[Paper Review] Gauged cooling of topological excitations and emergent fermions on quantum simulators

Gilad Kishony, Mark S. Rudner+2 more|arXiv (Cornell University)|Oct 24, 2023

Quantum many-body systems4 citations

TL;DR

This paper proposes a gauged cooling protocol that efficiently removes non-local topological excitations—such as domain walls in the quantum Ising model—by coupling the system to a Z₂ gauge field that acts as a local reservoir. The method enables efficient cooling of both ferromagnetic and paramagnetic phases by allowing local operations to remove individual domain walls via gauge transformations, overcoming the inefficiency of conventional bath-based cooling in topological systems.

ABSTRACT

Simulated cooling is a robust method for preparing low-energy states of many-body Hamiltonians on near-term quantum simulators. In such schemes, a subset of the simulator's spins (or qubits) are treated as a ``bath,'' which extracts energy and entropy from the system of interest. However, such protocols are inefficient when applied to systems whose excitations are highly non-local in terms of the microscopic degrees of freedom, such as topological phases of matter; such excitations are difficult to extract by a local coupling to a bath. We explore a route to overcome this obstacle by encoding of the microscopic degrees of freedom into those of the quantum simulator in a non-local manner. To illustrate the approach, we show how to efficiently cool the ferromagnetic phase of the quantum Ising model, whose excitations are domain walls, via a ``gauged cooling'' protocol in which the Ising spins are coupled to a $Z_2$ gauge field that simultaneously acts as a reservoir for removing excitations. We show that our protocol can prepare the ground states of the ferromagnetic and paramagnetic phases equally efficiently. The gauged cooling protocol naturally extends to (interacting) fermionic systems, where it is equivalent to cooling by coupling to a fermionic bath via single-fermion hopping.

Motivation & Objective

• To address the inefficiency of conventional simulated cooling in preparing ground states of topological phases, where non-local excitations like domain walls resist local bath coupling.
• To overcome the limitation that single domain walls cannot be removed by local operations in systems like the ferromagnetic Ising model.
• To develop a protocol that enables efficient cooling of both topologically ordered and trivial phases using a gauge field as a dynamic reservoir.
• To extend the approach to fermionic systems by showing equivalence to cooling via a fermionic bath with single-fermion hopping.

Proposed method

• The system's spins are coupled to a Z₂ gauge field on the bonds, transforming the Ising model into a gauge theory with emergent gauge symmetry.
• The gauge field degrees of freedom (τj) serve as a local bath: their Zeeman field is adiabatically reduced, allowing them to flip in response to domain walls.
• When a domain wall is present, the gauge field on the corresponding bond flips to lower the energy, effectively removing the excitation via a local operation.
• The final state is equivalent to the original ground state up to a gauge transformation, which can be reconstructed after measurement of the gauge field.
• The protocol is extended to fermionic systems by mapping the gauge field to a fermionic bath, where single-fermion hopping implements cooling.
• Numerical simulations use stochastic Schrödinger equations with noise modeling via depolarizing channels, and a fictitious spin is introduced to efficiently simulate non-local Pauli errors.

Experimental results

Research questions

• RQ1 Can a local cooling protocol efficiently remove non-local topological excitations such as domain walls in the Ising model?
• RQ2 How does coupling to a Z₂ gauge field enable local removal of individual domain walls despite their non-local nature?
• RQ3 Can the gauged cooling protocol achieve exponential cooling rates in topological phases, matching the efficiency seen in trivial phases?
• RQ4 Is the protocol generalizable to interacting fermionic systems through an equivalence to fermionic bath coupling?
• RQ5 How does the inclusion of noise affect the protocol’s performance, and can it be simulated efficiently on large systems?

Key findings

• The gauged cooling protocol achieves exponential cooling rates in both the ferromagnetic and paramagnetic phases of the quantum Ising model, overcoming the power-law slowdown seen in conventional local cooling.
• Domain walls are removed via local operations on the gauge field, with the final state being gauge-equivalent to the original ground state, enabling efficient ground state preparation.
• The protocol is equivalent to cooling via a fermionic bath with single-fermion hopping in interacting fermionic systems, extending its applicability beyond spin models.
• Numerical simulations demonstrate robustness to depolarizing noise, with the use of a fictitious spin enabling efficient simulation of non-local Pauli errors on large systems.
• The evolution of the single-particle density matrix under measurement depends only on the matrix itself, not higher-order correlations, enabling efficient classical simulation of the quantum dynamics.