Low-depth fermion routing without ancillas

TL;DR

This work shows that arbitrary fermion permutations under the Jordan-Wigner mapping can be implemented in depth $O(\log^2 N)$ without ancillas by decomposing permutations into staircase layers and realizing each layer with polylog-depth CZ-fanouts and SWAPs. It further extends these ideas to product-preserving ternary-tree fermion-to-qubit encodings, establishing $O(\log^2 N)$-depth state-and-operator mappings between encodings, and providing practical circuits for inter-encoding transformations. The results imply polylogarithmic-depth primitives for fermionic simulation, including the fermionic fast Fourier transform (FFFT) and single-Trotter-step time evolution, with ancilla-assisted improvements potentially reducing depth to $O(\log N)$. Together, these contributions offer substantial depth reductions for fermionic quantum computation on all-to-all connected hardware and yield meaningful bounds for more realistic constrained architectures. Overall, the work advances efficient fermion routing beyond ancilla-based schemes and broadens applicability to a wide class of fermionic encodings, with direct impact on quantum simulation of materials and chemistry.

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.

Figures (15)

• Figure 1: (a) The $\mathop{\mathrm{CZ}}\nolimits$-fanout gate (left) is defined as a circuit consisting of one $\mathop{\mathrm{CZ}}\nolimits$ gate between the control qubit (filled circle) and each target qubit (open circles). (b) The ordinary ($\mathop{\mathrm{CNOT}}\nolimits$) fanout is related to the $\mathop{\mathrm{CZ}}\nolimits$-fanout through conjugating each target qubit by Hadamards.
• Figure 2: The circuit for performing the swap of modes $i$ and $j$ in the Jordan-Wigner encoding. The two $\mathop{\mathrm{CZ}}\nolimits$-fanouts in the final equality are generated by commuting each $\mathop{\mathrm{CZ}}\nolimits$ gate to the front of the $\mathop{\mathrm{SWAP}}\nolimits$ circuit, and then collapsing the $\mathop{\mathrm{SWAP}}\nolimits$ operators into a single $\mathop{\mathrm{SWAP}}\nolimits$ of modes $i$ and $j$.
• Figure 3: In this example, each element (indicated by a circle) is labeled with its target position under a permutation $\sigma$. The first step of implementing this permutation using staircase permutations is shown. Red marked elements have destinations across the dashed midpoint of the array, and are arranged in pairs and swapped in one staircase permutation step. Following this staircase step, the algorithm is run recursively on each side of the midpoint, treating the left and right halves separately.
• Figure 4: The effect of conjugating the targets of a $\mathop{\mathrm{CZ}}\nolimits$-fanout with the parity ($P$) gate, as defined in Eq. \ref{['eq:parity']}. The fanout, which has control $0$ and targets $\{i, \dots, j\}$, becomes two $\mathop{\mathrm{CZ}}\nolimits$ gates, one $\mathop{\mathrm{CZ}}\nolimits_{0, i - 1}$ and another $\mathop{\mathrm{CZ}}\nolimits_{0, j}$. If the first qubit in $P$ is $i$, then the $\mathop{\mathrm{CZ}}\nolimits_{0, i - 1}$ gate is not present.
• Figure 5: An example of a contiguous-range $\mathop{\mathrm{CZ}}\nolimits$-fanout circuit. When transformed into the parity basis, the $\mathop{\mathrm{CZ}}\nolimits$-fanouts turn into a sequence of individual $\mathop{\mathrm{CZ}}\nolimits$ gates.

Theorems & Definitions (30)

• Theorem 1
• Corollary 1
• Definition 1: Fermion-to-qubit map
• Definition 2: Fermion permutation
• Definition 3: Fermionic swap
• Theorem 2
• Lemma 1
• proof
• Lemma 2
• proof