Algebraic Compression of Quantum Circuits for Hamiltonian Evolution: Part Two

Generating efficient circuits for time evolution is an important step in furthering the use of quantum computers as a tool for simulating many-body physics, which involves evolving states and computing their overlaps. For certain Hamiltonians, including often-studied ones such as the transverse field Ising model, using compression methods that rely on local algebraic relations between quantum gates [1,2] the time evolution may be significantly shortened into a circuit with a depth independent of simulation time. While powerful, the compression method is still missing several key features: it can only compress 1D chains, and cannot compress the controlled evolution that is necessary for computing wave function overlaps. Here, we present an advanced algebraic compression algorithm that overcomes these hurdles. First, we develop the necessary mathematics to enable compression of free fermionic evolution on any lattice topology, enabling embedding mismatching topologies on available quantum computers. Secondly, by extending the algebraic structure to controlled gates, we enable compression of controlled evolution with the addition of O($n$) controlled gates. In both cases, the compression for $n$ qubits results in a circuit with O($n$) depth and O($n^2$) gates. We demonstrate these developments in the contexts of two-dimensional free fermionic evolution and in the evaluation of the Zak phase in the Creutz model. These developments now enable the full range of necessary capabilities for simulating free fermionic and corresponding spin models, e.g., the 1D TFIM, TFXY, and Kitaev type models. [1] E. Kökcü et al., Phys. Rev. A 105, 032420 (2021). [2] D. Camps et al., SIAM Journal on Matrix Analysis and Applications, 43:3, pp. 1084-1108 (2022).