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Any Clifford+T circuit can be controlled with constant T-depth overhead

Isaac H. Kim, Tuomas Laakkonen

TL;DR

This work shows that controlled versions of Clifford and Clifford+T circuits can be synthesized with dramatically reduced non-Clifford resources: controlled CNOTs need at most $n - c$ Toffoli gates with Toffoli-depth $O(1)$ without ancillas, and depth $1$ with $2n-1$ ancillas via measurement-based uncomputation. These techniques extend to broader Clifford circuits, yielding constant $T$-depth, and to Clifford+T circuits with $T$-depth $O(D)$ and $T$-count $O(C+n)$, all without ancillas; with ancillas, depth constants can improve further. The paper also presents a catalytic method to implement arbitrary-angle $Z$-rotations with a universal catalyst of size $O( icefrac{1}{ ext{epsilon}})$ qubits, achieving $T$-depth $1$, and discusses practical implications for fault-tolerant synthesis and parallelization along with open questions on catalyst size and exact-angle catalysis. Additional results include constant-depth Hadamard/swap-test variants and a rigorous approximate-optimality proof for the central Toffoli-count bound via unitary stabilizer nullity. Overall, these findings offer a path to near-optimal, depth-efficient controlled Clifford and Clifford+T circuits with clear guidance for catalyst-based rotation synthesis.

Abstract

Since an n-qubit circuit consisting of CNOT gates can have up to $Ω(n^2/\log{n})$ CNOT gates, it is natural to expect that $Ω(n^2/\log{n})$ Toffoli gates are needed to apply a controlled version of such a circuit. We show that the Toffoli count can be reduced to at most n. The Toffoli depth can also be reduced to O(1), at the cost of 2n Toffoli gates, even without using any ancilla or measurement. In fact, using a measurement-based uncomputation, the Toffoli depth can be further reduced to 1. From this, we give two corollaries: any controlled Clifford circuit can be implemented with O(1) T-depth, and any Clifford+T circuit with T-depth D can be controlled with T-depth O(D), even without ancillas. As an application, we show how to catalyze a rotation by any angle up to precision $ε$ in T-depth exactly 1 using a universal $\lceil\log_2(8/ε)\rceil$-qubit catalyst state.

Any Clifford+T circuit can be controlled with constant T-depth overhead

TL;DR

This work shows that controlled versions of Clifford and Clifford+T circuits can be synthesized with dramatically reduced non-Clifford resources: controlled CNOTs need at most Toffoli gates with Toffoli-depth without ancillas, and depth with ancillas via measurement-based uncomputation. These techniques extend to broader Clifford circuits, yielding constant -depth, and to Clifford+T circuits with -depth and -count , all without ancillas; with ancillas, depth constants can improve further. The paper also presents a catalytic method to implement arbitrary-angle -rotations with a universal catalyst of size qubits, achieving -depth , and discusses practical implications for fault-tolerant synthesis and parallelization along with open questions on catalyst size and exact-angle catalysis. Additional results include constant-depth Hadamard/swap-test variants and a rigorous approximate-optimality proof for the central Toffoli-count bound via unitary stabilizer nullity. Overall, these findings offer a path to near-optimal, depth-efficient controlled Clifford and Clifford+T circuits with clear guidance for catalyst-based rotation synthesis.

Abstract

Since an n-qubit circuit consisting of CNOT gates can have up to CNOT gates, it is natural to expect that Toffoli gates are needed to apply a controlled version of such a circuit. We show that the Toffoli count can be reduced to at most n. The Toffoli depth can also be reduced to O(1), at the cost of 2n Toffoli gates, even without using any ancilla or measurement. In fact, using a measurement-based uncomputation, the Toffoli depth can be further reduced to 1. From this, we give two corollaries: any controlled Clifford circuit can be implemented with O(1) T-depth, and any Clifford+T circuit with T-depth D can be controlled with T-depth O(D), even without ancillas. As an application, we show how to catalyze a rotation by any angle up to precision in T-depth exactly 1 using a universal -qubit catalyst state.
Paper Structure (16 sections, 18 theorems, 30 equations, 1 figure, 2 tables)

This paper contains 16 sections, 18 theorems, 30 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. CNOT Circuits as Binary Matrices
  4. Controlled Unitaries
  5. Controlled CNOT Circuits
  6. Approximately optimal Toffoli-count
  7. Reducing the Toffoli-depth to $O(1)$ without ancillas
  8. Reducing Toffoli-depth to one with $O(n)$ ancillas
  9. Generalizations
  10. Controlled Clifford Circuits
  11. Controlling unitaries with constant $T$-depth
  12. Applications
  13. Catalyzing arbitrary-angle rotations
  14. Variations on the swap test
  15. Discussion
  16. ...and 1 more sections

Key Result

Theorem 1

Given an $n \times n$ matrix $A$ over an arbitrary field $\mathbb{F}$, there exists $k$ polynomials $f_1, f_2, \dots, f_k \in \mathbb{F}[x]$, and an invertible matrix $S \in GL_n(\mathbb{F})$ such that where $C_{f_i}$ is the companion matrix corresponding to $f_i = a_{i0} + a_{i1}x + a_{i2}x^2 + \cdots + a_{i,d_i-1}x^{d_i-1} + x^{d_i}$. Moreover, we have $f_i(x) = q_i(x)^k$ where each $q_i(x)$ is

Figures (1)

  • Figure 1: A high-level overview of the circuit that implements $c(U)$ with Toffoli-depth $1$, where $U$ is a CNOT circuit. Here, $U_D$ implements $|a\rangle|\vec{x}\rangle|0\rangle \to |a\rangle|\vec{x}\rangle|D\vec{x}\rangle$ while $\vec{u}$ and $\vec{v}$ are binary vectors that depend on the measurement outcome. The only non-Clifford part of this circuit is the $n$ Toffolis, highlighted in grey, which are trivially parallelizable.

Theorems & Definitions (34)

  • Theorem 1: Generalized Jordan Normal Form
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • ...and 24 more