Trading T gates for dirty qubits in state preparation and unitary synthesis

TL;DR

This paper introduces a quantum algorithm that optimally balances the use of clean and dirty qubits to significantly reduce T-gate costs in state preparation and unitary synthesis, crucial for efficient quantum computation.

Contribution

It presents a tunable trade-off scheme using both clean and dirty qubits to minimize T-gates in state and unitary synthesis, with proven optimality and quadratic improvements over previous methods.

Findings

Reduces T-gate count by up to a quadratic factor compared to prior approaches.

Provides an optimal trade-off between space (qubits) and T-gate complexity.

Establishes a T-efficient quantum oracle for classical data.

Abstract

Efficient synthesis of arbitrary quantum states and unitaries from a universal fault-tolerant gate-set e.g. Clifford+T is a key subroutine in quantum computation. As large quantum algorithms feature many qubits that encode coherent quantum information but remain idle for parts of the computation, these should be used if it minimizes overall gate counts, especially that of the expensive T-gates. We present a quantum algorithm for preparing any dimension-$N$ pure quantum state specified by a list of $N$ classical numbers, that realizes a trade-off between space and T-gates. Our scheme uses $\mathcal{O}(\log{(N/\epsilon)})$ clean qubits and a tunable number of $\sim(\lambda\log{(\frac{\log{N}}{\epsilon})})$ dirty qubits, to reduce the T-gate cost to $\mathcal{O}(\frac{N}{\lambda}+\lambda\log{\frac{N}{\epsilon}}\log{\frac{\log{N}}{\epsilon}})$. This trade-off is optimal up to logarithmic factors, proven through an unconditional gate counting lower bound, and is, in the best case, a quadratic improvement in T-count over prior ancillary-free approaches. We prove similar statements for unitary synthesis by reduction to state preparation. Underlying our constructions is a T-efficient circuit implementation of a quantum oracle for arbitrary classical data.