Thermodynamic-limit dispersion relations on trapped-ion quantum hardware

This paper demonstrates the feasibility of computing thermodynamic-limit ground-state energies and quasi-particle dispersions for the transverse-field Ising model using a 20-qubit trapped-ion quantum processor within a numerical linked-cluster expansion framework, while addressing the challenges of noise amplification during non-linear classical post-processing through novel techniques like the CX-test.

The Gist

The Big Picture: Solving a Giant Puzzle with Tiny Pieces

Imagine you are trying to understand how a massive, infinite crowd of people behaves. You can't possibly watch everyone at once; the crowd is too big, and the interactions are too complex.

In the world of physics, this "infinite crowd" is called the thermodynamic limit. Scientists want to know how particles interact in an infinite material, but classical computers (the kind we use today) hit a wall when trying to simulate these huge, strongly connected systems. They get stuck in the math.

This paper describes a new way to solve this problem using a quantum computer (a special machine that uses the laws of quantum physics to calculate). However, instead of trying to simulate the whole infinite crowd at once—which is impossible for today's small quantum computers—the researchers used a clever strategy called Numerical Linked-Cluster Expansion (NLCE).

The Analogy:
Think of the infinite crowd as a giant mosaic. Instead of trying to paint the whole thing at once, the researchers paint tiny, separate tiles (small clusters of particles). They then use a special mathematical recipe to stitch these tiles together to predict what the whole infinite picture looks like.

The Challenge: The "Noise" in the Room

The researchers used a real quantum computer (a 20-qubit trapped-ion machine) to paint these tiny tiles. But there's a catch: current quantum computers are "noisy." It's like trying to paint a masterpiece in a room where the lights are flickering, and the wind is blowing the paint around.

The specific problem they tackled was that their mathematical recipe requires non-linear post-processing.

• Simple Analogy: Imagine you measure the temperature of a cup of coffee. That's a simple number. But if you need to calculate the square root of that temperature, or divide one measurement by another, small errors in your initial measurement get blown up massively.
• The Paper's Claim: The researchers asked: "Is our quantum computer accurate enough to give us numbers that won't explode when we do these tricky math operations later?"

The Tools They Used

To make this work, they combined a few different techniques:

1. The "Cluster Solver" (VQE vs. ASP):
   To paint the tiny tiles, they used two different methods:

   • VQE (Variational Quantum Eigensolver): Think of this as a student taking a test. The computer tries a solution, gets graded, learns from the mistake, and tries again until it gets the best answer.
   • ASP (Adiabatic State Preparation): Think of this as slowly turning a dial. You start with a simple system and very slowly change it into the complex one you want.
   • Result: The "student" (VQE) did a better job than the "slow dial" (ASP) on this specific hardware, likely because the slow dial took too long and got too confused by the noise.

2. The "PCAT" (The Glue):
   Once they had the data from the tiny tiles, they needed to glue them together so they wouldn't fall apart. They used a method called PCAT.

   • The Metaphor: Imagine you have two separate Lego structures. If you just tape them together, they might wobble. PCAT is a special glue that ensures the combined structure acts exactly like the two separate structures would if they were part of a giant, infinite Lego set. It involves some heavy math (matrix inversion and square roots) that amplifies any noise.

3. The "CX-Test" (A Smarter Way to Measure):
   Usually, to get the data needed for this math, scientists use a standard measurement tool called the "Hadamard test." The authors realized this tool was too heavy and complicated for their noisy machine.

   • The Innovation: They invented a simpler tool they call the CX-test.
   • The Analogy: If the Hadamard test is like using a heavy, industrial crane to lift a feather, the CX-test is like using a pair of tweezers. It's much lighter, faster, and less likely to knock things over, but it still gets the job done. They found that for their specific math problem, they didn't need to measure a "global" value, so they could skip the heavy lifting entirely.

What They Found

The team tested this on a model called the Transverse-Field Ising Model (a standard test for quantum physics) in three different shapes: a line, a line with a twist, and a ladder.

• The Good News: In many cases, the noisy quantum computer produced data that, after the tricky math was applied, looked very close to the correct answer. The "CX-test" worked well, and the "VQE" method was robust enough to handle the noise.
• The Bad News: The "slow dial" method (ASP) was too deep and noisy to work well yet. Also, when they tried to break a specific symmetry in the physics (adding a longitudinal field), the math became so sensitive to noise that the current computer couldn't see the tiny corrections needed.
• The Conclusion: The paper proves that current quantum hardware is just barely good enough to handle these complex, non-linear math problems. The results aren't perfect, but they are "recognizable" and approach the right behavior.

The Bottom Line

The researchers successfully demonstrated that you can use a small, noisy quantum computer to solve problems about infinite systems, provided you use a smart "tile-based" strategy (NLCE) and a lighter measurement tool (CX-test).

They showed that while today's machines are still a bit shaky, they are reaching a point where they can provide data accurate enough for complex classical math to turn it into useful scientific predictions. It's a proof-of-concept that we are on the right track to using quantum computers for real-world physics problems that classical computers simply cannot solve.

Technical Summary

Technical Summary: Thermodynamic-limit dispersion relations on trapped-ion quantum hardware

Problem Statement
Computing properties of quantum many-body systems in the thermodynamic limit is a central challenge in condensed matter physics, particularly for strongly correlated systems where classical methods face fundamental limitations. While quantum computers offer a potential pathway, current Noisy Intermediate-Scale Quantum (NISQ) devices are constrained by qubit counts and gate fidelities, restricting direct calculations to small system sizes far from the thermodynamic limit. Furthermore, extracting thermodynamic-limit properties often requires non-linear classical post-processing (such as matrix inversion and square roots) of quantum expectation values. A critical open question is whether current hardware can provide expectation values with sufficient accuracy to withstand these non-linear operations without the errors being amplified to the point of rendering the results useless.

Methodology
The authors propose and implement a framework combining Numerical Linked-Cluster Expansions (NLCE) with Quantum Algorithms (QA), termed NLCE+QA. This approach computes ground-state energies and one quasi-particle (1QP) dispersions in the thermodynamic limit using data from small clusters.

• Framework: The method decomposes extensive quantities into contributions from connected clusters. To ensure the convergence of the NLCE, the effective Hamiltonian must be "cluster-additive." The authors employ the Projective Cluster-Additive Transformation (PCAT) to enforce this property. PCAT involves constructing a modified unitary that decouples the 1QP sector from other sectors while preserving the product structure of states on disconnected clusters.
• Quantum Algorithms: Two algorithms are explored for preparing the necessary states on the quantum hardware:
  1. Adiabatic State Preparation (ASP): A unitary transform connecting the undressed Hamiltonian eigenstates to the dressed ones.
  2. Variational Quantum Eigensolver (VQE): A hybrid approach using a Hamiltonian Variational Ansatz (HVA) trained classically to block-diagonalize the Hamiltonian.
• Measurement Schemes:
  • Ground State Energy: Computed using Sample-based Quantum Diagonalization (SQD).
  • Matrix Elements: The authors introduce the CX-test, a novel alternative to the standard Hadamard test. The CX-test measures overlap and Hamiltonian matrix elements using controlled-X gates, avoiding the need for deep controlled-unitary circuits. Crucially, in the PCAT context, the global prefactor required by the CX-test cancels out in the final effective Hamiltonian calculation, eliminating the need for the expensive Hadamard test entirely.
• Hardware: Experiments were conducted on a 20-qubit trapped-ion Quantum Processing Unit (QPU) (AQT Marmot) at the Leibniz Supercomputing Centre.

Key Contributions

1. First Hardware Implementation: This work presents the first hardware implementation of the NLCE+VQE framework and introduces the NLCE+ASP framework for computing 1QP dispersions in the thermodynamic limit.
2. CX-test: The introduction of the CX-test as a more efficient alternative to the Hadamard test for computing polynomial numbers of matrix elements in the context of PCAT, significantly reducing circuit depth.
3. Error Analysis: A detailed investigation into the interplay between shot noise, hardware depolarization noise, and the non-linear post-processing inherent in PCAT (matrix inversion/square roots). The authors demonstrate that non-linear functions amplify noise, creating an upward bias in dispersion curves.
4. Model Studies: The framework is applied to the Transverse-Field Ising Model (TFIM) in three regimes:
   • 1D chain (pure TFIM).
   • 1D chain with a longitudinal field (TFIM+LF), which breaks $Z_2$ symmetry and requires the full PCAT correction.
   • 2D ladder geometry.

Results

• Performance: The VQE-based approach generally outperformed ASP on current hardware due to the shallower circuit depth required. The ASP circuits were found to be too deep for the current noise levels, leading to significant deviations from analytic solutions.
• Accuracy: For the 1D TFIM in the polarized phase (sufficiently far from the quantum critical point), the experimental dispersions obtained from the QPU approached the expected analytic behavior. The results were more accurate than noisy simulations using reference depolarization rates, suggesting the actual device error rates were lower than the reference values.
• Limitations:
  • As the system approaches the quantum critical point ($J/h \to 1$) or includes a longitudinal field, the error increases. The non-linear post-processing amplifies errors, and the full PCAT correction (required for TFIM+LF) could not be decisively tested because the correction terms were below the sampling noise floor ($\Delta_{ij} \ll 1/\sqrt{M}$).
  • The ladder geometry (2D) showed larger errors than the 1D chain, attributed to higher qubit connectivity and the system being closer to its critical point relative to the cluster size used.
• Noise Sensitivity: The study confirms that non-linear post-processing is highly sensitive to noise. While the dispersion relations were recognizable, they did not perfectly overlap with analytic solutions, indicating that current hardware precision is at the threshold of viability for such methods.

Significance and Claims
The paper claims that its work opens the door to performing thermodynamic-limit computations with spatial correlations on quantum devices. It demonstrates that, in limited regimes (specifically the polarized phase of the TFIM), the non-linear post-processing pipeline is viable on current hardware, producing recognizable dispersion relations.

The authors assert that quantum hardware precisions have reached a stage where it is meaningful to develop new algorithms involving non-linear classical post-processing of quantum data. They conclude that while current results are not yet beyond classical reach, a reduction in error rates by a factor of 10 to 100 would enable accurate computation of dispersion relations using ASP and allow for the testing of more complex corrections (like the full PCAT for symmetry-broken systems). The work highlights that hybrid approaches pairing quantum measurements with non-linear classical post-processing offer a promising path forward as hardware improves and the transition to fault-tolerance broadens.