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Theoretical and Applied Cybersecurity

Toffoli gate implementation method based on Margolus gate on four or more qubits

Andrii Tereshchenko, V. K. Zadiraka

Published Dec 16, 2024• 0 citations
Publisher

Abstract

This paper considers the method of building the Toffoli Gate based on the Margolus Gate on four or more qubits. In the first part of the considered method, the modification of the Margolus gate on four or more qubits is suggested. In the second part of the method, the modification of the Margolus gate is transformed into the modification of the Toffoli gate implementation using step-by-step phase rotation compensations. The phase rotation compensation for an N-qubit quantum circuit can be performed with N successive steps, where at each step the gates with phase rotation pi/2^(s-1) are added, where s is the step number, starting from one. The compensation phase requires 2 two-qubit gates and 2N-1 one-qubit gates.

DOI
10.20535/tacs.2664-29132024.1.306056

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This paper considers the method of building the Toffoli Gate based on the Margolus Gate on four or more qubits. In the first part of the considered method, the modification of the Margolus gate on four or more qubits is suggested. In the second part of the method, the modification of the Margolus gate is transformed into the modification of the Toffoli gate implementation using step-by-step phase rotation compensations. The phase rotation compensation for an N-qubit quantum circuit can be performed with N successive steps, where at each step the gates with phase rotation pi/2^(s-1) are added, where s is the step number, starting from one. The compensation phase requires 2 two-qubit gates and 2N-1 one-qubit gates.
Unitary operations are expressed in the quantum circuit model as a finite sequence of elementary gates, such as controlled-not gates and single qubit gates. We prove that the simplified Toffoli gate by Margolus, which coincides with the Toffoli gate up to a single change of sign, cannot be realized with less than three controlled-not gates. If the circuit is implemented with three controlled-not gates, then at least four additional single qubit gates are necessary. This proves that the implementation suggested by Margolus is optimal.
A simplified Toffoli gate coincides with the Toffoli gate except that the result is allowed to differ on one computational basis state by a phase factor. We prove that the simplified Toffoli gate implementation by Margolus is optimal, in the sense that it attains a lower bound of {\em three} controlled-not gates, and subject to that, a sharp lower bound of {\em four} single-qubit gates. We also discuss optimal implementations of other simplified Toffoli gates, and explain why the phase factor −1-1−1 invariably occurs in such implementations.
Single-step implementations of multiqubit gates are generally believed to provide a simpler design, a faster operation, and a lower decoherence. For three coupled qubits interacting with a photon field, a realizable scheme for a single-step Toffoli gate is investigated. We find that the three-qubit system can be described by four effective modified Jaynes-Cummings models in the states of two control qubits. Within the rotating-wave approximation, the modified Jaynes-Cummings models are shown to be reduced to the conventional Jaynes-Cummings models with renormalized couplings between qubits and photon fields. A single-step Toffoli gate is shown to be realizable with tuning of the four characteristic oscillation periods that satisfy a commensurate condition. Possible values of the system parameters are estimated for a single-step Toffoli gate. Further, we discuss our single-step Toffoli-gate operation errors due to imperfections in the system parameters; numerical calculations show that a Toffoli gate with high fidelity can be obtained by adjusting pairs of the photon-qubit and qubit-qubit coupling strengths. In addition, a decoherence effect on the Toffoli-gate operation due to a thermal reservoir is discussed.
Multiple-controlled Toffoli gates are extensively used in quantum and reversible circuits. However, very limited attempt has been made for primitive quantum gate realizations of these gates. In this paper, we present realization of three-qubit Toffoli gate requiring five primitive quantum gates. We then propose realization of four-qubit Toffoli gate requiring 13 primitive quantum gates and one ancilla input. Finally, we propose realizations of (n + 1)-qubit (n > 3 control lines) Toffoli gates using (⌈n/2⌉ + 1)-qubit and (⌊n/2⌋ + 1)-qubit Toffoli gates and ancilla inputs, which make a trade off between quantum cost and number of ancilla inputs. Our proposed method outperforms the best result so far presented in the literature in terms of quantum cost and number of ancilla inputs.
We present improved circuits for the Toffoli gate and the control-swap (Fredkin) gate using three and four global two-qubit gates, respectively. This is a nearly double speedup compared to the conventional circuits, which require five (for Toffoli) and seven (for Fredkin) local two-qubit gates. We apply the same approach to construct the conditional four-qubit phase gate by seven global two-qubit gates. We also present construction of the Toffoli and Fredkin gates with five global gates in systems with nearest-neighbor interactions. Our constructions do not employ ancilla qubits or ancilla internal states and are particularly well suited for ion qubits and for circuit QED systems, where the entangling operations can be implemented by global addressing.
Building generic gates from a restricted gate set is a difficult but important problem, especially in the noisy regime where only a limited set of noise-resistant gates are available, e.g., fault-tolerant Clifford gates (generated by Hadamard, phase, and gates) and fault-tolerant Toffoli gates. The Toffoli is also often used as a building block of many algorithms and will need to be constructed if not directly available. This makes Clifford + Toffoli an attractive gate set for building generic gates. In this Letter we give a simple and efficient algorithm for building an approximate single-qubit rotation using only Clifford + Toffoli , which in turn enables any generic single-qubit unitary. An important difference compared to earlier attempts is that the use of the Toffoli allows us to use simple rounding as opposed to a complicated approximation algorithm. The resulting gate array does not rely on repeatedly applying a fixed rotation, but immediately applies a rotation R θ * that is ε -close to the desired rotation R θ , with a success probability strictly greater than 1 / 2 . It can be rerun upon failure, giving an expected number of repetitions strictly less than 2, an expected Toffoli count strictly less than 4 ⌈ log 1 ε ⌉ , an expected gate depth strictly less than 4 ⌈ log 1 ε ⌉ + 6 , and uses 2 ⌈ log 1 ε ⌉ ancillas. The small circuit depth of our construction enables low-noise gates on existing quantum computational devices, and allows for arbitrary precision using a very modest number of ancillas. Published by the American Physical Society 2024