Qubit-efficient variational algorithm for nuclear structure

This paper compares three qubit-mapping strategies within the Variational Quantum Eigensolver (VQE) framework to study the ground states of $^{10}$B and $^{12}$C nuclei, demonstrating that the Slater determinant (SD) mapping yields the highest accuracy on quantum hardware while the charge-symmetry adapted (cSD) mapping offers superior qubit efficiency for scaling to complex nuclei.

The Gist

Imagine you are trying to solve a massive, incredibly complex jigsaw puzzle. This puzzle represents the "ground state" (the most stable, lowest-energy arrangement) of an atomic nucleus, specifically for elements like Boron-10 and Carbon-12. In the world of physics, figuring out how these tiny particles arrange themselves is like trying to find the single perfect picture among billions of wrong possibilities.

Traditionally, scientists use powerful classical computers to solve this, but as the puzzle gets bigger (more particles), the number of possible arrangements explodes so fast that even the best supercomputers get stuck. This is where Quantum Computing comes in. It's like having a magical new type of puzzle solver that can look at many possibilities at once.

This paper is about testing three different "strategies" or "maps" to translate this nuclear puzzle into a language that a quantum computer can understand. The researchers used a method called VQE (Variational Quantum Eigensolver), which is essentially a trial-and-error process where the computer tweaks its settings until it finds the best solution.

Here is a breakdown of the three strategies they tested, using simple analogies:

The Three Strategies (Mappings)

Think of the quantum computer's "qubits" (its basic units of information) as seats on a bus. The goal is to get all the puzzle pieces (the nuclear states) onto the bus efficiently.

1. The "One-Seat-Per-Piece" Strategy (SD Mapping)

• How it works: Imagine you have 26 puzzle pieces. In this strategy, you assign one specific seat on the bus for every single puzzle piece. If you have 26 pieces, you need 26 seats.
• The Pros: It's very straightforward. The "rules" for how the pieces interact are simple, so the computer doesn't have to do much heavy lifting to calculate the answer. It's like having a very clear, simple instruction manual.
• The Cons: It uses a lot of seats (qubits). If your puzzle gets bigger, you might run out of seats on the bus.
• The Result: When tested on real quantum hardware, this method was the most accurate, missing the perfect answer by only 0.21%. It was the most reliable runner.

2. The "Split-Team" Strategy (pnSD Mapping)

• How it works: This strategy tries to save space by splitting the puzzle into two teams: "Protons" and "Neutrons." Instead of giving every single piece its own seat, it groups them. For the Boron puzzle, this reduced the need from 26 seats down to 20.
• The Pros: It saves space on the bus (fewer qubits).
• The Cons: The instructions for how these teams interact become incredibly complicated and messy. The computer has to perform a huge number of complex steps (gates) to figure out the answer. It's like trying to coordinate a dance between two teams where everyone has to follow a very long, confusing script.
• The Result: Because the instructions were so complex and the hardware is currently a bit "noisy" (like a room with a lot of background chatter), this method struggled the most, with errors around 8.88%.

3. The "Magic Compression" Strategy (cSD Mapping)

• How it works: This is the most innovative approach. Instead of giving every piece a seat, the researchers used a clever trick to "compress" the whole puzzle. They took the 26 pieces and squeezed them into a format that only needed 5 seats (qubits).
• The Pros: It is incredibly efficient with space. It allowed them to study a larger, more complex puzzle (Carbon-12) that would have been impossible to fit on the bus with the other two methods.
• The Cons: Because they squeezed the puzzle so tight, the "instruction manual" became very long and complex. The computer has to adjust many more knobs (parameters) to find the right answer.
• The Result: It performed reasonably well (about 3.37% error for Boron and 6.82% for Carbon). While not as accurate as the first method, it proved that you can solve much bigger problems with very few resources.

The Experiment and Results

The researchers ran these strategies on two types of "test tracks":

1. A Perfect Simulator: A noiseless computer simulation where everything works perfectly.
2. Real Quantum Hardware: They used an actual quantum computer (IBM's ibm_fez) and a noisy simulator that mimics real-world imperfections.

Key Findings:

• Noise is the Enemy: Real quantum computers are currently "noisy," meaning they make small mistakes. The more complex the instructions (like in the pnSD strategy), the more these mistakes add up.
• Error Correction: They used a technique called "Zero-Noise Extrapolation" (ZNE). Imagine taking a blurry photo, taking it again with the camera slightly more blurry, and then using math to guess what the sharp photo would have looked like. This helped clean up the results.
• The Winner: For the smaller puzzle (Boron-10), the "One-Seat-Per-Piece" (SD) strategy was the champion, getting the answer almost perfectly even on real hardware.
• The Future Hope: The "Magic Compression" (cSD) strategy showed great promise. Even though it wasn't the most accurate for the small puzzle, it proved that we can tackle much larger, more complex nuclei (like Carbon-12) without needing a bus with hundreds of seats.

The Bottom Line

This paper is a "stress test" for different ways to talk to quantum computers about atomic nuclei.

• If you want maximum accuracy right now on small problems, use the straightforward SD mapping.
• If you want to solve bigger, harder problems with limited quantum resources, the cSD mapping is the most efficient tool, even if it requires more complex tuning.

The authors conclude that while no single method is perfect yet, the "Magic Compression" (cSD) approach is a promising path forward for solving complex nuclear physics problems on the quantum computers we have today.

Technical Summary

Technical Summary: Qubit-efficient variational algorithm for nuclear structure

Problem Statement
The study of atomic nuclei using the nuclear shell model faces a significant computational bottleneck: the Hilbert space size grows combinatorially with the number of nucleons, rendering many systems intractable for classical computers. While quantum computing offers a pathway to simulate these strongly entangled states efficiently, current Noisy Intermediate-Scale Quantum (NISQ) devices are limited by qubit count and circuit depth. Previous work by the authors demonstrated that mapping many-body configurations directly to qubits could yield low-depth circuits but at the cost of requiring a large number of qubits, limiting scalability. This work addresses the trade-off between qubit efficiency and circuit depth by comparing three distinct mapping strategies for calculating nuclear ground states using the Variational Quantum Eigensolver (VQE).

Methodology
The authors investigate the ground states of two mid-p-shell nuclei, $^{10}\text{B}$ ($3^+$) and $^{12}\text{C}$ ($0^+$), using the CKpot effective interaction. They compare three strategies for mapping the many-particle basis (Slater Determinants, SDs) to the qubit Hilbert space:

1. Slater Determinant (SD) Mapping: Each of the $N_{SD}$ many-particle states is mapped to a single qubit. For $^{10}\text{B}$, this results in a 26-qubit system. The Hamiltonian is converted to Pauli strings where each qubit represents a full SD, naturally accounting for antisymmetry. This approach yields a simple, low-depth trial wavefunction but requires the maximum number of qubits.
2. Proton-Neutron Slater Determinant (pnSD) Mapping: The SDs are decomposed into proton and neutron components. Unique proton SDs and neutron SDs are assigned to separate qubits. For $^{10}\text{B}$, this reduces the system to 20 qubits (10 proton + 10 neutron). The Hamiltonian is mapped using the Jordan-Wigner transformation. This reduces qubit count but increases circuit depth due to the need for controlled excitation gates and a larger number of commuting Pauli groups.
3. Compact Slater Determinant (cSD) Mapping: This strategy aims for maximum qubit efficiency. The Hamiltonian matrix dimension is padded to the nearest power of two (e.g., 26 $\to$ 32 for $^{10}\text{B}$, 51 $\to$ 64 for $^{12}\text{C}$) and converted to a Pauli string Hamiltonian using Qiskit. This results in a 5-qubit system for $^{10}\text{B}$ and a 6-qubit system for $^{12}\text{C}$. The trial wavefunction uses a hardware-efficient ansatz (alternating layers of $RY$ and CNOT gates) rather than physics-inspired excitation operators. This minimizes qubits and gate count but requires a significantly higher number of variational parameters.

The VQE optimization was performed using the SLSQP optimizer on a noiseless statevector simulator to determine optimal parameters. These fixed-parameter circuits were then executed on a noisy simulator (IBM's FakeFez backend) and real quantum hardware (ibm_fez). Zero-Noise Extrapolation (ZNE) was applied to mitigate errors by scaling the noise level of two-qubit gates.

Key Results

• Resource Counts: The SD mapping required 26 qubits for $^{10}\text{B}$ but only 4 commuting Pauli groups for energy measurement. The pnSD mapping required 20 qubits but 530 commuting groups. The cSD mapping required only 5 qubits for $^{10}\text{B}$ and 6 for $^{12}\text{C}$, with 122 and 365 commuting groups, respectively.
• Accuracy on Hardware:
  • For $^{10}\text{B}$, the SD mapping yielded the most accurate post-error-mitigated result on hardware, with a percent error of 0.21% relative to the exact shell model result.
  • The cSD mapping for $^{10}\text{B}$ resulted in a 3.37% error.
  • The pnSD mapping for $^{10}\text{B}$ showed a 8.88% error.
  • For $^{12}\text{C}$, which was computationally prohibitive for SD and pnSD mappings on classical simulators, the cSD mapping achieved a 6.82% error on hardware.
• Wavefunction Fidelity: For the cSD mapping, the fidelity of the VQE wavefunction sampled on hardware (without error mitigation) was 0.942 for $^{10}\text{B}$ and 0.915 for $^{12}\text{C}$ compared to the exact shell model wavefunctions.

Significance and Claims
The paper claims that while the SD mapping offers superior accuracy on current hardware due to lower circuit depth and fewer commuting groups, the cSD mapping is the most promising strategy for scaling to complex nuclei. The cSD approach demonstrates the ability to define the ground state of $^{12}\text{C}$ using only 6 qubits, a task that would require 51 qubits for the SD mapping and 30 for pnSD, placing the latter beyond the reach of standard classical statevector simulators.

The authors conclude that the cSD mapping, despite requiring more variational parameters, holds promise for describing complex nuclear systems across different mass regions on NISQ hardware due to its qubit efficiency. They also note that the SD and pnSD mappings remain valuable for future quantum devices and for initial state preparation in other quantum algorithms where individual Slater determinants can be controlled by single parameters. The work establishes a comparative framework for selecting mapping strategies based on the specific constraints of qubit availability, circuit depth, and the computational cost of classical simulation.