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Topological Quantum Compilation Using Mixed-Integer Programming

Pavel Rytir, Phillip C. Burke, Christos Aravanis, Jiri Vala, Jakub Marecek

TL;DR

This work formalizes quantum gate compilation for topological quantum computing as a Mixed-Integer Quadratically Constrained Quadratic Program (MIQCQP) and demonstrates its applicability to non-semisimple Ising anyon systems. By introducing fixed-depth circuit variables, leakage constraints, and a quadratic objective measuring fidelity on the computational subspace, the authors leverage McCormick linearization to obtain tractable linear relaxations and global optimality guarantees. They further integrate the Cartan decomposition and Makhlin invariants to optimize over local equivalence classes, enabling efficient targeting of CNOT-like and perfect-entangler gates via the non-semisimple gate alphabet that includes a leakage-suppressed entangling gate $CPHASE$. Experimental results show that while exact CNOT was not reachable within modest depths for the full alphabet, the local-equivalence and perfect-entangler targets can be achieved with relatively shallow circuits, often driven by repeated application of the entangling gate. Overall, the approach provides a principled, globally optimal route to explicit gate construction in topological hardware and underscores the practical viability of universal braiding-based quantum computation in non-semisimple Ising TQFTs.

Abstract

We introduce the Mixed-Integer Quadratically Constrained Quadratic Programming framework for the quantum compilation problem and apply it in the context of topological quantum computing. In this setting, quantum gates are realized by sequences of elementary braids of quasiparticles with exotic fractional statistics in certain two-dimensional topological condensed matter systems, described by effective topological quantum field theories. We specifically focus on a non-semisimple version of topological field theory, which provides a foundation for an extended theory of Ising anyons and which has recently been shown by Iulianelli et al., Nature Communications {\bf 16}, 6408 (2025), to permit universal quantum computation. While the proofs of this pioneering result are existential in nature, the mixed integer programming provides an approach to explicitly construct quantum gates in topological systems. We demonstrate this by focusing specifically on the entangling controlled-NOT operation, and its local equivalence class, using braiding operations in the non-semisimple Ising system. This illustrates the utility of the Mixed-Integer Quadratically Constrained Quadratic Programming for topological quantum compilation.

Topological Quantum Compilation Using Mixed-Integer Programming

TL;DR

This work formalizes quantum gate compilation for topological quantum computing as a Mixed-Integer Quadratically Constrained Quadratic Program (MIQCQP) and demonstrates its applicability to non-semisimple Ising anyon systems. By introducing fixed-depth circuit variables, leakage constraints, and a quadratic objective measuring fidelity on the computational subspace, the authors leverage McCormick linearization to obtain tractable linear relaxations and global optimality guarantees. They further integrate the Cartan decomposition and Makhlin invariants to optimize over local equivalence classes, enabling efficient targeting of CNOT-like and perfect-entangler gates via the non-semisimple gate alphabet that includes a leakage-suppressed entangling gate . Experimental results show that while exact CNOT was not reachable within modest depths for the full alphabet, the local-equivalence and perfect-entangler targets can be achieved with relatively shallow circuits, often driven by repeated application of the entangling gate. Overall, the approach provides a principled, globally optimal route to explicit gate construction in topological hardware and underscores the practical viability of universal braiding-based quantum computation in non-semisimple Ising TQFTs.

Abstract

We introduce the Mixed-Integer Quadratically Constrained Quadratic Programming framework for the quantum compilation problem and apply it in the context of topological quantum computing. In this setting, quantum gates are realized by sequences of elementary braids of quasiparticles with exotic fractional statistics in certain two-dimensional topological condensed matter systems, described by effective topological quantum field theories. We specifically focus on a non-semisimple version of topological field theory, which provides a foundation for an extended theory of Ising anyons and which has recently been shown by Iulianelli et al., Nature Communications {\bf 16}, 6408 (2025), to permit universal quantum computation. While the proofs of this pioneering result are existential in nature, the mixed integer programming provides an approach to explicitly construct quantum gates in topological systems. We demonstrate this by focusing specifically on the entangling controlled-NOT operation, and its local equivalence class, using braiding operations in the non-semisimple Ising system. This illustrates the utility of the Mixed-Integer Quadratically Constrained Quadratic Programming for topological quantum compilation.

Paper Structure

This paper contains 18 sections, 34 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Mixed Integer Programming Formulation
  3. Quantum Compilation Problem
  4. Program Variables
  5. Program Constraints
  6. Program Objective Function
  7. MIQCQP Problem Formulation
  8. MIQCQP Problem Relaxation and Linearization
  9. Local Equivalence Classes
  10. Distance to the Local Equivalence Class MIQCQP Problem Formulation
  11. Solving the MIQCQP Programs
  12. Ising anyons from a non-semisimple Topological Quantum Field Theory
  13. Experimental Results
  14. CNOT gate compilation
  15. Local CNOT equivalence gate compilation
  16. ...and 3 more sections

Figures (1)

  • Figure 1: (a) Distance of powers of the entangling gate (Eq. \ref{['eq_entanglingMatrix']}), to the perfect entanglers (Eq. \ref{['eq_PE_dist']}) plotted versus power $L$. (b) Invariants of matrices from panel (a) plotted in $g_1,g_2,g_3$ space (up to $L=35$). The data is plotted alongside the perfect entangler section of the Weyl chamber, highlighted in red.