*This report aims to introduce quantum computing students at a high level to recent advancements
in quantum gravity research conducted on quantum computers. I am a student who only
learned these topics recently, but I have tried to explain as much as I understand.*

# Introduction

For almost a century physicists have attempted to connect the physics of relativity, how large masses shape space and time, with quantum mechanics, the laws of physics on the atomic scale. String theory is one proposition that holds much promise. It is a theory that unites quantum mechanics and gravity. However, there is little way to verify its findings experimentally. One significant obstacle in trying to verify string theory is the extremely limited ability to measure the force of gravity on subatomic particles.

A group of scientists decided to take a different approach and explore the AdS/CFT conjecture in string theory, which proposes that quantum systems with lots of entanglement lead to quantum gravity. In 2022, researchers published a paper in Nature **[1]** detailing an experiment conducted on a Google’s Sycamore quantum computer which claimed to have created a quantum mechanical system with wormhole-like properties, an object typically associated with lots of gravitational effects. This was an unexpected publication because scientists thought that it would be impossible to simulate a wormhole for at least another decade. Many scientists have agreed that in order to indisputably prove that a wormhole was created on a quantum computer, the computer would need at least 100 qubits to simulate the entrance and exit of a traversable wormhole. The number of qubits is not the only issue. The number of quantum gates necessary to configure the proper interactions between particles grows exponentially. In order to describe the physics of the two 100 qubit entangled systems of particles, millions of quantum gates are necessary **[2]**. A full scale simulation is far out of reach for today’s quantum computers.

The recent paper does not attempt to simulate a full-blown wormhole, but a sparsified model capable of being run on today’s quantum computers. Using machine learning, the researchers identified the terms of the Hamiltonian, the operator that describes the energy levels of qubits, that contributed most to wormhole dynamics. The researchers claimed that even with only 9 qubits and 164 two qubit gates they were able to observe key characteristics of a wormhole. One of which is known as size-winding.

This claim has seen met with much pushback regarding the characteristics of the learned Hamiltonians. Another group of scientists **[3]** have rebutted that the scrambling and unscrambling behavior was not because of wormhole dynamics, but because of commutativity of the chosen Hamiltonian terms. The machine learning algorithm purposefully rewarded Hamiltonians that contributed the most to wormhole dynamics to ease simulation, but also resulted in commuting terms. The other group argues that the model does not accurately capture the chaotic scrambling dynamics of an actual wormhole.

Despite the inconsistencies that need to be resolved, this moment marks a significant moment in quantum gravity research. This is one of the first attempts at demonstrating experimental evidence for quantum gravity. Even though the methods have not been undeniably demonstrated, they have opened the door to future experimental probing of quantum gravity.

# Wormholes

A traversable wormhole can be understood as a connection between two points in space. This creates a shorter path between two locations than if you were to travel the full length.

At the moment, wormholes are a theoretical construct. One reason why this may be the case is that traversable wormholes are unstable and collapse in on themselves shortly after appearing. It has been shown theoretically that if matter with negative energy, known as “exotic matter”, were to exist, it could prop open a wormhole and allow it to be traversable **[4]**. However, it is not yet known if exotic matter is possible.

While large amounts of matter with negative energy may not exist, on the quantum mechanical level, there are momentary violations of the Null Energy condition **[4]**. This condition states that, the energy density along any macroscopic region must be greater than or equal to 0. So a violation of this condition means that there are momentary pockets of negative energy experienced at the quantum scale which could be exploited for quantum-scale wormholes.

# Quantum Gravity

The study of holography in the context of string theory allows physicists to view space-time and gravity as a duality, that the two are essentially equivalent. A specific duality known as the AdS/CFT correspondence conjecture is a bridge for physicists to understand anti-de Sitter (AdS) spaces and conformal field theories (CFT). The current argument of the AdS/CFT correspondence is that large quantum systems with enough entanglement can result in emergent behavior such as gravity.

Anti-de Sitter space is one of three solutions to Einstein’s field equations which dictates the shape of space-time depending on the distribution of mass. AdS is a particular space used to formulate a quantum theory of gravity. Meanwhile conformal field theories, specifically the quantum field theories, are an extremely successful set of theories that describe the behavior of subatomic particles.

Solution to Einstein’s equations of space-time depends on the cosmological constant of the universe **[5]**. When the constant is positive, a positive constant curvature of space exists in the universe (like a sphere), known as de-Sitter Space. This is the universe that we live in. Then there is Minkowsi space which is a plane with no curvature as a result of zero cosmological constant. And finally, there is anti-de Sitter (AdS) space which has a constant negative curvature (hyperbolic plane).

This is the first visible limitation of the current string theory formulation of quantum gravity. So far, the current formulation of the AdS/CFT correspondence only works for AdS space and not the type of space that reality actually is (de Sitter). So the wormhole created in this paper is not a real wormhole in this sense, but it does allow the transportation of quantum information.

# SYK model

Not every quantum system has a gravitational dual. Only a few quantum systems have been proven to have one. One such system is known as the Sachdev-Ye-Kitaev (SYK) model. The SYK model has been shown to be an exactly solvable problem that demonstrates quantum many-body chaos. Many body chaos is a sign of quantum gravity because it is closely related to how fast gravity can scramble information.

There is a fundamental limit to how fast information can spread and scramble throughout a system. Black holes are a phenomenon that reaches this upper limit of scrambling as a result of its extreme relativistic effects. The SYK model has been shown to also reach the universal speed limit, which makes it an attractive candidate for quantum gravity research **[6]**. It was theorized that by linking together two SYK models, you can create a traversable wormhole **[6]**. Compared to most other nascent applications of a quantum computer, one major advantage of quantum gravity research is that models like SYK do not have to be fine-tuned in order to have appreciable results on a noisy quantum computer **[7]**.

An interesting side note is that the only difference between thermodynamic entropy and Shannon entropy is the unit of measure. Thermodynamic entropy is measured in terms of energy divided by temperature while Shannon entropy is measured in bits. This is why we can talk about “information” in the context of quantum systems and black holes, since they are analogous in this regard.

# A Primer on Schrodinger’s Equation

In quantum computing we concern ourselves with operators that do not depend on time. If we had a computer whose calculation depended on time, this would result in non-deterministic computation. This is generally an undesirable quality and especially relevant to quantum computers since they appear to exploit random processes for computation.

Quantum mechanical processes depend on time. This is the relation that Schrodinger’s equation conveys. It is a partial differential equation that predicts how a quantum system $\ket{\psi}$ will evolve.

$$i\hbar \frac{\partial}{\partial t}\ket{\psi(t)}=\hat{H}\ket{\psi(t)}$$

To make sense of this Schrodinger’s equation, we must examine what is known as the Hamiltonian. It is a linear operator that describes the total kinetic $(\hat{T})$ and potential energy $(\hat{V})$ of a quantum system.

$$\hat{H}=\hat{T}+\hat{V}$$

A more intuitive understanding of the Hamiltonian arises when looking at it from its spectral decomposition.

$$\hat{H}=\sum_{\psi \in \text{energy levels}} E_{\psi} \ket{\psi} \bra{\psi} $$

Where $E_{\psi}$ is the energy level of a particular outcome state $\ket{\psi}$.

The eigenvectors of the Hamiltonian represent linearly independent states of a quantum system, ones we can actually observe, such as $\ket{0}$ and $\ket{1}$. Meanwhile, the eigenvalue represents the total energy of the quantum system for each independent state/eigenvector. If you apply the Hamiltonian operator to one of its eigenvectors $\ket{\Psi}$, by definition, this is the same as scaling an observable state by a constant factor.

$$\hat{H}\ket{\Psi} =E_{\Psi}\ket{\Psi} \text{ where } E_{\psi} \text{ is the energy for observable } \ket{\Psi}$$

This indicates that if the particle is at a superposition of eigenstates, $\ket{\Psi}=\ket{\psi_{1}}+\ket{\psi_{2}}$, applying the Hamiltonian will give the total energy of the superposition.

$$\hat{H}\ket{\Psi}=\hat{H}\ket{\psi_{1}}+\hat{H}\ket{\psi_{2}}=\text{ total energy}$$

So looking at Schrodinger’s equation again, it says the change in a quantum state at any given time is proportional to the total energy of the state at that time.

$$i\hbar \frac{\partial}{\partial t}\ket{\psi(t)}=\hat{H}\ket{\psi(t)}$$

The solutions to Schrodinger’s equations are waves that change with time, known as the wave function. There are in fact solutions to Schrodinger’s equation that do not depend on time. The quantum states that satisfy this criteria are the eigenvectors of a Hamiltonian that are constant **[8]**. These are known as stationary states and they are the solutions to the time-independent Schrodinger’s equation **[9]**. The particle itself is not stationary. It is constantly fluctuating as time passes, but the probability distribution of the particle’s properties like spin, position, velocity, remains constant. The wave function that describes a particle over time looks like a standing wave.

If we solve Schrodinger’s equation using the time-independent solution, we see that the state of a particle does change with time, only along the phase.

$$\begin{aligned} i\hbar \frac{\partial}{\partial t}\ket{\Psi}& = \hat{H}\ket{\Psi} \\ \frac{\partial}{\partial t}\ket{\Psi}& = \frac{1}{i\hbar} E_{\Psi}\ket{\Psi} \\ \frac{\partial}{\partial t}\ket{\Psi} &= -\frac{i}{\hbar} E_{\Psi}\ket{\Psi} \\ \ket{\Psi(t)} &=e^{-iE_{\psi}t/\hbar}\ket{\Psi(0)} \end{aligned}$$

We can verify that the probability distribution has no relationship with time despite the state itself changing with time. Let’s say we are representing the wave function of a particle whose observable is its position $x$. This would be denoted $\ket{\Psi(x, t)}$. If the Hamiltonian is constant the state of the particle still changes with time, but the probability of the particle being at position $x$ is

$$\begin{aligned}
P(x)&=\braket{\Psi(x,t) | \Psi(x,t)}=\ket{\Psi(x,t)}\ket{\Psi(x,t)}^{*} \\
&= (e^{-iE_{\psi}t/\hbar})(e^{iE_{\psi}t/\hbar})\braket{\Psi(x,0) | \Psi(x,0)} \\
&=\braket{\Psi(x,0) | \Psi(x,0)}
\end{aligned}$$
tum states over time (red and blue are the real and imaginary values of the state) and the probability distribution (black). Adapted from **[8]**

Using Schrodinger’s equation for evolving a time independent quantum state, we will define the time evolution operator $\hat{U}(t)=\exp({-i\hat{H}t/\hbar})$. Or the reverse time evolution operator $\exp({i\hat{H}t/\hbar})$.

# Traversable Wormhole Circuit

The process for creating a circuit that has wormhole behavior is as follows **[7]**.

**1) Prepare the TFD state**

First, prepare $n$ qubits on the “left” and $n$ qubits on the “right” for a total of $2n$ qubits. The left $\ket{E_{j}}_{L}$ and right $\ket{E_j}_R$ systems are initially entangled to the thermofield double (TFD) state.

$$\ket{TFD}=\frac{1}{\sqrt{ Tr(e^{-\beta H}) }} \sum_{j \in \text{energy levels}}e^{-\beta E_{j}/2} \ket{E_j}_L\otimes \overline{\ket{E_j}_R}$$

The TFD is used because it allows treating a mixed state $\rho=e^{-\beta H}$ as a pure state $\ket{TFD}\bra{TFD}$ in a larger dimensional system **[10]**. If we trace out the right system, so we only pay attention to the left system $\rho_{L}=tr_{R}(\ket{TFD}\bra{TFD})$, we get our original mixed state $\rho=e^{-\Beta H}$ . This means that there is fundamentally no difference between the higher dimensional pure state and the lower dimensional mixed state. When considering the gravitational dual, the TFD state is used because of its ability to be injected with negative energy **[7]**.

**2) Move back in time and insert the message**

We first devolve the left qubit system backwards in time using $exp(i\hat{H}t/\hbar}$ the reverse time evolution operator $\exp(i \hat{H}_Lt_L)$. This is so that we can mimic inserting our message in the past.

We have a $m$ qubit message $\psi_{in}$ (where $m \ll n$ ) which we inject into the left system. This means we can throw away/ignore the original qubits and replace them with our message qubits using the SWAP gate.

**3) Evolve the right system forward in time**

Next, we evolve the right system forward in time using $\exp({-i\hat{H}t/\hbar})$. This has the effect of scrambling the information of the qubits onto the total qubits of the left system.

**4) Couple the left and the right systems after insertion**

Now we focus on qubits $n-m$ that were not replaced with the message on both the left and the right system, known as the carrier qubits, $B_L$ and $B_R$ respectively). For those qubits only, apply $\exp(igV)$ where

$$V=\frac{1}{n-m} \sum_{i\in \text{ carrier qubits}} (\sigma_z)_{i}^L (\sigma_z)_i ^R$$

**5) Evolve the right system forward in time**

Apply the forward time evolution to the right system. This is what allows the message to cross the boundary of the “right” side of the wormhole and emerge unscrambled.

# Penrose Diagram Interpretation

The Penrose diagram is a diagram to visualize space and time and is often used to understand the behavior of wormholes and black holes. Along the x-axis is the spatial dimension while the time dimension is along the y-axis. Moving to the top-right represents an infinitely far future in space and time, while moving to the bottom-left is an infinitely distant past.

The Penrose diagram indicates what futures are reachable and which ones aren’t. The fundamental assumption is that nothing can travel faster than light. If we traced out every point in space that was reachable by a photon across time, this would create a “light cone”. Anything on the interior of the light cone is a place/moment in space-time that is considered reachable from the present. However, anything outside this light cone is not physically possible to be reached, or else you would have to travel faster than the speed of light to get there.

In general, light is represented by a 45 degree ray. Any angle greater than 45 degrees means you travel slower than light. Any angle less than 45 degrees means you travel faster than light.

The Penrose diagram presented by the paper depicts a wormhole with two openings, a left (L) and a right (R). The right horizon (45 degree line) represents the untraversable barrier between the left wormhole and the right wormhole.

The red star indicating message insertion is equivalent to steps (1-3) in the traversable wormhole circuit. Step 4, which is coupling the left and the right side, is equivalent to the negative energy line in the diagram. This coupling results in a negative energy shockwave which allows the message to cross the boundary/horizon of the right opening of the wormhole.

If we inserted the message into our system and didn’t create a negative energy shockwave by coupling with $exp(igV)$, this would have kept the particle on the left opening of the wormhole, eventually hitting the singularity (top border) and disappearing (Fig. 6).

# Quantum Teleportation versus Quantum Wormhole

This “teleportation” of information might sound like the standard quantum teleportation protocol. After all, we are still transferring quantum information. There is, however, a subtle distinction between the two. What makes the traversable wormhole protocol different is that it does not require any post-processing for the sending or receiving party **[7]**. The message simply appears onto the receiving end of the system. We do not require Alice to communicate classically to Bob what controlled operations to apply.

# Conclusion

The interest in this protocol doesn’t come from the pure quantum interpretation of this experiment. It is possible to generate other such quantum circuits with similar properties of scrambling and unscrambling. What makes this particular configuration interesting is that this quantum system has a known gravitational dual according to String Theory. This system can be visualized as two entangled black holes such that there is a traversable wormhole between the two. It’s also important to make clear that a real wormhole was not actually created in the lab. This was only a highly tailored quantum system that was selected because of its interesting consequences theoretically.

This report only scratches the surface with the work that is being done in this field. It was quite challenging to research this report and I hope to have conveyed the complexities and depth accurately. It is likely in the coming years we will continue to hear more about wormholes and their quantum duals as quantum computing hardware continues to improve. It is a very exciting time to be involved with quantum gravity research and I am excited to see what else comes into fruition.

**If any inaccuracies are spotted or you have some feedback, please contact me here**

# References

- D. Jafferis et al., “Traversable wormhole dynamics on a quantum processor,” Nature, vol. 612, no. 7938, Art. no. 7938, Dec. 2022, doi: 10.1038/s41586-022-05424-3.
- N. Wolchover, “Physicists Create a Holographic Wormhole Using a Quantum Computer,” Quanta Magazine, Nov. 30, 2022. https://www.quantamagazine.org/physicists-create-a-wormhole-using-a-quantum-computer-20221130/ (accessed Apr. 23, 2023).
- B. Kobrin, T. Schuster, and N. Y. Yao, “Comment on ‘Traversable wormhole dynamics on a quantum processor.’” arXiv, Feb. 15, 2023. doi: 10.48550/arXiv.2302.07897.
- “Wormhole,” Wikipedia. Mar. 18, 2023. Accessed: May 04, 2023. [Online]. Available: https://en.wikipedia.org/w/index.php?title=Wormhole&oldid=1145369264
- “Anti-de Sitter space,” Wikipedia. May 03, 2023. Accessed: May 04, 2023. [Online]. Available: https://en.wikipedia.org/w/index.php?title=Anti-de_Sitter_space&oldid=1153047383
- I. Fadelli and Phys.org, “Researchers gather numerical evidence of quantum chaos in the Sachdev-Ye-Kitaev model.” https://phys.org/news/2021-02-numerical-evidence-quantum-chaos-sachdev-ye-kitaev.html (accessed Apr. 29, 2023).
- A. R. Brown et al., “Quantum Gravity in the Lab: Teleportation by Size and Traversable Wormholes.” arXiv, Feb. 01, 2021. doi: 10.48550/arXiv.1911.06314.
- “Stationary state,” Wikipedia. Nov. 05, 2022. Accessed: May 03, 2023. [Online]. Available: https://en.wikipedia.org/w/index.php?title=Stationary_state&oldid=1120227666
- “Schrödinger equation,” Wikipedia. Feb. 05, 2023. Accessed: May 04, 2023. [Online]. Available: https://en.wikipedia.org/w/index.php?title=Schr%C3%B6dinger_equation&oldid=1137575570
- Tom Hartman, “Eternal Black Holes and Entanglement,” Quantum Gravity and Black Holes. http://www.hartmanhep.net/topics2015/ (accessed May 04, 2023).
- “Penrose diagram,” Wikipedia. Mar. 22, 2023. Accessed: May 04, 2023. [Online]. Available: https://en.wikipedia.org/w/index.php?title=Penrose_diagram&oldid=1146133345
- Understanding Black Holes with Penrose Diagrams, (Dec. 17, 2018). Accessed: Apr. 30, 2023. [Online Video]. Available: https://www.youtube.com/watch?v=4aId5rHdcdM
- “Light cone,” Wikipedia. Mar. 10, 2023. Accessed: May 04, 2023. [Online]. Available: https://en.wikipedia.org/w/index.php?title=Light_cone&oldid=1143868744

# Holographic Wormholes on a Quantum Circuit

## Inducted May 9, 2023

An introductory report for quantum computing students