Solution of the Equation-of-Motion Phonon Method eigenvalue problems on the D-Wave quantum annealer

This paper presents a hybrid quantum-classical algorithm combining quantum annealing and classical deflation to iteratively solve large-scale eigenvalue problems from the Equation-of-Motion Phonon Method on D-Wave quantum hardware, demonstrating both the potential and current limitations of near-term quantum devices for nuclear many-body theory.

The Gist

Imagine you are trying to solve a massive, incredibly complex puzzle. In the world of nuclear physics, this puzzle is figuring out how protons and neutrons (nucleons) behave inside an atomic nucleus. The "pieces" of this puzzle are arranged in a giant mathematical grid called a Hamiltonian matrix. The bigger the nucleus, the more pieces there are, and the grid becomes so huge that even the world's fastest supercomputers struggle to find all the solutions (called eigenvalues and eigenvectors) in a reasonable time.

This paper presents a new way to solve these puzzles by teaming up a classical computer with a special type of quantum computer called a quantum annealer (specifically, a D-Wave machine).

Here is a breakdown of their approach using simple analogies:

1. The Problem: A Mountain Range of Solutions

Think of the energy states of a nucleus as a vast, foggy mountain range.

• The Goal: You want to find the lowest valley (the ground state) and then all the other valleys and peaks (excited states) in order.
• The Old Way (Classical Computers): Traditional algorithms are like a hiker who carefully checks every single step, one by one. They are good, but when the mountain range gets huge, the hiker gets tired, runs out of time, or gets stuck in a local dip thinking it's the bottom.
• The Quantum Way (Quantum Annealing): Imagine a magical fog that can instantly "feel" the shape of the entire mountain range at once. A quantum annealer is like a hiker who can tunnel through the fog to find the lowest points much faster than a human could.

2. The Strategy: Turning the Puzzle into a Binary Game

Quantum annealers don't understand complex math equations directly. They speak a simpler language: 0s and 1s (like a light switch being off or on).

• The Translation (QUBO): The authors had to translate the complex nuclear physics equations into a "Quadratic Unconstrained Binary Optimization" (QUBO) problem. Think of this as converting a complex recipe into a simple checklist of "on/off" switches. The quantum machine then tries different combinations of switches to find the one that gives the best (lowest energy) result.

3. The Innovation: Peeling an Onion (Deflation)

The biggest challenge is that quantum annealers are currently best at finding just one solution (the absolute lowest valley). But scientists need the entire list of solutions, not just the first one.

• The Solution: The authors created a "hybrid" method.
  1. Step 1: They use the quantum annealer to find the first solution (the lowest energy).
  2. Step 2: They use a classical computer to perform a "deflation." Imagine you found the lowest valley in the mountain range. To find the next lowest valley, you temporarily fill the first one with concrete so the hiker can't go back there.
  3. Step 3: They send the "filled-in" map back to the quantum annealer to find the next lowest spot.
  4. Repeat: They keep peeling the onion layer by layer until they have found the whole spectrum of solutions.

4. The Results: Speed and Accuracy

The team tested this method on a real quantum computer (D-Wave Advantage) and compared it to a standard classical simulation (Simulated Annealing).

• The Race: They set up a race between the "Quantum Hiker" and the "Classical Hiker" to solve puzzles of different sizes.
• The Outcome:
  • For small puzzles, both were okay.
  • For larger, more complex puzzles, the Quantum Hiker was significantly faster. In some cases, the classical method took hundreds of steps to get close to the answer, while the quantum method got there in just a few dozen steps.
  • The quantum method reached a higher level of precision (accuracy) much quicker.

5. The Catch: Not All Tools Work for Every Job

The paper also tested three different ways to "fill in" the valleys (deflation techniques):

• Hotelling and Orthogonal Projection: These worked well. They successfully helped the quantum machine find the next solution without messing up the math.
• Householder: This method worked great for simple puzzles but started to break down when the puzzles got complex (specifically for "Generalized" eigenvalue problems). It was like trying to use a sledgehammer to fix a watch; it worked for the big picture but introduced errors that made the later steps inaccurate.

Summary

The paper doesn't claim to have solved nuclear physics forever. Instead, it proves that near-term quantum computers (the ones we have right now, which are noisy and imperfect) can be useful partners. By combining the speed of quantum annealing for finding the best answers with the reliability of classical computers for organizing the search, they created a method that is faster and more accurate than using classical computers alone for these specific, massive nuclear puzzles.

It's a proof-of-concept that shows we are moving closer to using quantum machines for real-world physics problems, even before we have perfect, error-free quantum computers.

Technical Summary

Technical Summary: Solution of the Equation-of-Motion Phonon Method Eigenvalue Problems on the D-Wave Quantum Annealer

Problem Statement
The nuclear many-body problem involves solving eigenvalue equations for Hamiltonian matrices that grow exponentially with the number of nucleons, often reaching dimensions that are computationally prohibitive for exact diagonalization. While classical iterative methods (e.g., Lanczos, Davidson) exploit sparsity, they face storage bottlenecks for large-scale calculations. Quantum computing offers a potential alternative; however, the Quantum Phase Estimation (QPE) algorithm, while theoretically powerful, requires fault-tolerant hardware currently unavailable. Consequently, there is a need for near-term strategies capable of addressing large-scale eigenvalue problems on Noisy Intermediate-Scale Quantum (NISQ) devices. Specifically, this work addresses the challenge of computing the full eigenspectrum (not just the ground state) for both Standard (SEVP) and Generalized (GEVP) eigenvalue problems arising from the Equation of Motion Phonon Method (EMPM) in nuclear structure theory.

Methodology
The authors propose a hybrid quantum-classical algorithm that integrates quantum annealing with classical matrix deflation techniques. The core methodology involves three main components:

1. QUBO Formulation: The continuous optimization of the Rayleigh quotient is discretized and mapped to a Quadratic Unconstrained Binary Optimization (QUBO) problem. Real variables representing the eigenvector components are approximated using binary vectors with a specific precision vector. This allows the eigenvalue problem to be solved by finding the global minimum (or maximum magnitude) of the resulting quadratic form on a quantum annealer.
2. Iterative Descent and Initial Guess: The algorithm employs a two-phase approach. First, an initial guess for the eigenvector is generated by solving a shifted QUBO problem. Second, an iterative descent phase refines this solution by minimizing a local quadratic approximation of the objective function (involving the gradient and Hessian) encoded as a QUBO. The discretization grid is progressively scaled down to increase precision.
3. Matrix Deflation for Full Spectrum: To extract the full eigenspectrum rather than just the dominant eigenpair, the authors implement a sequential deflation strategy. After a quantum annealer identifies an eigenpair, classical deflation techniques are applied to the Hamiltonian matrix to "peel away" the influence of the found state, allowing the next iteration to target the subsequent eigenstate. Three deflation methods are evaluated:
   • Hotelling Deflation: A subtractive method ($A' = A - \lambda vv^T$).
   • Orthogonal Projection: A subtractive method projecting the matrix onto the subspace orthogonal to found eigenvectors.
   • Householder Deflation: An orthogonalization-based method using Householder reflections to transform the matrix.

For Generalized Eigenvalue Problems (GEVPs), the Rayleigh quotient is modified to include the metric matrix $B$, and the deflation steps are adapted to preserve the $B$-inner product structure where possible.

Key Results
The method was benchmarked on Hamiltonian matrices derived from EMPM calculations for the $^4$He nucleus, utilizing the D-Wave Advantage System 6.4 (5,614 qubits) and classical Simulated Annealing (SA) for comparison.

• Ground State Performance: Quantum Annealing (QA) demonstrated superior convergence compared to SA. For larger matrix dimensions (e.g., $d=25$) and higher bit resolutions ($b=4$), QA achieved machine-precision accuracy ($10^{-8}$) in approximately 20–30 iterations, whereas SA required significantly more iterations (up to 1,200) or failed to reach the same accuracy. The performance gap widened as matrix size increased.
• Full Spectrum and Deflation:
  • SEVP: All three deflation methods (Hotelling, Orthogonal Projection, Householder) successfully recovered the full eigenspectrum with high accuracy (up to 13 correct digits for $b=2$ and 8 for $b=4$). Householder deflation was the most computationally efficient in terms of execution time due to its strategy of reducing the active submatrix size.
  • GEVP: While Hotelling and Orthogonal Projection methods maintained accuracy, Householder deflation proved unstable for GEVPs. The repeated application of Householder transformations distorted the symmetry of the metric matrix $B$ and amplified rounding errors, leading to numerical deterioration and failure to converge for closely spaced eigenvalues.

Significance and Claims
The paper claims to provide a practical strategy for exploiting near-term quantum hardware in nuclear structure calculations. The primary significance lies in demonstrating that a hybrid QUBO-based framework, combined with classical deflation, can systematically reconstruct the full eigenspectrum of realistic nuclear Hamiltonians, moving beyond the limitation of ground-state-only calculations common in current variational quantum algorithms.

The authors highlight that the approach successfully bridges the gap between quantum algorithms and noisy hardware by recasting eigenvalue problems into QUBO formulations solvable by existing annealers. While the study is modest regarding the scale of problems (limited by current qubit counts and connectivity), it establishes the viability of quantum annealing for spectral estimation in nuclear physics. The work suggests that future implementations could further reduce classical overhead by encoding the deflation step itself as a QUBO problem, aiming for a fully quantum annealing-based spectral solver.