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Sub-Riemannian geometry of measurement based quantum computation

Lukas Hantzko, Arnab Adhikary, Robert Raussendorf

TL;DR

This work reframes measurement-based quantum computation in symmetry-constrained quantum phases as a problem in sub-Riemannian geometry. By modeling MBQC gates as elements generated by a fixed, phase-determined set of generators, the authors connect logical error to the Carnot–Carathéodory distance $d_{ m CC}(e,U)$ along a horizontal geodesic, and prove an asymptotic error bound $\epsilon \le \frac{1}{N}\left(\frac{1}{\sigma^2}-1\right) d_{ m CC}^2(e,U) + \mathcal{O}(N^{-2})$ for implementing a unitary $U$ with $N$ nontrivial measurements and order parameter $\sigma$. The proof combines a local-error analysis with a Lie–Trotter–Suzuki discretization of a minimizing geodesic, linking resource scaling to geodesic length and energy, and showing that MBQC-native gate sets can yield more efficient implementations than conventional Euler-angle decompositions. The results provide a principled, geometric framework to optimize MBQC across computational phases of quantum matter, with extensions to subsystem symmetries, 2D architectures, and higher-dimensional gate sets, and suggest practical numerical strategies (e.g., fast marching) for finding geodesics in large systems.

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.

Sub-Riemannian geometry of measurement based quantum computation

TL;DR

This work reframes measurement-based quantum computation in symmetry-constrained quantum phases as a problem in sub-Riemannian geometry. By modeling MBQC gates as elements generated by a fixed, phase-determined set of generators, the authors connect logical error to the Carnot–Carathéodory distance along a horizontal geodesic, and prove an asymptotic error bound for implementing a unitary with nontrivial measurements and order parameter . The proof combines a local-error analysis with a Lie–Trotter–Suzuki discretization of a minimizing geodesic, linking resource scaling to geodesic length and energy, and showing that MBQC-native gate sets can yield more efficient implementations than conventional Euler-angle decompositions. The results provide a principled, geometric framework to optimize MBQC across computational phases of quantum matter, with extensions to subsystem symmetries, 2D architectures, and higher-dimensional gate sets, and suggest practical numerical strategies (e.g., fast marching) for finding geodesics in large systems.

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.

Paper Structure

This paper contains 7 sections, 4 theorems, 38 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Result
  4. Methods
  5. Scaling
  6. Simplest Example
  7. Conclusion and Outlook

Key Result

theorem 1

Consider a resource state with a translation invariant computational order parameter $\sigma$ and executable generators $\mathcal{G}$. Let $A$ be the Lie hull of the executable generators $\mathcal{G}$ and $G$ be the corresponding closed Lie subgroup. Let $N$ be the number of sites with non-trivial Therein, $\, \text{\upshape d}_{\mathrm{CC}}(e,U)$ is the sub-Riemannian distance (Def. definition:

Figures (4)

  • Figure 1: Numerical example for confirmation of the error scaling for a Pauli $X$ rotation. We compute the error numerically (purple) and using our bound (Lemma \ref{['lem:error_est_rot_sequence']}) (green). As mentioned in the text, the bound is tight for all numerical experiments using also different executable generators. Using the Frobenius or the Trace Norm instead, the bound is not tight for combinations of different generators. Parameters: $\sigma = 0.9$, $N = 500$
  • Figure 2: The sub-Riemannian geodesic (Example \ref{['example:SubRiemannianGeodesics']}) for a rotation in the $y$ direction (Purple). For that curve we compare the Lie-Trotter-Suzuki product formula (Green), which is a curve with piecewise constant control. To the right, the control functions $c_z(t)$ (purple) and $c_x(t)$ (green) are depicted.
  • Figure 3: We compare the error norm $\epsilon$ for our implementation of a $y$ rotation (purple) to the theoretical upper limit (green) given by theorem \ref{['thm:implemented_error']}. We see that the error satisfies the bound. Parameters: $\sigma = 0.9$, $N = 500$
  • Figure 4: We compare the different Bounds for a $Y$-rotation in the $\mathrm{SU}(2)$ with $X$ and $Z$ as horizontal directions. Riemannian distance or arclength and scaling for the Euler angles (Blue). The Sub-Riemannian distance that is important to our error bound (Black). Bounds with $d_\infty$ given by the box-ball theorem (Red).

Theorems & Definitions (16)

  • definition 1: Sub-Riemannian Geodesics, Distance
  • theorem 1: Implemented Error
  • remark 1: Local Error
  • proof
  • lemma 1: Error bounds for rotation sequences
  • proof
  • remark 2: Numerical Observation
  • remark 3: Curves and Error
  • lemma 2: Error and Energy
  • proof
  • ...and 6 more