[Paper Review] On an implementation of the Solovay-Kitaev algorithm

Attila Nagy|ArXiv.org|Jun 8, 2006

advanced mathematical theories2 references13 citations

TL;DR

This paper presents a polynomial-time, space-efficient implementation of the Solovay-Kitaev algorithm for quantum gate synthesis by replacing the exponential preparatory stage with heuristic methods that trade product length for reduced time and memory. It introduces a novel decomposition method for unitaries into group commutators without matrix diagonalization, enabling scalable approximation of arbitrary unitaries in SU(d) with polylogarithmic depth and controlled error growth.

ABSTRACT

In quantum computation we are given a finite set of gates and we have to perform a desired operation as a product of them. The corresponding computational problem is approximating an arbitrary unitary as a product in a topological generating set of $SU(d)$. The problem is known to be solvable in time $polylog(1/ε)$ with product length $polylog(1/ε)$, where the implicit constants depend on the given generators. The existing algorithms solve the problem but they need a very slow and space consuming preparatory stage. This stage runs in time exponential in $d^2$ and requires memory of size exponential in $d^2$. In this paper we present methods which make the implementation of the existing algorithms easier. We present heuristic methods which make a time-length trade-off in the preparatory step. We decrease the running time and the used memory to polynomial in $d$ but the length of the products approximating the desired operations will increase (by a factor which depends on $d$). We also present a simple method which can be used for decomposing a unitary into a product of group commutators for $2

Motivation & Objective

• Address the exponential computational cost of the classical Solovay-Kitaev preparatory stage in SU(d) for d > 2.
• Reduce the time and memory complexity of generating initial covering sets from exponential to polynomial in d.
• Develop a heuristic method to construct approximate group coverings with increased product length but feasible runtime.
• Provide an alternative decomposition of unitaries into commutators that avoids matrix diagonalization, improving compatibility with symbolic computation systems.
• Enable scalable, practical implementation of the Solovay-Kitaev algorithm for multi-qubit quantum circuits.

Proposed method

• Replace the exponential-time generation of product sets Γ with heuristic, polynomial-time methods that build initial coverings in time and space polynomial in d.
• Adopt the Shrinking, Telescoping, and Zooming techniques from the original Solovay-Kitaev framework to iteratively refine approximations.
• Use a decomposition of traceless Hermitian matrices into commutators of bounded-norm Hermitian operators via a diagonal matrix G with entries G_{j,j} = -(d-1)/2 + (j-1).
• Construct unitary decompositions via exponentiation of these Hermitian generators, ensuring error bounds via operator norm inequalities.
• Apply Lemma 4.1 to decompose off-diagonal and diagonal parts of Hamiltonians into commutators, avoiding diagonalization by conjugating with Fourier or Hadamard matrices when d < 256 and 4|d.
• Use the triangle inequality and operator norm properties to bound the error propagation in nested commutators and exponentials.

Experimental results

Research questions

• RQ1 Can the exponential preparatory stage of the Solovay-Kitaev algorithm be replaced with a polynomial-time alternative without sacrificing approximation quality?
• RQ2 How can the decomposition of a unitary into group commutators be achieved without requiring matrix diagonalization?
• RQ3 What is the trade-off between the length of the approximating product and the runtime/memory cost of the preparatory phase?
• RQ4 To what extent can the initial covering quality be improved to reduce the error growth in recursive approximation steps?
• RQ5 Can the algorithm be made practical for d > 2 by avoiding numerically expensive operations like diagonalization in symbolic systems?

Key findings

• The preparatory stage is reduced from exponential to polynomial time and space in d, at the cost of increasing the product length by a factor depending on d.
• The error bound for the n-th iteration satisfies εₙ ≤ c_approx · εₙ₋₁^(3/2), with c_approx = 16(d(d−1)/2)^(1/2) + d + 8(d(d−1)/2)^(3/2), ensuring polylogarithmic convergence.
• The length of the approximating product grows as lₙ = 8·lₙ₋₁, leading to O(log(1/ε)) depth for target precision ε.
• The method avoids matrix diagonalization by using conjugation with Fourier or Hadamard matrices for d < 256 and 4|d, enabling compatibility with systems like GAP.
• The algorithm maintains the theoretical polylogarithmic scaling in 1/ε, with the constant factor in the product length increased due to the heuristic preparatory phase.
• The method is scalable: the trade-off between runtime and product length can be tuned by adjusting the quality of the initial covering.