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Unitary synthesis with fewer T gates

Xinyu Tan

TL;DR

This paper tackles the problem of compiling an arbitrary $n$-qubit unitary into a Clifford+T circuit with minimal $T$-gates. It introduces a recursive cosine-sine decomposition to express any unitary as a product of $2^n-1$ multi-controlled $k$-qubit unitaries and then groups blocks that share target qubits to reduce cost. By extending the diagonal-unitary synthesis method of Gosset, Kothari, and Wu to multi-controlled unitaries and carefully balancing ancilla and gate counts, the authors achieve a breakthrough upper bound of $O\left(2^{4n/3} \, L^{2/3} + 2^n L\right)$ $T$-gates, where $L = n + \log(1/\varepsilon)$. They also show a complementary bound with a tradeoff parameter $k$ giving $O\left(2^{(3n-k)/2} \sqrt{L}\right)$ $T$-gates and $O\left(2^{(n+k)/2} \sqrt{L}\right)$ ancillae for suitable choices of $k$, highlighting that large ancilla resources could approach the conjectured $2^n$ scaling. Together, these results provide evidence that the optimal $T$-gate count for arbitrary unitary synthesis could be as low as $2^n$ under favorable ancilla budgets and offer a concrete path toward more practical fault-tolerant quantum compilation. The work has implications for quantum simulation and state-preparation primitives, where reducing $T$-gates is particularly impactful on hardware platforms with costly magic-state distillation.

Abstract

We present a simple algorithm that implements an arbitrary $n$-qubit unitary operator using a Clifford+T circuit with T-count $O(2^{4n/3} n^{2/3})$. This improves upon the previous best known upper bound of $O(2^{3n/2} n)$, while the best known lower bound remains $Ω(2^n)$. Our construction is based on a recursive application of the cosine-sine decomposition, together with a generalization of the optimal diagonal unitary synthesis method by Gosset, Kothari, and Wu to multi-controlled $k$-qubit unitaries.

Unitary synthesis with fewer T gates

TL;DR

This paper tackles the problem of compiling an arbitrary -qubit unitary into a Clifford+T circuit with minimal -gates. It introduces a recursive cosine-sine decomposition to express any unitary as a product of multi-controlled -qubit unitaries and then groups blocks that share target qubits to reduce cost. By extending the diagonal-unitary synthesis method of Gosset, Kothari, and Wu to multi-controlled unitaries and carefully balancing ancilla and gate counts, the authors achieve a breakthrough upper bound of -gates, where . They also show a complementary bound with a tradeoff parameter giving -gates and ancillae for suitable choices of , highlighting that large ancilla resources could approach the conjectured scaling. Together, these results provide evidence that the optimal -gate count for arbitrary unitary synthesis could be as low as under favorable ancilla budgets and offer a concrete path toward more practical fault-tolerant quantum compilation. The work has implications for quantum simulation and state-preparation primitives, where reducing -gates is particularly impactful on hardware platforms with costly magic-state distillation.

Abstract

We present a simple algorithm that implements an arbitrary -qubit unitary operator using a Clifford+T circuit with T-count . This improves upon the previous best known upper bound of , while the best known lower bound remains . Our construction is based on a recursive application of the cosine-sine decomposition, together with a generalization of the optimal diagonal unitary synthesis method by Gosset, Kothari, and Wu to multi-controlled -qubit unitaries.

Paper Structure

This paper contains 14 sections, 14 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Proof overview
  3. The tradeoff between T gates and ancillae
  4. Applications
  5. Lower bound
  6. Notations
  7. Preliminaries
  8. Cosine-sine decomposition
  9. A naïve implementation using O(2³ⁿᐟ² · n¹ᐟ²) T gates
  10. Lower the T-count
  11. Notation
  12. Implementing multi-controlled k-qubit unitaries
  13. Proof of Theorem 1.1
  14. Proof of Lemma 2.1

Key Result

Theorem 1.1

Let $\varepsilon > 0$ and set $L = n+\log(1/\varepsilon)$. Then any $U\in \mathrm{U}(2^n)$ can be $\varepsilon$-approximated by a Clifford+T circuit using In particular, for any positive integer $k \leq (n - \log_2 L)/3$, $U$ can be $\varepsilon$-approximated by a Clifford+T circuit using

Theorems & Definitions (25)

  • Theorem 1.1: Main result
  • Definition 1.2: Clifford+T approximation
  • Theorem 1.3: GKW24
  • Lemma 2.0: Composition error bound
  • Lemma 2.1: Generating all monomials
  • proof
  • Lemma 2.2: T-count for Boolean function oracles, LKS24
  • Lemma 2.3: Single-qubit Clifford+T approximation, RS16
  • Definition 3.1: Multi-controlled unitaries
  • Theorem 3.2: Special case of the CS decomposition
  • ...and 15 more