Compact fermion to qubit mappings

Abstract

Mappings between fermions and qubits are valuable constructions in physics. To date only a handful exist. In addition to revealing dualities between fermionic and spin systems, such mappings are indispensable in any quantum simulation of fermionic physics on quantum computers. The number of qubits required per fermionic mode, and the locality of mapped fermionic operators strongly impact the cost of such simulations. We present a fermion to qubit mapping that outperforms all previous local mappings in both the qubit to mode ratio and the locality of mapped operators. In addition to these practically useful features, the mapping bears an elegant relationship to the toric code, which we discuss. Finally, we consider the error mitigating properties of the mapping-which encodes fermionic states into the code space of a stabilizer code. Although there is an implicit tradeoff between low weight representations of local fermionic operators, and high distance code spaces, we argue that fermionic encodings with low-weight representations of local fermionic operators can still exhibit error mitigating properties which can serve a similar role to that played by high code distances. In particular, when undetectable errors correspond to "natural" fermionic noise. We illustrate this point explicitly both for this encoding and the Verstraete-Cirac encoding.

Optimal local unitary encoding circuits for the surface code

“…Fermion to qubit mappings are essential for simulating fermionic systems using quantum computers, and an encoding circuit for such a mapping is an important subroutine in many quantum simulation algo-rithms. We now show how we can use our encoding circuits for the surface code to construct encoding circuits that prepare fermionic states in the compact mapping [19], a fermion to qubit mapping that is especially efficient for simulating the Fermi-Hubbard model. A fermion to qubit mapping defines a representation of fermionic states in qubits, as well as a representation of each fermionic operator in terms of Pauli operators.…”

“…Several methods have been proposed for mapping geometrically local fermionic operators to geometrically local qubit operators [9,19,27,47,48,50,54], all of which introduce auxiliary qubits and encode fermionic Fock space into a subspace of the full nqubit system, defined as the +1-eigenspace of elements of a stabiliser group S. Mappings that have this property as referred to as local.…”

“…We will now focus our attention on a specific local mapping, the compact mapping [19], since its stabiliser group is very similar to that of the surface code.…”

“…As we will see, this close connection to the surface code allows us to use the encoding circuits we have constructed for the surface code to encode fermionic states in the compact mapping. The compact mapping maps nearest-neighbour hopping (a † i a j + a † j a i ) and Coulomb (a † i a i a † j a j ) terms to Pauli operators with weight at most 3 and 2, respectfully, and requires 1.5 qubits for each fermionic mode [19]. Rather than mapping individual fermionic creation and annihilation operators, the compact mapping instead defines a representation of the fermionic edge (E jk ) and vertex (V j ) operators, defined as…”

“…Another advantage of using a unitary encoding circuit is that it does not require the use of ancillas to measure stabilisers, therefore providing a more qubit efficient method of preparing topologically ordered states (2× fewer qubits are required to prepare a surface code state of a given lattice size). Finally, we show how our unitary encoding circuits for the planar code can be used to construct O(L) depth circuits to encode a Slater determinant state in the compact mapping [19], which can be used for the simulation of fermionic systems on quantum computers.…”