Orbital-optimized spin-adapted multistate contracted VQE for excited states and properties on quantum hardware

This paper introduces the orbital-optimized multistate contracted VQE (oo-MC-VQE) method, which utilizes spin-adapted operators to efficiently compute ground and excited states along with their properties on quantum hardware while balancing accuracy and circuit complexity through a linear parameter scaling with the number of states.

The Gist

Imagine you are trying to tune a massive, complex orchestra to play a perfect symphony. In the world of chemistry, the "orchestra" is a molecule, and the "music" is the way its electrons move and interact. To understand how a molecule absorbs light (which gives us colors and drives things like photosynthesis), scientists need to calculate the exact notes these electrons play.

For a long time, doing this for molecules with many electrons has been like trying to solve a puzzle that gets exponentially harder the more pieces you add. Classical computers (the ones we use today) eventually hit a wall and can't solve these puzzles for complex molecules.

This paper introduces a new way to solve these puzzles using quantum computers, which are special machines designed to handle this kind of complexity naturally. Here is a simple breakdown of what the authors did and found:

1. The Problem: Tuning Many Notes at Once

Usually, scientists try to tune the orchestra to play just one note (the ground state) perfectly. But to understand how a molecule reacts to light, they need to know about many different notes (excited states) at the same time.

• The Challenge: If you try to tune the orchestra for 10 different songs simultaneously, the instructions (the computer circuit) get incredibly long and complicated. If the instructions are too long, the quantum computer gets confused by "noise" (static or errors), and the music falls apart.
• The Trade-off: You need a complex circuit to get an accurate answer, but a complex circuit is more likely to fail on current noisy machines.

2. The Solution: A Smart, Symmetrical Conductor

The authors developed a new method called oo-MC-VQE. Think of this as a "smart conductor" for the quantum orchestra.

• Spin-Adapted: In quantum chemistry, electrons have a property called "spin" (like spinning tops). The authors built their method so that the conductor always keeps the spinning tops spinning in the correct, symmetrical way. This prevents the music from getting "out of tune" due to symmetry errors.
• Orbital Optimized: They also let the conductor rearrange the seating chart of the musicians (the orbitals) to make the music sound better before even starting the complex tuning.
• Multistate Contracted: Instead of trying to tune 10 songs with 10 separate, massive instruction manuals, they found a way to use one shared, efficient set of instructions that works for all the songs at once.

3. The Discovery: Linear Growth

One of the biggest questions was: If I want to calculate 10 states instead of 1, do I need 10 times more computer power?

• The Finding: The authors found that the answer is surprisingly simple. The amount of computer "effort" (circuit parameters) needed grows linearly. If you double the number of states you want to calculate, you roughly just double the length of the instruction manual. It doesn't explode into an impossible size. This is great news because it means the method is scalable.

4. The Real-World Test: Playing on a Noisy Stage

The authors didn't just simulate this on a perfect computer; they actually ran their method on real quantum hardware (IBM quantum computers).

• The Setup: They tested two small molecules: Formaldehyde (a common chemical) and a trihydrogen cation ($H_3^+$).
• The Noise Issue: Real quantum computers are like a stage with a loud crowd and flickering lights. The results were messy without help.
• The Fix: They used "error mitigation" techniques. Imagine this as a sound engineer using software to filter out the crowd noise and flickering lights after the performance.
• The Result:
  • For Formaldehyde, the method worked quite well. Even with the noise, they could clearly see the "peaks" in the absorption spectrum (the colors the molecule absorbs).
  • For $H_3^+$, the noise was a bigger problem, shifting the results significantly. The authors noted this was because this specific molecule's math is more sensitive to noise (like a delicate instrument that goes out of tune easily).
  • Key Takeaway: While the numbers weren't perfect on the real machines, the shape of the results was correct. They could still see the main features of the molecule's behavior.

Summary

The paper shows that by using a smart, symmetrical approach, scientists can calculate the behavior of excited electrons in molecules using current, imperfect quantum computers. They proved that calculating multiple states doesn't require an impossible amount of resources, and with some "noise-canceling" tricks, they can get useful chemical insights from real quantum devices today.

What they did NOT claim:
The paper does not claim this method can immediately design new solar panels, cure diseases, or create new materials. It strictly focuses on proving the method works for calculating spectra on quantum hardware. Any future applications are implied by the general field but are not specific claims of this study.

Technical Summary

Technical Summary: Orbital-optimized spin-adapted multistate contracted VQE

Problem Statement
The robust and balanced computation of electronically excited states and their transition properties remains a significant challenge, particularly in regimes involving strong electronic reorganization, near-degeneracies, or multiconfigurational character. While linear-response (LR) methods are successful in many contexts, they treat excited states as perturbations of a ground-state reference, leading to conceptual limitations near conical intersections and in strongly correlated systems. Furthermore, existing variational quantum eigensolver (VQE) approaches for excited states, such as subspace-search VQE (SS-VQE) and multistate contracted VQE (MC-VQE), often face a trade-off between the accuracy of describing an increasing number of electronic states and the expressibility (circuit complexity) of the underlying ansatz. Additionally, current quantum hardware is limited by noise and decoherence, which restricts circuit depth and necessitates the development of methods that are both efficient and compatible with error mitigation techniques.

Methodology
The authors introduce the orbital-optimized multistate contracted variational quantum eigensolver (oo-MC-VQE) method, which incorporates spin-adapted operators to compute ground and excited states, as well as state-specific and transition properties.

• Spin Adaptation: To ensure the spin symmetry of reference states is conserved throughout the VQE optimization, the method utilizes a spin-adapted operator pool. Specifically, the authors employ singlet spin-adapted single excitation operators and pair-double excitation operators. This avoids the need for spin penalty terms and ensures the correct spin symmetry of all included states without introducing redundant parameters.
• Multistate Contracted Framework: The method optimizes multiple electronic states simultaneously using a state-averaged energy functional. Each state is constructed using a fixed, parameter-free state-preparation circuit (initializing orthogonal basis states) followed by a parameterized correlator circuit (using a Unitary Product State, or UPS, ansatz) that introduces electron correlation. The optimization targets the state-averaged energy, followed by a classical diagonalization of the multistate Hamiltonian matrix to resolve individual state energies and wavefunctions.
• Orbital Optimization: The framework is extended to include orbital optimization (oo-MC-VQE). Orbital rotations are treated as a classical cost function, allowing for the simultaneous optimization of circuit parameters ($\theta$) and orbital rotation parameters ($\kappa$). The authors demonstrate that for the specific ansatz used, single excitation circuit parameters adjacent to the orbital parameterization are redundant and can be removed, reducing circuit complexity.
• Matrix Element Evaluation: To evaluate matrix elements between different states (required for the multistate Hamiltonian and transition properties), the authors employ a superposition decomposition technique. This approach avoids deep state-preparation circuits for multi-determinant states by evaluating simpler superpositions of two determinants, thereby reducing the number of CNOT gates at the cost of increased expectation value measurements.
• Error Mitigation: The implementation integrates Mansatz0, an ansatz-based readout and gate error mitigation technique. For multistate calculations involving superpositions of reference states, the authors propose an extension (M0+) that characterizes errors for all unique multi-determinant reference states, though they note the exponential scaling of this approach.

Key Contributions

1. Spin-Adapted oo-MC-VQE: The development of a spin-adapted formulation for orbital-optimized state-averaged VQE, ensuring correct spin symmetry conservation across multiple states.
2. Scaling Analysis: An explicit investigation into the trade-off between accuracy and circuit complexity. The authors find that the number of circuit parameters required to obtain accurate results scales approximately linearly with the number of states included in the state-averaging.
3. Hardware Implementation: An explicit quantum-circuit implementation of the method, including the reduction of CNOT gates via determinant superposition and the integration of error mitigation techniques.
4. Real-Device Demonstration: Execution of the method on real quantum hardware (IBM ibm marrakesh and ibm aachen) to compute absorption spectra for benchmark molecular systems (formaldehyde and $H_3^+$).

Results

• Circuit Complexity: Simulations on LiH, $H_2O$, and $NH_3$ using the ADAPT and oo-ADAPT ansatzes show that the number of parameters required increases roughly linearly with the number of states. For example, achieving an error below $10^{-4}$ Hartree for $H_2O$ required 27, 32, 24, and 30 parameters per state for 1, 2, 3, and 4 states, respectively.
• Orbital Optimization Impact: While orbital optimization improves convergence in the early stages of the ADAPT procedure, it has a negligible effect on the final number of required circuit parameters for tighter convergence. In some cases (e.g., $NH_3$ with 3 or 4 states), folding single excitations into orbital optimization had a detrimental effect on the parameter count.
• Quantum Hardware Performance:
  • Simulated Noise: On emulated hardware using IBM noise models, error mitigation techniques (M0 and M0+) produced results nearly identical to noiseless simulations. The $H_3^+$ system was more adversely affected by noise than formaldehyde, attributed to a higher condition number of the multistate Hamiltonian matrix.
  • Real Hardware: Calculations on real devices showed that while error-mitigated results preserved the qualitative trends and primary features of the absorption spectra, quantitative accuracy was significantly lower than on simulated devices. The discrepancy highlights limitations in current noise models regarding time-dependent noise and crosstalk. However, the method successfully extracted meaningful chemical insight despite hardware noise.

Significance and Claims
The paper claims that the oo-MC-VQE method provides a balanced description of electronic structure suitable for challenging scenarios like conical intersections, addressing limitations inherent in linear-response approaches. The authors emphasize that the linear scaling of circuit parameters with the number of states suggests the method is viable for targeting multiple excited states on near-term quantum hardware. While acknowledging that quantitative accuracy remains challenging on present-day devices, the work demonstrates that meaningful chemical insight, such as absorption spectra, can be extracted when combining the method with appropriate error mitigation techniques. The study serves as a proof-of-concept for executing multistate, spin-adapted, orbital-optimized calculations on real quantum devices.