# The Gauge Picture of Quantum Dynamics

Kevin Slagle  Department of Electrical and Computer Engineering, Rice
University, Houston, Texas 77005 USA  Department of Physics, California
Institute of Technology, Pasadena, California 91125, USA  Institute for
Quantum Information and Matter and Walter Burke Institute for Theoretical
Physics, California Institute of Technology, Pasadena, California 91125, USA

###### Abstract

Although local Hamiltonians exhibit local time dynamics, this locality is not
explicit in the Schrödinger picture in the sense that the wavefunction
amplitudes do not obey a local equation of motion. In this work, we modify the
Schrödinger picture such that geometric locality is explicit. To do this, we
replace the wavefunction with a collection of _local wavefunctions_
$\ket{\psi_{J}}$, one for each patch of space $J$. The collection of spatial
patches is chosen to cover the space; e.g. we could choose the patches to be
nearest-neighbor sites on a lattice. The Hilbert spaces associated with
neighboring pairs of spatial patches $I$ and $J$ are related to each other by
dynamical unitary transformations $U_{IJ}$. The equations of motion for the
local wavefunctions $\ket{\psi_{J}}$ and connections $U_{IJ}$ are explicitly
local in space. We call this picture of quantum dynamics the _gauge picture_
since it exhibits a local gauge symmetry. The local dynamics of a single
spatial patch is related to the interaction picture, where the interaction
Hamiltonian consists of only nearby Hamiltonian terms. We can also generalize
the explicit locality to include locality in local charge and energy
densities.

Locality is of fundamental importance in theoretical physics. The observable
physics of quantum dynamics is local if the Hamiltonian is geometrically
local111Geometric locality is not to be confused with the weaker notion of
$k$-locality, for which an operator is $k$-local if it acts on at most $k$
qubits _anywhere_ in space., i.e. if the Hamiltonian is a sum of local
operators. An operator (other than the Hamiltonian) is said to be local if it
only acts on a small region of space. But locality is not explicit in the
Schrödinger picture because the wavefunction is global in the sense that it
can not be associated with any local region of space, and the time dynamics of
the wavefunction globally depends on all Hamiltonian terms. On the other hand,
locality is explicit in the Heisenberg picture, for which the time dynamics of
local operators only depends on nearby local operators (when the Hamiltonian
is local). Deutsch and Hayden (2000) This motivates us to ask: Is it possible
to modify the Schrödinger picture to make locality explicit in the equations
of motion?

We find that the answer is yes if we allow for not just one wavefunction, but
a collection of _local wavefunctions_ $\ket{\psi_{I}}$, each associated with a
local patch of space. The Hilbert spaces associated with nearby patches are
related by unitary transformations $U_{IJ}$, which are also dynamical.
Locality is explicit in the sense that the equations of motion [Eq. (19)] for
the dynamical variables ($\ket{\psi_{I}}$ and $U_{IJ}$) only depends on nearby
Hamiltonian terms and nearby dynamical variables.

##  I Schrödinger to Heisenberg Picture

To warm up, we first derive the Heisenberg picture from the Schrödinger
picture. In the Schrödinger picture, the wavefunction evolves according to

$\partial_{t}\Ket{\psi^{\text{S}}(t)}=-iH^{\text{S}}(t)\Ket{\psi^{\text{S}}(t)}$
(1)

where $H^{\text{S}}(t)$ is the Hamiltonian (and we set $\hbar=1$). We use
superscripts “S” and “H” to respectively label time-dependent variables in the
Schrödinger and Heisenberg pictures.

To derive the Heisenberg picture, consider applying a time-dependent unitary
transformation to the wavefunction and operators:

$\displaystyle\Ket{\psi^{\text{H}}(t)}$
$\displaystyle=U^{\dagger}(t)\Ket{\psi^{\text{S}}(t)}$ (2) $\displaystyle
A^{\text{H}}(t)$ $\displaystyle=U^{\dagger}(t)A^{\text{S}}(t)U(t)$

Equation (2) defines the meaning of the “H” superscript for all operators and
wavefunctions. This unitary transformation has the essential property that it
does not affect expectation values:

$\Braket{\psi^{\text{H}}(t)}{A^{\text{H}}(t)}{\psi^{\text{H}}(t)}=\Braket{\psi^{\text{S}}(t)}{A^{\text{S}}(t)}{\psi^{\text{S}}(t)}$
(3)

Let $G^{\text{S}}(t)$ be the Hermitian operator such that

$\partial_{t}U(t)=-iG^{\text{S}}(t)U(t)$ (4)

Or equivalently, $\partial_{t}U(t)=-iU(t)G^{\text{H}}(t)$ where
$G^{\text{H}}(t)=U^{\dagger}(t)G^{\text{S}}(t)U(t)$ is defined by Eq. (2). The
time derivatives of the new wavefunction and operators are

$\begin{array}[]{ccccl}\partial_{t}\Ket{\psi^{\text{H}}(t)}&=&-iH^{\text{H}}(t)\Ket{\psi^{\text{H}}(t)}&+&iG^{\text{H}}(t)\Ket{\psi^{\text{H}}(t)}\vspace{.15cm}\\\
\partial_{t}A^{\text{H}}(t)&=&(\partial_{t}A^{\text{S}})^{\text{H}}(t)&+&i[G^{\text{H}}(t),A^{\text{H}}(t)]\end{array}$
(5)

where
$(\partial_{t}A^{\text{S}})^{\text{H}}(t)=U^{\dagger}(t)\partial_{t}A^{\text{S}}(t)U(t)$
[Eq. (2)].

To obtain the Heisenberg picture, we simply choose

$G^{\text{H}}(t)=H^{\text{H}}(t)$ (6)

with the initial condition $U(0)=\mathbbl{1}$ at $t=0$, where $\mathbbl{1}$
denotes the identity operator. This choice makes the wavefunction
$\ket{\psi^{\text{H}}}$ constant in time, while operators evolve in time. If
the Hamiltonian is time-independent, then the Hamiltonian is the same in the
Schrödinger and Heisenberg pictures; i.e. $H^{\text{S}}(t)=H^{\text{H}}(t)$ if
$\partial_{t}H^{\text{S}}=0$.

###  I.1 Locality

A nice feature of the Heisenberg picture is that if the Hamiltonian is local,
then the time evolution of observables is _explicitly_ local. A local
Hamiltonian has the form

$H=\sum_{J}H_{J}$ (7)

where each $H_{J}$ only acts on a finite patch of space $J$. We use capital
letters, $I$, $J$, and $K$, to denote patches of space.

Now consider the time evolution of a local operator in the Heisenberg picture:

$\partial_{t}A_{I}^{\text{H}}(t)=i[H^{\text{H}}(t),A_{I}^{\text{H}}(t)]+(\partial_{t}A_{I}^{\text{S}})^{\text{H}}(t)$
(8)

Throughout this work, $A_{I}$ denotes a local operator that acts within the
spatial patch $I$ (when viewed in the Schrödinger picture), and similar for
$B_{J}$, etc. Due to locality, most Hamiltonian terms cancel out in the first
term. The result is an explicitly local Heisenberg equation of motion:

$\partial_{t}A_{I}^{\text{H}}(t)=i[H_{\langle
I\rangle}^{\text{H}}(t),A_{I}^{\text{H}}(t)]+(\partial_{t}A_{I}^{\text{S}})^{\text{H}}(t)$
(9)

where
$(\partial_{t}A_{I}^{\text{S}})^{\text{H}}(t)=U^{\dagger}(t)\,(\partial_{t}A_{I}^{\text{S}})^{\text{S}}(t)\,U(t)$
[Eq. (2)]. $H_{\langle I\rangle}$ is a sum over nearby Hamiltonian terms:

$H_{\langle I\rangle}=\sum_{J}^{J\cap I\neq\emptyset}H_{J}$ (10)

$\sum_{J}^{J\cap I\neq\emptyset}$ denotes a sum over patches $J$ that have
overlap with patch $I$. Note that the local Hamiltonian terms
$H_{I}^{\text{H}}(t)$ also evolve according to Eq. (9).

Locality is explicit in this local Heisenberg picture because the time
evolution of each local operator $A_{I}^{\text{H}}(t)$ only depends on nearby
time-dependent operators $H_{J}^{\text{H}}(t)$. Locality is not explicit in
Schrödinger’s picture since the time evolution of the wavefunction globally
depends on all Hamiltonian terms, and there is no sense in which the
wavefunction (or parts of it) can be associated with a point in space.

##  II Gauge Picture

We now want to obtain a local picture of quantum dynamics that features time-
dependent wavefunctions and time-independent operators. To do this, we first
choose a set of local patches of space that cover the entire space (or
lattice); see Fig. 1 for an example. Then for each patch of space, we apply a
time-dependent unitary transformation $U_{I}(t)$ to the Heisenberg picture
wavefunction and local operators:

$\displaystyle\big{|}\psi_{I}\big{\rangle}$
$\displaystyle=U_{I}\big{|}\psi^{\text{H}}\big{\rangle}$ (11) $\displaystyle
A_{I}^{\text{G}}$ $\displaystyle=U_{I}A_{I}^{\text{H}}U_{I}^{\dagger}$

We thus obtain a separate _local wavefunction_ $\ket{\psi_{I}}$ for each patch
of space. The second line transforms local operators (including $H_{I}$ and
$H_{\langle I\rangle}$) from the Heisenberg picture into a new picture of
quantum dynamics. The “G” superscript labels time-dependent operators that
evolve within this picture. We omit the “G” superscript for $\ket{\psi_{I}}$
since the $\ket{\psi_{I}}$ notation is not used in other pictures; thus, there
should be no confusion. To further reduce clutter, we also suppress the
“$(t)$” notation for time-dependent variables.

Figure 1: An example of a chain of qubits (black dots) and spatial patches
(colored ovals) consisting of pairs of neighboring qubits. A local
wavefunction $\ket{\psi_{I}}$ is associated with each patch $I$, and the
Hilbert spaces associated with neighboring patches are related by unitary
connections $U_{IJ}$.

After applying this unitary transformation, correlation functions within a
single patch become
$\braket{\psi^{\text{H}}}{A_{I}^{\text{H}}}{\psi^{\text{H}}}=\braket{\psi_{I}}{A_{I}^{\text{G}}}{\psi_{I}}$.
Correlation functions of operator products acting on multiple patches becomes

$\displaystyle\Braket{\psi^{\text{H}}}{A_{I}^{\text{H}}\quad\;\;\,B_{J}^{\text{H}}\quad\;\;\;C_{K}^{\text{H}}}{\psi^{\text{H}}}$
(12) $\displaystyle=$
$\displaystyle\Braket{\psi_{I}\,}{A_{I}^{\text{G}}U_{IJ}B_{J}^{\text{G}}U_{JK}C_{K}^{\text{G}}}{\psi_{K}}$

where we define

$U_{IJ}=U_{I}U_{J}^{\dagger}$ (13)

A _connection_ $U_{IJ}$ always appears between variables associated with
different spatial patches. Equation (11) implies that the following identities
hold for all $t$:

$\displaystyle U_{IJ}\ket{\psi_{J}}$ $\displaystyle=\ket{\psi_{I}}$ (14)
$\displaystyle U_{IJ}U_{JK}$ $\displaystyle=U_{IK}$

and $U_{IJ}^{\dagger}=U_{JI}$ and $U_{II}=\mathbbl{1}$. Therefore, $U_{IJ}$
serves the role of a connection that transports the local wavefunctions
between different spatial patches. But the connections are trivial in the
sense that there is no curvature or nontrivial holonomy (e.g.
$U_{IJ}U_{JK}U_{KI}=\mathbbl{1}$).

Let $G_{I}(t)$ be the Hermitian operator such that

$\partial_{t}U_{I}=-iG_{I}U_{I}$ (15)

Then the time derivatives of the local wavefunctions and operators are

$\displaystyle\partial_{t}\ket{\psi_{I}}$
$\displaystyle=-iG_{I}\ket{\psi_{I}}$ (16)
$\displaystyle\partial_{t}A_{I}^{\text{G}}$
$\displaystyle=i[H^{\text{G}}_{\langle
I\rangle}-G_{I},A_{I}^{\text{G}}]+(\partial_{t}A_{I}^{\text{S}})^{\text{G}}$

This follows from $\partial_{t}\ket{\psi^{\text{H}}}=0$ and Eqs. (9) and (11).
$(\partial_{t}A_{I}^{\text{S}})^{\text{G}}=U_{I}(\partial_{t}A)^{\text{H}}U_{I}^{\dagger}$
is defined from Eq. (11).

A nice choice for $U_{I}$ at $t=0$ and $G_{I}$ is

$\displaystyle G_{I}$ $\displaystyle=H^{\text{G}}_{\langle I\rangle}$ (17)
$\displaystyle U_{I}(0)$ $\displaystyle=\mathbbl{1}$

since this makes local operators $A_{I}^{\text{G}}$ (and Hamiltonian terms)
equal to Schrödinger picture operators:

$\displaystyle A_{I}^{\text{G}}$ $\displaystyle=A_{I}^{\text{S}}$
$\displaystyle H_{I}^{\text{G}}$ $\displaystyle=H_{I}^{\text{S}}$ (18)

The local wavefunctions and operators evolve as follows:

$\boxed{\begin{aligned} \partial_{t}\ket{\psi_{I}}&=-iH^{\text{G}}_{\langle
I\rangle}\ket{\psi_{I}}\\\ \partial_{t}U_{IJ}&=-iH^{\text{G}}_{\langle
I\rangle}U_{IJ}+iU_{IJ}H^{\text{G}}_{\langle J\rangle}\end{aligned}}$ (19)

$H^{\text{G}}_{\langle I\rangle}$ is defined from Eqs. (10) and (11) and can
be expressed as

$\boxed{H^{\text{G}}_{\langle I\rangle}=\sum_{J}^{J\cap
I\neq\emptyset}U_{IJ}H_{J}^{\text{G}}U_{JI}}$ (20)

Although the Schrödinger and Heisenberg pictures are linear differential
equations, this picture involves _non-linear_ differential equations for
$U_{IJ}$. The $t=0$ initial conditions are

$\displaystyle\big{|}\psi_{I}(0)\big{\rangle}$
$\displaystyle=\big{|}\psi^{\text{S}}(0)\big{\rangle}$ (21) $\displaystyle
U_{IJ}(0)$ $\displaystyle=\mathbbl{1}$

Note that one does not need to keep track of a $U_{IJ}$ for every pair of $I$
and $J$; only considering overlapping pairs of spatial patches suffices. If we
want to describe the dynamics of a mixed state, then we could take
$\ket{\psi^{\text{S}}(0)}$ to be a purification of the mixed state.

Thus, we were able to make Schrödinger’s picture explicitly local by
introducing a local wavefunction $\ket{\psi_{I}}$ for each patch of space and
unitary connections $U_{IJ}$ that map between the Hilbert spaces of the
different patches. Locality is explicit in the sense that the time evolution
of $\ket{\psi_{I}}$ and $U_{IJ}$ only depends on nearby variables
($\ket{\psi_{I}}$, $U_{IJ}$, and $H_{J}$).

Note that in this picture of quantum dynamics, correlation functions [e.g. Eq.
(12)] should be invariant under the following local gauge symmetry:

$\displaystyle\ket{\psi_{I}}$ $\displaystyle\to\Lambda_{I}\ket{\psi_{I}}$
$\displaystyle U_{I}$ $\displaystyle\to\Lambda_{I}U_{I}$ $\displaystyle
A_{I}^{\text{G}}$
$\displaystyle\to\Lambda_{I}A_{I}^{\text{G}}\Lambda_{I}^{\dagger}$
$\displaystyle U_{IJ}$
$\displaystyle\to\Lambda_{I}U_{IJ}\Lambda_{J}^{\dagger}$ (22)

where $\Lambda_{I}$ are arbitrary unitary transformations. For example,
$\braket{\psi_{I}}{B_{J}^{\text{G}}}{\psi_{I}}$ and
$\braket{\psi_{I}}{A_{I}^{\text{G}}B_{J}^{\text{G}}}{\psi_{J}}$ are not
invariant under this transformation and are therefore not equal to
$\braket{\psi^{\text{H}}}{B_{J}^{\text{H}}}{\psi^{\text{H}}}$ or
$\braket{\psi^{\text{H}}}{A_{I}^{\text{H}}B_{J}^{\text{H}}}{\psi_{J}^{\text{H}}}$
in the Heisenberg (or Schrödinger) picture. Therefore, the local wavefunctions
$\ket{\psi_{I}}$ are local in the sense that they can only be used to
calculate local expectation values of local operators $A_{I}$ acting within
the associated spatial patch, unless we make use of connections $U_{IJ}$ to
connect different patches. Due to the appearance of this gauge symmetry, we
call this picture of quantum dynamics the _gauge picture_.

As an alternative to Eq. (20), local dynamics can also be calculated by
integrating

$\partial_{t}U_{I}=-iH_{\langle I\rangle}^{\text{G}}U_{I}$ (23)

[from Eq. (15)] with initial condition $U_{I}(0)=\mathbbl{1}$. Equation (23)
can be rewritten more explicitly as

$\partial_{t}U_{I}=-i\sum_{J}^{J\cap
I\neq\emptyset}U_{I}(U_{J}^{\dagger}H_{J}^{\text{G}}U_{J})$ (24)

Since wavefunctions are constant in the Heisenberg picture, Eq. (11) implies
that the time-evolved local wavefunction is

$\big{|}\psi_{I}(t)\big{\rangle}=U_{I}(t)\big{|}\psi^{\text{S}}(0)\big{\rangle}$
(25)

###  II.1 Solving the Equations of Motion

Let us derive an expression for $U_{I}$ in terms of just the Hamiltonian.
Simplifying Eq. (24) using
$H_{J}^{\text{G}}=U_{J}H_{J}^{\text{H}}U_{J}^{\dagger}$ [Eq. (11)] and
$H_{\langle I\rangle}=\sum_{J}^{J\cap I\neq\emptyset}H_{J}$ [Eq. (10)] yields

$\displaystyle\partial_{t}U_{I}$ $\displaystyle=-iU_{I}H_{\langle
I\rangle}^{\text{H}}$ $\displaystyle=-i\widetilde{U}_{I}^{\dagger}H_{\langle
I\rangle}^{\text{S}}U$ (26)

The second line follows from $H_{\langle
I\rangle}^{\text{H}}=U^{\dagger}H_{\langle I\rangle}^{\text{S}}U$ in Eq. (2)
after we define $\widetilde{U}_{I}=UU_{I}^{\dagger}$. The derivative of
$\widetilde{U}_{I}=UU_{I}^{\dagger}$ is

$\partial_{t}\widetilde{U}_{I}=-i\left(H^{\text{S}}-H_{\langle
I\rangle}^{\text{S}}\right)\widetilde{U}_{I}$ (27)

where $H^{\text{S}}-H_{\langle I\rangle}^{\text{S}}$ is the sum of Hamiltonian
terms that do not overlap with patch $I$. Equation (27) follows from Eq. (26)
and

$\partial_{t}U(t)=-iH^{\text{S}}(t)U(t)$ (28)

[from Eqs. (4) and (6)].

We now have linear differential equations for $U(t)$ and
$\widetilde{U}_{I}(t)$ [Eqs. (28) and (27)]. For time-independent
Hamiltonians, we can solve these differential equations:

$\displaystyle U(t)$ $\displaystyle=e^{-iHt}$ (29)
$\displaystyle\widetilde{U}_{I}(t)$ $\displaystyle=e^{-i(H-H_{\langle
I\rangle}^{\text{S}})t}$

Once we have calculated $U$ and $\widetilde{U}_{I}$, we can obtain $U_{I}$
from:

$U_{I}=\widetilde{U}_{I}^{\dagger}U$ (30)

which follows from the definition of $\widetilde{U}_{I}=UU_{I}^{\dagger}$.
Equation (30) is useful when we want to calculate $U_{I}$, $\ket{\psi_{I}}$
[from Eq. (11)], or $U_{IJ}$ [from Eq. (13)] without caring about explicitly
local equations of motion.

$U(t)$ describes the time evolution of the Schrödinger picture wavefunction:
$\ket{\psi^{\text{S}}(t)}=U(t)\ket{\psi^{\text{S}}(0)}$. Therefore, Eq. (30)
tells us that $\widetilde{U}_{I}$ describes how much the gauge picture has
deviated from the Schrödinger picture. Indeed, the wavefunctions of the two
pictures are related by $\widetilde{U}_{I}$:

$\ket{\psi_{I}}=\widetilde{U}_{I}^{\dagger}\ket{\psi^{\text{S}}}$ (31)

This identity follows from comparing Eq. (25) to
$\ket{\psi^{\text{S}}(t)}=U(t)\ket{\psi^{\text{S}}(0)}$. The role of
$\widetilde{U}_{I}$ is to cancel out the effects of distant Hamiltonian terms
so that $U_{I}$ evolves locally. If we apply the gauge transformation (22)
with $\Lambda_{I}=\widetilde{U}_{I}$, then all gauge picture variables
transform back into the Schrödinger picture:
$\ket{\psi_{I}}\to\ket{\psi^{\text{S}}}$, $U_{I}\to U$, and
$U_{IJ}\to\mathbbl{1}$. Local operators $A_{I}^{\text{G}}=A_{I}^{\text{S}}$
are already equal in the two pictures, and they are invariant since they
commute with $\widetilde{U}_{I}$.

###  II.2 Generalized Hamiltonians

It is also possible to handle Hamiltonian terms that are not supported on a
single spatial patch. This is useful when we want the spatial patches to
consist of just single lattice site.

Consider a Hamiltonian

$H=\sum_{J\cdots K}\sum_{\mu\cdots\nu}h_{J\cdots
K}^{\mu\cdots\nu}\;\tau^{\mu}_{J}\cdots\tau^{\nu}_{K}$ (32)

that consists of a sum $\sum_{J\cdots K}$ over multiple spatial patches
$J\cdots K$. The $h_{J\cdots K}^{\mu\cdots\nu}$ are (generically time-
dependent) real coefficients, and $\tau^{\mu}_{J}$ denotes an operator (in the
Schrödinger or gauge picture) indexed by $\mu$ with support on patch $J$. For
example, we could take the patches to consist of a single qubit, and the
$\tau^{\mu}_{J}$ could be Pauli operators. For local Hamiltonians, $h_{J\cdots
K}^{\mu\cdots\nu}$ will only be nonzero if the patches $J\cdots K$ are close
together.

Equation (20) then generalizes to

$\displaystyle H_{\langle I\rangle}^{\text{G}}=\sum_{J\cdots K}^{(J\cup\cdots
K)\cap I\neq\emptyset}\sum_{\mu\cdots\nu}\,$ $\displaystyle h_{J\cdots
K}^{\mu\cdots\nu}$ (33)
$\displaystyle(U_{IJ}\tau^{\mu}_{J}U_{JI})\cdots(U_{IK}\tau^{\nu}_{K}U_{KI})$

where $\sum_{J\cdots K}^{(J\cup\cdots K)\cap I\neq\emptyset}$ sums over
spatial patches $J\cdots K$ such that the union $J\cup\cdots K$ has nontrivial
overlap with patch $I$. The gauge picture equations of motion [Eq. (19) or
(23)] can then be applied using Eq. (33) [instead of Eq. (20)]. This can be
shown by rederiving the local Heisenberg and gauge pictures for the
Hamiltonian in Eq. (32) [instead of Eq. (7)].

##  III Quantum Circuits

###  III.1 Example

(a) Schrödinger: $\ket{\psi^{\text{S}}(t_{\text{f}})}=U\ket{\psi_{0}}$ (b)
Heisenberg: $A^{\text{H}}(t_{\text{f}})=UA(0)\,U^{\dagger}$

(c) local Heisenberg: $A_{I}^{\text{H}}(t_{\text{f}})=UA_{I}(0)U^{\dagger}$

Figure 2: Dynamical variables used in the standard pictures of quantum
dynamics after the time evolution of a quantum circuit. (a) The Schrödinger
picture wavefunction $\ket{\psi^{\text{S}}(t_{\text{f}})}=U\ket{\psi_{0}}$
after the quantum circuit $U=U(t_{\text{f}})$ is applied to the initial
wavefunction $\ket{\psi_{0}}$. Each trapezoid depicts a unitary operator
acting on two qubits from time $t=\tau-1$ to $\tau$, where $\tau$ is the
number inside the trapezoid. The circuit $U$ acts until $t_{\text{f}}=5$, and
consists of the composition of these unitaries. Qubits are numbered on the
right. (b) A time-evolved operator
$A^{\text{H}}(t_{\text{f}})=UA(0)\,U^{\dagger}$ in the Heisenberg picture.
Trapezoids reflected horizontally denote the inverses of the associated
2-qubit unitaries. (c) For a local operator $A_{I}^{\text{H}}(t)$, many of the
unitaries cancel with their inverse. Here, $A_{I}(0)$ acts on a spatial patch
$I=(7,8)$ consisting of two qubits.

=

(a) $U_{I}(t_{\text{f}})$

(b) $U_{J}^{\dagger}(t_{\text{f}})$

=

(c) $U_{IJ}(t_{\text{f}})$

Figure 3: Dynamical variables in the gauge picture after the same quantum
circuit as in Fig. 2. (a) The unitary $U_{I}(t_{\text{f}})$ with $I=(7,8)$. We
also pictorially show that $\widetilde{U}_{I}^{\dagger}U=U_{I}$ [Eq. (30)]
after many unitaries outside the red light cone cancel with their inverse.
$\widetilde{U}_{I}$ is the same as $U$, but with all unitaries (or Hamiltonian
terms) that overlap with the spatial patch $I$ removed. The local wavefunction
follows from $\ket{\psi_{I}}=U_{I}\ket{\psi_{0}}$. (b) The unitary
$U_{J}^{\dagger}(t_{\text{f}})$ with $J=(8,9)$. (c) The connection
$U_{IJ}(t_{\text{f}})=U_{I}(t_{\text{f}})U_{J}^{\dagger}(t_{\text{f}})$, which
is also equal to $\widetilde{U}_{I}^{\dagger}\widetilde{U}_{J}$.

In this section, we study how unitary operators act in the gauge picture. As
an instructive example, in Figs. 2 and 3 we depict dynamical variables used in
different pictures of quantum dynamics after the time evolution of a quantum
circuit. The quantum circuit consists of a composition of unitary operators
that each act on pairs of neighboring qubits. Equivalently, the circuit could
result from a time-dependent Hamiltonian evolution.

Similar to operators in the local Heisenberg picture (Fig. 2c), the unitaries
$U_{I}$ and $U_{IJ}$ (Fig. 3a and 3c) in the gauge picture also grow as time
evolves. Figs. 2c and 3 demonstrate the explicit locality of the local
variables in the Heisenberg and gauge pictures: over a short time evolution,
these local variables are not affected by anything far away. Fig. 3a
demonstrates that this locality is achieved via the composition
$\widetilde{U}_{I}^{\dagger}U$, where $\widetilde{U}_{I}^{\dagger}$ cancels
out distant unitary evolution.

###  III.2 Unitary Operators

Consider the action of many local unitary operators $u_{I}$ on the Schrödinger
picture wavefunction:

$\ket{\psi^{\text{S}}}\to\prod_{I}u_{I}\ket{\psi^{\text{S}}}$ (34)

Assume that each unitary $u_{I}$ acts within patch $I$ and that all unitaries
commute (e.g. as they do for each layer of unitaries in Fig. 2):

$[u_{I},u_{J}]=0$ (35)

The local gauge picture variables transform as follows:

$\displaystyle\ket{\psi_{I}}$ $\displaystyle\to u_{\langle
I\rangle}\ket{\psi_{I}}$ (36) $\displaystyle U_{IJ}$ $\displaystyle\to
u_{\langle I\rangle}U_{IJ}u_{\langle J\rangle}^{\dagger}$ $\displaystyle
U_{I}$ $\displaystyle\to u_{\langle I\rangle}U_{I}$

In analogy to $H_{\langle I\rangle}^{\text{G}}$, we define

$u_{\langle I\rangle}=\prod_{J}^{J\cap I\neq\emptyset}U_{IJ}u_{J}U_{JI}$ (37)

where $\prod_{J}^{J\cap I\neq\emptyset}$ denotes a product over all patches
that overlap with patch $I$. Note that
$U_{IJ}=U_{I}U_{J}^{\dagger}=\widetilde{U}_{I}^{\dagger}\widetilde{U}_{J}$
since $U_{I}=\widetilde{U}_{I}^{\dagger}U$ [Eq. (30)], which implies that
$U_{IJ}u_{J}U_{JI}=\widetilde{U}_{I}^{\dagger}u_{J}\widetilde{U}_{I}$ since
$u_{J}$ and $\widetilde{U}$ commute. Therefore, all of the $U_{IJ}u_{J}U_{JI}$
terms in Eq. (37) commute, and $u_{\langle I\rangle}$ can also be expressed as

$u_{\langle I\rangle}=\widetilde{U}_{I}^{\dagger}\left(\prod_{J}^{J\cap
I\neq\emptyset}u_{J}\right)\widetilde{U}_{I}$ (38)

To derive Eq. (36), imagine that the local unitaries $u_{I}=e^{-iH_{I}t_{0}}$
are applied using a time-independent Hamiltonian $H=\sum_{I}H_{I}$ with
commuting local terms (i.e. $[H_{I},H_{J}]=0$). Then we can use Eqs. (28) and
(27) to calculate how $U$ and $\widetilde{U}_{I}$ transform:

$\displaystyle U$ $\displaystyle\to\big{(}\textstyle\prod_{J}u_{J}\big{)}\,U$
(39) $\displaystyle\widetilde{U}_{I}$ $\displaystyle\to\left(\prod_{J}^{J\cap
I=\emptyset}u_{J}\right)\widetilde{U}_{I}$

where $\prod_{J}^{J\cap I=\emptyset}$ denotes a product over all patches that
do not overlap with patch $I$. This allows us to calculate the transformation

$U_{I}\to\widetilde{U}_{I}^{\dagger}\left(\prod_{J}^{J\cap
I\neq\emptyset}u_{J}\right)U=u_{\langle I\rangle}U_{I}$ (40)

from $U_{I}=\widetilde{U}_{I}^{\dagger}U$ [Eq. (30)]. The other
transformations in Eq. (36) are extracted from
$\ket{\psi_{I}(t)}=U_{I}(t)\ket{\psi^{\text{S}}(0)}$ [Eq. (25)] and
$U_{IJ}=U_{I}U_{J}^{\dagger}$ [Eq. (13)].

##  IV Measurements

In the Schrödinger picture, a measurement results in a nonlocal collapse of
the wavefunction. Most generally, a measurement process can be specified in
terms of a collection of Kraus operators $E^{(k)}$, whose squares sum to the
identity:

$\sum_{k}E^{(k)\dagger}E^{(k)}=\mathbbl{1}$ (41)

For example, we could take the Kraus operators to be projection operators
$E^{(k)}=\ket{k}\bra{k}$ that project onto the different eigenstates of a
Hermitian operator $\sum_{k}\lambda_{k}\ket{k}\bra{k}$. After the measurement,
the wavefunction changes to

$\Ket{\psi^{\text{S}}}\to P_{k}^{-1/2}E^{(k)}\Ket{\psi^{\text{S}}}$ (42)

with probability

$P_{k}=\braket{\psi^{\text{S}}}{E^{(k)\dagger}E^{(k)}}{\psi^{\text{S}}}$ (43)

Note that if all measurement outcomes are averaged over, the expectation value
of an operator after a measurement is not affected due to Eq. (41).

In the gauge picture, we can consider the result of a local measurement
specified by a collection of local Kraus operators $E_{I_{0}}^{(k)}$ that act
on a spatial patch $I_{0}$. Similar to above, the squared Kraus operators must
sum to the identity:

$\sum_{k}E_{I_{0}}^{(k)\dagger}E_{I_{0}}^{(k)}=\mathbbl{1}$ (44)

After the measurement, the local wavefunction at patch $I_{0}$ changes to

$\big{|}\psi_{I_{0}}\big{\rangle}\to\big{|}\psi^{(k)}_{I_{0}}\big{\rangle}=P_{k}^{-1/2}E_{I_{0}}^{(k)}\big{|}\psi_{I_{0}}\big{\rangle}$
(45)

with probability

$P_{k}=\big{\langle}\psi_{I_{0}}\big{|}E_{I_{0}}^{(k)\dagger}E_{I_{0}}^{(k)}\big{|}\psi_{I_{0}}\big{\rangle}$
(46)

The connections $U_{IJ}$ are not affected. In order to maintain consistency
with the other spatial patches, all other local wavefunctions must transform
as well:

$\big{|}\psi_{J}\big{\rangle}\to
U_{JI_{0}}\big{|}\psi^{(k)}_{I_{0}}\big{\rangle}$ (47)

Equation (31) can be used to show that these equations are consistent with the
Schrödinger picture. Equation (47) demonstrates that the wavefunction collapse
is nonlocal in the gauge picture.

##  V Local Interaction Picture

In this section, we equate variables in the gauge picture to variables in the
interaction picture with interacting Hamiltonian $H_{\langle J\rangle}$.

In the interaction picture, the Hamiltonian

$H=H_{0}+H_{1}$ (48)

is divided into interacting $H_{1}$ and non-interacting $H_{0}$ parts. The
interaction picture wavefunction and operators are defined as

$\displaystyle\Ket{\psi^{\text{I}}}$
$\displaystyle=U_{0}^{\dagger}\Ket{\psi^{\text{S}}}$ (49) $\displaystyle
A^{\text{I}}$ $\displaystyle=U_{0}^{\dagger}A^{\text{S}}U_{0}$

where $U_{0}(t)$ is the solution to

$\partial_{t}U_{0}=-iH_{0}^{\text{S}}U_{0}$ (50)

with $t=0$ initial condition $U_{0}(0)=\mathbbl{1}$. The utility of the
interaction picture is that wavefunctions evolve via the interacting
Hamiltonian while operators evolve via the non-interacting Hamiltonian:

$\displaystyle\partial_{t}\Ket{\psi^{\text{I}}}$
$\displaystyle=-iH_{1}^{\text{I}}\Ket{\psi^{\text{I}}}$ (51)
$\displaystyle\partial_{t}A^{\text{I}}$
$\displaystyle=+i[H_{0}^{\text{I}},A^{\text{I}}]+(\partial_{t}A^{\text{S}})^{\text{I}}$

where $H_{0}^{\text{I}}=U_{0}^{\dagger}H_{0}^{\text{S}}U_{0}$ and
$H_{1}^{\text{I}}=U_{1}^{\dagger}H_{1}^{\text{S}}U_{1}$ also obey the second
line, and
$(\partial_{t}A^{\text{S}})^{\text{I}}=U_{0}^{\dagger}\partial_{t}A^{\text{S}}U_{0}$
[as defined by Eq. (49)]. If $H_{0}^{\text{S}}$ is time-independent, then
$H_{0}^{\text{I}}=H_{0}^{\text{S}}$.

Consider a fixed spatial patch $J$, and let the interaction Hamiltonian be

$H_{1}=H_{\langle J\rangle}$ (52)

Then $\widetilde{U}_{J}=U_{0}$ are equal since they obey the same equations of
motion, $\partial_{t}\widetilde{U}_{J}=-i(H^{\text{S}}-H_{\langle
J\rangle}^{\text{S}})\widetilde{U}_{J}=-iH_{0}^{\text{S}}\widetilde{U}_{J}$
[by Eqs. (27), (48), and (52)] and $\partial_{t}U_{0}=-iH_{0}^{\text{S}}U_{0}$
[Eq. (50)], with the same initial conditions
$\widetilde{U}_{J}(0)=U_{0}(0)=\mathbbl{1}$. Inserting
$\widetilde{U}_{J}=U_{0}$ into $U_{J}=\widetilde{U}_{J}^{\dagger}U$ [Eq. (30)]
yields

$U_{J}=U_{0}^{\dagger}U$ (53)

We thus arrive at an equivalence between the gauge picture on patch $J$ and
this interaction picture:

$\displaystyle\big{|}\psi_{J}\big{\rangle}$
$\displaystyle=\big{|}\psi^{\text{I}}\big{\rangle}$ (54) $\displaystyle
A_{J}^{\text{G}}$ $\displaystyle=A_{J}^{\text{S}}=A_{J}^{\text{I}}$

The first line follows from
$\ket{\psi_{J}}=U_{J}\ket{\psi^{\text{H}}}=U_{J}U^{\dagger}\ket{\psi^{\text{S}}}=U_{0}^{\dagger}\ket{\psi^{\text{S}}}=\ket{\psi^{\text{I}}}$
[using Eqs. (11), (2), (53), and (49)]. The equalities in the second line
respectively follow from Eq. (18) and
$A_{J}^{\text{I}}=U_{0}^{\dagger}A_{J}^{\text{S}}U_{0}$ [Eq. (49)] since
$U_{0}$ and $A_{J}^{\text{S}}$ commute when $H_{1}=H_{\langle J\rangle}$.

The above relation makes sense: $\ket{\psi_{J}}$ and $\ket{\psi^{\text{I}}}$
respectively evolve via $H_{\langle J\rangle}^{\text{G}}$ and
$H_{1}^{\text{I}}$; therefore $\ket{\psi_{J}}=\ket{\psi^{\text{I}}}$ since
$H_{\langle J\rangle}=H_{1}$. Operators in the interaction picture evolve via
the non-interacting Hamiltonian $H_{0}^{\text{I}}$, which commutes with
operators supported on $J$; therefore operators $A_{J}$ supported on $J$ are
equal in the Schrödinger and interaction pictures.

Unlike the gauge picture, the above interaction picture is not explicitly
local since $H_{1}^{\text{1}}$ evolves via a global Hamiltonian
$H_{0}^{\text{I}}$. However, the gauge picture could be viewed as many
simultaneous interactions pictures, one for each spatial patch $J$ and
interaction Hamiltonian $H_{1}=H_{\langle J\rangle}$, but with modified
equations of motion to obtain explicitly local dynamics.

##  VI Generalized Locality

In this section, we generalize the notion of locality in the gauge picture.
The underlying role of a spatial patch is to define a subspace of operators
that only act within the spatial patch. Thus, we can generalize the notion of
spatial patches to subspaces of operators. That is, instead of letting capital
$I$, $J$, and $K$ denote spatial patches, we could instead let them denote
operator subspaces. The above equations that applied for local operators
$A_{I}$ are generalized to apply to any operator in the subspace $I$. We can
therefore generalize the notion of geometric locality to locality between
different subspaces of operators, which might e.g. correspond to different
energy or charge densities.

We say that two operator subspaces ($I$ and $J$) do not commute if there
exists a pair of operators ($A$ and $B$), with one from each subspace ($A\in
I$ and $B\in J$), such that the pair does not commute ($[A,B]\neq 0$). The
notion of overlapping spatial patches generalizes to non-commuting operator
subspaces. Therefore in Eq. (20), the sum $\sum_{J}^{J\cap I\neq\emptyset}$
generalizes to a sum over all operator subspaces $J$ that do not commute with
subspace $I$. Similarly, the sum $\sum_{J\cdots K}^{(J\cup\cdots K)\cap
I\neq\emptyset}$ in Eq. (33) generalizes to a sum over all subspaces $J\cdots
K$ where any one of these subspaces does not commute with subspace $I$.

Despite the abstractness of this generalization, it does have physical
applications. For example, bosonic Hamiltonians (e.g. the Bose-Hubbard model)
consisting of bosonic creation and annihilation operators typically exhibit a
sense of locality in boson number. That is, for many bosonic Hamiltonians, the
action of a Hamiltonian term typically only changes the boson numbers at each
site by a small amount. Therefore, we could consider choosing the operator
subspaces $I_{i,n}$ to consist of the subspace of operators that act at site
$i$ and only on states with boson number $n$ or $n+1$ at site $i$. To be
concrete, let $\ket{n^{\prime}}_{i}\bra{n}_{i}$ denote a boson operator that
projects a wavefunction onto the subspace with $n$ bosons at site $i$ and then
changes the boson number at site $i$ to $n^{\prime}$. With this notation, the
boson number operator at site $i$ could be expressed as
$\hat{n}_{i}=\sum_{n=0}^{\infty}n\ket{n}_{i}\bra{n}_{i}$ and the annihilation
operator as $\hat{b}_{i}=\sum_{n=1}^{\infty}n\ket{n-1}_{i}\bra{n}.$ Then in
this example, the subspace $I_{i,n}$ consists of the operators spanned by
$\ket{n}_{i}\bra{n}_{i}$, $\ket{n}_{i}\bra{n+1}_{i}$,
$\ket{n+1}_{i}\bra{n}_{i}$, and $\ket{n+1}_{i}\bra{n+1}_{i}$. In the resulting
gauge picture, the operator subspaces $I_{i,n}$ and local wavefunctions
$\ket{\psi_{I_{i,n}}}$ are indexed by position $i$ and local boson number $n$,
which results in equations of motion that are explicitly local in both space
and local boson number.

We could similarly use the total energy or local energy density to specify the
operator subspaces. More generally, we can obtain an extra dimension of
locality whenever a model has a conserved (or approximately conserved) charge
or energy. For example, consider a collection of spatial patches and a local
Hamiltonian $H=\sum_{I}H_{I}$ where each Hamiltonian term $H_{I}$ has many
(possibly infinitely many) energy levels. Then we could define operator
subspaces $\widetilde{I}_{I,E}$ that consist of the subspace of operators that
act within the spatial patch $I$ and only on eigenstates of $H_{I}$ with
energy between $E\pm\Delta E$ (with $\Delta E$ chosen to be sufficiently large
that we can capture the necessary operators). In this example, the gauge
picture exhibits explicit locality in both space and local energy density.

##  VII Conclusion

Having different pictures for understanding the same physics has frequently
proved to be invaluable. Furthermore, it is useful when a fundamental property
of the physics is explicit in the equations, e.g. as in a Lorentz-invariant
Lagrangian. To this end, we studied a picture of quantum dynamics that makes
locality explicit in the equations of motions for local wavefunctions.

There are several intriguing future directions. (1) It is not clear how to
describe quantum channels or the Lindblad master equation in the gauge
picture. The obstruction is that we used a unitary operator
($\widetilde{U}_{I}$) to cancel out distant unitary dynamics in order to
achieve locality; but unitary operators can not cancel out the non-unitary
dynamics of quantum channels or the Lindbladian. (2) In another work Slagle
(2022), we show that truncating $\ket{\psi_{I}}$ and $U_{IJ}$ to smaller
Hilbert spaces yields a new kind of tensor network ansatz Orús (2014) with
promising applications for simulating quantum mechanics. (3) Is there utility
to studying gauge picture connections with non-trivial curvature or holonomy?
(4) Does the gauge picture have useful applications for understanding exotic
dynamical many-body phenomena, such as out-of-time-ordered correlation
functions Swingle (2018); García-Mata  _et al._ (2022), many body localization
Nandkishore and Huse (2015); Abanin  _et al._ (2019), or the speed of
information propagation Nachtergaele and Sims (2011); Roberts and Swingle
(2016); Wang and Hazzard (2020)?

###### Acknowledgements.

We thank Lesik Motrunich, Sagar Vijay, Gunhee Park, and Garnet Chan for
helpful conversations. K.S. was supported by the Walter Burke Institute for
Theoretical Physics at Caltech; and the U.S. Department of Energy, Office of
Science, National Quantum Information Science Research Centers, Quantum
Science Center.

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