Joel Klassen

The limiting constraint on simulating materials on near-term quantum hardware is the requisite circuit depths and qubit numbers, with current estimates placing them well beyond near-term capabilities. A critical subroutine of simulation algorithms is implementing a layer of unitary evolutions by each local term in the Hamiltonian. In this work we develop a new quantum algorithm which dramatically reduces the estimated cost of material simulations using this subroutine, improving circuit depths by up to 6 orders of magnitude for Strontium Vanadate, for example. We achieve this by introducing a fermionic encoding that leverages the locality of materials Hamiltonians describing an active space in the Wannier basis. This design generates quantum circuits whose depth is independent of the system’s size.

In this work we present a method for generating a fermionic encoding tailored to a set of target fermionic operators and to a target hardware connectivity. Our method uses brute force search, over the space of all encodings which map from Majorana monomials to Pauli operators, to find an encoding which optimizes a target cost function. In contrast to earlier works in this direction, our method searches over an extremely broad class of encodings which subsumes all known second quantized encodings that constitute algebra homomorphisms. In order to search over this class, we give a clear mathematical explanation of how precisely it is characterized, and how to translate this characterization into constructive search criteria. A benefit of searching over this class is that our method is able to supply fairly general optimality guarantees on solutions. A second benefit is that our method is, in principal, capable of finding more efficient representations of fermionic systems when the set of fermionic operators under consideration are faithfully represented by a smaller quotient algebra. Given the high algorithmic cost of performing the search, we adapt our method to handle translationally invariant systems that can be described by a small unit cell that is less costly. We demonstrate our method on various pairings of target fermionic operators and hardware connectivities. We additionally show how our method can be extended to find error detecting fermionic encodings in this class.

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 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. Local fermionic encodings are a class of fermion to qubit mappings that map local fermionic operators to local qubit operators. We present a novel local fermionic encoding -- called the compact encoding -- which outperforms all previous local fermionic encodings in both the qubit to mode ratio, and the locality of mapped operators. We demonstrate how this encoding may be applied to any uniform tiling of degree 4 or less, for example square and hexagonal lattices, and to a 3d cubic lattice. In order to characterize the encoding on various lattices we clarify the group theoretic structure underlying the design of such encodings. We also illuminate an elegant relationship between the compact encoding and the toric code, and show how the compact encoding may be understood as condensing the fermionic excitations of the toric code into its code space.