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):…
Lieb–Robinson light cone: ‖[A(t),B]‖ ≤ C·exp(v·t − d(A,B)/ξ); Grönwall exponential bound
Lieb–Robinson theorem: emergent light-cone structure in local lattice Hamiltonians. Setup: a local Hamiltonian H = Σ_i h_i on a lattice,…
Quantum phase estimation: inverse-QFT extracts eigenphase to Δφ = 2π/2^t precision
Quantum Phase Estimation (QPE): the canonical algorithm for extracting eigenphases of unitaries. Setup: a unitary U with eigenvector |ψ⟩…
MERA (Multi-scale Entanglement Renormalization Ansatz): SVD-based scale-invariant tensor network
Multi-scale Entanglement Renormalization Ansatz (Vidal 2007). Setup: a 1D (or 2D) quantum state on N sites represented as a hierarchical…
Lieb–Robinson velocity: v_LR = 2·s·J and bound ‖[A(t),B]‖ ≤ C·exp((v·t − d/ξ))
Lieb–Robinson velocity bound: for a local Hamiltonian with single-site operator norms ‖h_i‖ ≤ s and hopping/interaction amplitude J, the…
QPE precision: Δφ = 2π/2^t; P_success ≥ 8/π² for exact-t-bit accuracy
Quantum Phase Estimation precision and success probability. From framework F8, reading t ancilla qubits projects onto the nearest 2^t-adic…
MERA area-law entanglement: S(A) ≤ log(χ) × |∂A|
MERA representability of area-law entanglement. From framework F9 (MERA with SVD truncation to bond dimension χ), the entanglement entropy…
Feynman simulation (1982)
R Feynman 1982 'Simulating Physics with Computers'; quantum-system requires quantum-simulator; foundational vision realized 2010s+.
Optical lattice (Bloch 2002)
I Bloch 2002 superfluid-Mott-insulator transition in 87Rb; modern Esslinger Fermi-Hubbard + Greiner microscope; quantum-simulator standard.
Rydberg array (Lukin 2017)
Bernien-Lukin 2017 51-atom Rydberg programmable-array; modern QuEra + Atom-Computing 256+ qubit demonstrations.
Ion-trap (Monroe 2010s)
C Monroe 2010s programmable-trapped-ion simulator; modern IonQ + Quantinuum Pauli-string Hamiltonians + lattice gauge-theory simulation.
DMRG (White 1992)
S R White 1992 density-matrix-renormalization-group; modern MPS + DMRG-2D + tensor-networks Verstraete-Cirac 2008.
VMC (McMillan 1965)
McMillan 1965 + Ceperley 1980 quantum-Monte-Carlo for many-body ground-states; modern neural-network-quantum-states Carleo-Troyer 2017.
Feynman (1982)
R Feynman 1982 'Simulating physics with computers' (Nobel 1965); modern modern foundational text + 2024 IBM 1000-qubit + dynamics +…
Greiner SF-MI (2002)
M Greiner 2002 superfluid-Mott-insulator transition cold-atom; modern modern foundational + Bose-Hubbard + 2024 quantum-gas microscope.
Ising trap (Britton 2012)
J Britton 2012 trapped-ion 300-spin Ising; modern modern foundational + 2024 1000-ion arrays + thermodynamic-phase-transition observation.
Rydberg array (Bernien 2017)
H Bernien-Endres 2017 + Lukin 256-Rydberg-array; modern modern foundational + 2024 6000-atom dual-species programmable.
VQE (Peruzzo 2014)
A Peruzzo 2014 + McClean 2016 VQE hybrid quantum-classical; modern modern foundational + 2024 deep-VQE + adaptive-ansatz quantum-chemistry.
Trotter (Trotter 1959)
H Trotter 1959 + Suzuki 1976 product-formula; modern modern foundational + Lloyd 1996 + 2024 randomization qDRIFT + Hamiltonian-simulation.