2017/09/09 by Steven B. Damelin, Damelin, S. B., B. A. W. Mode +1
Computer Science · #11E12 #11K36 #28A78 #65D32 #68Q12 #81P68 #FOS: Mathematics #FOS: Physical sciences #Quantum Algebra (math.QA) #Quantum Computing Algorithms and Architecture #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.1709.03007
openalex publication_date 2017/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The problem of finding good approximations of arbitrary 1-qubit gates is identical to that of finding a dense group generated by a universal subset of SU(2) to approximate an arbitrary element of SU(2). The Solovay-Kitaev Theorem is a well-known theorem that guarantees the existence of a finite sequence of 1-qubit quantum gates approximating an arbitrary unitary matrix in SU(2) within specified accuracy ε > 0. In this note we study a quantitative description of this theorem in the following sense. We will work with a universal gate set T, a subset of SU(2) such that the group generated by the elements of T is dense in SU(2). For ε > 0 small enough, we define tε as the minimum reduced word length such that every point of SU(2) lies within a ball of radius ε centered at the points in the dense subgroup generated by T. For a measure of efficiency on T, which we denote K(T), we prove the following theorem: Fix a δ in [0, (2)/(3)]. Choose f: (0, ∞) → (1, ∞) satisfying limε→ 0+\dfraclog(f(tε))tε exists with value 0. Assume that the inequality ε \leqslant f(tε)⋅ 5^\frac-tε6-3δ holds. Then K(T) \leqslant 2-δ. Our conjecture implies the following: Let ν(5^tε) denote the set of integer solutions of the quadratic form: x12+x22+x32+x42=5^tε. Let M≡ MS3(N) denote the covering radius of the points N=ν(5^tε)∪ν(5^tε-1) on the sphere S3 in ℝ4. Then M ∼ f(log N)N(-1)/(6-3δ). Here N≡ N(ε)=6⋅5^tε-2.