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Bravyi-Kitaev Superfast simulation of fermions on a quantum computer

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Abstract

Present quantum computers often work with distinguishable qubits as their computational units. In order to simulate indistinguishable fermionic particles, it is first required to map the fermionic state to the state of the qubits. The Bravyi-Kitaev Superfast (BKSF) algorithm can be used to accomplish this mapping. The BKSF mapping has connections to quantum error correction and opens the door to new ways of understanding fermionic simulation in a topological context. Here, we present the first detailed exposition of BKSF algorithm for molecular simulation. We provide the BKSF transformed qubit operators and report on our implementation of the BKSF fermion-to-qubits transform in OpenFermion. In this initial study of the hydrogen molecule, we have compared BKSF, Jordan-Wigner and Bravyi-Kitaev transforms under the Trotter approximation. We considered different orderings of the exponentiated terms and found lower Trotter errors than previously reported for Jordan-Wigner and Bravyi-Kitaev algorithms. These results open the door to further study of the BKSF algorithm for quantum simulation.

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... Generalization of the transformation to higher dimensions using lattice gauge fields have also been proposed [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Such exact bosonization mappings are also important form the quantum simulation perspective, for they address the possibility of simulating a quantum many-body fermionic systems using a bosonic quantum computer [21][22][23][24][25][26]. Symmetries, however, may not remain manifest under the transformation. ...
... The ηχ approach is inspired by earlier works provid-ing lattice bosonization recipes through the introduction of Majorana fermions [7,8,23]. It is also closely related to the approach developed in Refs. ...
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We discuss a scheme for performing Jordan-Wigner transformation for various lattice fermion systems in two and three dimensions which keeps internal and spatial symmetries manifest. The correspondence between fermionic and bosonic operators is established with the help of auxiliary Majorana fermions. The current construction is applicable to general lattices with even coordination number and an arbitrary number of fermion flavors. The approach is demonstrated on the single-orbital square, triangular and cubic lattices for spin-1/2 fermions. We also discuss the relation to some quantum spin liquid models.
... However, there is a mapping with which locally-interacting Fermion and qubit lattices can be related: the Verstraete-Cirac transform (VCT) [23] also known as Auxiliary Fermion Mapping [24][25][26], can be regarded as a manipulation of the Jordan-Wigner transform, in which additional auxiliary particles are added, hence the name. Other works on Fermion-to-qubit mappings [25,[27][28][29] are based on two transforms proposed by Bravyi and Kitaev in [30]. Firstly, there is what is commonly known as the Bravyi-Kitaev transform, that, compared to the Jordan-Wigner transform, exhibits an up to exponential improvement on the number of qubits that each fermionic interaction term acts on. ...
... So far, we still have not left the even-parity subspace, but we might have systems to solve that are populated by odd numbers of Fermions. In [29], it is suggested to add another mode to the system that is however not coupled to any other term in the Hamiltonian. From the original concept of the BKSF it is however not clear how this mode is brought into the system, since all qubits correspond to couplings of modes in the Hamiltonian, which here do not exist. ...
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... Constructing all generators directly in their fermionic representation, will result in automatic gradient evaluation schemes independent of the underlying qubit encoding, allowing flexible adaption of new encodings [44,45] and improved compiling strategies into quantum gates [46]. In analogy to the qubit construction in Eq. (29) the nullspace projector of a general single, double or n-fold fermionic excitation (5) can be constructed as ...
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