Multiple-control Toffoli gates are basic building blocks for reversible and quantum circuits. 3-bit Peres gate can be considered as a pair of 3-bit and 2-bit Toffoli gates, which can be implemented with fewer elementary quantum gates (what defines a metric called quantum cost) than the total number of elementary quantum gates required by each of Toffoli gates implemented separately. Due to this property, Peres gates are often used for quantum cost reduction of reversible circuits. This paper introduces a generalization of the Peres gate with n > 2 control signals. For such generalized gates corresponding quantum circuits are presented. The realizations have quantum cost of n2and require no ancillary signals. The proposed designs are built from elementary quantum gates, which consist of NOT, controlled NOT, controlled `square-roots-of-NOT' (known as controlled V/V+gates) and controlled `higher roots-of-NOT' (up to 2n-1-roots-of-NOT). The obtained quantum circuits are not entangled. The proposed designs are used to construct Toffoli gates of n control signals with quantum cost of 2n2- 2n + 1 and using no ancillary signals. This is an improvement in comparison with 2n+1- 3 and 48n2+ O(n), which are quantum costs of the known designs of the multiple-control Toffoli gate without ancillary signals.
The minimization of Quantum Cost (QC), specifically the number of Toffoli gates, is a primary objective in the synthesis of reversible arithmetic circuits. For decades, the Peres Full Adder has been considered the benchmark for minimal realizations, requiring 2 Toffoli gates and a Quantum Cost of 12. In this note, we propose a novel reversible Full Adder circuit that utilizes only one Toffoli gate and four CNOT gates. The proposed design achieves a Quantum Cost of 9, representing a 25% reduction compared to the Peres structure, while maintaining a competitive logical depth of 4. We rigorously verify the logic and demonstrate that the reduction is achieved through the automated synthesis framework recently proposed by the authors.
We design a circuit structure with linear depth to implement an
n-qubit Toffoli gate. The proposed construction uses a quadratic-size circuit that consists of elementary two-qubit controlled-rotation gates around the
x axis and uses no ancilla qubit. Circuit depth remains linear in quantum technologies with finite-distance interactions between qubits. The suggested construction is related to the long-standing construction by A. Barenco et al. [Phys. Rev. A 52, 3457 (1995)], which uses a quadratic-size, quadratic-depth quantum circuit for an
n-qubit Toffoli gate.
The family of
n-bit Toffoli gates, with the two-bit Toffoli gate as the figurehead, are of great interest in quantum information as they can be used as universal gates and in quantum error correction, among other things. We present a single-step implementation of arbitrary
n-bit Toffoli gates (up to a local change of basis), based on resonantly driving a single qubit that has a strong Ising coupling to
n other qubits. The setup in the two-qubit case turns out to be identical to the universal Barenco gate. The gate time and error are, in theory, independent of the number of control qubits, scaling better than conventional circuit decompositions. We note that our assumptions, namely, strongly coupling
n+1 qubits and a driving frequency that scales with
n, may break down for large systems. Still, our protocol could enhance the capabilities of intermediate-scale quantum computers, and we discuss the prospects of implementing our protocol on trapped ions, Rydberg atoms, and superconducting circuits. Simulations of the latter platform show that the Toffoli gate with two control bits attains fidelities of above 0.98 even in the presence of decoherence. We also show how similar ideas can be used to make a series of controlled-not gates in a single step. We show how these can speed up the implementation of quantum error correcting codes and we simulate the encoding steps of the three-qubit bit-flip code and the seven-qubit Steane code.
We present two constructions for the Toffoli gate which substantially reduce resource costs in fault-tolerant quantum computing. The first contribution is a Toffoli gate requiring Clifford operations plus only four $T=\mathrm{exp}(i\ensuremath{\pi}{\ensuremath{\sigma}}^{z}/8)$ gates, whereas conventional circuits require seven
T gates. An extension of this result is that adding
n control inputs to a controlled gate requires
4n T gates, whereas the best prior result was
8n. The second contribution is a quantum circuit for the Toffoli gate which can detect a single ${\ensuremath{\sigma}}^{z}$ error occurring with probability
p in any one of eight
T gates required to produce the Toffoli gate. By postselecting circuits that did not detect an error, the posterior error probability is suppressed to lowest order from
4p (or
7p, without the first contribution) to
28p2 for this enhanced construction. In fault-tolerant quantum computing, this construction can reduce the overhead for producing logical Toffoli gates by an order of magnitude.
Quantum circuit synthesis is the procedure of automatically generating quantum circuits to represent specified functions. A common gate in quantum circuits is the reversible Toffoli gate, a type of generalized controlled NOT operation. There are physical barriers to implementing large quantum gates. Large Toffoli gates can be decomposed into equivalent sets of smaller, quantum elementary gates. The cost of a quantum circuit can be measured by counting the number of elementary gates in the circuit after all gates have been decomposed. Traditionally this decomposition is done independently for each gate in the circuit. This thesis identifies pairs of gates that, if decomposed together, result in fewer total elementary gates than they would otherwise. These improvements are incorporated into a simple decomposition algorithm which manipulates the circuit in order to search for such pairs. The decomposition algorithm is compared to a naive implementation, and the resulting gate costs are presented and compared.