New Quantum Decomposition Method Optimizes Circuit Construction, Reduces CNOT Gates

A new optimized quantum block ZXZ decomposition method has been proposed that constructs more optimal quantum circuits than the quantum Shannon decomposition (QSD), the most optimal decomposition method since 2006. The new method allows a general 3-qubit gate to be decomposed using 19 CNOT gates instead of 20.

For general n-qubit gates, the proposed decomposition generates circuits with fewer CNOT gates than the best known exact decomposition algorithm. This efficiency could potentially lead to more efficient quantum computing operations, reducing the time and error rates of two-qubit gates, which are currently worse than for single qubit gates.

What is the New Optimized Quantum BlockZXZ Decomposition Method?

The paper discusses a new optimized quantum blockZXZ decomposition method that results in the construction of more optimal quantum circuits than the quantum Shannon decomposition (QSD) can achieve. The QSD has been the most optimal decomposition method since 2006. The proposed decomposition method allows a general 3-qubit gate to be decomposed using 19 CNOT gates rather than 20. For general n-qubit gates, the proposed decomposition generates circuits that have fewer CNOT gates than the best known exact decomposition algorithm.

The process of implementing arbitrary quantum operations on real quantum hardware requires a matrix to be translated into elementary gate operations. This is a complex task and has been the focus of much research over the years into methods for performing such translation using quantum gate decomposition. An important target of gate decomposition methods is to minimize the number of CNOT gates required to implement a given unitary matrix. This is essential because the time and error rates of two-qubit gates are an order of magnitude worse than for single qubit gates in current quantum hardware.

It has been proven that any exact decomposition of an arbitrary n-qubit gate requires at least 1.44n^3-n+1 CNOT gates. This limit has been achieved for one-qubit gates, which do not require any CNOTs and can be decomposed into a sequence of three rotation gates. The minimum number of CNOTs to implement arbitrary two-qubit gates is three, but less CNOTs are necessary when the gate meets certain conditions. For arbitrary three qubit gates, there is no algorithm that results in the minimum 14 CNOTs, but algorithms exist to decompose them into 64, 22, 40, 24, 26, 8 or 20 CNOTs.

How Does the New Decomposition Method Compare to Previous Approaches?

For the decomposition of quantum gates of arbitrary size, there exist several approaches. In 1995, Barenco et al. showed that any unitary operator on n-qubits can be constructed using at most O(n^3/4n) two-qubit gates by using the standard QR decomposition based on Givens rotations. This CNOT count has been improved over the years by use of Gray codes and gate cancellations to O(1.24n) CNOT gates. Another approach to unitary decomposition has been to use Cosine-Sine Decomposition (CSD). This was combined with Singular Value Decomposition (SVD) in 2006 to construct Quantum Shannon Decomposition (QSD). With QSD, an n-qubit unitary gate can be decomposed into at most 2.3 * 4.84n^3-2.2n^4+3 CNOTs.

The new unitary decomposition method combines blockZXZ decomposition with the optimizations for quantum Shannon decomposition. With this decomposition method, an arbitrary n-qubit operator can be decomposed into at most 2.2 * 4.84n^3-2.2n^5+3 CNOT gates. This is 4n^2+13 less than the best previously published work. More specifically, a general three-qubit operator can be constructed with at most 19 qubits, which is currently the least known for any exact decomposition method.

What is the Significance of the New Decomposition Method?

The significance of the new decomposition method lies in its efficiency. The proposed method results in the construction of more optimal quantum circuits than the quantum Shannon decomposition (QSD) can achieve. The QSD has been the most optimal decomposition method since 2006. The proposed decomposition method allows a general 3-qubit gate to be decomposed using 19 CNOT gates rather than 20. For general n-qubit gates, the proposed decomposition generates circuits that have fewer CNOT gates than the best known exact decomposition algorithm.

The process of implementing arbitrary quantum operations on real quantum hardware requires a matrix to be translated into elementary gate operations. This is a complex task and has been the focus of much research over the years into methods for performing such translation using quantum gate decomposition. An important target of gate decomposition methods is to minimize the number of CNOT gates required to implement a given unitary matrix. This is essential because the time and error rates of two-qubit gates are an order of magnitude worse than for single qubit gates in current quantum hardware.

How Does the New Decomposition Method Work?

The new unitary decomposition method combines blockZXZ decomposition with the optimizations for quantum Shannon decomposition. With this decomposition method, an arbitrary n-qubit operator can be decomposed into at most 2.2 * 4.84n^3-2.2n^5+3 CNOT gates. This is 4n^2+13 less than the best previously published work. More specifically, a general three-qubit operator can be constructed with at most 19 qubits, which is currently the least known for any exact decomposition method.

The paper is organized as follows: it starts with the notation and gate definitions, then shows the decomposition of uniformly controlled rotations. It continues with the full decomposition, and the optimizations and the resulting gate count are shown. The paper ends with the conclusion.

What are the Implications of the New Decomposition Method?

The implications of the new decomposition method are significant for the field of quantum computing. The proposed method results in the construction of more optimal quantum circuits than the quantum Shannon decomposition (QSD) can achieve. The QSD has been the most optimal decomposition method since 2006. The proposed decomposition method allows a general 3-qubit gate to be decomposed using 19 CNOT gates rather than 20. For general n-qubit gates, the proposed decomposition generates circuits that have fewer CNOT gates than the best known exact decomposition algorithm.

This efficiency in the decomposition of quantum gates could potentially lead to more efficient quantum computing operations, reducing the time and error rates of two-qubit gates, which are an order of magnitude worse than for single qubit gates in current quantum hardware. This could potentially lead to advancements in the field of quantum computing, making it more practical and efficient.