[Paper Review] Trading inverses for an irrep in the Solovay-Kitaev theorem

Adam Bouland, Māris Ozols|DROPS (Schloss Dagstuhl – Leibniz Center for Informatics)|Dec 28, 2017

Electronic and Structural Properties of Oxides30 references3 citations

TL;DR

This paper presents a new version of the Solovay-Kitaev theorem that replaces the requirement for a universal quantum gate set to be closed under inversion with the condition that it contains an irreducible representation of any finite group G. By leveraging group representation theory and orthogonality of irreps, the authors achieve polylogarithmic compilation overhead, offering partial progress toward a full inverse-free Solovay-Kitaev theorem for universal quantum computation.

ABSTRACT

The Solovay-Kitaev theorem states that universal quantum gate sets can be exchanged with low overhead. More specifically, any gate on a fixed number of qudits can be simulated with error $ε$ using merely $\mathrm{polylog}(1/ε)$ gates from any finite universal quantum gate set $\mathcal{G}$. One drawback to the theorem is that it requires the gate set $\mathcal{G}$ to be closed under inversion. Here we show that this restriction can be traded for the assumption that $\mathcal{G}$ contains an irreducible representation of any finite group $G$. This extends recent work of Sardharwalla et al. [arXiv:1602.07963], and applies also to gates from the special linear group. Our work can be seen as partial progress towards the long-standing open problem of proving an inverse-free Solovay-Kitaev theorem [arXiv:quant-ph/0505030, arXiv:0908.0512].

Motivation & Objective

• To resolve the long-standing open problem of proving a Solovay-Kitaev theorem without requiring gate sets to be closed under inversion.
• To extend the applicability of the Solovay-Kitaev theorem to gate sets that may not contain inverses, by replacing this condition with a representation-theoretic alternative.
• To provide a constructive algorithm for approximating arbitrary unitary gates using finite universal gate sets containing irreducible representations of finite groups.
• To improve the theoretical foundation of quantum circuit compilation by reducing reliance on inverse operations, which are often impractical in physical implementations.

Proposed method

• Replaces the standard inverse-closure requirement in the Solovay-Kitaev theorem with the assumption that the gate set contains an irreducible representation of a finite group G.
• Uses the orthogonality of irreducible representations to achieve error cancellation in gate approximations, analogous to the commutator-based error suppression in the original theorem.
• Employs a function f(U) that averages over group elements in the irrep to map ε-close-to-identity operators to O(ε²)-close-to-identity operators, enabling iterative refinement.
• Applies a volume argument to bound the initial ε₀-net size, showing ε₀ scales as 1/d! for d-dimensional systems, leading to ℓ₀ = Ω(d³ log d) initial sequence length.
• Adapts the Solovay-Kitaev algorithm framework to work with irrep-averaged error reduction instead of group commutators.
• Extends the result to special linear group gates and discusses potential generalization to other Lie groups with perfect Lie algebras.

Experimental results

Research questions

• RQ1 Can the inverse-closure requirement in the Solovay-Kitaev theorem be replaced by a representation-theoretic condition without sacrificing polylogarithmic compilation overhead?
• RQ2 Is it possible to achieve the same error suppression in gate approximation using irreducible representations instead of group commutators or inverses?
• RQ3 Can the ε₀-net size in the compilation process be improved when using irreps, especially in high-dimensional systems?
• RQ4 Do specific orderings of group elements in the irrep averaging process lead to higher-order error cancellation (e.g., ε³ instead of ε²)?
• RQ5 Can the inverse-free Solovay-Kitaev theorem be generalized to arbitrary connected Lie groups with perfect Lie algebras, as done for the inverse-closed case?

Key findings

• The paper establishes a new version of the Solovay-Kitaev theorem that replaces inverse-closure with the presence of an irreducible representation of any finite group G in the gate set.
• The compilation overhead remains polylogarithmic in 1/ε, specifically O(log³·⁹⁷(1/ε)), matching the best-known bounds of the original theorem.
• The error reduction mechanism relies on averaging over irreducible representations, which enables O(ε²) error suppression without requiring inverses.
• The initial ε₀-net size scales as ε₀ = Θ(1/d!), leading to an initial sequence length ℓ₀ = Ω(d³ log d), which is worse than the ℓ₀ = Ω(d²) in the inverse-closed case.
• For the 2D irrep of S₃, specific group element orderings lead to O(ε³) error terms, suggesting potential improvements in the exponent via structured averaging.
• The result extends to gates in the special linear group and opens pathways toward generalizing the theorem to broader classes of Lie groups.