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The Encoding Gauge of Fermionic Variational Quantum Algorithms

S. M. Yousuf Iqbal Tomal, Abdullah Al Shafin

Published Jan 14, 2026• 0 citations
Publisher

Abstract

Whether a fermionic variational quantum algorithm offers a genuine quantum advantage depends on whether its outputs are classically hard to compute. We show that for such algorithms classical simulability is not an intrinsic property of the circuit. It depends on the fermion-to-qubit encoding. Standard encodings are related by Clifford transformations, so they form a gauge group, and we classify the standard simulability resources by how they transform under it. The Pauli weight, the causal-cone width, and the entanglement that sets the cost of tensor-network contraction are gauge-covariant: a generic interacting Hamiltonian whose truncated Pauli-propagation cost grows linearly in the system size under the Jordan-Wigner encoding becomes logarithmic under a tree encoding, and we exhibit the resulting change of verdict at a fixed physical ansatz. The dynamical-Lie-algebra dimension and the non-stabilizerness are gauge-invariant. From this split we prove that the encoding-optimized simulation cost isbounded by the invariant resources, so genuine, encoding-robust hardness requires both the dynamical Lie algebra and the magic to be large; the covariant weight is necessary but never sufficient. These routes are sufficient certificates of simulability, not a complete characterization, so the floor is an upper bound and a full classification of representation-invariant cost functionals is left open. We confirm both invariant simulation routes numerically, and we show that the dynamical-Lie-algebra dimension separates simulable from hard instances where the Pauli weight cannot; we also construct an explicit one-parameter interacting family for which all three certificate routes fail at once, so the floor is achieved. We then establish a sharp dichotomy with trainability. Because the cost landscape is encoding-invariant, the gradient and the barren-plateau variance are exactly gauge-invariant, which we verify to machine precision; we show further that this invariance follows from faithfulness rather than Clifford-equivalence and is robust even to symmetry tapering, so trainability can be altered only by changing what the dynamics explores, such as restricting the ansatz, never by relabeling the qubits. One may therefore re-encode a fermionic circuit to make it easier to simulate, but never to make it easier to train. These results reframe advantage claims for fermionic ansatze: an advantage that isnot gauge-invariant is an artifact of the encoding

DOI
10.5281/zenodo.21059512

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Whether a fermionic variational quantum algorithm offers a genuine quantum advantage depends on whether its outputs are classically hard to compute. We show that for such algorithms classical simulability is not an intrinsic property of the circuit. It depends on the fermion-to-qubit encoding. Standard encodings are related by Clifford transformations, so they form a gauge group, and we classify the standard simulability resources by how they transform under it. The Pauli weight, the causal-cone width, and the entanglement that sets the cost of tensor-network contraction are gauge-covariant: a generic interacting Hamiltonian whose truncated Pauli-propagation cost grows linearly in the system size under the Jordan-Wigner encoding becomes logarithmic under a tree encoding, and we exhibit the resulting change of verdict at a fixed physical ansatz. The dynamical-Lie-algebra dimension and the non-stabilizerness are gauge-invariant. From this split we prove that the encoding-optimized simulation cost isbounded by the invariant resources, so genuine, encoding-robust hardness requires both the dynamical Lie algebra and the magic to be large; the covariant weight is necessary but never sufficient. These routes are sufficient certificates of simulability, not a complete characterization, so the floor is an upper bound and a full classification of representation-invariant cost functionals is left open. We confirm both invariant simulation routes numerically, and we show that the dynamical-Lie-algebra dimension separates simulable from hard instances where the Pauli weight cannot; we also construct an explicit one-parameter interacting family for which all three certificate routes fail at once, so the floor is achieved. We then establish a sharp dichotomy with trainability. Because the cost landscape is encoding-invariant, the gradient and the barren-plateau variance are exactly gauge-invariant, which we verify to machine precision; we show further that this invariance follows from faithfulness rather than Clifford-equivalence and is robust even to symmetry tapering, so trainability can be altered only by changing what the dynamics explores, such as restricting the ansatz, never by relabeling the qubits. One may therefore re-encode a fermionic circuit to make it easier to simulate, but never to make it easier to train. These results reframe advantage claims for fermionic ansatze: an advantage that isnot gauge-invariant is an artifact of the encoding
Abstract Whether a fermionic variational quantum algorithm offers genuine advantage depends on whether its outputs resist classical computation. We show that classical simulability is encoding-dependent, not intrinsic to the circuit. Standard encodings are Clifford-related, forming a gauge group, and we classify resources by how they transform under it. Pauli weight, causal-cone width, and the entanglement setting tensor-network cost are gauge-covariant: an interacting Hamiltonian's truncated Pauli-propagation cost is linear under Jordan-Wigner but logarithmic under a tree encoding, changing the verdict at fixed ansatz. Dynamical-Lie-algebra dimension and non-stabilizerness are gauge-invariant. We prove the encoding-optimized simulation cost is bounded by the invariant resources, so encoding-robust hardness requires a large Lie algebra and large magic; covariant weight is necessary but never sufficient. These routes are sufficient certificates, not a complete characterization, so the floor is an upper bound; a full classification of representation-invariant costs remains open. We confirm both invariant routes numerically, show Lie-algebra dimension separates simulable from hard instances where weight cannot, and construct a one-parameter family where all three certificates fail at once, achieving the floor. We then establish a sharp dichotomy with trainability: because the cost landscape is encoding-invariant, gradient and barren-plateau variance are exactly gauge-invariant to machine precision. This follows from faithfulness, not Clifford-equivalence, and survives symmetry tapering, so trainability shifts only by changing what the dynamics explores, such as restricting the ansatz, never by relabeling qubits. One may re-encode a circuit to ease simulation but never to ease training. These results reframe advantage claims: an advantage that is not gauge-invariant is an artifact of the encoding.
Efficient classical simulation has matured to a critical component of the quantum computing stack, driving hardware validation, algorithm design, benchmarking, and the study of structured quantum dynamics. Lie-algebraic simulation (g\mathfrak{g}g-sim) offers a compelling approach: it represents Heisenberg-picture dynamics in the adjoint space whose dimension is set by the dynamical Lie algebra (DLA) governing the circuit, enabling efficient simulation of expectation values whenever the DLA grows only polynomially with system size. Despite this promise, existing applications of g\mathfrak{g}g-sim have been confined to free-fermionic settings. It has therefore remained unclear if the method can be applied to other structured circuit families, especially when their generators have large Pauli expansions, and hence whether Lie-algebraic simulability presents a genuinely broader paradigm than free fermions. In this work, we resolve this question by identifying additional non-trivial families of polynomial-dimensional DLAs and introducing symmetry-adapted bases that make the required adjoint-space preprocessing tractable. In particular, we develop an explicit Pauli orbit representation for permutation-equivariant dynamics, enabling efficient processing of cubic-dimensional algebras despite exponential Pauli support, and a modified generalized Gell--Mann representation for bounded Hamming-weight (U(1)U(1)U(1)-equivariant) dynamics, yielding polynomial simulation costs on fixed excitation sectors. Together with streamlined routines for free-fermionic algebras, these constructions significantly broaden the practical scope of g\mathfrak{g}g-sim as a unifying simulation tool for structured quantum circuits. Numerical benchmarks confirm favorable preprocessing scaling and validate large-scale proof-of-concept simulations beyond the reach of state-vector simulation.
Standard fermion-to-qubit encodings (Jordan–Wigner, Bravyi–Kitaev, the Bravyi–Kitaev tree, parity, ternary tree) are related by Clifford transformations, so they form a gauge group acting on the classical-simulability problem. This code demonstrates the central split: the Pauli weight and causal-cone width are gauge-covariant (re-encoding can lower them exponentially), while the dynamical-Lie-algebra dimension and the non-stabilizerness (magic) are gauge-invariant. Trainability — the cost landscape, the gradient, and the barren-plateau variance — is gauge-invariant to machine precision.
Scalable parameterized quantum circuits for machine learning face three obstacles: barren plateaus, no guarantee the learned function class is classically hard, and prohibitive circuit evaluations per gradient step. We propose the unitary brick-wall: a kkk-particle fermionic architecture for nearest-neighbor hardware, combining Reconfigurable Beam Splitters with interleaved single-qubit phases and a non-Gaussian magic-state encoding, where kkk trades classical hardness against training cost. Trainable. The brick-wall has dynamical Lie algebra u(n)\mathfrak{u}(n)u(n) and is surjective onto U(n)U(n)U(n) via Givens rotations, enabling Haar initialization. Two-body correlator readouts achieve gradient variance Θ(k3/n5)Θ(k^3/n^5)Θ(k3/n5), polynomial in nnn throughout n−2k=Ω(n)n-2k=Ω(n)n−2k=Ω(n). Expressive. Hardness is controlled by kkk: best-known classical sampling runs in time 2Θ(k)2^{Θ(k)}2Θ(k), worst-case #P-hardness holds from k=nεk=n^εk=nε, and the average-case machinery of Fermion Sampling applies at k=Θ(n)k=Θ(n)k=Θ(n). The sampler is input-dependent, so costs are paid per query: at our operating point k=100k=100k=100 one query needs ∼1015\sim 10^{15}∼1015 operations, beyond commercial query rates, and at k=120k=120k=120 an exascale machine sustains under one query per second. Efficient. A multi-layer parallel parameter-shift rule computes all O(n2)O(n^2)O(n2) gradients from 4kn4kn4kn circuit evaluations per step, a factor n/kn/kn/k reduction over the standard rule's 4n24n^24n2, growing linearly with nnn at fixed kkk. The unitary butterfly targets all-to-all hardware, with 2log⁡n2\log n2logn layers of parallel gates and (3/2)nlog⁡n(3/2)n\log n(3/2)nlogn parameters, similar hardness guarantees, and 4klog⁡n4k\log n4klogn evaluations per step -- the same factor-n/kn/kn/k reduction. Its trainability holds at two levels: absence of exponential barren plateaus is unconditional, while the sharp Θ(k3/n5)Θ(k^3/n^5)Θ(k3/n5) rate needs a conjectured scalar comparison with the two-particle Haar second moment.
We investigate the utility of the locality-preserving Derby-Klassen (DK) fermion-to-qubit mapping [arXiv:2003.06939] for variational quantum simulation of two-dimensional ttt-VVV and Fermi-Hubbard models. The DK mapping preserves the locality of fermionic interactions with an enlarged Hilbert space, thereby requiring additional constraints that define the physical sector. We incorporate these constraints directly into a Hamiltonian Variational Ansätz (HVA) through Clifford-gate state preparation and use the Variational Quantum Eigensolver (VQE) to show that the low-energy properties of the resulting qubit Hamiltonian are accurately reproduced. We further exploit particle-number conservation inherent in the ansätz to resolve distinct symmetry sectors and reliably access degenerate states. Consequently, we benchmark the DK-HVA against Jordan-Wigner-based variational circuits at nonzero chemical potential, where particle-hole symmetry and the associated half-filled sign-free condition are absent. Finally, we demonstrate the advantage of locality-preserving mappings in higher-dimensional fermionic systems, where the conventional Jordan-Wigner (JW) transformation generates increasingly long Pauli strings and corresponding circuit overheads. We further extend the framework to the spinful Fermi-Hubbard model and identify a tradeoff between fermionic-mode placement and the locality of hopping and on-site interaction terms. These results establish a practical framework combining locality-preserving fermion-to-qubit mappings, constraint-preserving Clifford state preparation, and symmetry-preserving variational circuits, that trades auxiliary-qubit and stabilizer-preparation overhead for reduced operator nonlocality, providing a practical route toward resource-efficient quantum simulation of higher-dimensional interacting fermionic systems.
Abstract Barren plateaus are the main obstacle to training quantum machine-learning models. On a barren plateau, the cost-function gradient shrinks exponentially as the circuit grows, and training stalls. Today the only reliable test is to measure it. That means sampling many gradients for every candidate circuit, which is expensive. We ask a different question. Can a classical model predict trainability from the circuit architecture alone? The architecture here means qubit count, depth, ansatz type, entanglement pattern, entangler gate, and cost-observable locality. We test this directly. We use an exact statevector simulator to build a dataset of \Nrows{} random variational circuits. For each circuit we estimate the gradient variance Var[∂C/∂θ]\mathrm{Var}[\partial C/\partial\theta]Var[∂C/∂θ] from \Samples{} random parameter vectors, using the parameter-shift rule. This variance is the label. A gradient-boosted regressor then predicts log⁡10Var\log_{10}\mathrm{Var}log10​Var with $R^2 = \RsqHold{}$ on held-out architectures. A classifier separates barren from trainable circuits with an area under the ROC curve of \AucHold{}. We also test transfer to larger circuits. We train only on small circuits and test on the two largest unseen sizes (n=11,12n=11,12n=11,12). The model still reaches $R^2 = \RsqExtrap{}$. Permutation importance then recovers established theory. Cost-observable locality and entanglement structure matter most. We make no quantum-advantage claim. This is classical machine learning about quantum circuits. It offers a cheap way to screen ans\"atze, and it shows which design choices cause untrainability.