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Quantum state preparation with optimal T-count

David Gosset, Robin Kothari, Kewen Wu

TL;DR

This work establishes the optimal asymptotic T-count for quantum state preparation and diagonal unitary synthesis under Clifford+$T$ gates with ancillas. The authors derive a tight upper bound of $O\left(\sqrt{2^n\log(1/\varepsilon)}+\log(1/\varepsilon)\right)$ T gates and ancillas for arbitrary $n$-qubit states, and prove a matching adaptive Clifford+$T$ lower bound, thereby closing a long-standing gap from previous results. They generalize to diagonal unitaries, and provide practical corollaries such as batched synthesis of multiple single-qubit unitaries and mass production of $U^{\otimes m}$ with near-linear T-cost in $m$. The paper introduces a coarse-to-fine state-approximation strategy combining diagonal-unitary synthesis, LCU, and amplitude amplification, and refines diagonal synthesis via a boolean-oracle-based construction. Overall, the results offer near-optimal resource estimates and enable parallelized and scalable quantum state and diagonal-unitary synthesis, with implications for magic-state costs and decompositions of quantum operations.

Abstract

How many T gates are needed to approximate an arbitrary $n$-qubit quantum state to within error $\varepsilon$? Improving prior work of Low, Kliuchnikov, and Schaeffer, we show that the optimal asymptotic scaling is $Θ\left(\sqrt{2^n\log(1/\varepsilon)}+\log(1/\varepsilon)\right)$ if we allow ancilla qubits. We also show that this is the optimal T-count for implementing an arbitrary diagonal $n$-qubit unitary to within error $\varepsilon$. We describe applications in which a tensor product of many single-qubit unitaries can be synthesized in parallel for the price of one.

Quantum state preparation with optimal T-count

TL;DR

This work establishes the optimal asymptotic T-count for quantum state preparation and diagonal unitary synthesis under Clifford+ gates with ancillas. The authors derive a tight upper bound of T gates and ancillas for arbitrary -qubit states, and prove a matching adaptive Clifford+ lower bound, thereby closing a long-standing gap from previous results. They generalize to diagonal unitaries, and provide practical corollaries such as batched synthesis of multiple single-qubit unitaries and mass production of with near-linear T-cost in . The paper introduces a coarse-to-fine state-approximation strategy combining diagonal-unitary synthesis, LCU, and amplitude amplification, and refines diagonal synthesis via a boolean-oracle-based construction. Overall, the results offer near-optimal resource estimates and enable parallelized and scalable quantum state and diagonal-unitary synthesis, with implications for magic-state costs and decompositions of quantum operations.

Abstract

How many T gates are needed to approximate an arbitrary -qubit quantum state to within error ? Improving prior work of Low, Kliuchnikov, and Schaeffer, we show that the optimal asymptotic scaling is if we allow ancilla qubits. We also show that this is the optimal T-count for implementing an arbitrary diagonal -qubit unitary to within error . We describe applications in which a tensor product of many single-qubit unitaries can be synthesized in parallel for the price of one.

Paper Structure

This paper contains 17 sections, 31 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Applications
  3. Diagonal unitary synthesis with optimal T-count
  4. Batched synthesis
  5. Mass production
  6. Quantum state preparation with optimal T-count
  7. Approximating a state to constant error: Lemma \ref{['lem:apx_state_syn_hd4']}
  8. Approximating a state to eps error: Lemma \ref{['lem:apx_coeff_simple']}
  9. Case $T\le j^*$.
  10. Case $T>j^*$.
  11. Flagging and exact amplitude amplification: Lemma \ref{['lem:state_circuit']}
  12. Lower bounds
  13. State preparation
  14. Unitary synthesis
  15. Canonical form of Clifford circuits with magic states and Pauli postselection
  16. ...and 2 more sections

Key Result

Theorem 1.1

Any $n$-qubit state can be prepared up to error $\varepsilon$ by a Clifford+${T}$ circuit starting with the all-zeros state using ${T}$ gates and ancillas.This construction uses $O(2^n\log(1/\varepsilon))$ Clifford gates. Furthermore, no Clifford+${T}$ circuit (even with measurements and adaptivity) can use asymptotically fewer ${T}$ gates.

Theorems & Definitions (66)

  • Theorem 1.1: Quantum state preparation with optimal $\Tgate$-count
  • Theorem 1.2: Diagonal unitary synthesis with optimal $\Tgate$-count
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1: Diagonal unitary synthesis with optimal $\Tgate$-count
  • Lemma 2.1: low2018trading
  • Remark 2.2
  • Lemma 2.3: selinger2015efficientross2016optimal
  • proof : Proof of \ref{['thm:diagonal_unitary_T-count']}
  • Corollary 2.5
  • ...and 56 more