Classification of Schmidt-rank-two multipartite unitary gates by singular number

Abstract

The multipartite unitary gates are called genuine if they are not product unitary operators across any bipartition. We mainly investigate the classification of genuine multipartite unitary gates of Schmidt rank two, by focusing on the multiqubit scenario. For genuine multipartite (excluding bipartite) unitary gates of Schmidt rank two, there is an essential fact that their Schmidt decompositions are unique. Based on this fact, we propose a key notion named as singular number to classify the unitary gates concerned. The singular number is defined as the number of local singular operators in the Schmidt decomposition. We then determine the accurate range of singular number. For each singular number, we formulate the parametric Schmidt decompositions of genuine multiqubit unitary gates under local equivalence. Finally, we extend the study to three-qubit diagonal unitary gates due to the close relation between diagonal unitary gates and Schmidt-rank-two unitaries. We start with discussing two typical examples of Schmidt rank two, one of which is a fundamental three-qubit unitary gate, i.e., the CCZ gate. Then we characterize the diagonal unitary gates of Schmidt rank greater than two. We show that a three-qubit diagonal unitary gate has Schmidt rank at most three, and present a necessary and sufficient condition for such a unitary gate of Schmidt rank three. This completes the characterization of all genuine three-qubit diagonal unitary gates.

arXiv comment23 pages, 2 figures

References in corpus (28)

• Elementary gates for quantum computation
• Three qubits can be entangled in two inequivalent ways
• Persistent entanglement in arrays of interacting particles
• Parallel implementation of high-fidelity multi-qubit gates with neutral atoms
• Generalized Schmidt decomposition and classification of three-quantum-bit states
• Optimal Creation of Entanglement Using a Two--Qubit Gate
• Optimal local implementation of non-local quantum gates
• Implementation of a Toffoli Gate with Superconducting Circuits
• Realization of the quantum Toffoli gate with trapped ions
• Distributed Quantum Computation Based-on Small Quantum Registers
• Entanglement in Quantum Information Theory
• Universal fault-tolerant quantum computation with only transversal gates and error correction
• Quantum dynamics as a physical resource
• Randomized Benchmarking of Multi-Qubit Gates
• Simplified instantaneous non-local quantum computation with applications to position-based cryptography
• Entanglement in mutually unbiased bases
• Tensor Rank and Stochastic Entanglement Catalysis for Multipartite Pure States
• Optimal conversion of non--local unitary operations
• Recovering quantum gates from few average gate fidelities
• Efficient implementation of bipartite nonlocal unitary gates using prior entanglement and classical communication
• The Tensor Rank of the Tripartite State ${|W⟩}^{\otimes n}$}
• Nonlocal and controlled unitary operators of Schmidt rank three
• Decomposition of bipartite and multipartite unitary gates into the product of controlled unitary gates
• All unitaries having operator Schmidt rank 2 are controlled unitaries
• Entanglement cost and entangling power of bipartite unitary and permutation operators
• On the Schmidt-rank-three bipartite and multipartite unitary operator
• Entangling and assisted entangling power of bipartite unitary operations
• Entangling power of two-qubit unitary operations

Cited by in corpus (2)

• Multipartite entangling power by von Neumann entropy
• The detection power of real entanglement witnesses under local unitary equivalence