Near-optimal quantum circuit construction via Cartan decomposition

Abstract

We show the applicability of the Cartan decomposition of Lie algebras to quantum circuits. This approach can be used to synthesize circuits that can efficiently implement any desired unitary operation. Our method finds explicit quantum circuit representations of the algebraic generators of the relevant Lie algebras allowing the direct implementation of a Cartan decomposition on a quantum computer. The construction is recursive and allows us to expand any circuit down to generators and rotation matrices on individual qubits, where through our recursive algorithm we find that the generators themselves can be expressed with controlled-not (CNOT) and SWAP gates explicitly. Our approach is independent of the standard CNOT implementation and can be easily adapted to other cross-qubit circuit elements. In addition to its versatility, we also achieve near-optimal counts when working with CNOT gates, achieving an asymptotic cnot cost of $\frac{21}{16}{4}^n$ for $n$ qubits.

References in corpus (12)

• Quantum Machine Learning
• Elementary gates for quantum computation
• An introduction to quantum machine learning
• Challenges and Opportunities in Quantum Machine Learning
• Optimal Quantum Circuits for General Two-Qubit Gates
• Quantum Circuits for General Multiqubit Gates
• Minimal Universal Two-qubit Quantum Circuits
• A universal quantum circuit for two-qubit transformations with three CNOT gates
• Efficient decomposition of quantum gates
• Logical-qubit operations in an error-detecting surface code
• A Verified Optimizer for Quantum Circuits
• Universal Fault-Tolerant Quantum Computing with Stabiliser Codes

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