Gaussian Backend

QuantumCircuitsMPS.jl also ships a fermionic Gaussian (free-fermion / FLO — "fermionic linear optics") backend for circuits built entirely out of Gaussian-preserving gates and parity measurements. Instead of an MPS, a dense state vector, or a stabilizer tableau, the state is stored as a 2L×2L real antisymmetric Majorana covariance matrix Γ, and gate/measurement application is a Schur-complement tensor contraction on Γ rather than any operation on a Hilbert-space vector. apply!, track!, record!, and simulate! all work exactly as on the other three backends; only the SimulationState(...) constructor call changes.

When to use it: circuits built entirely out of fermionic Gaussian unitaries and parity measurements — measurement-induced phase transitions in free-fermion (class-D/DIII) circuits, entanglement transitions with free fermions, and the staggered class-DIII Majorana-chain protocol of Pan et al. Polynomial-time simulation (O(L³) per gate), no truncation error, exact for any circuit depth.

When not to use it: any circuit that needs a non-Gaussian gate — Hadamard, CNOT, HaarRandom, RandomClifford, generic Projection, or anything from the qubit-gate vocabulary. Use backend=:mps (see MPS Backend) or backend=:statevector (see State Vector Backend) for those. The Gaussian backend is fermionic-mode-only (local_dim=2); arbitrary spin-S sites (see Arbitrary Spin-S Support) are MPS/state-vector-only.

Quick Example

using QuantumCircuitsMPS

state = SimulationState(L=32, bc=:periodic, backend=:gaussian,
    rng=RNGRegistry(gates_spacetime=1, gates_realization=2, born_measurement=3, state_init=4))
initialize!(state, ProductState(binary_int=0))          # vacuum
apply!(state, GaussianHaar(), Bricklayer(:odd))          # random Gaussian circuit
apply!(state, Measure(:Z), SingleSite(5))                # on-site occupation parity measurement
apply!(state, BondParity(), AdjacentPair(3))             # bond parity measurement
track!(state, :entropy => EntanglementEntropy(cut=16))
record!(state)

entropies = state.observables[:entropy]
println("Final entropy: $(entropies[end])")

Physics: GaussianHaar() draws an independent Haar-random O(4)/O(2) rotation for each bond (from :gates_realization) and conjugates it directly onto the Majorana covariance matrix — no dense Hilbert-space unitary is ever built. Measure(:Z) and BondParity() are projective parity measurements of, respectively, the on-site occupation iγ_{2i-1}γ_{2i} and the bond parity spanning two adjacent sites; both collapse Γ via the same fermionic-linear-optics contraction kernel used for the unitary case.

What a Gaussian State Is

A pure fermionic Gaussian state on L modes is completely characterized (no exponentially large amplitude vector needed) by its 2L×2L real antisymmetric Majorana covariance matrix

Γ[a,b] = (i/2) ⟨[γ_a, γ_b]⟩ ,

where γ_1, …, γ_{2L} are the Majorana operators (γ_{2i-1} = c_i + c_i†, γ_{2i} = i(c_i† - c_i) for fermionic mode i, with a Jordan–Wigner string on lower-indexed modes). A pure Gaussian state satisfies the invariant

Γ² = -I

exactly (the eigenvalues of iΓ are all ±1). Every gate and measurement in this backend updates Γ while preserving this invariant to machine precision; a re-purification step (purify! — eigen-clamp iΓ's spectrum back to ±1) fires automatically whenever floating-point drift exceeds state.backend.purify_tol (default 1e-10).

Supported Gates

Any gate outside this set — Hadamard, CNOT, HaarRandom, RandomClifford, SWAP, PhaseGate, PauliY, PauliZ, CZ, Projection, SpinSectorProjection — has no covariance-matrix representation and raises an informative error rather than being silently approximated:

using QuantumCircuitsMPS

state = SimulationState(L=4, bc=:open, backend=:gaussian,
    rng=RNGRegistry(gates_spacetime=1, gates_realization=2, born_measurement=3, state_init=4))
initialize!(state, ProductState(binary_int=0))
apply!(state, CNOT(), AdjacentPair(1))

ArgumentError: Gaussian backend only supports fermionic Gaussian operations (GaussianHaar, PauliX, Measure(:Z), BondParity, Reset). Received: CNOT. Please switch to backend=:mps or backend=:statevector for non-Gaussian gates.

Measure(:Z) and Reset() do not get their own _apply_single!/execute! overrides on the fermionic-mode granularity: Measure flows through the generic engine, which calls this backend's born_probability + _measure_single_site!; Reset composes Measure with PauliX. BondParity gets a dedicated execute! override (see Backend Interface Contract) because it measures two sites at once — outside the single-site born_probability contract.

Majorana Chain (site_type="Majorana")

A second site granularity turns each site into one Majorana mode rather than one fermionic mode (2 Majoranas). This models the class-DIII monitored Majorana-chain circuit of Pan, Shapourian, Jian et al. directly, with no new gate types: GaussianHaar on 2 Majorana sites is exp(θγ_aγ_b), θ ~ U[0,2π) (Haar on SO(2) ≅ U(1) is exactly the uniform-angle rotation), and BondParity on 2 Majorana sites is the iγ_iγ_{i+1} parity measurement — the staggered odd/even-link protocol maps directly onto Bricklayer(:odd)/Bricklayer(:even).

using QuantumCircuitsMPS

mstate = SimulationState(L=8, bc=:periodic, backend=:gaussian, site_type="Majorana",
    rng=RNGRegistry(gates_spacetime=1, gates_realization=2, born_measurement=3, state_init=4))
initialize!(mstate, ProductState(binary_int=0))
apply!(mstate, GaussianHaar(), Bricklayer(:odd))    # odd links: exp(θγγ), θ~U[0,2π)
apply!(mstate, BondParity(), Bricklayer(:even))     # even links: iγγ parity measurement
size(mstate.backend.corr)   # (8, 8) — Γ is L×L, not 2L×2L, on the Majorana chain

Key facts:

• site_type="Majorana" requires even L (a pure Gaussian state needs an even number of Majoranas): odd L throws ArgumentError at construction.
• Γ is L×L (one Majorana per site) instead of 2L×2L.
• ProductState bit patterns have length L÷2: bit k sets the parity sign of the consecutive Majorana pair (γ_{2k-1}, γ_{2k}) (dimerized vacuum when all bits are 0).
• Rejected on the Majorana chain (informative ArgumentError, each naming "Majorana"): PauliX, Measure(:Z), Reset, Magnetization. There is no single-Majorana occupation or ⟨Z⟩ — parity lives on a pair.
• EntanglementEntropy, MutualInformation, and TripartiteMutualInformation work unchanged (the site→Majorana index mapping is the identity on this granularity, and arbitrary/wrapped site subsets are still supported).

Physics sanity check on the dimerized vacuum (ProductState(binary_int=0), all pairs unoccupied): even cuts see zero entanglement (the cut falls between dimerized pairs), odd cuts split a pair and see log(2)/2 nats (half a fermion's worth), and two dimerized-paired Majorana sites have MI = log(2):

using QuantumCircuitsMPS

mstate = SimulationState(L=8, bc=:periodic, backend=:gaussian, site_type="Majorana",
    rng=RNGRegistry(gates_spacetime=1, gates_realization=2, born_measurement=3, state_init=4))
initialize!(mstate, ProductState(binary_int=0))
EntanglementEntropy(cut=2, base=ℯ)(mstate)   # ≈ 0.0
EntanglementEntropy(cut=1, base=ℯ)(mstate)   # ≈ log(2)/2 ≈ 0.3466
MutualInformation([1], [2])(mstate)          # ≈ log(2) ≈ 0.6931

Supported and Rejected Observables

Every rejection is an informative ArgumentError naming the observable and suggesting backend=:mps/:statevector — never a raw field-access crash:

using QuantumCircuitsMPS

state = SimulationState(L=4, bc=:open, backend=:gaussian,
    rng=RNGRegistry(gates_spacetime=1, gates_realization=2, born_measurement=3, state_init=4))
initialize!(state, ProductState(binary_int=0))
StringOrder(1, 4)(state)

ArgumentError: StringOrder is not supported on the Gaussian backend: its formula requires spin-1 Sz-expectation-string MPO/MPS contractions, which have no native fermionic-Gaussian (covariance-matrix) representation. Please use backend=:mps or backend=:statevector for StringOrder.

Example: Reproducing the Class-DIII Phase Diagram

examples/gaussian_example.ipynb reproduces Fig. 1b of Pan et al., "Topological Modes in Monitored Quantum Dynamics": the staggered class-DIII monitored Majorana chain, mutual information vs. measurement probability p, at demo system sizes finishing in a few minutes. It uses the Majorana-chain granularity documented above with Bricklayer(:odd)/Bricklayer(:even) staggering.

Initialization

Three ways to prepare a Gaussian state, all via initialize!:

using QuantumCircuitsMPS, LinearAlgebra

mkstate() = SimulationState(L=4, bc=:open, backend=:gaussian,
    rng=RNGRegistry(gates_spacetime=1, gates_realization=2, born_measurement=3, state_init=4))

# 1. Vacuum: all modes unoccupied
s1 = mkstate(); initialize!(s1, ProductState(binary_int=0))
s1.backend.corr[1, 2]   # +1.0 (unoccupied convention)

# 2. Arbitrary occupation pattern (site 1 = MSB, same convention as every other backend)
s2 = mkstate(); initialize!(s2, ProductState(bitstring="0101"))   # sites 2, 4 occupied
s2.backend.corr[3, 4]   # -1.0 (occupied)

# 3. Haar-random pure Gaussian state
s3 = mkstate(); initialize!(s3, RandomGaussianState())
norm(s3.backend.corr * s3.backend.corr + I) < 1e-10   # true — purity holds

On the Majorana chain (site_type="Majorana"), ProductState bit patterns have length L÷2 (one bit per dimerized Majorana pair — see the Majorana Chain section above) and RandomGaussianState draws O ∈ SO(L) instead of SO(2L).

Conventions

• Mode ↔ Majorana mapping: fermionic mode i (1-indexed) ↔ Majorana indices (2i-1, 2i). On the Majorana chain, site i IS Majorana i directly.
• Occupation sign: Γ[2i-1,2i] = +1 ↔ mode i unoccupied; Γ[2i-1,2i] = -1 ↔ occupied — i.e. ⟨c_i†c_i⟩ = (1 - Γ[2i-1,2i])/2. Verified empirically against the Python GTN reference implementation's get_C_f.
• Measurement outcome: outcome = 0 ↔ unoccupied/parity +1 result on Γ; outcome = 1 ↔ occupied/parity -1 result. ProductState bit 1 ↔ occupied, matching this convention.
• Exact-Haar sampler note: haar_orthogonal draws Haar-random SO(n) matrices exactly via QR decomposition of a Ginibre matrix (sign-fixed, det-corrected) — this is a deliberate departure from the Python GTN reference implementation, whose get_O uses expm of a random skew-symmetric matrix (an approximation the GTN codebase's own notes flag as not proven exactly Haar). All Gaussian-backend randomness (GaussianHaar, RandomGaussianState) uses the exact sampler.

Cross-Backend RNG Reproducibility

Like the other three backends, every measurement (Measure(:Z) and BondParity()) consumes exactly one :born_measurement draw per measured site/bond — even for a deterministic outcome, where the draw is discarded (the REDUNDANT-DRAW CONTRACT; see the Backend Interface Contract). Random gate content (GaussianHaar) draws from :gates_realization; random initial states (RandomGaussianState) draw from :state_init. Same seeds ⇒ bitwise-identical Γ and identical measurement records, within the Gaussian backend:

using QuantumCircuitsMPS

mkrng(k) = RNGRegistry(gates_spacetime=k, gates_realization=k+10, born_measurement=k+20, state_init=k+30)

sA = SimulationState(L=6, bc=:open, backend=:gaussian, rng=mkrng(7))
initialize!(sA, ProductState(binary_int=0))
apply!(sA, GaussianHaar(), Bricklayer(:odd))

sB = SimulationState(L=6, bc=:open, backend=:gaussian, rng=mkrng(7))
initialize!(sB, ProductState(binary_int=0))
apply!(sB, GaussianHaar(), Bricklayer(:odd))

sA.backend.corr == sB.backend.corr   # true — bitwise identical

References

• The fermionic-linear-optics / Majorana-covariance-matrix formalism this backend implements: [1].
• The Haar-random SO(2n) Gaussian-unitary ensemble and the general contraction formula: [2]; conventions follow [3].
• The class-DIII staggered monitored Majorana-chain protocol reproduced by examples/gaussian_example.ipynb: [3].