[Paper Review] Universal Translationally-Invariant Hamiltonians

Stephen Piddock, Johannes Bausch|arXiv (Cornell University)|Jan 22, 2020

Quantum many-body systems27 references4 citations

TL;DR

This paper constructs a translationally-invariant universal quantum simulator on a 2D qudit lattice with nearest-neighbor interactions, using history state and tiling techniques to embed a target Hamiltonian's spectrum into the simulator's low-energy subspace. The key result is that any local Hamiltonian can be simulated to arbitrary precision using only translationally-invariant couplings, implying that complex phenomena like many-body localization can emerge in translationally-invariant systems.

ABSTRACT

In this work we extend the notion of universal quantum Hamiltonians to the setting of translationally-invariant systems. We present a construction that allows a two-dimensional spin lattice with nearest-neighbour interactions, open boundaries, and translational symmetry to simulate any local target Hamiltonian---i.e. to reproduce the whole of the target system within its low-energy subspace to arbitrarily-high precision. Since this implies the capability to simulate non-translationally-invariant many-body systems with translationally-invariant couplings, any effect such as characteristics commonly associated to systems with external disorder, e.g. many-body localization, can also occur within the low-energy Hilbert space sector of translationally-invariant systems. Then we sketch a variant of the universal lattice construction optimized for simulating translationally-invariant target Hamiltonians. Finally we prove that qubit Hamiltonians consisting of Heisenberg or XY interactions of varying interaction strengths restricted to the edges of a connected translationally-invariant graph embedded in $\mathbb{R}^D$ are universal, and can efficiently simulate any geometrically local Hamiltonian in $\mathbb{R}^D$.

Motivation & Objective

• To establish the existence of a universal simulator Hamiltonian that is translationally invariant, despite open boundaries.
• To demonstrate that translationally-invariant systems can simulate arbitrary local Hamiltonians, including those with disorder or localization.
• To construct a universal simulator using history state and tiling techniques to encode computation and target Hamiltonian parameters.
• To prove that Heisenberg or XY interactions on a translationally-invariant graph in R^D can efficiently simulate any geometrically local Hamiltonian in R^D.
• To show that the simulation precision scales polynomially with system size and inverse error, while maintaining translational invariance in interactions.

Proposed method

• Construct a 2D qudit lattice with open boundaries and translationally-invariant local and nearest-neighbor interactions.
• Use a tiling construction to encode lattice dimensions W and H as binary numbers on the left and bottom edges.
• Define a history state Hamiltonian that simulates a quantum Turing machine starting from the lower-left corner, winding across a marked triangular and square region.
• Implement single-qubit rotations via Solovay-Kitaev approximation to encode target Hamiltonian parameters (e.g., coupling strengths) into the system.
• Embed the target Hamiltonian as an effective low-energy theory via perturbation theory, using a projector Hamiltonian that projects onto the desired subspace.
• Apply perturbative gadgets to map the history state Hamiltonian to a universal simulator Hamiltonian H_sim = Δ₁H₁ + Δ₂H₂ with tunable scaling parameters Δ₁, Δ₂.

Experimental results

Research questions

• RQ1 Can a translationally-invariant Hamiltonian simulate any local quantum Hamiltonian to arbitrary precision?
• RQ2 Can many-body localization—typically associated with disordered systems—emerge in a translationally-invariant system?
• RQ3 Is it possible to construct a universal simulator using only nearest-neighbor, translationally-invariant interactions on a 2D lattice?
• RQ4 What is the minimal resource cost (lattice size, local dimension) required to simulate a given target Hamiltonian with translationally-invariant couplings?
• RQ5 Can Heisenberg or XY interactions on a translationally-invariant graph in R^D simulate any geometrically local Hamiltonian in R^D?

Key findings

• A universal translationally-invariant simulator Hamiltonian H_sim exists on a 2D qudit lattice with nearest-neighbor interactions and open boundaries.
• The simulator can simulate any local target Hamiltonian H_target to precision ϵ using a lattice size W×H = poly(Δ, 1/η, 1/ϵ, J_max) × 2^poly(n), where n is the number of qudits in H_target.
• The scaling parameters are Δ₁ = W⁵H⁵ and Δ₂ = WH, ensuring the correct energy scale for the effective target Hamiltonian.
• The construction uses history state techniques and tiling to encode lattice dimensions and target coupling parameters into the system.
• The effective low-energy Hamiltonian approximates H_target with error ‖H̃_target − H_target‖ ≤ ϵ/2, achieved via perturbative gadgets and Solovay-Kitaev approximation.
• Heisenberg or XY interactions on a translationally-invariant graph in R^D can efficiently simulate any geometrically local Hamiltonian in R^D, proving universality for such interaction types.