Mathematical Background

This section is here, more to fix the convention of Jordan-Wigner transform and Majorana operators that we use here, and less to explain the full theory behind those. For the latter, we, once again, recommend Classical simulation of noninteracting-fermion quantum circuits.

We define the Majorana operators $γ_i$ via

\[ γ_{2i-1} = ∏_{j=1}^{i-1} (-Z_j) X_i \qquad \textrm{and} \qquad γ_{2i} = -∏_{j=1}^{i-1} (-Z_j) Y_i.\]

This implies the normal fermionic creation and annihilation operators are given by

\[ c_j = \frac{1}{2} (γ_{2j-1} + iγ_{2j}) \quad \textrm{and} \quad c_j^† = \frac{1}{2} (γ_{2j-1} - iγ_{2j})\]

and products of two Majorana operators are of the form

\[ σ_i \left(∏_{i<j<k} -Z_j \right) σ_k \quad \textrm{or} \quad Z_i\]

with $σ_i, σ_k ∈ \{X, Y\}$.

Any unitary that takes all Majorana operators to a linear combination of Majorana operators under conjugation, i.e. that satisfies

\[ U γ_i U^† = R_i^j γ_j\]

with some $R ∈ O(2n)$ is a FLO unitary. In particular, if a unitary is of the form

\[ U = e^{-iθH}\]

with

\[ H = \frac{i}{4} \sum_{i,j} H^{ij} γ_i γ_j\]

it is a FLO unitary with $R ∈ SO(2n)$.

But note, that not all FLO unitaries are of that form. For example, $X_i$ is also a FLO gate since it either commutes or anti-commutes with all Majorana operators, but the associated matrix $R$ always has determinant $-1$.

Calculating the expectation values of hamiltonians like the one above when evolving the vacuum state with FLO circuits is efficiently possible. First evolve the Hamiltonian in the Heisenber picture to

\[ UHU^† = \frac{i}{4} R^{m}_{i} R^{n}_{j} H^{ij} γ_{m} γ_{n} =: \frac{i}{4} \tilde H^{mn} γ_{m} γ_{n}.\]

and then compute the expectation value

\[\begin{aligned} ⟨ψ|UHU^†|ψ⟩ &= \frac{i}{4} \tilde H^{mn} ⟨Ω|γ_{m} γ_{n}|Ω⟩ \\ &= - \frac{1}{2} ∑_{i} \tilde H^{2i-1,2i} \\ &= - \frac{1}{2} ∑_{i} R^{2i-1}_{m} R^{2i}_{n} H^{mn} \\ \end{aligned}.\]

From the first to second line one needs to carefully think which of the $⟨Ω|γ_{m} γ_{n}|Ω⟩$ are zero and which cancel each other out due to the anti-symmetry of $H^{mn}$.