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The rotation of subspaces by a chosen angle is a fundamental quantumcomputing operation, with applications in error correction and quantumalgorithms such as the Quantum Approximate Optimization Algorithm, theVariational Quantum Eigensolver and the quantum singular value transformation.Such rotations are usually implemented at the hardware level viamultiple-controlled-phase gates, which lead to large circuit depth whendecomposed into one- and two-qubit gates. Here, we propose a fast,high-fidelity way to implement such operations via topological quantum walks,where a sequence of single-qubit $z$ rotations of an ancilla qubit areinterleaved with the evolution of a system Hamiltonian in which a matrix $A$ isembedded. The subspace spanned by the left or right singular vectors of $A$with non-zero singular values is rotated, depending on the state of theancilla. This procedure can be implemented in superconducting qubits, ion-trapsand Rydberg atoms with star-type connectivity, significantly reducing the totalgate time required compared to previous proposals.

Efficient multi-qubit subspace rotations via topological quantum walks