}_{x}\ket{\Psi}\equiv\sum_{y}R_{yx}\hat{n}_{y}\ket{\Psi}.$
(74)

Following similar arguments used to simplify the Hamiltonian bound, Eq. (73)
can be rewritten as

$\displaystyle\frac{1}{L^{d}}\sum_{xya}R_{xa}R_{ay}e^{iq(x-y)}\leq|e^{-2\Delta}|.$
(75)

It is useful to define $\tilde{R}_{xy}\equiv\sum_{a}R_{xa}R_{ay}$, which is
itself a symmetric, doubly-stochastic matrix corresponding to the walk on the
circuit defined by $(\hat{T}_{k}^{U_{l}})^{\dagger}\hat{T}_{k}^{U_{l}}$. For
small $q$ and sufficiently short-ranged $\tilde{R}$,

$\displaystyle\frac{1}{L^{d}}\sum_{xy}\tilde{R}_{xy}e^{iq(x-y)}$
$\displaystyle=\frac{1}{L^{d}}\sum_{xy}\tilde{R}_{xy}\cos\left(q(x-y)\right)$
$\displaystyle\simeq
1-q^{2}\underbrace{\left(\frac{1}{2L^{d}}\sum_{xy}(x-y)^{2}\tilde{R}_{xy}\right)}_{\tilde{C}}+~{}O(q^{4})$
(76)

where $\tilde{C}$ is a geometry dependent constant. Plugging this into Eq.
(75) and selecting the longest wavelength available, leads to

$\displaystyle\text{Re}(\Delta)\leq\frac{C}{L^{2}}$ (77)

for another constant $C$. In Appendix D, we explicitly compute the bound
including constants for the case of 2-body brick-layer circuits in $d=1$ and
obtain a bound $\frac{4\pi^{2}}{L^{2}}\geq\text{Re}(\Delta)$.

##  VII Convergence Time to Approximate Unitary $k$-Designs

The results of Sec. VI show that diffusive processes govern the relaxation of
$\hat{T}_{k}^{U_{t}}$ to $\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}$ with a rate
$\Delta\sim 1/L^{2}$. In this section, we convert this into a bound on the
convergence time, $\tau$, at which $\hat{T}_{k}^{U_{t}}$ becomes approximately
indistinguishable from a unitary $k$-designLow (2010),

$\displaystyle||\hat{T}_{k}^{U_{t}}-\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}||_{\diamond}\leq\varepsilon$
(78)

for some error tolerance $\varepsilon$, where $||\cdot||_{\diamond}$ denotes
the diamond norm Aharonov  _et al._ (1998). In general, $\hat{T}_{k}^{U_{t}}$
is not Hermitian and its left and right eigenspaces may differ. However, it
still admits an eigendecomposition

$\displaystyle\hat{T}_{k}^{U_{t}}=\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}+\sum_{i}\hat{C}_{i}e^{-t\Delta_{i}}$
(79)

where $\hat{C}_{i}$ are non-Hermitian projectors and the “energies”
$\Delta_{i}$ have positive real part. In this expansion, the Hermitian
$\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}$ projects onto the ground space as a
consequence of the frustration-freeness of the model.

The diamond norm is bounded by the simpler $2$-norm Hunter-Jones (2019);
Watrous (2018),

$\displaystyle||\hat{T}_{k}^{U_{t}}-\hat{T}_{k}^{U(\mathcal{H}|\hat{N})}||^{2}_{\diamond}$
$\displaystyle\leq{\mathcal{D}}^{2k}||\sum_{i}\hat{C}_{i}e^{-t\Delta_{i}}||^{2}_{2}$
$\displaystyle\leq{\mathcal{D}}^{2k}\sum_{i}||\hat{C}_{i}||^{2}_{2}e^{-2t\Delta_{i}}$
(80)

This right hand side can be interpreted as a partition function of a system
with degeneracies given by $||\hat{C}_{i}||^{2}_{2}$ and energies
$\Delta_{i}$.

Naively, the convergence is controlled by the lowest-energy term. This is
bounded by single excitation of the Goldstone modes at the smallest allowed
momentum $q\sim 2\pi/L$. For specificity, we work in the remainder of this
section on the $d$-dimensional square lattice with periodic boundary
conditions, although these results are readily generalized to other lattices
at the expense of a few $O(1)$ constants. On the square lattice, there are
$2kd$ degenerate states with momentum $2\pi/L$ and associated gap
$\Delta_{q}=q^{2}$. Plugging this estimate into (78), we find

$\displaystyle{\mathcal{D}}^{2k}\times 2kd\times
e^{-2\tau\left(\frac{2\pi}{L}\right)^{2}}$ $\displaystyle\leq\varepsilon^{2}.$
(81)

To interpret this result, it is useful to make the dependence on system size
explicit by writing ${\mathcal{D}}=Q^{L^{d}}$, where $Q=\sum_{n}{d^{(1)}_{n}}$
is the local Hilbert space dimension. The bound on the convergence time from
Eq. (81) is then

$\displaystyle\tau\geq\frac{kL^{d+2}}{4\pi^{2}}\ln(Q)+\frac{L^{2}}{8\pi^{2}}\ln(\frac{2kd}{\varepsilon^{2}}).$
(82)

The cautious reader may worry that the large low energy density of states for
the bosonic Goldstone modes might lead to parametric corrections increasing
the right hand side of Eq. (82). This is not the case, as might be expected
because $\tau\sim L^{d+2}\gg 1/\Delta$. A slightly more detailed estimate of
the statistical mechanics of $k$ independently populated Goldstone modes (see
App. E) confirms this result.

It is instructive to compare this result with that of Ref. Hunter-Jones, 2019,
which computed the convergence time of random unitary circuits without a
conservation law. There it was found that $\tau$ scales as $O(Lk)$ for a
system of $L$ qudits in $d=1$ which should be contrasted with $O(L^{3}k)$ in
the presence of diffusive modes.

##  VIII Discussion

We have shown that local random circuits with number conservation form an
approximate $k$-design on the full group of number-conserving unitaries in
depth $\tau\gtrsim kL^{d+2}$. We expect the bound on $\tau$ is tight as it
simply reflects the diffusive bottleneck of charge transport in the random
circuit. This immediately points to a geometric approach to speed up the
convergence given a fixed large number of qudits, $V$ – embed the circuit in
higher spatial dimension, where $\tau\gtrsim kV^{1+2/d}$. If long-range gates
are available, then a random circuit on an expander graph corresponds to the
$d\to\infty$ limit, where $\tau\gtrsim kV$ up to logarithmic corrections.

One well-known application of unitary $k$-designs is to measuring the Rényi
entropy of quantum states van Enk and Beenakker (2012); Vermersch  _et al._
(2018); Elben  _et al._ (2018); Brydges  _et al._ (2019). The entropy can be
computed from the statistics of measurements after evolution by Haar random
unitaries. True Haar unitaries are challenging to apply experimentally, but
may be approximated by sufficiently deep random local circuits. Our results
indicate that number conservation, as arises for example in systems of
ultracold atoms/molecules, necessitates significantly longer evolution times
($\sim L^{d+2}$) as compared to what has been discussed in the literature to
date.

Technically, our results rely on the indistinguishability of moments of the
Haar measure on the full group of number-conserving unitaries,
$U(\mathcal{H}|\hat{N})$, and the group generated by $b$-body circuits,
$U_{b}(\mathcal{H}|\hat{N})$. We showed that moments up to $k_{c}=L^{d}$ are
identical. It is clear that $k_{c}$ is much larger than $L^{d}$ for $b>2$ or
for qudits with more than one state in the maximal or minimal local number
sector; either of these conditions leads to super-extensive $k_{c}$. Future
work could tighten these bounds.

The perspective afforded by spontaneous symmetry breaking in the replica model
suggests several immediate generalizations. First, random circuits obeying
discrete symmetry groups lead to replica models with more intricate but
nonetheless gapped discrete symmetry breaking phases. These ought to converge
as $k$-designs on an $O(L^{0})$ time scale, as in the case without symmetry.
Circuits obeying non-Abelian continuous symmetry groups, such as $G=SU(2)$,
would map to spontaneous symmetry breaking phases with $z=2$ frustration-free
Goldstone modes. Accordingly, we expect diffusion of non-Abelian charge
remains the bottleneck. In this case, however, the global group generated by
$b$-body circuits is a more complicated proper subgroup of $U(\mathcal{H}|G)$
than in the $U(1)$ case Marvian (2022) and it remains an open question to sort
out what order moments distinguish them Marvian  _et al._ (2021).

###### Acknowledgements.

The authors are grateful to J. Chalker, V. Khemani and A. Potter for
stimulating discussions. This work was supported by the Air Force Office of
Scientific Research through grant No. FA9550-16-1-0334 (M.O.F.) and by the
National Science Foundation through the awards DMR-1752759 (S.N.H., M.O.F. and
A.C.) and PHY-1752727 (C.R.L.).

##  Appendix A Moment Operator Decomposition From Schur-Weyl Duality

In this appendix, we show that the moment operator expansion

$\hat{T}_{k}^{U(\mathcal{H})}=\sum_{\sigma\tau}V^{\sigma\tau}\ket{\sigma}\bra{\tau}$
(83)

follows from a well-known expansion of quantum channels in permutation
operators. We begin by briefly reviewing that expansion, which is derived in
more detail in Ref. Roberts and Yoshida (2017).

Consider a Hilbert space $\mathcal{H}$, and let $\hat{A}$ be a linear operator
on the replicated Hilbert space $\mathcal{H}^{\otimes k}$. Let
$\Phi_{k}(\hat{A})$ denote the $k$-fold twirl of $\hat{A}$, defined as

$\Phi_{k}(\hat{A})=\int dU\left(U^{\otimes k}\right)^{\dagger}AU^{\otimes k}$
(84)

Due to shift invariance of the Haar measure, $\Phi_{k}(\hat{A})$ commutes with
all replicated unitary operators of the form $V^{\otimes k}$. This implies, as
a consequence of Schur-Weyl duality, that $\Phi_{k}(\hat{A})$ can be expanded
in terms of permutation operators
$\hat{\sigma}=\ket{\sigma(\boldsymbol{i})}\bra{\boldsymbol{i}}$,

$\displaystyle\Phi_{k}(\hat{A})$
$\displaystyle=\sum_{\sigma\tau}V^{\sigma\tau}\Tr[\hat{\tau}\hat{A}]\hat{\sigma}$
(85)

Diagramatically (taking $k=3$ for simplicity),

$\displaystyle\Phi_{k}(\hat{A})$ $\displaystyle=\int dU\left(\leavevmode\hbox
to51.97pt{\vbox to35.97pt{\pgfpicture\makeatletter\hbox{\hskip
1.97829pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}{}\pgfsys@moveto{-1.77829pt}{32.00935pt}\pgfsys@lineto{49.79233pt}{32.00935pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-1.77829pt}{17.78297pt}\pgfsys@lineto{49.79233pt}{17.78297pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-1.77829pt}{3.5566pt}\pgfsys@lineto{49.79233pt}{3.5566pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.6,0.6,1}\pgfsys@color@rgb@fill{0.6}{0.6}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{7.11319pt}{28.45276pt}\pgfsys@moveto{7.11319pt}{28.45276pt}\pgfsys@lineto{7.11319pt}{35.56595pt}\pgfsys@lineto{14.22638pt}{35.56595pt}\pgfsys@lineto{14.22638pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{35.56595pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,1,0.6}\pgfsys@color@cmyk@fill{0}{0}{0.4}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{35.56595pt}{28.45276pt}\pgfsys@moveto{35.56595pt}{28.45276pt}\pgfsys@lineto{35.56595pt}{35.56595pt}\pgfsys@lineto{42.67914pt}{35.56595pt}\pgfsys@lineto{42.67914pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{42.67914pt}{35.56595pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.6,0.6,1}\pgfsys@color@rgb@fill{0.6}{0.6}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{7.11319pt}{14.22638pt}\pgfsys@moveto{7.11319pt}{14.22638pt}\pgfsys@lineto{7.11319pt}{21.33957pt}\pgfsys@lineto{14.22638pt}{21.33957pt}\pgfsys@lineto{14.22638pt}{14.22638pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{21.33957pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,1,0.6}\pgfsys@color@cmyk@fill{0}{0}{0.4}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{35.56595pt}{14.22638pt}\pgfsys@moveto{35.56595pt}{14.22638pt}\pgfsys@lineto{35.56595pt}{21.33957pt}\pgfsys@lineto{42.67914pt}{21.33957pt}\pgfsys@lineto{42.67914pt}{14.22638pt}\pgfsys@closepath\pgfsys@moveto{42.67914pt}{21.33957pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.6,0.6,1}\pgfsys@color@rgb@fill{0.6}{0.6}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{7.11319pt}{7.11319pt}\pgfsys@lineto{14.22638pt}{7.11319pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{7.11319pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,1,0.6}\pgfsys@color@cmyk@fill{0}{0}{0.4}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{35.56595pt}{0.0pt}\pgfsys@moveto{35.56595pt}{0.0pt}\pgfsys@lineto{35.56595pt}{7.11319pt}\pgfsys@lineto{42.67914pt}{7.11319pt}\pgfsys@lineto{42.67914pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{42.67914pt}{7.11319pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,0.6,0.6}\pgfsys@color@rgb@fill{1}{0.6}{0.6}\pgfsys@invoke{
}{}\pgfsys@moveto{19.91695pt}{0.0pt}\pgfsys@moveto{19.91695pt}{0.0pt}\pgfsys@lineto{19.91695pt}{35.56595pt}\pgfsys@lineto{29.87537pt}{35.56595pt}\pgfsys@lineto{29.87537pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{29.87537pt}{35.56595pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{22.11838pt}{14.17186pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\hat{A}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right)$ (86)
$\displaystyle=\sum_{\sigma\tau}V^{\sigma\tau}\left(\leavevmode\hbox
to18.18pt{\vbox to35.97pt{\pgfpicture\makeatletter\hbox{\hskip
1.97829pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}{}\pgfsys@moveto{-1.77829pt}{32.00935pt}\pgfsys@lineto{16.00467pt}{32.00935pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-1.77829pt}{17.78297pt}\pgfsys@lineto{16.00467pt}{17.78297pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-1.77829pt}{3.5566pt}\pgfsys@lineto{16.00467pt}{3.5566pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{3.5566pt}{0.0pt}\pgfsys@moveto{3.5566pt}{0.0pt}\pgfsys@lineto{3.5566pt}{35.56595pt}\pgfsys@lineto{10.66978pt}{35.56595pt}\pgfsys@lineto{10.66978pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{10.66978pt}{35.56595pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\hat{\sigma}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
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} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{35.56595pt}\pgfsys@lineto{7.11319pt}{35.56595pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{7.11319pt}{35.56595pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,0.6,0.6}\pgfsys@color@rgb@fill{1}{0.6}{0.6}\pgfsys@invoke{
}{}\pgfsys@moveto{12.80376pt}{0.0pt}\pgfsys@moveto{12.80376pt}{0.0pt}\pgfsys@lineto{12.80376pt}{35.56595pt}\pgfsys@lineto{22.76218pt}{35.56595pt}\pgfsys@lineto{22.76218pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{22.76218pt}{35.56595pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.00519pt}{14.17186pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\hat{A}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.77881pt}{14.17186pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\hat{\tau}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(87)

With this result in hand, the claimed expansion for moment operators (83)
follows from index rearrangements, which we carry out diagramatically. Instead
of thinking of the channel $\Phi_{k}$ acting on an operator
$\hat{A}=\sum_{\boldsymbol{i},\bar{\boldsymbol{i}}}A_{\boldsymbol{i},\bar{\boldsymbol{i}}}\ket{\boldsymbol{i}}\bra{\bar{\boldsymbol{i}}}$,
consider the moment operator $\hat{T}_{k}^{U(\mathcal{H})}$ acting on the
corresponding state
$\ket{A}\equiv\sum_{\boldsymbol{i},\bar{\boldsymbol{i}}}\hat{A}_{\boldsymbol{i},\boldsymbol{\bar{i}}}\ket{\boldsymbol{i}}\ket{\boldsymbol{\bar{i}}}\in\left(\mathcal{H}\otimes\mathcal{H}^{*}\right)^{k}$,

$\displaystyle\hat{T}_{k}^{U(\mathcal{H})}\ket{A}$ $\displaystyle=\int
dU\left(~{}\leavevmode\hbox to40.85pt{\vbox
to78.65pt{\pgfpicture\makeatletter\hbox{\hskip 1.97829pt\lower-42.87914pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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} {}{{}}{}
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
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}\definecolor{pgffillcolor}{rgb}{1,1,0.6}\pgfsys@color@cmyk@fill{0}{0}{0.4}{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,0.6,0.6}\pgfsys@color@rgb@fill{1}{0.6}{0.6}\pgfsys@invoke{
}{}\pgfsys@moveto{21.33957pt}{-42.67914pt}\pgfsys@moveto{21.33957pt}{-42.67914pt}\pgfsys@lineto{21.33957pt}{35.56595pt}\pgfsys@lineto{35.56595pt}{35.56595pt}\pgfsys@lineto{35.56595pt}{-42.67914pt}\pgfsys@closepath\pgfsys@moveto{35.56595pt}{35.56595pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.36942pt}{-6.0566pt}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{}\right).$ (88)

The state $\hat{T}_{k}^{U(\mathcal{H})}\ket{A}$ is related to the operator
$\Phi_{k}(\hat{A})$ by reindexing,

$\displaystyle\bra{\boldsymbol{i},\bar{\boldsymbol{i}}}\hat{T}_{k}^{U(\mathcal{H})}\ket{A}$
$\displaystyle=\bra{\bar{\boldsymbol{i}}}\Phi_{k}(\hat{A})\ket{\boldsymbol{i}}.$
(89)

Diagrammatically, this reindexing manifests as a simple manipulation of the
incoming and outgoing indices,

$\displaystyle\int dU\left(~{}\leavevmode\hbox to44.86pt{\vbox
to78.65pt{\pgfpicture\makeatletter\hbox{\hskip 1.97829pt\lower-42.87914pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}{}\pgfsys@moveto{-1.77829pt}{32.00935pt}\pgfsys@lineto{28.45276pt}{32.00935pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-1.77829pt}{-10.66978pt}\pgfsys@lineto{28.45276pt}{-10.66978pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-1.77829pt}{17.78297pt}\pgfsys@lineto{28.45276pt}{17.78297pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-1.77829pt}{-24.89616pt}\pgfsys@lineto{28.45276pt}{-24.89616pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-1.77829pt}{3.5566pt}\pgfsys@lineto{28.45276pt}{3.5566pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-1.77829pt}{-39.12254pt}\pgfsys@lineto{28.45276pt}{-39.12254pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.6,0.6,1}\pgfsys@color@rgb@fill{0.6}{0.6}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{7.11319pt}{28.45276pt}\pgfsys@moveto{7.11319pt}{28.45276pt}\pgfsys@lineto{7.11319pt}{35.56595pt}\pgfsys@lineto{14.22638pt}{35.56595pt}\pgfsys@lineto{14.22638pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{35.56595pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,1,0.6}\pgfsys@color@cmyk@fill{0}{0}{0.4}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{7.11319pt}{-14.22638pt}\pgfsys@moveto{7.11319pt}{-14.22638pt}\pgfsys@lineto{7.11319pt}{-7.11319pt}\pgfsys@lineto{14.22638pt}{-7.11319pt}\pgfsys@lineto{14.22638pt}{-14.22638pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{-7.11319pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.6,0.6,1}\pgfsys@color@rgb@fill{0.6}{0.6}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{7.11319pt}{14.22638pt}\pgfsys@moveto{7.11319pt}{14.22638pt}\pgfsys@lineto{7.11319pt}{21.33957pt}\pgfsys@lineto{14.22638pt}{21.33957pt}\pgfsys@lineto{14.22638pt}{14.22638pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{21.33957pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,1,0.6}\pgfsys@color@cmyk@fill{0}{0}{0.4}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{7.11319pt}{-28.45276pt}\pgfsys@moveto{7.11319pt}{-28.45276pt}\pgfsys@lineto{7.11319pt}{-21.33957pt}\pgfsys@lineto{14.22638pt}{-21.33957pt}\pgfsys@lineto{14.22638pt}{-28.45276pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{-21.33957pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.6,0.6,1}\pgfsys@color@rgb@fill{0.6}{0.6}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{7.11319pt}{7.11319pt}\pgfsys@lineto{14.22638pt}{7.11319pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{7.11319pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,1,0.6}\pgfsys@color@cmyk@fill{0}{0}{0.4}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{7.11319pt}{-42.67914pt}\pgfsys@moveto{7.11319pt}{-42.67914pt}\pgfsys@lineto{7.11319pt}{-35.56595pt}\pgfsys@lineto{14.22638pt}{-35.56595pt}\pgfsys@lineto{14.22638pt}{-42.67914pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{-35.56595pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,0.6,0.6}\pgfsys@color@rgb@fill{1}{0.6}{0.6}\pgfsys@invoke{
}{}\pgfsys@moveto{21.33957pt}{-42.67914pt}\pgfsys@moveto{21.33957pt}{-42.67914pt}\pgfsys@lineto{21.33957pt}{35.56595pt}\pgfsys@lineto{28.45276pt}{35.56595pt}\pgfsys@lineto{28.45276pt}{-42.67914pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{35.56595pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.6,0.6}\pgfsys@color@rgb@stroke{1}{0.6}{0.6}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0.6}{0.6}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,0.6,0.6}{}{{
{\pgfsys@beginscope\pgfsys@setlinewidth{0.32pt}\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{}
{}{}{} {}{}{}
\pgfsys@moveto{-1.19998pt}{1.59998pt}\pgfsys@curveto{-1.09998pt}{0.99998pt}{0.0pt}{0.09999pt}{0.29999pt}{0.0pt}\pgfsys@curveto{0.0pt}{-0.09999pt}{-1.09998pt}{-0.99998pt}{-1.19998pt}{-1.59998pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}\pgfsys@moveto{28.45276pt}{-39.12254pt}\pgfsys@curveto{36.33012pt}{-39.12254pt}{42.67914pt}{-32.77353pt}{42.67914pt}{-25.35616pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{42.67914pt}{-25.35616pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{}\right)$
$\displaystyle\leftrightarrow\int dU\left(~{}\leavevmode\hbox to64.42pt{\vbox
to35.97pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{32.00935pt}\pgfsys@lineto{21.33957pt}{32.00935pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{35.56595pt}{32.00935pt}\pgfsys@lineto{64.0187pt}{32.00935pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{17.78297pt}\pgfsys@lineto{21.33957pt}{17.78297pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{35.56595pt}{17.78297pt}\pgfsys@lineto{64.0187pt}{17.78297pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{3.5566pt}\pgfsys@lineto{21.33957pt}{3.5566pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{35.56595pt}{3.5566pt}\pgfsys@lineto{64.0187pt}{3.5566pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.6,0.6,1}\pgfsys@color@rgb@fill{0.6}{0.6}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{7.11319pt}{28.45276pt}\pgfsys@moveto{7.11319pt}{28.45276pt}\pgfsys@lineto{7.11319pt}{35.56595pt}\pgfsys@lineto{14.22638pt}{35.56595pt}\pgfsys@lineto{14.22638pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{35.56595pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,1,0.6}\pgfsys@color@cmyk@fill{0}{0}{0.4}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{49.79233pt}{28.45276pt}\pgfsys@moveto{49.79233pt}{28.45276pt}\pgfsys@lineto{49.79233pt}{35.56595pt}\pgfsys@lineto{56.90552pt}{35.56595pt}\pgfsys@lineto{56.90552pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{56.90552pt}{35.56595pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.6,0.6,1}\pgfsys@color@rgb@fill{0.6}{0.6}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{7.11319pt}{14.22638pt}\pgfsys@moveto{7.11319pt}{14.22638pt}\pgfsys@lineto{7.11319pt}{21.33957pt}\pgfsys@lineto{14.22638pt}{21.33957pt}\pgfsys@lineto{14.22638pt}{14.22638pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{21.33957pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,1,0.6}\pgfsys@color@cmyk@fill{0}{0}{0.4}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{49.79233pt}{14.22638pt}\pgfsys@moveto{49.79233pt}{14.22638pt}\pgfsys@lineto{49.79233pt}{21.33957pt}\pgfsys@lineto{56.90552pt}{21.33957pt}\pgfsys@lineto{56.90552pt}{14.22638pt}\pgfsys@closepath\pgfsys@moveto{56.90552pt}{21.33957pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope {}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.6,0.6,1}\pgfsys@color@rgb@fill{0.6}{0.6}{1}\pgfsys@invoke{
}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{7.11319pt}{7.11319pt}\pgfsys@lineto{14.22638pt}{7.11319pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{7.11319pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,1,0.6}\pgfsys@color@cmyk@fill{0}{0}{0.4}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{49.79233pt}{0.0pt}\pgfsys@moveto{49.79233pt}{0.0pt}\pgfsys@lineto{49.79233pt}{7.11319pt}\pgfsys@lineto{56.90552pt}{7.11319pt}\pgfsys@lineto{56.90552pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{56.90552pt}{7.11319pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope {}{{}}{} {}{} {}{} {}{}
{}{} {}{} {}{} {}{} {}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,0.6,0.6}\pgfsys@color@rgb@fill{1}{0.6}{0.6}\pgfsys@invoke{
}{}\pgfsys@moveto{21.33957pt}{0.0pt}\pgfsys@lineto{21.33957pt}{35.56595pt}\pgfsys@lineto{28.45276pt}{35.56595pt}\pgfsys@lineto{28.45276pt}{7.11319pt}\pgfsys@lineto{35.56595pt}{7.11319pt}\pgfsys@lineto{35.56595pt}{35.56595pt}\pgfsys@lineto{42.67914pt}{35.56595pt}\pgfsys@lineto{42.67914pt}{0.0pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{}\right).$ (90)

Applying the operator expansion of Eq. (85) to $\Phi_{k}(\hat{A})$ yields,

$\displaystyle\Phi(\hat{A})$
$\displaystyle=\sum_{\sigma\tau}V^{\sigma\tau}\left(\leavevmode\hbox
to18.18pt{\vbox to35.97pt{\pgfpicture\makeatletter\hbox{\hskip
1.97829pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}{}\pgfsys@moveto{-1.77829pt}{32.00935pt}\pgfsys@lineto{16.00467pt}{32.00935pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-1.77829pt}{17.78297pt}\pgfsys@lineto{16.00467pt}{17.78297pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-1.77829pt}{3.5566pt}\pgfsys@lineto{16.00467pt}{3.5566pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{3.5566pt}{0.0pt}\pgfsys@moveto{3.5566pt}{0.0pt}\pgfsys@lineto{3.5566pt}{35.56595pt}\pgfsys@lineto{10.66978pt}{35.56595pt}\pgfsys@lineto{10.66978pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{10.66978pt}{35.56595pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.3354pt}{14.17186pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\hat{\sigma}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right)\Tr[\leavevmode\hbox
to48.41pt{\vbox to35.97pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{32.00935pt}\pgfsys@lineto{21.33957pt}{32.00935pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{35.56595pt}{32.00935pt}\pgfsys@lineto{48.01402pt}{32.00935pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{17.78297pt}\pgfsys@lineto{21.33957pt}{17.78297pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{35.56595pt}{17.78297pt}\pgfsys@lineto{48.01402pt}{17.78297pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{3.5566pt}\pgfsys@lineto{21.33957pt}{3.5566pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{35.56595pt}{3.5566pt}\pgfsys@lineto{48.01402pt}{3.5566pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{7.11319pt}{35.56595pt}\pgfsys@lineto{14.22638pt}{35.56595pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{35.56595pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.892pt}{14.17186pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\hat{\tau}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{} {}{} {}{} {}{}
{}{} {}{} {}{} {}{} {}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,0.6,0.6}\pgfsys@color@rgb@fill{1}{0.6}{0.6}\pgfsys@invoke{
}{}\pgfsys@moveto{21.33957pt}{0.0pt}\pgfsys@lineto{21.33957pt}{35.56595pt}\pgfsys@lineto{28.45276pt}{35.56595pt}\pgfsys@lineto{28.45276pt}{7.11319pt}\pgfsys@lineto{35.56595pt}{7.11319pt}\pgfsys@lineto{35.56595pt}{35.56595pt}\pgfsys@lineto{42.67914pt}{35.56595pt}\pgfsys@lineto{42.67914pt}{0.0pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}]$ (91)

Reorganizing the indices in a manner analogous to Eq. (89) leads to the
desired result Eq. (83),

$\displaystyle\hat{T}_{k}^{U(\mathcal{H})}\ket{A}$
$\displaystyle=\sum_{\sigma\tau}V^{\sigma\tau}\left(\leavevmode\hbox
to15.28pt{\vbox to39.52pt{\pgfpicture\makeatletter\hbox{\hskip
5.53488pt\lower-21.53957pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{-5.33488pt}{16.00467pt}\pgfsys@lineto{7.11319pt}{16.00467pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{-5.33488pt}{-5.33488pt}\pgfsys@lineto{7.11319pt}{-5.33488pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-5.33488pt}{8.89148pt}\pgfsys@lineto{7.11319pt}{8.89148pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-5.33488pt}{-12.44807pt}\pgfsys@lineto{7.11319pt}{-12.44807pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-5.33488pt}{1.77829pt}\pgfsys@lineto{7.11319pt}{1.77829pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-5.33488pt}{-19.56126pt}\pgfsys@lineto{7.11319pt}{-19.56126pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-21.33957pt}\pgfsys@moveto{0.0pt}{-21.33957pt}\pgfsys@lineto{0.0pt}{17.78297pt}\pgfsys@lineto{7.11319pt}{17.78297pt}\pgfsys@lineto{7.11319pt}{-21.33957pt}\pgfsys@closepath\pgfsys@moveto{7.11319pt}{17.78297pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.69954pt}{-3.93106pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\sigma$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right)\Tr[\leavevmode\hbox
to48.41pt{\vbox to35.97pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{32.00935pt}\pgfsys@lineto{21.33957pt}{32.00935pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{35.56595pt}{32.00935pt}\pgfsys@lineto{48.01402pt}{32.00935pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{17.78297pt}\pgfsys@lineto{21.33957pt}{17.78297pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{35.56595pt}{17.78297pt}\pgfsys@lineto{48.01402pt}{17.78297pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{3.5566pt}\pgfsys@lineto{21.33957pt}{3.5566pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{35.56595pt}{3.5566pt}\pgfsys@lineto{48.01402pt}{3.5566pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{7.11319pt}{35.56595pt}\pgfsys@lineto{14.22638pt}{35.56595pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{14.22638pt}{35.56595pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.892pt}{14.17186pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\hat{\tau}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{} {}{} {}{} {}{}
{}{} {}{} {}{} {}{} {}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,0.6,0.6}\pgfsys@color@rgb@fill{1}{0.6}{0.6}\pgfsys@invoke{
}{}\pgfsys@moveto{21.33957pt}{0.0pt}\pgfsys@lineto{21.33957pt}{35.56595pt}\pgfsys@lineto{28.45276pt}{35.56595pt}\pgfsys@lineto{28.45276pt}{7.11319pt}\pgfsys@lineto{35.56595pt}{7.11319pt}\pgfsys@lineto{35.56595pt}{35.56595pt}\pgfsys@lineto{42.67914pt}{35.56595pt}\pgfsys@lineto{42.67914pt}{0.0pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}]$
$\displaystyle=\sum_{\sigma\tau}V^{\sigma\tau}\left(\leavevmode\hbox
to41.3pt{\vbox to39.52pt{\pgfpicture\makeatletter\hbox{\hskip
5.53488pt\lower-21.53957pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}} {}{{}}{}
{}{}{}\pgfsys@moveto{-5.33488pt}{16.00467pt}\pgfsys@lineto{7.11319pt}{16.00467pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-5.33488pt}{-5.33488pt}\pgfsys@lineto{7.11319pt}{-5.33488pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-5.33488pt}{8.89148pt}\pgfsys@lineto{7.11319pt}{8.89148pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-5.33488pt}{-12.44807pt}\pgfsys@lineto{7.11319pt}{-12.44807pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-5.33488pt}{1.77829pt}\pgfsys@lineto{7.11319pt}{1.77829pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-5.33488pt}{-19.56126pt}\pgfsys@lineto{7.11319pt}{-19.56126pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-21.33957pt}\pgfsys@moveto{0.0pt}{-21.33957pt}\pgfsys@lineto{0.0pt}{17.78297pt}\pgfsys@lineto{7.11319pt}{17.78297pt}\pgfsys@lineto{7.11319pt}{-21.33957pt}\pgfsys@closepath\pgfsys@moveto{7.11319pt}{17.78297pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.69954pt}{-3.93106pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\sigma$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{16.00467pt}\pgfsys@lineto{28.45276pt}{16.00467pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{-5.33488pt}\pgfsys@lineto{28.45276pt}{-5.33488pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{8.89148pt}\pgfsys@lineto{28.45276pt}{8.89148pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{-12.44807pt}\pgfsys@lineto{28.45276pt}{-12.44807pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{1.77829pt}\pgfsys@lineto{28.45276pt}{1.77829pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{-19.56126pt}\pgfsys@lineto{28.45276pt}{-19.56126pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{-21.33957pt}\pgfsys@moveto{14.22638pt}{-21.33957pt}\pgfsys@lineto{14.22638pt}{17.78297pt}\pgfsys@lineto{21.33957pt}{17.78297pt}\pgfsys@lineto{21.33957pt}{-21.33957pt}\pgfsys@closepath\pgfsys@moveto{21.33957pt}{17.78297pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,0.6,0.6}\pgfsys@color@rgb@fill{1}{0.6}{0.6}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{-21.33957pt}\pgfsys@moveto{28.45276pt}{-21.33957pt}\pgfsys@lineto{28.45276pt}{17.78297pt}\pgfsys@lineto{35.56595pt}{17.78297pt}\pgfsys@lineto{35.56595pt}{-21.33957pt}\pgfsys@closepath\pgfsys@moveto{35.56595pt}{17.78297pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.59721pt}{-3.93106pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\tau$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right)$ (92)
$\displaystyle\hat{T}_{k}^{U(\mathcal{H})}$
$\displaystyle=\sum_{\sigma\tau}V^{\sigma\tau}\left(\leavevmode\hbox
to32.41pt{\vbox to39.52pt{\pgfpicture\makeatletter\hbox{\hskip
5.53488pt\lower-21.53957pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}} {}{{}}{}
{}{}{}\pgfsys@moveto{-5.33488pt}{16.00467pt}\pgfsys@lineto{7.11319pt}{16.00467pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-5.33488pt}{-5.33488pt}\pgfsys@lineto{7.11319pt}{-5.33488pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-5.33488pt}{8.89148pt}\pgfsys@lineto{7.11319pt}{8.89148pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-5.33488pt}{-12.44807pt}\pgfsys@lineto{7.11319pt}{-12.44807pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-5.33488pt}{1.77829pt}\pgfsys@lineto{7.11319pt}{1.77829pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-5.33488pt}{-19.56126pt}\pgfsys@lineto{7.11319pt}{-19.56126pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-21.33957pt}\pgfsys@moveto{0.0pt}{-21.33957pt}\pgfsys@lineto{0.0pt}{17.78297pt}\pgfsys@lineto{7.11319pt}{17.78297pt}\pgfsys@lineto{7.11319pt}{-21.33957pt}\pgfsys@closepath\pgfsys@moveto{7.11319pt}{17.78297pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.69954pt}{-3.93106pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\sigma$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{16.00467pt}\pgfsys@lineto{26.67445pt}{16.00467pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{-5.33488pt}\pgfsys@lineto{26.67445pt}{-5.33488pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{8.89148pt}\pgfsys@lineto{26.67445pt}{8.89148pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{-12.44807pt}\pgfsys@lineto{26.67445pt}{-12.44807pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{1.77829pt}\pgfsys@lineto{26.67445pt}{1.77829pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{-19.56126pt}\pgfsys@lineto{26.67445pt}{-19.56126pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.8,0.8,0.8}\pgfsys@color@gray@fill{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{-21.33957pt}\pgfsys@moveto{14.22638pt}{-21.33957pt}\pgfsys@lineto{14.22638pt}{17.78297pt}\pgfsys@lineto{21.33957pt}{17.78297pt}\pgfsys@lineto{21.33957pt}{-21.33957pt}\pgfsys@closepath\pgfsys@moveto{21.33957pt}{17.78297pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.59721pt}{-3.93106pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\tau$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right)=\sum_{\sigma\tau}V^{\sigma\tau}\ket{\sigma}\bra{\tau}.$
(93)

##  Appendix B Generalized Weingarten Functions and the Moment Operator for
$U(\mathcal{H})$

In this appendix, we present the explicit formulae for the matrix elements
$V^{\sigma\tau}$ of the moment operator

$\displaystyle\hat{T}_{k}^{U(\mathcal{H})}=\sum_{\sigma\tau}V^{\sigma\tau}\ket{\sigma}\bra{\tau}.$
(94)

in the subspace spanned by the permutation states $\ket{\sigma}$. For
$k\leq{\mathcal{D}}$, the matrix elements $V^{\sigma\tau}$ are the well-known
Weingarten functions. For $k>{\mathcal{D}}$, the permutation states are
linearly dependent, and the expansion is not unique. This regime is often
ignored in the physics literature.

The matrix elements $V^{\sigma\tau}$ are determined by the overlaps of the
permutation states $\ket{\sigma}$ onto which $\hat{T}_{k}^{U(\mathcal{H})}$
projects. The overlap matrix $D^{\tau\sigma}=\innerproduct{\tau}{\sigma}$
counts cycles in the relative permutation $\sigma\tau^{-1}$,

$\displaystyle D^{\tau\sigma}$
$\displaystyle=\sum_{\boldsymbol{i},\bar{\boldsymbol{i}}}\innerproduct{\tau}{\boldsymbol{i},\bar{\boldsymbol{i}}}\innerproduct{\boldsymbol{i},\bar{\boldsymbol{i}}}{\sigma}$
$\displaystyle=\sum_{\boldsymbol{i},\bar{\boldsymbol{i}}}\delta[\bar{\boldsymbol{i}}=\tau(\boldsymbol{i})]\delta[\bar{\boldsymbol{i}}=\sigma(\boldsymbol{i})]$
$\displaystyle=\prod_{c\in\textrm{Cycles}(\sigma\tau^{-1})}{\mathcal{D}}$ (95)

Pictorially,

$\displaystyle\innerproduct{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\tau}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\sigma}$
$\displaystyle=\leavevmode\hbox to40.23pt{\vbox
to74.38pt{\pgfpicture\makeatletter\hbox{\hskip-8.3357pt\lower-3.04523pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}{}\pgfsys@moveto{19.91663pt}{62.59511pt}\pgfsys@lineto{36.98802pt}{62.59511pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{19.91663pt}{51.21417pt}\pgfsys@lineto{36.98802pt}{51.21417pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{19.91663pt}{39.83325pt}\pgfsys@lineto{36.98802pt}{39.83325pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{19.91663pt}{28.45232pt}\pgfsys@lineto{36.98802pt}{28.45232pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{19.91663pt}{17.07138pt}\pgfsys@lineto{36.98802pt}{17.07138pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{19.91663pt}{5.69046pt}\pgfsys@lineto{36.98802pt}{5.69046pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\pgfsys@stroke@opacity{0.1}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.1}\pgfsys@invoke{
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