Scattering phase shift in quantum mechanics on quantum computers

This paper investigates the feasibility of extracting infinite-volume scattering phase shifts on quantum computers using a one-dimensional trapped system model, demonstrating that while the method yields accurate results with two qubits on IBM hardware, it currently fails with three qubits due to gate operation and thermal relaxation errors.

The Gist

Imagine you are trying to understand how two billiard balls bounce off each other. In the real world, you'd just watch them collide in a huge, empty room. But in the quantum world, things are tricky. You can't just "watch" them easily, and if you try to simulate this on a supercomputer, the math gets so heavy it breaks the machine.

This paper is about a team of scientists trying to solve this problem using Quantum Computers—machines that use the weird rules of quantum physics to do calculations. They are testing a new method to figure out how particles scatter (bounce) off each other, but they are doing it in a very small, controlled "playground" first.

Here is the breakdown of their journey, using some everyday analogies:

1. The Goal: The "Infinite Room" Problem

In physics, we want to know how particles behave in an infinite universe (an infinite room). But quantum computers (and even supercomputers) can only handle a finite box (a small room with walls).

• The Problem: When you put a particle in a small box, it bounces off the walls. This changes how it moves, making it look like it's behaving differently than it would in an infinite room. It's like trying to hear a whisper in a small, echoey bathroom; the sound bounces around and gets distorted.
• The Solution: The authors use a clever mathematical trick called the Integrated Correlation Function (ICF). Think of this as a special "noise-canceling headphone" algorithm. It takes the messy, bouncing data from the small box and mathematically filters out the "wall echoes" to reveal what the particle would have done in the infinite room.

2. The Test Drive: A One-Dimensional World

Before trying this on complex real-world particles (like protons), they tested it on a 1D Quantum Model.

• The Analogy: Imagine a bead sliding on a single wire. It can only move left or right. It's the simplest possible version of a particle.
• The Interaction: They added a "contact potential," which is like a tiny, invisible bump on the wire. When the bead hits the bump, it scatters. Because this is a simple math problem, they already knew the exact answer. This allowed them to check if their quantum computer was telling the truth.

3. The Hurdle: The "Jittery" Signal

Here is where it gets tricky. When they ran the simulation in real-time (simulating time moving forward), the data didn't look smooth.

• The Analogy: Imagine trying to take a photo of a hummingbird's wing. If your camera shutter is too slow, the image is a blur. If it's too fast, you get a jittery, vibrating mess. The quantum data was vibrating so fast (oscillating) that it was hard to read the actual signal.
• The Fix: They proposed two ways to smooth out the jitter:
  1. The "Magic Lens" ($E + i\epsilon$): Adding a tiny bit of imaginary math to the energy to blur the sharp edges of the data, making it easier to read.
  2. The "Imaginary Room" ($L \to iL$): Instead of simulating a real box, they simulated a box made of "imaginary numbers." This turns the vibrating, chaotic signal into a smooth, calm curve that is much easier to analyze.

4. The Hardware Reality Check: The "Noisy Kitchen"

This is the most critical part of the paper. They tried to run these calculations on real IBM Quantum Computers (the kind available to the public today).

• The 2-Qubit Success: When they used 2 qubits (the basic units of quantum memory, like bits in a regular computer), it worked! The results matched the theory perfectly. It was like successfully baking a single cookie in a kitchen.
• The 3-Qubit Failure: When they added just one more qubit (making it 3 total), the experiment collapsed. The results became random noise.
  • Why? Quantum computers today are like a kitchen with a very shaky table, a wobbly oven, and a chef who gets tired after 5 seconds.
  • The Culprits:
    1. Gate Errors: Every time the computer performs an operation (a "gate"), there's a small chance it makes a mistake. With 3 qubits, you need more operations, and the mistakes pile up like a Jenga tower falling over.
    2. Thermal Relaxation: Quantum states are fragile. They are like a spinning top; if you don't keep spinning it, it falls over. The "top" in the quantum computer falls over (loses its data) because the machine isn't cold or stable enough to keep it going long enough to finish the calculation.

5. The Conclusion: A Stepping Stone

The authors conclude that while the math (the recipe) is perfect and the theory works, the hardware (the kitchen) isn't ready yet.

• The Takeaway: They successfully built the blueprint for how to extract scattering data from quantum computers. They proved that with 2 qubits, it's possible. But to do it with 3 or more, we need quantum computers that are much more stable and less "noisy."
• The Future: This work is a "training wheels" experiment. Once the hardware improves, this same method can be used to study complex nuclear physics, helping us understand how atoms stick together or how stars explode, all by simulating them on a quantum computer.

In a nutshell: They built a perfect map for a treasure hunt (the math), tested it on a small island (2 qubits) and found the treasure. But when they tried to sail to a bigger island (3 qubits), their boat sank because the ocean was too rough (hardware errors). They now know exactly what kind of boat they need to build for the next trip.

Technical Summary

Here is a detailed technical summary of the paper "Scattering phase shift in quantum mechanics on quantum computers" by Peng Guo et al.

1. Problem Statement

The extraction of infinite-volume scattering phase shifts from finite-volume simulations is a fundamental challenge in nuclear and particle physics, particularly for Lattice Quantum Chromodynamics (LQCD). Traditional methods rely on determining the discrete energy spectrum of a system in a finite box and applying quantization conditions (e.g., the Lüscher method). However, these approaches face significant hurdles:

• Computational Cost: They require high-performance computing and often struggle with the "signal-to-noise" (S/N) problem, especially at large Euclidean times.
• Real-Time Dynamics: Classical computers cannot efficiently simulate real-time quantum dynamics due to the sign problem and exponential scaling.
• Quantum Hardware Limitations: While quantum computers offer a pathway to simulate real-time dynamics, current Noisy Intermediate-Scale Quantum (NISQ) devices suffer from gate errors, thermal relaxation, and limited qubit counts, making the extraction of physical observables difficult.

The authors aim to investigate the feasibility of using Integrated Correlation Functions (ICF) to extract scattering phase shifts directly from real-time quantum simulations on current hardware, bypassing the need for explicit energy spectrum determination.

2. Methodology

The study employs a one-dimensional (1D) quantum mechanical model with a contact interaction potential ($V(x) = V_0\delta(x)$), which possesses exact analytic solutions for validation.

Theoretical Framework: ICF Formalism

The core method relates the integrated correlation function of a trapped system to the infinite-volume scattering phase shift $\delta(\epsilon)$ via a weighted integral:
$$C(t) - C_0(t) \xrightarrow{trap \to \infty} \frac{it}{\pi} \int_0^\infty d\epsilon \, \delta(\epsilon) e^{-i\epsilon t}$$
where $C(t) = \text{Tr}[e^{-i\hat{H}t}]$ is the integrated correlation function.

• Real-Time Challenge: In real-time evolution, $C(t)$ exhibits rapid oscillations due to the discrete energy spectrum of the finite trap, making direct extraction of $\delta(\epsilon)$ difficult.
• Post-Data Analysis Solutions: To mitigate oscillations, the authors propose two analytical transformations:
  1. $E + i\epsilon$ Prescription: Adding a small imaginary part to the energy to smooth poles.
  2. $L \to iL$ Rotation: Rotating the box size to the imaginary axis. This transforms the Hamiltonian into a non-Hermitian form but effectively smooths the correlation function, allowing for direct phase shift extraction via the Friedel formula (Krein's theorem):
     $$\text{Tr}\left[\frac{1}{E-\hat{H}} - \frac{1}{E-\hat{H}_0}\right] \xrightarrow{trap \to \infty} -\frac{d}{dE} \ln T(E)$$
     where $T(E)$ is the transmission amplitude related to the phase shift.

Quantum Implementation

• Discretization: The 1D space is discretized into $N$ grid points, mapped to $\Gamma = \log_2 N$ qubits.
• Hamiltonian Mapping: The Hamiltonian is split into kinetic and potential terms.
  • Coordinate Basis: Mapped using tight-binding models and Z-gate insertions for the potential.
  • Momentum Basis: Mapped using diagonal matrices and constant matrices, utilizing X-gate insertions.
• Circuit Construction:
  • Time evolution $e^{-i\hat{H}t}$ is implemented using Trotterization.
  • The Integrated Correlation Function $C(t)$ is computed using the Hadamard test with an ancilla qubit to measure the trace (sum of diagonal elements).
• Hardware Testing: The circuits were executed on IBM Quantum Processors (QPUs) using 1, 2, and 3 qubits (including the ancilla).

3. Key Contributions

1. Validation of ICF on Quantum Hardware: The paper provides the first realistic test of the ICF formalism on NISQ devices, demonstrating the workflow from Hamiltonian construction to phase shift extraction.
2. Post-Processing Techniques: It rigorously evaluates and compares the $E+i\epsilon$ and $L \to iL$ rotation methods for handling the fast oscillatory nature of real-time correlation functions, showing that $L \to iL$ offers superior smoothing for phase shift extraction.
3. Hardware Error Analysis: A comprehensive noise analysis isolates the dominant error sources. The study identifies that two-qubit gate errors and thermal relaxation ($T_1, T_2$) are the primary bottlenecks, causing rapid decoherence that destroys the signal even with error mitigation techniques like mthree.
4. Scalability Roadmap: The authors provide a clear pathway for extending these circuits to scalar field theories (e.g., $\phi^4$ theory) by adapting the qubit basis and interaction terms.

4. Results

• Single-Qubit (1 Regular + 1 Ancilla): The results on IBM hardware showed good agreement with exact analytic solutions. The simple circuit depth allowed the signal to survive the noise.
• Two-Qubit (2 Regular + 1 Ancilla): Results were consistent with exact solutions on simulators but failed completely on actual IBM hardware. The measured expectation values collapsed rapidly toward zero, becoming indistinguishable from random noise after just a few Trotter steps.
  • Error Mitigation: Standard readout error mitigation (mthree) provided no improvement, confirming the errors were not merely measurement artifacts but dynamic decoherence.
  • Noise Simulation: Simulations using Qiskit Aer revealed that:
    • Readout errors (<5%) and single-qubit gate errors (<0.1%) had negligible impact.
    • Two-qubit gate errors (0.25%–1%) and thermal relaxation were the dominant factors. The circuit depth required for Trotterization exceeded the coherence times of current devices.
• Phase Shift Extraction: While the $L \to iL$ method theoretically allows for direct extraction, the current hardware noise levels prevent the accumulation of sufficient coherent evolution time to resolve the phase shift for systems larger than 2 qubits.

5. Significance and Outlook

• Feasibility Limit: The study concludes that while the ICF formalism is theoretically sound and pedagogically valuable, current quantum hardware is not yet capable of simulating scattering processes for systems requiring more than 2 regular qubits due to the "circuit depth vs. coherence time" trade-off.
• Hardware Requirements: To make this approach viable, two-qubit gate errors must be reduced to ~0.01% or lower, and coherence times must be significantly extended (or gate durations shortened to ~50 ns).
• Algorithmic Improvements: The authors suggest that higher-order Suzuki-Trotter decompositions or Qubitization techniques might reduce circuit depth and improve fidelity.
• Future Applications: The framework established here serves as a baseline for more complex simulations, such as few-body scattering in $\phi^4$ scalar field theory, which is a stepping stone toward Lattice QCD applications.

In summary, the paper successfully demonstrates the theoretical viability of the ICF method for scattering on quantum computers but highlights the severe practical limitations imposed by current NISQ hardware noise, specifically two-qubit gate fidelity and thermal relaxation.