Quantum Randomized Subspace Iteration

TL;DR

The paper introduces Quantum Randomized Subspace Iteration (QRSI), a parallel quantum algorithm for efficiently resolving degenerate eigenspaces, demonstrated on topological codes and random Hamiltonians.

Contribution

QRSI is a novel parallel method that spans degenerate eigenspaces using random unitaries, improving state preparation in quantum systems with degeneracies.

Findings

QRSI recovers all topological ground states of the toric code.

QRSI effectively identifies degenerate eigenspaces in random Hamiltonians.

Guarantees hold under weaker conditions than full Haar randomness.

Abstract

Resolving degenerate quantum eigenspaces - including topologically ordered ground states and frustrated magnets - requires preparing high-fidelity states that span every direction of the target manifold. Existing variational and projective algorithms do not naturally cover a multi-dimensional degenerate subspace without sequential orthogonality constraints. We introduce the quantum randomized subspace iteration (QRSI), a fully parallel construction that conjugates the Hamiltonian by independent random unitaries across as many branches as the degeneracy g, then invokes any chosen eigenstate-preparation primitive on each branch. The target subspace is identified from the resulting ensemble via standard subspace estimation, either classically through the coefficient matrix or on hardware through Gram-matrix measurements. We prove that the construction spans the full eigenspace almost surely and preserves the spectral gap exactly on every branch. For practical use, we show that these guarantees hold whenever the random rotations satisfy an anti-concentration condition over the degenerate manifold, substantially weaker than full Haar randomness. We demonstrate QRSI on the toric code, recovering all four topological ground states, and on random Hamiltonians with planted degeneracies.