[Paper Review] Quantum-gate decomposer

Ken Nakanishi, Takahiko Satoh+1 more|arXiv (Cornell University)|Sep 27, 2021

Quantum Computing Algorithms and Architecture3 references4 citations

TL;DR

This paper presents an optimized parameterized quantum circuit method for decomposing multi-qubit gates like CCZ and CCCZ into efficient sequences of CZ and single-qubit gates, achieving a CZ-depth of only four for CCZ on square-shaped connectivity and 17 CZ gates for CCCZ on T-shaped connectivity. The approach uses sequential rotation optimization to minimize circuit depth and two-qubit gate count, significantly improving efficiency and fidelity on NISQ-era quantum processors with sparse connectivity.

ABSTRACT

Efficient decompositions of multi-qubit gates are essential in NISQ applications, where the number of gates or the circuit depth is limited. This paper presents efficient decompositions of CCZ and CCCZ gates, typical multi-qubit gates, under several qubit connectivities. We can construct the CCZ gate with only four CZ-depth when the qubit is square-shaped, including one auxiliary qubit. In T-shaped qubit connectivity, which has no closed loop, we can decompose the CCCZ gate with 17 CZ gates. While previous studies have shown a CCCZ gate decomposition with 14 CZ gates for the fully connected case, we found only four connections are sufficient for 14 CZ gates' implementation. The search for constraint-sufficient decompositions is aided by an optimization method we devised to bring the parameterized quantum circuit closer to the target quantum gate. We can apply this scheme to decompose any quantum gates, not only CCZ and CCCZ. Such decompositions of multi-qubit gates, together with the newly found CCZ and CCCZ decompositions, shorten the execution time of quantum circuits and improve the accuracy of complex quantum algorithms on near future QPUs.

Motivation & Objective

• To develop a scalable optimization method for decomposing arbitrary multi-qubit quantum gates into sequences of primitive gates suitable for NISQ-era quantum processors.
• To minimize circuit depth and two-qubit gate count—critical for reducing execution time and noise—under realistic qubit connectivity constraints such as square-shaped and T-shaped layouts.
• To enable efficient gate decomposition for quantum algorithms on superconducting QPUs with limited two-qubit gate connectivity, where gate fidelity and execution time are major constraints.
• To provide a general-purpose framework that can replace or enhance existing parameter optimization in quantum gate decomposition, applicable to any gate and gate set.

Proposed method

• A sequential optimization algorithm is used to adjust rotation angles in a parameterized quantum circuit to maximize the overlap with the target unitary, measured via the trace fidelity |Tr[V_T^† V P]|^2.
• The method iteratively updates one rotation gate at a time, using a closed-form solution for the optimal angle based on trace components, ensuring monotonic improvement of the objective function.
• The algorithm exploits the cyclic property of the trace to efficiently update the intermediate matrix M_n, avoiding recomputation from scratch at each step and reducing computational cost.
• The optimization is applied to various parameterized circuit templates to search for minimal-depth and low-count decompositions of CCZ and CCCZ gates under different qubit connectivities.
• The approach is general and not restricted to CZ gates; it can be adapted to any two-qubit primitive gate set by modifying the circuit template and target unitary.
• The method enables systematic exploration of gate decompositions under connectivity constraints, identifying optimal or near-optimal solutions for specific hardware architectures.

Experimental results

Research questions

• RQ1 Can we find a parameterized quantum circuit optimization method that efficiently decomposes multi-qubit gates like CCZ and CCCZ under realistic qubit connectivity constraints?
• RQ2 What is the minimal CZ-depth and two-qubit gate count required to decompose CCZ and CCCZ gates on square-shaped and T-shaped qubit connectivities?
• RQ3 How does the proposed optimization method compare in efficiency and accuracy to existing methods such as steepest descent or simulated annealing for gate decomposition?
• RQ4 Can the proposed method be generalized to decompose arbitrary quantum gates beyond CCZ and CCCZ, and to different primitive gate sets?
• RQ5 Is the known decomposition of CCCZ with 14 CZ gates optimal, and can it be achieved with fewer connections than the fully connected case?

Key findings

• The CCZ gate can be decomposed into a circuit with only four CZ-depth when using a square-shaped qubit connectivity layout and one auxiliary qubit.
• On a T-shaped qubit connectivity (acyclic, no closed loop), the CCCZ gate is decomposed using 17 CZ gates, significantly fewer than previous methods under fully connected settings.
• The study reveals that only four connections are sufficient to implement a CCCZ gate with 14 CZ gates, challenging the assumption that full connectivity is required for minimal gate counts.
• The proposed optimization method reduces computational complexity by efficiently updating the trace matrix using cyclic trace properties, enabling faster convergence than brute-force re-evaluation.
• The method successfully identifies efficient decompositions for CCZ and CCCZ gates across various connectivity types, demonstrating its general applicability to arbitrary quantum gates.
• The optimization framework can be used to verify the optimality of existing decompositions and to discover new, more efficient gate sequences for specific quantum hardware platforms.