Hybrid QPE-Ansatz Strategy for Reliable Excited-State Variational Quantum Deflation

This paper proposes a spin-filtering variational quantum deflation (sfVQD) scheme that combines a symmetry-preserving ansatz with a shallow quantum phase estimation routine to efficiently suppress spin contamination in NISQ-era excited-state calculations without requiring costly explicit spin evaluations.

The Gist

Imagine you are trying to find the perfect recipe for a specific dish (let's say, a Singlet Cake) in a massive, chaotic kitchen. However, the kitchen is full of other dishes that look almost identical but have a slightly different ingredient mix (like Triplet Cakes).

In the world of quantum computing, finding these specific "recipes" (quantum states) is the job of an algorithm called VQE (Variational Quantum Eigensolver). But here's the problem: the kitchen is noisy (it's a "Noisy Intermediate-Scale Quantum" or NISQ device), and the chefs (algorithms) often get confused, mixing up the Singlet Cake with the Triplet Cake. This is called spin contamination.

This paper introduces a clever new strategy called sfVQD (Spin-Filtering Variational Quantum Deflation) to solve this mess. Here is how it works, broken down into simple concepts:

1. The Problem: The "Confused Chef"

Traditionally, to find an excited state (a cake that isn't the basic one), the algorithm tries to find a solution that is "different" from the ones it already found. It does this by adding a penalty if the new recipe looks too much like an old one.

• The Issue: As the algorithm tries to find more complex cakes (higher excited states), the errors pile up. The chef starts mixing up the Singlet and Triplet cakes because they look so similar in the noisy kitchen. The result is a "Frankenstein" cake that isn't a real Singlet or a real Triplet.

2. The First Fix: The "Strict Ingredient List" (SSP Ansatz)

The authors first built a better "recipe book" (called an Ansatz).

• The Old Way: The recipe book allowed the chef to mix ingredients freely, which was efficient but risky.
• The New Way (SSP): They created a Symmetry-Preserving recipe book. It forces the chef to keep the number of "Spin-Up" and "Spin-Down" ingredients strictly separate.
• The Result: This is like telling the chef, "You can only use 2 red eggs and 2 blue eggs." This stops the chef from making a completely wrong dish, but it doesn't guarantee the exact right flavor (total spin) because you can still mix those 2 red and 2 blue eggs in different ways to get different results.

3. The Second Fix: The "Magic Taste-Tester" (The QPE Screening)

This is the paper's big innovation. Even with the strict ingredient list, the chef might still make a "Triplet-flavored" cake when you wanted a "Singlet."

• The Solution: Instead of checking the whole cake (which is slow and expensive), they use a Magic Taste-Tester (a small extra helper called an Ancilla).
• How it works:
  1. Before the chef finishes the cake and serves it, the Magic Taste-Tester spins the cake on a specific axis (like spinning a top).
  2. If the cake is the right type (Singlet), it spins smoothly and lands in a "Green Zone."
  3. If the cake is the wrong type (Triplet), it wobbles and lands in a "Red Zone."
• The Filter: If the tester lands in the Red Zone, the computer immediately throws that attempt away and doesn't waste time measuring the energy of the bad cake. It only keeps the "Green Zone" attempts.

4. Why This is a Game-Changer

Think of it like a bouncer at a club:

• Old Method: You let everyone in, let them dance, and then check their ID at the end. If they are fake, you kick them out, but you've already wasted time and energy on the party.
• New Method (sfVQD): The bouncer (the Magic Taste-Tester) checks the ID at the door before they even enter the dance floor. If they don't have the right ticket, they never get in.

The Benefits:

1. Cleaner Results: The final "cakes" (quantum states) are pure Singlets or pure Triplets, not messy mixtures.
2. Faster: By throwing out bad attempts early, the computer doesn't waste energy calculating the energy of wrong states.
3. NISQ-Friendly: It doesn't require building a massive, complex machine (deep circuit). The "Magic Taste-Tester" is small and simple, making it perfect for today's imperfect quantum computers.

Summary

The authors combined a strict recipe book (SSP) with a quick, early-warning filter (QPE screening). This hybrid approach ensures that when quantum computers try to find excited states, they don't get lost in a fog of wrong answers. They filter out the "imposter" states early, saving time and delivering physically meaningful results that scientists can actually trust.

Technical Summary

1. Problem Statement

The paper addresses a critical challenge in calculating excited states on Noisy Intermediate-Scale Quantum (NISQ) devices using the Variational Quantum Eigensolver (VQE): spin contamination.

• The Context: While VQE is effective for ground states, finding excited states typically requires methods like Variational Quantum Deflation (VQD). VQD finds the $k$-th excited state by minimizing energy while enforcing orthogonality to all previously found lower-energy states.
• The Challenge:
  1. Spin Mixing: In the absence of strict symmetry constraints, variational ansätze (especially hardware-efficient ones) often converge to states that are mixtures of different spin multiplicities (e.g., a singlet state contaminated by triplet components).
  2. Cost of Correction: Explicitly enforcing spin purity by measuring the total spin operator $\langle \hat{S}^2 \rangle$ is computationally expensive, requiring $O(n^2)$ measurement overhead due to Pauli string decomposition.
  3. Error Accumulation: In sequential VQD, numerical errors in lower states propagate to higher states. If the variational space allows spin mixing, the optimizer may exploit "contaminated" intermediate states to find lower energies, leading to physically meaningless results.
  4. Ansatz Limitations: Deep, symmetry-preserving ansätze (like Unitary Coupled Cluster) are too resource-heavy for NISQ, while shallow, hardware-efficient ansätze often violate physical symmetries.

2. Methodology

The authors propose a hybrid strategy called spin-filtering Variational Quantum Deflation (sfVQD). This approach combines a symmetry-preserving ansatz with a lightweight, ancilla-assisted screening routine based on a shallow Quantum Phase Estimation (QPE) protocol.

A. Symmetry-Preserving (SSP) Ansatz

• Base: The authors start with a Symmetry-Preserving (SP) ansatz, which is hardware-efficient and conserves the total number of particles.
• Modification: They construct an $\hat{S}_z$-conserving SSP ansatz by applying SP operations independently on the $\alpha$ (spin-up) and $\beta$ (spin-down) qubit registers.
• Effect: This fixes the number of $\alpha$ and $\beta$ electrons ($n_\alpha$ and $n_\beta$), thereby conserving the $z$-component of spin ($\hat{S}_z$).
• Limitation: While $\hat{S}_z$ is fixed, the total spin quantum number $S$ is not uniquely determined. States with different $S$ but the same $m_z$ can still coexist, leading to residual spin contamination.

B. $\hat{S}_x$-Based Screening (QPE Routine)

To filter out unwanted spin states without measuring $\hat{S}^2$, the authors introduce a shallow QPE-like routine:

• Principle: Under the assumption of a spin-free Hamiltonian (ignoring spin-orbit coupling), $\hat{S}_x$ commutes with the Hamiltonian $\hat{H}$.
• Mechanism:
  1. A controlled rotation $e^{i\theta \hat{S}_x}$ is applied to the state using an ancilla register.
  2. Since $\hat{S}_x$ is diagonal in the $|S, m_x\rangle$ basis, this operation encodes the phase $e^{i m_x \theta}$ onto the ancilla.
  3. A shallow QPE circuit (controlled rotations + inverse QFT) extracts the eigenvalue $m_x$ into the ancilla register.
• Screening Logic:
  • For a target state with fixed $m_z$, the probability of measuring a specific $m_x$ depends on the total spin $S$.
  • If the target is a state with minimal spin $S = |m_z|$, the probability of measuring $|m_x| > |m_z|$ is zero.
  • Filtering: If the ancilla readout yields $|m_x| > |m_z|$, the shot is flagged as "out-of-sector" (spin contaminated). These shots are either discarded or heavily penalized in the cost function before the expensive Hamiltonian measurement occurs.

C. Computational Protocol

• Hybrid Workflow: The VQD cost function is augmented with overlap penalties (for orthogonality) and the spin-filtering penalty.
• Shot-Based vs. Statevector:
  • In real hardware (shot-based), shots failing the spin check are discarded early to save resources.
  • In simulation, an extended Hamiltonian $\hat{H}_{ext}$ is constructed to mathematically penalize invalid spin sectors, shifting their effective energy upward.

3. Key Contributions

1. Novel Hybrid Strategy: The integration of a symmetry-preserving ansatz (SSP) with a shallow, ancilla-assisted QPE discriminator. This avoids the $O(n^2)$ overhead of measuring $\langle \hat{S}^2 \rangle$.
2. Modular Screening: The screening module is independent of the variational ansatz, making it compatible with other excited-state methods (like SSVQE) and other symmetry constraints (e.g., point-group symmetries).
3. Resource Efficiency:
   • Circuit Depth: The screening routine is shallow, utilizing native two-qubit controlled-rotation gates.
   • Ancilla Count: The number of ancilla qubits scales logarithmically with system size ($n_{anc} \sim \log_2(\text{spin range})$), requiring very few qubits (e.g., 3 qubits for a quintet state).
4. Error Mitigation: By filtering out spin-contaminated states early, the method reduces the propagation of numerical errors in sequential VQD, improving the fidelity of higher excited states.

4. Results

The method was tested on LiH and BeH$_2$ molecules across various geometries (symmetric and asymmetric stretches) using the STO-3G basis set.

• Spin Purity:
  • VQD/SP (Standard): Showed significant spin contamination ($\langle \hat{S}^2 \rangle$ deviated from ideal values).
  • VQD/SSP (Fixed $m_z$): Improved purity but still exhibited deviations for higher excitations or distorted geometries.
  • sfVQD/SSP: Achieved essentially correct $\langle \hat{S}^2 \rangle$ values (near ideal singlet/triplet values) regardless of the ansatz depth or geometry.
• Separation of Manifolds: The method successfully separated singlet and triplet manifolds that were mixed in conventional VQD approaches.
• Computational Cost:
  • sfVQD reduced the number of required orthogonality checks by a factor of ~2 (vs. VQD/SSP) and ~4 (vs. VQD/SP) for the singlet manifold of BeH$_2$, as the search space was confined to the correct spin sector.
  • Trade-off: The screening introduced a "rougher" optimization landscape. In some cases, VQD/SSP converged more smoothly because it could traverse "shortcut" states with incorrect spin characters that sfVQD blocked. However, sfVQD yielded physically correct results where VQD/SSP failed.

5. Significance

• NISQ Compatibility: The paper provides a practical route to obtaining physically meaningful excited states on current noisy hardware without requiring deep circuits or prohibitive measurement overheads.
• Beyond Spin: The framework demonstrates that conserved quantities (like spin) can be enforced via "symmetry-oriented sensing" rather than direct projection. This suggests a generalizable method for enforcing other symmetries (e.g., molecular point-group irreducible representations) in variational algorithms.
• Robustness: By preventing the optimizer from exploiting contaminated intermediate states, the method ensures that the resulting excited states possess the correct quantum-mechanical characters, a crucial requirement for reliable quantum chemistry simulations.

In summary, the authors present sfVQD as a modular, efficient, and robust solution to the spin contamination problem in NISQ-era excited-state calculations, successfully balancing physical validity with hardware constraints.