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Algebraic Compression of Free Fermionic Quantum Circuits: Particle Creation, Arbitrary Lattices and Controlled Evolution

Efekan Kökcü, Daan Camps, Lindsay Bassman Oftelie, Wibe A. de Jong, Roel Van Beeumen, A. F. Kemper

TL;DR

The paper tackles efficient simulation of time evolution for free-fermion–mapped Hamiltonians on quantum hardware by extending algebraic compression from nearest-neighbor systems to long-range hopping and particle-number-changing processes. It introduces Q-blocks and Q-diamonds, enabling fixed-depth circuit compression for controlled evolutions and creation/annihilation terms, and combines them with fermionic swap gates to realize arbitrary lattice connectivity. Key results include compression of long-range fermionic dynamics on 2D lattices, adiabatic state preparation incorporating creation operators, and Zak-phase calculations for Creutz-Hubbard models using Hadamard-test circuits implemented on contemporary hardware and emulators. The work broadens the class of Hamiltonians amenable to efficient quantum simulation and provides practical tooling (F3C) to translate these theoretical constructs into compact transpiled circuits for near-term devices.

Abstract

Recently we developed a local and constructive algorithm based on Lie algebraic methods for compressing Trotterized evolution under Hamiltonians that can be mapped to free fermions. The compression algorithm yields a circuit which scales linearly in the number of qubits, has a depth independent of evolution time and compresses time-dependent Hamiltonians. The algorithm is limited to simple nearest-neighbor spin interactions and fermionic hopping. In this work, we extend our methods to compress evolution with long-range fermionic hopping, thereby enabling the embedding of arbitrary lattices onto a chain of qubits for fermion models. Moreover, we show that controlled time evolution, as well as fermion creation and annihilation operators can also be compressed. We demonstrate our results by adiabatically preparing the ground state for a half-filled fermionic chain, simulating a $4 \times 4$ tight binding model on ibmq washington, and calculating the topological Zak phase on a Quantinuum H1-1 trapped-ion quantum computer. With these new developments, our results enable the simulation of a wider range of models of interest and the efficient compression of subcircuits.

Algebraic Compression of Free Fermionic Quantum Circuits: Particle Creation, Arbitrary Lattices and Controlled Evolution

TL;DR

The paper tackles efficient simulation of time evolution for free-fermion–mapped Hamiltonians on quantum hardware by extending algebraic compression from nearest-neighbor systems to long-range hopping and particle-number-changing processes. It introduces Q-blocks and Q-diamonds, enabling fixed-depth circuit compression for controlled evolutions and creation/annihilation terms, and combines them with fermionic swap gates to realize arbitrary lattice connectivity. Key results include compression of long-range fermionic dynamics on 2D lattices, adiabatic state preparation incorporating creation operators, and Zak-phase calculations for Creutz-Hubbard models using Hadamard-test circuits implemented on contemporary hardware and emulators. The work broadens the class of Hamiltonians amenable to efficient quantum simulation and provides practical tooling (F3C) to translate these theoretical constructs into compact transpiled circuits for near-term devices.

Abstract

Recently we developed a local and constructive algorithm based on Lie algebraic methods for compressing Trotterized evolution under Hamiltonians that can be mapped to free fermions. The compression algorithm yields a circuit which scales linearly in the number of qubits, has a depth independent of evolution time and compresses time-dependent Hamiltonians. The algorithm is limited to simple nearest-neighbor spin interactions and fermionic hopping. In this work, we extend our methods to compress evolution with long-range fermionic hopping, thereby enabling the embedding of arbitrary lattices onto a chain of qubits for fermion models. Moreover, we show that controlled time evolution, as well as fermion creation and annihilation operators can also be compressed. We demonstrate our results by adiabatically preparing the ground state for a half-filled fermionic chain, simulating a tight binding model on ibmq washington, and calculating the topological Zak phase on a Quantinuum H1-1 trapped-ion quantum computer. With these new developments, our results enable the simulation of a wider range of models of interest and the efficient compression of subcircuits.
Paper Structure (20 sections, 2 theorems, 82 equations, 12 figures)

This paper contains 20 sections, 2 theorems, 82 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Notation and recap
  3. Notation for circuit elements
  4. Block Definition
  5. TFIM and TFXY Block Mappings
  6. New Block Mapping Results
  7. Fermionic Swap Gate and Long-Range Hoppings
  8. Fermion Annihilation and Creation
  9. New type of block: Q-Block
  10. Q-block and Q-compression
  11. $Q$-block mapping for Controlled Free Fermions
  12. $Q$-block mapping for Controlled Free Fermions with Creation
  13. Discussion and Outlook
  14. Fermion Creation with Definite Momentum
  15. $Q$-block Mappings for 1-D Spin Models with Periodic Boundary Condition
  16. ...and 5 more sections

Key Result

Theorem 1

A diamond of height $n$ can be merged with any $B$-block $B_i$ with $i=1,2,...,n$ and $Q$-block $Q$, For odd values of $n$, merging $Q$ with the diamond requires only one $Q$-fusion, while merging $B_1$ requires $n-1$ turnovers and $1$ fusion. For even values of $n$, merging $Q$ with the diamond requires $n-1$ turnovers and $1$ fusion, while merging $B_1$ requires $1$ fusion. Merging $B_i$ with $

Figures (12)

  • Figure 1: Summary of the results from Refs. kokcu2022algebraiccamps2022algebraic that we will use in this work. Panels (a-c) illustrate the block properties listed in \ref{['def:block']}. Panel (d) illustrates the triangle structure and how it can absorb a $B$-block via the $B$-block properties. Panel (e) is an illustration of the usage of $B$-block properties to transform a triangle into a shallower square structure. Panels (f) and (g) illustrate two different realizations or representations of the abstract $B$-block objects as circuit elements, i.e. $B$-block mappings, for transverse field Ising model (TFIM) and transverse field XY (TFXY) model. Note that $B$-block mappings are not limited to TFIM and TFXY, and more examples can be found in Refs. kokcu2022algebraiccamps2022algebraic.
  • Figure 2: (a) An illustration of compression of long range hoppings. Any non-local fermion hopping can be written in terms of nearest neighbor hoppings as shown in \ref{['eq:long_range_hopping']}, which then can be compressed into a TFXY square. The other panels illustrate the simulation results from ibm_washington of a free-fermion on a two-dimensional (2D) lattice. Specifically: (b) Schematic of the 16-site, 2D lattice, color-coded by the distance $M$ each lattice site is from the reference lattice site, labeled '0'. The fermion is initialized at reference lattice site '0', and is allowed to evolve freely in time. (c) The topology of the qubits in the ibmq_washington quantum processing unit (QPU) is shown, along with how the lattice sites from the schematic in panel (b) are mapped to the particular 16 qubits used in these simulations. (d) The occupation number at each distance $M$ versus time with no disorder in the lattice. This leads to ballistic transport of the fermion. The left-hand plot shows results from the QPU, while the right-hand plot shows results from a noise-free quantum simulator. (e) The occupation number at each distance $M$ versus time with random disorder in the lattice. This leads to Anderson localization of the fermion. The left-hand plot shows results from the QPU, while the right-hand plot shows results from a noise-free quantum simulator.
  • Figure 3: In panel (a), we show the block mapping that covers the free fermions with creation. By using these, we can build fermionic swaps, and carry the creation-annihilation operator on the first qubit to any other qubit. Panel (b) show the complete set of blocks for 4 qubits. In panel (c), we show the final circuit representing the square structure with this particular block mapping, that can be obtained by using the compression theorems. In this form, the circuit requires $2(n-1)(n+1) = 2(n^2-1) = O(2n^2)$ CNOT gates. By using the relation given in panel (d) on every other $X-XX$ vertical stripe, certain $XX$ gates can be grouped as shown in panel (e). Then using the relation \ref{['eq:tfim2tfxy']}, these groups can be transformed into free fermionic gates. After this simplification, the number CNOTs is reduced to $(n-1)(n+2) = O(n^2)$, which is approximately half of the CNOT count of the circuit in panel (c).
  • Figure 4: Numerical results for the adiabatic state preparation for the 1D chain via fermionic compression. (a) Instantaneous eigenstates and the result of the compressed time evolution with the parameters discussed in the text. Inset: a close-up view near the end of the evolution, with the instantaneous eigenvalues of $\mathcal{H}_0$ (orange dashed lines) and $\mathcal{H}=\mathcal{H}_0 + \mathcal{H}'$ (gray lines). (b) Time evolution of the chemical potential $\mu(t)$ and symmetry-breaking field $\lambda(t)$.
  • Figure 5: (a-c) $Q$-block properties given in \ref{['def:q-block']}. (d-e) The diamond structure defined in \ref{['def:q-diamond']} with heights $n=5$ and $n=6$. Notice that the alternating nature of $B_1$ and $Q$ in the diamond makes the top right corner of the odd height and even height diamonds differ from each other. This leads to different compression sequences for even-odd height diamonds.
  • ...and 7 more figures

Theorems & Definitions (9)

  • Definition 1: $B$-block kokcu2022algebraiccamps2022algebraic
  • Definition 2: $Q$-block
  • Definition 3: diamond
  • Theorem 1: $Q$-compression
  • proof
  • Definition 4: $P$-Block
  • Definition 5: $P$-Diamond
  • Theorem 2: $P$-compression
  • proof