Sub-Riemannian geometry of measurement based quantum computation

Abstract

The computational power of quantum phases of matter with symmetry can be accessed through local measurements, but what is the most efficient way of doing so? In this work, we show that minimizing operational resources in measurement-based quantum computation on subsystem symmetric resource states amounts to solving a sub-Riemannian geodesic problem between the identity and the target logical unitary. This reveals a geometric structure underlying MBQC and offers a principled route to optimize quantum processing in computational phases.

9 pages, 4 figures. This is the Accepted Manuscript version of an article accepted for publication in Phys. Rev. A. The American Physical Society is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. This Accepted Manuscript is published under a CC BY licence. The Version of Record is available online at 10.1103/zq4l-vwbz

References in corpus (26)

• Time Optimal Control in Spin Systems
• Quantum Computation as Geometry
• Restrictions on Transversal Encoded Quantum Gate Sets
• Sub-Riemannian Geometry and Time Optimal Control of Three Spin Systems: Quantum Gates and Coherence Transfer
• Braid Topologies for Quantum Computation
• Symmetry-protected phases for measurement-based quantum computation
• Higher Order Decompositions of Ordered Operator Exponentials
• Optimal control, geometry, and quantum computing
• Quantum computation on the edge of a symmetry-protected topological order
• Identifying phases of quantum many-body systems that are universal for quantum computation
• Krotov: A Python implementation of Krotov's method for quantum optimal control
• A computationally universal phase of quantum matter
• Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter
• Universal quantum computation using fractal symmetry-protected cluster phases
• Resource quality of a symmetry-protected topologically ordered phase for quantum computation
• Quantum computational renormalization in the Haldane phase
• Symmetry-protected topological phases with uniform computational power in one dimension
• Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices
• Efficient Product Formulas for Commutators and Applications to Quantum Simulation
• Universal measurement-based quantum computation in a one-dimensional architecture enabled by dual-unitary circuits
• Measurement-based quantum computation in finite one-dimensional systems: string order implies computational power
• Classification of measurement-based quantum wire in stabilizer PEPS
• Approximating exponentials of commutators by optimized product formulas
• Optimally generating $\mathfrak{su}(2^N)$ using Pauli strings
• Counter-intuitive yet efficient regimes for measurement based quantum computation on symmetry protected spin chains
• Monte Carlo approach for finding optimally controlled quantum gates with differential geometry