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# Algebraic Compression of Free Fermionic Quantum Circuits: Particle Creation, Arbitrary Lattices and Controlled Evolution

Mar 2023

Recently we developed a local and constructive algorithm based on Liealgebraic methods for compressing Trotterized evolution under Hamiltonians thatcan be mapped to free fermions. The compression algorithm yields a circuitwhich scales linearly in the number of qubits, is fixed depth for forarbitrarily long evolution times and is applicable to time dependentHamiltonians, although is limited to simple nearest-neighbor spin interactionsand fermionic hopping. In this work, we extend the algorithm to compresscircuits simulating evolution with long-range spin interactions and fermionichopping, thereby enabling embedding of arbitrary lattices onto a chain ofqubits. Moreover, we show that controlled time evolution, as well as fermioncreation and annihilation operators can also be compressed. We demonstrate ourresults by adiabatically preparing the ground state for a half-filled fermionicchain, and simulating a $4 \times 4$ tight binding model on ibmq_washington.With these new developments, our results enable the simulation of a wider rangeof models of interest and the efficient compression of subcircuits.

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