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On Golden Gates and Discrepancy

Steven B. Damelin, Qiwei Liang, B. A. W. Mode

Published Jun 18, 2015• 0 citations
OpenAlex

Abstract

A 1-qubit gate is defined as an element of the unitary group $U(2)$. 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 $PSU(2)$ to approximate an arbitrary element of $U(2)$. Notably, this problem has been recently studied by Sarnak in [8]. 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 an arbitrary $\varepsilon > 0$, we define $t_{\varepsilon}$ as the minimum reduced word length such that every point of $SU(2)$ lies within a ball of radius $\varepsilon$ centered at the points in the dense subgroup generated by $T$. We define a measure of efficiency on $T$, denoted $K(T)$, construct a $t_{\varepsilon}'$ such that most points in $PSU(2)$ have optimally efficient approximation and we conjecture that $\varepsilon \leq f(t_{\varepsilon})5^{-t_{\varepsilon}/4}$ for a function $f:(0,\infty)\to (1,\infty)$ satisfying: \begin{equation*} \lim_{ t_{\varepsilon} \to \infty} log(f(t_{\varepsilon}))/t_{\varepsilon} \end{equation*} exists with value $0$. Our conjecture can be formulated to mean the following: Let $\nu(5^{t_{\varepsilon}})$ denote the set of integer solutions of the quadratic form: $x_1^2+x_2^2+x_3^2+x_4^2=5^{t_{\varepsilon}}$. Let $M\equiv M_{S^3}(\mathcal{N})$ denote the covering radius of the points $\mathcal{N}=\nu(5^{t_{\varepsilon}})\cup\nu(5^{t_{\varepsilon}-1})$ on the sphere $S^{3}$ in $\mathbb{R}^{4}$. Then $M \sim f(\log N)N^{-1/4}$. Here $N\equiv N(\varepsilon)=6\cdot5^{t_{\varepsilon}}-2$. This is a weaker conjecture than that proposed in [2], namely: $M\geq N^{{-1/4}+o(1)}$. We prove that this conjecture implies that $K(T)=\dfrac{4}{3}$. This answers a question of Sarnak in [8].

Related papers

A Note on a Quantitative Form of the Solovay-Kitaev Theorem

Steven B. Damelin, B. A. W. Mode

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 $\varepsilon > 0$. 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 $\varepsilon > 0$ small enough, we define $t_{\varepsilon}$ as the minimum reduced word length such that every point of $SU(2)$ lies within a ball of radius $\varepsilon$ 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 $[0, \frac{2}{3}]$. Choose $f: (0, \infty) \rightarrow (1, \infty)$ satisfying $\lim_{\varepsilon\to 0+}\dfrac{\log(f(t_{\varepsilon}))}{t_{\varepsilon}}$ exists with value $0$. Assume that the inequality $\varepsilon \leqslant f(t_{\varepsilon})\cdot 5^{\frac{-t_{\varepsilon}}{6-3δ}}$ holds. Then $K(T) \leqslant 2-δ$. Our conjecture implies the following: Let $ν(5^{t_{\varepsilon}})$ denote the set of integer solutions of the quadratic form: $x_1^2+x_2^2+x_3^2+x_4^2=5^{t_{\varepsilon}}$. Let $M\equiv M_{S^3}(\mathcal{N})$ denote the covering radius of the points $\mathcal{N}=ν(5^{t_{\varepsilon}})\cupν(5^{t_{\varepsilon}-1})$ on the sphere $S^{3}$ in $\mathbb{R}^{4}$. Then $M \sim f(\log N)N^{\frac{-1}{6-3δ}}$. Here $N\equiv N(\varepsilon)=6\cdot5^{t_{\varepsilon}}-2$.

arXiv:1709.03007•Sep 9, 2017•1 citation

On a Problem of Construction of Efficient Universal Sets of Quantum Gates: A preliminary report

Qingzhong Liang, Steven B. Damelin

A 1-qubit gate is defined as an element of the unitary group $U(2)$, the group of complex valued matrices whose conjugate transposes equal their inverses. In quantum computing, it is important to be able to find good approximations of 1-qubit gates by using a dense group generated by a universal subset of the projective special unitary group $G=PSU(2)$. Here, $PSU(2)$ is the group formed by taking all elements of $U(2)$ with determinant 1 and taking the quotient of this subgroup by the group $\{I,-I\}$ where $I$ is the identity matrix. In this http URL [1], Sarnak recently studied this problem motivated by developing a continued fraction type algorithm for this problem. The continued fraction algorithm is the fastest known algorithm to approximate real numbers by the dense subset of rational numbers. In this report, we study the efficiency of the approximation of $G$ by using a dense group generated by a universal subset of $G$. The measure of the efficiency $K(g)$ of an arbitrary universal subset $g$ is defined in [1] and satisfies $K(g)\geqslant 1$. $g$ is more efficient when $K(g)$ is closer to $1$. We address some open problems posed by Sarnak in [1]. This report forms the basis of the shorter preprint [LD].

•Jun 18, 2015•0 citations

On a Problem of Sarnak Related to Constructing Efficient Universal Sets of Quantum Gates

Qingzhong Liang, Steven B. Damelin

Constructing an efficient universal subset of quantum gates can be done by finding an efficient universal subset of $G=PSU(2)$ (or $SU(2)$). Following [Letter], we define the efficiency $K$ of a universal subset of $G$. We focus on two different universal subsets $S$ and $T$, and estimate their efficiencies. We show that the efficiency of the first set is bounded by $\frac{4}{3}\leqslant K(S)\leqslant 2$. Also, based on a conjecture about covering radius on $3$-sphere, we show the efficiency of the second set is $K(T)=\frac{4}{3}$, based on a question of Sarnak.

•Jun 18, 2015•0 citations

Elementary Quantum Gates from Lie Group Embeddings in $U(2^n)$: Geometry, Universality, and Discretization

Antonio Falco, Daniela Falco-Pomares, Hermann G. Matthies

In the standard circuit model, elementary gates are defined relative to a chosen tensor factorization and are therefore extrinsic to the ambient group $U(2^n)$. Writing $N=2^n$, we introduce an \emph{intrinsic descriptor layer} in $U(N)$ by declaring as primitive the motions inside faithful embedded copies of $SU(2)$ (phase-free), together with a phase-inclusive $U(2)$ variant. We describe the embedding landscape $\Emb(SU(2),U(N))$ as a finite union of $U(N)$-homogeneous strata indexed by isotypic multiplicities, with stabilizers given by centralizers, and we isolate a canonical \emph{two-level sector} parameterized by $\Gr_2(\C^N)$ up to a $PSU(2)$ gauge. Equipping $U(N)$ with the Hilbert--Schmidt bi-invariant metric, each embedded subgroup is totally geodesic, yielding a variational characterization of elementary motions via minimal-norm logarithms. On the constructive side, we prove phase-free universality in $SU(N)$ from two-level primitives using QR/Givens factorizations together with explicit diagonal generation, and we obtain full universality in $U(N)$ by explicit abelian phase bookkeeping (equivalently, via the $U(2)$ two-level dictionary). Finally, we formalize a modular finite-alphabet compilation interface: any approximation routine in $SU(2)$ (e.g.\ Solovay--Kitaev) can be lifted through two-level embeddings to yield $U(N)$-level synthesis with global operator-norm error control.

quant-pharXiv:2601.17936•Jan 27, 2026•0 citations

Golden Gates: Solovay-Kitaev, Fibonacci Anyons, and the Golden Ratio — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: The Solovay-Kitaev theorem guarantees efficient approximation of any single-qubit unitary using finite gate sets; its optimal implementations ("golden gates") are deeply tied to algebraic number theory and, for Fibonacci anyons, to the golden ratio. | MATH: The theorem states: For any finite set \(S \subset SU(2)\) generating a dense subgroup, any \(U \in SU(2)\) can be \(\varepsilon\)-approximated in \(O(\log^c(1/\varepsilon))\) gates, \(c \approx 3.97\) (improved to \(c=1\) for certain sets). Exact synthesis for Fibonacci anyons uses the braid group \(B_3\) representation where the fusion space dimension grows as \(F_n\) (Fibonacci numbers). The key constant: the golden ratio \(\varphi = (1+\sqrt{5})/2 = 1.618...\), with its inverse \(\varphi^{-1} = 0.618...\), and the related \(\varphi^{-2} = 0.382...\). The Fibonacci anyon braid matrices have entries in \(\mathbb{Z}[\varphi]\), the ring of integers of \(\mathbb{Q}(\sqrt{5})\). | CONNECTION: Direct and explicit. The Fibonac Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Oct 19, 2026•0 citations

Universality of Single-Qudit Gates

Adam Sawicki, Katarzyna Karnas

We consider the problem of deciding if a set of quantum one-qudit gates $$\mathcal {S}=\{g_1,\ldots ,g_n\}\subset G$$ is universal, i.e. if $${<}\mathcal {S}{>}$$ is dense in G, where G is either the special unitary or the special orthogonal group. To every gate g in $$\mathcal {S}$$ we assign the orthogonal matrix $$\mathrm {Ad}_g$$ that is image of g under the adjoint representation $$\mathrm {Ad}:G\rightarrow SO(\mathfrak {g})$$ and $$\mathfrak {g}$$ is the Lie algebra of G. The necessary condition for the universality of $$\mathcal {S}$$ is that the only matrices that commute with all $$\mathrm {Ad}_{g_i}$$ ’s are proportional to the identity. If in addition there is an element in $${<}\mathcal {S}{>}$$ whose Hilbert–Schmidt distance from the centre of G belongs to $$]0,\frac{1}{\sqrt{2}}[$$ , then $$\mathcal {S}$$ is universal. Using these we provide a simple algorithm that allows deciding the universality of any set of d-dimensional gates in a finite number of steps and formulate a general classification theorem.

arXiv:1609.05780•Annales Henri Poincaré•Aug 10, 2017•35 citations
A 1-qubit gate is defined as an element of the unitary group U(2)U(2)U(2). 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 PSU(2)PSU(2)PSU(2) to approximate an arbitrary element of U(2)U(2)U(2). Notably, this problem has been recently studied by Sarnak in [8]. We will work with a universal gate set TTT, a subset of SU(2)SU(2)SU(2) such that the group generated by the elements of TTT is dense in SU(2)SU(2)SU(2). For an arbitrary ε>0\varepsilon > 0ε>0, we define tεt_{\varepsilon}tε​ as the minimum reduced word length such that every point of SU(2)SU(2)SU(2) lies within a ball of radius ε\varepsilonε centered at the points in the dense subgroup generated by TTT. We define a measure of efficiency on TTT, denoted K(T)K(T)K(T), construct a tε′t_{\varepsilon}'tε′​ such that most points in PSU(2)PSU(2)PSU(2) have optimally efficient approximation and we conjecture that ε≤f(tε)5−tε/4\varepsilon \leq f(t_{\varepsilon})5^{-t_{\varepsilon}/4}ε≤f(tε​)5−tε​/4 for a function f:(0,∞)→(1,∞)f:(0,\infty)\to (1,\infty)f:(0,∞)→(1,∞) satisfying:
\begin{equation*}
\lim_{ t_{\varepsilon} \to \infty} log(f(t_{\varepsilon}))/t_{\varepsilon}
\end{equation*} exists with value 000. Our conjecture can be formulated to mean the following: Let ν(5tε)\nu(5^{t_{\varepsilon}})ν(5tε​) denote the set of integer solutions of the quadratic form: x12+x22+x32+x42=5tεx_1^2+x_2^2+x_3^2+x_4^2=5^{t_{\varepsilon}}x12​+x22​+x32​+x42​=5tε​. Let
M≡MS3(N)M\equiv M_{S^3}(\mathcal{N})M≡MS3​(N) denote the covering radius of the points N=ν(5tε)∪ν(5tε−1)\mathcal{N}=\nu(5^{t_{\varepsilon}})\cup\nu(5^{t_{\varepsilon}-1})N=ν(5tε​)∪ν(5tε​−1) on the sphere S3S^{3}S3 in R4\mathbb{R}^{4}R4.
Then M∼f(log⁡N)N−1/4M \sim f(\log N)N^{-1/4}M∼f(logN)N−1/4. Here N≡N(ε)=6⋅5tε−2N\equiv N(\varepsilon)=6\cdot5^{t_{\varepsilon}}-2N≡N(ε)=6⋅5tε​−2.
This is a weaker conjecture than that proposed in [2], namely: M≥N−1/4+o(1)M\geq N^{{-1/4}+o(1)}M≥N−1/4+o(1).
We prove that this conjecture implies that K(T)=43K(T)=\dfrac{4}{3}K(T)=34​. This answers a question of Sarnak in [8].
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)SU(2)SU(2) to approximate an arbitrary element of SU(2)SU(2)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)SU(2)SU(2) within specified accuracy ε>0\varepsilon > 0ε>0. In this note we study a quantitative description of this theorem in the following sense. We will work with a universal gate set TTT, a subset of SU(2)SU(2)SU(2) such that the group generated by the elements of TTT is dense in SU(2)SU(2)SU(2). For ε>0\varepsilon > 0ε>0 small enough, we define tεt_{\varepsilon}tε​ as the minimum reduced word length such that every point of SU(2)SU(2)SU(2) lies within a ball of radius ε\varepsilonε centered at the points in the dense subgroup generated by TTT. For a measure of efficiency on T, which we denote K(T)K(T)K(T), we prove the following theorem: Fix a δδδ in [0,23][0, \frac{2}{3}][0,32​]. Choose f:(0,∞)→(1,∞)f: (0, \infty) \rightarrow (1, \infty)f:(0,∞)→(1,∞) satisfying lim⁡ε→0+log⁡(f(tε))tε\lim_{\varepsilon\to 0+}\dfrac{\log(f(t_{\varepsilon}))}{t_{\varepsilon}}limε→0+​tε​log(f(tε​))​ exists with value 000. Assume that the inequality ε⩽f(tε)⋅5−tε6−3δ\varepsilon \leqslant f(t_{\varepsilon})\cdot 5^{\frac{-t_{\varepsilon}}{6-3δ}}ε⩽f(tε​)⋅56−3δ−tε​​ holds. Then K(T)⩽2−δK(T) \leqslant 2-δK(T)⩽2−δ. Our conjecture implies the following: Let ν(5tε)ν(5^{t_{\varepsilon}})ν(5tε​) denote the set of integer solutions of the quadratic form: x12+x22+x32+x42=5tεx_1^2+x_2^2+x_3^2+x_4^2=5^{t_{\varepsilon}}x12​+x22​+x32​+x42​=5tε​. Let M≡MS3(N)M\equiv M_{S^3}(\mathcal{N})M≡MS3​(N) denote the covering radius of the points N=ν(5tε)∪ν(5tε−1)\mathcal{N}=ν(5^{t_{\varepsilon}})\cupν(5^{t_{\varepsilon}-1})N=ν(5tε​)∪ν(5tε​−1) on the sphere S3S^{3}S3 in R4\mathbb{R}^{4}R4. Then M∼f(log⁡N)N−16−3δM \sim f(\log N)N^{\frac{-1}{6-3δ}}M∼f(logN)N6−3δ−1​. Here N≡N(ε)=6⋅5tε−2N\equiv N(\varepsilon)=6\cdot5^{t_{\varepsilon}}-2N≡N(ε)=6⋅5tε​−2.
A 1-qubit gate is defined as an element of the unitary group U(2)U(2)U(2), the group of complex valued matrices whose conjugate transposes equal their inverses. In quantum computing, it is important to be able to find good approximations of 1-qubit gates by using a dense group generated by a universal subset of the projective special unitary group G=PSU(2)G=PSU(2)G=PSU(2). Here, PSU(2)PSU(2)PSU(2) is the group formed by taking all elements of U(2)U(2)U(2) with determinant 1 and taking the quotient of this subgroup by the group {I,−I}\{I,-I\}{I,−I} where III is the identity matrix. In this http URL [1], Sarnak recently studied this problem motivated by developing a continued fraction type algorithm for this problem. The continued fraction algorithm is the fastest known algorithm to approximate real numbers by the dense subset of rational numbers. In this report, we study the efficiency of the approximation of GGG by using a dense group generated by a universal subset of GGG. The measure of the efficiency K(g)K(g)K(g) of an arbitrary universal subset ggg is defined in [1] and satisfies K(g)⩾1K(g)\geqslant 1K(g)⩾1. ggg is more efficient when K(g)K(g)K(g) is closer to 111. We address some open problems posed by Sarnak in [1]. This report forms the basis of the shorter preprint [LD].
Constructing an efficient universal subset of quantum gates can be done by finding an efficient universal subset of G=PSU(2)G=PSU(2)G=PSU(2) (or SU(2)SU(2)SU(2)). Following [Letter], we define the efficiency KKK of a universal subset of GGG. We focus on two different universal subsets SSS and TTT, and estimate their efficiencies. We show that the efficiency of the first set is bounded by 43⩽K(S)⩽2\frac{4}{3}\leqslant K(S)\leqslant 234​⩽K(S)⩽2. Also, based on a conjecture about covering radius on 333-sphere, we show the efficiency of the second set is K(T)=43K(T)=\frac{4}{3}K(T)=34​, based on a question of Sarnak.
Elementary Quantum Gates from Lie Group Embeddings in U(2n)U(2^n)U(2n): Geometry, Universality, and Discretization
In the standard circuit model, elementary gates are defined relative to a chosen tensor factorization and are therefore extrinsic to the ambient group U(2n)U(2^n)U(2n). Writing N=2nN=2^nN=2n, we introduce an \emph{intrinsic descriptor layer} in U(N)U(N)U(N) by declaring as primitive the motions inside faithful embedded copies of SU(2)SU(2)SU(2) (phase-free), together with a phase-inclusive U(2)U(2)U(2) variant. We describe the embedding landscape $\Emb(SU(2),U(N))$ as a finite union of U(N)U(N)U(N)-homogeneous strata indexed by isotypic multiplicities, with stabilizers given by centralizers, and we isolate a canonical \emph{two-level sector} parameterized by $\Gr_2(\C^N)$ up to a PSU(2)PSU(2)PSU(2) gauge. Equipping U(N)U(N)U(N) with the Hilbert--Schmidt bi-invariant metric, each embedded subgroup is totally geodesic, yielding a variational characterization of elementary motions via minimal-norm logarithms. On the constructive side, we prove phase-free universality in SU(N)SU(N)SU(N) from two-level primitives using QR/Givens factorizations together with explicit diagonal generation, and we obtain full universality in U(N)U(N)U(N) by explicit abelian phase bookkeeping (equivalently, via the U(2)U(2)U(2) two-level dictionary). Finally, we formalize a modular finite-alphabet compilation interface: any approximation routine in SU(2)SU(2)SU(2) (e.g.\ Solovay--Kitaev) can be lifted through two-level embeddings to yield U(N)U(N)U(N)-level synthesis with global operator-norm error control.
FINDING: The Solovay-Kitaev theorem guarantees efficient approximation of any single-qubit unitary using finite gate sets; its optimal implementations ("golden gates") are deeply tied to algebraic number theory and, for Fibonacci anyons, to the golden ratio. | MATH: The theorem states: For any finite set S⊂SU(2)S \subset SU(2)S⊂SU(2) generating a dense subgroup, any U∈SU(2)U \in SU(2)U∈SU(2) can be ε\varepsilonε-approximated in O(log⁡c(1/ε))O(\log^c(1/\varepsilon))O(logc(1/ε)) gates, c≈3.97c \approx 3.97c≈3.97 (improved to c=1c=1c=1 for certain sets). Exact synthesis for Fibonacci anyons uses the braid group B3B_3B3​ representation where the fusion space dimension grows as FnF_nFn​ (Fibonacci numbers). The key constant: the golden ratio φ=(1+5)/2=1.618...\varphi = (1+\sqrt{5})/2 = 1.618...φ=(1+5​)/2=1.618..., with its inverse φ−1=0.618...\varphi^{-1} = 0.618...φ−1=0.618..., and the related φ−2=0.382...\varphi^{-2} = 0.382...φ−2=0.382.... The Fibonacci anyon braid matrices have entries in Z[φ]\mathbb{Z}[\varphi]Z[φ], the ring of integers of Q(5)\mathbb{Q}(\sqrt{5})Q(5​). | CONNECTION: Direct and explicit. The Fibonac Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
We consider the problem of deciding if a set of quantum one-qudit gates S={g1,…,gn}⊂G\mathcal {S}=\{g_1,\ldots ,g_n\}\subset GS={g1​,…,gn​}⊂G is universal, i.e. if <S>{<}\mathcal {S}{>}<S> is dense in G, where G is either the special unitary or the special orthogonal group. To every gate g in S\mathcal {S}S we assign the orthogonal matrix Adg\mathrm {Ad}_gAdg​ that is image of g under the adjoint representation Ad:G→SO(g)\mathrm {Ad}:G\rightarrow SO(\mathfrak {g})Ad:G→SO(g) and g\mathfrak {g}g is the Lie algebra of G. The necessary condition for the universality of S\mathcal {S}S is that the only matrices that commute with all Adgi\mathrm {Ad}_{g_i}Adgi​​ ’s are proportional to the identity. If in addition there is an element in <S>{<}\mathcal {S}{>}<S> whose Hilbert–Schmidt distance from the centre of G belongs to ]0,12[]0,\frac{1}{\sqrt{2}}[]0,2​1​[ , then S\mathcal {S}S is universal. Using these we provide a simple algorithm that allows deciding the universality of any set of d-dimensional gates in a finite number of steps and formulate a general classification theorem.