Measuring out-of-time-order correlators on a quantum computer based on an irreversibility-susceptibility method

This paper presents the first experimental demonstration of the irreversibility-susceptibility method (ISM) for measuring out-of-time-order correlators (OTOCs) on a trapped-ion quantum computer, alongside a comparative analysis of ISM with the rewinding time and weak-measurement methods to evaluate their effectiveness in probing quantum information scrambling.

The Gist

The Big Picture: The "Spaghetti" of Information

Imagine you drop a single drop of bright red ink into a glass of clear water. At first, you can see exactly where the drop is. But as time passes, the water swirls, and the ink spreads out until the entire glass is a uniform pink color. You can't find the original drop anymore; the information about its specific location has been scrambled throughout the whole glass.

In the quantum world, this happens with information. When a tiny piece of data (like the spin of one particle) interacts with its neighbors, it doesn't just disappear; it gets tangled up with the entire system so thoroughly that looking at just one part of the system tells you nothing about the original data. This process is called Quantum Information Scrambling.

Scientists want to measure exactly how fast and how well this scrambling happens. To do this, they use a mathematical tool called the OTOC (Out-of-Time-Ordered Correlator). Think of the OTOC as a "scramble-meter."

The Problem: The "Time-Travel" Paradox

Measuring this "scramble-meter" is incredibly hard. In a normal experiment, you watch a movie forward. But to measure the OTOC, the math requires you to watch the movie forward, then backward, then forward again, all while checking specific details.

In the real world, you can't easily run a quantum system backward in time. It's like trying to un-mix a cake batter or un-break an egg. Doing this on a quantum computer usually introduces so many errors that the result is garbage.

The Solution: Three New Ways to Measure

The authors of this paper didn't just try to force the computer to run backward. Instead, they tested three different strategies (protocols) to measure the scramble-meter on a real quantum computer (a trapped-ion machine called Reimei).

Think of these three methods as three different ways to figure out how messy a room got without having to clean it up and re-mess it up perfectly.

1. The Rewinding Time Method (RTM)

• The Analogy: Imagine you have a magic camera that can record a video, then play it backward perfectly, then forward again.
• How it works: This method tries to literally reverse the clock on the quantum computer. It runs the system forward, applies a "twist," runs it backward, and checks if the system returns to its original state.
• The Catch: It's very sensitive. If the "rewind" button is even slightly sticky (due to computer errors), the measurement gets distorted. In the paper, this method worked well at the start but got "noisy" and inaccurate as time went on, especially in complex systems.

2. The Weak-Measurement Method (WMM)

• The Analogy: Imagine trying to see how much a spiderweb is vibrating without touching it. If you poke it hard, you change the vibration. So, you use a very gentle breeze (a "weak" touch) to feel the vibration.
• How it works: Instead of a full backward time-travel, this method uses a series of tiny, gentle "nudges" (measurements) on the system. It collects a lot of these tiny nudges to build a picture of the scrambling.
• The Catch: It avoids the hard "rewind" problem, but because the nudges are so gentle, you have to do the experiment thousands of times to get a clear signal. In the paper, this method worked well early on but started drifting upward (getting too high) in its readings later on.

3. The Irreversibility-Susceptibility Method (ISM) — The Star of the Show

• The Analogy: Imagine you drop a glass on the floor. It shatters. You can't put it back together perfectly (that's irreversibility). This method asks: "How hard is it to fix the glass after it was dropped?" The harder it is to fix, the more "scrambled" the information was.
• How it works: This is the new method the authors are proud of. Instead of trying to reverse time, they ask: "If we disturb the system slightly and then try to recover it, how much information is lost?" They measure the "difficulty" of recovery.
• The Result: This was the first time this method was tested on a real quantum computer with a complex system (a chain of magnetic spins). It was surprisingly robust! It didn't get confused by the complexity of the system as much as the other two methods.
• The Catch: Because it relies on those "gentle nudges" (weak interactions), the signal is very faint. It's like trying to hear a whisper in a noisy room. The data had a lot of "static" (statistical noise), so the lines on their graphs were a bit wobbly, but the average result was very accurate.

What Did They Find?

The researchers ran these three methods on a simulated chain of magnetic atoms (an XXZ spin chain) to see how they behaved.

1. They all worked: They proved that current quantum computers are powerful enough to study these complex, chaotic phenomena.
2. They behaved differently: This is the big discovery. Even though they were measuring the same thing, the three methods gave slightly different results as time went on.
   • The Rewinding method underestimated the scrambling later on.
   • The Weak-Measurement method overestimated it.
   • The Irreversibility method stayed true to the theory but had "wobbly" data due to noise.

Why Does This Matter?

This paper is like a "User Manual" for the future of quantum physics.

• Validation: It shows that we can use quantum computers to study "Quantum Chaos" (how information spreads), which is crucial for understanding black holes, superconductors, and the fundamental nature of reality.
• The Toolkit: It tells scientists, "Hey, if you want to measure scrambling, you have three tools. Here is the pro and con of each."
  • Use Rewinding if you need a quick, direct look but the system is simple.
  • Use Weak-Measurement if you want to avoid time-reversal but can handle lots of data collection.
  • Use Irreversibility if you want a robust, thermodynamic approach, even if you need to run the experiment many times to clear the noise.

In short, the authors successfully tested a new way to measure how quantum information gets lost in the shuffle, proving that even with today's imperfect quantum computers, we can start to understand the deep secrets of how the universe scrambles its data.

Technical Summary

1. Problem Statement

The Out-of-Time-Ordered Correlator (OTOC) is a fundamental diagnostic tool for quantifying quantum information scrambling and diagnosing quantum chaos in many-body systems. It measures how local information spreads irreversibly across a system, analogous to the Lyapunov exponent in classical chaos.

However, experimentally measuring the OTOC is notoriously difficult because its definition involves time-reversed evolution (backward time evolution), which is hard to implement with high fidelity on noisy, intermediate-scale quantum (NISQ) hardware. While various protocols exist (e.g., interferometry, weak measurements), there is a lack of comprehensive experimental comparison of these methods on real hardware, particularly regarding how the choice of protocol influences the measured results in non-trivial many-body systems.

2. Methodology

The authors conducted experiments on the Quantinuum reimei trapped-ion quantum computer to evaluate the OTOC of an XXZ spin-1/2 chain prepared in a thermal Gibbs state. They implemented and compared three distinct protocols:

A. System Setup

• Hamiltonian: A 1D Heisenberg XXZ model in a transverse magnetic field ($H$) with 4 qubits ($n=4$).
• State Preparation: Since the Gibbs state ($\rho = e^{-\beta H}/Z$) is a mixed state, the authors used a Variational Quantum Algorithm (VQA). This algorithm prepares a purified Thermofield Double (TFD) state on an enlarged system (Target + Ancilla) by minimizing the Helmholtz free energy, avoiding direct entropy measurement.
• Observables: Local Pauli-x operators ($W(0) = V(0) = \sigma_x^1$) were used to probe the scrambling dynamics.

B. The Three Protocols

1. Rewinding Time Method (RTM):

   • Principle: Uses an interferometric setup with an ancillary control qubit. It creates a superposition of two evolution paths: one where the operator $V$ acts before time evolution $U$, and another where it acts after.
   • Mechanism: Requires forward evolution ($U$), backward evolution ($U^\dagger$), and controlled operations. The OTOC is extracted by measuring the expectation values of Pauli operators on the control qubit.
   • Challenge: Requires precise implementation of time reversal ($U^\dagger$), which is sensitive to gate errors.

2. Weak-Measurement Method (WMM):

   • Principle: Relates the OTOC to a sequence of weak measurements and time evolutions, circumventing the need for explicit time reversal.
   • Mechanism: Involves a sequence of four weak interactions (coupling the system to ancillas) interleaved with forward and backward time evolutions. The OTOC is derived from the weighted average of measurement outcomes.
   • Advantage: Can utilize projective measurements if operators are dichotomic (like Pauli matrices), reducing the need for strictly "weak" coupling, though it requires many measurement shots.

3. Irreversibility-Susceptibility Method (ISM):

   • Principle: A novel approach proposed in a companion letter, linking the OTOC to the irreversibility of a quantum process. It quantifies how difficult it is to recover an initial state after a weak perturbation and a scrambling process.
   • Mechanism:
     1. Apply a weak interaction $U_{V,\theta}$ on the system and an ancilla.
     2. Apply the scrambling process $W(\tau)$.
     3. Apply a recovery map (inverse of the weak interaction).
     4. Measure the distinguishability (irreversibility) of the ancilla state.
   • Key Feature: The OTOC is proportional to the second-order term of the generated irreversibility ($\lim_{\theta \to 0} \frac{\delta^2}{\theta^2}$). It requires only a single final measurement on the ancilla, avoiding complex interferometry.

3. Key Contributions

• First Experimental Demonstration of ISM: This paper presents the first experimental realization of the Irreversibility-Susceptibility Method on a trapped-ion quantum computer for a non-trivial many-body system (the XXZ chain).
• Comprehensive Comparative Analysis: The study provides the first side-by-side experimental comparison of RTM, WMM, and ISM on the same hardware and system, revealing that measured OTOC values depend on the measurement protocol used.
• Gibbs State Preparation: Successfully demonstrated the preparation of thermal Gibbs states on a quantum computer using a variational algorithm, a prerequisite for studying finite-temperature dynamics.

4. Results

The experiments were performed for various anisotropy parameters ($\Delta$) and inverse temperatures ($\beta$).

• General Agreement: All three methods showed good overall agreement with theoretical predictions and noiseless simulations, validating the feasibility of OTOC measurement on current hardware.
• Method-Dependent Deviations:
  • RTM: At early times, results matched theory well. However, at later times, for larger anisotropy ($\Delta = 0.5, 0.7, 0.9$), the measured OTOC deviated downward from theoretical predictions. This is likely due to errors in the backward time evolution ($U^\dagger$).
  • WMM: Similar to RTM, early-time results were accurate. However, at later times for larger $\Delta$, the values deviated upward.
  • ISM: The mean values tracked the theoretical predictions well across all $\Delta$ values. However, the standard deviation was significantly larger than in the other methods.
• Noise Analysis: The large variance in ISM is intrinsic to the method. Because the signal scales quadratically with the weak coupling strength ($\theta^2$) while shot noise scales as $1/\sqrt{N}$, the Signal-to-Noise Ratio (SNR) is inherently suppressed. In this experiment ($\theta=0.4, N=1000$), statistical noise masked subtle parameter dependencies that were visible in RTM and WMM.

5. Significance and Conclusion

• Validation of Protocols: The work validates RTM, WMM, and ISM as practical tools for exploring quantum chaos on near-term hardware, despite their respective limitations (gate fidelity for RTM, shot noise for ISM).
• New Insight into Scrambling: The discovery of method-dependent behaviors in the measured OTOC is a critical finding. It suggests that the "true" OTOC of a system may be obscured by the specific noise characteristics and approximations inherent to the measurement protocol.
• Future Directions: The authors conclude that to fully leverage these methods, future work must:
  1. Scale up to larger qubit numbers to probe the onset of many-body chaos more directly.
  2. Improve the fidelity of variational Gibbs state preparation.
  3. Implement advanced error mitigation techniques.
  4. Increase measurement shots significantly for ISM to overcome the $\theta^2$ suppression of the signal.

This paper establishes a practical framework for future experimental investigations into quantum information scrambling, highlighting both the potential and the current limitations of NISQ devices in measuring fundamental quantum chaos metrics.