Efficient quantum simulation for translationally invariant systems

TL;DR

This work develops and demonstrates a framework for efficient quantum simulation of translationally invariant lattice systems by exploiting discrete spatiotemporal symmetry in quantum circuits. It introduces tileable circuit concepts (basis graphs, basis circuits, and patches), formalizes their routing via SMT-based transpilation (QuanTile), and shows how Trotterization preserves tileability across multiple models. By applying specialized routing, the method dramatically reduces depth overhead and SWAP counts, enabling large-scale simulations on near-term devices and fault-tolerant architectures alike. The approach is illustrated across diverse models (Rule 54, Rokhsar-Kivelson, Fermi-Hubbard, Kogut-Susskind QED, and more), and is complemented by a basis-graph database, benchmarking against standard routers, and a clear pathway to broader compilation tasks in quantum computation.

Abstract

Discrete translational symmetry plays a fundamental role in condensed matter physics and lattice gauge theories, enabling the analysis of systems that would otherwise be intractable. Despite this, many open problems remain. Quantum simulation promises to offer new insights, but progress is often limited by device connectivity constraints, which lead to prohibitively long computation times. We extend the use of spatial symmetry from the systems to be simulated to the quantum circuits simulating them. One application is that it becomes possible to efficiently and optimally alleviate device connectivity constraints algorithmically. This leads to reductions in quantum computational time by several orders of magnitude even for moderate system sizes, making such simulations feasible, with even greater relative gains for larger systems. This substantially enhances the capabilities of quantum computers in the simulation of condensed matter systems and lattice gauge theories, even before hardware improvements. Our work forms the basis for using spatial symmetry of quantum circuits in other areas of quantum computation, such as in the design and implementation of quantum error correcting codes.

Figures (13)

• Figure 1: (a) Basis circuit (bold edges) for the quantum simulation of the Kogut-Susskind model of 2D QED. Dashed lines delineate the circuit cells, with coordinates in gray, and seed numbers within the vertices. The Jordan-Wigner transformation breaks 2D translational symmetry, which we circumvent by using the compact encoding derby2021compactclinton2024towards, leading to the red auxiliary qubits. (b) Qudit mobility zone. The logical qudits of the basis circuit (black worldlines) are free to move by the action of SWAPs (omitted) along the edges in a patch of $2\delta+1$ by $2\delta+1$ hardware cells (here, $\delta=1$ and edges are likewise omitted). The logical qudits are acted upon by single-qudit (dark blue dot) and two-qudit (dark orange edge) physical gates. In this example, the physical circuit (black lines, dark blue dots, and dark orange edges) is repeated to form a $3\times1$ physical circuit patch. These physical basis circuits interact; a two-qudit gate from a translated copy of the physical basis circuit (wavy orange line) acts between a logical qudit from the original basis circuit and a logical qudit from a translated copy (green line). (c) Benchmarking results comparing our method's implementation ("QuanTile", dashed lines) kattemolle2024quantile with Qiskit's AIRouter qiskit2024quantum (solid lines).
• Figure 2: Gate scheduling. (a) A basis circuit is specified as a list of gates. Dashed lines delineate the central cell of the basis circuit. In this example, the basis circuit is defined on a 1D chain. (b) Treating the basis circuit as a standard circuit results in incorrect gate scheduling; the depicted circuit leads to gate collisions when the circuit is used to generate a circuit patch. (c) The correct scheduling method is to first impose periodic boundary conditions on the circuit. In this picture, time runs radially outward in integer steps. Gates are assigned times accordingly. (d) Periodic boundary conditions are lifted, yielding the correctly scheduled basis circuit.
• Figure 3: DAG representation of the circuit in Fig. \ref{['fig:scheduling']}c. Source nodes are white and sink nodes are gray. Each gate node also displays the qudits it acts on as an ordered list. This information is necessary; otherwise, it would be impossible to determine, for example, in the case of a CNOT gate, which of the two incoming edges represents the control qudit and which represents the target qudit.
• Figure 4: Hybrid encoding of the 2D Fermi-Hubbard model. The green lines depict the 2D square lattice on which the original Hamiltonian [Eq. (\ref{['eq:FH']})] is defined. The orbitals [labelled $i$ in Eq. (\ref{['eq:FH']})] reside on the intersections of the green lines (one orbital per intersection). Each orbital consists of two spin-orbitals [labelled ${\sigma}$ in Eq. (\ref{['eq:FH']})]. The spin-up orbitals are associated with qubits placed at nodes 1--4 inside the central unit cell. The spin-down orbitals arise in altered copies of this central cell, as explained later. A Pauli operator close to a qubit indicates it acts on that qubit and operators of the same color connected by a (multi)edge act simultaneously. Operators of the type $E_{{i{\sigma}},{j{\sigma}}}V_{j{\sigma}}$ from Eq. (\ref{['eq:FHev']}) are mapped to the blue Pauli operators, while those of the type $V_{i{\sigma}}E_{{i{\sigma}},{j{\sigma}}}$ are mapped to the orange operators, and operators of the type $V_{i\uparrow}V_{i\downarrow}$ are mapped to the red operators. Operators of the type $V_{i{\sigma}}$ (not shown) map to $Z_{i{\sigma}}$, a single-qubit Pauli-$Z$ operator at $i{\sigma}$. These operators hit only those qubits already having a $Z$. The full encoding of a patch of $n$ by $m$ orbitals is obtained by repeating the shown structure in $n$ by $m$ unit cells, where the arrows (and the Pauli operators attached to them) are reversed in spatially alternating unit cells (which we call the even and odd cells). The even cells encode the spin-up orbitals, whereas the odd cells encode the spin-down orbitals. Additionally, a minus sign is added to both the reversed bottom left and top reversed top right multi-edge. Finally, the boundaries need to be patched with spin-orbitals so that only complete orbitals arise.
• Figure 5: Basis circuit for the implementation of the time evolution along the Fermi-Hubbard Hamiltonian [Eq. (\ref{['eq:FH']})], up to Trotter error, with $\alpha_T=\frac{T}{2}\frac{t}{r}$ and $\alpha_U=-\frac{U}{4}\frac{t}{r}$. The circuit comprises 16 general 2-qubit gates and has a depth of 15. The shown circuit is for the even cells, the circuit for the odd cells is obtained by sending $\alpha_T\rightarrow -\alpha_T$ for gates 1,4,8,11. The circuit is tileable.

Theorems & Definitions (2)

• Theorem 1
• proof