Optimal ancilla-free Clifford+T approximation of z-rotations

TL;DR

The paper tackles optimal ancilla-free synthesis of arbitrary $z$-rotations using Clifford+$T$ circuits. It introduces a fast probabilistic algorithm that is optimal given a factoring oracle, and near-optimal without factoring by solving a pair of coupled problems: a 2D grid problem and a Diophantine equation, via enumerating candidates $u\in\mathbb{D}[\omega]$ and solving $t^{\dagger}t=1-u^{\dagger}u$. The core technical advance is an efficient, iterative grid-transformation approach (grid operators) that normalizes convex regions to upright shapes, enabling guaranteed enumeration with polylogarithmic time in $1/\varepsilon$. The results yield a typical $T$-count of $3\log_2(1/\varepsilon)+O(\log\log(1/\varepsilon))$ and demonstrate practical performance against previous methods, with optimality proven in the presence of a factoring oracle and near-optimality otherwise. This framework significantly improves exact and approximate synthesis for the Clifford+$T$ gate set and offers a blueprint for phase-agnostic variants and related gate sets.

Abstract

We consider the problem of approximating arbitrary single-qubit z-rotations by ancilla-free Clifford+T circuits, up to given epsilon. We present a fast new probabilistic algorithm for solving this problem optimally, i.e., for finding the shortest possible circuit whatsoever for the given problem instance. The algorithm requires a factoring oracle (such as a quantum computer). Even in the absence of a factoring oracle, the algorithm is still near-optimal under a mild number-theoretic hypothesis. In this case, the algorithm finds a solution of T-count m + O(log(log(1/epsilon))), where m is the T-count of the second-to-optimal solution. In the typical case, this yields circuit approximations of T-count 3log_2(1/epsilon) + O(log(log(1/epsilon))). Our algorithm is efficient in practice, and provably efficient under the above-mentioned number-theoretic hypothesis, in the sense that its expected runtime is O(polylog(1/epsilon)).

Figures (6)

• Figure 1: The real grid for two different intervals $B$. In both cases, the interval $B$ is shown in green, and grid points are shown as black dots.
• Figure 2: The complex grid for three different convex sets $B$. In each case, the set $B$ is shown in green, and grid points are shown as black dots. (a) $B=[-1,1]^2$. (b) $B=\{(x,y)\mid x^2+y^2\leqslant 2\}$. (c) $B=\{(x,y)\mid 6x^2 + 16xy + 11y^2 \leqslant 2\}$.
• Figure 3: Grid problems for upright and non-upright sets
• Figure 4: (a) The grid problem for two sets $A$ and $B$. (b) The grid problem with $G(A)$ and $G^{\bullet}(B)$. Note that the solutions of (a), which are the grid points in the set $A$, are in one-to-one correspondence with the solutions of (b), which are the grid points in the set $G(A)$.
• Figure 5: Part (a) shows the grid points and grid lines (of slope $r\in\mathbbQ(\sqrt 2)$) for a convex set $B$. Part (b) shows the set $B$ and the dual grid lines (of slope $r^\bullet$), all of whose $y$-intercepts lie in an interval $B'$.

Theorems & Definitions (186)

• Definition 3.1
• Definition 3.2
• Remark 3.3
• Definition 3.4
• Definition 3.5
• Remark 3.6
• Definition 4.1
• Example 4.2
• Definition 4.3
• Lemma 4.4