# Variational Monte Carlo Study of Symmetric Mass Generation in a Bilayer
Honeycomb Lattice Model

Wanda Hou  Yi-Zhuang You  Department of Physics, University of California at
San Diego, La Jolla, CA 92093, USA

###### Abstract

We investigate a bilayer honeycomb lattice model of spin-1/2 fermions at half-
filling with local Heisenberg coupling of fermion spins across the two layers.
Using variational Monte Carlo (VMC) simulation, we demonstrate that the system
undergoes a direct transition from a Dirac semimetal phase to a symmetric
gapped phase, known as symmetric mass generation (SMG), as the Heisenberg
coupling strength is increased. The transition does not involve spontaneous
symmetry breaking or topological order and has been proposed as an example of
the fermionic deconfined quantum critical point (fDQCP). Our simulation shows
that a fermionic parton bilinear mass opens at the transition point while all
symmetries are still preserved thanks to the quantum fluctuations introduced
by the correlation factor in the variational wave function. From the
simulation data, we extract the critical exponent $\nu=0.96\pm 0.03$ and the
fermion scaling dimension $\Delta_{c}=1.31\pm 0.04$ at the SMG critical point,
which are consistent with the field theoretical prediction of fDQCP in (2+1)D.
These findings support the hypothesis that the fermion fractionalizes at the
SMG critical point.

##  I Introduction

Symmetric mass generation (SMG) [1] is a mechanism by which gapless Dirac
fermions acquire a gap in their excitation spectrum without breaking symmetry
or developing topological order. The gap-opening transition, known as the SMG
transition, involves non-perturbative interaction effects among the fermions
and does not require any fermion bilinear mass condensation, in contrast to
the conventional Higgs mechanism [2, 3, 4, 5, 6]. Numerical simulations [7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17] have shown that the SMG transition can be a
direct and continuous quantum critical point in various models, which is
intriguing because it does not fit the Landau-Ginzburg-Wilson paradigm, as the
phases on either side are not distinguished by any symmetry-breaking order
parameter. It has been suggested [18, 19] that the SMG critical point is
described by a _fermionic_ version of the deconfined quantum critical point
(DQCP) [20, 21, 22, 23], in which fermions _fractionalize_ into deconfined
bosonic and fermionic partons at (and only at) the critical point. However,
numerical evidence for the fermion fractionalization is still lacking.

The goal of this study is to understand the universal properties of the SMG
transition and provide evidence for the fermion fractionalization hypothesis.
To this end, we examine a bilayer honeycomb lattice model of interacting
fermions and investigate the SMG transition using numerical and analytical
methods. On the numerical side, we propose a variational ansatz for the
quantum many-body ground state of the fermion system and use the variational
Monte Carlo (VMC) approach to identify and simulate the SMG transition. On the
theoretical side, we develop a quantum field theory description of the SMG
critical point as a fermionic DQCP (fDQCP) and predict its universal
properties using renormalization group (RG) analysis. Our numerical results
are in good agreement with the analytic predictions, supporting the
fractionalization of fermions at the SMG transition.

##  II Lattice Model

We investigated a (2+1)D example of SMG. The model is defined on an A-A
stacking bilayer honeycomb lattice, as illustrated in Fig. 1(a), with each
site $i$ hosting four complex fermion modes, denoted as $c_{il\sigma}$, where
$l=1,2$ labels the two layers and $\sigma=\uparrow,\downarrow$ labels the two
spin components. The model is described by the following Hamiltonian

$H=-t\sum_{\langle
ij\rangle,l,\sigma}(c^{\dagger}_{il\sigma}c_{jl\sigma}+\text{h.c.})+J\sum_{i}{\bm{S}}_{i1}\cdot{\bm{S}}_{i2},$
(1)

where $\langle ij\rangle$ stands for the nearest neighboring pairs of sites
$i$ and $j$ on the honeycomb lattice. There is no interlayer fermion hopping.
The two layers are only coupled by the antiferromagnetic Heisenberg
interaction $J>0$.
$\bm{S}_{il}=\frac{1}{2}c^{\dagger}_{il\alpha}{\bm{\sigma}}^{\alpha\beta}c_{il\beta}$
denotes the spin operator at site-$i$ layer-$l$, with
${\bm{\sigma}}=(\sigma_{x},\sigma_{y},\sigma_{z})$ being the Pauli matrices.
The relative interaction strength $J/t$ is the only tuning parameter of this
model.

Figure 1: (a) The bilayer honeycomb lattice model with intra-layer hopping $t$
and inter-layer Heisenberg coupling $J$. (b) The phase diagram as $J/t$
varies. (c) On-site fermion configurations of different inter-layer
$\mathrm{U}(1)^{-}$ charges $q_{i}^{-}$.

Consider the fermion system at half-filling (with zero chemical potential).
When $J/t=0$, the model describes two decoupled layers of graphene, which
produces eight gapless Dirac fermions (two valleys $\times$ two layers
$\times$ two spins) at low energy. The gapless _Dirac semimetal_ phase is
stable against small $J/t$ perturbation, as the local interaction $J$ is
irrelevant for (2+1)D Dirac fermions under RG. When $J/t\to\infty$, the inter-
layer Heisenberg interaction acts on each honeycomb site independently, which
leads to a unique gapped ground state
$\bigotimes_{i}(c^{\dagger}_{i1\uparrow}c^{\dagger}_{i2\downarrow}-c^{\dagger}_{i1\downarrow}c^{\dagger}_{i2\uparrow})|0\rangle$
as the direct product of inter-layer spin-singlet states. Such an insulating
state will be called an SMG insulator, as it does not break any symmetry
(either lattice or internal) of the model. Given that any band insulator in
this half-filled bilayer honeycomb lattice model would necessarily break some
symmetry (see Appendix A for symmetry analysis), the interaction effect is
essential in achieving the SMG insulator. Therefore, as $J/t$ is increased, we
expect a phase diagram as shown in Fig. 1(b), where an _SMG transition_
happens at some critical $J_{c}$ (to be determined), separating the Dirac
semimetal and the SMG insulator. The model could be relevant to the twisted
bilayer graphene [24, 25], which has the same Dirac fermion content at low
energy, although the interaction may be different. Nevertheless, investigating
this toy model is instructive for the future experimental realization of SMG
physics in correlated materials.

##  III VMC Simulation

###  III.1 Method

The (2+1)D SMG transition has been numerically observed [7, 11] in similar
models by quantum Monte Carlo (QMC) simulations. The results suggest that the
SMG transition is direct and continuous, but the critical exponent $\nu$ (of
the correlation length $\xi\sim|J-J_{c}|^{-\nu}$ scaling) and the fermion
operator scaling dimension were not determined yet. To investigate these
universal properties of the SMG transition, we proposed a variational Monte
Carlo (VMC) approach [26, 27], based on a variational wave function motivated
by the parton-Higgs theory [18]. Ref. 18 proposes a non-Abelian gauge theory
that involves a more complicated confinement mechanism to enter the SMG phase.
This work simplifies the field theory, reducing the gauge group to the Abelian
group, which allows the direct application of known RG analysis.

Compared to the QMC approach, although the VMC approach is biased by the
variational ansatz, it can be pushed to relatively larger system sizes with
fewer computational resources. Moreover, the variational wave function
provides a unique insight that was not available by other methods, i.e., the
mean-field mass. This approach allows us to effectively observe the gap
opening of the fermionic parton, which deepens our understanding of the
underlying parton physics across the SMG transition.

The construction of the variational state starts with a mean-field (free-
fermion) Hamiltonian $H_{\text{MF}}[\lambda]=-t\sum_{\langle
ij\rangle,l}(c_{il}^{\dagger}c_{jl}+\text{h.c.})+\lambda\sum_{i}(-)^{i}(c_{i1}^{\dagger}c_{i2}+\text{h.c.})$
(with $c_{il}=[c_{il\uparrow},c_{il\downarrow}]^{\intercal}$ including both
spins, and spin indices are summed over in $H_{\text{MF}}$ implicitly). It
introduces an inter-layer hopping term $\lambda$ with a stagger sign
$(-)^{i}=\pm$ between $A/B$ sublattices on the honeycomb lattice. This stagger
sign is important to make the $\lambda$ term to be a mass term that gaps out
the Dirac fermions on the mean-field level. $\lambda$ will be treated as a
variational parameter. Consider the ground state
$|\Psi_{\text{MF}}[\lambda]\rangle$ of $H_{\text{MF}}[\lambda]$ followed by a
soft projection (correlation factor) that depends on the configuration of
inter-layer charge difference
$q_{i}^{-}:=c_{i1}^{\dagger}c_{i1}-c_{i2}^{\dagger}c_{i2}$ on every site, as
exemplified in Fig. 1(c), the variational many-body state then takes the
following form, similar to a Slater-Jastrow wave function [28]

$|\Psi[\lambda,V]\rangle\propto\mathrm{e}^{-V[\\{q_{i}^{-}\\}]}|\Psi_{\text{MF}}[\lambda]\rangle,$
(2)

where the Jastrow-like energy functional
$V[\\{q_{i}^{-}\\}]=\sum_{i}V(q_{i}^{-})$ will be variational optimized as
well. The objective is to optimize the varitional state by minimizing the
energy expectation value $\langle\Psi[\lambda,V]|H|\Psi[\lambda,V]\rangle$
given the physical Hamiltonian $H$ in Eq. (1).

The variational state in Eq. (2) is designed to reproduce the exact ground
states in both the strong interaction ($J/t\to\infty$) and the weak
interaction ($J/t\to 0$) limits. In the strong interaction limit, the extreme
SMG state corresponds a product of inter-layer singlets, which can be obtained
by taking the limit of strong inter-layer hopping ($\lambda/t\to\infty$)
followed by the hard projection
$\mathrm{e}^{-V[\\{q_{i}^{-}\\}]}\propto\prod_{i}\delta_{q_{i}^{-}=0}$ that
imposes $q_{i}^{-}=0$ on every site. On the other hand, the exact free-fermion
ground state at $J/t=0$ is achieved by turning off $\lambda$ and $V$. This
design ensures that the variational state accurately captures the behavior of
the system in both limits.

For intermediate $J/t$, we evaluated the variational energy
$\langle\Psi[\lambda,V]|H|\Psi[\lambda,V]\rangle$ by the VMC sampling
technique (see Appendix B for algorithm details) and minimized it using the
PyTorch library [29] with the Adam optimizer [30]. This allows us to optimize
the variation state by leveraging the auto-differentiation capabilities of
PyTorch. To verify the accuracy of our algorithm, we conducted a benchmark
test with exact diagonalization in a single unit cell (consisting of two
sites), which shows that our variational ground state can achieve a fidelity
of at least $0.99$ and has a relative energy excess of at most $10^{-4}$
across all values of $J/t$.

Figure 2: (a) Ground state energy $E_{\text{GS}}$ and (b) its first-order
derivative $\partial_{J}E_{\text{GS}}$ with respect to the interaction
strength $J$. Each dot in (a) represents an estimation of $E_{\text{GS}}$ from
one VMC simulation.

###  III.2 Numerical Results

In our study, we set the energy unit to be $t=1$ and plotted the ground state
energy $E_{\text{GS}}$ as a function of the interaction strength $J$ in Fig.
2(a). To estimate $E_{\text{GS}}$ at each value of $J$, we conducted multiple
VMC simulations with different initial random seeds and fit a smooth curve to
the data. The first-order derivative $\partial_{J}E_{\text{GS}}$ was then
computed based on the fitted energy curve, as shown in Fig. 2(b). The
calculations were performed on lattices of $L\times L$ unit cells (totally
$2L^{2}$ sites) for $L=6,7,\cdots,10$. We observed that all curves cross at a
single point, which we identified as the SMG critical point $J_{c}=3.48$. By
rescaling the horizontal axis to $(J-J_{c})L^{1/\nu}$, we were able to
collapse all curves onto a single curve when the exponent was tuned to
$\nu=0.96\pm 0.03$, as shown in the inset of Fig. 2(b). This determines the
power-law scaling of the correlation length $\xi\sim|J-J_{c}|^{-\nu}$ near the
SMG transition.

Figure 3: Fermion-fermion correlation $C(r)$ for (a) $J\leq J_{c}$ and (b)
$J>J_{c}$, computed on the $L=16$ lattice. (c) Fermion scaling dimension
$\Delta_{c}$ for $J\leq J_{c}$.

To investigate the fermion scaling dimension at the critical point, we
measured the fermion-fermion correlation function
$C(r):=\langle\Psi[\lambda,V]|c_{il}^{\dagger}c_{jl}|\Psi[\lambda,V]\rangle$
between two sites $i$ and $j$ as a function of their separation distance
$r=|{\bm{r}}_{i}-{\bm{r}}_{j}|$ on a lattice of size $L=16$. The correlation
function $C(r)$ exhibits a power-law behavior $C(r)=1/r^{2\Delta_{c}}$ for
$J\leq J_{c}$, as plotted in Fig. 3(a) with the log-log scale. The exponent
$\Delta_{c}$ corresponds to the scaling dimension of the fermion operator
$c_{il}$. We found that the scaling dimension in the Dirac semimetal phase (at
$J=2<J_{c}$ for example) is $\Delta_{c}=1.00\pm 0.03$, which is the same as
that of the free Dirac fermion in (2+1)D, indicating that the local
interaction $J$ is perturbatively irrelevant in this phase. However, at the
SMG critical point ($J=J_{c}$), we observed a distinct scaling dimension of
$\Delta_{c}=1.31\pm 0.04$, which suggests a different RG fixed point. In the
gapped phase (at $J=7>J_{c}$ for example), we verified that $C(r)$ becomes
short-ranged and decays exponentially as expected, as plotted in Fig. 3(b)
with the log-linear scale. This clearly signifies that the SMG transition is a
semimetal-insulator (gapless to gapped) transition.

The enlarged fermion scaling dimension at the SMG critical point indicates the
fractionalization of fermions. To validate this observation more
systematically, we extracted the fermion scaling dimension $\Delta_{c}$ by
fitting the power-law correlation at different values of $J\leq J_{c}$ for two
system sizes $L=12,16$, as shown in Fig. 3(c). As the system size increases
(from $L=12$ to $L=16$), we observed the trend that $\Delta_{c}$ approaches 1
for $J<J_{c}$ but seems to converge to some finite value above 1 at $J=J_{c}$.
Limited by the available system sizes and the data quality, we were unable to
extrapolate our results to infinite system size reliably. Nevertheless, the
evidence clearly requires a different conformal field theory (CFT) description
of the SMG critical point distinct from the free fermion CFT.

Figure 4: (a) The inter-layer hopping strength $\lambda$ in the mean-field
Hamiltonian $H_{\text{MF}}[\lambda]$. (b) The gap of Jastrow energy $\Delta
V=V(\pm 2)-V(0)$ between $q^{-}=\pm 2$ and $q^{-}=0$ configurations,
characterizing the strength of the soft projector. (c) The inter-layer local
fermion coherence (tunneling amplitude). (d) The inter-layer local spin-spin
correlation. The results are obtained by finite-size scaling to $L\to\infty$
based on $L=5,\cdots,10$ VMC simulations (see Appendix C).

To better understand the SMG mechanism, we investigated how the variational
parameters change across the SMG transition. We found that the staggered
inter-layer hopping parameter $\lambda$ in the mean-field Hamiltonian
$H_{\text{MF}}[\lambda]$ turns on continuously near the transition, as in Fig.
4(a), resulting in a fermion gap opening in the mean-field state
$|\Psi_{\text{MF}}[\lambda]\rangle$. This gap is sufficient to make the
fermion-fermion correlation $C(r)$ short-ranged and generate a mass for all
fermion excitations in the SMG insulator phase. However, such kind of mass
generation comes with the spontaneous breaking of the layer
$\mathrm{U}(1)^{-}$ global symmetry that is associated with the conservation
of the total inter-layer charge difference $Q^{-}=\sum_{i}q_{i}^{-}$. Under
the $\mathrm{U}(1)^{-}$ symmetry, the inter-layer hopping term
$c_{i1}^{\dagger}c_{i2}$ transforms as a charge-2 operator. On the mean-field
level, a finite $\lambda$ will induce a finite $\langle
c_{i1}^{\dagger}c_{i2}\rangle$ expectation value on the mean-field state
$|\Psi_{\text{MF}}[\lambda]\rangle$, which breaks the $\mathrm{U}(1)^{-}$
symmetry to its $\mathbb{Z}_{2}^{-}$ subgroup.

Fortunately, the correlation factor $\mathrm{e}^{-V[\\{q_{i}^{-}\\}]}$ helps
to restore the $\mathrm{U}(1)^{-}$ symmetry without closing the fermion
excitation gap. Recall that the Jastrow-like energy function
$V[\\{q_{i}^{-}\\}]=\sum_{i}V(q_{i}^{-})$ was written as a sum of on-site
potentials. We found that the local potential difference $\Delta V=V(\pm
2)-V(0)$ increases across the SMG transition, as shown in Fig. 4(b). The
possitive $\Delta V$ favors the $\mathrm{U}(1)^{-}$ neutral ($q_{i}^{-}=0$)
over the $\mathrm{U}(1)^{-}$ charged ($q_{i}^{-}=\pm 2$) configurations on
every site (see Fig. 1(c) for examples). As the potential difference $\Delta
V$ increases, the correlation factor $\mathrm{e}^{-V[\\{q_{i}^{-}\\}]}$
suppresses local $\mathrm{U}(1)^{-}$ charge fluctuations, effectively
introducing $\mathrm{U}(1)^{-}$ gauge (phase) fluctuations [31, 32, 33] due to
the quantum mechanical uncertainty relation between the charge and phase
observables. As a result, the $\langle c_{i1}^{\dagger}c_{i2}\rangle$
expectation value is disordered by the gauge fluctuation, and only those
$\mathrm{U}(1)^{-}$ symmetric observables, such as
$\langle{\bm{S}}_{i1}\cdot{\bm{S}}_{i2}\rangle$, survive the gauge
fluctuation, as clearly shown in Fig. 4(c,d). This establishes the $J>J_{c}$
phase as a _symmetric gapped_ phase, where the fermion mass (gap) is generated
by the condensation of multi-fermion operators (such as
${\bm{S}}_{i1}\cdot{\bm{S}}_{i2}$) instead of two-fermion operators (such as
$c_{i1}^{\dagger}c_{i2}$).

##  IV Field Theory Analysis

The above discussion motivates the fermion fractionalization hypothesis, which
states that the physical fermion $c_{il}$ fractionalizes into deconfined
bosonic $b_{il}$ and fermionic $f_{il}$ partons at (and only at) the SMG
critical point (hence an fDQCP), as

$c_{il}=\begin{bmatrix}c_{il\uparrow}\\\
c_{il\downarrow}\end{bmatrix}=b_{il}^{\dagger}\begin{bmatrix}f_{il\uparrow}\\\
f_{il\downarrow}\end{bmatrix}=b_{il}^{\dagger}f_{il},$ (3)

for each layer $l=1,2$ separately. A parton mean-field analysis can be found
in Appendix D. The fermionic parton $f_{il}$ still carries the
$\mathrm{SU}(2)$ spin quantum number and hops on the honeycomb lattice.
However, the separate charge conservation symmetry in each layer $l$, denoted
as $\mathrm{U}(1)_{l}$, is now acting on the bosonic parton $b_{il}$ only.
Note that the previously mentioned layer $\mathrm{U}(1)^{-}$ symmetry is a
subgroup of the larger $\mathrm{U}(1)_{1}\times\mathrm{U}(1)_{2}$ symmetry.

The partons $b_{il},f_{il}$ are redundant descriptions of the physical fermion
$c_{il}$, because the following transformations have no physical effect on Eq.
(3)

$\tilde{\mathrm{U}}(1)_{l}:b_{il}\to\mathrm{e}^{\mathrm{i}\theta_{il}}b_{il},f_{il}\to\mathrm{e}^{\mathrm{i}\theta_{il}}f_{il}.$
(4)

They are identified as the emergent gauge group
$\tilde{\mathrm{U}}(1)_{1}\times\tilde{\mathrm{U}}(1)_{2}$ arising from the
fermion fractionalization. The low-energy physics of the SMG transition in the
bilayer honeycomb model can then be described by the following field theory in
the parton language

$\mathcal{L}=\sum_{l=1,2}|(\partial-\mathrm{i}(a_{l}-A_{l}))b_{l}|^{2}+r|b_{l}|^{2}+\bar{f}_{l}\,\gamma\cdot(\partial-\mathrm{i}a_{l})f_{l},$
(5)

where $l=1,2$ labels the two layers. Within each layer $l$, the theory
describes a single-flavor ($N_{b}=1$) bosonic scalar field $b_{l}$ coupled to
a four-flavor ($N_{f}=4$) fermionic spinor field $f_{l}$ through the
$\tilde{\mathrm{U}}(1)_{l}$ gauge field $a_{l}$. The four internal flavors of
$f_{l}$ stand for the spin and valley degrees of freedom inherited from the
physical fermion $c_{l}$, and $\gamma$ denotes the $\gamma$ matrices in the
Dirac spinor space. $A_{l}$ is the background gauge field that track the
physical $\mathrm{U}(1)_{l}$ symmetry.

In this field theory Eq. (5), the SMG transition is driven by tuning the only
parameter $r$:

  * •

When $r<0$, the scalar fields $b_{l}$ condense, pinning the gauge fields
$a_{l}$ to the symmetry background fields $A_{l}$ through the Higgs mechanism,
such that the fermionic partons $f_{l}$ regain the
$\mathrm{U}(1)_{1}\times\mathrm{U}(1)_{2}$ symmetry charges and restore the
gapless physical fermions in the weakly-interacting Dirac semimetal phase.

  * •

When $r>0$, the scalar fields $b_{l}$ are gapped and decoupled from the
theory. Driven by the gauge interaction, the fermionic partons can
spontaneously develop a parton Higgs mass
$\lambda(\bar{f}_{1}f_{2}+\text{h.c.})$, which gaps out all fermions and Higgs
the $\tilde{\mathrm{U}}(1)_{1}\times\tilde{\mathrm{U}}(1)_{2}$ gauge structure
down to the diagonal $\tilde{\mathrm{U}}(1)^{+}$ 111The $\mathbb{Z}_{2}$
subgroup of $\tilde{\mathrm{U}}(1)^{-}$ is shared with
$\tilde{\mathrm{U}}(1)^{+}$, so the Higgsing does not lead to $\mathbb{Z}_{2}$
topological order as well.. Then the diagonal $\tilde{\mathrm{U}}(1)^{+}$
gauge field automatically confines at low energy by the Polyakov mechanism
[35], leading to a trivially-gapped SMG phase. Since $A_{l}$ does not couple
to the parton Higgs field $\lambda$, the
$\mathrm{U}(1)_{1}\times\mathrm{U}(1)_{2}$ symmetry remains unbroken even if
$\lambda$ has condensed.

The SMG critical point is therefore described by the field theory Eq. (5) at
$r=0$. The state of various fields across the transition is summarized in Tab.
1

| Dirac semimetal | SMG transition | SMG insulator
---|---|---|---
$c_{l}$ | gapless | fractionalized | gapped
$b_{l}$ | condensed ($r<0$) | critical ($r=0$) | gapped ($r>0$)
$f_{l}$ | gapless ($\lambda=0$) | gapless ($\lambda=0$) | gapped ($\lambda\neq 0$)
$a_{l}$ | Higgsed | deconfined |  $a^{-}$ Higgsed
$a^{+}$ confined
Table 1: State of physical fermion $c_{l}$, bosonic $b_{l}$ and fermionic
$f_{l}$ parton, and gauge fields $a_{l}$ across SMG transition.
$a^{\pm}=a_{1}\pm a_{2}$ are linear combinations of $a_{l}$.

To estimate the scaling dimension $\Delta_{c}$ of the physical fermion
$c_{l}\sim b_{l}^{\dagger}f_{l}$, we extended the quantum electrodynamics
(QED) theory to general $N_{b}$ bosonic flavors and $N_{f}$ fermionic flavors
and used the result of RG analysis [36, 37] by the large-$N_{b},N_{f}$
expansion, which predicted

$\Delta_{c}=\frac{3}{2}+\frac{2}{3\pi^{2}N_{b}}-\frac{40}{3\pi^{2}(N_{b}+N_{f})}+\cdots.$
(6)

Extrapolating the result to $(N_{b},N_{f})=(1,4)$, Eq. (6) predicts
$\Delta_{c}\simeq 1.3>1$, which is consistent with our VMC simulation. This
provides supportive evidence for the fermion fractionalization hypothesis and
establishes the SMG transition as an fDQCP.

##  V Summary

We developed a VMC approach to simulate a bilayer honeycomb lattice model and
investigated the critical behavior of the SMG transition in this model. We
tested the fermion fractionalization hypothesis by measuring the fermion
scaling dimension $\Delta_{c}=1.31\pm 0.04$ at the critical point and showing
that the result is consistent with the prediction for an fDQCP,
$\Delta_{c}\simeq 1.3>1$. The fDQCP field theory indicates that the mean-field
parameter $\lambda$ in the variational state should be interpreted as the
Higgs mass of the fermion parton as $\lambda(\bar{f}_{1}f_{2}+\text{h.c.})$.
The VMC simulation explicitly reveals how the parton Higgs mass $\lambda$ is
generated across the SMG transition in Fig. 4(a), providing a deeper
understanding of the SMG mechanism from the perspective of the parton Higgs
theory [18, 38]. Note that the parameter $\lambda$ is not accessible in other
numerical approaches because it is not a gauge-neutral physical observable.
This demonstrates the unique advantage of the VMC approach in studying SMG
physics. We expect the methodology to apply to more general SMG phenomena in
higher dimensions [39, 40, 41, 42, 43, 38, 44] or Fermi liquids [45, 46],
which could have broader implications for the lattice regularization of chiral
fermions [47, 48, 49, 50, 51, 52, 53, 16], the strong-CP problem [54], and the
pseudo-gap phenomenon [55, 56, 57, 58, 59, 60, 61, 62].

###### Acknowledgements.

We acknowledge the helpful discussions with Tarun Grover, Juven Wang, Da-Chuan
Lu, and Meng Zeng. We thank Peiyuan Wang for independently checking our
calculations in Appendix D. The National Science Foundation supported this
research under Grant DMR-2238360.

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##  Appendix A Symmetry Analysis

The bilayer honeycomb lattice model

$H=-t\sum_{\langle
ij\rangle,l,\sigma}(c^{\dagger}_{il\sigma}c_{jl\sigma}+\text{h.c.})+J\sum_{i}{\bm{S}}_{i1}\cdot{\bm{S}}_{i2},$
(7)

has an
$\mathrm{U}(1)_{1}\times\mathrm{U}(1)_{2}\times\mathrm{SU}(2)\times\mathbb{Z}_{2}^{S}$
internal symmetry together with the honeycomb lattice symmetry. Each symmetry
and its corresponding transformation to the fermionic operator is listed in
the Tab. 2.

Internal symmetries
---
conservation | symmetry group | transformation
intra-layer charge conservation | $\mathrm{U}(1)_{1}\times\mathrm{U}(1)_{2}$ | $c_{il}\rightarrow e^{-\mathrm{i}\theta_{l}}c_{il}$
inter-layer spin conservation | $\mathrm{SU}(2)$ | $c_{il}\rightarrow e^{-\frac{\mathrm{i}}{2}\bm{\theta}\cdot\bm{\sigma}}c_{il}$
sub-lattice charge conjugation | $\mathbb{Z}_{2}^{S}$ |  $c_{il}\rightarrow(-)^{i}c_{il}$ $\mathrm{i}\rightarrow-\mathrm{i}$
Table 2: Internal symmetry of the bilayer honeycomb model.

where $c_{il}=[c_{il\uparrow},c_{il\downarrow}]^{\intercal}$. The sub-lattice
charge-conjugation symmetry $\mathbb{Z}_{2}^{S}$ let the fermion modes on one
sub-lattice pick up a minus sign while keeping the fermion modes on the other
sub-lattice unchanged, then take the complex conjugate of both the fermion
mode and the imaginary unit $\mathrm{i}$. Other than the internal symmetries
listed above, the system also has honeycomb lattice symmetries such as
translation, rotation, reflection etc. Most importantly, the
$\mathbb{Z}_{2}^{S}$ and translation symmetry together rule out the
possibility of adding any fermionic bilinear terms to gap out the Dirac
fermions. Therefore the gapless Dirac semimetal phase is protected by the
symmetries at the free-fermion level.

This statement can be proven more explicitly as follows. When $J=0$, the free
hopping Hamiltonian on honeycomb lattice gives rise to the graphene band
structure that produces $2\times 2\times 2=8$ (two layers, two spins and two
valleys) gapless Dirac fermions $\psi_{Ql\sigma}$ at low energy. They can be
described by the low-energy effective field theory Lagrangian

$\mathcal{L}=\sum_{Ql\sigma}\bar{\psi}_{Ql\sigma}\gamma^{\mu}\partial_{\mu}\psi_{Ql\sigma}$
(8)

where $Q=K,K^{\prime}$ labels the two valleys from fermion doubling and
$\gamma^{\mu}=(\sigma^{2},\sigma^{1},\sigma^{3})$,
$\bar{\psi}_{Ql\sigma}=\psi^{\dagger}_{Ql\sigma}\gamma^{0}$.

Following the existing literature, it is convenient to use the Majorana basis
in Hamiltonian formulation, such that eight complex Dirac fermions can be
described by an effective Hamiltonian with matrices of size $32\times 32$

$\begin{split}H&=\int\mathrm{d}^{2}\bm{x}\chi^{\intercal}h_{\times 32}\chi,\\\
h_{\times
32}&=\sum_{a=1}^{2}\mathrm{i}\partial_{a}\alpha^{a}+\sum_{b}^{5}m_{b}\beta^{b}\end{split}$
(9)

where $\alpha^{a}$ are symmetric matrices, $\beta^{b}$ are anti-symmetric
matrices and all the $\alpha^{a},\beta^{b}$ anti-commute with each other. Here
the $\chi^{\intercal}m_{b}\beta^{b}\chi$ terms represent all the possible
bilinear mass terms that can be added to the effective theory. The strategy is
to list all possible mass terms and then test if they are invariant under the
required symmetries. The complete Hamiltonian with all possible bilinear mass
terms is

$\begin{split}h_{\times
32}&=\mathrm{i}\partial_{1}\sigma^{30000}+\mathrm{i}\partial_{2}\sigma^{13000}+m_{1}\sigma^{20000}\\\
&+m_{2}\sigma^{11120}+m_{3}\sigma^{11200}+m_{4}\sigma^{11320}+m_{5}\sigma^{12000}\end{split}$
(10)

where $\sigma^{ab\cdots}=\sigma^{a}\otimes\sigma^{b}\otimes\cdots$ are the
Pauli matrices.

Among the
$\mathrm{U}(1)_{1}\times\mathrm{U}(1)_{2}\times\mathrm{SU}(2)\times\mathbb{Z}_{2}^{S}$
symmetries and lattice symmetries we need to concern, it turns out just the
translation symmetry plus $\mathbb{Z}_{2}^{S}$ can already rule out all the
bilinear mass terms. Following the existing literature, the translation
symmetry and $\mathbb{Z}_{2}^{S}$ can be combined to create an emergent anti-
unitary symmetry $\mathbb{Z}_{4}^{TF}$ at low-energy:
$\psi_{Ql\sigma}\rightarrow\mathrm{i}\gamma^{0}\psi^{\dagger}_{Ql\sigma},\mathrm{i}\rightarrow-\mathrm{i}$.
Translating this symmetry transformation into Majorana basis becomes

$\begin{split}\mathbb{Z}_{4}^{TF}:\psi_{Ql\sigma}=\begin{pmatrix}\psi_{Ql\sigma
1}\\\ \psi_{Ql\sigma 2}\end{pmatrix}&\rightarrow\begin{pmatrix}-\psi_{Ql\sigma
2}^{\dagger}\\\ \psi_{Ql\sigma
1}^{\dagger}\end{pmatrix},\mathrm{i}\rightarrow-\mathrm{i}\\\
\chi_{Ql\sigma}=\begin{pmatrix}\chi_{Ql\sigma 11}\\\ \chi_{Ql\sigma 12}\\\
\chi_{Ql\sigma 21}\\\ \chi_{Ql\sigma
22}\end{pmatrix}&\rightarrow\begin{pmatrix}-\mathrm{i}\chi_{Ql\sigma 21}\\\
\mathrm{i}\chi_{Ql\sigma 22}\\\ \mathrm{i}\chi_{Ql\sigma 11}\\\
-\mathrm{i}\chi_{Ql\sigma
12}\end{pmatrix},\mathrm{i}\rightarrow-\mathrm{i}\end{split}$ (11)

the symmetry transformation in Majorana basis can be written as
$\mathbb{Z}_{4}^{TF}:\chi_{Ql\sigma}\rightarrow
M\cdot\chi_{Ql\sigma},\mathrm{i}\rightarrow-\mathrm{i}$, where
$M=\sigma^{23000}$. Now we can verify the dynamical part of the Hamiltonian is
invariant under $M^{\dagger}\cdot\mathrm{i}\sigma^{a}\cdot
M,\mathrm{i}\rightarrow-\mathrm{i},(a=30000,13000)$ but fails after acting on
every bilinear mass terms $M^{\dagger}\cdot\sigma^{b}\cdot
M,\mathrm{i}\rightarrow-\mathrm{i},(b=20000,11120,11200,11320,12000)$.

However, the system can still have an on-site Heisenberg interaction while
preserving the internal symmetries, results in a symmetric gap state at strong
coupling limit. Therefore, one can expect a SMG transition between both sides.
Since there lacks a symmetry-breaking order parameter to distinguish the
phases on both sides, such exotic transition is beyond the Landau-Ginzburg-
Wilson paradigm and is interesting to study.

##  Appendix B Variational Monte Carlo Algorithm

###  B.1 Variational ansatz

The VMC simulation is based on a mean-field ansatz

$H_{\text{MF}}=-t\sum_{\langle ij\rangle
l\sigma}c^{\dagger}_{il\sigma}c_{jl\sigma}+\lambda\sum_{i,\sigma}(-)^{i}c^{\dagger}_{i1\sigma}c_{i2\sigma}+\text{h.c.}.$
(12)

The mean-field Hamiltonian $H_{\text{MF}}$ can be solved exactly. Letting
$|\Psi_{\text{MF}}[\lambda]\rangle$ be the ground-state of $H_{\text{MF}}$, we
assume that the true many-body ground state of the original Hamiltonian Eq.
(7) can be modeled by $|\Psi_{\text{MF}}[\lambda]\rangle$ followed by a soft
projection.

$\begin{split}|\Psi[\lambda,V]\rangle=&\mathcal{P}[V]|\Psi_{\text{MF}}[\lambda]\rangle\\\
\mathcal{P}[V]&=\mathcal{P}_{\text{global}}\exp\Big{(}{-\sum_{i}V(q_{i})}\Big{)}\end{split}$
(13)

where $q_{i}=(q_{i}^{+},q_{i}^{-})$ is the local charge vector specified by
on-site state configurations, defined by

$q_{i}^{\pm}:=\sum_{l\sigma}(\pm)^{l}c^{\dagger}_{il\sigma}c_{il\sigma}.$ (14)

In general, to model the true ground state, $V(q_{i})$ is parameterized by a
neural network (a multi-layer feed-forward model) and then optimized by deep
learning. In our case, since the Hamiltonian is invariant under charge
$\mathrm{U}(1)^{+}$ and layer $\mathrm{U}(1)^{-}$ symmetries,

$\begin{split}&Q^{\pm}:=\sum_{i}q_{i}^{\pm}=\sum_{il\sigma}(\pm)^{l}c^{\dagger}_{il\sigma}c_{il\sigma},\\\
&[H,Q^{\pm}]=0.\end{split}$ (15)

The many-body ground state must be in the global charge neutral $Q^{\pm}=0$
subspace, so a global projection operator $\mathcal{P}_{\text{global}}$ is
introduced to project to the $Q^{\pm}=0$ subspace.

###  B.2 Sampling through Markov chain

Following the existing literature, the mean-field wave function can be
computed by an unitary transformation of $H_{\text{MF}}$. Denote the diagonal
form of mean-field Hamiltonian as
$H_{\text{MF}}=\sum_{\mu}\epsilon_{\mu}\gamma^{\dagger}_{\mu}\gamma_{\mu}$ and
eigenstates $|\phi_{\mu}\rangle=\gamma^{\dagger}_{\mu}|0\rangle$, the explicit
form of the unitary matrix of system with $L$ sites and four flavours
($\sigma=\uparrow,\downarrow l=1,2$) is

$U=\begin{pmatrix}\langle 1|\phi_{1}\rangle&\dots&\langle
1|\phi_{4L}\rangle\\\ \vdots&\ddots&\vdots\\\ \langle
4L|\phi_{1}\rangle&\dots&\langle 4L|\phi_{4L}\rangle\end{pmatrix}.$ (16)

The wave-function amplitude can be computed by a Slater determinant

$\begin{split}&\langle x|\Phi_{\text{MF}}\rangle=\langle
x_{1},\dots,x_{N_{e}}|\gamma^{\dagger}_{1},\dots,\gamma^{\dagger}_{N_{e}}|0\rangle\\\
&=\det\begin{pmatrix}\langle x_{1}|\phi_{1}\rangle&\dots&\langle
x_{1}|\phi_{N_{e}}\rangle\\\ \vdots&\ddots&\vdots\\\ \langle
x_{N_{e}}|\phi_{1}\rangle&\dots&\langle
x_{N_{e}}|\phi_{N_{e}}\rangle\end{pmatrix}=\det\mathbf{D}.\end{split}$ (17)

where $N_{e}$ is the electron number at half filling in our model. Introducing
the auxiliary matrix $\mathbf{M}$, the wave-function ratio can be obtained as

$\begin{split}\frac{\langle x^{\prime}|\Phi_{\text{MF}}\rangle}{\langle
x|\Phi_{\text{MF}}\rangle}&=\frac{\det\mathbf{D^{\prime}}}{\det\mathbf{D}}=\sum_{m}D^{-1}_{m\beta}M_{lm}=W_{l\beta},\\\
M&=\begin{pmatrix}\langle 1|\phi_{1}\rangle&\dots&\langle
1|\phi_{N_{e}}\rangle\\\ \vdots&\ddots&\vdots\\\ \langle
4L|\phi_{1}\rangle&\dots&\langle 4L|\phi_{N_{e}}\rangle\end{pmatrix},\\\
\mathbf{W}&=\mathbf{M\cdot D^{-1}}.\end{split}$ (18)

where the configuration $|x^{\prime}\rangle$ is achieved by moving the
$x_{\beta}$ electron on to site $l$. The key point that reduces the
computational complexity is that the $\mathbf{W}$ matrix can be updated base
on what we already have.

$W^{\prime}_{h\alpha}=W_{h\alpha}-\frac{W_{h\beta}}{W_{l\beta}}(W_{l\alpha}-\delta_{\alpha\beta}).$
(19)

Using this update rule, the wave-function amplitude can be computed with least
effort along sampling. The samples are collected through Markov chain, such
that the algorithm is proceeded in the following way:

  1. 1.

Staring from a configuration $|x\rangle$, a new configuration is generated by
transition probability $T(x^{\prime}|x)$.

  2. 2.

Compute probability ratio $A(x^{\prime}|x)=|\frac{p(x^{\prime})}{p(x)}|$, if
$A(x^{\prime}|x)>1$, accept new configuration; if $A(x^{\prime}|x)<1$, accept
new configuration with probability of $A(x^{\prime}|x)$.

  3. 3.

The sampling process is then iterated till equilibrium.

After collecting the configurations along sampling, the physical quantity
$O(x)$ and probability $p(x)$ of each configuration can be computed to obtain
expectation values and the objective function. Finally, the parameters of the
model ($\lambda$ in the mean-field ansatz and several other parameters in the
potential function $V$) will be updated by backward propagation through the
objective function.

###  B.3 Compute expectation value and the objective function

Let ${|x\rangle}_{x\in X}$ be a complete basis of the many-body Hilbert space
(with total electric charge and total valley charge neutral). The expectation
value of any observable $O$ can be written as

$\langle
O\rangle=\frac{\langle\Psi|O|\Psi\rangle}{\langle\Psi|\Psi\rangle}=\sum_{x}O(x)p(x),$
(20)

with

$\begin{split}O(x)=\frac{\langle\Psi|O|x\rangle}{\langle\Psi|x\rangle}&=\sum_{x^{\prime}}\frac{\langle\Psi|x^{\prime}\rangle}{\langle\Psi|x\rangle}\langle
x^{\prime}|O|x\rangle,\\\
p(x)&=\frac{|\langle\Psi|x\rangle|^{2}}{\sum_{x}|\langle\Psi|x\rangle|^{2}}.\end{split}$
(21)

To sample from $p(x)$, following the Markov chain with the transition
probability

$p(x^{\prime}|x)=\frac{p(x^{\prime})}{p(x)}=\left|\frac{\langle\Psi|x^{\prime}\rangle}{\langle\Psi|x\rangle}\right|^{2}.$
(22)

The many-body basis $|x\rangle$ is chosen to be the eigen-basis of the
projection operator $\mathcal{P}[g]$, such that the amplitude ratio can be
evaluated as

$\frac{\langle\Psi|x^{\prime}\rangle}{\langle\Psi|x\rangle}=\frac{\langle\Psi_{\text{MF}}[u]|x^{\prime}\rangle}{\langle\Psi_{\text{MF}}[u]|x\rangle}\frac{\langle
x^{\prime}|\mathcal{P}[E]|x^{\prime}\rangle}{\langle
x|\mathcal{P}[E]|x\rangle}.$ (23)

The Mote Carlo expectation value of the physical quantity is approximated by

$\langle O\rangle=\sum_{x}O(x)p(x)\sim\frac{1}{N}\sum_{x\in\mathcal{S}}O(x)$
(24)

where $\mathcal{S}\in X$ is a multi-subset of all configurations. This
approximation lies on the fact that, through Markov chain sampling, the number
of visits of a configuration $|x\rangle$ is proportional to its probability
$p(x)$. The objective function to miminize is

$\begin{split}\mathcal{L}&=\sum_{x}(\langle H\rangle-H(x))\log
p_{\theta}(x)\\\ \langle
H\rangle&=\sum_{x}H(x)p(x)\sim\frac{1}{N}\sum_{x\in\mathcal{S}}H(x)\end{split}$
(25)

where $\theta$ denotes the set ofs all parameters of the variational model.

##  Appendix C Finite-Size Scaling

Here we provide examples of extrapolating numerical results of finite spacial
system size $(L=5,6,7,8,9,10)$ to one of infinite size. The results of these
extrapolations are used to make Fig. 4.

Figure 5: Extrapolation of Fig. 4(a) Figure 6: Extrapolation of Fig. 4(b)
Figure 7: Extrapolation of Fig. 4(d)

##  Appendix D Mean-Field Analaysis

The fermion fractionalization theory can be used to understand the SMG
transition. Consider writing an physical fermion mode on each site as a
combination of bosonic parton and fermionic parton

$c_{il}=\begin{bmatrix}c_{il\uparrow}\\\
c_{il\downarrow}\end{bmatrix}=b_{il}\begin{bmatrix}f_{il\uparrow}\\\
f_{il\downarrow}\end{bmatrix}=b_{il}f_{il}$ (26)

The original Hamiltonian Eq. (7) can be rewritten as

$H=-t\sum_{\langle ij\rangle
l\sigma}b^{\dagger}_{il}b_{jl}f^{\dagger}_{il\sigma}f_{jl\sigma}+h.c.+J\sum_{i}b^{\dagger}_{i1}b_{i1}b^{\dagger}_{i2}b_{i2}\bm{S}^{f}_{i1}\bm{S}^{f}_{i2}$
(27)

where the fermionic parton spin has the form
$\bm{S}^{f}_{il}=\frac{1}{2}f^{\dagger}_{il\alpha}\sigma^{\alpha\beta}f_{il\beta}$.
The partons are redundant descriptions of the original physical fermion, such
that the following transformation has no physical effect

$\begin{split}&b_{il}\rightarrow\mathrm{e}^{-\mathrm{i}\theta_{il}}b_{il}\\\
&f_{il}\rightarrow\mathrm{e}^{\mathrm{i}\theta_{il}}f_{il}\end{split}$ (28)

An emergent gauge group
$\tilde{\mathrm{U}}(1)_{1}\times\tilde{\mathrm{U}}(1)_{2}$ arises from the
fermion fractionalization, the gift of this redundant description is that one
can assign the original $\mathrm{U}(1)_{l}$ layer charge onto bosonic parton
and write down a bilinear term for fermionic parton without breaking any
original symmetry. The charge assignment for physical fermions and
bosonic/fermionic partons are specified in Tab. 3.

| $\tilde{\mathrm{U}}(1)_{1}$ | $\tilde{\mathrm{U}}(1)_{2}$ | $\mathrm{U}(1)_{1}$ | $\mathrm{U}(1)_{2}$ | $\mathrm{SU}(2)$
---|---|---|---|---|---
$c_{i1}$ | 0 | 0 | 1 | 0 | 2
$c_{i2}$ | 0 | 0 | 0 | 1 | 2
$m_{i}$ | 0 | 0 | 1 | -1 | 1
$b_{i1}$ | -1 | 0 | 1 | 0 | 1
$b_{i2}$ | 0 | -1 | 0 | 1 | 1
$f_{i1}$ | 1 | 0 | 0 | 0 | 2
$f_{i2}$ | 0 | 1 | 0 | 0 | 2
$M_{i}$ | 1 | -1 | 0 | 0 | 1
Table 3: Charge assignment of physical fermion and bosonic/fermionic parton.
$m_{i}$ and $M_{i}$ are the bilinear mass of physical fermion and fermionic
parton respectively.

We present a mean-field analysis of the fractionalized Hamiltonian Eq. (27).
The Hamiltonian can be split into bosonic parton part and fermionic parton
part under mean-field approximation, such that $H_{MF}=H_{b}+H_{f}$. The
Heisenberg interaction $\bm{S}_{i1}\bm{S}_{i2}$ can be further simplified as
spin-exchange interaction $f^{\dagger}_{i2}f_{i1}f^{\dagger}_{i1}f_{i2}$
between layers, which plays the same role of spin-spin interaction with only
$XY$ components. The bosonic parton Hilbert space is simplified as the Hilbert
space of quantum rotor denoted as
$b^{\dagger}\sim\mathrm{e}^{\mathrm{i}\theta_{il}}$.

$\langle\mathrm{e}^{\mathrm{i}\theta_{il}}\rangle$ | $\phi$
---|---
$\langle\mathrm{e}^{-\mathrm{i}\theta_{i1}}\mathrm{e}^{\mathrm{i}\theta_{i1}}\mathrm{e}^{-\mathrm{i}\theta_{i2}}\mathrm{e}^{\mathrm{i}\theta_{i2}}\rangle$ | $\psi$
$\langle f^{\dagger}_{il\sigma}f_{jl\sigma}\rangle$ | $u$
$\langle f^{\dagger}_{i2}f_{i1}\rangle$ | $(-)^{i}2M$
Table 4: Mean-field change of variables

After some change of variables, the mean-field bosonic Hamiltonian becomes on-
site and the mean-field fermionic Hamiltonian is quadratic, such that each can
be solved self consistently.

$\begin{split}H_{b}&=\sum_{i}-6t\phi
u\mathrm{e}^{\mathrm{i}\theta_{il}}+h.c.+12t|\phi|^{2}u-J\mathrm{e}^{-\mathrm{i}\theta_{i1}}\mathrm{e}^{\mathrm{i}\theta_{i1}}\mathrm{e}^{-\mathrm{i}\theta_{i2}}\mathrm{e}^{\mathrm{i}\theta_{i2}}|M|^{2},\\\
H_{f}&=-t\sum_{\langle ij\rangle
l\sigma}|\phi|^{2}f^{\dagger}_{il\sigma}f_{jl\sigma}+h.c.+\frac{J}{2}\psi\sum_{i}(|M|^{2}-(-)^{i}Mf^{\dagger}_{i1}f_{i2}-(-)^{i}M^{*}f^{\dagger}_{i2}f_{i1}),\end{split}$
(29)

where $|\phi|^{2}$ and $|M|^{2}$ play the role of order parameters in bosonic
parton and fermionic parton respectively, and the factor of $6=3\times 2$ in
$H_{b}$ is from the number of bonds on the honeycomb lattice for each site.
The ground-state energy per site for each part is

$\begin{split}E_{b}&=\frac{1}{2}(-J|M|^{2}-24tu|\phi|^{2}-\sqrt{J^{2}|M|^{4}+576t^{2}u^{2}|\phi|^{2}}),\\\
E_{f}&=\frac{1}{N}\sum^{BZ}_{\bm{k}}J\psi|M|^{2}-\sqrt{J^{2}\psi^{2}|M|^{2}+4t^{2}|\phi|^{4}|f(\bm{k})|^{2}}.\end{split}$
(30)

In each part, $|\phi|^{2}$ and $|M|^{2}$ are computed by minimizing $E_{b}$
and $E_{f}$ respectively, then the expectation values can be computed self-
consistantly as
$u=\frac{1}{N}\sum^{BZ}_{\bm{k}}\frac{t|\phi|^{2}|f(\bm{k})|^{2}}{3\sqrt{J^{2}\psi^{2}|M|^{2}+4t^{2}|\phi|^{4}|f(\bm{k})|^{2}}}$
and
$\psi=\frac{1}{2}(1+\frac{J|M|^{2}}{\sqrt{J^{2}|M|^{4}+576t^{2}u^{2}|\phi|^{2}}})$.

The ground-state optimization is implemented by:

  1. 1.

Input $u$ and $|M|^{2}$ to $E_{b}$ then minimize $E_{b}\big{|}_{(u,|M|^{2})}$
to get $|\phi|^{2}$, $\psi$.

  2. 2.

Input $|\phi|^{2}$ and $\psi$ to $E_{f}$ then minimize
$E_{f}\big{|}_{(|\phi|^{2},\psi)}$ to get $|M|^{2}$, $u$.

  3. 3.

Iterate till equilibrium.

The result is shown in Fig. 8.

Figure 8: Mean-field result and phase diagram.

  * •

In the weak coupling limit, the fermion site-hopping expectation value
recovers the result of the Dirac semimetal $u\sim-0.26$, the inter-layer
hopping amplitude $|M|^{2}$ is $0$, and bosonic parton is at super-fluid state
$|\phi|^{2}>0$. This corresponds to the gapless Dirac semimetal phase.

  * •

In the strong coupling limit, the fermionic parton opens up inter-layer
hopping order $|M|^{2}>0$, and the bosonic parton is at gap state
$|\phi|^{2}=0$. Although the fermionic parton bilinear mass is condensed

$H_{\text{M}}=\sum_{i}M_{i}f^{\dagger}_{i1}f_{i2}+h.c.,$ (31)

it does not break any symmetry, but only to Higgs down the gauge group. The
resulting state is symmetric and gapped (for both bosonic and fermionic
partons), which corresponds to the SMG insulator phase.

  * •

However, the mean-field theory also predicts an intermediate phase, where
$|M|^{2}>0$ and $|\phi|^{2}>0$, then $\phi M=m$ will combine into the physical
fermion bilinear mass term $m$, which breaks the physical
$\mathrm{U}(1)_{1}\times\mathrm{U}(1)_{2}$ symmetry. This corresponds to a
spontaneous symmetry breaking (SSB) phase, where the system develops inter-
layer exciton order. Nevertheless, as we have shown in the main text, the more
reliable VMC simulation rules out such an intermediate phase.

Figure 9: Zoom-in version of the bosonic order parameter

In the mean-field theory, the gapless-SSB transition is continuous, see Fig.
9, while the SSB-SMG transition is of first-order (which is obvious from the
jump of mean-field parameters).


# Non-local computation of quantum circuits with small light cones

Kfir Dolev  Stanford University  Sam Cree  Stanford University

###### Abstract

The task of non-local quantum computation requires implementation of a unitary
on $n$ qubits between two parties with only one round of communication,
ideally with minimal pre-shared entanglement. We introduce a new protocol that
makes use of the fact that port-based teleportation costs much less
entanglement when done only on a small number of qubits at a time. Whereas
previous protocols have entanglement cost independent of the unitary or
scaling with its complexity, the cost of the new protocol scales with the
_non-locality_ of the unitary. Specifically, it takes the form $\sim n^{4V}$
with $V$ the maximum volume of a past light cone in a circuit implementing the
unitary. Thus we can implement unitary circuits with $V\sim O(1)$ using
polynomial entanglement, and those with $V\sim\operatorname{polylog}(n)$ using
quasi-polynomial entanglement. For a general unitary circuit with $d$ layers
of $k$-qubit gates $V$ is at most $k^{d}$, but if geometric locality is
imposed it is at most polynomial in $d$. We give an explicit class of
unitaries for which our protocol’s entanglement cost scales better than any
known protocol. We also show that several extensions can be made without
significantly affecting the entanglement cost – arbitrary local pre- and post-
processing; global Clifford pre- and post-processing; and the addition of a
polynomial number of auxiliary systems.

###### Contents

  1. 1 Introduction
  2. 2 Background
    1. 2.1 Non-local quantum computation
    2. 2.2 Teleportation primitives
    3. 2.3 Best known protocol for general unitary
  3. 3 The protocol and its cost
    1. 3.1 The protocol
    2. 3.2 Entanglement cost
    3. 3.3 Regime of efficiency
    4. 3.4 Distinction from other protocols
  4. 4 Enhancements
    1. 4.1 Local pre/post-processing
    2. 4.2 Clifford sandwich
    3. 4.3 Auxiliary systems
  5. 5 Conclusion
  6. A Protocol error bounds

##  1 Introduction

Non-local quantum computation (NLQC) is a quantum task with both practical
applications and relevance to fundamental physics [1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12]. Practically, it is a fundamental primitive for position-based
quantum cryptography [13, 3]. In fundamental physics, it appears to have deep
connection with the holographic principle [14, 15].

The NLQC task involves two individuals that are given a quantum state, and
must apply to it a given quantum computation with only one round of
communication. Critically, it can be shown that for some unitaries the two
individuals can only accomplish this if they share a certain amount of
entanglement ahead of time [14]. Knowing exactly how much entanglement is
necessary is the crux of determining whether a large class of position-based
cryptography schemes is viable.

Currently, linear lower bounds [14, 16] and exponential upper bounds [4] are
known for the number of shared Bell pairs needed as a function of the number
of qubits the computation acts on. However, if the structure of the
computation is taken into account the upper bounds may be tightened. For
example, there are protocols that need only entanglement exponential in the
T-depth or T-count of the unitary [17], and others that can efficiently
perform a certain class of unitaries related to classical complexity classes
[18] and [19]. Furthermore, arguments from high energy physics are suggestive
that the entanglement cost upper bound should be polynomial in the circuit
depth for any unitary [20].

In this work we introduce a new protocol capable of efficiently performing
NLQC for a new class of unitaries. Our protocol is a generalization of the one
introduced in Ref. [4], which relies on a form of quantum teleportation known
as _port teleportation_ [21]. Unlike standard quantum teleportation, in which
a teleported system is encrypted by the presence of an unknown Pauli operator,
in port-based teleportation the teleported system is encrypted in that it
appears in one of many systems or “ports”, the identity of which is unknown to
the receiver. Unitaries can be easily implemented on a port-teleported system
by transversally applying the unitary in question to all possible ports,
whereas after standard teleportation the receiver would need to first undo an
unknown Pauli encryption. However, easier implementation of unitaries comes at
the cost of requiring entanglement exponential in the number of qubits being
teleported, which is the origin of the exponential cost of the protocol in
Ref. [4]. By contrast, our protocol makes use of the circuit structure of a
unitary, allowing us to port-teleport only a small batch of qubits at a time.

At a high level, our protocol consists of Alice and Bob iterating through the
gates of the circuit, teleporting back and forth the batch of qubits on which
each gate acts. Alice always normal-teleports to Bob, while Bob always port-
teleports to Alice. In this way Bob always knows the identity of the correct
ports, and thus can port-teleport the right batch of systems to Alice.
Meanwhile Alice always knows the identity of the Pauli encryptions, and thus
can remove them to perform computations. However, because Alice does not know
the identity of the correct port, she must normal-teleport all ports back to
Bob, resulting in her obtaining Pauli keys that depend on the port, and thus
she does not know which key to use when receiving the system at a later step.
We deal with this issue via the re-indexing technique of [22], i.e. by having
Bob port-teleport the system through a shared resource labeled by the correct
port. Thus when Bob port-teleports a new batch of qubits to Alice, they must
have set up a resource state for each combination of port values of all the
port-based teleportations in the causal past of these qubits. The number of
port teleportation resource states required to do this is proportional to an
exponential of the number of such teleportations, which is the number of gates
in the past light cone of the current gate. Furthermore, the size of the port
teleportation resource state scales exponentially with $k$, the number of
qubits each gate in the circuit acts on. This results in a total entanglement
cost scaling like $n^{4V}$, with $V$ the volume of the largest past light cone
in the circuit – that is, the total number of qubits acted on by gates in the
causal past of a given gate, where the same qubit can be counted multiple
times.

In the worst case, the past light cone volume scales as $k^{d}$ with $k$ the
number of qubits each gate in the circuit acts on, and $d$ the depth of the
circuit. Since the base of the exponent scales with $n$, functions
implementable with only polynomial entanglement must have $O(1)$ layers of
$O(1)$-sized gates. However, the number of efficiently implementable unitaries
is much larger if we mildly relax the notion of “efficient” to include quasi-
polynomial entanglement – that is, entanglement that scales as an exponential
of a polylogarithmic function. This class grows only mildly faster than
polynomials, and still much slower than exponential functions111For example,
$n^{m}$ is larger than $\exp(\log(n)^{2})$ until $n=e^{m}$.. One class of
unitaries that our protocol efficiently performs (i.e. using a quasi-
polynomial number of Bell pairs) is constant-depth circuits of polylogarithmic
sized gates.

In section 2, we formally introduce the problem of NLQC and the primitives of
normal and port-based teleportation, and review the protocol of [4] that ours
generalizes. We present the protocol in section 3 and analyze its performance,
calculating and analyzing the entanglement cost and comparing with existing
protocols. In section 4, we give several enhancements to the protocol that
increase the class of unitaries it can perform efficiently: (i) arbitrary
local pre- and post-processing, whereby Alice and Bob are free to act on their
initial states before and after the protocol; (ii) arbitrary global Clifford
pre- and post-processing; and (iii) use of a polynomial number of auxiliary
registers.

##  2 Background

###  2.1 Non-local quantum computation

A non-local quantum computation (NLQC) task involves two parties, Alice and
Bob. At the start, Alice holds a quantum systems $A$ and $R_{A}$ while Bob
holds $B$ and $R_{B}$. The systems $AB$ collectively contain $n$ qubits, and
for simplicity we assume each party has $n/2$ qubits. They are also
potentially entangled with a reference state $R$ that neither of them hold.
These start out in a state $\ket{\psi}_{ABR}\otimes\ket{\phi}_{R_{A}R_{B}}$.
The state $\ket{\psi}_{ABR}$ is unknown to Alice and Bob, but they are free to
choose the “resource” state $\ket{\phi}_{R_{A}R_{B}}$. Alice and Bob are told
of a unitary $U_{AB}$ acting on the system $AB$, which they are to perform in
the following restricted manner. First, Alice performs a quantum channel of
her choice $\mathcal{N}_{AR_{A}\rightarrow A^{\prime}M_{A}}$ on the systems
she holds to produce an output system $A^{\prime}$ that she keeps and a
message system $M_{A}$ that she sends to Bob. Similarly Bob performs
$\mathcal{N}_{BR_{B}\rightarrow B^{\prime}M_{B}}$ to obtain $B^{\prime}$ and
$M_{B}$. Critically, the message systems do not interact during the exchange,
which is why the computation is “non-local”. Alice then performs a channel
$\mathcal{N}_{A^{\prime}M_{B}\rightarrow A}$ and Bob the channel
$\mathcal{N}_{B^{\prime}M_{A}\rightarrow B}$. They succeed in the task if the
final state of $ABR$ is $U_{AB}\ket{\psi}_{ABR}$. For brevity, we henceforth
omit the reference state $R$ and refer to $AB$ as a pure state; however our
results still apply in the more general setting.

###  2.2 Teleportation primitives

We make heavy use of two types of quantum teleportation: normal teleportation
[23] and port-based teleportation [21].

######  Primitive 1.

Normal teleportation

  1. 1.

Setup: Alice holds an $n_{A}$-qubit system $A$ which, together with some
reference system $R$ is in a state $\ket{\psi}_{AR}$. Alice and Bob each hold
one of a maximally entangled pair of $2^{n_{A}}$ dimensional systems
$A^{\prime}B^{\prime}$.

  2. 2.

Alice performs some joint measurement on $A$ and $A^{\prime}$, receiving the
identity of an $n_{A}$-qubit Pauli operator $P$ drawn uniformly from all such
operators.

  3. 3.

Bob’s half of the maximally entangled state, together with the reference
system, is then in the state $P\ket{\psi}_{B^{\prime}R}$, where $P$ acts on
the $B^{\prime}$ subsystem.

######  Primitive 2.

Port-based teleportation

  1. 1.

Setup: Alice holds an $n_{A}$-qubit system $A$ which, together with some
reference system $R$ is in a state $\ket{\psi}_{AR}$. Alice and Bob each hold
one half of $N$ maximally entangled pairs of $2^{n_{A}}$ dimensional systems
$\\{A^{\prime}_{x}B^{\prime}_{x}\\}$.

  2. 2.

Alice performs some joint measurement on $A$ and $\\{A^{\prime}_{x}\\}$,
receiving a random number $x^{*}\in[N]$222We use the notation $[N]$ to denote
the set $\\{1,\ldots,N\\}.$.

  3. 3.

Bob’s half of the $x^{*}$th entangled system, together with the reference
system, is then close to the state $\ket{\psi}_{B_{x^{*}}R}$, with error
decreasing with $N$.

Specifically, if
$\mathcal{E}:\mathcal{B}(A)\rightarrow\mathcal{B}(B^{\prime}_{x^{*}})$ is the
CPTP map describing the error channel, then from [4] we have

$\displaystyle||\mathcal{E}-\mathcal{I}_{\mathbb{C}^{2^{\scriptstyle{n}_{\\!\text{\tiny{$A$}}}}}}||_{\diamond}\leq
4\frac{2^{2n_{A}}}{\sqrt{N}}.$ (1)

We say that the system $A$ has been _port-teleported_.

###  2.3 Best known protocol for general unitary

The best known NLQC protocol for a generic unitary is due to Beigi and Koenig
in Ref. [4], and proceeds as follows.

######  Protocol 3.

Beigi-Koenig

  1. 1.

Alice normal-teleports $A$ to Bob, receiving Pauli correction $P$.

  2. 2.

Bob port-teleports the $n$-qubit system $AB$ to Alice, recieving port key
$x^{*}$.

  3. 3.

Alice then applies $P^{\dagger}$ to the $A$ part of all the ports on her side.
She then applies $U$ to every port.

  4. 4.

_Communication Round:_ Alice sends the $B$ side of every port to Bob, and Bob
sends $x^{*}$ to Alice. Both sides then throw away all systems not coming from
the $x^{*}$ port, completing the protocol.

From eq. 1 we see that this protocol requires
$N=16\frac{2^{4n}}{\epsilon^{2}}$ in order to obtain some constant error
$\epsilon$.

##  3 The protocol and its cost

###  3.1 The protocol

Because of the important role that circuit decompositions of a unitary plays
in our protocol, we carefully define some formal terminology for use
throughout.

######  Definition 4.

Let a _circuit decomposition_ of $U$ be
$\\{U_{j}^{i}\\}_{i\in[d],j\in[n_{i}]}$, meaning $U=U^{d}\cdots U^{1}$, with
each $U^{i}$ being a product of $n_{i}$ gates $U^{i}=U^{i}_{1}\cdots
U^{i}_{n_{i}}$ each with disjoint support on $k_{i,j}$ qubits. Denote the
support333By support we mean the set of qubits that the operator acts on. of
$U^{i}_{j}$ as $S_{j}^{i}=\operatorname{supp}(U^{i}_{j})$, the collection of
which we refer to as the circuit structure
$\\{S_{j}^{i}\\}_{i\in[d],j\in[n_{i}]}$.

We will see that the circuit structure completely specifies the entanglement
cost of our protocol. Specifically, it depends on causal properties of the
circuit structure that we characterize as follows.

For low-depth circuits, a given gate acts on qubits whose state only depends
on a small number of previous gates (as well as the initial state), sometimes
called the past light cone. We call the subsystems corresponding to these
gates in the past light cone the _ancestors_ of the subsystem in question,
defined formally as follows, and depicted in fig. 1.

######  Definition 5.

For a circuit structure $\\{S_{j}^{i}\\}_{i\in[d],j\in[n_{i}]}$, we denote the
_parents_ $\operatorname{par}(i,j)$ of a subsystem $S_{j}^{i}$ at circuit
layer $i$ and with subsystem label $j$, to be the set of all subsystems from
the previous layer $S_{j^{\prime}}^{i-1}$ with nonzero overlap, i.e.
$S_{j}^{i}\cap S_{j^{\prime}}^{i-1}\neq\emptyset$. The _ancestors_
$\operatorname{anc}(i,j)$ of $S_{j}^{i}$ is then the set of all parents,
parents’ parents, and so on; formally, it is defined recursively as

$\displaystyle\operatorname{anc}(i,j)=\operatorname{par}(i,j)\cup\operatorname{anc}(\operatorname{par}(i,j))$

Finally, we refer to the union of a subsystem with its ancestors as its
_family_ , $\operatorname{fam}(i,j)=\operatorname{anc}(i,j)\cup\\{(i,j)\\}$.

Figure 1: Various sets of interest for gate $(4,1)$. Blue: its family. Yellow:
its ancestors. Red: its parents.

Now we have the notation needed to formally state the general protocol. In
this protocol, steps 2-5 are looped over many times – once per gate in the
circuit structure. Some parts of these steps are only relevant on iterations
corresponding to gates beyond the first layer of the circuit, and are reliant
on later steps that occurred in previous iterations. To aid the reader in such
cases, we separately express the simpler form of those steps in the first
layer.

######  Protocol 6.

Setup: $N$ is fixed to be a constant that controls the accuracy of the
protocol. We also have a fixed circuit structure $\\{S_{j}^{i}\\}$
corresponding to a circuit decomposition $\\{U_{j}^{i}\\}$ of the unitary to
be implemented, $U$. For each $(i,j)$, Alice and Bob allocate a set of
$2N^{|\operatorname{fam}(i,j)|}$ maximally entangled pairs of
$2^{k_{i,j}}$-dimensional systems, labelled by
$(i,j,x_{\operatorname{fam}(i,j)},m)$. Here the notation
$x_{\operatorname{fam}(i,j)}$ denotes a set of labels
$x^{i^{\prime}}_{j^{\prime}}\in[N]$ for each
$(i^{\prime},j^{\prime})\in\operatorname{fam}(i,j)$, and we use similar
notation throughout the protocol. The label $m\in\\{A,B\\}$ specifies whether
the system is used for port-based teleportation from Bob to Alice $(B)$ or for
normal teleportation from Alice to Bob $(A)$. We refer to Alice’s share of an
entangled pair $(i,j,x_{\operatorname{fam}(i,j)},m)$ as
$A(i,j,x_{\operatorname{fam}(i,j)},m)$, and similarly Bob’s share as
$B(i,j,x_{\operatorname{fam}(i,j)},m)$. A further $n/2$ qubits are allocated
for an initial teleportation from Alice to Bob. The protocol then proceeds as
follows.

  1. 1.

Alice normal-teleports her $n/2$ qubits to Bob, resulting in him holding state
$P\ket{\psi}$ with $P$ a Pauli known by Alice. It remains to implement
$UP^{\dagger}$. Begin by setting $i=j=1$.

  2. 2.

First layer: Bob port-teleports $S_{j}^{1}$ to Alice through the $N$ systems
labelled $B(1,j,x_{j}^{1},B)$, from which he gets a key $x_{j}^{1*}\in[N]$.
Subsequent layers: Bob port-teleports $S_{j}^{i}$ to Alice through the $N$
systems labelled $B(i,j,(x_{j}^{i},x^{*}_{\operatorname{anc}(i,j)}),B)$444This
is $N$ ports since only $x_{j}^{i}$ varies; the label for each ancestor is
fixed by the keys of previous port-based teleportations., from which he gets a
key $x_{j}^{i*}\in[N]$. This is depicted in fig. 2.

  3. 3.

First layer: To each of the $N$ systems labeled $A(1,j,x_{j}^{i},B)$, Alice
applies $U^{1}_{j}P^{\dagger}|_{S_{j}^{1}}$555For a Pauli string $P^{\prime}$,
$P^{\prime}|_{S^{i}_{j}}$ means the string obtained by keeping only the part
of the string supported on ${S^{i}_{j}}$..
Subsequent layers: To each of the $N^{|\operatorname{fam}(i,j)|}$ systems
labeled $A(i,j,x_{\operatorname{fam}(i,j)},B)$, Alice applies
$U^{i}_{j}\displaystyle{\prod_{(i^{\prime},j^{\prime})\in\operatorname{par}(i,j)}P^{\dagger}_{i^{\prime},j^{\prime}}(x_{\operatorname{fam}(i^{\prime},j^{\prime})})|_{S^{i}_{j}}}$666Note
that the notation
$P^{\dagger}_{i^{\prime},j^{\prime}}(x_{\operatorname{fam}(i^{\prime},j^{\prime})})$
is defined in step 4; it is not used until the second layer.; that is, she
looks up the normal-teleportation key corresponding to the values of the
variables in $x_{\operatorname{fam}(i^{\prime},j^{\prime})}\subset
x_{\operatorname{fam}(i,j)}$ to decrypt each port.

  4. 4.

Alice normal-teleports each system back to Bob using the entangled pair
labelled $A(i,j,x_{\operatorname{fam}(i,j)},A)$, receiving a separate key for
each port. She labels these keys according to the corresponding port label,
$x_{\operatorname{fam}(i,j)}$. She then has a lookup function
$P_{i,j}(x_{\operatorname{fam}(i,j)})$ that gives her the appropriate Pauli to
be used in subsequent instances of step 3.

  5. 5.

Bob discards all the systems $B(i,j,x_{\operatorname{fam}(i,j)},A)$ not
labelled $x^{*}_{\operatorname{fam}(i,j)}$. Repeat from step 2 for all $j$
from $1$ to $n_{i}$, and then for all $i$ from $1$ to $d$.

  6. 6.

_Communication Round:_ Bob sends Alice’s $n/2$ qubits back to her, and Alice
and Bob send all keys to one another.

  7. 7.

Both players fully decrypt their systems to approximately obtain the state
$U\ket{\psi}_{AB}$.

Figure 2: Port-based teleportation step of the protocol for various gates. The
gate under consideration is highlighted in yellow, and the arrows labelled PT
denote the result of port-based teleportation. Top left: only the resources
for performing one port-based teleportation are needed for gates in the first
layer. Bob port-teleports the two qubits the gate acts on, $S^{1}_{1}$, and
learns that they have arrived on Alice’s end at port number $x^{1*}_{1}=2$.
Top right: Bob port-teleports the two qubits in $S^{1}_{2}$ and learns they
arrive at port $x^{1*}_{2}=1$. Bottom: the systems on which this gate acts
have already been port-teleported before, and Alice needs to know the value of
the correct port for each of the previous teleportations in order to know
which Pauli keys to apply. For this reason they set up resources for every
possible combination of values. Bob then port-teleports the qubits $S^{2}_{1}$
through the resource labeled by $x^{1}_{1}=x^{1*}_{1}=1$ and
$x^{1}_{2}=x^{1*}_{2}=1$.

In appendix A we show that the diamond norm between the channel $\mathcal{N}$
implemented by this protocol and the desired unitary channel $\mathcal{U}$ is
bounded as

$\displaystyle\|\mathcal{N}-\mathcal{U}\|_{\diamond}\leq\epsilon,$ (2)

so long as we choose

$N\geq\left(\frac{4}{\epsilon}\sum_{i,j}2^{2k_{i,j}}\right)^{2}.$ (3)

###  3.2 Entanglement cost

Let us determine the entanglement cost of this protocol, ignoring the
$\frac{n}{2}$ term from the initial normal teleportation from Alice to Bob,
which is subleading in all regimes of interest. We need
$2N^{|\operatorname{fam}(i,j)|}$ sets of maximally entangled pairs of
$k_{i,j}$-qubit systems for each combination of $i$ and $j$. Thus, our total
entanglement cost as measured in Bell pairs is

$\displaystyle E$
$\displaystyle=2\sum_{i,j}k_{i,j}N^{|\operatorname{fam}(i,j)|}.$ (4)

We emphasize that this depends only on the circuit structure $\\{S_{j}^{i}\\}$
and not any other properties of the unitary.

For simplicity we henceforth assume that $k$ is constant. We also define the
maximal past light cone volume as $V:=k\max_{i,j}|\operatorname{fam}(i,j)|$.
Using $V$ as an upper bound for $|\operatorname{fam}(i,j)|$ and substituting
the expression for $N$ in eq. 3 gives

$\displaystyle E$ $\displaystyle\leq 2nd\left(\frac{4nd}{\epsilon
k}2^{2k}\right)^{\frac{2V}{k}}.$ (5)

This is essentially dominated by the scaling

$\displaystyle E$ $\displaystyle\lesssim O\left(n^{4V}\right),$ (6)

where we treat $\epsilon$ as a constant and ignore sub-leading dependencies in
each variable. Here we have also made use of $V\geq d$ and used $n2^{2k}\leq
n^{2k}$ which is always true for large $n$ regardless of how $k$ might scale
with $n$. These inequalities only mildly weaken the bound in eq. 6.

Firstly, if the circuit has generic all-to-all interactions with $d$ layers of
$k$-local gates, then $V=\sum_{i=1}^{d}k^{i}\sim k^{d}$, and so $E$ is bounded
by

$\displaystyle E$ $\displaystyle\sim O\quantity(n^{4k^{d}}).$ (7)

In the case that the circuit has a geometrically local structure in
$D$-dimensional Euclidean space, then $V=\alpha kd^{D+1}$ for some constant
$\alpha$, and so

$\displaystyle E$ $\displaystyle\sim O\quantity(n^{4\alpha kd^{D+1}}).$ (8)

###  3.3 Regime of efficiency

From the expression eqs. 8 and 7, we can make the following conclusions. A
unitary $U$ can be computed non-locally with polynomial entanglement if it can
be implemented by a circuit with $k\sim O(1)$ and $d\sim O(1)$. Furthermore,
it can be computed with _quasi-polynomial_ entanglement if any of the
following hold:

  1. 1.

$k\sim\operatorname{polylog}n$, $d\sim O(1)$,

  2. 2.

$k\sim O(1)$, $d\sim\log\log n$, or

  3. 3.

$k\sim\operatorname{polylog}n$, $d\sim\operatorname{polylog}n$, and the
circuit has geometrically local structure in $D$ spatial dimensions.

Without geometric locality but with $k\sim\operatorname{polylog}n$ and
$d\sim\log\log n$, the entanglement scales as
$\exp\exp\operatorname{polylog}\log n$, which can be thought of as “quasi-
quasi-polynomial”, and is still much slower than exponential growth.

Consider the classes described by cases 1 and 3 above. To the authors’
knowledge, there has been essentially no work studying circuits with such
“medium” sized gates, and thus new methods are needed to characterize these
classes of unitaries. They are both strictly contained in those unitaries
implementable by quasi-polynomial depth circuits of $O(1)$-local gates. This
follows by appealing to Ref. [24], which shows that a $k$-qubit gate can be
decomposed into $4^{k}$ layers of $2$-qubit gates.

One can also use non-rigorous counting arguments of the sort found in Ref.
[25] to characterize the size of such a class. These arguments characterize
the “number of unitaries” in a set as the ratio of the volume of the set
according to some metric to the volume of a ball of radius $\epsilon$.
Including combinatorial factors, one can show that the log-number of distinct
circuits777Note that the number of distinct circuits is larger than the number
of distinct unitaries implemented by such a circuit; however for $d$ less than
exponential in $n$ these numbers are believed to be similar [25]. with $d$
layers of $k$-qubit gates on $n$ total qubits scales roughly as

$\displaystyle\log|\operatorname{circ}(n,k,d)|\sim
nd\left[\log(n)+4^{k}\right].$ (9)

This indicates that, for example, the asymptotic scaling of the log number of
unitaries implementable in class 1 above is comparable to that of the
unitaries implementable by quasi-polynomial-depth circuits of $2$-qubit gates.

###  3.4 Distinction from other protocols

We will now show that the class of unitaries that our protocol can efficiently
implement contains unitaries not achievable by other protocols. As we have
already seen, the protocol of Ref. [4] requires entanglement exponential in
the number of qubits regardless of the unitary. Thus our protocol outperforms
this one for circuits with sufficiently small light cones.

The other protocol for general unitaries that outperforms the one from Ref.
[4] in some regimes is that of Ref. [17]. There they provide two protocols
whose entanglement cost is determined by Clifford + T circuit decompositions.
One protocol scales as $n2^{k_{t}}$ where $k_{t}$ is the number of $T$ gates
in the circuit. The other scales as $(68n)^{d_{t}}$ where $d_{t}$ is the
number of circuit layers containing a $T$ gate, i.e. the $T$-depth. We will
show that our protocol can efficiently implement a unitary with $k_{t}$ and
$d_{t}$ polynomial in $n$, a regime where the aforementioned protocols perform
poorly.

The simplest example of such a unitary is one that factorizes into unitaries
acting on $O(\log n)$ qubits, with each of the smaller unitaries having near-
maximal $T$-depth. This is exponential in the number of qubits acted on [24],
so $k_{t}\sim d_{t}\sim\operatorname{poly}(n)$. If we ran our protocol on the
total unitary, treating each smaller unitary as a $k\sim\log n$-qubit gate,
then the depth will be $1$ and our protocol costs $\sim n^{\log n}$, a quasi-
polynomial.

Note however that in this example our protocol does nothing but perform the
protocol of Ref. [4] many times in parallel. To obtain a less trivial example,
add a second layer of $\log n$ sized gates which act across the supports of
the original small unitaries. Choosing the gates of the second layer to have
$O(1)$ $T$-count, we can be assured that the total unitary remains in the
inefficient regime of the protocols in [17]. For this new unitary our protocol
still requires only a quasi-polynomial amount of entanglement.

##  4 Enhancements

There are several complementary actions Alice and Bob can take in order to re-
express the unitary they perform in a less costly manner. Firstly, they can
apply local operations on their systems before and after the protocol.
Secondly, they can apply arbitrary Cliffords at the beginning and end of the
circuit. Finally, they can add auxiliary systems. We show how each of these is
done, and give examples where the latter is useful.

###  4.1 Local pre/post-processing

Let Alice and Bob re-express $U$ as $(W_{A}^{\prime}\otimes
W_{B}^{\prime})U_{AB}(W_{A}\otimes W_{B})$ for some $n/2$-qubit unitaries
$W_{A},W_{B},W_{A}^{\prime},$ and $W_{B}^{\prime}$ and $n$-qubit unitary
$U_{AB}$. They act with $W_{A}$ and $W_{B}$ on the systems they respectively
hold at the beginning of the protocol, and with $W_{A}^{\prime}$ and
$W_{B}^{\prime}$ at the end of it. By running the protocol on $U_{AB}$ they
then obtain the same result as though they had run it on $U$ without pre/post-
processing. Crucially, the entanglement cost will depend on the circuit
structure of $U_{AB}$, which may be more conducive to our protocol than that
of $U$.

###  4.2 Clifford sandwich

Suppose that $U$ has a circuit decomposition $C_{f}U^{\prime}C_{i}$ with
$C_{i},C_{f}$ being $n$-qubit Cliffords and $U^{\prime}$ an $n$-qubit unitary
that has a circuit decomposition with structure $\\{S^{i}_{j}\\}$. Then we can
construct an NLQC protocol that accomplishes $U$ with the same cost as our
original protocol takes to accomplish any unitary via a circuit structure
$\\{S^{i}_{j}\\}$. To do so we run the protocol as though to implement
$U^{\prime}$ via this circuit structure, but modified as follows.

Between steps 2 and 3, i.e. right after Alice has teleported her $\frac{n}{2}$
qubits to Bob, Bob applies the Clifford $C_{i}$ to his combined system. This
means he now holds the state $QC_{i}\ket{\psi}$ where
$Q=C_{i}PC_{i}^{\dagger}$. Note that $Q$ is a Pauli, and that Alice knows its
identity since she knows both $C_{i}$ and $P$. If Alice and Bob then run the
remainder of the original protocol but replacing $P$ with $Q$, they will have
implemented $U^{\prime}C_{i}$.

To additionally implement $C_{f}$, the following modification is added right
before step 6, the communication round. At this point Bob holds the state

$\displaystyle\mathcal{P}(\vec{x}^{*})U^{\prime}C_{i}\ket{\psi}_{AB},$ (10)

where $\mathcal{P}$ is a Pauli valued function of $\vec{x}$, all indices of
the port teleportations Bob has performed throughout the protocol. Alice
posses knowledge of $\mathcal{P}$ but not $\vec{x}^{*}$, and vice versa for
Bob. Nonetheless, Bob can apply $C_{f}$ to the state, and the communication
round can commence as in the original protocol. They then both possess
knowledge of $\mathcal{P}(\vec{x}^{*})$, and can jointly apply
$C_{f}\mathcal{P}(\vec{x}^{*})C_{f}^{\dagger}$, since this is a Pauli and thus
factorizes between the $A$ and $B$ systems. This completes the protocol.

Local pre/post-processing can straightforwardly be performed before and after
the Clifford gates.

###  4.3 Auxiliary systems

Alice and Bob can make use of Auxiliary systems by generalizing the pre/post-
processing step to make use of isometries rather than unitaries. More
precisely, suppose $U$ can be written as

$\displaystyle
U=(\mathcal{W}_{A\rightarrow\tilde{A}}\otimes\mathcal{W}_{B\rightarrow\tilde{B}})U_{\tilde{A}\tilde{B}}(\mathcal{V}_{A\rightarrow\tilde{A}}\otimes\mathcal{V}_{B\rightarrow\tilde{B}})$
(11)

where $\mathcal{V}_{A\rightarrow\tilde{A}}$ and
$\mathcal{V}_{B\rightarrow\tilde{B}}$ are isometries,
$\mathcal{W}_{A\rightarrow\tilde{A}}$ and
$\mathcal{W}_{B\rightarrow\tilde{B}}$ are channels, and
$U_{\tilde{A}\tilde{B}}$ is a unitary. Then the isometries and channels can be
absorbed into local pre- and post-processing, and the NLQC protocol performed
on $\tilde{U}$. The entanglement cost will depend only on the circuit
structure needed to implement $U_{\tilde{A}\tilde{B}}$. Provided the number of
auxiliary systems added is polynomial in $n$, and $d$ and $k$ are unchanged,
the regime of efficiency will remain the same.

When would using auxiliary systems be useful? We will consider two examples:
locality-preserving unitaries, and the embedding of the protocol in Ref. [19].

It is well known that locality-preserving unitaries do not always admit a
circuit decomposition that reflects their causal structure [26]. For example,
the unitary that shifts all $n$ qubits in a closed 1-d chain by one site in
the same direction requires a circuit whose depth grows proportionally to the
number of qubits. However, using the techniques of Ref. [26], $n$ auxiliary
qubits can be introduced in the manner of eq. 11 to implement this unitary
with a circuit of depth $2$ and gate size $2$, which our protocol can do
efficiently. More generally these techniques can be used to decompose any
locality-preserving unitary.

As a second example, we will consider the protocol introduced in [19] for the
task of $f-$routing, a subclass of NLQC in which Alice receives a qubit and a
classical string $x$, Bob receives a classical string $y$, and their task is
to return the qubit to Alice or Bob depending on the value of a Boolean
function $f(x,y)$, subject to the same constraints as general NLQC. Their
protocol can be written in the form of eq. 11 with $\\{\tilde{S^{i}_{j}}\\}$
having $O(1)$ depth and $O(1)$-sized gates, and thus can be embedded into our
protocol while still performed efficiently. In their protocol, auxiliary
systems are introduced when Alice’s input qubit is encoded into a secret
sharing scheme. Each of these is routed to Alice or Bob depending on the value
of another Boolean function that depends only on an $O(1)$ subset of the
classical bits in $x$ and $y$, and so can be implemented as an $O(1)$-sized
gate. These gates can be parallelized by making as many copies of $x$ and $y$
in the pre-processing stage as needed for the cases when the routing of
several different shares depends on the same bits. In particular, only a
polynomial number of additional copies are needed because the construction of
Ref. [19] requires that the number of gates be polynomial in $n$.

##  5 Conclusion

As with NLQC in general, our results are relevant for both theory and
practice. On the practical side, our result can be considered as a new class
of attacks on position-based cryptography, which is based on the premise that
an attacker does not have access to a sufficient amount of entanglement in
order to perform a computation. If the computation is in the class we have
described, however, the amount required is quasi-polynomial in the system size
and so may not be large enough to be prohibitive.

Our results may also be of interest for modular quantum computing
architectures (see e.g. [27]), which are composed of several parallelized
computational components capable of performing computation on some fixed
number of qubits with constraints on the communication between them (e.g. due
to spatial separation). Our results show that such architectures can at least
perform computations in the class we have described with only one round of
communication.

On the theoretical side, our results are in line with a prediction made about
quantum information from studying holography, that polynomial-depth circuits
should also be implementable with efficient entanglement cost. This further
establishes the deep connection between holography and NLQC. In return,
understanding how NLQC can be performed efficiently may give insight into how
dynamics occur in holography, which have been notoriously difficult to capture
in discrete models.

Future work is needed to better understand the classes of efficiently
computable unitaries described in section 3.3, and how they relate to more
familiar complexity classes of unitaries. In particular, it remains to be
found which, if any, polynomial-depth circuits of two-qubit gates are outside
the regime of efficiency of our protocol when accounting for the extensions of
section 4. Finally, it may be possible to find a better method to deal with
the issue of the port dependence of Pauli keys, as this is the greatest
bottleneck of our protocol.

## Acknowledgement

We thank Alex May for suggesting the approach of exploiting a unitary’s
decomposition into a circuit of medium-sized gates to make more efficient use
of port teleportation. We also thank Nick Hunter-Jones, Adam Brown and Adam
Bouland for useful discussions. SC is supported by a graduate fellowship award
from Knight-Hennessy Scholars at Stanford University.

##  Appendix A Protocol error bounds

Let $\mathcal{N}$ be the channel applied to an input state of the protocol in
3 for a unitary $U$ decomposed into $\\{U^{i}_{j}\\}$ with circuit structure
$\\{S_{i}^{j}\\}$, with gate size $|S_{i}^{j}|=k_{i,j}$. Let
$\mathcal{U}:=U\cdot U^{\dagger}$. We would like to determine an upper bound
on $||\mathcal{N}-\mathcal{U}||_{\diamond}$.

First we need to write down the explicit form of $\mathcal{N}$. To do so
define the following quantities

  * •

$\mathcal{E}^{i}_{j}$ is the error channel defined by port-teleporting
$S_{i}^{j}$ using $N$ maximally entangled $k_{i,j}$ qubit states, and keeping
the correct port.

  * •

$\mathcal{E}_{i}:=\mathcal{E}^{i}_{1}\cdots\mathcal{E}^{i}_{n_{i}}$

  * •

$\mathcal{U}_{i}:=U^{i}\cdot U^{i\dagger}$ where $U^{i}=U^{i}_{1}\cdots
U^{i}_{n_{i}}$. Thus $\mathcal{U}=\mathcal{U}_{d}\cdots\mathcal{U}_{1}$.

  * •

$\mathcal{P}_{i}:=P^{i}\cdot P^{i\dagger}$ where
$P^{i}=\prod_{(i^{\prime},j^{\prime})\in\operatorname{par}(i,j)}P_{i^{\prime},j^{\prime}}(x^{*}_{\operatorname{fam}(i^{\prime},j^{\prime})})|_{S^{i}_{j}}$
for $i\geq 1$ and $P^{0}=P$.

  * •

$\mathcal{E}^{\prime}_{i}=\mathcal{P}_{i-1}^{\dagger}\mathcal{E}_{i}\mathcal{P}_{i-1}$

Then we can write $\mathcal{N}$ as

$\displaystyle\mathcal{N}=\mathcal{U}_{d}\mathcal{E}^{\prime}_{d}\cdots\mathcal{U}_{1}\mathcal{E}^{\prime}_{1}.$

We will need to make use of the following properties of the diamond norm (see
section 3.3 of Watrous’ book where it is referred to as the "completely
bounded trace norm").

Let $\Phi_{0}$,$\Phi_{1}$,$\Psi_{0}$,$\Psi_{1}$ be CPTP maps from
$\mathcal{B}(A)$ to $\mathcal{B}(A)$ for some quantum system $A$. Then

$\displaystyle||\Psi_{1}\Psi_{0}-\Phi_{1}\Phi_{0}||_{\diamond}\leq$
$\displaystyle||\Psi_{0}-\Phi_{0}||_{\diamond}+||\Psi_{1}-\Phi_{1}||_{\diamond}$
(12)

$\displaystyle||\Phi_{0}||_{\diamond}=1$ (13)

If $\Phi_{0}$, and $\Phi_{1}$ are instead general linear maps then we have

$\displaystyle||\Phi_{0}\otimes\Phi_{1}||_{\diamond}=||\Phi_{0}||_{\diamond}||\Phi_{1}||_{\diamond}$
(14)

We can use equation A to get

$\displaystyle||\mathcal{N}-\mathcal{U}||_{\diamond}$
$\displaystyle\leq\sum_{i=1}^{d}||\mathcal{U}_{i}\mathcal{E}^{\prime}_{i}-\mathcal{U}_{i}||_{\diamond}$
$\displaystyle\leq\sum_{i=1}^{d}||\mathcal{E}^{\prime}_{i}-\mathcal{I}_{\mathbb{C}^{2^{n}}}||_{\diamond}$
$\displaystyle\leq\sum_{i=1}^{d}||\mathcal{P}_{i-1}^{\dagger}\mathcal{E}_{i}\mathcal{P}_{i-1}-\mathcal{P}_{i-1}^{\dagger}\mathcal{P}_{i-1}||_{\diamond}$
$\displaystyle\leq\sum_{i=1}^{d}||\mathcal{P}_{i-1}^{\dagger}\mathcal{E}_{i}-\mathcal{P}_{i-1}^{\dagger}||_{\diamond}$
$\displaystyle\leq\sum_{i=1}^{d}||\mathcal{E}_{i}-\mathcal{I}_{\mathbb{C}^{2^{n}}}||_{\diamond}$
$\displaystyle\leq
d||\mathcal{E}_{i}-\mathcal{I}_{\mathbb{C}^{2^{n}}}||_{\diamond}.$

where in the last line we used that the argument of the sum does not depend on
$i$. Writing
$\mathcal{E}_{i}:=\mathcal{E}^{i}_{1}\cdots\mathcal{E}^{i}_{n_{i}}$ and again
using equation A we get

$\displaystyle||\mathcal{N}-\mathcal{U}||_{\diamond}\leq\sum_{i,j}||\mathcal{E}^{i}_{j}-\mathcal{I}_{\mathbb{C}^{2^{n}}}||_{\diamond}.$

Finally, use the fact that
$\mathcal{E}^{i}_{j}=\mathcal{E}^{i}_{j}|_{S^{i}_{j}}\otimes\mathcal{I}_{\mathbb{C}^{2^{n-k_{i,j}}}}$
and equations 14, and 13 to get

$\displaystyle||\mathcal{N}-\mathcal{U}||_{\diamond}\leq\sum_{i,j}||\mathcal{E}^{i}_{j}|_{S^{i}_{j}}-\mathcal{I}_{\mathbb{C}^{2^{k_{i,j}}}}||_{\diamond}.$

From [4] we have that

$\displaystyle||\mathcal{E}^{i}_{j}|_{S^{i}_{j}}-\mathcal{I}_{\mathbb{C}^{2^{k}}}||_{\diamond}\leq
4\frac{2^{2k}}{\sqrt{N}}.$

Thus the bound we desire is

$\displaystyle||\mathcal{N}-\mathcal{U}||_{\diamond}\leq
4\sum_{i,j}\frac{2^{2k_{i,j}}}{\sqrt{N}}.$

If we want the norm to be less than $\epsilon$, then we can choose

$\displaystyle
N=\frac{16}{\epsilon^{2}}\left(\sum_{i,j}2^{2k_{i,j}}\right)^{2}.$ (15)

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