Adiabatic quantum state preparation in integrable models

This paper proposes and validates an adiabatic quantum algorithm that efficiently prepares both ground and arbitrary high-energy eigenstates of integrable models, such as the XXZ Heisenberg and Richardson-Gaudin chains, with a circuit depth that scales polynomially with system size by leveraging local conserved quantities and the thermodynamic Bethe ansatz.

The Gist

Imagine you are trying to organize a massive, chaotic library. In this library, every book represents a specific state of a quantum system (like a collection of tiny magnets called spins). Most libraries are so messy that finding a specific book takes forever, or you might need a supercomputer that runs for billions of years just to figure out where it is.

However, some special libraries are Integrable Models. These are like libraries with a secret, perfect filing system. The books are arranged according to strict mathematical rules, making them theoretically easier to understand. But even with this perfect system, finding a specific "high-energy" book (an excited state) is still incredibly hard for classical computers.

This paper proposes a clever new way to find these books using a Quantum Computer. Here is the breakdown of their idea, using simple analogies:

1. The Problem: The "Mountain" of States

Usually, when we want to prepare a specific quantum state on a computer, we use a method called the Adiabatic Algorithm. Think of this like slowly pushing a ball up a hill.

• The Goal: You want the ball to end up at the very bottom of a valley (the ground state). This is easy because the ball naturally wants to roll down.
• The Problem: The authors want to prepare excited states. In the quantum world, these are like specific spots high up on the mountain.
• The Obstacle: Usually, the "valleys" for these high-up spots are so close together that they are practically touching. If you try to push the ball slowly, it gets confused and rolls into the wrong valley. This is why standard methods fail for excited states.

2. The Solution: Building a Custom "Trap"

The authors realized that because these models are "integrable" (they have hidden rules), we can cheat. Instead of just pushing the ball up a generic hill, they propose building a custom-made trap specifically for the book you want.

• The "Parent Hamiltonian" (The Custom Trap): Imagine you want to find a specific book. Instead of searching the whole library, you build a cage that only fits that one book. If you put any other book in the cage, it doesn't fit and gets pushed out.
• How they do it: They use the library's secret filing system (the "conserved quantities" or "charges"). They create a mathematical equation that says: "The energy is zero only if the book matches this specific set of rules. If it doesn't match, the energy is high."
• The Result: The specific state you want becomes the lowest energy point (the bottom of the valley) in this new, custom-built landscape.

3. The Journey: The Slow Walk

Now that they have built this custom trap where the target state is the "ground state" (the easiest place to be), they use the standard Adiabatic Algorithm again.

• They start with a simple, easy-to-find state (like a book in a completely empty room).
• They slowly morph the room into their custom trap.
• Because the target state is now the "bottom of the valley" in this new landscape, the quantum computer can smoothly slide into it without getting confused by the other states.

4. The Proof: Does it Work?

The authors tested this idea on two types of quantum models:

• The XY Chain (The Easy Test): This is like a library with no interacting books (they don't affect each other). They proved mathematically that their method works perfectly and quickly.
• The Richardson-Gaudin Models (The Hard Test): This is a library where the books interact and push each other around. This is much harder. They ran computer simulations and found that even with these interactions, their "custom trap" method still worked efficiently. The time it took to find the state grew only polynomially (like $N^2$ or $N^3$) rather than exponentially (like $2^N$).

5. The Catch (The XXZ Model)

They also tried this on the famous XXZ model. Here, the "filing system" (the conserved charges) is so complex that building the custom trap would require a cage made of an impossibly huge number of parts. It's like trying to build a cage out of a billion tiny Lego bricks; it's theoretically possible, but practically impossible to build in a reasonable time. So, their method doesn't work for every model, but it works for a very important class of them.

Summary

In short, the authors found a way to rearrange the landscape of a quantum problem.

1. Identify the specific state you want.
2. Build a custom mathematical "cage" (Parent Hamiltonian) using the model's hidden rules, making that specific state the easiest one to find.
3. Slowly transform the system into this cage.
4. Result: You can prepare complex, high-energy quantum states efficiently, which was previously thought to be nearly impossible for interacting systems.

This is a big step forward because it suggests that quantum computers might be able to simulate complex materials and chemical reactions much faster than we thought, provided we know how to build the right "cages" for the states we want to study.

Technical Summary

1. Problem Statement

The preparation of high-energy eigenstates of interacting many-body systems on quantum computers is a significant challenge.

• Generic Hamiltonians: For generic interacting systems, preparing eigenstates typically requires quantum circuits with depth scaling exponentially with the number of qubits ($N$).
• Integrable Models: While integrable models (e.g., XXZ Heisenberg chain, Richardson-Gaudin models) possess rich mathematical structures (Bethe ansatz) and conserved quantities, existing quantum algorithms for their eigenstates are either:
  • Variational: Requiring exponential resources or suffering from barren plateaus.
  • Limited: Efficient only for ground states or a small subset of eigenstates.
  • Non-adiabatic: Relying on explicit integrability constructions that often fail for arbitrary excited states.
• The Gap: It was previously unclear if the adiabatic algorithm, typically restricted to ground states due to exponentially closing gaps in excited spectra, could be adapted to efficiently prepare arbitrary excited states of interacting integrable models.

2. Methodology

The authors propose a protocol combining the adiabatic theorem with the conserved quantities (charges) inherent to integrable models.

A. Standard Adiabatic Review (Ground States)

The paper first reviews the standard adiabatic algorithm for preparing ground states in the XXZ Heisenberg chain.

• Protocol: Start with the ground state of a non-interacting limit ($\Delta=0$) and adiabatically evolve to an interacting point ($\Delta \in (0,1)$).
• Efficiency: Using the Thermodynamic Bethe Ansatz (TBA), they demonstrate that the energy gap between the ground state and the first excited state in a fixed magnetization sector scales as $O(1/N)$.
• Result: This ensures the circuit depth is polynomial in $N$, outperforming previous methods that relied on explicit integrability constructions for ground states.

B. Proposed Protocol for Arbitrary Eigenstates

To prepare arbitrary excited states $|v\rangle$, the authors construct a Parent Hamiltonian $H_v(g)$:
$$H_v(g) = \sum_{k=1}^{N} \left( Q^{(k)}(g) - q^{(k)}_v(g) \right)^2$$
Where:

• $Q^{(k)}(g)$ are the extensive set of local conserved charges (integrals of motion) of the integrable model.
• $q^{(k)}_v(g)$ are the specific eigenvalues of these charges for the target state $|v\rangle$.
• Mechanism: The term $(Q^{(k)} - q^{(k)}_v)^2$ acts as an energy penalty. The state $|v\rangle$ is the unique ground state of $H_v(g)$ with zero energy. All other states have higher energy.
• Adiabatic Path: The algorithm starts at a parameter $g^*$ (usually a non-interacting point where $|v\rangle$ is a simple product state) and evolves to the target Hamiltonian parameters.

C. Complexity Conditions

For the protocol to be efficient (polynomial circuit depth), three conditions must be met:

1. Sparsity: The parent Hamiltonian must be decomposable into $S = O(\text{poly}(N))$ Pauli strings.
2. Gap Scaling: The gap of the parent Hamiltonian $\delta(g)$ must not close faster than $O(1/\text{poly}(N))$ along the adiabatic path.
3. Classical Computability: The eigenvalues $q^{(k)}_v(g)$ must be computable classically in polynomial time.

3. Key Results and Analysis

A. Non-Interacting XY Spin Chain

The authors rigorously prove the protocol's efficiency for the XY chain.

• Construction: They map the model to free fermions. The charges are occupation numbers $c^\dagger_p c_p$.
• Gap: The parent Hamiltonian constructed from these charges has a constant gap $\delta = 1$ (independent of $N$) because distinct eigenstates differ in at least one mode occupation.
• Implementation: The charges can be expressed as $O(N^2)$ Pauli strings with bounded coefficients.
• Conclusion: All eigenstates of the XY chain can be prepared with $O(\text{poly}(N))$ circuit depth.

B. Interacting Richardson-Gaudin (RG) Models

The core contribution is applying this to the interacting spin-1/2 XXX Richardson-Gaudin models.

• Charges: The eigenstates are uniquely specified by the eigenvalues of $N$ commuting charges $Q^{(k)}$. These satisfy quadratic Bethe equations.
• Classical Computation: The authors propose a numerical method (iterative optimization/Taylor expansion) to solve the Bethe equations for $q^{(k)}_v(g)$ along the path. While a rigorous proof of polynomial-time classical solvability is open, numerical evidence suggests it is efficient for many states.
• Gap Analysis:
  • Weak Coupling ($g \to 0$): Perturbation theory shows the gap is $O(1)$.
  • Strong Coupling ($g \to \infty$): The gap is proven to scale as $O(1/N)$.
  • Intermediate Coupling: Rigorous bounds are difficult, but numerical simulations for system sizes up to $N=14$ show that the minimal gap between any two eigenstates scales as $O(1/N)$ across the entire adiabatic path.
• Circuit Depth: Since the charges $Q^{(k)}$ involve $O(N^3)$ Pauli strings and the gap scales as $O(1/N)$, the total circuit depth is polynomial in $N$.

C. Limitations: The XXZ Model

The authors discuss why this specific protocol does not directly apply to the XXZ model for all eigenstates.

• Bottleneck: The $k$-th conserved charge in the XXZ model involves $O(N^{2k})$ Pauli strings. To uniquely specify an arbitrary state, one needs $O(N)$ charges.
• Result: Constructing the full parent Hamiltonian would require $O(\exp(N))$ terms, leading to exponential circuit depth.
• Potential Workaround: Truncating the sum to $O(\log N)$ charges could target low-energy states in specific sectors, but not arbitrary high-energy states.

4. Significance and Contributions

1. New Paradigm for Excited States: The paper demonstrates that the adiabatic algorithm, traditionally viewed as unsuitable for excited states due to spectral crowding, can be highly effective for integrable models when combined with conserved quantities.
2. Polynomial Scaling for Interacting Models: It provides the first proposal for preparing arbitrary eigenstates of interacting integrable models (specifically RG models) with polynomial circuit depth, overcoming the exponential scaling expected for generic Hamiltonians.
3. Parent Hamiltonian Construction: The introduction of the "parent Hamiltonian" based on the squared difference of conserved charges is a novel and powerful tool for state preparation in quantum simulation.
4. Numerical Validation: Extensive numerical evidence supports the $O(1/N)$ gap scaling in RG models, suggesting that the "spectral crowding" problem is mitigated by the structure of the conserved charges.
5. Future Directions: The work highlights the importance of the classical complexity of solving Bethe equations and suggests that Smale's $\alpha$-theory could be used to rigorously prove the gap scaling in future work.

Conclusion

Lutz et al. successfully bridge the gap between integrability and quantum simulation. By leveraging the extensive set of conserved charges to construct a tailored parent Hamiltonian, they show that adiabatic evolution can efficiently prepare arbitrary eigenstates of interacting integrable models, provided the classical computation of Bethe roots remains efficient. This offers a promising route for quantum simulation of complex many-body phenomena beyond ground states.