⚛️ quantum physics

Low-Depth Initial-State Preparation for Ground-State Energy Estimation of Two-Dimensional Strongly Correlated Systems

Guided by classical simulations, this paper proposes a low-depth quantum circuit strategy for preparing ground states of two-dimensional Hubbard models by embedding approximate Heisenberg states and applying Schrieffer-Wolff-derived charge-fluctuation gates, enabling scalable parameter transfer to large lattices with an estimated preparation cost of approximately $10^5$ T gates.

Original authors: Ryo Watanabe, Keisuke Fujii

Published 2026-09-15

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CC BY 4.0

Original authors: Ryo Watanabe, Keisuke Fujii

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

In the quest to understand how matter behaves at its most fundamental level, physicists often turn to models that describe how electrons interact within a solid. One of the most persistent challenges in this field is predicting the lowest energy state of a system, a value that determines the stability and properties of materials like superconductors or magnets. For decades, classical computers have struggled with these calculations because the number of possible arrangements for electrons grows so rapidly that it overwhelms even the most powerful supercomputers. Quantum computers offer a potential solution, as they can naturally mimic these complex interactions. However, for a quantum computer to find the correct answer, it must first be fed a starting point—an initial state—that is already somewhat close to the true solution. If this starting point is too far off, the computer wastes time or fails to find the answer at all. The difficulty lies in creating these starting points efficiently, especially for large, two-dimensional grids of atoms where electrons are tightly coupled and behave in highly correlated ways.

A team of researchers has now designed a method to create these crucial starting points using very shallow, or short, quantum circuits. Their work focuses on a specific model of electron interaction known as the Hubbard model, which describes electrons hopping between sites on a grid while repelling each other. The researchers realized that when the repulsion between electrons is very strong, the system behaves almost like a simpler model called the Heisenberg model, which deals only with the spins of the electrons rather than their movement. They constructed a quantum circuit that first prepares an approximate state for this simpler spin model. This initial state is built by linking pairs of electrons into "singlets," a specific type of quantum connection where the spins of two particles cancel each other out. By arranging these links in a specific pattern across the grid and then applying a sequence of operations that allow the electrons to fluctuate slightly in their positions, the team created a circuit that closely mimics the complex behavior of the full system.

The team tested this approach by simulating the circuits on classical computers to see how well they performed. They began by optimizing the settings of their circuit on a small grid of four by four sites. Remarkably, when they took the exact same settings and applied them to much larger grids, up to ten by ten sites, the circuits continued to perform well without needing any further adjustment. This "parameter transfer" suggests that the physical principles guiding the small system hold true for larger ones, a finding that is vital because simulating larger systems directly is currently impossible for classical computers. To verify the quality of their results, the researchers compared their circuit states against highly accurate reference states generated by other advanced simulation techniques. They found that for a ten by ten grid, their method produced a starting state that was sufficiently close to the true ground state to be useful for future quantum calculations.

To ensure the method works for the full Hubbard model, the researchers added a specific set of operations designed to capture the rare moments when electrons briefly occupy the same spot or leave a spot empty. These operations, derived from a mathematical transformation known as the Schrieffer–Wolff transformation, act as a correction to the simpler spin-based state. When they tested this combined circuit on the small four by four grid, where they could calculate the exact answer, they found that adding these correction steps significantly improved the accuracy of the result. The final circuit, which combines the spin preparation with these charge-fluctuation corrections, is designed to be "hardware-friendly," meaning it uses a minimal number of steps and relies on connections that are likely to be available on future quantum machines.

The researchers calculated the resources required to run this circuit on a fault-tolerant quantum computer, which is a theoretical machine capable of correcting its own errors. They estimated that preparing the state for a ten by ten grid would require approximately 100,000 specific logic operations known as T gates. This number is considered low in the context of quantum computing, falling well within the range of what is expected to be achievable in the near future, a regime sometimes referred to as the "megaquop" era. By demonstrating that a simple, shallow circuit can generate a high-quality starting state for a complex, strongly correlated system, the team has provided a practical pathway for using quantum computers to solve long-standing problems in condensed matter physics. Their work suggests that by leveraging the relationship between simpler and more complex models, scientists can bypass the need for deep, error-prone circuits and move closer to simulating real-world materials.

Technical Summary: Low-Depth Initial-State Preparation for Ground-State Energy Estimation of Two-Dimensional Strongly Correlated Systems

Problem Statement
Quantum algorithms for ground-state energy estimation, such as Quantum Phase Estimation (QPE), require an initial state with non-negligible fidelity to the true ground state. The probability of successfully extracting the ground-state energy scales with this fidelity ( $F_0$ ); if $F_0$ decays exponentially with system size, the algorithm becomes intractable due to the required number of repetitions ( $1/F_0$ ). While Variational Quantum Eigensolvers (VQE) offer a hybrid approach, they face challenges in gradient evaluation costs and the classical simulability of shallow circuits. Furthermore, compiling accurate classical approximations (e.g., Matrix Product States) into quantum circuits often results in excessive circuit depth, which is prohibitive for early fault-tolerant quantum computers (FTQCs) where resources like logical qubits and magic states are limited. The central challenge is designing shallow, resource-efficient quantum circuits that prepare high-fidelity initial states for strongly correlated systems, specifically the two-dimensional (2D) Hubbard model, without relying on deep optimization loops or classical intractability.

Methodology
The authors propose a strategy to construct shallow quantum circuits by leveraging the strong-coupling connection between the 2D Heisenberg model and the half-filled Hubbard model via the Schrieffer–Wolff (SW) transformation. The methodology proceeds in three main stages:

Heisenberg Circuit Construction:
- Ansatz: The circuit targets the $S=0$ (singlet) subspace of the antiferromagnetic Heisenberg model. The initial state is a product of singlet dimers ( $| \Phi \rangle$ ) defined on a perfect matching of the lattice.
- Dynamics: The state is evolved using layers of exponentiated SWAP ( $e$ SWAP) gates. These gates respect SU(2) symmetry, ensuring the state remains in the singlet subspace.
- Structure: The lattice bonds are partitioned into four disjoint subsets ( $B_{x0}, B_{x1}, B_{y0}, B_{y1}$ ). The circuit applies $e$ SWAP gates in a specific sequence ( $B_{y1} \to B_{y0} \to B_{x1} \to B_{x0}$ ) across multiple blocks ( $N_b$ ). This structure mimics the resonating-valence-bond (RVB) physics of the Heisenberg ground state.
- Fermionic Mapping: Spin states are mapped to the singly occupied fermionic subspace using a local unitary embedding ( $V$ ) that introduces ancillary qubits.
Hubbard Extension via SW Gates:
- To capture charge fluctuations (doublons and holons) present in the Hubbard model at finite $U/t$ , the Heisenberg circuit is "dressed" with charge-fluctuation gates.
- These gates are derived directly from the first-order SW generator ( $\Omega$ ), approximated via a Suzuki–Trotter decomposition into two-site unitaries ( $W_{\langle i,j \rangle}$ ).
- The final variational ansatz combines the Heisenberg evolution ( $U_B$ ) and the charge-fluctuation block ( $W_B$ ), parameterized by rotation angles $\theta$ and $\phi$ .
Fidelity Estimation and Benchmarking:
- Parameter Transfer: Parameters optimized on a small $4 \times 4$ lattice are transferred to larger lattices ( $6 \times 6$ to $10 \times 10$ ) without further optimization.
- Simulation: Circuit states are simulated using Projected Entangled Pair States (PEPS) with belief propagation (BP)-guided simple updates.
- Fidelity Bounds: Since the exact ground state is unknown for large systems, the authors estimate a lower bound on the ground-state fidelity ( $F_0$ $F_{0}$ ). This is achieved by:
  - Calculating the fidelity ( $F_{ref}$ ) between the circuit state and a Variational Monte Carlo (VMC) reference state.
  - Bounding the VMC state's fidelity to the exact ground state ( $F_{ref,0}$ ) using the reference energy, the exact ground-state energy (from Stochastic Series Expansion), and the singlet excitation gap ( $\Delta_0$ ).
  - Applying the triangle inequality for Fubini–Study angles to derive a rigorous lower bound for $F_0$ .

Key Contributions and Results

Effective Parameter Transfer: For the Heisenberg model, parameters optimized on a $4 \times 4$ lattice yield robust fidelity lower bounds for lattices up to $10 \times 10$ without re-optimization. Specifically, for $N_b=3$ , the estimated ground-state fidelity lower bound remains positive ( $\approx 0.028$ ) even for the $10 \times 10$ lattice, suggesting the initial state is suitable for amplitude amplification.
Hubbard Fidelity Improvement: Exact calculations on a $4 \times 4$ lattice demonstrate that incorporating the SW-derived charge-fluctuation gates significantly improves the fidelity to the Hubbard ground state. For $N_b=4$ , the fidelity increases from $\approx 0.44$ (Heisenberg only) to $\approx 0.89$ (with optimized charge angles).
Resource Efficiency: The authors estimate the state-preparation cost for a $10 \times 10$ lattice. Assuming Clifford+ $T$ synthesis with a per-rotation error of $10^{-5}$ , the total cost is on the order of $10^5$ $T$ gates. This places the preparation cost well within the "megaquop" regime of early FTQCs and significantly lower than the costs associated with subsequent ground-state energy estimation algorithms.
Scalability: The circuit depth scales linearly with the number of lattice bonds ( $|B|$ ), and the parameter sharing strategy allows the construction of circuits for large systems based on small-system optimization.

Significance and Claims
The paper claims to offer a viable route to low-cost initial-state preparation for ground-state energy estimation in strongly correlated systems. By utilizing the physical connection between the Heisenberg and Hubbard models, the authors demonstrate that shallow circuits can achieve non-negligible ground-state fidelity on lattices beyond the reach of exact classical diagonalization. The results suggest that these circuits are attractive candidates for the early-FTQC regime, where minimizing circuit depth and $T$ -gate counts is critical. The work supports the hypothesis that small-lattice simulations can effectively guide the design of quantum circuits for larger, classically intractable systems, provided the parameters are transferable and the circuit depth is kept shallow. The authors explicitly note that while their bounds are rigorous, the actual fidelities may be higher, and future work could involve deeper Hamiltonian variational ansatzes or more accurate simulation techniques (e.g., full-update PEPS) to further refine these estimates.

Technical Summary: Low-Depth Initial-State Preparation for Ground-State Energy Estimation of Two-Dimensional Strongly Correlated Systems

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