Using a controlled quantum system to simulate the dynamics or ground state of another quantum system that is intractable classically. Feynman 1982 proposal. Two paradigms: (1) analog quantum simulators — engineering a physical system…
quantum-simulation
Analog quantum simulator: engineered H_device ≈ H_target
Feynman 1982 conceptual proposal; Jaksch-Zoller 1998 concrete Hubbard-model-in-optical-lattice proposal; Greiner 2002 Mott-superfluid…
Tensor-network ansatz: MPS / PEPS / MERA obey the area law
Tensor-network states (MPS/TT, PEPS, MERA) represent quantum many-body wavefunctions as contractions of small local tensors. Matrix…
VQE: minimise ⟨ψ(θ)|H|ψ(θ)⟩ over a parametric quantum circuit
Peruzzo-McClean-Shadbolt-Yung-Love-Aspuru-Guzik-O'Brien 2014 variational quantum eigensolver (VQE): a hybrid quantum-classical algorithm…
Trotter-Suzuki first-order error: ||e^{−i(A+B)t} − (e^{−iAt/n}e^{−iBt/n})^n|| ≤ t²·||[A,B]||/(2n)
Trotter 1959 / Suzuki 1976: for non-commuting Hamiltonians H = A + B, exp(−iHt) is approximated by the product formula (exp(−iA t/n)…
DMRG truncation: ε_trunc = Σ_{k>χ} λ_k² (sum of discarded Schmidt weights)
White 1992 DMRG / Schollwöck 2005 MPS view: at every bond of the MPS, the Schmidt decomposition |ψ⟩ = Σ_k λ_k |L_k⟩|R_k⟩ (Σ_k λ_k² = 1)…
Optimal Hamiltonian-simulation query complexity: O(τ + log(1/ε)/log log(1/ε))
Berry-Childs-Cleve-Kothari-Somma 2015 lower bound and Low-Chuang 2017 matching upper bound (quantum signal processing / qubitisation):…