Abstract
The mapping of fermionic states onto qubit states, as well as the mapping of fermionic Hamiltonian into quantum gates enables us to simulate electronic systems with a quantum computer. Benefiting the understanding of many-body systems in chemistry and physics, quantum simulation is one of the great promises of the coming age of quantum computers. Interestingly, the minimal requirement of qubits for simulating Fermions seems to be agnostic of the actual number of particles as well as other symmetries. This leads to qubit requirements that are well above the minimal requirements as suggested by combinatorial considerations. In this work, we develop methods that allow us to trade-off qubit requirements against the complexity of the resulting quantum circuit. We first show that any classical code used to map the state of a fermionic Fock space to qubits gives rise to a mapping of fermionic models to quantum gates. As an illustrative example, we present a mapping based on a nonlinear classical error correcting code, which leads to significant qubit savings albeit at the expense of additional quantum gates. We proceed to use this framework to present a number of simpler mappings that lead to qubit savings with a more modest increase in gate difficulty. We discuss the role of symmetries such as particle conservation, and savings that could be obtained if an experimental platform could easily realize multi-controlled gates.
Highlights
Simulating quantum systems on a quantum computer is one of the most promising applications of small scale quantum computers [1]
We first show that any classical code used to map the state of a fermionic Fock space to qubits gives rise to a mapping of fermionic models to quantum gates
We show that for any encoding e : Z2⊗N → Z2⊗n there exists a mapping of Fermionic models to quantum gates
Summary
Simulating quantum systems on a quantum computer is one of the most promising applications of small scale quantum computers [1]. ΝN where νj ∈ {0, 1} indicates the presence (νj = 1) or absence (νj = 0) of a spinless fermionic particle at orbital j [24] Such a mapping e : Z2⊗N → Z2⊗n is called an encoding [25]. We need a way to simulate the dynamics of fermions on these N orbitals These dynamics can be modeled entirely in terms of the annihilation and creation operators cj and c†j that act on the fermionic Fock space as c†im c†i1. Mappings of the operators cj to qubits typically use the Pauli matrices X, Z, and Y acting on one qubit, characterized by their anticommutation relations [Pi, Pj ]+ = 2δijI for all Pi ∈ P = {X, Z, Y } An example of such a mapping is the Jordan-. We remark that instead of looking for a mapping for individual operators cj we may instead opt to map pairs (or higher order terms) of such operators at once, or even look to represent sums of such operators
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