Local, Expressive, Quantum-Number-Preserving VQE Ansatze for Fermionic Systems

I. INTRODUCTION

• A good VQE entangler circuit is expected to balance seven criteria, including efficient state representation, local realizable gates, trainability, symmetry preservation, and sufficient-depth universality.The authors identify exact target-state representation as generally requiring exponential depth.Source: §I · p. 1
• Existing ansatz recipes generally satisfy only subsets of these criteria; the authors identify generalized swap network k-UpCCGSD [17] as a notable exception.They became aware of that nearest-neighbor reformulation after posting their first manuscript version.Source: §I · p. 1
• The proposed Jordan–Wigner entangler recipes preserve relevant particle-number and spin quantum numbers gate by gate, while evidence for trainability remains partly future work.The authors note that contamination or variational collapse into other quantum-number sectors can degrade wavefunction quality.Source: §I · p. 1
• Figure 1’s alternating nearest-neighbor SU(4) gate tessellation provides inspiration for the fermionic quantum-number-preserving fabrics developed here.The authors distinguish their approach from [17] in qubit ordering and in its emphasis on quantum-number symmetry.Source: §I · p. 1, Figure 1 · p. 2

II. GATE FABRICS

• Alternating layers of universal two-qubit SU(4) gates can span SU(2^N) at a finite, N-dependent depth, though practical circuits may be shallower.At shorter depths, the authors note that a useful approximation to physically relevant subgroup states may still be achievable.Source: §II · p. 2
• A gate fabric is a tessellation of gates over N qubits arranged in layers, with a repeated local pattern.The paper highlights locality and simplicity as defining features.Source: §II · p. 2
• Its stated design criteria are a single k-qubit, l-parameter gate type, linear locality, finite-depth universality within the target subgroup, and commutation with that subgroup’s symmetry operators.The gate element’s parameters and support size are independent of N.Source: §II · p. 2

III. GATE FABRICS FOR FERMIONS UNDER THE JORDAN-WIGNER MAPPING

• The target subgroup consists of real unitaries that commute with the alpha- and beta-electron number operators and total spin squared.The construction uses spatial orbitals represented by alpha and beta spin orbitals in an interleaved Jordan–Wigner qubit ordering.Source: §III · p. 2
• The proposed Q gate is a two-parameter, four-qubit element combining a spin-adapted spatial orbital rotation with a diagonal pair-exchange gate.The simpler Q-type fabric grew from related, more complete constructions described in Appendix D.Source: §III · pp. 2–3
• The gates individually preserve all relevant quantum numbers, and numerical tests indicate universality in most quantum-number irreducible representations, with some high-spin Q-type exceptions.The authors leave a rigorous sector-by-sector universality proof beyond the scope of the work.Source: §III · p. 3
• The paper supplies elementary-gate decompositions and parameter-shift rules that compute gate-parameter gradients using at most four distinct circuits without increasing circuit depth, gate count, or qubit number.The rules are also extended to a broader variety of gates.Source: §III · p. 3

IV. NUMERICAL DEMONSTRATIONS

• The p-benzyne and naphthalene tests use (6e, 6o) and (10e, 10o) active spaces, respectively, corresponding to 12 and 20 qubits.The target states are ground states with Nα = Nβ and S = 0.Source: §IV · p. 3
• The simulations initialize an uncorrelated product state and compare two parameter initializations using L-BFGS, without shot noise or decoherence.The noiseless setup permits analytical expectation values and gradients in classical statevector simulations.Source: §IV · p. 3
• At 50–80 parameters, p-benzyne reaches chemical accuracy; at 180 parameters, its error is ~10^-10 Eh relative to FCI, numerically indicating universality.The 180-parameter circuit also extinguishes the FCI zero-amplitude tail.Source: §IV · pp. 3–4
• At 800–1,000 parameters, naphthalene reaches chemical accuracy despite 19,404 configuration state functions, without reaching universality at the explored depths.Deeper fabrics require fewer L-BFGS epochs to converge to chemical accuracy.Source: §IV · p. 4
• Strategy A appears to avoid optimization plateaus and, for deeper circuits, speeds convergence below chemical accuracy.This strategy uses Π = QNPOR(π).Source: §IV · p. 5
• The 20-qubit, 1,080-parameter ansatz decomposes to depth 2,761 without Π gates, versus approximately 57,600 for a naive UCCSD Trotter step.The 12-qubit, 110-parameter ansatz has depth 507 without Π gates.Source: §IV · p. 5

• Hardware-efficient ansatze are local but do not preserve particle- or spin-quantum-number symmetries, which may complicate finding low-lying states in a target irrep, particularly in larger active spaces.Source: §V · pp. 5–6
• UCCSD, k-UpCCGSD, ADAPT-VQE, and Jastrow-Factor VQE can be symmetry-adapted or potentially universal, but as described require nonlocal gates, and many use heterogeneous layers.The generalized swap-network reformulation of k-UpCCGSD appears to realize it as a local fabric of four-qubit gates [17].Source: §V · p. 6
• Orbital-rotation fabrics and pUCCD map to fabrics requiring only M qubits, but do not reach FCI universality as parameter depth increases.Source: §V · p. 6
• Other proposals preserve only total particle number, omit spin-squared symmetry, or use nonlocal circuits; the paper also points to alternative local, symmetry-preserving fabrics built from generic four-qubit FCI gates.The alternative gates are decomposable into QNPPX, QNPOR, single-particle/hole substitution, and pair-break gates.Source: §V · p. 6

• The proposed two-parameter four-qubit Q gate combines a spatial orbital rotation QNPOR(ϕ) with a diagonal pair exchange QNPPX(θ), and the combined fabric appears to attain FCI universality at sufficient depth.Source: §VI · p. 6
• At intermediate depths, the fabric appears pragmatically expressive in ground-state energy convergence tests on strongly correlated molecular systems.Source: §VI · p. 6
• The Q-type fabric appears universal in most quantum-number irreps, including nonzero-spin or unequal-spin-population cases, with a few high-spin edge cases addressable using F-type gates.Source: §VI · p. 7
• The authors leave noise robustness, optimization methods, barren-plateau mitigation, relative-property performance, and error scaling as open questions.Source: §VI · p. 7

Appendix A: Symmetry-Constrained Subgroups of

• For real operators, replacing SU(4) gates with SO(4) gates in the alternating nearest-neighbor fabric yields a fabric for SO(2N).Source: Appendix A · p. 10, Figure 4 · p. 10
• A two-qubit gate fabric can preserve Hamming weight yet fail to span H(2N) at any depth; a three-qubit construction spans H(2N) at sufficient depth.The three-qubit construction uses cascading 6-parameter gates acting on qubit triples.Source: Figure 5 · p. 10, Figure 6 · p. 11

Appendix C: The Specific Case of M = 2 in F(22M)

• The M = 2 Fock-space enumeration records each determinant's Jordan-Wigner basis state, α and β populations, CSF status, and spin when it is a CSF.Source: Appendix C · p. 12, §1 · p. 12, Table I · p. 13

3. Configuration State Functions (CSFs)

• Configuration state functions (CSFs) are sparse linear combinations of Slater determinants that are eigenstates of S^2; the + combination yields an S = 0 singlet, while the − combination yields an S = 2 triplet.The complete real, orthonormal set for F(22*2) contains 16 CSFs: 5 singlets, 8 doublets, and 3 triplets.Source: §3 · pp. 12–13

4. Quantum Number Irreps

• Quantum-number irreps group simultaneous eigenstates with fixed Nα, Nβ, and S, and the appendix lists their dimensions and a convention for their CSF bases.Source: §4 · p. 13, Table II · p. 13

5. Full Configuration Interaction (FCI) States

• FCI states remain flexible within a target irrep; complete special-orthogonal rotations on its CSF basis can preserve the quantum numbers.For M = 2, enforcing S^2 symmetry across related irreps reduces the FCI gate fabric from seven parameters to five.Source: §5 · p. 14

Appendix D: Gate Fabric for F(22M) via M = 2 FCI Gates

• A fabric of five-parameter four-qubit F gates preserves quantum-number symmetry, becomes universal for F(22M) at sufficient parameter depth, and is expressive at intermediate depths.Its representation power and numerical convergence were similar to those of the Q fabric, which the authors chose to emphasize as conceptually simpler.Source: Appendix D · p. 14

1. Decomposition of ˆF into Simple Gate Elements

• The F gate's five parameters implement one-particle and one-hole rotations plus a three-parameter SO(3) rotation among singlet CSFs.The appendix identifies five component operations: QNP1p, QNP1h, QNPPX, QNPPBU, and QNPPBL.Source: §1 · p. 14, Figure 7 · p. 15

2. Simplifications of the ˆF Gate Fabric

• Removing a redundant rotation reduces the F fabric from five to four parameters; for most irreps, a three-parameter version suffices, and the main-text Q fabric reduces this to two.For edge-case irreps, the required one-particle versus one-hole gate depends on whether the irrep has more particles or holes.Source: §2 · pp. 15–16

4. Variants of the QNP fabrics

• Prepending a fixed quantum-number-preserving gate is proposed as a possible way to improve trainability or intermediate-depth expressiveness.The explored choices include identity and QNPOR(π).Source: §4 · p. 16

Appendix E: Explicit decompositions of the quantum number preserving gates

• The appendix supplies elementary decompositions for the introduced quantum-number-preserving gates, although the component gates used to build QNP1p and QNP1h are not individually quantum-number preserving.Source: Appendix E · p. 16

3. Orbital Rotations

• The QNPOR gate applies the same Givens rotation in the α and β spaces and can be arranged in a fabric that implements arbitrary orbital rotations.A classical QR decomposition can relate the orbital-rotation matrix to the fabric parameters.Source: §3 · p. 16, Figure 8 · p. 17

1. Pair exchange gate QNPPX(θ)

• For QNPPX, expanding controlled-Y rotations changes the reported depth from 16 to 18 while reducing the two-qubit gate count from 18 to 14.Source: §1 · p. 17

2. Two orbital Givens rotation gate QNPOR

• The adjacent-orbital QNPOR decomposition has depth 5 and four CNOT gates; the appendix also describes OFSWAP as two fermionic swaps, each a SWAP followed by controlled Z.Source: §2 · p. 18, §5 · p. 21

1. Shift tuning

• Shift tuning generalizes the standard two-term rule by leaving the shift angle free, with α = π/2 recovering the original choice.The tuning concept was independently discovered in [37] and introduced in PennyLane [35].Source: §1 · p. 22

a. Reducing the gate count

• For θ not a multiple of π, choosing α = −θ makes one evaluation f(0), so the identity gate U(0) can be skipped.Neighboring gates may also merge, as in QAOA circuits.Source: a. Reducing the gate count · p. 22

2. Four-term parameter-shift rule

• Generators with spectrum {−a+c, c, a+c} satisfy the four-term rule; it applies to the introduced quantum-number-preserving gates except spin-adapted QNPOR.The spin-adapted QNPOR gate can instead be differentiated analytically through its G gates and the chain rule.Source: Appendix F · p. 21, §2 · p. 22
• For multiple controls and QNP gates, chain-rule differentiation of gate decompositions can use fewer circuit evaluations than applying the four-term rule directly.The passage specifies applying the two-term rule to the gate decomposition.Source: §2 · p. 23

a. Relation to other four-term rule

• At optimal shifts, the two four-term rules reach the same minimum variance, despite using different numbers of distinct circuits.Thus, for a fixed circuit-execution budget, their estimated-derivative quality is the same.Source: a. Relation to other four-term rule · p. 23
• The rule in [36] is preferable when circuits and observables are real-valued, V± are available, and simulation cost depends on distinct circuits.Otherwise, the paper favors its four-term rule for quantum computers because it requires slightly fewer gates with the same circuit executions.Source: a. Relation to other four-term rule · p. 23

b. Impossibility of some further shift rules

• A direct extension of the four-term rule to further generator-power conditions is not possible, although other shifted-expectation derivative schemes are not excluded.The examined generalized condition already implies Q^3 = Q, so the four-term rule applies.Source: b. Impossibility of some further shift… · p. 24

3. Minimizing the variance

• Under the constant-variance assumption, all discussed two- and four-term rules reach variance V/N at a fixed measurement budget, making them equally expensive on quantum devices.The four-term rules also attain this minimum variance with optimal shifts.Source: §3 · p. 24

1. Computational Basis State Amplitudes

• With 110 parameters, the VQE fabric approximates most FCI computational-basis amplitudes but has too few parameters to reproduce them all exactly.The states are plotted in a consistent ordering to support direct amplitude comparison.Source: §1 · p. 24
• The consistent ordering in Figure 9 allows direct comparison of individual VQE and FCI amplitudes.Figure 3b orders each trace separately, which shows tail shapes but prevents direct comparison between individual amplitudes.Source: §1 · p. 24
• Figure 9 labels each computational-basis state by seniority, with the first state identified as the seniority-0 Hartree–Fock determinant.The figure plots the true ground state and one optimized VQE state.Source: Figure 9 · p. 25

2. Numerical universality demonstration for Haar random states

• Spatial point-group symmetry complicates molecular tests and appears to worsen numerical optimization convergence; random XYZ perturbations from D2h to C1 yielded better convergence.Some leftmost gates can be inactive because they rotate orbitals or exchange pairs within occupied or virtual Hartree–Fock subspaces.Source: §2 · p. 25
• The random-state tests use Haar-random statevectors within selected irreps and optimize the fabric to maximize overlap with L-BFGS and noise-free analytical gradients.The tests omit the Π-based pre-mixing initialization enhancement.Source: §2 · p. 25
• As parameter count crosses the number of CSFs, overlap error sharply falls near machine epsilon, indicating onset of universality in the tested irreps.Quantum-number symmetries remain preserved to at least machine epsilon; fabrics above the universality threshold show strongly geometric convergence to machine epsilon.Source: §2 · p. 25
• The tests are artificial but remove external artifacts present in molecular cases, providing stronger evidence of universality and quantum-number preservation for the tested space.Figure 10 shows half-filled cases with M = 4 and M = 6; its instances are described as representative of generic cases.Source: §2 · p. 25, Figure 10 · p. 26

3. Non-universal edge cases

• The Q-type fabric is non-universal when the reduced unconstrained irrep has only holes or only particles, except when a single orbital remains.In those spaces, QNPPX gates act trivially and cannot explore new configurations.Source: §3 · p. 26
• Non-universal irreps appear to grow roughly as O(M), versus roughly O(M^3) total irreps, and occur at the Hilbert-space edge.These cases have severe high-spin constraints and are described as likely polynomially tractable classically, physically uninteresting, or both.Source: §3 · pp. 26–27
• The authors report that moderately low-spin, roughly half-filled cases almost surely lie in irreps where the Q-type fabric is universal.The claim is about the likely placement of interesting cases, not a guarantee for every such irrep.Source: §3 · p. 27
• The five-parameter F-type fabric avoids the observed edge-case non-universalities and was numerically universal in all tested cases.The authors present it as an alternative for addressing Q-type edge cases.Source: §3 · p. 27
• Numerical tests distinguish non-universal from universal irreps by overlap discrepancies of order 10^-2 versus below 10^-13, respectively.Tables III and IV list the non-universal irreps for M = 4 and M = 6.Source: §3 · p. 27, Table III · p. 27, Table IV · p. 27
• For M = 4, six of 35 unique irreps are non-universal; for M = 6, 24 of 84 are non-universal, all due to high-spin constraints.The listed irreps have total dimensions 36 and 400, respectively; other irreps were numerically universal to essentially machine precision.Source: Table III · p. 27, Table IV · p. 27