Compiling the 2D Fermi-Hubbard ground-state energy estimation algorithm for active volume quantum architectures

AI summary

Overview

Research area: Fault-tolerant quantum computing, specifically architecture-aware circuit compilation and resource estimation for quantum simulation of the two-dimensional Fermi-Hubbard model.
Technical level: Advanced.
Scope: This paper compiles the 2D Fermi-Hubbard ground-state energy estimation algorithm for active-volume quantum architectures and reports reductions in active volume, Toffoli count, and projected runtime for lattice sizes from (L=4) to (L=20).

What This Paper Is About

Early fault-tolerant quantum computers will have severely limited logical qubits and execution time, so estimating and reducing the true cost of algorithms matters. Conventional resource estimates often use (T)-count or circuit volume, which can misrepresent cost on architectures with limited non-local connectivity. The paper addresses this gap by compiling the Fermi-Hubbard ground-state energy estimation algorithm directly for an active-volume architecture, where the optimization target accounts for both Clifford and non-Clifford execution cost.

Key Contributions

1. First active-volume-aware compilation of an FTQC lattice Hamiltonian algorithm. The authors compile the 2D Fermi-Hubbard ground-state energy estimation algorithm using quantum phase estimation and Trotterized time evolution for square lattices from (L=4) to (L=20), and state that no active-volume-aware compilation of an FTQC lattice Hamiltonian algorithm had been reported previously.
2. Up to (3.9\times) active volume reduction. The proposed compilation reduces active volume by up to (3.9\times) compared with prior work optimized for non-Clifford cost. Separately, conventional circuit volume overestimates the baseline active volume by up to (12.1\times) for the (20\times20) lattice.
3. State-of-the-art Toffoli counts. For the (20\times20) lattice, the circuits reduce the Toffoli count by (3.37\times) relative to Campbell [20] and (1.96\times) relative to Kan et al. [52], which the authors describe as the lowest reported Toffoli count for this benchmark to date.
4. Runtime and space-time analysis. Using the block scheduler from Ref. [41], the authors translate compilation improvements into runtime estimates on an illustrative photonic implementation. For (L=8), the improved circuit is (73\times) faster than the baseline, with a runtime of (4.57,\mathrm{s}) rather than (332,\mathrm{s}).

Main Findings

• Circuit volume is a poor cost proxy for this architecture: For the (20\times20) lattice, conventional circuit volume exceeds the baseline active volume by (12.1\times), showing that (T)-count-oriented volume estimates substantially overstate execution cost on an active-volume architecture.
• Compilation delivers a further (3.9\times) active volume reduction: The improvement holds across (L=4) to (20), with the largest reported reduction at (3.9\times). After compilation, total active volume remains nearly constant across the lattice sizes considered, at around (4\times10^{7}) logical blocks.
• Both Clifford and non-Clifford costs are reduced: The largest absolute reduction comes from the Clifford contribution, because the baseline was optimized primarily for (T)-count. For the (20\times20) lattice, the Clifford and non-Clifford contributions approach an approximately (1:1) split.
• Improvements are distributed across subroutines: The largest subroutine-level reduction occurs in the fermionic swap network, but appreciable reductions also appear in other subroutines. The optimized circuit has a more balanced distribution of active volume than the baseline, where the fermionic swap network increasingly dominates.
• Toffoli counts improve despite a conservative comparison: The authors explicitly compile the controlled QPE implementation and restrict the number of queries rather than scaling a single query. For the (20\times20) lattice, they report (3.37\times) and (1.96\times) reductions against Refs. [20] and [52], respectively, and an approximately (2\times) reduction for the (L=20) case.
• Runtime improves dramatically for matched qubit budgets: For (L=8), the improved circuit uses 220 logical circuit qubits versus 214 for the baseline, corresponding to 110 versus 107 workspace qubits, a (2.8%) increase. Despite this small difference, the improved circuit runs in (4.57,\mathrm{s}) versus (332,\mathrm{s}), a (73\times) speedup attributed mostly to compilation.
• Batching offers a favorable space-time trade-off: Partitioning parallel rotations into four batches reduces the logical qubit requirement for the (L=20) lattice by nearly (25%) while increasing active volume by only around (3%).
• Sine-window QPE reduces query counts: At an example point where the product of QPE energy error and evolution time equals (0.05), sine-windowed QPE requires 32 queries, whereas the entanglement-free QPE variant requires 96.
• Favorable scaling is recovered after compilation: Under the extensive error target of (0.0051tL_xL_y), the gate cost per Trotter step grows with lattice size but fewer Trotter steps are needed, so overall gate cost remains approximately constant. The baseline active volume did not show this behavior because increasing Clifford overhead dominated; the improved compilation largely restores it.
• Runtime can be traded against qubit count: For the (20\times20) circuit, increasing logical qubit capacity reduces runtime until the reaction-limited regime is reached, revealing an approximately order-of-magnitude space-time trade-off between qubit count and runtime.

Methodology in Plain English

The authors study the single-band 2D Fermi-Hubbard model on an (L\times L) square lattice with periodic boundary conditions, using hopping amplitude (t=1) and interaction strength (u=8). Their target is the ground-state energy to an extensive error of (0.0051tL_xL_y), so that the error in energy density stays fixed as the lattice grows.

They estimate the ground-state energy with quantum phase estimation applied to a Trotterized time-evolution operator. The Hamiltonian is split into three parts: the on-site interaction term and two disjoint sets of hopping terms on plaquettes. A second-order Suzuki-Trotter formula approximates the time evolution. The paper compares a baseline circuit optimized for (T)-count with a new compilation optimized for active volume.

The active volume metric counts the computational volume of logical operations in units of logical blocks. For example, a CNOT costs 4 logical blocks and a Toffoli costs 47 logical blocks. The active-volume architecture separates data qubits from reusable workspace qubits and provides limited non-local connectivity, with each qubit connected to (O(\log N)) others. This reduces idle volume and routing overhead compared with strictly local architectures.

The compilation improvements include adopting sine-window QPE, using directionally controlled time-evolution unitaries, reordering Suzuki-Trotter terms and merging consecutive evolutions, implementing efficient controlled and directionally controlled Hamming weight phasing with phase fix-ups, reducing the fermionic SWAP network, designing active-volume-efficient gates using oriented ZX diagrams, batching rotations to trade qubits against depth, and re-optimizing the division of the error budget.

All circuits are implemented in the open-access PsiQuantum software package, which compiles the circuits, reports gate counts, and computes active volume using extended lookup tables from Ref. [62]. The authors explicitly include the control structure required for QPE. They exclude initial-state preparation, window-state preparation, and the inverse QFT from their cost accounting, noting that the latter two are negligible compared with the query cost, while initial-state preparation is an active research area. Runtime estimates use the block scheduler from Ref. [41] on an illustrative photonic implementation of the active-volume architecture, with the machine’s logical qubit capacity set to (1.5\times) the circuit’s peak data-qubit requirement to avoid the reaction-limited regime.

Why This Matters

This work shows that architecture-aware compilation can change the practical cost of a flagship quantum simulation application by a large factor, not just by a constant overhead. It provides the first active-volume resource estimates for a lattice Hamiltonian algorithm and demonstrates co-design across algorithm, architecture, compilation, and execution scheduling. The results argue that (T)-count alone is insufficient for early fault-tolerant planning, because Clifford costs and architecture-specific execution constraints can dominate or substantially alter the resource picture.

Real-world applications:

• Correlated materials modeling: The Fermi-Hubbard model is a stepping stone toward cuprate, pnictide, and extended lattice models relevant to understanding strongly correlated electron systems.
• Quantum chemistry and many-body simulation: The paper frames lattice Hamiltonian simulation as part of the broader effort to simulate many-body systems and chemistry on quantum computers.
• Fault-tolerant architecture design: The results provide a benchmark for evaluating photonic, neutral-atom, and superconducting architectures with long-range couplers, as well as LDPC and concatenated-code approaches

Authors’ abstract

As quantum computing enters the early fault-tolerant era, circuit compilation choices will increasingly depend on details of the underlying architecture rather than solely optimizing for generic proxies such as non-Clifford count. We present an active-volume-aware compilation of the ground-state energy estimation algorithm for the two-dimensional Fermi-Hubbard model using quantum phase estimation and Trotterized time evolution. The proposed compilation reduces the active volume across $L\times L$ square lattices with $L=4$ to $20$, achieving up to a $3.9\times$ reduction over prior work optimized for non-Clifford cost. As a by-product of these compilation improvements, the resulting circuits also achieve state-of-the-art Toffoli counts, with a ~$2\times$ reduction for the $L=20$ case. Lastly, the active volume architecture and recent execution scheduling advances provide a means of translating these reduction trends into runtime. This demonstrates the increasing importance of architecture-aware compilation for practical early fault-tolerant quantum computing.