⚛️ quantum physics

Symmetry-guided quantum state preparation: Branched-Subspaces Adiabatic Preparation (B-SAP)

This paper introduces Branched-Subspaces Adiabatic Preparation (B-SAP), a hybrid quantum algorithm that combines Variational Quantum Algorithms and Adiabatic Preparation with group-theoretic symmetries and classical post-processing to efficiently prepare low-energy eigenstates of many-body Hamiltonians with polynomial circuit depth scaling.

Original authors: Davide Cugini, Giacomo Guarnieri, Mario Motta, Dario Gerace

Published 2026-01-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Davide Cugini, Giacomo Guarnieri, Mario Motta, Dario Gerace

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to find the perfect route through a massive, foggy mountain range to reach a specific valley (the "ground state" or lowest energy level of a quantum system). This is a fundamental task for quantum computers, which promise to solve complex problems in chemistry and physics. However, getting there is notoriously difficult.

The paper introduces a new navigation strategy called Branched-Subspaces Adiabatic Preparation (B-SAP). To understand why it's special, let's look at the two old ways of doing this and why they struggle.

The Old Ways: Two Flawed Maps

The "Guess-and-Check" Method (Variational Quantum Algorithms):
Imagine trying to find the valley by randomly guessing a path, checking how low you are, and then tweaking your route based on that feedback.
- The Problem: The map is so huge that you might get stuck in a "barren plateau"—a flat area where no matter which way you turn, the ground feels exactly the same. You can't tell if you're getting closer or further away, so you stop learning. Also, you need to guess the right type of path to start with; if your guess is bad, you'll never find the valley.
The "Slow Walk" Method (Adiabatic Preparation):
Imagine starting at a known, easy-to-reach hill and slowly walking toward the target valley, changing the landscape very gradually so you never lose your footing.
- The Problem: Sometimes, as you walk, two different paths (energy levels) cross each other. If you are walking slowly along one path and it crosses another, you might accidentally slide onto the wrong path. In complex systems, these "crossroads" happen constantly, causing you to end up in the wrong valley or a confusing mix of valleys.

The New Solution: B-SAP (The "Branching" Strategy)

The authors, Davide Cugini and colleagues, propose a hybrid method that combines the best of both worlds while avoiding their pitfalls. They use a clever trick involving symmetry (think of it as the "rules of the terrain").

Here is how B-SAP works, using a simple analogy:

1. Start with a "Super-Degenerate" Hill
Instead of starting on a simple, unique hill (like the old "Slow Walk" method), B-SAP starts on a giant, flat plateau where many different paths look identical at the start.

Why? In the old method, if paths cross later, you get lost. In B-SAP, the authors intentionally start with a landscape where paths are already mixed together in a known way. They know exactly where they are on this plateau.

2. The "Branching" Trick
As they slowly walk toward the target valley, the landscape changes. Because of the specific rules (symmetries) they chose for their starting point, the paths don't cross each other; instead, they branch out.

The Metaphor: Imagine a single wide river that slowly splits into smaller, distinct streams. In the old method, streams would crash into each other (crossings). In B-SAP, the streams separate cleanly. This means you never accidentally jump from your intended path to a wrong one.

3. The "Smart Guide" (The Quantum Circuit)
Before starting the slow walk, the algorithm uses a small, smart quantum circuit to "tune" exactly which spot on the starting plateau you want to be in.

The Innovation: Because the starting plateau is highly structured (based on math called group theory), the algorithm doesn't need to guess blindly. It only needs to adjust a few knobs (parameters) to pick the right "branch." This avoids the "barren plateau" problem because the search space is much smaller and smarter.

4. The Final Stretch
Once the system is tuned and the slow walk begins, the paths naturally separate. The algorithm then uses a classical computer to analyze the results and fine-tune the knobs to ensure they land exactly on the specific target state (whether it's the lowest energy state or an excited one).

What Did They Prove?

The team tested this new method on a famous model called the XYZ Heisenberg model (a way of simulating how tiny magnets interact in a line).

The Result: They successfully prepared the lowest energy states and even higher "excited" states with very high accuracy.
The Efficiency: The complexity of their circuit grew only polynomially (a manageable, steady growth) as they added more particles (qubits). This is a huge improvement over methods that grow exponentially (which become impossible very quickly).
The Comparison: When they compared B-SAP to the standard "Slow Walk" method:
- For the lowest energy state, both worked well, but B-SAP was slightly better.
- For the next energy level (the first excited state), the standard method completely failed because the paths crossed. B-SAP, however, succeeded because its "branching" strategy prevented the paths from getting mixed up.

In a Nutshell

The paper presents a new navigation tool for quantum computers. Instead of blindly guessing a path or slowly walking through a maze of crossing roads, B-SAP starts on a known, structured platform and uses the rules of symmetry to ensure the roads naturally split apart as you travel. This allows the computer to find specific quantum states (both the calmest and the more energetic ones) efficiently, without getting lost in the fog or stuck in dead ends.

The authors validated this on a simulator with up to 10 qubits, showing it works across a wide variety of conditions, making it a promising candidate for future quantum hardware.

Technical Summary: Branched-Subspaces Adiabatic Preparation (B-SAP)

Problem Statement
Quantum state preparation is a fundamental task for simulating complex many-body systems in physics, chemistry, and data science. Current approaches face significant trade-offs. Variational Quantum Algorithms (VQAs) rely on parameterized ansatzes and classical optimization, which often suffer from "barren plateaus" (exponentially vanishing gradients), poor scalability due to the need for exponentially large parameter spaces, and convergence issues related to local minima. Conversely, Adiabatic Preparation (AP) evolves an initial state under a time-dependent Hamiltonian to reach a target state. While conceptually distinct, AP is highly sensitive to energy spectral gaps; if energy levels cross or gaps close exponentially with system size, the adiabatic theorem fails to guarantee the preparation of a specific eigenstate, often resulting in an unknown superposition of degenerate states. These limitations hinder the simultaneous preparation of ground and excited states, which is crucial for understanding optical spectra, charge-transfer processes, and reaction intermediates.

Methodology: Branched-Subspaces Adiabatic Preparation (B-SAP)
The authors propose B-SAP, a hybrid algorithm that integrates the structural strengths of VQAs and AP while mitigating their respective weaknesses through group-theoretic structures and classical post-processing.

Symmetry-Guided Initialization: Unlike conventional AP, which typically selects an initial Hamiltonian ( $H_0$ ) with a minimal symmetry group to avoid degeneracy, B-SAP selects an $H_0$ whose symmetry group contains the symmetry group of the target Hamiltonian ( $H_T$ ) as a subgroup. Consequently, the initial energy spectrum exhibits high degeneracy.
Subspace Exploration via Parametrized Gates: The algorithm identifies a degenerate eigenspace $S(B_n)$ $S (B_{n})$ of $H_0$ $H_{0}$ corresponding to an energy level $E_n$ $E_{n}$ . Instead of using a generic ansatz, B-SAP constructs a specific set of parametrized unitary gates $\{G_j(\alpha_j)\}$ ${G_{j} (α_{j})}$ that act as endomorphisms on this subspace. These gates are derived from the generators of the unitary group $U(d_n)$ $U (d_{n})$ (where $d_n$ $d_{n}$ is the subspace dimension), reduced to the coset space $U(d_n)/U(d_{n-1})$ $U (d_{n}) / U (d_{n - 1})$ to ensure the mapping is bijective for a fixed reference state.
- Crucially, the number of trainable parameters scales as $2d_n - 1$ (or $2(d_n-1)$ if global phase is irrelevant). Since $d_n$ scales polynomially with system size $L$ for the models considered, the parameter count avoids the exponential scaling that leads to barren plateaus.
Adiabatic Evolution with Branching: The system undergoes adiabatic evolution from $H_0$ to $H_T$ . Because the symmetry group of $H(s)$ either remains constant or decreases (symmetries are broken, not introduced), the irreducible representations (irreps) of the symmetry group either remain unchanged or split (branch). This "branching" structure prevents the merging of distinct energy levels that causes level crossings in conventional AP. The protocol ensures that the spectral gap condition is only required between distinct eigenspaces, rather than between every individual eigenstate, relaxing the constraints for successful preparation.
Classical Post-Processing (MC-VQE): To optimize the variational parameters without suffering from optimization bottlenecks, the authors employ the Multistate-Contracted Variational Quantum Eigensolver (MC-VQE).
- Instead of optimizing a single state, the algorithm prepares overlap states $|\Psi^p_{ml}\rangle$ (superpositions of basis states) on the quantum register.
- It measures matrix elements of the target Hamiltonian $H_T$ in the evolved basis, constructing a $d_n \times d_n$ matrix $[H_T]^{(n)}_{ml}$ .
- This matrix is diagonalized classically to obtain the eigenvalues and the coefficients of the target eigenvectors.
- The variational parameters are then tuned classically to maximize the overlap between the prepared state and the classically derived target vector. This avoids the need for a quantum circuit during the final minimization step and provides a known global minimum for the loss function.

Key Contributions

Hybrid Framework: The paper introduces a novel algorithm that bypasses the need for heuristic ansatzes (common in VQAs) and the strict non-crossing gap requirements (common in AP).
Group-Theoretic Construction: It utilizes the representation theory of the Hamiltonian's symmetry group to construct a parametrized circuit that explores a specific degenerate subspace efficiently, ensuring the number of parameters scales polynomially with system size.
Robustness to Level Crossings: By designing the adiabatic path such that symmetries are broken rather than introduced, the method prevents the formation of accidental degeneracies that typically plague adiabatic evolution, allowing for the preparation of excited states.
Classical Optimization Bypass: The integration of MC-VQE allows for the extraction of excited state information via classical diagonalization of a small matrix, reducing the reliance on difficult quantum-classical optimization loops.

Results
The authors validated B-SAP using the one-dimensional XYZ Heisenberg model with periodic boundary conditions on a quantum simulator (Qiskit) for system sizes up to $L=10$ qubits.

Accuracy: The algorithm successfully prepared the ground state and the first four excited states across a broad range of coupling constants ( $J_x, J_y, J_z$ ).
Error Metrics: For the ground state, the error (infidelity) was negligible compared to random state overlap ( $2^{-L}$ ) in most parameter regions, even with a minimal number of Trotter steps ( $L/2$ ). For excited states, errors generally remained below 10% and often below the percent level.
Scaling: The circuit depth required scales polynomially with the system size $L$ .
Comparison: When compared to standard AP on the same model and parameters:
- Standard AP performed comparably for the ground state but failed completely for the first excited state in most parameter regions due to energy level crossings.
- B-SAP successfully prepared the excited states where standard AP failed, demonstrating its ability to navigate level crossings via the branching of degenerate subspaces.

Significance and Claims
The paper claims that B-SAP offers a promising route for quantum state preparation on near-term quantum computing platforms by balancing accuracy, resource efficiency, and robustness.

Near-Term Applicability: The polynomial scaling of circuit depth and parameters makes the algorithm suitable for devices with limited coherence times and connectivity.
Excited State Access: The method addresses a critical gap in current capabilities by enabling the reliable preparation of higher-order excited states, which are essential for determining optical spectra and chemical reactivity but are typically inaccessible via standard adiabatic or variational methods due to level crossings and optimization difficulties.
Future Outlook: The authors suggest that the approach could be extended to quantum simulations of complex materials, quantum chemistry, and field-theoretical systems, with future work focusing on integrating error mitigation techniques for deployment on real hardware. The paper does not claim to solve the general problem of all quantum state preparation but positions B-SAP as a specific, effective solution for many-body Hamiltonians with known symmetries.

The Old Ways: Two Flawed Maps

The New Solution: B-SAP (The "Branching" Strategy)

What Did They Prove?

In a Nutshell

Technical Summary: Branched-Subspaces Adiabatic Preparation (B-SAP)

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