# Debating the Reliability and Robustness of the Learned Hamiltonian in the
Traversable Wormhole Experiment

Galina Weinstein  Reichman University, The Efi Arazi School of Computer
Science, Herzliya; University of Haifa, The Department of Philosophy, Haifa,
Israel.

###### Abstract

The paper discusses Daniel Jafferis et al.’s _Nature_ publication on
”Traversable wormhole dynamics on a quantum processor.” The experiment
utilized Google’s Sycamore quantum processor to simulate a sparse SYK model
with a learned Hamiltonian. A debate ensued when Bryce Kobrin, Thomas
Schuster, and Norman Yao raised concerns about the learned Hamiltonian’s
reliability, which Jafferis and the team addressed. Recently, there has been
an update in the wormhole experiment saga. In an attempt to rescue the
commuting Hamiltonian from its inevitable fate of being invalidated, a recent
paper by Ping Gao proposed a creative solution to reinvigorate the concept
within the context of teleportation through wormholes. This paper delves into
the ongoing debate and the recent endeavor to address the comments made by
Kobrin et al. I remain skeptical about the efforts to address Kobrin et al.’s
challenges. By its nature, a commuting Hamiltonian does not exhibit chaotic
behavior like non-commuting Hamiltonians. Moreover, it’s always essential to
assess the sensitivity of the Hamiltonian to noise to understand its practical
feasibility for the real-world Sycamore processor.

##  1 Introduction

The experiment of Daniel Jafferis et al. published in _Nature_ on ”Traversable
wormhole dynamics on a quantum processor” [7] garnered significant attention.
The experiment harnessed the computational power of Google’s Sycamore quantum
processor to simulate a sparse SYK model with the aid of a learned
Hamiltonian.111The Sachdev-Ye-Kitaev (SYK) model serves as a theoretical
framework for the onset of quantum chaos and holography. The SYK model was
initially proposed in condensed matter physics by Subir Sachdev, Jinwu Ye, and
Alexei Kitaev. The model’s large $N$ limit exhibits emergent scale invariance,
resembling conformal symmetry, due to many $N$ interacting Majorana fermions.
As the number of fermions increases, the system displays maximally chaotic
behavior, characterized by chaotic dynamics and universal scaling properties
of correlation functions. The SYK model demonstrates non-Fermi liquid behavior
and possesses a low-temperature entropy that matches the Bekenstein-Hawking
entropy of black holes. A notable feature of the SYK model is the all-to-all
interactions between Majorana fermions resulting from random and uncorrelated
interaction terms. This feature accounts for the model’s chaotic behavior and
other unconventional phenomena observed in the system. This achievement
presented exciting prospects for exploring traversable wormholes through
quantum means. In Sections 2 and 3, I introduce the experiment and delve into
its intricacies and nuances.

However, as I show in section 4, the paper’s progress was not without its
share of controversy. Bryce Kobrin, Thomas Schuster, and Norman Yao raised
valid concerns about the reliability [5] of the learned Hamiltonian, prompting
a heated debate within the scientific community and popular media [11].
Nevertheless, Jafferis and the team diligently addressed these concerns,
attempting to salvage the credibility of their approach [8]. Another paper,
authored by Ping Gao, was subsequently published in response to Kobrin et
al.’s challenges [4]. Like Jafferis and his research team, Gao attempted to
demonstrate that a commuting Hamiltonian is reliable from the standpoint of
teleportation through a traversable wormhole.

In section 4, I aim to analyze the multifaceted debate surrounding Jafferis et
al.’s work and analyze the recent efforts undertaken to confront the critiques
put forth by Kobrin et al. I delve into the intricacies of the learned
Hamiltonian’s reliability and the ingenious proposal that seeks to revitalize
the notion of commuting Hamiltonians in the context of holographic wormholes.
Through this exploration, I seek to shed light on the current state of affairs
in quantum wormhole dynamics and the exciting potential that lies ahead.

In section 5, I discuss robustness [5]. The learned Hamiltonian might lack
robustness against noise and errors in the actual conditions in the Sycamore
processor, such as fluctuations in gate parameters and environmental noise. As
a result, it could produce inaccurate results on the Sycamore processor. I
then apply Ian Hacking’s notion of material models, which enables scientists
to gain insights and improve their understanding of phenomena while suggesting
enhancements to experiments, to Jafferis et al.’s experiment. Hacking
emphasizes that such models are not exact representations of reality. In light
of Hacking’s ideas, it is important to question whether the experiment on the
Sycamore quantum processor can be regarded as a Planckian wormhole in real
physical space. I show that this holds significant implications for the
realistic interpretation of the phrase ”quantum circuit as wormholes” and
calls into question the necessity to simulate the model on a quantum
processor.

##  2 Teleportation through a traversable wormhole

Adam Brown, Leonard Susskind, and a team of scientists expound in their paper
titled ”Quantum Gravity in the Lab” that they explore a potential avenue
through which quantum gravity can be experimentally examined. Specifically,
they investigate a particular entangled state—one that is feasible to create
in an atomic physics laboratory — and examine the consequences of introducing
a message into the system in a specific manner. One approach to understanding
this phenomenon is employing the Schrödinger equation through brute force.
However, the researchers argue that the phenomenon can be elucidated much more
straightforwardly within the framework of holographic quantum gravity. They
propose an explanation involving a traversable wormhole connecting two black
holes, simplifying the understanding of the observed effects [1].

The two black holes or eternal black hole’s left and right external bulk
regions are linked through a wormhole. They are dual to two identical copies
of the original _Conformal Field Theory_ (CFT) in the _thermofield double_
(TFD) state, as proposed by Juan Maldacena [12]. The TFD state represents an
entangled pure state between these two copies of the quantum system (CFT):

$\frac{1}{\sqrt{Z_{\beta}}}\ket{\text{TFD}_{\beta}}=e^{-\beta(H_{L}+H_{R})}\ket{n}_{L}\otimes\ket{n}_{R},$
(1)

where $\ket{n}_{L}$ and $\ket{n}_{LR}$ are the energy eigenstates of the right
system and the left system, respectively.222System $R$ is introduced to mimic
system $L$ and ensure maximal entanglement between the two systems. We can
effectively treat the mixed state as pure by working with the TFD state
instead of the thermal density matrix. A thermal state is described by a
density matrix: $\rho=e^{-\beta H}$, where $\beta=1/k_{BT}$ is the inverse
temperature, with $k_{B}$ being the Boltzmann constant and $T$ the
temperature. This density matrix represents a mixed state. To purify $\rho$,
we introduce an additional Hilbert space that is isomorphic to the original
Hilbert space of the system. Instead of working with the thermal density
matrix $\rho$, we consider the pure state $\ket{\psi}$ in the enlarged Hilbert
space of the $\ket{\text{TFD}}$. In essence, the thermofield double formalism
enlarges the Hilbert space and introduces an entangled state that encodes the
statistical properties of the thermal state. Let’s consider the reduced
density matrix of the system L, denoted as $\rho_{L}$. To obtain $\rho_{L}$,
we trace out (disregard) system R from the TFD state. In other words, we
consider subsystem L in isolation and ignore any correlations and entanglement
with subsystem R. If we compare $\rho_{L}$ with the density matrix in
$e^{-\beta H_{L}}$, where $H_{L}$ is the Hamiltonian of the system L for
subsystem L in a thermal state, we can see that they have the same form. When
we restrict our attention to subsystem L, the $\ket{\text{TFD}}$ state appears
indistinguishable from a thermal state described by $e^{-\beta H_{L}}$. This
similarity implies that the reduced state of subsystem L when the complete
system is in $\ket{\text{TFD}}$ resembles a mixed, thermal state described by
$e^{-\beta H_{L}}$.

The entangled state of the CFT subsystems manifests as a connected geometry in
the bulk, represented by the wormhole.333This connection suggests that
entanglement is not merely an abstract concept but has a corresponding
geometrical interpretation in the bulk spacetime. A wormhole links the
entangled pairs in the CFT. Two copies of the CFT in TFD state, equation (1),
would be dual to the two-sided eternal black hole, and an eternal black hole
can be thought of as having two separate black holes connected by a wormhole.
This dual possibility made Gao, Jafferis, and Aron C. Wall realize they could
create a teleportation model through a traversable wormhole [3], [2]. Brown,
Susskind, et al. discovered that a process called _unscrambling_ comes after
scrambling in a wormhole.

The discovery of the scrambling and unscrambling process has significantly
enhanced the possibility of realizing a quantum teleportation protocol in the
lab called _teleportation-by-size_ : a qubit (message) is scrambled on the
left side of the wormhole. Since the two sides, right and left of the
wormhole, are coupled, the qubit is unscrambled and pops out on the right
side. In the dual gravitational interpretation, teleportation-by-size leads to
the interesting conclusion that a particle can pass through the holographic
wormhole. Two essential things enable traversability: the two sides of the
wormhole must be entangled before sending the information, and the two sides
must be coupled after sending the message [1].

Brown, Susskind, and their team introduced an Ansatz known as _size winding_
to describe the transmission of a signal through a semi-classical holographic
wormhole. Initially, they described size winding from a boundary perspective
and then extended its application to the traversable wormholes within the
bulk.

Around the _scrambling time_ concerning the SYK model, they inserted a thermal
operator denoted as $P$ into the left boundary (corresponding to the left side
$L$).444The scrambling time refers to the characteristic timescale within the
system during which information becomes highly mixed and spread out throughout
the system. The growth of the operator’s size serves as a fundamental
manifestation of quantum chaos and the system’s complexity. Notably, the
distribution of operator sizes exhibits a winding pattern in the clockwise
direction. A coupling is applied to examine the influence of coupling between
the two subsystems, $L$ and $R$. This coupling leads to the unwinding of the
complex winding of the operator size distribution and results in winding the
size distribution in the opposite direction, effectively reversing the winding
direction. Consequently, the thermal operator $P$ from the left side becomes
accurately mapped to its counterpart on the right side. Size-winding behavior
in thermal operators near the scrambling time has been demonstrated for the
SYK model, and there is a conjecture that this phenomenon could also be
present in other holographic systems.

Furthermore, _perfect-size winding_ is necessary for traversable wormhole
behavior. It specifically occurs in the ground state, where the temperature is
at its lowest possible point (essentially zero temperature through the
wormhole). In this context, the ground state of a pair of coupled SYK models
closely resembles the TFD state. It has been anticipated that systems with a
holographic dual would exhibit this perfect size winding behavior [1], [13].

As a consequence of the _teleportation-by-size_ protocol and the _perfect-size
winding_ Ansatz, a team of researchers led by Jafferis suggested a quantum
circuit that draws inspiration from the quantum circuit proposed by Brown,
Susskind, and their team for the concept of wormhole teleportation. The
researchers aimed to develop a quantum circuit that emulates a holographic
wormhole. Specifically, the quantum circuit is designed to be dual to a semi-
classical holographic traversable wormhole. The following outlines the primary
steps of the circuit[1] and [9]:

Two identical copies of the quantum system - SYK models - are prepared: a
system of $N$ qubits on the left (side $L$) and a system of $N$ qubits on the
right (side $R$). The two subsystems are initially entangled in the TFD state.

Register $P$ holds a reference qubit entangled with a qubit $q$ (the message)
on register $Q$; both are inserted into the wormhole.555Several components are
involved in the described quantum circuit: registers $P$, $Q$, $L$, $R$, and
$T$, SWAP gates, unitary operators, and interaction across the left and right
subsystems with coupling operators. Quantum teleportation within a wormhole
aims to replicate the teleportation process observed in quantum mechanics.
Recall that teleportation involves transferring the state of a qubit from one
location to another without physically moving the qubit itself. In
teleportation through a traversable wormhole, specifically utilizing the
Google Sycamore chip, a reference qubit is needed to establish entanglement
with the qubit on register $Q$. The reference qubit allows the transfer of
quantum information from the qubit on register $Q$ to the corresponding qubit
situated on the opposite side of the wormhole. This process resembles a
typical teleportation protocol found in quantum mechanics.

1) The first step involves evolving all the qubits on side $L$ (register $L$)
backward in time by applying the inverse of the time-evolution operator
($\exp^{-iHt}$).666The notion of ”running time backward” does not refer to
physically reversing the flow of time. Instead, it is a conceptual approach to
describe the sequence of events in the teleportation process through a
wormhole. Running time backward in this context means we consider the qubit
injection event to occur at a negative time ($-t_{0}$) relative to the
coupling interaction at $t=0$. This choice of assigning negative values to the
time parameter is merely a convention to consistently describe the sequence of
events. A qubit $q$ is injected, namely, swapped into the left side $L$ at the
time $t=-t_{0}$.777The SWAP gates achieve qubit injection and arrival readout
in the teleportation protocol. The SWAP gates enable the qubit on register $Q$
to enter the wormhole and the resulting qubit on register $T$ to be read out
at the appropriate times.

2) Subsequently, we progress by evolving register $L$ forward in time,
utilizing the time-evolution operator ($\exp^{-iHt}$).888The time evolution
operators are implemented using a series of single qubit gates and the
controlled $Z$ gate. Consequently, $q$ becomes rapidly scrambled and entangled
with the carrier qubits present in subsystem $L$.

3) Step 3 involves weakly coupling side $L$ to side $R$ at $t=0$, employing
the coupling operator $\exp^{i\mu V}$, where the operator $V$ is defined in
equation (6).

This coupling occurs suddenly, connecting all the qubits on side $L$ with
those on side $R$.999This interaction, along with the time evolution, is
implemented using a Trotter step. The single Trotter step involves breaking
down the exponential evolution operator into smaller operations, typically
achieved through single-qubit and controlled gates [8], [9], [16].

4) Finally, in Step 4, we evolve all the qubits on side $R$ forward in time
using the time-evolution operator ($\exp^{-iHt}$). At a later time $t=t_{1}$,
we measure side $R$ (register $T$). Remarkably, the qubit $q$ reappears on the
side $R$ unscrambled, requiring no decoding. The message is effortlessly
refocused on side $R$.

To measure the entanglement of the qubits, one computes the mutual information
$I_{PT}(t)$:101010The mutual information $I_{PT}$ measures the correlation
shared between the two systems. The $I_{PT}$ tracks the flow of information
between the sender and the receiver during this process. It quantifies how
much information is gained about the qubit’s state as a function of time. A
peak in $I_{PT}$ within a certain time window indicates the occurrence of
quantum teleportation. This peak represents when the left side obtains the
maximum information about the qubit’s state, suggesting that the qubit has
successfully teleported through the wormhole.

$I_{PT}(t)=S_{P}(t)+S_{T}(t)-S_{PT}(t),$ (2)

where $S$ is the von Neumann entropy.

Brown, Susskind et al. and Jafferis et al. identified two distinct mechanisms
through which the quantum circuit can accomplish teleportation [1],[7]:

1\. _Low-temperature teleportation, near the scrambling time_. This regime
involves teleportation through the wormhole and is applicable to Hamiltonians
with a holographic dual. There are two transmission mechanisms associated with
low-temperature teleportation, with a strong dependence on the sign of $\mu$
[1]:

a) If $\mu<0$, the qubit $q$ experiences a time advance and reappears on side
$R$, representing traversable wormhole teleportation. In this case, a peak in
the signal $I_{PT}(t)$ is reached around $t$ ($t$ being approximately the
scrambling time), indicating quantum teleportation within the time window when
the wormhole is traversable. The $LR$ coupling applied between the two sides
of the wormhole allows for its traversability. If $\mu<0$, the coupling
operator generates a negative energy shockwave in the bulk, modifying the
wormhole’s geometry and enabling traversability. The traversing qubit
experiences a Shapiro time advance upon encountering the pulse of negative
energy shockwave, causing it to emerge on the wormhole’s other side ($R$) at
$t=t_{1}$.

b) On the contrary, in the case where $\mu>0$, the qubit becomes entangled
with the qubits on side $L$, but it remains scrambled, indicating scrambling
teleportation. This scenario involves a reversal of the sign of the coupling
interaction, resulting in a positive energy shockwave leading to the
generation of a positive energy shockwave that draws the qubit toward the
singularity instead of repelling it. Consequently, the qubit is inevitably
drawn to the singularity, preventing its emergence on the wormhole’s right
side ($R$).

2\. _High-temperature teleportation, for $t>$ the scrambling time_. This
mechanism, unexpected from gravity, does not involve signals traversing a
geometric wormhole.

Jafferis and his team opted to examine the hypothesis of many-body
teleportation by simulating the described quantum circuit on the quantum
device called _Sycamore_ developed by Google [7].

In particular, the SYK model is only dual to teleportation through a semi-
classical holographic wormhole in the low-temperature limit, and in this
regime, it exhibits perfect-size winding. However, this relationship holds
specifically for scenarios involving large $N$ Majorana fermions interacting
with a large number ($q$) of other Majorana fermions, effectively allowing for
the teleportation of $q$ fermions. This presented a considerable challenge as
simulating the dense SYK model on a noisy quantum device was deemed
impractical. In response to the aforementioned findings, the team of scholars
led by Jafferis embarked on a project to simplify the SYK model.

##  3 Simplification and sparsification

Certain simplifications were made. To study the emergence of gravitational
behavior at small $N$ and a few terms of the Hamiltonian, Jafferis and his
team considered the Hamiltonian dual to a two-dimensional Anti-de Sitter
(AdS2) space. This dual relationship allowed them to relate the properties of
the SYK model to gravitational phenomena in AdS2. The Hamiltonian dual to
nearly AdS2 space was expressed in terms of the left Hamiltonian ($H_{L}$) and
right Hamiltonian ($H_{R}$), as well as the Majorana fermions and the coupling
constant $J_{ijkl}$ associated with the interaction between the left and right
sectors. The left Hamiltonian ($H_{L}$) and right Hamiltonian ($H_{R}$) refer
to the Hamiltonians of the left and right CFTs, respectively. In the context
of the Hamiltonian dual to nearly AdS2 space:

$H_{L}=\sum_{1\leq i<j<k<l\leq
N}J_{ijkl}\psi_{L}^{i}\psi_{L}^{j}\psi_{L}^{k}\psi_{L}^{l},\quad
H_{R}=\sum_{1\leq i<j<k<l\leq
N}J_{ijkl}\psi_{R}^{i}\psi_{R}^{j}\psi_{R}^{k}\psi_{R}^{l},$ (3)

$\quad J_{ijkl}\sim N(0,\sigma^{2})$,

the coupling constant $J_{ijkl}$ is considered a random variable drawn from a
Gaussian distribution [15], [16].111111It is commonly assumed to follow a
normal distribution with zero mean and variance of $\sigma^{2}$: $J_{ijkl}\sim
N(0,\sigma^{2})$. This assumption of randomness in the coupling constant
$J_{ijkl}$ arises from the chaotic behavior of the underlying quantum system,
which is a key feature of the AdS2/CFT correspondence.

The large-$N$ SYK model dual to nearly-AdS2 is expected to exhibit perfect-
size winding behavior in the low-temperature limit.

Afterward, Jafferis and his team posed a fundamental question: What is the
most straightforward Hamiltonian that preserves the gravitational physics
observed in the dense SYK model? Their response to this inquiry is presented
in their paper published in _Nature_ : ”Our numerical simulation shows that
$N=10$ [with $210$ terms] is sufficient to produce such traversable wormhole
behavior” [7]. The $N=10$ SYK model represents a system composed of $10$
interacting Majorana fermions. Jafferis and his team demonstrated that the
$N=10$, $q=4$ dense SYK model is sufficient to exhibit the size winding
behavior and, more broadly, the phenomenon of teleportation through a
wormhole. Their findings revealed that the ground state of this model closely
resembles a TFD state, highlighting the system’s potential for emulating
wormhole teleportation. They simulated the $N=10$ SYK model on a classical
computer and collected data that provided insights into perfect-size winding.
Once the simulation was complete, the collected data was analyzed to
investigate whether the system exhibited the desired teleportation through a
wormhole-like behavior in the low-temperature regime.

But simulating this SYK model on a noisy quantum processor is very challenging
due to the complexity of the model and the required number of qubits and
gates.121212Simulating the described $N=10$ SYK model on the Sycamore quantum
computer is currently not feasible for the following reasons. The $N=10$ SYK
model requires $10$ Majorana fermions, typically represented using $20$
qubits. Each Majorana fermion corresponds to two qubits. In a quantum
processor, qubits are usually represented using two-level quantum systems,
such as the energy levels of a superconducting circuit. However, in the case
of Majorana fermions, they are represented using two qubits. The two qubits
correspond to the two possible states of the Majorana fermion, which are
related to each other in a non-local manner. These qubits encode the
information about the occupation of the Majorana fermion. The Sycamore
processor with $72$ qubits cannot accommodate the required qubits. The SYK
model requires a large number of gate operations. However, the Sycamore
processor has a limited number of gate operations before the quantum states
become too noisy due to errors. Moreover, implementing the complex SYK
interactions using the native gate set of the Sycamore processor is not
straightforward. As the number of qubits and gates increases, the circuit
depth grows, and the susceptibility to noise and errors also increases. The
$N=10$ SYK model would require a substantial circuit depth, which is likely
beyond the capabilities of the Sycamore processor. Given these limitations,
simulating an $N=10$ SYK model on the Google Sycamore quantum chip is
currently not viable. Hence, for simulating the $N=10$ SYK model, Jafferis and
his research team used classical computational methods.

The team subsequently used classical machine learning to reproduce the
teleportation behavior of the $N=10$ SYK model with only a few Hamiltonian
terms [see equation (9), section 4].131313A kind of approximation called
_Trotterization_ was applied to optimize the procedure. Many couplings
$J_{j_{1}\ldots j_{4}}$ in equation (3) were set to zero to obtain the sparse
Hamiltonian. This sparsity was introduced to simplify the model and reduce its
computational complexity. The sparsification procedure reduced the $N=10$ SYK
Hamiltonian with $210$ terms to a large population of candidate sparse
Hamiltonians thought to preserve the gravitational properties of the original
model (see equations in section 4).

Jafferis and his research team aimed to simplify the $N=10$, $q=4$ SYK model
by reducing it to a Hamiltonian with the smallest $N$ and the most reduced
number of terms while preserving the essential characteristics of the original
model, specifically the gravity-like effects. They utilized machine learning
techniques to acquire a Hamiltonian operator with only a few non-zero
terms.141414Using machine learning techniques on a classical computer, the
team trained an analog of a neural network to sparsify the SYK model with
$N=10$ and $210$ terms, backpropagated over the Hamiltonian coefficients, and
applied regularization, interpreting the Hamiltonian coefficients as neural
network weights. Each term in the $N=10$ SYK Hamiltonian was treated as a
feature, and the corresponding coefficient was seen as the weight of that
feature. Each data point in the dataset represented a specific configuration
of the Hamiltonian terms. A neural network was constructed that mimics the
structure of the SYK model. The network had an input layer with neurons
corresponding to each term in the original Hamiltonian. These neurons serve as
the inputs to the neural network. The training process involved adjusting the
neural network’s weights (Hamiltonian coefficients) to sparsify the model.
This was done using backpropagation, where the errors were propagated from the
output layer back to the input layer, and the weights were updated
accordingly. The training objective was to find a set of weights that reduce
the $N=10$ SYK model to an $N=7$ model with five terms. To encourage sparsity,
weight regularization techniques were applied. These techniques introduced a
penalty term encouraging the model to have fewer non-zero weights. This helped
select the most relevant terms and discard the less significant ones. Terms
with low weights were removed. The Hamiltonian coefficients were interpreted
as the neural network weights once the training process was complete. The non-
zero coefficients indicate the terms selected in the reduced $N=7$ model.

The following $N=7$ model, referred to as the _learned Hamiltonian_ ,
represents the result consisting of five terms:

$H_{L,R}=-0.36\psi^{1}\psi^{2}\psi^{4}\psi^{5}+0.19\psi^{1}\psi^{3}\psi^{4}\psi^{7}$
(4)

$-0.71\psi^{1}\psi^{3}\psi^{5}\psi^{6}+0.22\psi^{2}\psi^{3}\psi^{4}\psi^{6}+0.49\psi^{2}\psi^{3}\psi^{5}\psi^{7}.$

According to the team’s claim, this model successfully captures the
fundamental characteristics of the original $N=10$ SYK model while
substantially reducing its overall complexity [8], [9].

But the sparsification process resulted in an $N=7$ Hamiltonian with only
commuting terms. Since the reduced $N=7$ learned Hamiltonian consists solely
of commuting terms, other scientists suggested that the specific features
related to the gravitational behavior were not accurately captured in the
simplified model [11], as discussed in section 4.

Reproducing the gravitational behavior of a traversable wormhole using the
sparse $N=7$ SYK Hamiltonian is challenging. The success of sparsification
depends on various factors, including the details of the machine learning
procedure, the quality and quantity of training, and the underlying
assumptions and approximations made during the modeling process. The machine-
learning procedure uses a reduced number of Hamiltonian terms to mimic the
teleportation behavior through a wormhole of the dense $N=10$ SYK model. This
results in a Hamiltonian, denoted as $H_{L,R}$, involving specific operator
combinations. When the teleportation protocol is implemented using this
Hamiltonian, which is holographically dual to gravity, the process is
described from the perspective of a particle traveling through the wormhole.

As demonstrated by equations (3) and (4), the Hamiltonian is doubled to yield
left $H_{L}$ and right $H_{R}$ Hamiltonians, each simulating the SYK model and
each containing seven Majorana fermions on their respective sides. In
addition, the wormhole teleportation protocol incorporates a pair of entangled
qubits, namely, a reference qubit that is entangled with the injected qubit.
Consequently, the entire circuit involves a total of nine qubits. The five
terms in the Hamiltonian describe the interactions between these qubits,
forming the $9$-qubit circuit for the protocol [7].151515Considering two SYK
models, each with $N=7$, and the need for two additional qubits for
teleportation, the total number of qubits used on the Google Sycamore chip
would be seven qubits for the left SYK model, seven qubits for the right SYK
model, and two qubits for the teleportation process. Therefore, in this
scenario, the total number of qubits used on the Sycamore chip would be
$7+7+2=16$ qubits.

To determine the ground state of the learned Hamiltonian (equation 4), the
researchers employed a hybrid classical-quantum algorithm known as the
_Variational Quantum Eigensolver_ (VQE). They utilized the VQE algorithm to
the following Hamiltonian:161616Initializing the wormhole at $t=0$ with
$\ket{\text{TFD}}$ using a VQE algorithm is done on the Google Sycamore chip
by preparing the individual qubits in the state $\ket{0}$, and then entangling
them appropriately using a sequence of controlled gates and single-qubit
rotations.

$H_{TFD}=H_{L}+H_{R}+i\mu V,$ (5)

where $V$ is the interaction term:

$V=\sum_{j}\psi{{}^{j}_{L}}\psi{{}^{j}_{R}},$ (6)

and $\mu$ represents the coupling interaction.

The researchers’ findings reveal that the ground state of $H_{TFD}$ closely
approximates the TFD state.

Equation (5) bears a resemblance to the Hamiltonian corresponding to the
traversable wormhole equivalent, known as the _eternal traversable wormhole
Hamiltonian_ , denoted as $H_{tot}$:

$H_{tot}=H_{L}+H_{R}+i\mu^{\prime}V.$ (7)

This similarity is a consequence of the ground-state behavior of the coupled
Hamiltonian.

The team of scholars demonstrated that the learned Hamiltonian, represented by
equation 4, displays similar scrambling and thermalization behavior to the
original $N=10$ SYK model. This behavior is characterized by the two-point and
four-point functions, where the two-point function indicates thermalization
time, and the four-point function indicates scrambling time.171717The two-
point function is a correlation function that tracks the relationship between
two observables as time progresses in a quantum system. In thermalizing
quantum systems, correlations between observables diminish over time until
thermal equilibrium is reached, and this duration is the thermalization time.
The four-point correlation function probes scrambling behavior in chaotic
quantum systems. In such systems, the four-point function deviates
significantly from its initial value and rapidly increases, indicating the
spread of entanglement, and the time at which it reaches a characteristic
value is referred to as the scrambling time.

In the left side $H_{L}$, the transmitted fermions undergo thermalization and
scrambling, as evidenced by the decay of two-point correlators and out-of-time
ordered correlators (OTOC). The averages of these correlators over Majorana
operators were plotted, demonstrating a correspondence between the curves of
the $N=10$ SYK model and the learned Hamiltonian [7], [9].

It is worth noting that the large-$N$ SYK model exhibits a significant
difference between the thermalization time and the scrambling time, with the
scrambling time occurring just before the onset of chaotic behavior. However,
the scrambling time is approximately equal to the thermalization time for the
$N=10$ SYK model and the learned Hamiltonian represented by equation (4). This
observation is consistent with wormhole-like teleportation properties.

Jafferis et al. first investigated the low-temperature limit and verified that
the behavior of the learned Hamiltonian aligns with the original $N=10$ SYK
model for both $\mu=-12$ and $\mu=+12$. This correspondence was illustrated
through the similarity in the curves of the two models. Notably, the
similarity was evident in the peak positions of the $N=10$ SYK model and the
learned Hamiltonian [see equation 4]. For a fixed injection time of
$-t_{0}=-2.8$, it was anticipated that $I_{PT}(t)$ would reach its peak around
the scrambling time $t_{*}$ ($t_{0}\approx t_{1}\approx t_{*}$) for $\mu=-12$
(the interaction) due to the negative energy shockwave making the wormhole
traversable. Through a simulation carried out on the Sycamore quantum
processor with gate errors, a peak in $I_{PT}(t)$ was observed solely for
$\mu=-12$, suggesting the occurrence of teleportation.

The outcomes also revealed an asymmetry in $I_{PT}(t)$ when considering
opposite signs of $\mu$, specifically, a peak for $\mu=-12$ and a trough for
$\mu=+12$ (corresponding to teleportation occurring by means of scrambling
instead of through the wormhole, consistent with theoretical expectations).
This observation demonstrated qualitative consistency with the numerical
simulation performed on classical computers using the $N=10$ SYK model.

Furthermore, the researchers demonstrated that in the high-temperature regime,
non-gravitational teleportation occurred, and there was no size-winding effect
[7].

Multiple noisy simulations were conducted, and a noteworthy trend emerged
regarding the behavior of traversable wormhole teleportation. It was
consistently observed that $I_{PT}(t)$ exhibited a peak around the scrambling
time across various noisy runs. On the other hand, when coherent noise was
present, it was more likely to lead to random fluctuations in $I_{PT}(t)$. The
above signatures were verified on classical computers, confirming that the
quantum system dynamics were consistent with a quantum gravity interpretation
and the holographic principle [15]. Based on these findings, Jafferis and his
team concluded: Teleportation was more pronounced when the interaction
introduced a negative energy shockwave than a positive one. The distinct
asymmetric signature observed in the results aligns with the physical
interpretation that the qubit underwent teleportation through the wormhole
[7].

Graphs were presented that show curves with their coarse-grained SYK model
preserving key properties of the traversable wormhole physics: perfect size
winding, coupling interaction on either side of the wormhole that is
consistent with a negative energy shock wave, a Shapiro time delay, causal
time-order of signals emerging from the wormhole, and scrambling and
unscrambling [7].181818In addition to teleporting a single qubit from the left
side to the right side, the researchers conducted another experiment where a
qubit was sent from the right side to the left side. This resulted in what
they termed ”time-ordered teleportation,” which they interpreted as an
indication of gravitational teleportation. The process involved swapping a
qubit $Q$ into the left side ($L$) at time $-t_{0}$, and simultaneously
swapping a qubit $R$ into the right side ($R$). At the time $t_{1}$, the team
performed measurements and compared the two processes. In this experiment, the
causal time ordering played a crucial role. Qubits inserted earlier emerged
while qubits inserted later would pop out later, providing evidence supporting
the gravitational interpretation. However, sending one qubit into $L$ and
another into $R$ with the expectation of them meeting in the middle resulted
in a slight delay. This phenomenon was identified as a ”Shapiro time delay,”
wherein a qubit injected into $R$ would interact with a qubit injected into
$L$ [7], [16].

##  4 Examining the controversy: the reliability of the learned Hamiltonian

A few months after the publication of Jafferis et al.’s _Nature_ paper, a
comment was published by three researchers led by Norman Yao. Their comment
raised concerns about the validity of Jafferis et al.’s $5$-term commuting
learned Hamiltonian, as described in equation (4). They identified multiple
flaws and challenged its reliability [11]. Kobrin et al. focus on evaluating
the reliability of the sparsification method and the learned Hamiltonian. The
central aim of Yao and his team’s inquiry was to establish whether this
equation aligns with the gravitational dynamics (specifically, a qubit
emerging from a traversable wormhole) of the original SYK model. In other
words, they were interested in determining whether the Hamiltonian could be
considered a dependable model for simulating the teleportation of a qubit
through a semi-classical wormhole [14].

Recall from section 3 that certain simplifications were made to the dense
$N=10$ SYK model during the sparsification process to reduce its complexity.
These idealizations were instrumental in obtaining a sparser representation of
the model while preserving its essential gravitational properties. However,
sparsification comes with a trade-off between model complexity and accuracy.
While a simpler model may be suitable for implementation on the Google
Sycamore quantum processor, it may not fully capture all the intricate details
of the original dense SYK model. Some physical effects might be overlooked or
underestimated in the process. Additionally, the sparsified Hamiltonian has
limitations in its range of validity. Its predictions and behavior may only be
reliable within certain conditions. To address these potential issues,
Jafferis et al. carefully analyzed the reliability and applicability of the
sparsified learned Hamiltonian in their experiment. They demonstrated that the
behavior of the mutual information $I_{PT}(t)$ between the $N=10$ model and
the learned Hamiltonian remains almost the same, indicating that the
sparsified model captures relevant information despite its simplifications. By
thoroughly examining the agreement between the sparsified learned Hamiltonian
and the $N=10$ SYK model, they sought to validate the reliability of their
simplified representation within the experimental setup.

But some critics argued that the measures taken were insufficient to tackle
the issues adequately. Kobrin et al. discovered a gap in equation (4),
asserting that it fails to encompass both the global physics seen in typical
SYK models and the complete characteristics of gravitational physics. In other
words, the equation does not fully represent the broader aspects of ordinary
SYK models nor adequately capture the intricacies associated with
gravitational physics.

According to Kobrin et al., the learned Hamiltonian falls short in
representing the dense SYK model in the following aspects [11]:

1\. The teleportation signal only resembles the $N=10$ SYK model used in the
machine-learning training process for certain Majorana operators. The
complexity of the teleportation through the wormhole and the associated
dynamics in the $N=10$ SYK model is influenced by the interactions among
different operators and the specific Hamiltonian terms. While effective for
the trained operators, the machine-learning procedure may not fully capture
the intricacies of all operators in the system, leading to limited resemblance
to the behavior of the $N=10$ SYK model. During the machine-learning
procedure, the training is conducted on teleportation involving specific
fermions, namely $\psi^{1}$ and $\psi^{2}$. Jafferis and his team evaluate the
interaction on these two qubits numerically, involving the time evolution
operator $\exp^{iHt}$, which yields results demonstrating asymmetry for
opposite signs of $\mu$ (where $\mu=12$ represents scrambling teleportation
and $\mu=-12$ represents wormhole teleportation). This finding indicates a
qualitative agreement of equation (4) with the numerical simulation of the
$N=10$ SYK model.

However, Kobrin et al. argue that the observed teleportation signal,
characterized by a negative energy shockwave, causal time-ordering of
teleported signals, and a Shapiro time delay, as well as perfect size winding,
is specific to the fermions $\psi^{1}$ and $\psi^{2}$, which were involved in
the training process. The agreement between equation (4) and the $N=10$ SYK
model is observed for this trained pair of fermions. Yet, these distinctive
characteristics are not observed when the teleportation protocol is attempted
with other fermions not included in the training procedure.

Kobrin et al. specifically identified an issue with the untrained fermions
$\psi^{4}$ and $\psi^{7}$. They observed that these fermions display poor size
winding at $t_{0}\approx 2.8$, which is the time of the wormhole teleportation
signal or injection time. In simpler terms, the size winding and teleportation
behavior, involving $\mu=-12$ and at $t_{0}\approx 2.8$, are exclusively
observed for the trained Majorana fermions $\psi^{1}$ and $\psi^{2}$, and not
for any other fermions that were not part of the training process.

To generate the Hamiltonian, the machine-learning procedure is specifically
trained on the behavior of two operators, $\psi^{1}$ and $\psi^{2}$.
Consequently, the Hamiltonian is optimized to capture the teleportation
dynamics involving these specific operators. However, its performance might
not generalize well to other operators, such as $\psi^{3}$ and $\psi^{4}$, as
the Hamiltonian is tailored to the training data of $\psi^{1}$ and $\psi^{2}$.
Thus, the behavior of untrained operators might not be accurately represented
in the generated Hamiltonian.

2\. Equation (4) comprises solely commuting terms, but a crucial distinction
exists in the structure of time-evolved operators between fully-commuting
models, such as equation (4), and non-commuting models, like the original SYK
model. The original SYK model, with non-commuting Hamiltonian terms, exhibits
distinctive properties such as information scrambling, strong entanglement,
rapid thermalization, and chaotic behavior. The non-commutative nature of the
Hamiltonian terms plays a crucial role in generating these characteristics. In
this model, interactions between Majorana fermions occur all-to-all, allowing
any pair of fermions to interact. On the other hand, if we consider a fully
commuting Hamiltonian, where each term commutes with every other term, the
system demonstrates more regular and integrable behavior. As acknowledged in
their paper, Jafferis and his research team state that the Hamiltonian in
question with five terms consists of all commuting terms [8].

Kobrin et al. examine two alternatives to equation (4):

1) In Jafferis et al.’s _Nature_ paper, they demonstrate that the machine-
learning procedure used to derive equation (4) actually generates a $6$-term
Hamiltonian. Unlike equation (4), this $6$-term Hamiltonian does not consist
entirely of commuting terms [7]:

$H_{L,R}=-0.35\psi^{1}\psi^{2}\psi^{3}\psi^{6}+0.11\psi^{1}\psi^{2}\psi^{3}\psi^{8}$
(8)

$-0.17\psi^{1}\psi^{2}\psi^{4}\psi^{7}-0.67\psi^{1}\psi^{3}\psi^{5}\psi^{7}+0.38\psi^{2}\psi^{3}\psi^{6}\psi^{7}-0.05\psi^{2}\psi^{5}\psi^{6}\psi^{7}.$

Equation (8) represents a weakly perturbed version of a fully commuting
Hamiltonian.

However, Kobrin et al. identified a flaw in this Hamiltonian. They demonstrate
that the key observations made about equation (4) are also applicable to
equation (8). In other words, the issues and concerns raised in relation to
equation (4) carry over to the perturbed equation (8). Specifically, the
teleportation signal does not resemble the $N=10$ SYK model for untrained
operators. However, the size winding behavior is similar to that observed in
equation (4). Notably, there is a distinct difference between equation (8) and
the $N=10$ SYK model regarding the timescale during which $\psi^{1}$ and
$\psi^{2}$ were trained. For equation (8), the teleportation signal displays
an initial peak, followed by significant revivals as time progresses, unlike
the $N=10$ SYK model.

Jafferis and his team assert that ”the learned Hamiltonian scrambles and
thermalizes similarly to the original SYK model as characterized by the four-
and two-point correlators” [7]. However, Kobrin et al. present a different
perspective. Both the SYK model and equation (4) display decay in their two-
point functions. In the case of the SYK model, this decay aligns with the
expected quantum thermalization. However, for equation (4), the individual
two-point correlators exhibit strong revivals as time progresses. According to
Kobrin et al., this behavior suggests that the training procedure was
unsuccessful and unreliable. They also emphasize that such discrepancies were
not demonstrated or discussed in the _Nature_ paper. Consequently, they
concluded that the apparent agreement between equation (4) and the SYK model
is, in reality, an artifact resulting from averaging over the two-point and
four-point correlation functions [11].

2) In their _Nature_ paper, Jafferis et al. employ an alternative machine-
learning approach to create another Hamiltonian featuring $N=10$ and $8$
terms. The specific objective of this Hamiltonian is to maximize the
distinction in the teleportation signal between $-\mu$ (associated with
wormhole teleportation) and $+\mu$ (related to scrambling teleportation) [7].

$H_{L,R}=0.60\psi^{1}\psi^{3}\psi^{4}\psi^{5}+0.72\psi^{1}\psi^{3}\psi^{5}\psi^{6}$
(9)

$+0.49\psi^{1}\psi^{5}\psi^{6}\psi^{9}+0.49\psi^{1}\psi^{5}\psi^{7}\psi^{8}$

$+0.64\psi^{2}\psi^{4}\psi^{8}\psi^{10}-0.75\psi^{2}\psi^{5}\psi^{7}\psi^{8}+0.58\psi^{2}\psi^{5}\psi^{7}\psi^{10}-0.53\psi^{2}\psi^{7}\psi^{8}\psi^{10}.$

Jafferis and his team explain that as the number of terms increases in
equation (9), the number of gates needed for Trotterization also rises
linearly. This implies that the gate count required to implement the
Hamiltonian must at least double. Unfortunately, the fidelity of circuits
exponentially decreases with the number of gates and qubits, and the
experimentally measured fidelity was already below half of the noiseless
fidelity. Consequently, Jafferis and his team conclude that equation (9)
”cannot provide a stronger teleportation signal when experimentally measured”
and does not exhibit perfect size winding but rather a slightly damped one
[7].

However, despite the technical explanation being essentially correct, there
are dissenting opinions, as Kobrin et al. provide an alternative
interpretation, leading to different perspectives on the matter. Kobrin et al.
identify a concern with small-size fully-commuting Hamiltonians consisting of
only a few terms, including equation (9). Despite being non-commuting, they
comment that this particular Hamiltonian shows clearer indications of
thermalization over a longer time scale, approximately $t\approx 30$. Unlike
the expected perfect-size winding, the teleportation signal for this
Hamiltonian displays a single peak structure for nearly all operators. The
phenomenon of perfect-size winding is generally observed in fully commuting
Hamiltonians with only a few terms. However, it does not persist in larger
fully-commuting or non-commuting systems. Kobrin et al. demonstrate that
introducing random numerical coefficients in front of the terms in equation
(4), or considering random commuting terms, also yields a perfect-size
winding. Thus, Kobrin et al. conclude that perfect-size winding in small-size
fully-commuting Hamiltonians with only a few terms is likely a side effect of
the sparsification method and does not imply that equation (4) is
holographically dual to gravity [11].

Kobrin et al. conclude that the reported perfect-size winding in the _Nature_
paper is contingent on the small size of the system and equation (4).
Therefore, unlike the $N=10$ SYK model, equation (4) does not undergo
thermalization, and any agreement in thermalization behavior between the two
is merely an artifact [11]. In other words, the observed similarity in
thermalization is not a genuine characteristic of equation (4), but rather a
consequence of the specific conditions in which it was examined.

Kobrin et al. are raising concerns that the machine-learning procedure used by
the authors may not have been entirely reliable and unbiased. In essence, they
question the validity and robustness of the machine-learning procedure,
implying that the obtained results might not be entirely trustworthy. In other
words, the assumption of perfect-size winding in the learned Hamiltonian (for
$N=7$) may not be accurate due to the influence of bias introduced during the
machine-learning process. They argue that introducing bias among the trained
operators could have influenced the results, potentially leading to
inaccuracies or misleading conclusions [11]. The presence of this bias may
have affected the results, casting uncertainty on the validity of the observed
perfect-size winding in relation to teleportation through the wormhole. Their
main point is that the observed perfect-size winding in [7] is contingent on
the small size of the system, which is the fundamental aspect they emphasize.
Moreover, these fully-commuting models contradict other essential features of
holography, thermalization, complexity, and chaos. This raises doubts about
whether the observed perfect-size winding is genuinely connected to the
gravitational picture in a significant way.

The criticism focuses on the reliability and appropriateness of the fully-
commuting Hamiltonians in reproducing a dual gravitational behavior. The
gravitational picture itself is not being questioned or challenged; rather,
the scrutiny is directed toward whether these Hamiltonians can effectively
model the well-established features of holography that are assumed to be true.
In essence, the criticism is centered on the capability of the learned
Hamiltonian equation (4) to accurately capture the expected holographic
properties, given the specific nature of the fully-commuting Hamiltonians
under consideration.

One month later, Jafferis and his team published a comment in response to the
concerns raised by Kobrin et al. In their response, they presented a
compelling solution to defend their learned Hamiltonian, equation (4), as
expected. Jafferis et al. found an elegant way to counter the main arguments
put forth by Kobrin et al. To properly examine Jafferis et al.’s response, we
must analyze its specific points and arguments. It may shed light on how they
addressed the criticisms raised by Kobrin et al. and how they defended their
learned Hamiltonian, equation (4). A thorough examination of their response
could provide a clearer understanding of their perspective and whether they
could effectively address the concerns raised by Kobrin et al.

The comment argued that the single-sided Hamiltonian $H_{L}$ in equation (4)
consists of commuting terms and does not thermalize later after teleportation
due to the recurrence of these commuting terms. Jafferis et al. found a way,
in the context of the eternal traversable wormhole Hamiltonian $H_{tot}$ to
rebut this claim: evolving under $H_{tot}$ [equation (7)] from the _Nature_
paper, shows that equation (7) exhibits operator growth and thermalizes at
high temperatures after teleportation. A single Trotter step of time evolution
under the Hamiltonian $H_{tot}$ can be understood as being equivalent to the
teleportation quantum circuit described in the _Nature_ paper [8].

Jafferis et al. further found another counterclaim stating the following: In
the context of the gravitational interpretation of thermalization rates, each
fermion corresponds to a distinct mass. When a fermion is inserted onto the
TFD and then time-evolved under a single-sided Hamiltonian $H_{L}$, it leads
to operator growth, where lighter fermions experience slower growth. This
phenomenon is attributed to the fact that the wave packet of a lighter fermion
is more spread out, causing its two-point function to decay at a slower rate.
Jafferis et al., in their work [8], reach the conclusion that the two-point
function described by equation (7) decays for all fermions, signifying
thermalization. As a result, they note that the fermions exhibiting the
slowest operator growth in $H_{L}$ are precisely the same fermions that
thermalize the slowest in $H_{tot}$, as indicated by equation (7). In equation
(7), Jafferis et al. observed that all fermions exhibit significant size
winding within the time range $2\gtrsim t\lesssim 5$. On the other hand,
Kobrin et al.’s comment only focused on analyzing size winding at a specific
time point, approximately $t_{0}\approx 2.8$, and claimed it to be an
artifact. Jafferis et al. examined a counterfactual scenario to gain deeper
insights, revealing meaningful gravitational behavior. They demonstrated that
even at later times, specifically in the range $5\gtrsim t\lesssim 10$, the
wormhole teleportation phenomena persist despite the introduction of a strong
non-commuting perturbation represented by equation (10). Although the late-
time dynamics are now governed by a noncommuting Hamiltonian, the system’s
behavior during teleportation at $t_{0}\approx 2.8$ remains unchanged and
consistent with the expected gravitational signature [8].

Kobrin et al. assert that equation (4) displays a bias, resulting in favorable
size winding behavior only for the two fermions implemented in their
experiment [11]. Furthermore, they stress that particular emphasis is placed
on a specific pair of input operators, namely, $\psi^{1}$ and $\psi^{2}$. In
other words, Jafferis et. all discovered a representation within the sparse
$N=7$ Hamiltonian that replicated the gravitational behavior demonstrated by
the $N=10$ model when these specific operators were considered. The $N=7$
Hamiltonian mimics the dynamics and properties observed in the $N=10$ model,
specifically concerning the behavior of these selected operators, $\psi^{1}$
and $\psi^{2}$. However, Jafferis and his team challenge this claim and
present a compelling physical justification for their findings. They observe
that fermions exhibiting slower thermalization in the eternal traversable
wormhole Hamiltonian [equation (7)] demonstrate significant size winding at
later times. This behavior is attributed to the varying masses of the
fermions: lighter fermions undergo slower thermalization and consequently take
longer to traverse the wormhole. Specifically, fermions $\psi^{4}$ and
$\psi^{7}$ are the slowest to thermalize, leading to their achievement of size
winding at slightly later times.

The argument put forth by Jafferis and his team may initially seem peculiar,
as it relies on physical reasoning. In contrast, the criticism raised by
Kobrin et al. concerning the two operators, specifically fermions $\psi^{3}$
and $\psi^{4}$, is grounded in considerations derived from machine learning
and sparsification techniques. Jafferis and his team’s explanation is rooted
in the physical behavior of the system, where they observe that fermions with
slower thermalization achieve size winding at later times, corresponding to
different masses across fermions. This reasoning delves into the fundamental
physics underlying the phenomena. On the other hand, Kobrin et al.’s criticism
revolves around machine learning and sparsification aspects. They point out
that the machine-learning procedure, focused on specific operators such as
$\psi^{1}$ and $\psi^{2}$, might not effectively generalize to untrained
operators like $\psi^{3}$ and $\psi^{4}$. This limitation could affect the
accuracy of the generated Hamiltonian in capturing the behavior of these
untrained fermions during teleportation. Therefore, these two arguments stem
from different perspectives, with Jafferis and his team’s approach drawing on
physics-based reasoning. In contrast, Kobrin et al.’s critique centers around
the machine learning and sparsification aspects of the problem.

Jafferis and his team further point out that the fermions identified in Kobrin
et al.’s comment as having poor size winding at $t_{0}\approx 2.8$ also
experience slower thermalization and operator growth. However, their size
winding occurs later at around $t_{0}\approx 4$ instead of the previously
assumed time of $t_{0}\approx 2.8$ as mentioned in the comment. Jafferis and
his team interpret this delay as being consistent with the notion that these
fermions take longer to traverse the wormhole. In conclusion, Jafferis and his
team establish that, eventually, all fermions achieve size winding, regardless
of initial differences, indicating a consistent pattern of behavior in the
system [8].

As a reminder, Kobrin et al. discovered an issue with small-size fully-
commuting Hamiltonians that consist of only a few terms. They argued that the
perfect-size winding observed in those systems is likely a side effect and
does not indicate a holographic duality to gravity [11]. Jafferis et al.’s
next rebuttal against Kobrin et al.’s arguments concerns the assertion that
the commuting structure of equation (4) is not directly linked to the size
winding properties and dynamics. Jafferis and his team introduced a
significant non-commuting term, denoted as $H_{1}$, to the original equation
(4), which they refer to as $H_{0}$. Therefore, the perturbed Hamiltonian can
be expressed as follows:

$H_{0}+H_{1}=H_{0}+0.3\psi^{1}\psi^{2}\psi^{3}\psi^{5}.$ (10)

Jafferis and his team straightforwardly addressed Kobrin et al.’s issue by
proposing that equation (10) exhibits sufficient size winding, leading to an
asymmetric teleportation signal. According to Jafferis et al., introducing the
non-commuting term, $H_{1}$, generates a behavior that resembles that of the
SYK model. Despite thermalizing later, the perturbed Hamiltonian exhibits
comparable size winding and teleportation behavior with a parameter value of
$\mu=-12$ and occurs approximately at $t_{0}\approx 2.8$. Jafferis et al.
found that the non-commuting perturbation $H_{1}$ has little impact on the
physics during teleportation, indicating that the commuting structure is not
essential for the presence of gravitational physics. Furthermore, similar to
the $N=10$ SYK model, the two fermions, $\psi^{1}$ and $\psi^{2}$, traversing
the wormhole do not display significant revivals over time after the initial
peak. This observation supports their argument and suggests that the behavior
exhibited by equation (10) aligns with the expected gravitational physics [8].

Jafferis et al. believed that they had effectively addressed the concern
raised by Kobrin et al. They thought they had successfully resolved the issue.

As previously mentioned, Kobrin et al. suggested that size winding might
result from the sparsification process rather than an inherent characteristic
of the Hamiltonians. Moreover, the perfect-size winding is commonly observed
in fully commuting Hamiltonians at smaller system sizes [11].

In response to this criticism, a recent paper by Gao proposed that a large-$N$
commuting SYK model, with $q=4$, demonstrates ”pseudo-holographic” features,
including size winding at high temperatures and a teleportation mechanism that
exhibits several similarities to the semiclassical traversable wormhole
teleportation but operates in distinct parameter regimes. The paper also
argued that a commuting SYK model with $N=8$, $q=4$, and six terms also
displays size winding, namely, that the small $N$ Hamiltonian is nearly
identical to the large $N$ case. Gao writes that upon closer examination of
his model and the learned Hamiltonian, it becomes apparent that the underlying
mechanism is not solely attributed to size winding. Remember that Kobrin et
al. proposed a Hamiltonian specifically optimized for achieving the best
teleportation fidelity for a particular operator $\psi_{1}$. However, for
other operators $\psi_{i}$, the quality of size-winding diminishes. In
contrast, Gao comments that his $N=8$ model performs equally well for all
operators $\psi_{i}$, exhibiting a consistent and average level of
teleportation efficiency. Furthermore, he claims that the thermalization of
his Hamiltonian is nearly complete [4].

As mentioned in Sections 3 and 4, high-temperature teleportation occurs for
values of $t$ greater than the scrambling time, this mechanism, unexpected
from gravity, does not involve signals traversing a geometric wormhole.
Finally, Jafferis et al. demonstrated that in the high-temperature regime,
non-gravitational teleportation occurred, but there was no size-winding effect
[7]. In fact, Gao points out that upon closer examination of his model, the
size winding in the large $N$ limit differs significantly from the size
winding in an ordinary SYK model [4]. As such, it remains unclear how Kobrin
et al.’s criticism is refuted in light of this current state of affairs.

There appears to be a complication. Gao notes that the size winding observed
in the large $N$ model differs significantly from that in a typical SYK model.
The operator size distribution in his model is narrowly peaked, indicating
that it does not undergo rapid scrambling like a holographic model would [4].
Rapid scrambling is associated with chaotic behavior. Now, when Gao mentions
that the operator size distribution in his model is ”narrowly peaked” and
”does not scramble as fast as a holographic model,” he is suggesting that the
information in his model does not spread and get entangled as quickly as it
would in a holographic model, such as one with properties analogous to a semi-
classical holographic wormhole. Instead, the information remains more
localized. In other words, Gao’s model possesses ”pseudo-holographic” (or non-
holographic) features and has different properties than the holographic
models.

I, therefore, remain skeptical about the large-$N$ and the $N=8$ commuting
Hamiltonians. Given that the large $N$ model does not display significant
scrambling and chaotic behavior, an $N=8$ model derived from it would likely
also share similar characteristics. Reducing the number of degrees of freedom
from a large $N$ model to $N=8$ simplifies the system and makes it more
tractable. But it seems to me that this reduction also leads to a loss of some
complex behaviors observed in the original large $N$ model, including rapid
scrambling and chaotic dynamics. In this case, the $N=8$ model is not expected
to show rapid scrambling and chaotic dynamics.

Kobrin et al.’s criticism is based on the notion that it is unlikely for a
commuting Hamiltonian at larger system sizes to exhibit perfect-size winding,
similar to what is expected from a semi-classical traversable wormhole arising
from a non-commuting Hamiltonian at low temperatures. The key point here is
that the semi-classical traversable wormhole, associated with large-$N$ non-
commuting Hamiltonians in the context of holography, involves fast scrambling
and non-trivial quantum dynamics, which are typically absent in commuting
systems known for their lack of complexity and chaotic fast scrambling. In
general, a commuting Hamiltonian cannot exhibit chaotic behavior, at least not
in the same way as non-commuting Hamiltonians do, because the system’s time
evolution is integrable.

##  5 Assessing reliability in noisy quantum systems

Jafferis and his team’s quantum circuit was run on the Google Sycamore
processor, which is susceptible to noise. Indeed, Jafferis et al. conducted a
simulation involving inherent noise on the Sycamore quantum processor.

As said in section 3, Jafferis et al. simplified the $N=10$ SYK model using
machine learning techniques while retaining its crucial gravitational
properties. They implemented the sparse model through a quantum circuit that
employed $164$ controlled $Z$ gates 191919A two-qubit gate introduces
entanglement between the qubits, i.e., operates on two qubits simultaneously.
and $295$ single-qubit gates.202020A gate that allows for individual qubit
operations. While a more complex model than the learned Hamiltonian could have
offered enhanced accuracy, it would also entail a higher number of gates and,
consequently, a higher error rate. To enhance the accuracy of the simulation,
Jafferis et al. conducted XEB calibration (Xmon’s Error Model and
Benchmarking). They addressed readout errors212121Inaccuracies in measuring
the quantum state of qubits., specifically focusing on the $9$ qubits region
of the Sycamore quantum processor. The Sycamore chip comprises $72$ qubits;
however, not all qubits exhibit the same levels of noise and errors. Hence,
Jafferis et al. carefully selected the $9$ least noisy qubits on the chip for
their experimentation. They then applied the calibration tools to these $9$
qubits.

Despite the inherent errors and noise present in the Sycamore chip, Jafferis
et al. successfully achieved a teleportation signal on the 9 noisy qubits, as
reported in their _Nature_ paper. In their paper, Jafferis et al. reported
that their approach, which involved using machine learning to simplify the
quantum model while preserving crucial gravitational properties and combining
it with calibration techniques, enabled the successful execution of quantum
teleportation. According to Jafferis et al., their method proved effective
even on qubits characterized by significant noise levels.

However, skeptics could challenge the experiment by proposing that coherent
errors cause the observed teleportation signal. As said above, the Sycamore
processor is prone to errors and can introduce noise and uncertainty into the
experimental results, making it difficult to discern the true quantum
teleportation signal. In their _Nature_ paper, Jafferis and his team rebut
this claim. They have reported their findings, stating that ”device noise is
primarily influenced by an incoherent channel, making it improbable for
coherent errors to imitate the traversable wormhole signal.” On the other
hand, they noted that ”coherent errors are largely overshadowed by the genuine
teleportation signal” [7]. Jafferis et al. contend that despite the presence
of coherent errors, the observable effects of the genuine quantum
teleportation signal are more prominent. In light of this situation, they
assert that they have effectively discerned and differentiated the true
teleportation signal from the surrounding background noise.

If Kobrin et al. deemed the learned Hamiltonian problematic, it could pose
challenges in differentiating between coherent noise and the genuine quantum
teleportation signal. In such a scenario, accurately identifying the true
signal from the background noise becomes more complex, and the observable
effects become less pronounced and more ambiguous. The ability to distinguish
between the two may be compromised due to the issues with the learned
Hamiltonian, leading to uncertainties and potential limitations in Jafferis et
al.’s findings on quantum teleportation through a holographic wormhole.

Moreover, during the training process of the Hamiltonian, the system’s
behavior was approximated using simplified models that cannot fully capture
all the complexities of the actual Sycamore processor. Even if the training
process relies on noisy simulations rather than idealized models of the
Sycamore processor, discrepancies and errors inevitably arise. The problem is
that the learned Hamiltonian probably performs well in simulations but fails
to capture the true behavior of the Sycamore process due to the noise inherent
in the processor not being accounted for during training. Consequently, the
trained model might not be as accurate when deployed on the Sycamore because
of gate parameter fluctuations and environmental noise. As a result, it could
produce inaccurate or inconsistent results on the Sycamore processor.

Thus, due to the presence of noise, we cannot have complete confidence in the
reliability of the processor. Quantum gates are susceptible to errors stemming
from imperfect control over quantum systems and imprecise calibration.
Additionally, quantum measurements can introduce noise and disturb the quantum
state, affecting subsequent computations. Achieving high confidence in the
Sycamore processor poses significant challenges. Noise introduces
uncertainties that limit the reliability of the Sycamore processor [10].

Considering these challenges and the wormhole experiment using only $9$ qubits
without aiming to demonstrate quantum supremacy, opting for a classical
simulation could have been preferable. We can observe why using a classical
simulation is preferable by applying Ian Hacking’s ideas to the wormhole
experiment.

According to Hacking, the term ”model” can sometimes refer to a material model
constructed in a laboratory, essentially a simulation of an experiment. For
instance, scientists might find it helpful to build a desktop model to gain
clearer insights into a particular phenomenon [6].

In the case of the experiment performed by Jafferis et al., they utilized the
Sycamore, a superconducting quantum processor consisting of transmons
(Josephson junctions). By running the experiment on the $9$ qubits, Jafferis
et al. studied how the qubit is transferred from the left side to the right
side, thus deepening their understanding of the teleportation-through-a-
wormhole phenomenon. Implementing the simulation of teleportation between the
left and right sides involves precise control and manipulation of the $9$
qubits (or transmons). It requires carefully designed quantum gates and
methods to prepare, entangle, and measure the qubits appropriately.

According to Hacking, using the material model allows scientists to obtain
accurate inputs and outputs, leading to a better understanding of the
phenomenon and potentially generating ideas for enhancing experiments.
However, Hacking emphasizes that such a model is not an exact representation
of reality; it is not a literal depiction of how things truly exist [6].

In other words, the experiment by Jafferis et al. involves running a quantum
teleportation simulation using $9$ qubits in a superconducting quantum
processor. But it’s essential to understand that such a model does not
accurately represent reality; namely, Jafferis et al.’s simulation does not
literally represent an actual teleportation through a traversable wormhole
event. The quantum teleportation process, as simulated in the laboratory, does
not mean that physical particles are instantaneously moving between the left
and right sides of the processor. A (superconducting) qubit is not a particle.
It typically comprises several physical devices and components working
together to represent and manipulate quantum information (the Josephson
junction222222Which allows the qubit to exhibit quantum coherence and behave
as a two-level quantum system., microwave pulses to manipulate the qubit’s
quantum state, etc.). In other words, the model serves as a tool for
investigation and exploration, but it should not be confused with a genuine
manifestation of a traversable wormhole.

## Acknowledgement

This work is supported by ERC advanced grant number 834735.

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