Catalytic $z$-rotations in constant $T$-depth

Abstract: We show that the $T$-depth of any single-qubit $z$-rotation can be reduced to $3$ if a certain catalyst state is available. To achieve an $\epsilon$-approximation, it suffices to have a catalyst state of size polynomial in $\log(1/\epsilon)$. This implies that $\mathsf{QNC}^0_f/\mathsf{qpoly}$ admits a finite universal gate set consisting of Clifford+$T$. In particular, there are catalytic constant $T$-depth circuits that approximate multi-qubit Toffoli, adder, and quantum Fourier transform arbitrarily well. We also show that the catalyst state can be prepared in time polynomial in $\log (1/\epsilon)$.

• The paper achieves a T-depth of 3 for any single-qubit z-rotation by using a catalyst state with a size that scales polynomially in log(1/ε).
• It introduces an efficient method based on Clifford and T gates along with primitive polynomial techniques over GF(2^n) to reduce circuit depth.
• The approach enables constant T-depth execution for critical quantum operations, such as Toffoli gates and Fourier transforms, fostering robust fault-tolerant circuits.

Catalytic $z$-rotations in constant $T$-depth

The paper "Catalytic $z$-rotations in constant $T$-depth" by Isaac H. Kim offers advancements in the efficiency of quantum gate implementations, focusing on achieving $z$-rotations with reduced $T$-depth using catalyst states. The central thesis of the paper is the demonstration that with a suitable catalyst state, any single-qubit $z$-rotation can attain a $T$-depth of 3, with implications extending to wider quantum computations such as Toffoli gates, adders, and the quantum Fourier transform in fault-tolerant quantum circuits.

The considered model of quantum computation discussed in this paper is based on using Clifford and $T$ gates, where minimizing the $T$-depth—rather than the $T$-count—becomes critical under the surface code approach for fault-tolerant quantum computation. The trade-off between spatial and temporal resources reveals that computation time is ultimately constrained by the $T$-depth, urging a need for optimization within a fixed circuit depth.

Main Contributions and Results

1. Catalytic $T$-depth Reduction: The paper presents a method where the $z$-rotation for any single-qubit can be achieved at a $T$-depth of 3, contingent upon a pre-prepared catalyst state. The size of this catalyst state is polynomial in $\log(1/\epsilon)$ for an $\epsilon$-approximation, which indicates its reasonable scalability and efficiency.
2. Universal Gate Set: The work establishes that the planned strategy feeds into a finite universal gate set for quantum computations, namely, the Clifford+$T$ set. This result enriches the understanding of quantum circuits and their functionality within $\mathsf{QNC}^0_f/\mathsf{qpoly}$, certifying their viability for a broad spectrum of quantum operations with uniform resource metrics.
3. Primitive Polynomial Method: A detailed algorithmic discussion highlights the usage of special polynomials over $\mathrm{GF}(2^n)$ and the construction of a rotation operation through Clifford gates. Notably, this includes a refined approach to using primitive polynomials to enable constant-depth implementations.
4. Efficient Catalyst State Preparation: The methodology for preparing the necessary catalyst states is presented in a way that utilizes polynomial-time quantum processes. This includes performing efficient operations based on discrete logarithms and utilizing Frobenius endomorphism, which allows for scalable and practical preparation routines complementing the rotation process.

Implications

The proposed method introduces robust improvements to how single-qubit rotations are approached in quantum computing. With wider implications, it is suggested that important subroutines—such as universal quantum computation using Toffoli, adders, and Fourier transforms—can be executed in constant $T$-depth using catalysts, thus circumventing traditional bounds found in classical counterparts and showcasing quantum supremacy in depth reduction under certain scenarios.

Discussion and Future Work

The paper sets the stage for ongoing discourse in quantum computing regarding the optimal use of catalyst states in reducing circuit overhead. It presents open questions, such as whether the methods can be adapted using only clean or dirty ancillas, and whether the approach can be extended to more processes and applications. Additionally, future work is suggested to improve exact implementations for wider classes of quantum transformations and strengthen the theoretical underpinnings surrounding the interactions of clean and catalytic states in quantum operations.

Overall, this paper contributes significantly to quantum computational theory by further revealing the potential of catalyst states in reducing circuit complexity and enhancing performance, offering pathways for optimized quantum algorithms in practical applications.