Generalized Superfast Encoding for Quantum Simulations

• Generalized Superfast Encoding (GSE) is an advanced method that maps fermionic systems to qubit Hamiltonians with optimized locality and built-in error detection.
• It leverages explicit Majorana-mode constructions, graph-theoretic structures, and custom Pauli-operator mappings to reduce circuit depth and resource overhead.
• GSE supports scalable error-mitigation protocols and efficient hardware implementations for quantum chemistry and strongly correlated models.

Generalized Superfast Encoding (GSE) is an advanced framework for mapping fermionic quantum many-body systems to qubit Hamiltonians, designed to optimize locality, error-robustness, and computational overhead in quantum simulations, particularly for quantum chemistry and strongly correlated models. GSE generalizes and improves upon previous edge-centric encodings by leveraging explicit Majorana-mode constructions, graph-theoretic auxiliary structures, and custom Pauli-operator mappings to realize simulator Hamiltonians with tunable Pauli weight and embedded quantum error detection or correction properties. This enables both lower circuit depth and built-in resilience to noise with minimal resource overhead, establishing GSE as a scalable and hardware-efficient fermion-to-qubit mapping for near-term molecular and lattice simulations (Setia et al., 2018, Hagge et al., 2023, Brown et al., 12 Nov 2025).

1. Algebraic Foundations and Operator Mapping

GSE begins by recasting the physical fermionic Hamiltonian—typically expressed in second quantization—as a polynomial in Majorana operators $\{c_{2j},c_{2j+1}\}$: $$c_{2j} = a_j + a_j^\dagger,\quad c_{2j+1} = -i(a_j - a_j^\dagger),\quad j=1,\ldots,m.$$ The fermionic even operator algebra is generated by vertex operators $B_j = -i c_{2j}c_{2j+1}$ and edge operators $A_{jk} = -i c_{2j}c_{2k}$. Any two-body fermionic interaction on interaction graph $G=(V,E)$ of maximum degree $d$ can be represented as a function of $\{B_j, A_{jk}\}$ (Setia et al., 2018, Brown et al., 12 Nov 2025).

GSE constructs local Majorana-mode analogues $\{\gamma_{i,p}\}$ at each vertex, with $d(i)$ logically independent Majoranas stored in $d(i)/2$ qubits per vertex. Pauli-string representatives $\tilde{B}_i$ and $\tilde{A}_{jk}$ are given by: $$\tilde{B}_i = (-i)^{d(i)/2}\gamma_{i,1}\cdots \gamma_{i,d(i)}, \qquad \tilde{A}_{jk} = \epsilon_{jk} \gamma_{j,p} \gamma_{k,q},$$ where the assignment of $\gamma$ labels aligns uniquely with edges, and orientation $\epsilon_{jk} = \pm1$ is arbitrary per edge (Setia et al., 2018, Hagge et al., 2023). On the codespace—defined soon—the operator algebra exactly mirrors the fermionic one, including all commutation and anticommutation relations, except for fermionic cycles, which map to codeword stabilizers.

2. Stabilizer Construction and Code Properties

To enforce the requisite fermionic loop algebra, GSE uses stabilizer generators corresponding to all independent cycles in $G$: $$S_\zeta = i^{|\zeta|} \prod_{(u\to v)\in \zeta} \tilde{A}_{u,v}.$$ The codespace is the simultaneous $+1$ eigenspace of all $S_\zeta$ (Setia et al., 2018, Hagge et al., 2023). For an $m$-vertex, $|E|$-edge, Eulerian connected $G$, this encodes $k = m-1$ logical qubits into $n = |E|$ physical qubits. The global fermion-parity condition is imposed via an additional stabilizer $\prod_i \tilde{B}_i = I$.

GSE admits variants with different coding-theoretic properties. If $G$ is 3-connected, of even degree $d \ge 6$, and with at most two edges between any pair, the encoding achieves code distance $d_{code}\geq 3$, enabling single-qubit error correction. The explicit construction assigns local Majorana operators so that every logical and stabilizer operator maintains constant or logarithmic Pauli weight (Setia et al., 2018).

For lower-degree graphs or where error correction is infeasible, a Fenwick-tree Majorana mapping minimizes operator weight $O(\log d)$, yielding “low-weight GSE” with only error-detection (code distance $d_{code}=2$ in typical configurations) (Setia et al., 2018, Hagge et al., 2023).

3. Graph-Theoretic Optimizations and Hardware Mapping

GSE generalizes beyond 1D chains or binary trees used in JW or Bravyi–Kitaev; it supports arbitrary interaction graphs and multi-edge (parallel edge) structures. For path optimization, GSE identifies minimal-Pauli-weight operator paths in the interaction graph for terms like $a_i^\dagger a_j$. Each such path $P$ yields a product of edge operators; the mapping chooses, for every term, the path $$P_{ij}^* = \arg\min_{P: i\to j}\,w(\tilde{A}_{ij}(P))$$ to minimize total Pauli weight (Brown et al., 12 Nov 2025).

Multi-edge graph extensions allow multiple parallel edges between the same fermionic modes, increasing code distance or detection strength by adding stabilizer loops. A key property is that—via cyclic Majorana assignment—each stabilizer remains of bounded (typically weight-6) Pauli weight, irrespective of code distance (Brown et al., 12 Nov 2025).

A recently introduced [[2N,N,2]] GSE variant is adapted to square-lattice and linear topologies: two qubits per fermionic mode, explicit local Majorana operators, and self-contained stabilizers conferring distance-2 error-detection or correction. Qubit layouts are optimized for minimal SWAP requirements and efficient orbital rotation compilation (depth $O(N)$) (Brown et al., 12 Nov 2025).

4. Error Detection, Correction, and Error-Mitigation Properties

One of GSE’s distinguishing features is the direct embedding of an error-detecting or -correcting stabilizer code compatible with the fermion-to-qubit mapping. In GSE, all single-qubit errors can be detected (code distance $d_{code}=2$) or, for appropriate graphs, corrected ($d_{code}\geq 3$), with stabilizers of constant or logarithmic Pauli weight (Setia et al., 2018, Hagge et al., 2023, Brown et al., 12 Nov 2025).

Formally, error detection proceeds by measuring all stabilizers (loop operators). Any (single-qubit) Pauli error anticommutes with at least one stabilizer and is therefore detected with probability 1; two-qubit errors are also detected with high probability ($p_a\simeq 1-1/(3m^2)$ on a square $m\times m$ lattice). The constructed logical operators always have weight $\geq 2$ (distance-2) or $\geq 3$ (distance-3) depending on the specific variant (Hagge et al., 2023).

By integrating error-mitigation via error detection directly in the encoding, resource overhead is moderate: added qubits per mode and moderate increases in two-qubit gate count and circuit depth (depth increase from 144 to 158 reported for a $4\times4$ model). The overhead is outweighed by the benefits of error mitigation when two-qubit gate fidelities exceed $99.2$–$99.9\%$, depending on system size (Hagge et al., 2023).

5. Measurement, Readout, and Clifford Stabilizer Protocols

Stabilizer measurement in GSE schemes originally used explicit ancilla circuits for each parity-check; multiple parallel loop (plaquette) checks can be performed in low depth via sequences of Pauli-controlled-Pauli gates (Hagge et al., 2023). Later advances realized Clifford-based global rotations that map all logical and stabilizer operators to the $Z$-basis, followed by simultaneous measurement of all physical qubits and post-selection on valid stabilizer outcomes. This approach avoids mid-circuit measurements and preserves code distance, enabling single-shot syndrome collection and improved energy estimation accuracy (Brown et al., 12 Nov 2025).

In practice, the entire projective measurement is simulated with Clifford stabilizer simulators (such as Stim), and the extracted Clifford circuit is inverted on the state-preparation circuit, ensuring all relevant logical observables and stabilizers are mapped directly into $Z$-basis measurements.

6. Physical Resource Scaling and Performance Benchmarks

Numerical and experimental benchmarks demonstrate that, for representative molecules and lattice Hamiltonians, GSE achieves lower Pauli weight and circuit depth than Jordan–Wigner (JW) or Bravyi–Kitaev Superfast Encoding (BKSF/OSE). For example, in propyne (19 orbitals):

• JW: average Pauli weight 12.85, max 38, depth $5.1\times 10^6$ gates (38 qubits).
• Path-optimized GSE: average Pauli weight 11.9, max 16, depth $1.2\times 10^6$ gates (342 qubits).

For $(\mathrm{H}_2)_2$ and $(\mathrm{H}_2)_3$ simulations under realistic noise models, GSE ($d=2,3$) recovers over 90% of true correlation energy with 50% or fewer accepted shots; JW recovers $\lesssim25\%$ correlation energy. Increasing code distance further improves energy error suppression (Brown et al., 12 Nov 2025).

On hardware (IBM Kingston, 8-orbital system), GSE with double qubit resources reduces RMSE by a factor of two compared to JW (Brown et al., 12 Nov 2025).

7. Comparison With Alternative Encodings and Regimes of Applicability

GSE improves upon original Superfast Encoding (OSE/BKSF) by enabling lower Pauli weight ($O(\log d)$ with Fenwick-tree mapping), constant-weight stabilizers for higher code distances, and hardware-compatible measurement protocols. Compared to Jordan–Wigner and Bravyi–Kitaev, GSE is more favorable when the target interaction graph is non-1D, of moderate to high degree, or when robust embedded error detection/correction is desired without excessive overhead (Setia et al., 2018, Hagge et al., 2023, Brown et al., 12 Nov 2025).

For systems with highly overlapping single-particle bases or dense two-body interactions, path optimization strategies within GSE are essential to maintain low Pauli weight and circuit depth. When applied to highly localized orbital lattices, GSE enables significant resource reductions and superior noise robustness (Brown et al., 12 Nov 2025).

Encoding	Qubit Overhead	Pauli Weight	Code Distance	Stabilizer Weight
JW	$M$	$O(M)$	1	—
BK	$M$	$O(\log M)$	1	—
OSE/BKSF	$O(md)$	$O(d)$	1 or 2	$O(d)$
GSE (error-detecting)	$O(md)$	$3$–$4$ (lattice)	2	$6$
GSE (error-correcting)	$O(md)$	$O(d)$	$\geq3$	$O(d)$
GSE (low-weight)	$O(md)$	$O(\log d)$	1–2	$O(\log d)$

• "Superfast encodings for fermionic quantum simulation" (Setia et al., 2018)
• "Error mitigation via error detection using Generalized Superfast Encodings" (Hagge et al., 2023)
• "Efficient and Noise-Resilient Molecular Quantum Simulation with the Generalized Superfast Encoding" (Brown et al., 12 Nov 2025)