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Cartan Decomposition of SU(2^n), Constructive Controllability of Spin systems and Universal Quantum Computing

Navin Khaneja, Steffen Glaser

TL;DR

Problem: synthesize any unitary on n qubits from simple gates. Approach: use Cartan decomposition of SU(2^n) with KAK-style recursions, Weyl orbits, and a product-operator basis to express arbitrary unitaries as sequences of one- and two-qubit gates; Demonstrates with a concrete two-qubit example and then extends recursively to n qubits in spin networks (NMR) with constructive controllability. Contributions: explicit parameterization, geometric interpretation, and time-optimal considerations for small cases; insights into the scalability and optimality of the design. Significance: provides a practical, geometry-based framework for designing universal quantum control in spin systems, with implications for pulse-sequence design and quantum computing architectures.

Abstract

In this paper we provide an explicit parameterization of arbitrary unitary transformation acting on n qubits, in terms of one and two qubit quantum gates. The construction is based on successive Cartan decompositions of the semi-simple Lie group, SU(2^n). The decomposition highlights the geometric aspects of building an arbitrary unitary transformation out of quantum gates and makes explicit the choice of pulse sequences for the implementation of arbitrary unitary transformation on $n coupled spins. Finally we make observations on the optimality of the design procedure.

Cartan Decomposition of SU(2^n), Constructive Controllability of Spin systems and Universal Quantum Computing

TL;DR

Problem: synthesize any unitary on n qubits from simple gates. Approach: use Cartan decomposition of SU(2^n) with KAK-style recursions, Weyl orbits, and a product-operator basis to express arbitrary unitaries as sequences of one- and two-qubit gates; Demonstrates with a concrete two-qubit example and then extends recursively to n qubits in spin networks (NMR) with constructive controllability. Contributions: explicit parameterization, geometric interpretation, and time-optimal considerations for small cases; insights into the scalability and optimality of the design. Significance: provides a practical, geometry-based framework for designing universal quantum control in spin systems, with implications for pulse-sequence design and quantum computing architectures.

Abstract

In this paper we provide an explicit parameterization of arbitrary unitary transformation acting on n qubits, in terms of one and two qubit quantum gates. The construction is based on successive Cartan decompositions of the semi-simple Lie group, SU(2^n). The decomposition highlights the geometric aspects of building an arbitrary unitary transformation out of quantum gates and makes explicit the choice of pulse sequences for the implementation of arbitrary unitary transformation on $n coupled spins. Finally we make observations on the optimality of the design procedure.

Paper Structure

This paper contains 7 sections, 11 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Cartan Decomposition and Riemannian Symmetric Spaces
  3. Product Operator Basis
  4. The Two Qubit Example
  5. How does $[SU(2)]^{\otimes n}$ sit inside $SU(2^n)$
  6. Generating Cartan subalgebras from Weyl orbits
  7. Conclusions and Future Work

Key Result

Theorem 1

If $\mathfrak{h}$ and $\mathfrak{h}'$ are two maximal Abelian subalgebras contained in $\mathfrak{m}$, then

Theorems & Definitions (28)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Definition 4
  • Definition 5
  • Example 1
  • Remark 2
  • ...and 18 more