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arXiv:quant-ph/0307177 · Physical Review A

Universal quantum circuit for two-qubit transformations with three controlled-NOT gates

Guifré Vidal, Christopher M. Dawson

Published Jan 8, 2004• 276 citations
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Abstract

We consider quantum circuits made of controlled-NOT (CNOT) gates and single-qubit unitary gates and look for constructions that minimize the use of CNOT gates. We show, by means of an explicit quantum circuit, that three CNOT gates are necessary and sufficient in order to implement an arbitrary unitary transformation of two qubits. We also identify the subset of two-qubit gates that can be performed with only two CNOT gates and provide a simple characterization for them.

DOI
10.1103/physreva.69.010301

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We consider quantum circuits made of controlled-NOT (CNOT) gates and single-qubit unitary gates and look for constructions that minimize the use of CNOT gates. We show, by means of an explicit quantum circuit, that three CNOT gates are necessary and sufficient in order to implement an arbitrary unitary transformation of two qubits. We also identify the subset of two-qubit gates that can be performed with only two CNOT gates and provide a simple characterization for them.
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Using a geometric approach, we derive the minimum number of applications needed for an arbitrary controlled-unitary gate to construct a universal quantum circuit. An analytic construction procedure is presented and shown to be either optimal or close to optimal. This result can be extended to improve the efficiency of universal quantum circuit construction from any entangling gate. In addition, for both the controlled-NOT (CNOT) and double-CNOT gates, we develop simple analytic ways to construct universal quantum circuits with three applications, which is the least possible for these gates.
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.
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++.
We address the problem of constructing quantum circuits to implement an arbitrary two-qubit quantum computation. We pursue circuits without ancilla qubits and as small a number of elementary quantum gates as possible. Our lower bound for worst-case optimal two-qubit circuits calls for at least 171717 gates: 151515 one-qubit rotations and 222 controlled-NOT (CNOT) gates. We also constructively prove a worst-case upper bound of 232323 elementary gates, of which at most four (CNOT gates) entail multiqubit interactions. Our analysis shows that synthesis algorithms suggested in previous work, although more general, entail larger quantum circuits than ours in the special case of two qubits. One such algorithm has a worst case of 616161 gates, of which 181818 may be CNOT 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, yyy-rotation, zzz-rotation, and phase gates.