Circuit Optimization for Universality Transformation

Yasuaki Nakayama; Yuki Takeuchi; Seiseki Akibue

Circuit Optimization for Universality Transformation

Yasuaki Nakayama, Yuki Takeuchi, Seiseki Akibue

Abstract

It is known that a computationally universal gate set $\{H,CCZ\}$ can be transformed to a strictly universal one $\{Λ(S), H\}$ using one maximally imaginary state $|+i\rangle$ and non-imaginary ancillary qubits. We succeed this transformation with a shorter circuit that eliminates non-imaginary ancillary qubits. We further extend this to the continuous gate-set setting, showing that any multi-qubit unitary can be exactly generated by real single-qubit unitary gates, $CCZ$ gates and $|+i\rangle$.

Circuit Optimization for Universality Transformation

Abstract

It is known that a computationally universal gate set

can be transformed to a strictly universal one

using one maximally imaginary state

and non-imaginary ancillary qubits. We succeed this transformation with a shorter circuit that eliminates non-imaginary ancillary qubits. We further extend this to the continuous gate-set setting, showing that any multi-qubit unitary can be exactly generated by real single-qubit unitary gates,

gates and

Paper Structure (5 theorems, 6 equations, 1 figure)

This paper contains 5 theorems, 6 equations, 1 figure.

Key Result

Theorem 1

The operator $e^{i \theta /2} R_{z}(\theta)$ can be constructed by using a real orthogonal matrix and the maximally imaginary state $\ket{+i}$ without using any non-imaginary ancillary qubit. The concrete construction is as follows: where $\ket{\psi}$ is an arbitrary state, and the concrete real orthogonal matrix $U$ is

Figures (1)

Figure 1: The circuit discovered by the previous work Takeuchi24 and the new ancilla-free circuit discovered in this paper to transform computational universality to strict one. $\mathbf{Top} \, \mathbf{and} \, \mathbf{Middle}:$ In the previous work Takeuchi24, $\Lambda(S)$ is generated by $\{H,CCZ\}$ with an ancilla $\ket{0}$. $S$ is generated by $\{ H,CCZ\}$ and a resource state $\ket{+i}$ with an ancilla $\ket{1}$, which is generated by the preparation of $\ket{100}$ from $\ket{000}$ by using $\{H,CCZ\}$. (Here we omit the circuit for generating $\ket{100}$. See the previous work Takeuchi24.) $\mathbf{Bottom}:$ In this note, we show the generation of $\Lambda(S)$ by two $CCZ$ gates without any ancillary qubit except for $\ket{+i}$. We can reduce the number of $CCZ$ gates by at least 75% compared to the previous work.

Theorems & Definitions (8)

Theorem 1
proof
Corollary 1
Proposition 1
Lemma 1
proof
Theorem 2
proof