arXiv:2510.05099

Low-depth fermion routing without ancillas

Nathan Constantinides, Jeffery Yu, Dhruv Devulapalli, Ali Fahimniya, Luke Schaeffer, Andrew M. Childs, Michael J. Gullans, Alexander Schuckert, Alexey Vyacheslavovich Gorshkov

Published Oct 6, 2025• 0 citations

arXiv HTML PDF

Abstract

Routing is the task of permuting qubits in such a way that quantum operations can be parallelized maximally, given constraints on the hardware geometry. When simulating fermions in the Jordan-Wigner encoding with qubits, a one-dimensional nearest-neighbor-connected geometry is effectively imposed on the system, independently of the underlying hardware, which means that naively, an $O(N)$ depth routing overhead is incurred. Recently, Maskara et al. [arXiv:2509.08898] demonstrated that this routing overhead can be reduced to $O(\log N)$ by decomposing general fermion routing into $O(\log N)$ interleave permutations of depth $O(1)$, using $Θ(N)$ ancillary qubits and employing measurements and feedforward. Here, we exhibit an alternative construction that achieves the same asymptotic performance. We also generalize the result in two ways. Firstly, we show that fermion routing can be performed in depth $O(\log^2 N)$ \emph{without} ancillas, measurements, or feedforward. Secondly, we construct efficient mappings with $O(\log^2 N)$ depth between all product-preserving ternary tree fermionic encodings, thereby showing that fermion routing in any such encoding can be done efficiently. While these results assume all-to-all connectivity, they also imply upper bounds for fermion routing in devices with limited connectivity by multiplying the fermion routing depth by the worst-case qubit routing depth.

DOI: 10.48550/arxiv.2510.05099

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arXiv:2510.05099

Low-depth fermion routing without ancillas

Nathan Constantinides, Jeffery Yu, Dhruv Devulapalli, Ali Fahimniya, Luke Schaeffer, Andrew M. Childs, Michael J. Gullans, Alexander Schuckert, Alexey Vyacheslavovich Gorshkov

Published Oct 6, 2025• 0 citations

arXiv HTML PDF

Abstract

O(N)

depth routing overhead is incurred. Recently, Maskara et al. [arXiv:2509.08898] demonstrated that this routing overhead can be reduced to

O(\log N)

by decomposing general fermion routing into

O(\log N)

interleave permutations of depth

O(1)

, using

Θ(N)

ancillary qubits and employing measurements and feedforward. Here, we exhibit an alternative construction that achieves the same asymptotic performance. We also generalize the result in two ways. Firstly, we show that fermion routing can be performed in depth

O(\log^2 N)

\emph{without} ancillas, measurements, or feedforward. Secondly, we construct efficient mappings with

O(\log^2 N)

depth between all product-preserving ternary tree fermionic encodings, thereby showing that fermion routing in any such encoding can be done efficiently. While these results assume all-to-all connectivity, they also imply upper bounds for fermion routing in devices with limited connectivity by multiplying the fermion routing depth by the worst-case qubit routing depth.

DOI: 10.48550/arxiv.2510.05099

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Succinct Fermion Data Structures

Joseph Carolan, Luke Schaeffer

Simulating fermionic systems on a quantum computer requires representing fermionic states using qubits. The complexity of many simulation algorithms depends on the complexity of implementing rotations generated by fermionic creation-annihilation operators, and the space depends on the number of qubits used. While standard fermion encodings like Jordan-Wigner are space optimal for arbitrary fermionic systems, physical symmetries like particle conservation can reduce the number of physical configurations, allowing improved space complexity. Such space saving is only feasible if the gate overhead is small, suggesting a (quantum) data structures problem, wherein one would like to minimize space used to represent a fermionic state, while still enabling efficient rotations. We define a structure which naturally captures mappings from fermions to systems of qubits. We then instantiate it in two ways, giving rise to two new second-quantized fermion encodings of $F$ fermions in $M$ modes. An information theoretic minimum of $\mathcal{I}:=\lceil\log \binom{M}{F}\rceil$ qubits is required for such systems, a bound we nearly match over the entire parameter regime. (1) Our first construction uses $\mathcal I+o(\mathcal I)$ qubits when $F=o(M)$ , and allows rotations generated by creation-annihilation operators in $O(\mathcal I)$ gates and $O(\log M \log \log M)$ depth. (2) Our second construction uses $\mathcal I+O(1)$ qubits when $F=Θ(M)$ , and allows rotations generated by creation-annihilation operators in $O(\mathcal I^3)$ gates. In relation to comparable prior work, the first represents a polynomial improvement in both space and gate complexity (against Kirby et al. 2022), and the second represents an exponential improvement in gate complexity at the cost of only a constant number of additional qubits (against Harrison et al. or Shee et al. 2022), in the described parameter regimes.Simulating fermionic systems on a quantum computer requires representing fermionic states using qubits. The complexity of many simulation algorithms depends on the complexity of implementing rotations generated by fermionic creation-annihilation operators, and the space depends on the number of qubits used. While standard fermion encodings like Jordan-Wigner are space optimal for arbitrary fermionic systems, physical symmetries like particle conservation can reduce the number of physical configurations, allowing improved space complexity. Such space saving is only feasible if the gate overhead is small, suggesting a (quantum) data structures problem, wherein one would like to minimize space used to represent a fermionic state, while still enabling efficient rotations. We define a structure which naturally captures mappings from fermions to systems of qubits. We then instantiate it in two ways, giving rise to two new second-quantized fermion encodings of $F$ fermions in $M$ modes. An information theoretic minimum of $\mathcal{I}:=\lceil\log \binom{M}{F}\rceil$ qubits is required for such systems, a bound we nearly match over the entire parameter regime. (1) Our first construction uses $\mathcal I+o(\mathcal I)$ qubits when $F=o(M)$, and allows rotations generated by creation-annihilation operators in $O(\mathcal I)$ gates and $O(\log M \log \log M)$ depth. (2) Our second construction uses $\mathcal I+O(1)$ qubits when $F=Θ(M)$, and allows rotations generated by creation-annihilation operators in $O(\mathcal I^3)$ gates. In relation to comparable prior work, the first represents a polynomial improvement in both space and gate complexity (against Kirby et al. 2022), and the second represents an exponential improvement in gate complexity at the cost of only a constant number of additional qubits (against Harrison et al. or Shee et al. 2022), in the described parameter regimes.

arXiv:2410.04015Leibniz international proceedings in informaticsOct 5, 20241 citation

O(N)

depth routing overhead is incurred. Recently, Maskara et al. [arXiv:2509.08898] demonstrated that this routing overhead can be reduced to

O(\log N)

by decomposing general fermion routing into

O(\log N)

interleave permutations of depth

O(1)

, using

Θ(N)

O(\log^2 N)

\emph{without} ancillas, measurements, or feedforward. Secondly, we construct efficient mappings with

O(\log^2 N)

O(\sqrt{N}\log N)

for

N

fermionic modes on 2D nearest-neighbor architectures. We present a fermionic permutation protocol tailored to 2D grid architectures that achieves the optimal

O(\sqrt{N})

depth with

O(N\sqrt{N})

nearest-neighbor gates and no ancilla qubits, measurements, or classical feedforward. For

N=L^2

modes, the CNOT depth is at most

10L+12

. This matches the

Ω(\sqrt{N})

lower bound, which holds even when

O(N)

ancillas and classical feedforward are permitted. We further construct an

O(\sqrt{N})

-depth transformation between the Jordan--Wigner, Bravyi--Kitaev, and Parity encodings on the 2D grid via a Hilbert-curve layout, extending our result to all three encodings. Benchmarks on a standalone fermionic permutation circuit, the fermionic fast Fourier transform, and Trotter simulation of a sparse SYK model demonstrate that the practical crossover is near the analytical depth-bound breakeven at

L=6

(

N=36

), followed by consistent reductions in circuit depth and spacetime volume and higher estimated no-fault probability in the early fault-tolerant regime, growing with system size.

N

fermionic modes in which each participates in at most a constant number

d

of interactions and study the circuit depth required to implement the Trotterized time evolution on qubit hardware with all-to-all connectivity. We introduce an encoding that augments each physical fermionic mode with a small number of auxiliary fermions, enabling the removal of Jordan--Wigner strings. Although the preparation of the auxiliary fermion state incurs an initial overhead, this state remains invariant under time evolution. As a result, long-time evolution can be implemented with asymptotically optimal circuit depth, reducing a previously multiplicative

O(\log N)

overhead to an additive overhead. Our results thus establish that the simulation of sparse fermionic models on qubit hardware matches the performance achievable on ideal fermionic hardware up to constant factors and

O(dN)

ancillary qubits.

Fermion lattices can be simulated by same-size qubit lattices with

\mathcal{O}(1)

interaction overhead

N

-site two-dimensional fermion lattice with no asymptotic overhead in the number of interactions and no space overhead. The primary overhead of our method is circuit depth, which on a qubit lattice matches that of fermionic swap networks, scaling as

\mathcal{O}(\sqrt{N})

, but reduces to

\mathcal{O}(\log N)

on reconfigurable qubit arrays and to

\mathcal{O}(1)

in lattice-surgery-based surface-code architectures. This is enabled by dynamically reorienting the Jordan-Wigner transformation to switch the lattice dimension along which locality is preserved. Furthermore, we study fermion routing, as required for the simulation of non-local interactions. When using qubit lattices, we reach resource scaling that asymptotically matches that of qubit routing, whilst on fully connected qubit devices, a depth scaling arbitrarily close to

\mathcal{O}(\log N)

is reached. This allows the fermionic fast Fourier transform to be implemented on qubit lattices with asymptotically optimal resource scaling under these locality constraints. Notably, all of our constructions naturally extend to

d

-dimensional lattices. Beyond scaling improvements, we show explicit examples of our method, including Fermi-Hubbard-model simulations of the square-, Lieb- and kagome lattice and the fermionic fast Fourier transform.

F

fermions in

M

modes. An information theoretic minimum of

\mathcal{I}:=\lceil\log \binom{M}{F}\rceil

qubits is required for such systems, a bound we nearly match over the entire parameter regime. (1) Our first construction uses

\mathcal I+o(\mathcal I)

qubits when

F=o(M)

, and allows rotations generated by creation-annihilation operators in

O(\mathcal I)

gates and

O(\log M \log \log M)

depth. (2) Our second construction uses

\mathcal I+O(1)

qubits when

F=Θ(M)

, and allows rotations generated by creation-annihilation operators in

O(\mathcal I^3)

gates. In relation to comparable prior work, the first represents a polynomial improvement in both space and gate complexity (against Kirby et al. 2022), and the second represents an exponential improvement in gate complexity at the cost of only a constant number of additional qubits (against Harrison et al. or Shee et al. 2022), in the described parameter regimes.