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9. √ By definition (see Eqs. 59) where a pair (n, j) refers to the nth qubit in the chain of qubits representing the k jth bosonic mode. The Pauli creation and annihilation operators σ± were previ† ¯ ously defined in Sec. 1. 60) j P P so its matrix representation in this basis is analogous to the matrix representation of ¯b†j in the basis of bosonic states. 9: Mapping of the bosonic state |φα , of a chain with 5 sites and 7 bosons (NP = 3), into a four-spin-1/2 (or four-qubit) state. The convention is |↑j ≡ |0j and |↓j ≡ |1j .

This is an isomorphic mapping between operator algebras and is independent of the Hamiltonian of the fermionic system to be simulated. Different Hamiltonians establish different connections (connectivity) between fermionic modes. Historically, Eqs. 36 correspond to lattices in one space dimension. Nevertheless, it is also valid for lattice systems in any dimension, when the set of modes j is countable. In particular, the set of all ordered p-tuples of integers can be placed in one-to-one correspondence with the set of integers.

54) However, if the operators b†j are restricted to the finite basis of states represented by {|n1 , n2 , · · · , nN with max(ni ) = NP }, that is, NP is the maximum number of bosons per site, they acquire the following matrix representation (see Eqs. 55) where the symbol ⊗ indicates the usual tensorial product between matrices, and the (NP + 1) × (NP + 1) dimensional matrices 1l and ˆb† are given by     0 0 0 ··· 0 0 1 0 0 ··· 0 0 1 0 ··· 0  1 √0 0 · · · 0 0    †   . 56)  , ˆb =  0 2 0 · · · 0 0 0 0 1 · · · 0 1l =  .