In order to demonstrate nontrivial quantum computations experimentally, such as the synthesis of arbitrary entangled states, it will be useful to understand how to decompose a desired quantum computation into the shortest possible sequence of one-qubit and two-qubit gates. We contribute to this effort by providing a method to construct an optimal quantum circuit for a general two-qubit gate that requires at most 3 controlled-NOT (CNOT) gates and 15 elementary one-qubit gates. Moreover, if the desired two-qubit gate corresponds to a purely real unitary transformation, we provide a construction that requries at most 2 CNOT and 12 one-qubit gates. We then prove that these constructions are optimal with respect to the family of CNOT, $y$ -rotation, $z$ -rotation, and phase gates.In order to demonstrate nontrivial quantum computations experimentally, such as the synthesis of arbitrary entangled states, it will be useful to understand how to decompose a desired quantum computation into the shortest possible sequence of one-qubit and two-qubit gates. We contribute to this effort by providing a method to construct an optimal quantum circuit for a general two-qubit gate that requires at most 3 controlled-NOT (CNOT) gates and 15 elementary one-qubit gates. Moreover, if the desired two-qubit gate corresponds to a purely real unitary transformation, we provide a construction that requries at most 2 CNOT and 12 one-qubit gates. We then prove that these constructions are optimal with respect to the family of CNOT, $y$-rotation, $z$-rotation, and phase gates.

arXiv:quant-ph/0308006Physical Review AMar 22, 2004442 citations

Optimal design of two-qubit quantum circuits

Farrokh Vatan, Colin P. Williams

In order to demonstrate non-trivial quantum computations experimentally, such as the synthesis of arbitrary entangled states, it will be useful to nderstand how to decompose a desired quantum computation into the shortest possible sequence of one-qubit and two-qubit gates. We contribute to this effort by providing a method to construct an optimal quantum circuit for a general two-qubit gate that requires at most 3 CNOT gates and 15 elementary one qubit gates. We then prove that these constructions are optimal with respect to the family of CNOT, y-rotation, z-rotation, and phase gates.In order to demonstrate non-trivial quantum computations experimentally, such as the synthesis of arbitrary entangled states, it will be useful to nderstand how to decompose a desired quantum computation into the shortest possible sequence of one-qubit and two-qubit gates. We contribute to this effort by providing a method to construct an optimal quantum circuit for a general two-qubit gate that requires at most 3 CNOT gates and 15 elementary one qubit gates. We then prove that these constructions are optimal with respect to the family of CNOT, y-rotation, z-rotation, and phase gates.

Jan 5, 20040 citations

QUANTUM CIRCUIT OPTIMIZATION (FOR 3 QUBITS)

International Journal of Advance Engineering and Research DevelopmentJun 30, 20150 citations

Entanglement dynamics and Page curves in random permutation circuits

Dávid Szász-Schagrin, Michele Mazzoni, Bruno Bertini +2 more

The characterization of ensembles of many-qubit random states and their realization via quantum circuits are crucial tasks in quantum-information theory. In this work, we study the ensembles generated by quantum circuits that randomly permute the computational basis, thus acting classically on the corresponding states. We focus on the averaged entanglement and present two main results. First, we derive generically tight upper bounds on the entanglement that can be generated by applying permutation circuits to arbitrary initial states. We show that the late-time “entanglement Page curves” are bounded in terms of the initial state participation entropies and their overlap with the “maximally antilocalized” state. Generally speaking, this result states that the quantum correlations generated by classical circuits are bounded in terms of some quantum property of the initial state (namely, the degree to which it can be written as a superposition of classical states). Second, comparing the averaged Rényi-2 entropies generated by ( 1 ) an infinitely deep random circuit of two-qubit gates and ( 2 ) global random permutations, we show that the two quantities are different for finite N but the corresponding Page curves coincide in the thermodynamic limit. We also discuss how these conclusions are modified by additional random phases or considering circuits of k -local gates with k ≥ 3 . Our results are exact and highlight the implications of classical features on entanglement generation in many-body systems.The characterization of ensembles of many-qubit random states and their realization via quantum circuits are crucial tasks in quantum-information theory. In this work, we study the ensembles generated by quantum circuits that randomly permute the computational basis, thus acting classically on the corresponding states. We focus on the averaged entanglement and present two main results. First, we derive generically tight upper bounds on the entanglement that can be generated by applying permutation circuits to arbitrary initial states. We show that the late-time “entanglement Page curves” are bounded in terms of the initial state participation entropies and their overlap with the “maximally antilocalized” state. Generally speaking, this result states that the quantum correlations generated by classical circuits are bounded in terms of some quantum property of the initial state (namely, the degree to which it can be written as a superposition of classical states). Second, comparing the averaged Rényi-2 entropies generated by ( 1 ) an infinitely deep random circuit of two-qubit gates and ( 2 ) global random permutations, we show that the two quantities are different for finite N but the corresponding Page curves coincide in the thermodynamic limit. We also discuss how these conclusions are modified by additional random phases or considering circuits of k -local gates with k ≥ 3 . Our results are exact and highlight the implications of classical features on entanglement generation in many-body systems.

arXiv:2505.06158Physical Review ResearchDec 19, 20253 citations

y

-rotation,

z

-rotation, and phase gates.