Improving Ground State Accuracy of Variational Quantum Eigensolvers with Soft-coded Orthogonal Subspace Representations

This paper proposes a Variational Quantum Eigensolver (VQE) approach that improves ground state accuracy by replacing hard-coded orthogonality constraints with soft-coded penalty terms, enabling shallower quantum circuits while maintaining high fidelity across benchmark spin-glass models.

The Gist

Imagine you are trying to find the lowest point in a vast, foggy mountain range. This is the job of a Variational Quantum Eigensolver (VQE). In the world of quantum computing, finding this "lowest point" (the ground state) is crucial for solving problems like designing new medicines or creating better batteries.

However, the fog is thick, and the terrain is tricky. The current methods often get stuck in small dips that look like the bottom but aren't the true lowest point.

This paper introduces a clever new trick to help the quantum computer find the true bottom faster and more accurately, using a method called Soft-Coded Orthogonal Subspace Representations.

Here is the breakdown using simple analogies:

1. The Old Way: The "Solo Hiker" vs. The "Rigid Team"

Traditionally, there are two main ways to tackle this problem:

• The Solo Hiker (Standard VQE): You send one explorer (a quantum circuit) to find the bottom. They are flexible, but they can easily get lost or settle for a shallow dip because they are looking at the problem from only one angle.
• The Rigid Team (Hard-Coded Subspace Methods like SSVQE/MCVQE): Instead of one hiker, you send a team of explorers. To make sure they don't all walk the same path and get stuck in the same spot, you tie them together with rigid steel ropes.
  • The Problem: These "steel ropes" (hard-coded orthogonality constraints) force the team to stay perfectly apart. While this sounds good, it makes the team stiff and clumsy. They can't move fluidly, and the "ropes" require a lot of extra energy and complex maneuvers (deep quantum circuits) to maintain. In the noisy, fragile world of current quantum computers (called NISQ), this extra complexity often causes the team to stumble before they even reach the bottom.

2. The New Way: The "Soft-Constraint Team"

The authors propose a third option: The Soft-Constraint Team.

Instead of tying the explorers with rigid steel ropes, you give them a gentle, invisible magnetic repulsion.

• How it works: You tell the explorers, "Please try to stay apart from each other, but don't worry if you get a little close." You add a small "penalty" to their score if they get too close, but you don't force them to be perfectly orthogonal (90 degrees apart) at every single step.
• The Result: Because they aren't tied down by rigid rules, the team can move much more freely and fluidly. They can explore the terrain with shallower, simpler paths (simpler quantum circuits).

3. Why This Matters: The "Shallow Circuit" Advantage

Think of quantum computers like a delicate glass sculpture. The longer you hold it up (the deeper the circuit), the more likely it is to crack due to noise and interference.

• The Rigid Team needs a very deep, complex circuit to maintain their strict formation. This increases the chance of the "glass" cracking (errors) before they finish.
• The Soft-Constraint Team can use a shallow, simple circuit. They don't need to be perfect at every step; they just need to be "mostly" apart. This allows them to reach the solution with less risk of breaking, making them much more reliable on today's imperfect quantum hardware.

4. The "Magic Finish"

Once the team has explored the area and found a good spot, they don't just pick the person who is lowest. Instead, they look at the entire group they formed.

• They take all the explorers' positions and mathematically mix them together (like blending colors) to create a "super-explorer" that is an even better approximation of the true bottom than any single person could find alone.
• The paper shows that this "Soft" method finds a much better "super-explorer" than the "Rigid" method, even when using simpler tools.

5. The Test Drive

The authors tested this on two difficult "mountain ranges":

1. The Ising Model: A standard, somewhat predictable mountain range.
2. The Spin-Glass Model: A chaotic, messy mountain range with random bumps and holes (very hard to navigate).

The Results:

• The Solo Hiker often got stuck.
• The Rigid Team did slightly better but struggled with the complexity of their own rules.
• The Soft-Constraint Team consistently found the deepest points, often with 97%+ accuracy, while using much simpler circuits. They were especially good at navigating the chaotic "Spin-Glass" terrain where other methods failed.

The Bottom Line

This paper suggests that in the world of quantum computing, flexibility is better than rigidity. By replacing strict, hard rules with gentle, "soft" penalties, we can guide quantum computers to find better solutions faster, using less energy, and with fewer errors. It's like teaching a team to dance: if you force them to hold a perfect pose, they might trip; if you just ask them to keep a little distance, they can move gracefully and find the rhythm much easier.

Technical Summary

1. Problem Statement

The Variational Quantum Eigensolver (VQE) is a leading algorithm for finding the ground state of quantum systems on Noisy Intermediate-Scale Quantum (NISQ) devices. However, standard VQE (using a single parameterized state) often struggles with:

• Expressibility limitations: Shallow circuits may not capture the complex entanglement required for accurate ground states.
• Non-monotonic convergence: The fidelity of the estimated ground state can decrease even as the cost function (energy) decreases, often due to the variational state getting "stuck" in regions overlapping with low-lying excited states rather than the true ground state.
• Subspace methods' constraints: Existing subspace-based approaches (like SSVQE and MCVQE) improve accuracy by searching a $K$-dimensional subspace. However, these methods enforce hard-coded orthogonality between the basis states via circuit structure. This constraint restricts the expressibility of the ansatz, often requiring deeper circuits to achieve high fidelity, which is detrimental on noisy hardware.

2. Methodology

The authors propose a novel Soft-coded Orthogonal Subspace Representation that relaxes the strict orthogonality constraints found in traditional subspace methods.

Core Concept

Instead of enforcing orthogonality through the circuit architecture (hard-coded), the authors enforce it via penalty terms in the cost function (soft-coded).

• Hard-coded (Baseline): Uses a single unitary $U(\theta)$ applied to $K$ orthogonal initial states. Orthogonality is guaranteed by design but limits the search space.
• Soft-coded (Proposed): Uses $K$ independent parameterized circuits $\{U_p(\theta^{(p)})\}$ applied to a single initial state $|0\rangle$. The states are not guaranteed to be orthogonal by construction.

Cost Function

The optimization minimizes a cost function $C_K(\theta)$ comprising two parts:

1. Average Energy: The sum of expectation values of the Hamiltonian $H$ for the $K$ states.
2. Orthogonality Penalty: A term proportional to the squared overlaps between distinct states, controlled by a hyperparameter $\beta$:
   $$C_K(\theta) = \sum_{p=0}^{K-1} \langle \psi_p | H | \psi_p \rangle + \beta \sum_{p<q} |\langle \psi_q | \psi_p \rangle|^2$$
   This allows the optimizer to explore a broader, more expressive parameter space while gently pushing states toward orthogonality.

Post-Processing (Diagonalization)

Once optimization converges:

1. Overlap Matrix: Unlike hard-coded methods where the overlap matrix $S$ is the identity, the soft-coded method requires estimating the overlap matrix elements $S_{pq} = \langle \psi_p | \psi_q \rangle$ using generalized Hadamard tests.
2. Generalized Eigenvalue Problem: The ground state is found by solving $H_{trc} \mathbf{c} = \lambda S \mathbf{c}$, where $H_{trc}$ is the Hamiltonian restricted to the subspace. The lowest eigenvalue $\lambda$ estimates the ground state energy, and the eigenvector $\mathbf{c}$ provides the linear combination of the $K$ states.

Key Assumptions

The study focuses on "agnostic" ansätze that:

• Do not inject prior knowledge of Hamiltonian symmetries (crucial for disordered systems like spin glasses).
• Keep the circuit structure fixed during optimization (no sequential growth of the subspace).

3. Key Contributions

1. Soft-Coded Orthogonality: Introduction of a subspace representation where orthogonality is a soft constraint in the cost function rather than a hard circuit constraint. This increases the expressibility of the variational search space.
2. Shallow Circuit Advantage: Demonstration that soft-coded representations achieve higher fidelities with shallower circuits compared to hard-coded subspace methods and standard VQE. This is critical for NISQ devices where circuit depth is limited by decoherence.
3. Mitigation of Non-Monotonicity: The method significantly reduces the non-monotonic behavior of fidelity during optimization, where standard VQE and hard-coded methods often degrade in accuracy despite lowering energy.
4. Benchmarking on Disordered Systems: Successful application to the Edwards-Anderson spin-glass model, a system with no exploitable symmetries, proving the method's robustness in low-symmetry regimes.

4. Results

The authors tested their method on two benchmark models using classical emulation (noise-free):

A. Transverse-Field Ising (TFI) Model ($3 \times 3$ lattice)

• Fidelity: Soft-ortho achieved fidelities >95% across all subspace dimensions ($K=2,4,8$) and circuit depths.
• Comparison: Standard VQE peaked at ~76% fidelity; Hard-ortho peaked at ~80%.
• Circuit Depth: Soft-ortho reached high accuracy with significantly fewer parameters (shallower circuits) than the other methods. Increasing $K$ beyond 2 yielded diminishing returns.
• Convergence: Soft-ortho showed a more direct, monotonic path to high fidelity, whereas VQE and Hard-ortho exhibited non-monotonic drops in fidelity as optimization progressed.

B. Edwards-Anderson (EA) Spin-Glass Model ($4 \times 4$ lattice)

• Challenge: This model features random couplings and lacks symmetries, making it a harder test case.
• Performance: Soft-ortho consistently outperformed both VQE and Hard-ortho across 10 different disorder realizations.
• Relative Gain: The "Relative Gain Factor" (ratio of infidelity reduction) for Soft-ortho was typically 2.5x to 4x better than VQE, whereas Hard-ortho showed negligible or no improvement over standard VQE.
• Robustness: Soft-ortho maintained high fidelity even in the most difficult disorder realizations where other methods struggled.

5. Significance and Conclusion

• NISQ Relevance: The primary significance is the ability to achieve higher accuracy with shallower circuits. By trading the strict circuit-level orthogonality for a cost-function penalty, the algorithm accesses a richer set of entangled states without increasing circuit depth, directly addressing the decoherence limits of current quantum hardware.
• Generalizability: The method does not rely on problem-specific symmetries, making it highly suitable for complex, disordered systems (like spin glasses) and general quantum chemistry problems where symmetries are unknown or difficult to encode.
• Future Directions: The authors suggest exploring adaptive schemes (dynamically changing $K$ or $\beta$), combining soft-ortho with adaptive ansatz design (like ADAPT-VQE), and investigating the impact of shot noise and hardware noise on the generalized eigenvalue step.

In summary, this work demonstrates that soft-coded orthogonality is a superior strategy for subspace-based VQE, offering a practical pathway to more accurate ground state estimation on near-term quantum devices by balancing expressibility and circuit depth.