4.11 Gaussian, fermionic, and matchgate compression
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4.11.1 Polynomial covariance representations
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Gaussian bosonic and fermionic states are specified by first and second moments rather than by all Fock-basis amplitudes. A bosonic covariance matrix has size polynomial in the number of modes, and a fermionic Gaussian state is represented by an antisymmetric Majorana covariance matrix. Gaussian unitaries act by symplectic or orthogonal transformations:
This is a compact internal encoding even when the corresponding Hilbert-space packet is exponentially large.
4.11.2 Hybrid symmetry and Gaussianity
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Number conservation, parity, translation symmetry, and point-group symmetry can further block the covariance matrices. A simulator can therefore combine two reductions:
The first removes unreachable sectors. The second compresses states within the remaining sectors.
4.11.3 Matchgate circuits
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Nearest-neighbor matchgate circuits have polynomial classical representations through fermionic Gaussian dynamics. Geometry matters: adding SWAP, fermionic SWAP, long-range gates, or a non-matchgate resource changes the simulation problem. A symmetry-collapse atlas should therefore record the gate library and connectivity as part of the model, not as implementation details.
4.11.4 Low-rank non-Gaussian corrections
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One research direction is to represent a weakly non-Gaussian state as
with small Gaussian rank \(R\), or as a Gaussian reference plus sparse packet corrections. A polynomial collapse requires \(R\) and the update cost to remain polynomial.
Open program or boundary
Gaussianity is an internal representation class, not a symmetry projector. A complete hybrid theorem must prove closure of the chosen compressed class under every allowed gate and bound the growth of any non-Gaussian residue.
4.11.5 Few-particle versus Gaussian routes
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Fixed-particle simulation and Gaussian simulation are independent. A fixed two-fermion state may be non-Gaussian but live in a polynomial-dimensional sector. A half-filled Gaussian state may live in an exponential number sector but have a polynomial covariance description. The two-axis model in Section 6.1 is designed to capture this distinction.