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arXiv:quant-ph/0411058 · Physical Review A

Conditions for optimal construction of two-qubit nonlocal gates

Yong-Sheng Zhang, Ming-Yong Ye, Guang‐Can Guo

Published Jun 24, 2005• 42 citations
arXivHTMLPDF

Abstract

Optimal implementation of quantum gates is crucial for designing a quantum computer. The necessary condition for optimal construction of a two-qubit unitary operation is obtained. It can be proved that the $B$ gate is the unique gate that can construct a two-qubit universal circuit with only two applications, i.e., this condition is also sufficient in the case of two applications of the elementary two-qubit gate. It is also shown that many perfect entanglers cannot simulate an arbitrary two-qubit gate with only three applications. We show how a two-qubit gate's construction power is related to the construction power of its mirror gate, and introduce a specific kind of two-qubit entangling gate, i.e., a super controlled gate, which can construct an arbitrary two-qubit gate by three applications.

DOI
10.1103/physreva.71.062331

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Optimal implementation of quantum gates is crucial for designing a quantum computer. The necessary condition for optimal construction of a two-qubit unitary operation is obtained. It can be proved that the BBB gate is the unique gate that can construct a two-qubit universal circuit with only two applications, i.e., this condition is also sufficient in the case of two applications of the elementary two-qubit gate. It is also shown that many perfect entanglers cannot simulate an arbitrary two-qubit gate with only three applications. We show how a two-qubit gate's construction power is related to the construction power of its mirror gate, and introduce a specific kind of two-qubit entangling gate, i.e., a super controlled gate, which can construct an arbitrary two-qubit gate by three applications.
We provide an analytic way to implement any arbitrary two-qubit unitary operation, given an entangling two-qubit gate together with local gates. This is shown to provide explicit construction of a universal quantum circuit that exactly simulates arbitrary two-qubit operations in SU(4). Each block in this circuit is given in a closed form solution. We also provide a uniform upper bound of the applications of the given entangling gates, and find that exactly half of all the controlled-unitary gates satisfy the same upper bound as the CNOT gate. These results allow for the efficient implementation of operations in SU(4) required for both quantum computation and quantum simulation.
Abstract In the circuit model of quantum computation, an entangling two-qubit gate and a set of single-qubit gates are used as universal gate set or basis gates for doing quantum computation. CNOT and iSWAP gates, the perfect entanglers that can create maximally entangled two-qubit states in one application, are broadly used entangling two-qubit basis gates in quantum computers. In this paper, we analyze the potentiality of B gate, an unexplored non-perfect entangler of the form, exp i π ( σ ˆ x ⨂ σ ˆ x ) 8 + i π ( σ ˆ y ⨂ σ ˆ y ) 16 , as an entangling two-qubit basis gate in quantum computers by studying its ability to generate other two-qubit gates. We derive a necessary condition for a two-qubit gate to be generated by n applications of B gate. Using this condition, we show that the gates that can be generated by two and three applications of B gate are contained in the 50% and 92.97% of the volume of Weyl chamber of two-qubit gates, respectively. We prove that two applications of B gate can generate both SWAP and SWAP
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, yyy-rotation, zzz-rotation, and phase gates.
Two applications of gates from the B gate equivalence class can generate all two-qubit gates. This local equivalence class is invariant under the mirror (multiplication with the SWAP gate) operation, inverse (Hermitian conjugate) operation, and the combined inverse and mirror operations. The last two symmetries are associated with the ability of a two-qubit gate to generate the two-qubit local gates and the SWAP gate in two applications. No single local equivalence class of two-qubit gates, except the B gate equivalence class, has these two symmetries. Only the planar regions of the Weyl chamber, describing the mirror operation, contain the local equivalence classes with either one of the two symmetries. We show that there exist one-parameter families of local equivalence classes on these planes, with and without the B gate equivalence class, such that each of them can be used to construct a parameterized universal two-qubit quantum circuit that involves only two nonlocal two-qubit gates. We also discuss the implementation of the gates from a few families of local equivalence classes on superconducting quantum computers for optimal generation of all two-qubit gates. We provide upper bounds on the number of two-qubit gates required to generate an arbitrary nnn-qubit gate for two families, each of which is conjectured to generate all two-qubit gates in two applications. We show that there exists a positive correlation between the area of the convex hull of the squared eigenvalues of the nonlocal part of a parameterized two-qubit gate and the fractional volume of the Weyl chamber covered in two applications of the parameterized two-qubit gate for two families of local equivalence classes.
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.
An important result from the mid nineties shows that any unitary evolution may be realized as a sequence of controlled-not and one-qubit gates. This work surveys especially efficient circuits in this library, in the special case of evolutions on two-quantum bits. In particular, we show that to construct an arbitrary two-qubit state from |00>, one CNOT gate suffices. To simulate an arbitrary two-qubit operator up to relative phases, two CNOTs suffice. To simulate an arbitrary two-qubit operator up to global phase, three CNOTs suffice. In each case, we construct an explicit circuit and prove optimality in the generic case. We also contribute a procedure to determine the minimal number of CNOT gates necessary to simulate a given two-qubit operator up to global phase. We use this procedure to discuss timing a given Hamiltonian to simulate the CNOT and to determine an optimal circuit for the two-qubit Quantum Fourier Transform. Our constructive proofs amount to circuit synthesis algorithms and have been coded in C++.