# Global-in-time semiclassical regularity for the Hartree-Fock equation

Jacky J. Chong  ,  Laurent Lafleche  Department of Mathematics, The University
of Texas at Austin, Austin, TX 78712, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Chiara Saffirio  Department of Mathematics and
Computer Science, University of Basel, 4051 Basel, Switzerland,
<EMAIL_ADDRESS>
###### Abstract.

For arbitrarily large times $T>0$, we prove the uniform in $\hbar$ propagation
of semiclassical regularity for the solutions to the Hartree-Fock equation
with singular interactions of the form $V(x)=\pm\left\lvert
x\right\rvert^{-a}$ with $a\in(0,\frac{1}{2})$. As a byproduct of this result,
we extend to arbitrarily long times the derivation of the Hartree-Fock and the
Vlasov equations from the many-body dynamics provided in [J. Chong, L.
Lafleche, C. Saffirio: arXiv:2103.10946 (2021)].

###### Key words and phrases:

Hartree-Fock equation, Hartree equation, semiclassical, regularity, singular
interaction.

###### 2010 Mathematics Subject Classification:

35Q55, 35B65, 82C10, 81Q20.

######  Table of Contents

  1. 1 Introduction and Main Result
  2. 2 Propagation of Moments
    1. 2.1 The Hartree Equation
    2. 2.2 The Hartree-Fock Equation
  3. 3 Proof of Theorem 1.1

##  1\. Introduction and Main Result

Consider the time-dependent Hartree-Fock equation

(1)
$i\hbar\,\partial_{t}\boldsymbol{\rho}=\left[{H_{\boldsymbol{\rho}},\boldsymbol{\rho}}\right]$

describing the evolution of a positive self-adjoint trace class operator
$\boldsymbol{\rho}=\boldsymbol{\rho}(t)$ acting on $L^{2}(\mathbb{R}^{3})$.
Here $\hbar=\frac{h}{2\pi}$ is the reduced Planck constant,
$\left[{\cdot,\cdot}\right]$ denotes the commutator $\left[{A,B}\right]=AB-
BA$, and $H_{\boldsymbol{\rho}}$ is the Hamiltonian operator given by

(2)
$H_{\boldsymbol{\rho}}=-\frac{\hbar^{2}\Delta}{2}+V_{\boldsymbol{\rho}}-h^{3}\,\mathsf{X}_{\boldsymbol{\rho}},$

where $V_{\boldsymbol{\rho}}$ is the mean-field potential and
$\mathsf{X}_{\boldsymbol{\rho}}$ is the exchange operator. The mean-field
potential is defined as the multiplication operator by the function
$V_{\boldsymbol{\rho}}(x)=(K*\rho)(x)$, where $K:\mathbb{R}^{3}\to\mathbb{R}$
is the potential associated to some two-body interaction and $\rho(x)$ is the
spatial density, defined as the rescaled diagonal of the integral kernel
$\boldsymbol{\rho}(\cdot,\cdot)$ of the operator $\boldsymbol{\rho}$, given by

$\rho(x)=\operatorname{diag}(\boldsymbol{\rho})(x):=h^{3}\boldsymbol{\rho}(x,x).$

The exchange operator is defined to be the operator with kernel

$\mathsf{X}_{\boldsymbol{\rho}}(x,y)=K(x-y)\,\boldsymbol{\rho}(x,y).$

In this work, we normalize $\boldsymbol{\rho}$ so that

(3)
$\operatorname{Tr}\\!\left(\boldsymbol{\rho}\right)=h^{-3},\qquad\text{and}\qquad\mathcal{C}_{\infty}:=\left\lVert\boldsymbol{\rho}\right\rVert_{\infty},$

where $\left\lVert\cdot\right\rVert_{\infty}$ is the operator norm, which is
possible since the spectrum of $\boldsymbol{\rho}$ is preserved by Equation
(1). With this scaling, we see that $\rho$ satisfies
$\int_{\mathbb{R}^{3}}\rho(x)\,\mathrm{d}x=h^{3}\operatorname{Tr}\\!\left(\boldsymbol{\rho}\right)=1$.
In the absence of $\mathsf{X}_{\boldsymbol{\rho}}$, we refer to Equation (1)
as the Hartree equation. All the results presented in this work hold for both
the Hartree and the Hartree-Fock equation.

In the case when $\hbar$ is fixed, say $\hbar=1$, the well-posedness theory
for the Hartree and the Hartree-Fock equations are well known. For the case of
the Hartree equation with Coulomb potential, one can find the proof of the
global-in-time well-posedness in $L^{2}$ and the propagation of higher $H^{s}$
regularity for the wave function in [5], which builds on the earlier works [9,
10, 12]. The case of density operators in Schatten spaces but with an infinite
trace was studied in [16]. In the case of the Hartree-Fock equation, well-
posedness in $H^{2}$ was proved in [3] for bounded interactions and then in
[6, 4] for more singular potentials including the case of the Coulomb
potential. However, these works do not provide satisfactory estimates when
$\hbar$ is small and tending towards zero. Obtaining uniform in $\hbar$
estimates is crucial for understanding, of course, the errors in the
semiclassical limit $\hbar\to 0$ as in [17, 13, 19, 20, 14, 15], but also to
create adapted numerical schemes [11] and to understand the joint mean-field
and semiclassical limit [2, 1, 18, 7].

In this paper, we are interested in proving the global-in-time propagation of
regularity uniformly in the semiclassical parameter $\hbar$ for solutions to
the Hartree-Fock equation (1) when the interaction potential $K$ is the
inverse power law potential

$K(x)=\dfrac{\pm 1}{\left\lvert x\right\rvert^{a}},\qquad\text{ with }\quad
a\in\left(0,\tfrac{1}{2}\right).$

In particular, $K\in L^{\mathfrak{b},\infty}(\mathbb{R}^{3})$, where
$\mathfrak{b}=\frac{3}{a+1}$ and $L^{\mathfrak{b},\infty}(\mathbb{R}^{3})$
denotes the weak $L^{\mathfrak{b}}$ space on $\mathbb{R}^{3}$. Our main
motivation is to extend the results of the local-in-time regularity obtained
in our previous paper [7] to global-in-time results, leading to the global-in-
time mean-field and semiclassical limits for fermions from the $N$ body
Schrödinger equation to the Hartree-Fock and Vlasov equation.

Before stating our main result, we introduce the function spaces that we will
be working with. First, we define the semiclassical phase space Lebesgue norms
by

$\left\lVert\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{p}}:=h^{\frac{3}{p}}\left\lVert\boldsymbol{\rho}\right\rVert_{p}=h^{\frac{3}{p}}\operatorname{Tr}\\!\left(\left\lvert\boldsymbol{\rho}\right\rvert^{p}\right)^{\frac{1}{p}}$

for $p\in[1,\infty]$, with the obvious modification for $p=\infty$. Here
$\left\lVert\boldsymbol{\rho}\right\rVert_{p}$ denotes the Schatten norm of
order $p$ and $\left\lvert A\right\rvert=\sqrt{A^{*}A}$ the absolute value of
the operator $A$ with adjoint $A^{*}$. Let $\boldsymbol{p}=-i\hbar\nabla$ be
the momentum operator and

$\mathsf{m}_{n}:=1+\left\lvert\boldsymbol{p}\right\rvert^{n},\qquad
n\in\mathbb{N}.$

The moment of order $n$ and the weighted semiclassical Lebesgue norms with the
operator weight $\mathsf{m}_{n}$ are given by

$M_{n}:=h^{3}\operatorname{Tr}\\!\left(\left\lvert\boldsymbol{p}\right\rvert^{n}\boldsymbol{\rho}\right)\qquad\text{and}\qquad\left\lVert\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{p}(\mathsf{m}_{n})}:=\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{p}}.$

In order to consider the quantum analogue of Sobolev norms, we introduce the
following operators

$\boldsymbol{\nabla}_{\\!x}\boldsymbol{\rho}:=\left[{\nabla,\boldsymbol{\rho}}\right]\qquad\text{and}\qquad\boldsymbol{\nabla}_{\\!\xi}\boldsymbol{\rho}:=\left[{\frac{x}{i\hbar},\boldsymbol{\rho}}\right].$

Then the semiclassical homogeneous Sobolev norms are defined by

$\left\lVert\boldsymbol{\rho}\right\rVert_{\dot{\mathcal{W}}^{1,p}}:=\sum_{k=1}^{3}\left(\left\lVert\boldsymbol{\nabla}_{x_{k}}\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{p}}+\left\lVert\boldsymbol{\nabla}_{\xi_{k}}\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{p}}\right)$

with corresponding inhomogeneous Sobolev norms given by
$\left\lVert\boldsymbol{\rho}\right\rVert_{\mathcal{W}^{1,p}}:=\left\lVert\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{p}}+\left\lVert\boldsymbol{\rho}\right\rVert_{\dot{\mathcal{W}}^{1,p}}$,
and the weighted semiclassical Sobolev norms with the operator weight
$\mathsf{m}_{n}$ by

$\left\lVert\boldsymbol{\rho}\right\rVert_{\mathcal{W}^{1,p}(\mathsf{m}_{n})}:=\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n}\right\rVert_{\mathcal{W}^{1,p}}.$

Our main result states the global-in-time propagation of the regularity in
terms of these norms.

######  Theorem 1.1.

Let $a\in\left(0,\frac{1}{2}\right)$, $n\in 2\mathbb{N}$ be an even integer
and $\boldsymbol{\rho}$ be a solution to the Hartree-Fock equation (1) with
initial datum
$\boldsymbol{\rho}^{\mathrm{in}}\in\mathcal{L}^{\infty}(\mathsf{m}_{n})$
satisfying (3) and such that

(4)
$\boldsymbol{\rho}^{\mathrm{in}}\in\mathcal{W}^{1,2}(\mathsf{m}_{n})\cap\mathcal{W}^{1,q}(\mathsf{m}_{n})$

for $q\in[2,\infty)$, and with moments of order strictly larger than
$\frac{3}{1-a}\left(n+a+1\right)$ bounded uniformly in $\hbar$. Then

(5) $\boldsymbol{\rho}\in
L^{\infty}_{\mathrm{loc}}(\mathbb{R}_{+},\mathcal{W}^{1,2}(\mathsf{m}_{n})\cap\mathcal{W}^{1,q}(\mathsf{m}_{n})\cap\mathcal{L}^{\infty}(\mathsf{m}_{n}))$

uniformly in $\hbar\in(0,1)$.

Notice that Theorem 1.1 extends to arbitrarily long times the local-in-time
theory studied in [7, Theorem 3.1] for $a\in(0,\frac{1}{2})$. As a corollary,
Theorem 1.1 entails the global-in-time derivation of the Hartree-Fock and the
Vlasov equations from the many-body Schrödinger equation in the mean-field
regime for mixed states, thus extending from local to global-in-time Theorem
3.2 and Theorem 3.3 in [7]. The crucial regularity conditions needed to
perform the joint mean-field and semiclassical limit in [7] when
$a\in(0,\frac{1}{2})$ are indeed of the form (5) with $q>\frac{6}{1+2a}$ (take
$p=2$ and $q=q_{1}$ in the beginning of [7, Section 10.1]).

As in [13, 14], the key ingredient to getting long-time estimates in this
paper is the use of quantum moments, used to bound the semiclassical weighted
Lebesgue norms. In this paper, we prove the global-in-time propagation of
quantum moments for the solution of the Hartree-Fock equation (1) when
$a\in(0,\frac{4}{5}]$ and show that if $a\in(0,\frac{1}{2})$ then the global-
in-time bound on the moments, combined with a Grönwall-type argument, proves
the propagation of regularity in weighted semiclassical Sobolev spaces,
uniformly in $\hbar$.

The paper is structured as follows: in Section 2, we recall the result
obtained in [13] about the global-in-time propagation of quantum moments for
the Hartree equation when $a\in(0,\frac{4}{5}]$ and extend it to solutions of
the Hartree-Fock equation (1), while Section 3 is devoted to the proof of
Theorem 1.1.

##  2\. Propagation of Moments

######  Theorem 2.1.

Let $a\in(0,\frac{4}{5}]$, $n\in 2\mathbb{N}$ and $\boldsymbol{\rho}$ be a
solution of the Hartree-Fock equation with initial condition
$\boldsymbol{\rho}^{\mathrm{in}}\in\mathcal{L}^{1}\cap\mathcal{L}^{\infty}$
with moments of order $n$ bounded uniformly in $\hbar$. Then there exists a
continuous function $\Phi\in C^{0}(\mathbb{R}_{+})$ independent of $\hbar$
such that for any $t\in\mathbb{R}_{+}$,

$M_{n}(t)\leq\Phi(t).$

######  Remark 2.1.

If $a\in(\frac{4}{5},2)$, then we can still get a short-time estimate when
$a\leq a_{n}=\frac{2\,n}{n+3}$ (see Remark 2.3). In particular, $a_{n}$ is
larger than $1$ as soon as $n\geq 3$. More generally, for any $a\in(0,2)$, the
propagation of moment of order $n$ holds for any even $n\geq
n_{a}=\frac{3\,a}{2-a}$.

######  Remark 2.2.

The proof of the theorem can be used to get an explicit function $\Phi(t)$. It
has a polynomial growth in time when $a<\frac{4}{5}$, and exponential growth
in time when $a=\frac{4}{5}$.

###  2.1. The Hartree Equation

In this section, we recall the main ingredients of the proof of the
propagation of moments for the Hartree equation obtained in [13, Theorem 3].
This provides us a guide for the extension to the case of the Hartree-Fock
equation addressed in Theorem 2.1.

Recall that for any density operator $\boldsymbol{\rho}$ (i.e. any positive
trace class operator with trace one), there exists $J\subset\mathbb{N}$, a
sequence of functions $\left(\psi_{j}\right)_{j\in J}$ orthonormal in
$L^{2}(\mathbb{R}^{3})$ and a positive summable sequence
$\left(\lambda_{j}\right)_{j\in J}$ such that $\boldsymbol{\rho}$ can be
written as

$\boldsymbol{\rho}=\sum_{j\in J}\lambda_{j}\ket{\psi_{j}}\bra{\psi_{j}}.$

For any even integer $n\in 2\mathbb{N}$, we define the moment density of order
$n$ by

(6) $\rho_{n}(x):=h^{3}\sum_{j\in
J}\lambda_{j}\left\lvert\boldsymbol{p}^{\frac{n}{2}}\psi_{j}(x)\right\rvert^{2}=\operatorname{diag}\\!\left(\boldsymbol{p}^{\frac{n}{2}}\boldsymbol{\rho}\cdot\boldsymbol{p}^{\frac{n}{2}}\right)(x)$

so that the moment of order $n$, previously defined, can be rewritten as
$M_{n}=\left\lVert\rho_{n}\right\rVert_{L^{1}}$. Notice we also have
$M_{n}=\left\lVert\sqrt{\boldsymbol{\rho}}\left\lvert\boldsymbol{p}\right\rvert^{\frac{n}{2}}\right\rVert_{\mathcal{L}^{2}}^{2}$.
With this notation, Inequality (38) in [13] reads

(7) $\left\lvert
h^{3}\operatorname{Tr}\\!\left(\tfrac{1}{i\hbar}\left[{V,\left\lvert\boldsymbol{p}\right\rvert^{n}}\right]\boldsymbol{\rho}\right)\right\rvert\leq
C\,M_{n}^{\frac{1}{2}}\sup_{\begin{subarray}{c}(j,k,l)\in\mathbb{N}^{3}\\\
j+k+l=n/2-1\end{subarray}}\left\lVert\rho_{2j}\right\rVert_{L^{\alpha}}^{\frac{1}{2}}\left\lVert\rho_{2k}\right\rVert_{L^{\beta}}^{\frac{1}{2}}\left\lVert\rho_{2l}\right\rVert_{L^{\gamma}}^{\frac{1}{2}},$

where

$\frac{1}{\alpha^{\prime}}+\frac{1}{\beta^{\prime}}+\frac{1}{\gamma^{\prime}}=\frac{1}{\mathfrak{b}}$

and $\alpha^{\prime},\,\beta^{\prime},\,\gamma^{\prime}$ are the Hölder
conjugates of $\alpha,\,\beta,\,\gamma$ respectively. The semiclassical
kinetic inequality [13, Theorem 6], which is a generalization of the Lieb-
Thirring inequality for the $n$th order moment density, tells us that for any
$(k,n)\in(2\mathbb{N})^{2}$ verifying $k\in[0,n]$ we have that

(8) $\displaystyle\left\lVert\rho_{k}\right\rVert_{L^{p}}$ $\displaystyle\leq
C\left\lVert\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{\infty}}^{\frac{1}{p^{\prime}}}M_{n}^{\frac{1}{p}}\qquad\text{
with }\qquad p=p_{n,k}:=\frac{3+n}{3+k}.$

Combining this inequality with Inequality (7) implies there exist some
positive constants $\Theta$, $\Theta_{0}$, and $\Theta_{2}$ such that the
following estimate (see [13, Inequality (44)])

$\left\lvert
h^{3}\operatorname{Tr}\\!\left(\tfrac{1}{i\hbar}\left[{V,\left\lvert\boldsymbol{p}\right\rvert^{n}}\right]\boldsymbol{\rho}\right)\right\rvert\leq
C\left\lVert\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{\infty}}^{\Theta_{2}}M_{n-2}^{\Theta_{0}}\,M_{n}^{\Theta},$

where $\Theta\leq 1$ when $a\leq\frac{4}{5}$. This leads to the boundedness of
moments by a Grönwall-type argument, together with the uniform boundedness of
$M_{2}$ due to the conservation of energy and an induction on $n$. Moreover,
when $a<\frac{4}{5}$, then $\Theta<1$, thus proving a polynomial growth in
time in this latter case.

###  2.2. The Hartree-Fock Equation

We now consider the Hartree-Fock equation. By estimating the exchange
(operator) term with a similar strategy, we get the analogue of [13, Theorem
3]. The analogue of Inequality (7) for the exchange term is given by the
following lemma.

######  Lemma 2.1.

Denote the components of $\boldsymbol{p}$ by $\boldsymbol{p}_{x_{l}}$ or
simply $\boldsymbol{p}_{l}$ for $l\in\set{1,2,3}$. Suppose $n\in\mathbb{N}$
and define $Q\subset[\frac{1}{2},\infty]^{4}$ the set of
$(q_{1},q_{2},q_{3},q_{4})$ verifying
$\frac{1}{q_{1}}+\frac{1}{q_{2}}\in(0,2)$,
$\frac{1}{q_{3}}+\frac{1}{q_{4}}\in(0,2)$ and

(9)
$\frac{1}{q_{1}^{\prime}}+\frac{1}{q_{2}^{\prime}}+\frac{1}{q_{3}^{\prime}}+\frac{1}{q_{4}^{\prime}}=\frac{2}{\mathfrak{b}}.$

Then there exists a constant $C$ independent of $\hbar$ such that

(10)
$\left\lvert\frac{h^{3}}{\hbar}\operatorname{Tr}\\!\left(\left[{h^{3}\mathsf{X}_{\boldsymbol{\rho}},\boldsymbol{p}_{l}^{n}}\right]\boldsymbol{\rho}\right)\right\rvert\leq
C\sup_{k_{1},\dots
k_{4}}\inf_{Q}\left\lVert\rho_{k_{1}}\right\rVert_{L^{q_{1}}}^{\frac{1}{2}}\left\lVert\rho_{k_{2}}\right\rVert_{L^{q_{2}}}^{\frac{1}{2}}\left\lVert\rho_{k_{3}}\right\rVert_{L^{q_{3}}}^{\frac{1}{2}}\left\lVert\rho_{k_{4}}\right\rVert_{L^{q_{4}}}^{\frac{1}{2}},$

where the supremum is taken over all the integers
$(k_{1},\dots,k_{4})\in\mathbb{N}^{4}$ such that
$k_{1}+\dots+k_{4}=2\left(n-1\right)$.

###### Proof.

Let us begin by making the observation that
$[\mathsf{X}_{\boldsymbol{\rho}},\boldsymbol{p}_{l}^{n}]\boldsymbol{\rho}$ is
trace class. By the Leibniz formula the kernel of
$[\mathsf{X}_{\boldsymbol{\rho}},\boldsymbol{p}_{l}^{n}]\boldsymbol{\rho}$ is
given by

$([\mathsf{X}_{\boldsymbol{\rho}},\boldsymbol{p}_{l}^{n}]\boldsymbol{\rho})(x,y)=-\sum^{n-1}_{k=0}\binom{n}{k}\int_{\mathbb{R}^{3}}\boldsymbol{p}_{x_{l}}^{n-k}K(x-z)\boldsymbol{p}_{x_{l}}^{k}\boldsymbol{\rho}(x,z)\boldsymbol{\rho}(z,y)\,\mathrm{d}z\\\
=\sum^{n-1}_{k=0}\sum_{\ell=0}^{n-k}\binom{n}{k}\binom{n-k}{\ell}(-1)^{n-k+1}\int_{\mathbb{R}^{3}}K(x-z)(\boldsymbol{p}_{x_{l}}^{k}\boldsymbol{p}_{z_{l}}^{\ell}\boldsymbol{\rho}(x,z))\boldsymbol{p}_{z_{l}}^{n-k-\ell}\boldsymbol{\rho}(z,y)\,\mathrm{d}z\\\
=:\sum^{n-1}_{k=0}\sum_{\ell=0}^{n-k}\binom{n}{k}\binom{n-k}{\ell}(-1)^{n-k+1}\mathbf{A}_{n,k,l}(x,y).$

It is clear that each $\mathbf{A}_{n,k,l}$ is a product of two Hilbert-Schmidt
operators provided $\boldsymbol{\rho}$ is sufficiently regular. Thus, it
follows that
$[\mathsf{X}_{\boldsymbol{\rho}},\boldsymbol{p}_{l}^{n}]\boldsymbol{\rho}$ is
indeed trace class.

Let
$I_{n}:=\operatorname{Tr}\\!\left(\left[{\mathsf{X}_{\boldsymbol{\rho}},\boldsymbol{p}_{l}^{n}}\right]\boldsymbol{\rho}\right)$.
Then, by [8, VI.7 Theorem 17], we may express the trace of
$\left[{\mathsf{X}_{\boldsymbol{\rho}},\boldsymbol{p}_{l}^{n}}\right]\boldsymbol{\rho}$
in terms of its kernel as follows

$I_{n}=\iint_{\mathbb{R}^{6}}K(x-y)\,\boldsymbol{\rho}(y,x)\,\boldsymbol{p}_{x_{l}}^{n}(\boldsymbol{\rho})(x,y)-\boldsymbol{p}_{x_{l}}^{n}\\!\left(K(x-y)\boldsymbol{\rho}(x,y)\right)\boldsymbol{\rho}(y,x)\,\mathrm{d}x\,\mathrm{d}y.$

Therefore, using the Leibniz formula and then diagonalizing the self-adjoint
compact operator $\boldsymbol{\rho}$ yields

$I_{n}=-\sum_{k=0}^{n-1}\binom{n}{k}\iint_{\mathbb{R}^{6}}(\boldsymbol{p}_{x_{l}}^{n-k}K)(x-y)\left(\boldsymbol{p}_{x_{l}}^{k}\boldsymbol{\rho}(x,y)\right)\boldsymbol{\rho}(y,x)\,\mathrm{d}x\,\mathrm{d}y\\\
=-\sum_{(\mathrm{i},j)\in
J^{2}}\sum^{n-1}_{k=0}\binom{n}{k}\lambda_{\mathrm{i}}\,\lambda_{j}\int_{\mathbb{R}^{3}}(\boldsymbol{p}_{l}^{n-k}K)*\\!\left(\overline{\psi_{\mathrm{i}}}\,\psi_{j}\right)\boldsymbol{p}_{l}^{k}\\!\left(\psi_{\mathrm{i}}\right)\overline{\psi_{j}},$

and so

$\displaystyle\frac{I_{n}}{i\hbar}$ $\displaystyle=\sum_{(\mathrm{i},j)\in
J^{2}}\sum^{n-1}_{k=0}\binom{n}{k}\lambda_{\mathrm{i}}\,\lambda_{j}\int_{\mathbb{R}^{3}}(\partial_{l}K)*\boldsymbol{p}_{l}^{n-1-k}\\!\left(\overline{\psi_{\mathrm{i}}}\,\psi_{j}\right)\boldsymbol{p}_{l}^{k}\\!\left(\psi_{\mathrm{i}}\right)\overline{\psi_{j}}.$

Let $I_{n,k}$ denote the integral in the above formula. We need to balance the
powers of $\boldsymbol{p}_{l}$ in $I_{n,k}$. For simplicity of notation, we
assume $n=2\,\widetilde{n}\in 2\,\mathbb{N}$. The case when $n$ is odd is
similar. Then, if $k\leq\widetilde{n}$, we have

$\displaystyle I_{n,k}$
$\displaystyle=\int_{\mathbb{R}^{3}}(\partial_{l}K)*\boldsymbol{p}_{l}^{\tilde{n}-1}\\!\left(\overline{\psi_{\mathrm{i}}}\,\psi_{j}\right)\boldsymbol{p}_{l}^{\tilde{n}-k}\\!\left(\boldsymbol{p}_{l}^{k}\\!\left(\psi_{\mathrm{i}}\right)\overline{\psi_{j}}\right)=\sum_{\ell=0}^{\tilde{n}-k}\sum_{\tilde{\ell}=0}^{\tilde{n}-1}\binom{\widetilde{n}-k}{\ell}\binom{\widetilde{n}-1}{\widetilde{\ell}}I_{n,k,\ell,\tilde{\ell}},$

where

$\displaystyle I_{n,k,\ell,\tilde{\ell}}$
$\displaystyle=\int_{\mathbb{R}^{3}}(\partial_{l}K)*\\!\left(\boldsymbol{p}_{l}^{\tilde{\ell}}(\overline{\psi_{\mathrm{i}}})\,\boldsymbol{p}_{l}^{\tilde{n}-1-\tilde{\ell}}(\psi_{j})\right)\boldsymbol{p}_{l}^{\tilde{n}-\ell}(\psi_{\mathrm{i}})\,\boldsymbol{p}_{l}^{\ell}(\overline{\psi_{j}}).$

Now notice that by the Cauchy-Schwarz inequality for sums and Definition (6)
for the moment density, we have that

$\displaystyle h^{2d}\sum_{(\mathrm{i},j)\in
J^{2}}\lambda_{\mathrm{i}}\,\lambda_{j}\,I_{n,k,\ell,\tilde{\ell}}$
$\displaystyle\leq\int_{\mathbb{R}^{3}}\left\lvert\partial_{l}K\right\rvert*\\!\left(\rho_{2\tilde{\ell}}^{\frac{1}{2}}\,\rho_{2(\tilde{n}-1-\tilde{\ell})}^{\frac{1}{2}}\right)\rho_{2(\tilde{n}-\ell)}^{\frac{1}{2}}\,\rho_{2\ell}^{\frac{1}{2}}$
$\displaystyle\leq
C_{K}\left\lVert\rho_{2\tilde{\ell}}\right\rVert_{L^{q_{1}}}^{\frac{1}{2}}\left\lVert\rho_{2(\tilde{n}-1-\tilde{\ell})}\right\rVert_{L^{q_{2}}}^{\frac{1}{2}}\left\lVert\rho_{2(\tilde{n}-\ell)}\right\rVert_{L^{q_{3}}}^{\frac{1}{2}}\left\lVert\rho_{2\ell}\right\rVert_{L^{q_{4}}}^{\frac{1}{2}},$

where the last inequality follows from the Hardy-Littlewood-Sobolev inequality
and the Hölder inequality. Similarly, when $k>\widetilde{n}$, we have that

$\displaystyle I_{n,k}:=$
$\displaystyle\,\int_{\mathbb{R}^{3}}\boldsymbol{p}_{l}^{k-\tilde{n}}\\!\left((\partial_{l}K)*\boldsymbol{p}_{l}^{n-k-1}\\!\left(\overline{\psi_{\mathrm{i}}}\,\psi_{j}\right)\overline{\psi_{j}}\right)\boldsymbol{p}_{l}^{\tilde{n}}\\!\left(\psi_{\mathrm{i}}\right)$
$\displaystyle=$
$\displaystyle\,\sum_{\ell=0}^{k-\tilde{n}}\sum_{\tilde{\ell}=0}^{\tilde{n}-1-\ell}\binom{k-\widetilde{n}}{\ell}\binom{\widetilde{n}-1-\ell}{\widetilde{\ell}}I_{n,k,\ell,\tilde{\ell}}$

where

$\displaystyle I_{n,k,\ell,\tilde{\ell}}$
$\displaystyle=\int_{\mathbb{R}^{3}}(\partial_{l}K)*\\!\left(\boldsymbol{p}_{l}^{\tilde{\ell}}(\overline{\psi_{\mathrm{i}}})\,\boldsymbol{p}_{l}^{\tilde{n}-1-\ell-\tilde{\ell}}(\psi_{j})\right)\boldsymbol{p}_{l}^{\ell}(\overline{\psi_{j}})\,\boldsymbol{p}_{l}^{\tilde{n}}(\psi_{\mathrm{i}}).$

This completes our proof of the lemma. ∎

Now combining the semiclassical kinetic interpolation inequality (8) for $n=2$
and another $n\in\mathbb{N}$, we get the following inequalities (See [14,
Propostion 3.2]).

######  Lemma 2.2.

Let $0\leq k\leq n$ and
$p_{n,k}^{\prime}=\left(\frac{n}{k}\right)^{\prime}\left(1+\frac{3}{n}\right)=\frac{3+n}{n-k}$.
Then for any $p\in[1,p_{n,k}]$, if $k\geq 2$ or if $p\geq
p_{2,0}=\frac{5}{3}$, it holds

$\left\lVert\rho_{k}\right\rVert_{L^{p}}\leq
C_{n,k}^{\frac{1}{p^{\prime}}}\,\left\lVert\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{\infty}}^{\frac{1}{p^{\prime}}}M_{2}^{\theta_{2}}\,M_{n}^{\theta_{n}}$

with $\theta_{2}=\frac{n-k}{n-2}-\frac{3+n}{n-2}\frac{1}{p^{\prime}}$ and
$\theta_{n}=\frac{k-2}{n-2}+\frac{5}{n-2}\frac{1}{p^{\prime}}$. If $k=0$ and
$p\leq p_{2,0}$ then

$\left\lVert\rho\right\rVert_{L^{p}}\leq
C^{\frac{1}{p^{\prime}}}\left\lVert\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{\infty}}^{\frac{1}{p^{\prime}}}M_{0}^{1-\frac{5}{2\,p^{\prime}}}\,M_{2}^{\frac{3}{2\,p^{\prime}}}.$

These two inequalities can be merged into a single inequality in terms of the
non-homogeneous moments
$1+M_{n}=h^{3}\operatorname{Tr}\\!\left(\boldsymbol{\rho}\,\mathsf{m}_{n}\right)$.

######  Corollary 2.1.

For any $n\geq 2$, $k\in[0,n]$ and $p\in[1,p_{n,k}]$, there exists a constant
$C>0$ such that for any compact operator $\boldsymbol{\rho}$ we have the
following estimate

(11) $\left\lVert\rho_{k}\right\rVert_{L^{p}}\leq
C\left\lVert\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{\infty}}^{\frac{1}{p^{\prime}}}\left(1+M_{2}\right)^{\theta_{2}}\left(1+M_{n}\right)^{\theta_{n}},$

with $\theta_{n}=\frac{k-2}{n-2}+\frac{5}{n-2}\frac{1}{p^{\prime}}$.

######  Remark 2.3.

If $(q_{j})_{j=1,\ldots,4}$ verify $q_{j}\in[1,p_{n,k_{j}}]$ and
$\sum_{j=1}^{4}k_{j}=2\left(n-1\right)$, then
$q^{\prime}_{j}\in[p^{\prime}_{n,k_{j}},\infty]$ so

$0\leq\sum_{j=1}^{4}\frac{1}{q^{\prime}_{j}}\leq\sum_{j=1}^{4}\frac{n-k_{j}}{n+3}=\frac{2\left(n+1\right)}{n+3}.$

Hence, we can find such a family verifying (9) as soon as
$\frac{2\left(n+1\right)}{n+3}\geq\frac{2}{\mathfrak{b}}$, or equivalently as
soon as $a\leq\frac{2\,n}{n+3}=:a_{n}$.

We can now complete the proof of Theorem 2.1 using a similar strategy as to
the one explained in Section 2.1 for the Hartree equation. Instead of
proceeding by induction and bounding the time derivative of $M_{n}$ by a
product involving $M_{n}$ and $M_{n-2}$, we directly estimate it by a product
involving $M_{n}$ and $M_{2}$. This method allows us to improve slightly the
result of [13, Theorem 3], even in the case of the Hartree equation, as it
allows us to propagate moments of high order locally in time for any
$a\in(0,2)$ while [13, Theorem 3] only covers the case $a\in(0,\frac{8}{7})$
in dimension $3$.

###### Proof of Theorem 2.1.

Taking the time derivative of moments and using the cyclicity of the trace
yields

$i\hbar\,\frac{\mathrm{d}}{\mathrm{d}t}\,\operatorname{Tr}\\!\left(\boldsymbol{\rho}\,\boldsymbol{p}_{l}^{n}\right)=-\operatorname{Tr}\\!\left(\left[{V_{\boldsymbol{\rho}},\boldsymbol{p}^{n}_{l}}\right]\boldsymbol{\rho}\right)-\operatorname{Tr}\\!\left(\left[{\mathsf{X}_{\boldsymbol{\rho}},\boldsymbol{p}^{n}_{l}}\right]\boldsymbol{\rho}\right).$

Then, by Inequality (7), which also holds for $\boldsymbol{p}_{l}^{n}$ in
place of $|\boldsymbol{p}|^{n}$, and Inequality (10), we deduce that

$\left\lvert
h^{3}\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{Tr}\\!\left(\boldsymbol{\rho}\,\boldsymbol{p}_{l}^{n}\right)\right\rvert\leq
C_{n,a}\sup_{k_{1}+\dots+k_{4}=2\left(n-1\right)}\inf_{Q}\left\lVert\rho_{k_{1}}\right\rVert_{L^{q_{1}}}^{\frac{1}{2}}\left\lVert\rho_{k_{2}}\right\rVert_{L^{q_{2}}}^{\frac{1}{2}}\left\lVert\rho_{k_{3}}\right\rVert_{L^{q_{3}}}^{\frac{1}{2}}\left\lVert\rho_{k_{4}}\right\rVert_{L^{q_{4}}}^{\frac{1}{2}}$

with the notations of Lemma 2.1. Using the interpolation formula (11) for each
of the terms in the right-hand side of the above inequality, we get

$\left\lvert\frac{h^{3}}{i\hbar}\operatorname{Tr}\\!\left(\left[{h^{3}\mathsf{X}_{\boldsymbol{\rho}},\boldsymbol{p}_{l}^{n}}\right]\boldsymbol{\rho}\right)\right\rvert\leq
C_{n,a}\left\lVert\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{\infty}}^{\frac{1}{\mathfrak{b}}}\left(1+M_{2}\right)^{\Theta_{2}}\left(1+M_{n}\right)^{\Theta_{n}}$

with $\Theta_{2}=\frac{n+1}{n-2}-\frac{n+3}{n-2}\frac{1}{\mathfrak{b}}$ and

$\Theta_{n}=\frac{2n-10}{2\left(n-2\right)}+\frac{5}{n-2}\frac{1}{\mathfrak{b}}=1+\frac{1}{n-2}\left(\frac{5}{\mathfrak{b}}-3\right).$

Moreover, by [7, Lemma 6.3], $M_{n}$ can be further bounded by

$\displaystyle
C^{-1}\sum_{l=1}^{3}\operatorname{Tr}\\!\left(\boldsymbol{\rho}\,(1+\boldsymbol{p}_{l}^{n})\right)\leq
M_{n}\leq
C\sum_{l=1}^{3}\operatorname{Tr}\\!\left(\boldsymbol{\rho}\,(1+\boldsymbol{p}_{l}^{n})\right)$

for some $C>0$. In particular

$\Theta_{n}\leq 1\iff\mathfrak{b}\geq\tfrac{5}{3}\iff a\leq\tfrac{4}{5}$

which yields the result by Grönwall’s Lemma. ∎

##  3\. Proof of Theorem 1.1

Notice that bounds on the moments imply bounds on the semiclassical Schatten
norms for any $p\in[2,\infty)$. More precisely, we have the following
proposition.

######  Proposition 3.1.

Let $\boldsymbol{\rho}$ be a positive trace class operator. Then, for any
$p\in[2,\infty)$, there exists a constant $C>0$ such that

$\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{p}}\leq
C\left\lVert\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{\infty}}^{\frac{1}{p^{\prime}}}\left(1+M_{np}\right)^{\frac{1}{p}}.$

###### Proof.

Using the fact that $\boldsymbol{\rho}$ is a bounded operator, we get that

$\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{p}}\leq\left\lVert\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{\infty}}^{\frac{1}{p^{\prime}}}\left\lVert\boldsymbol{\rho}^{\frac{1}{p}}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{p}}\leq\left\lVert\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{\infty}}^{\frac{1}{p^{\prime}}}\left\lVert\sqrt{\boldsymbol{\rho}}\,\mathsf{m}_{n}^{\frac{p}{2}}\right\rVert_{\mathcal{L}^{2}}^{\frac{2}{p}}=\left\lVert\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{\infty}}^{\frac{1}{p^{\prime}}}h^{3}\operatorname{Tr}\\!\left(\boldsymbol{\rho}\,\mathsf{m}_{n}^{p}\right)^{\frac{1}{p}},$

where the second inequality follows from the Araki-Lieb-Thirring inequality. ∎

For $p=\infty$, we control
$\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{\infty}}$
by means of a Grönwall argument. We will need the following commutator
estimates in the spirit of [13, 7].

######  Proposition 3.2 (Weighted commutator estimate).

Let $a\in(0,\frac{1}{2})$, $n\in\mathbb{N}$ and $q\in[1,\infty]$. Then for any
$\varepsilon\in(0,1)$ and $n_{1}>n+a+1$, there exists a constant $C>0$ such
that for every compact self-adjoint operators $\boldsymbol{\rho}$ and
$\boldsymbol{\mu}$

(12)
$\displaystyle\frac{1}{\hbar}\left\lVert\left[{E_{\boldsymbol{\rho}},\boldsymbol{p}_{\mathrm{j}}^{n}}\right]\boldsymbol{\mu}\right\rVert_{\mathcal{L}^{q}}$
$\displaystyle\leq
C\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n_{1}}\right\rVert_{\mathcal{L}^{r\pm\varepsilon}}\left\lVert\boldsymbol{\mu}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{q}},$
(13)
$\displaystyle\frac{1}{\hbar}\left\lVert\left[{V_{\boldsymbol{\rho}},\boldsymbol{p}_{\mathrm{j}}^{n}}\right]\boldsymbol{\mu}\right\rVert_{\mathcal{L}^{q}}$
$\displaystyle\leq
C\left(\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n_{1}}\right\rVert_{\mathcal{L}^{r\pm\varepsilon}}+M_{0}\right)\left\lVert\boldsymbol{\mu}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{q}},$

where $r=\tfrac{3}{1-a}$ and
$\left\lVert\cdot\right\rVert_{\mathcal{L}^{r\pm\varepsilon}}$ stands for the
norm

$\left\lVert\cdot\right\rVert_{\mathcal{L}^{r\pm\varepsilon}}:=\left\lVert\cdot\right\rVert_{\mathcal{L}^{r+\varepsilon}\cap\mathcal{L}^{r-\varepsilon}}=\left\lVert\cdot\right\rVert_{\mathcal{L}^{r+\varepsilon}}+\left\lVert\cdot\right\rVert_{\mathcal{L}^{r-\varepsilon}}.$

###### Proof of Proposition 3.2.

We first proceed as in [13, 7] and write

$\frac{1}{\hbar}\left\lVert\left[{E,\boldsymbol{p}_{\mathrm{j}}^{n}}\right]\boldsymbol{\mu}\,\right\rVert_{\mathcal{L}^{q}}\leq\sum_{\ell=0}^{n-1}C_{\ell}\left\lVert
g_{\ell}\,\,\boldsymbol{p}_{\mathrm{j}}^{n-1-\ell}\boldsymbol{\mu}\,\right\rVert_{\mathcal{L}^{q}}\leq\sum_{\ell=0}^{n-1}C_{\ell}\left\lVert
g_{\ell}\right\rVert_{L^{\infty}}\left\lVert\boldsymbol{\mu}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{q}},$

where $g_{\ell}$ is the function defined by
$g_{\ell}(x)=(\boldsymbol{p}_{\mathrm{j}}^{\ell}(\partial_{\mathrm{j}}E))(x)$
and $C_{\ell}=\sum_{k=\ell}^{n-1}\binom{k}{\ell}$. Noticing that

$\displaystyle g_{\ell}=-\partial_{\mathrm{j}}\nabla
K*(\boldsymbol{p}_{\mathrm{j}}^{\ell}\rho)=-\partial_{\mathrm{j}}\nabla
K*\operatorname{diag}\\!\left(\operatorname{ad}_{\boldsymbol{p}_{\mathrm{j}}}^{\ell}({\boldsymbol{\rho}})\right)$

where $\operatorname{ad}_{A}(X):=[A,X]$ and using the fact that
$\partial_{\mathrm{j}}\nabla K\in
L^{\frac{3}{a+2}+\epsilon}+L^{\frac{3}{a+2}-\epsilon}$ for any $\epsilon>0$
sufficiently small, then Young’s inequality yields

$\displaystyle\left\lVert g_{\ell}\right\rVert_{L^{\infty}}\leq
C_{K}\left\lVert\operatorname{diag}\\!\left(\operatorname{ad}_{\boldsymbol{p}_{\mathrm{j}}}^{\ell}\\!(\boldsymbol{\rho})\right)\right\rVert_{L^{r\pm\varepsilon}}$

for any $\varepsilon\in(0,1)$. By [7, Proposition 6.4 and Lemma 6.5], it
implies that

$\displaystyle\left\lVert g_{\ell}\right\rVert_{L^{\infty}}\leq
C_{K}\left\lVert\operatorname{ad}_{\boldsymbol{p}_{\mathrm{j}}}^{\ell}\\!\left({\boldsymbol{\rho}}\right)\mathsf{m}_{n_{0}}\right\rVert_{\mathcal{L}^{r\pm\varepsilon}}\leq
2^{\ell+1}\,C_{K}\left\lVert{\boldsymbol{\rho}}\,\mathsf{m}_{n_{0}+n-1}\right\rVert_{\mathcal{L}^{r\pm\varepsilon}}$

with $n_{0}>3/r^{\prime}$. This proves Inequality (12). When $E$ is replaced
by $V$, then we just replace $\nabla K$ by $K$ and so we need to find a
$L^{\infty}$ bound for the function

$\displaystyle
g_{\ell}=-\partial_{\mathrm{j}}K*(\boldsymbol{p}_{\mathrm{j}}^{\ell}\rho).$

When $\ell>0$, we just write
$g_{\ell}=i\hbar\,\partial_{\mathrm{j}}^{2}K*(\boldsymbol{p}_{\mathrm{j}}^{\ell-1}\rho)$
and then use the same estimates as for $E$. If $\ell=0$, then
$g_{\ell}=E_{j}=-\partial_{\mathrm{j}}K*\rho$ is bounded using Young’s
inequality by

$\left\lVert
g_{\ell}\right\rVert_{L^{\infty}}\leq\left\lVert\rho\right\rVert_{L^{\mathfrak{b}^{\prime}\pm\varepsilon}}\leq\left\lVert\rho\right\rVert_{L^{r}}+\left\lVert\rho\right\rVert_{L^{1}}$

since $\mathfrak{b}^{\prime}\leq r$. By [7, Proposition 6.4],
$\left\lVert\rho\right\rVert_{L^{r}}\leq\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n_{0}}\right\rVert_{\mathcal{L}^{r}}$
with $n_{0}>3/r^{\prime}=a+2$. ∎

######  Proposition 3.3.

Let $a\in\left(0,\frac{1}{2}\right)$ and $\boldsymbol{\rho}$ be a solution to
the Hartree-Fock equation (1). Assume
$\boldsymbol{\rho}^{\mathrm{in}}\in\mathcal{L}^{\infty}(\mathsf{m}_{n})$ for
some even $n\in 2\mathbb{N}$ with moments of order
$\left(n+a+1\right)\big{(}\frac{3}{1-a}+\varepsilon\big{)}$ bounded uniformly
in $\hbar$ for some $\varepsilon\in\,\big{(}0,1\big{)}$. Then

$\boldsymbol{\rho}\in
L^{\infty}_{\mathrm{loc}}(\mathbb{R}_{+},\mathcal{L}^{\infty}(\mathsf{m}_{n})).$

###### Proof.

By [7, Lemma 6.2] with $q\rightarrow\infty$, we have

(14)
$\frac{\mathrm{d}}{\mathrm{d}t}\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{\infty}}\leq\dfrac{1}{\hbar}\left\lVert\left[{V_{\boldsymbol{\rho}},\mathsf{m}_{n}}\right]\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{\infty}}+\dfrac{1}{\hbar}\left\lVert\left[{h^{3}\mathsf{X}_{\boldsymbol{\rho}},\mathsf{m}_{n}}\right]\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{\infty}}.$

Next, we use Proposition 3.2 to bound the first term on the right-hand side of
Inequality (14), leading to111We use the standard notation $x\lesssim y$ to
denote $x\leq Cy$ for some constant $C>0$ which is independent of $x$, $y$ and
$\hbar$.

$\dfrac{1}{\hbar}\left\lVert\left[{V_{\boldsymbol{\rho}},\mathsf{m}_{n}}\right]\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{\infty}}\lesssim\left(1+\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n_{1}}\right\rVert_{\mathcal{L}^{r\pm\varepsilon}}\right)\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{\infty}},$

where $r=\frac{3}{1-a}$. Since $\varepsilon\leq 1$ and $r\geq 3$, we see that
$r-\varepsilon\geq 2$ which allows us to apply Proposition 3.1 to get

(15)
$\dfrac{1}{\hbar}\left\lVert\left[{V_{\boldsymbol{\rho}},\mathsf{m}_{n}}\right]\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{\infty}}\lesssim\left(1+\left\lVert\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{\infty}}+M_{n_{1}\left(r+\varepsilon\right)}\right)\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{\infty}}.$

As for the second term on the right-hand side of (14), we apply [7,
Proposition 6.8] and use that if $\mathsf{m}_{n}=1+|{\boldsymbol{p}}|^{n}$
then

$\hbar^{\frac{3}{2}-a}\left\lVert\boldsymbol{\nabla}_{\\!x}\boldsymbol{\rho}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{p}}=\hbar^{\frac{1}{2}-a}\left\lVert\left[{\boldsymbol{p},\boldsymbol{\rho}}\right]\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{p}}\leq
2\,\hbar^{\frac{1}{2}-a}\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n+1}\right\rVert_{\mathcal{L}^{p}},$

whence

$\dfrac{1}{\hbar}\left\lVert\left[{h^{3}\mathsf{X}_{\boldsymbol{\rho}},\mathsf{m}_{n}}\right]\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{\infty}}\lesssim\hbar^{\frac{1}{2}-a}\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n+1}\right\rVert_{\mathcal{L}^{2}}\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{\infty}}.$

Proposition 3.1 allows us to bound
$\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n+1}\right\rVert_{\mathcal{L}^{2}}$
in terms of $M_{2\left(n+1\right)}$ and obtain the following inequality for
the left-hand side of Equation (14)

$\frac{\mathrm{d}}{\mathrm{d}t}\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{\infty}}\lesssim\left(1+\left\lVert\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{\infty}}+M_{n_{1}\left(r+\varepsilon\right)}+M_{2\left(n+1\right)}\right)\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{\infty}}$

that gives a bound on
$\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{\infty}}$
by means of Grönwall’s Lemma and the control on moments established in Theorem
2.1. ∎

Summarizing, Theorem 2.1 and the two propositions 3.1 and 3.3 imply the
following result.

######  Corollary 3.1.

Let $a\in\left(0,\frac{1}{2}\right)$ and $\boldsymbol{\rho}$ be a solution to
the Hartree-Fock equation (1). For $n\in 2\mathbb{N}$ and let
$\boldsymbol{\rho}^{\mathrm{in}}\in\mathcal{L}^{\infty}(\mathsf{m}_{n})$ with
moment of order $\left(n+a+1\right)\left(\frac{3}{1-a}+\varepsilon\right)$
bounded uniformly in $\hbar$ for some $\varepsilon\in\left(0,1\right)$. Then
for any $q\in[2,\infty]$, we have that

$\boldsymbol{\rho}\in
L^{\infty}_{\mathrm{loc}}(\mathbb{R}_{+},\mathcal{L}^{q}(\mathsf{m}_{n})).$

###### Proof of Theorem 1.1.

To propagate quantum Sobolev norms, we proceed as in [7, Section 6] and
consider the inequalities

$\frac{\mathrm{d}}{\mathrm{d}t}\left\lVert\boldsymbol{\nabla}_{\\!x}\boldsymbol{\rho}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{q}}\leq\dfrac{1}{\hbar}\left(\left\lVert\left[{V_{\boldsymbol{\rho}},\mathsf{m}_{n}}\right]\boldsymbol{\nabla}_{\\!x}\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{q}}+\left\lVert\left[{E_{\boldsymbol{\rho}},\mathsf{m}_{n}}\right]\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{q}}\right)+\dfrac{1}{\hbar}\left\lVert\left[{E_{\boldsymbol{\rho}},\boldsymbol{\rho}\,\mathsf{m}_{n}}\right]\right\rVert_{\mathcal{L}^{q}}\\\
+\dfrac{1}{\hbar}\left\lVert\left[{h^{3}\mathsf{X}_{\boldsymbol{\rho}},\mathsf{m}_{n}}\right]\boldsymbol{\nabla}_{\\!x}\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{q}}+\dfrac{1}{\hbar}\left\lVert\left[{h^{3}\mathsf{X}_{\boldsymbol{\nabla}_{\\!x}\boldsymbol{\rho}},\boldsymbol{\rho}}\right]\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{q}}$

and

$\frac{\mathrm{d}}{\mathrm{d}t}\left\lVert\boldsymbol{\nabla}_{\\!\xi}\boldsymbol{\rho}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{q}}\leq\dfrac{1}{\hbar}\left\lVert\left[{V_{\boldsymbol{\rho}},\mathsf{m}_{n}}\right]\boldsymbol{\nabla}_{\\!\xi}\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{q}}+\left\lVert\boldsymbol{\nabla}_{\\!x}\boldsymbol{\rho}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{q}}\\\
+\dfrac{1}{\hbar}\left\lVert\left[{h^{3}\mathsf{X}_{\boldsymbol{\rho}},\mathsf{m}_{n}}\right]\boldsymbol{\nabla}_{\\!\xi}\boldsymbol{\rho}\right\rVert_{\mathcal{L}^{q}}+\dfrac{1}{\hbar}\left\lVert\left[{h^{3}\mathsf{X}_{\boldsymbol{\nabla}_{\\!\xi}\boldsymbol{\rho}},\boldsymbol{\rho}}\right]\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{q}}$

and we define
$N_{q}=N_{q}(t):=\left\lVert\boldsymbol{\rho}\right\rVert_{\mathcal{W}^{1,q}(\mathsf{m}_{n})}$.
Now, when $q\leq\frac{6}{1+2a}$, [7, Proposition 6.5] yields an estimate of
the form

(16)
$\dfrac{1}{\hbar}\left\lVert\left[{E_{\boldsymbol{\rho}},\boldsymbol{\rho}\,\mathsf{m}_{n}}\right]\right\rVert_{\mathcal{L}^{q}}\leq\left\lVert\rho\right\rVert_{L^{r\pm\varepsilon}\cap
L^{2}}\left\lVert\boldsymbol{\nabla}_{\\!x}\\!\left(\boldsymbol{\rho}\,\mathsf{m}_{n}\right)\right\rVert_{\mathcal{L}^{q}}$

with $r=\frac{3}{1-a}$. Then, using the above inequality, Proposition 3.2 and,
similarly as in the proof of the previous proposition, [7, Proposition 6.8 and
Proposition 6.9] to bound the terms involving the exchange term, we obtain for
any $q\leq\frac{6}{1+2a}$

$\frac{\mathrm{d}}{\mathrm{d}t}N_{q}\lesssim\left(1+\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n_{1}}\right\rVert_{\mathcal{L}^{r\pm\varepsilon}}\right)N_{q}+\left\lVert\rho\right\rVert_{L^{r\pm\varepsilon}\cap
L^{2}}N_{q}\\\
+h^{\frac{1}{2}-a}\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n+1}\right\rVert_{\mathcal{L}^{2}}N_{q}+h^{3\left(\frac{1}{q}+\frac{1}{2}-\frac{1}{\mathfrak{b}}\right)}\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{\infty}}N_{2},$

with $n_{1}>n+a+1$. Notice that for $a\in(0,\frac{1}{2})$ the quantity
$\frac{1}{q}+\frac{1}{2}-\frac{1}{\mathfrak{b}}$ is positive for every
$q\in[2,\infty)$. Hence, for $h\in(0,1)$, we have that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(N_{2}+N_{q}\right)$
$\displaystyle\lesssim\left(1+\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n_{1}}\right\rVert_{\mathcal{L}^{r\pm\varepsilon}}+\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n}\right\rVert_{\mathcal{L}^{\infty}}+\left\lVert\boldsymbol{\rho}\,\mathsf{m}_{n+1}\right\rVert_{\mathcal{L}^{2}}\right)\left(N_{2}+N_{q}\right)$

which proves Theorem 1.1 for $q\leq\frac{6}{1+2a}$ by Corollary 3.1 and
Grönwall’s Lemma. The limitation on $q$ is due to the fact that we do not want
to have any $\nabla\rho$ in the right-hand side of inequality (16). But now
that we obtained the boundedness of $N_{2}$, we can propagate higher norms of
the form $N_{q_{2}}$ when $q_{2}>\frac{6}{1+2a}$. Indeed, [7, Proposition 6.5]
also yields for instance

$\dfrac{1}{\hbar}\left\lVert\left[{E_{\boldsymbol{\rho}},\boldsymbol{\rho}\,\mathsf{m}_{n}}\right]\right\rVert_{\mathcal{L}^{\infty}}\leq\left\lVert\rho\right\rVert_{L^{2}}^{1-\theta}\left\lVert\rho\right\rVert_{H^{1}}^{\theta}\left\lVert\boldsymbol{\nabla}_{\\!x}\\!\left(\boldsymbol{\rho}\,\mathsf{m}_{n}\right)\right\rVert_{\mathcal{L}^{\infty}}$

with $\theta=\frac{3}{\mathfrak{b}}-\frac{1}{2}=a+\frac{1}{2}\in(0,1)$, and
$\left\lVert\rho\right\rVert_{L^{2}}$ is controlled by $N_{2}$. All the
$q_{2}\in(2,\infty)$ follow in the same way finishing the proof of the
theorem. ∎

Acknowledgments. J.C. was supported by the NSF through the RTG grant DMS- RTG
184031. C.S. acknowledges the NCCR SwissMAP and the support of the SNSF
through the Eccellenza project PCEFP2_181153.

## References

  * [1] N. Benedikter, V. Jaksic, M. Porta, C. Saffirio, and B. Schlein.  Mean-Field Evolution of Fermionic Mixed States.  Communications on Pure and Applied Mathematics, 69(12):2250–2303, 2016.
  * [2] N. Benedikter, M. Porta, and B. Schlein.  Mean-field Evolution of Fermionic Systems.  Communications in Mathematical Physics, 331(3):1087–1131, Nov. 2014\.
  * [3] A. Bove, G. Da Prato, and G. Fano.  An existence proof for the Hartree–Fock time-dependent problem with bounded two-body interaction.  Communications in Mathematical Physics, 37(3):183–191, Sept. 1974\.
  * [4] A. Bove, G. Da Prato, and G. Fano.  On the Hartree–Fock time-dependent problem.  Communications in Mathematical Physics, 49(1):25–33, 1976.
  * [5] F. Castella.  L2 Solutions to the Schrödinger–Poisson System: Existence, Uniqueness, Time Behaviour, and Smoothing Effects.  Mathematical Models and Methods in Applied Sciences, 7(08):1051–1083, 1997.
  * [6] J. M. Chadam.  The time-dependent Hartree–Fock equations with Coulomb two-body interaction.  Communications in Mathematical Physics, 46(2):99–104, June 1976\.
  * [7] J. J. Chong, L. Lafleche, and C. Saffirio.  From Many-Body Quantum Dynamics to the Hartree–Fock and Vlasov Equations with Singular Potentials.  arXiv:2103.10946, pages 1–74, Mar. 2021.
  * [8] S. A. Gaal.  Linear analysis and representation theory.  Die Grundlehren der mathematischen Wissenschaften. Springer, Berlin, Heidelberg, 1973.
  * [9] J. Ginibre and G. Velo.  On a Class of non Linear Schrödinger Equations with non Local Interaction.  Mathematische Zeitschrift, 170(2):109–136, 1980.
  * [10] J. Ginibre and G. Velo.  The Global Cauchy Problem for the non Linear Schrödinger Equation Revisited.  Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 2(4):309–327, July 1985.
  * [11] F. Golse, S. Jin, and T. Paul.  On the Convergence of Time Splitting Methods for Quantum Dynamics in the Semiclassical Regime.  Foundations of Computational Mathematics, 21(3):613–647, June 2021\.
  * [12] R. Illner, P. F. Zweifel, and H. Lange.  Global Existence, Uniqueness and Asymptotic Behaviour of Solutions of the Wigner–Poisson and Schrodinger–Poisson Systems.  Mathematical Methods in the Applied Sciences, 17(5):349–376, Apr. 1994.
  * [13] L. Lafleche.  Propagation of Moments and Semiclassical Limit from Hartree to Vlasov Equation.  Journal of Statistical Physics, 177(1):20–60, Oct. 2019.
  * [14] L. Lafleche.  Global Semiclassical Limit from Hartree to Vlasov Equation for Concentrated Initial Data.  Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 38(6):1739–1762, Nov. 2021.
  * [15] L. Lafleche and C. Saffirio.  Strong Semiclassical Limit from Hartree and Hartree-Fock to Vlasov-Poisson Equation.  Analysis & PDE, accepted for publication:1–35, Oct. 2021.
  * [16] M. Lewin and J. Sabin.  The Hartree Equation for Infinitely Many Particles I. Well-Posedness Theory.  Communications in Mathematical Physics, 334(1):117–170, Feb. 2015\.
  * [17] P.-L. Lions and T. Paul.  Sur les mesures de Wigner.  Revista Matemática Iberoamericana, 9(3):553–618, 1993.
  * [18] M. Porta, S. Rademacher, C. Saffirio, and B. Schlein.  Mean Field Evolution of Fermions with Coulomb Interaction.  Journal of Statistical Physics, 166(6):1345–1364, 2017.
  * [19] C. Saffirio.  Semiclassical Limit to the Vlasov Equation with Inverse Power Law Potentials.  Communications in Mathematical Physics, 373(2):571–619, Mar. 2019\.
  * [20] C. Saffirio.  From the Hartree equation to the Vlasov–Poisson system: strong convergence for a class of mixed states.  SIAM Journal on Mathematical Analysis, 52(6):5533–5553, Jan. 2020\.


# Boundaries for the Honeycomb Code

Jeongwan Haah  Microsoft Quantum and Microsoft Research, Redmond, WA 98052,
USA  Matthew B. Hastings  Station Q, Microsoft Quantum, Santa Barbara, CA
93106-6105, USA  Microsoft Quantum and Microsoft Research, Redmond, WA 98052,
USA

###### Abstract

We introduce a simple construction of boundary conditions for the honeycomb
code [1] that uses only pairwise checks and allows parallelogram geometries at
the cost of modifying the bulk measurement sequence. We discuss small
instances of the code.

The recently introduced honeycomb code [1] is a code with dynamically
generated logical qubits that falls outside the usual stabilizer and subsystem
code formalism. This code uses only pairwise checks, which are a product of
Pauli matrices on two different qubits. The checks are measured in a
particular sequence which is broken up into “rounds” where in each round one-
third of the checks are measured, with the measurement pattern repeating every
three rounds.

Since the honeycomb code at any moment in the dynamics is in a state that is
virtually the same as the toric code state [2, 3], it is conceivable that a
full quantum architecture can be built out of the honeycomb code. One of the
potential advantages of the honeycomb code over the toric code as a basic
logical element in a quantum architecture is that the number of possible error
locations per unit spacetime volume in a honeycomb code implementation appears
to be smaller than that for the toric code. This is especially so when the
codes are implemented by one- and two-qubit Pauli measurements; compare [4]
and [5]. The fewer the number of ways errors might occur, the better the
performance should be.

The toric code has well-known boundary conditions [6, 7] (rough and smooth
[8]) with which one can implement a version of the code, called the surface
code, using nearest-neighbor interactions on a two-dimensional planar grid of
qubits. This is quite important because of a straightforward layout of many
logical qubits in a plane, making the surface code appealing for a quantum
architecture at scale. In contrast, the honeycomb code’s boundary conditions
were relatively poorly understood. The dynamics interchanges electric and
magnetic operators after every round, and hence, in order to construct a code
with boundaries rather than on a torus, it is necessary for the rough and
smooth boundary conditions to alternate every round. To this end, we sketched
[1] one solution where the code is shrunk after certain measurement rounds to
preserve the correct boundary conditions, and then periodically grown by using
some nonpairwise checks. However, a simpler solution is certainly desirable.
In Section 1, we present a simple solution for boundary conditions of the
honeycomb code which involves modifying the bulk measurement sequence. In
Section 2, we discuss surgery. In Section 3, we briefly consider tilings of
the torus to construct small codes and consider their distance. A detailed
performance comparison will require numerical simulation [9].

While this paper was in preparation, another planar realization of the
honeycomb code was given [10]. Unfortunately, the boundary conditions of [10]
make the protocol non-fault-tolerant. We will give some intuition for this at
the end of Section 1.6.

##  1 Boundary conditions

We begin by considering ways of “gapping” the instantaneous stabilizer group
(ISG) with boundaries in Section 1.1. Then in Section 1.3, we describe a
modified bulk measurement sequence to realize a dynamically generated logical
qubit on an annulus (or more generally on a multiply punctured disk). Finally,
in Section 1.4, we give a parallelogram geometry.

###  1.1 2-gon and 4-gon boundaries

Figure 1:  Left: armchair with 4-gons on both boundaries. Right: armchair with
2-gons on bottom boundary and 4-gons at top. These two figures show that we
can tune the height of the strip by using different boundary conditions. We
have displayed the boundary check operators explicitly.

The left side of Fig. 1 is the same as Fig. 7 of [1], with the addition of
some extra cells at the top to gap both boundaries. This shows one way to
introduce boundaries. Assume the left and right edges are continued further
and then joined to make an (topological) annulus. The geometry of the hexagons
may be termed an “armchair” geometry at the boundary, following terminology in
carbon nanotubes. We have added additional edges to create additional
plaquettes with four edges, that we term _4-gons_. Each 4-gon is a type 2
plaquette at the bottom, and the types of the edges are defined
correspondingly: an edge at the very bottom that belongs to a hexagon is type
2, and an edge at the very bottom that belongs to a 4-gon is type 0. The void
below the lattice can be regarded as a very large plaquette of type 1. At the
top, type 0 and type 2 are interchanged; the void above the lattice is
regarded type 1.

The right side of Fig. 1 shows an alternative. In this case we have added
_2-gons_ (also sometimes called bigons) at the bottom, each of which has only
two edges. These 2-gons are of type 1 in this case. Each added half-circle
edge is type 0, and the other edge of a 2-gon is type 2. Note that the
perpetual plaquette stabilizer on a 2-gon has weight $2$. Further, notice that
if we measure a check which is an edge of a 2-gon (either type 0 or type 2 in
this figure), then the two qubits in the 2-gon are in a Bell state, and so
disentangled from the others.

###  1.2 Embedded surface codes

It is routine [1] but is crucial to our construction to identify an embedded
surface code state with gapped boundary after each round. Every check projects
a pair of qubits into an effective qubit (${\mathbb{C}}^{2}$), and one can
draw a line segment orthogonal to the edge that covers the pair. These line
segments give a superlattice of effective qubits, and perpetual plaquette
stabilizers correspond to supervertex and superplaquette stabilizers on the
effective qubits. The result is drawn in Figs. 2 and 3; see the quadrants
labeled by 0, 1, and 2; an exception is given to step $0^{*}$ where an
effective qubit may be due to a surviving check operator from the previous
round.

In fact, there is a canonical choice of Clifford frame for each effective
qubit, implied by the superlattice stabilizers. Here, a Clifford frame is
nothing but a choice of logical operators (up to a sign) on the $[[2,1,1]]$
code defined by the check operator. Each superplaquette contains exactly one
plaquette of the original lattice, on which the perpetual stabilizer acts by
some single-qubit Pauli operator on each effective qubit that surrounds the
superplaquette. That single-qubit operator must be identified with
$Z_{\text{eff}}$ on the effective qubit. Likewise, each supervertex is
contained in exactly one original plaquette, on which the perpetual stabilizer
is a tensor product of several two-qubit operator on edges, which we identify
with $X_{\text{eff}}$ up to a sign. For $Z_{\text{eff}}$ in the bulk, a
simpler rule is that if a check operator is $a\otimes b$, then
$Z_{\text{eff}}=\pm(a\otimes I)=(I\otimes b)(a\otimes b)$.

The thickness of the annulus is chosen such that the top boundary conditions
of the embedded toric code state match the bottom ones. With a different
thickness it is possible to have 2-gons at both top and bottom.

###  1.3 Modification of bulk sequence

Figure 2:  At each round, the superlattice supports an embedded toric code
state with boundaries. _After round 0_ , each 4-gon at the bottom has two
edges of type 0, both of which are measured. Since there is a perpetual
plaquette stabilizer at each 4-gon, the instantaneous stabilizers on a 4-gon
project its four qubits down to ${\mathbb{C}}^{2}$, depicted by a slightly
longer vertical superedge. _After round 2_ , a similar projection is realized
at the top 4-gons of type 0. _After round 0 ∗_, the edges of type 0 at the
boundary (top and bottom) are _not_ measured, and therefore the type 2 check
at the boundary remains in the ISG, projecting its two qubits to
${\mathbb{C}}^{2}$. These projections make super-3-gons and super-5-gons at
the boundaries.  Figure 3:  Superlattices similar to those in Fig. 2. _After
round 0 ∗_, the checks of type 1 at the top boundary remain in the ISG,
rendering the superlattice to have 5-gons at the top boundary. The perpetual
plaquette stabilizer on each 2-gon projects the two qubits of the 2-gon to
${\mathbb{C}}^{2}$, rendering the superlattice to have 5-gons at the bottom
boundary. The product of the two type 1 checks positioned the lowest in the
figure around a “bay,” survives in the ISG and gives the supervertex
stabilizer. _After round 2_ , the type 2 check on the edge of a 2-gon and its
perpetual stabilizer make a Bell state, disentangling the two qubits of the
2-gon from the others. _After round 0_ , each 4-gon makes its four qubits into
one effective qubit, and each 2-gon makes a Bell state.

Under the choice of boundaries with all 4-gons as in the left side of Fig. 1,
if we measured type 2 checks and then type 0 checks, we would reveal the inner
logical operator, destroying the dynamically generated logical qubit. This may
be understood in two ways. One way is that the bulk dynamics interchanges
electric and magnetic operators every round, while the boundary conditions
after round 2 and round 0 are of the same type (both rough) rather than
alternating. Another way is to view the annulus on a topological
$2$-dimensional sphere so that the bottom edges form the boundary of a single
“very large plaquette” of type 1; measuring type 2 and then type 0 checks
infers the stabilizer of the very large plaquette. A similar problem arises
with the right side of Fig. 1. In this case, the “very large plaquette” is of
type 2, whose stabilizer (an inner logical operator) would be revealed if we
measured type 0 and then type 1 checks.

To have a dynamically generated logical qubit on an annulus, while using only
pairwise checks, we modify the bulk measurement sequence as follows. We
measure edges of type

$0,1,2,0^{*},2,1$ (1)

in order and repeating. Depending on the boundary geometry we may use a
sequence

$0^{*},1,2,0,2,1.$ (2)

They have period $6$, not $3$. The notation is that $0,1,2$ means measuring
all checks of the given type, but $0^{*}$ means measuring all type-0 checks
except those on a boundary. Thus we do not reveal the inner operator.

Both sequences infer all small plaquettes: for $(a,b,c)$ that is a permutation
of $(0,1,2)$, after measuring checks of type $a$ and type $b$ in succession,
we infer plaquette stabilizers of type $c$. In other words, we infer plaquette
stabilizers of type $2,2,0,1,1,0$ in sequence. This is independent of where we
have $0^{*}$ in the measurement sequence. See Figs. 2 and 3. The boundary
conditions in the embedded toric codes are alternating from round to round as
they should.

###  1.4 Parallelogram

To encode a single logical qubit on a parallelogram in the usual toric code,
one pair of opposite sides (top & bottom) of the parallelogram should have
smooth boundary conditions, while the other pair (left & right) should have
rough boundary conditions. In the honeycomb code, we have to make sure that
boundaries of a parallelogram alternate in time while neighboring sides have
opposite boundary conditions. One solution is depicted in Fig. 4. It is
straightforward to make the parallelogram larger. Note that the patch depicted
in Fig. 4 can fill the infinite plane of the honeycomb lattice of qubits with
no gaps in between parallelograms.

Figure 4:  A parallelogram patch of honeycomb code using 4- and 2-gon
boundaries. The construction of superlattice follows the same reasoning as in
Figs. 2 and 3. New phenomena occur at the corners. First, the top left qubit
participates in four checks, and there is no static plaquette stabilizer
associated with the two intersecting 2-gons; however, the three qubits in the
support of these two 2-gons are in some subspace ${\mathbb{C}}^{2}$ always
(except for the very first round). Second, in the lower left corner after
round 02 the perpetual plaquette stabilizer at the 2-gon and the type 1 check
from the previous round 1 project the three qubits covered by the curvy line
to a subspace ${\mathbb{C}}^{2}$. A symmetric situation (1$\leftrightarrow$2)
is observed in the top right corner at round 01.

A strategy to construct a parallelogram with armchair boundaries is as
follows. (i) We choose the height of a horizontal strip and use 2- or 4-gon
boundaries (as in Fig. 1) such that the top and bottom boundaries have the
same type of superlattice boundaries (rough and smooth) at round 1 and 2. (ii)
We truncate the strip on the left and use 2- or 4-gon boundary geometry such
that the superlattice boundary condition on the left is opposite to that on
the top and bottom at round 1 and 2. (iii) We truncate the strip on the right
such that the superlattice boundary condition is the same as that on the left.
Whether we use 2-gon or 4-gon boundary geometry on the right will be dictated
by the width of the parallelogram. (iv) By demanding the spatial alternation
of superlattice boundary conditions along the boundary, we determine which
kind of round 0 should be used in the measurement sequence

$0^{\prime},1,2,0^{\prime\prime},2,1$ (3)

where 0′ or 0′′ indicates that certain type 0 checks are not measured at the
boundaries at that round.

###  1.5 More general shapes

A general prescription to build a planar patch that encodes one logical qubit
is as follows. A qubit must participate in exactly $3$ checks except for four
qubits where the boundary condition of the embedded surface code changes. In
each portion of the boundary on which the qubits have degree $3$ only, the
types of the edges should alternate between two options. The bulk measurement
sequence is still $\ldots,0,1,2,0,2,1,\ldots$ that has period $6$. At round
$1$ and $2$, all checks of a given type is measured. At round $0$, some type
$0$ checks near the boundary are omitted, which can be easily identified by
demanding that the embedded surface code should have time-alternating boundary
conditions. In Fig. 4 there are three qubits that are acted on by only two
checks, and one qubit that is acted on by four checks. In Fig. 6 below we
construct a patch with two degree-$3$ qubits, and two degree-$4$ qubits.

###  1.6 Decoding graphs

#### Bulk.

We begin by considering the infinite honeycomb lattice with the period 6
sequence, $0,1,2,0,2,1$. Since there is no boundary, the round 0 is not
modified. We continue to use the simplified model [1] in which each check
$E\otimes E$ of weight $2$ at any location in spacetime may be followed by an
error $E$ on one of the qubits, but no other error occurs. This simplified
error model generates all possible error patterns including measurement
outcome flips. Type $a$ checks at a round and type $b$ checks at the next
round infer type $c$ plaquette stabilizers, where $a,b,c$ are distinct. So,
the sequence of plaquette types that are inferred (immediately after the check
operators are measured) is

$2,0,1,1,0,2$ (4)

that has period $6$.

An error $E$ after a type $a$ check and before a type $b$ check, flips a fresh
type $c$ plaquette stabilizer; by construction, $E$ anticommutes with the type
$b$ check that follows $E$ in time. All subsequent plaquette stabilizers are
inferred by checks that are measured after $E$, and therefore whether a given
plaquette stabilizer $P$ is flipped by $E$ is determined by the commutation
relation between $E$ and $P$.111 This conclusion is absurdly trivial in a
usual stabilizer code. However, in our consideration, some errors (that are
not first order in the current simplified error model) that commute with all
plaquette stabilizers, can cause nontrivial syndromes due to the timing when
the plaquette stabilizers are inferred. In other words, some error that
commutes with a plaquette stabilizer may still excite the syndrome node
associated with the plaquette stabilizer if the error is sandwiched between
the two measurements in time, by which the eigenvalue of the plaquette
stabilizer is inferred.  There are exactly two plaquette stabilizers that
anticommute with $E$.

We examine the plaquette stabilizers in spacetime that are inferred _after_ a
given error and mark them with $\star$ if they are the _first_ flip of the
plaquette stabilizer. Due to the spatial symmetry, we only have to look at the
plaquette types.

$\displaystyle\begin{array}[]{c|llllll|llllll|c}E\text{ after round
}0&2\star&0&1\star&1&0&2&2&0&1&1&0&2&\cdots\\\ E\text{ after round
}1&&0\star&1&1&0&2\star&2&0&1&1&0&2&\cdots\\\ E\text{ after round
}2&&&1\star&1&0\star&2&2&0&1&1&0&2&\cdots\\\ E\text{ after round
}0&&&&1\star&0&2\star&2&0&1&1&0&2&\cdots\\\ E\text{ after round
}2&&&&&0\star&2&2&0&1\star&1&0&2&\cdots\\\ E\text{ after round
}1&&&&&&2\star&2&0\star&1&1&0&2&\cdots\end{array}$ (11)

where $E$ is a single-qubit error that immediately follows a check at the
indicated round. This table allows us to construct a decoding graph. The nodes
are the plaquette stabilizers in spacetime, whose time coordinates are when
they are inferred. A node indicates if the associated plaquette stabilizer has
an eigenvalue that is _different_ from that of the same plaquette immediately
preceding in time. An error (in the current simplified noise model) causes
flips in the nodes indicated by $\star$, which happen to be two in number, so
the error gives an edge between those flipped nodes.

We see that there is no edge between time slices of an odd time coordinate
difference. If we collect all time slices with a fixed time coordinate parity,
the decoding graph is connected. Hence, there are two separate decoding
graphs. They are isomorphic due to the symmetry 1$\leftrightarrow$2\. The
decoding subgraph is not regular. Recalling the $2\pi/3$ rotation symmetry of
the honeycomb lattice, we see that in the subgraph containing nodes of the
first column in the table, type 2 nodes have degree $3$, type 0 nodes have
degree $6$, and type 1 nodes have degree $9$ ($6$ to the past and $3$ to the
future). The same matching graph argument as in [1] proves that there is a
fault tolerance accuracy threshold for this model that is nonzero.
Uncorrectable errors are precisely those that give a nontrivial homology cycle
in the decoding graph.

#### Annulus.

With boundaries at the top and bottom of a strip, there is a new type of
syndrome bit. From the decoding graph of the infinite bulk, the decoding graph
for a topological annulus can be obtained with the following modifications.

  * •

(Perpetual stabilizer) Keep bulk nodes that correspond to existing plaquettes
and edges among them.

  * •

(Transient stabilizer) Some check $c$ of type $a$ ($a=1,2$) at the boundary
near a $4$-gon survives at round $0$ that is sandwiched by two rounds of type
$a$ because we do not measure certain type-$0$ edges. Similarly, near a
$2$-gon, the product $c^{\prime}$ of the two type-$a$ checks on the “armrest”
of the “armchair” survives at round $0$. Such a check $c$ or a product
$c^{\prime}$ defines a transient stabilizer whose value is remeasured at the
subsequent round $a$. The consistency of these values is a syndrome bit. It is
important to keep track of these transient stabilizers, since they contribute
to the boundary conditions of the decoding graph, giving correct homology for
one logical qubit.

  * •

Add a terminal node for each fictitious “very large plaquette” beyond the
boundaries.

  * •

Connect the terminal nodes to the bulk nodes, as if it were a usual plaquette
neighboring the boundary plaquettes. The terminal node has high degree.

Uncorrectable errors are those that give a nontrivial homology cycle
_relative_ to the terminal nodes. The connected component of the decoding
graph which contains the new type of syndrome bits, has smooth boundaries and
is disconnected from the terminal nodes. The other subgraph has rough
boundaries if we delete the terminal nodes. This is analogous to the surface
code on a topological annulus where one component of the decoding graph
accounts for $X$ errors, and the other for $Z$ errors.

#### Non-fault-tolerance due to boundaries.

In introduction, we have remarked that the boundary construction of [10]
renders the code non-fault-tolerant. This phenomenon may seem subtle because
every ISG in the scheme of [10] defines a code with a large code distance.
However, a constant-weight fault configuration leading to a logical error
exists in spacetime, corresponding to the motion of an anyon with _finite_
velocity. In [10], the period $3$ measurement sequence is used for a whole
planar patch. In particular, the ISG with boundary included has period $3$ up
to signs. Consider an $e$-particle $x$ that enters into the bulk from a
boundary component at time $t$. Suppose that $x$ stays in the bulk for $3$
measurement rounds. Due to the nontrivial automorphism implemented by the
measurement sequence, the particle $x$ becomes $m$ at time $t+3$. However, the
boundary conditions of the ISG at time $t+3$, through which $x$ has entered,
is the same as those at time $t$, so the particle may exit, not through the
same spatial location, but only through a different boundary component. The
whole process may occur at a corner where the boundary conditions of ISG
changes, and therefore can be implemented by a constant-weight operator in
spacetime. On the other hand, nothing prevents for an $m$-particle $y$ at time
$t$ to stay near the corner indefinitely. The worldline of $y$ can therefore
link with that of $x$, suggesting that $x$ may implement a logical error.
Here, it is necessary that $x$ stays undetected to any decoder. It turns out
that this is indeed possible, as not all stabilizers are inferred every
measurement round.

#### Parallelogram.

The prescription for the annulus case applies verbatim; there are transient
stabilizers along the boundary, which are inferred whenever the check
measurement outcome is deterministic in the absence of errors. Note that in
contrast to [10], our boundary conditions evolve in time in a commensurate way
with the nontrivial anyon automorphism in the bulk. Any anyon that enters
through a boundary component $b$ may exit through either $b$ or the opposite
boundary component $b$.

##  2 Surgery

In this section we consider logical operations across neighboring
parallelograms. We present protocols to embed a qubit into a larger patch, and
to measure a product of logical operators, $\bar{X}\bar{X}$, $\bar{Z}\bar{Z}$,
$\bar{X}\bar{Z}$, or $\bar{Z}\bar{X}$, across two patches. Since we know how
to prepare an eigenstate of logical $\bar{X}$ or $\bar{Z}$ and to measure a
logical patch in $\bar{X}$ and $\bar{Z}$ basis [1], if supplemented with a
single-qubit logical gate $S={\mathrm{diag}}(1,i)$ or state
$\ket{0}+i\ket{1}$, our measurement portfolio implements all Clifford
operations.

Indeed, the joint $\bar{X}\bar{X}$ or $\bar{Z}\bar{Z}$ measurement, the
single-qubit preparation in $\bar{X}$ or $\bar{Z}$ basis, and the single-qubit
measurement in $\bar{X}$ or $\bar{Z}$ allow quantum teleportation. Logical
CNOT is possible by the joint and single-qubit measurements [11, §5]. The
group of single-qubit Clifford operations is generated by $S$ and the Hadamard
gate $H$. The $S$ gate is implemented by injecting $\ket{0}+i\ket{1}$ for
which we need the joint $\bar{Z}\bar{Z}$ and single-qubit $\bar{X}$
measurements. The effect of $H$ is to interchange $\bar{X}$ and $\bar{Z}$, but
our measurement portfolio is closed under such an interchange. Therefore, we
achieve Clifford completeness.

###  2.1 Growing patches

This step is used to prepare a patch containing a magic state
$\ket{0}+e^{i\pi/4}\ket{1}$. The prescription for a usual surface code [12] is
that we prepare data qubits that are going to be occupied by a larger patch in
an eigenstate of the to-be-grown part of the logical operators, and begin
stabilizer measurements. Since the logical operators are conserved operators
of the stabilizer measurements, the grown patch will have the original logical
information encoded. The same strategy is applicable for our honeycomb code.
We prepare ancillas in an eigenstate of the to-be-grown part of the
instantaneous logical operator, and begin our check measurement dynamics.

###  2.2 Logical $XX$ and $ZZ$ measurements

We bring two regular patches side by side. Then, we stop the measurements for
the patches and start the measurement sequence for the double sized patch. The
boundary 2-gons and 4-gons are dropped in between the patches and the “gap”
between the patches is filled with bulk hexagons and one 4-gon. The logical
outcome of the surgery is given by the product of the new plaquette
stabilizers. Whether the surgery implements logical $\bar{X}\otimes\bar{X}$ or
$\bar{Z}\otimes\bar{Z}$ measurement depends on whether the relative
displacement of the two patches is horizontal or vertical.

###  2.3 Logical $XZ$ and $ZX$ measurements

Figure 5:  An extended patch. The displayed patch occupies space that is worth
two parallelograms. Observe that along the bottom boundary there are both
smooth and rough sections. At the top right corner the type 0 hexagon is
disentangled from the rest in round 1 and 2.

Figure 6:  An extended patch of Fig. 5 is stitched with a parallelogram patch.
The qubits form a portion of honeycomb lattice and the hexagon labels are
inherited from the infinite honeycomb lattice. Unlike other surgery
operations, the upper extended patch and the lower regular patch run
measurement schedules that are opposite of each other, which is equivalent to
running the same sequence but with offset of 3 rounds. This offset
synchronization can be implemented by letting one of the patches idle for $3$
rounds. At the stitch, all the new edges are type 0, each of which is measured
at round 0. The outcome of the logical measurement is given by the product of
the plaquette stabilizers that depend on new type 0 edges. If this surgery
measures logical $\bar{Z}\otimes\bar{X}$, the symmetric version where the
extended patch is extended vertically measures logical
$\bar{X}\otimes\bar{Z}$.

We bring two regular patches so that they are diagonally separated in the
square lattice of regular patches. Then we merge the two patches via lattice
surgery depicted in Figs. 5 and 6. While the extension in Fig. 5 and the
merger in Fig. 6 do not have to be done separately, it is conceptually easier
to understand the full surgery by regarding it in those two steps, so we have
drawn two sets of figures. The extension step is no different from the usual
extension above. The merger is also similar; we stop the measurement sequences
for each patch and begin the depicted measurements. The measurement sequences
after the merger for the upper part and the lower part are opposite of each
other:

$\displaystyle\begin{array}[]{cccccccc}\text{upper:}&0^{2}&1&2&0^{1}&2&1&\cdots\\\
\text{lower:}&0^{1}&2&1&0^{2}&1&2&\cdots\end{array}$ (14)

This opposite sequence is introduced to cope with the time-alternating
boundary conditions of the embedded surface code; e.g., at round $1$, the
embedded surface code on the parallelogram has the smooth boundary at the top,
but the rough boundary at the lower bottom of the extended patch. It remains
open if the merger is possible using two-qubit Pauli measurements only without
introducing the opposite measurement sequence. The opposite sequence can be
most easily implemented by letting one of the patches idle for 3 rounds. For
example,

$\displaystyle\begin{array}[]{ccccccccccccccc}\text{upper:}&\cdots&0^{2}&1&2&0^{1}&2&1&0^{2}&1&2&0^{1}&2&1&\cdots\\\
\text{lower:}&\cdots&0^{2}&1&2&&&&0^{1}&2&1&0^{2}&1&2&\cdots\end{array}$ (17)

This is likely the best strategy if the idling qubit noise is low compared to
operation noise. Since there is no type 1 or 2 edge crossing the stitch, the
type of each edge is unambiguously determined by that of the pre-merger
patches.

##  3 Small codes on torus

In general, the honeycomb code can be defined for any trivalent graph $G$ for
which the faces can be $3$-colored.222 Of course, if the graph is not a
honeycomb, then the name “honeycomb code” may be less appropriate.  This is
the same condition needed to construct a color code [13]. Examples include the
$4.8.8$ and $4.6.12$ uniform tilings.

Since $G$ is trivalent, every face of the dual graph $G^{*}$ is a simplex
(i.e., a triangle). The $3$-coloring of the faces of $G$ gives a $3$-coloring
of vertices of $G^{*}$. Excluding some degenerate examples in which $G^{*}$
does not define a simplicial complex, the dual $G^{*}$ is a triangulation of
the torus. The smallest possible triangulation of the torus is given by taking
$G$ to be the Heawood graph which has $14$ vertices. However, famously, the
dual $G^{*}$ is $K_{7}$, the complete graph on $7$ vertices so we cannot
$3$-color its vertices.

A natural family of triangulations to consider is as follows. Consider an
infinite square grid with vertices labeled by elements of $\mathbb{Z}^{2}$.
Divide each square into two triangles by adding an edge from $(i,j)$ to
$(i+1,j+1)$ for all $i,j$. Call the resulting triangulation of the plane
$\Lambda$, and color each vertex by color $i+j\bmod 3$. Let $u=(a,b)$ and
$v=(c,d)$ with $a,b,c,d\in\mathbb{Z}$ and triangulate the torus by $\Lambda$
modulo $u,v$. As a consistency relation on the coloring, we require
$a+b=0\bmod 3$ and $c+d=0\bmod 3$. The number of qubits of the resulting code
is equal to the number of triangles, which is equal to twice the number of
vertices, and hence equals $2|ad-bc|$.

Figure 7:  Dual graph for $18$-qubit code on a torus. Left and right sides of
figure are identified, as are top and bottom sides.

In the particular case where

$\displaystyle\begin{pmatrix}a&b\\\ c&d\end{pmatrix}=\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\ell\quad\text{ where }\ell=0\bmod 3,$ (18)

we have $n=2\ell^{2}$ qubits. For $\ell=3$, this gives an $18$-qubit code. See
Fig. 7.

The notion of “distance” for codes with dynamically generated logical qubits
must be defined carefully. The reason is that a logical error can occur purely
as a result of measurement outcome flips without any single-qubit Pauli errors
occurring. For simplicity, we just consider the simplest definition of
distance: the minimum number of single-qubit Pauli errors which can produce a
logical error without causing a nonvanishing error syndrome. We claim that the
distance of this family of codes is $d=(4/3)\ell$, and so these codes have
$n=(9/8)d^{2}$ qubits (interestingly, the same factor of $9/8$ was found for
color codes in [14], and a $(9/8)d^{2}$ family of honeycomb codes was noted in
[5]). We will examine the decoding graph defined by single-qubit Pauli errors
and count the number of edges in a nontrivial homology cycle. It suffices to
consider one connected component of the decoding graph, because any single-
qubit Pauli error activates at most two edges that belong to the different
connected components.

With the bulk sequence $\ldots,0,1,2,\ldots$ of period $3$ as in [1], the
decoding graph is the $1$-skeleton of the simple cubic lattice, whose (111)
direction of the cubic lattice is our time direction. For a cycle in the $3$d
graph, there is a highest point. Since the $3$d graph does not have any
horizontal (orthogonal to the time axis) edge, the highest point must have two
edges that overlap with some square. We can remove the highest point by
changing the error cycle by the boundary of the square, reducing the elevation
of that point. The same can be done to the lowest point in the cycle. This
modification cannot increase the weight of the chain by construction. Continue
until we arrive at a cycle $C$ along which the time coordinate takes only two
values, because the boundary of a square assumes three values in time
coordinates. Without loss of generality, suppose these two values of time
coordinate correspond to rounds $0,2\bmod 3$. Then, the error chain
corresponds to a path on the edges of the triangulation (e.g., the edges of
Fig. 7) which avoids type 1 vertices. This is equivalent to investigating the
graph on which the embedded toric code after round 1 lives and counting the
number of edges in nontrivial homology cycles. Therefore, the code distance is
$(4/3)\ell$.

With the modified bulk sequence $\ldots,0,1,2,0,2,1,\ldots$ of period $6$, the
decoding graph is no longer the simple cubic lattice, but a similar argument
holds. There are degree-$3$ vertices whose edges are all pointing towards the
future or all towards the past. If an error cycle passes through a degree-$3$
vertex, then we can make the cycle avoid this vertex by changing the error
cycle with a square. Then, the resulting cycle does not visit any edge
connecting a degree-$3$ vertex and a degree-$9$ vertex, and hence lives on a
graph consisting of degree-$6$ vertices only. The argument above applies
again. In conclusion, the code family defined by Eq. 18 has code distance
$(4/3)\ell$ for both of the two versions of the bulk measurement sequence.

Related to our parallelogram is the following code family on torus.

$\displaystyle\begin{pmatrix}a&b\\\ c&d\end{pmatrix}=\begin{pmatrix}1&2\\\
-1&1\end{pmatrix}m\quad\text{ where }m=1,2,3,\ldots.$ (19)

The number of qubits is $n=6m^{2}$. The same reduction of cycles as above
holds for this family, and the code distance (with respect to single-qubit
Pauli errors) is $2m$. This family can realize a finer sequence of code
distances than Eq. 18.

## References

  * Hastings and Haah [2021] M. B. Hastings and J. Haah, “Dynamically generated logical qubits,” arXiv preprint (2021), 2107.02194 .
  * Kitaev [2006] A. Kitaev, “Anyons in an exactly solved model and beyond,” Annals of Physics 321, 2–111 (2006), cond-mat/0506438 .
  * Wootton [2015] J. R. Wootton, “A family of stabilizer codes for $D({\mathbb{Z}}_{2})$ anyons and Majorana modes,” J. Phys. A: Math. Theor. 48, 215302 (2015), arXiv:1501.07779 .
  * Chao  _et al._ [2020] R. Chao, M. E. Beverland, N. Delfosse, and J. Haah, “Optimization of the surface code design for majorana-based qubits,” Quantum 4, 352 (2020), 2007.00307 .
  * [5] C. Gidney, M. Newman, A. Fowler, and M. Broughton, “A fault-tolerant honeycomb memory,” arXiv:2108.10457 .
  * [6] S. B. Bravyi and A. Y. Kitaev, “Quantum codes on a lattice with boundary,” quant-ph/9811052 .
  * [7] M. H. Freedman and D. A. Meyer, “Projective plane and planar quantum codes,” quant-ph/9810055 .
  * Dennis  _et al._ [2002] E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory,” J. Math. Phys. 43, 4452–4505 (2002), quant-ph/0110143 .
  * [9] A. Paetznick et. al., in preparation.
  * [10] C. Vuillot, “Planar floquet codes,” arXiv:2110.05348 .
  * Gottesman [1999] D. Gottesman, “Fault-tolerant quantum computation with higher-dimensional systems,” Chaos Solitons Fractals 10, 1749–1758 (1999), arXiv:quant-ph/9802007 .
  * Li [2015] Y. Li, “A magic state’s fidelity can be superior to the operations that created it,” New J. Phys. 17, 023037 (2015), 1410.7808 .
  * Bombin and Martin-Delgado [2006] H. Bombin and M. A. Martin-Delgado, “Topological quantum distillation,” Physical review letters 97, 180501 (2006), quant-ph/0605138 .
  * Bombín and Martin-Delgado [2007] H. Bombín and M. A. Martin-Delgado, “Optimal resources for topological two-dimensional stabilizer codes: Comparative study,” Physical Review A 76, 012305 (2007), arXiv:quant-ph/0703272 .