[Paper Review] Super controlled gates and controlled gates in two-qubit gate simulations

Ming-Yong Ye, Yongsheng Zhang+1 more|arXiv (Cornell University)|Jul 15, 2004

Quantum Computing Algorithms and Architecture3 citations

TL;DR

This paper establishes a mirror gate theorem showing that the simulation power of a two-qubit entangling gate extends to its mirror gate, enabling universal two-qubit gate simulation. It proves that three applications of a super controlled gate, or four of certain controlled gates satisfying specific angle constraints, can simulate any two-qubit unitary using single-qubit rotations, providing a constructive framework for quantum circuit design with minimal entangling gate usage.

ABSTRACT

In two-qubit gate simulations an entangling gate is used several times together with single qubit gates to simulate another two-qubit gate. We show how a two-qubit gate's simulation power is related to the simulation power of its mirror gate. And we show that an arbitrary two-qubit gate can be simulated by three applications of a super controlled gate together with single qubit gates. We also give the gates set that can be simulated by n applications of a controlled gate in a constructive way. In addition we give some gates which can be used four times to simulate an arbitrary two-qubit gate.

Motivation & Objective

• To determine the simulation power of two-qubit entangling gates, particularly super controlled and controlled gates, in quantum circuit design.
• To address the challenge of minimizing entangling gate usage in universal two-qubit gate decomposition, crucial for experimental quantum computing.
• To develop a constructive method for identifying which two-qubit gates can be simulated using n applications of a given controlled gate.
• To establish a mirror gate theorem linking the simulation capability of a gate to its mirror counterpart, enabling broader applicability of simulation results.
• To identify specific gate families that can simulate any two-qubit unitary in four uses, offering practical alternatives to three-use super controlled gate schemes.

Proposed method

• Uses the canonical decomposition of two-qubit gates into the form $ U_{AB} = (U^A \otimes U^B) U_d(V^A \otimes V^B) $, where $ U_d(\alpha_1,\alpha_2,\alpha_3) = \exp(i\sum_{j=1}^3 \alpha_j \sigma_j^A \otimes \sigma_j^B) $, to analyze simulation power.
• Introduces the concept of a mirror gate: $ U_d(\alpha_1 + \pi/4, \alpha_2 + \pi/4, \alpha_3 + \pi/4) $ is locally equivalent to the mirror of $ U_d(\alpha_1,\alpha_2,\alpha_3) $, enabling symmetry-based simulation extension.
• Applies the SWAP gate as a transformation tool to derive the mirror gate theorem, showing that replacing a gate with its mirror allows simulation of the mirror’s target gate.
• Employs the ZYZ decomposition of single-qubit gates and controlled-Z type interactions to construct sequences that simulate arbitrary two-qubit gates.
• Derives conditions on gate parameters using trigonometric identities from the decomposition of controlled-Z and controlled-Y type interactions, leading to constraints on $ \cos(2a_1 \pm 2a_3) $ for four-use simulations.
• Constructs explicit simulation circuits using sequences of controlled gates and single-qubit rotations, with parameters chosen to satisfy eigenvalue matching and entangling power requirements.

Experimental results

Research questions

• RQ1 How is the simulation power of a two-qubit gate related to that of its mirror gate?
• RQ2 Can any two-qubit gate be simulated using only three applications of a super controlled gate and single-qubit operations?
• RQ3 What are the necessary and sufficient conditions on a controlled gate such that four applications can simulate any two-qubit unitary?
• RQ4 Which sets of two-qubit gates can be simulated using exactly n applications of a given controlled gate, and how can this be constructed explicitly?
• RQ5 What role does the magic basis play in characterizing the entangling power and simulation capability of two-qubit gates?

Key findings

• The mirror gate theorem establishes that if a gate $ U_d(\gamma_1,\gamma_2,\gamma_3) $ simulates $ U_d(\beta_1,\beta_2,\beta_3) $, then its mirror gate can simulate the mirror of $ U_d(\beta_1,\beta_2,\beta_3) $, enabling symmetry-based simulation extension.
• An arbitrary two-qubit gate can be simulated using exactly three applications of a super controlled gate $ U_d(\pi/4, \alpha_2, 0) $, combined with single-qubit gates, proving its universality for two-qubit simulation.
• A gate $ U_d(a_1/2, 0, a_3/2) $ can simulate any two-qubit gate in four uses if $ \cos(2a_1 + 2a_3) \leq -1/\sqrt{2} $ and $ \cos(2a_1 - 2a_3) \leq 1/\sqrt{2} $, providing a four-use universal gate family.
• The set of gates that can be simulated using $ n $ applications of a controlled gate $ U_d(\alpha, 0, 0) $ is constructively characterized via parameter constraints derived from trigonometric identities in the gate decomposition.
• The DCNOT gate is the mirror of the CNOT gate, and the SWAP gate is locally equivalent to $ U_d(\pi/4, \pi/4, \pi/4) $, confirming the geometric and algebraic consistency of the mirror concept.
• For controlled gates with $ a_1 = a_2 = 0 $, the simulation power is determined by the parameter $ b_3 $, and the gate $ e^{i a_3 \sigma_3^A \otimes \sigma_3^B} $ can simulate $ U_d(x/2, y/2, 0) $ with $ x, y $ constrained by trigonometric relations involving $ \cos(a_3)\cos(b_3) $ and $ \sin(a_3)\sin(b_3) $.