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arXiv:2005.12310 · IEEE International Conference on Quantum Computing and Engineering (QCE)

Quantum Circuits for Functionally Controlled NOT Gates

Mathias Soeken, Martin Roetteler

Published Oct 1, 2020• 4 citations
arXivHTMLPDF

Abstract

We generalize quantum circuits for the Toffoli gate presented by Selinger [1] and Jones [2] for functionally controlled NOT gates, i.e.,$X$gates controlled by arbitrary n- variable Boolean functions. Our constructions target the gate set consisting of Clifford gates and single qubit rotations by arbitrary angles. Our constructions use the Walsh-Hadamard spectrum of Boolean functions and build on the work by Schuch and Siewert [3] and Welch et al [4]. We present quantum circuits for the case where the target qubit is in an arbitrary state as well as the special case where the target is in a known state. Additionally, we present constructions that require no auxiliary qubits and constructions that have a rotation depth of 1.

DOI
10.1109/qce49297.2020.00052

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We generalize quantum circuits for the Toffoli gate presented by Selinger [1] and Jones [2] for functionally controlled NOT gates, i.e.,XXXgates controlled by arbitrary n- variable Boolean functions. Our constructions target the gate set consisting of Clifford gates and single qubit rotations by arbitrary angles. Our constructions use the Walsh-Hadamard spectrum of Boolean functions and build on the work by Schuch and Siewert [3] and Welch et al [4]. We present quantum circuits for the case where the target qubit is in an arbitrary state as well as the special case where the target is in a known state. Additionally, we present constructions that require no auxiliary qubits and constructions that have a rotation depth of 1.
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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
We design a circuit structure with linear depth to implement an nnn-qubit Toffoli gate. The proposed construction uses a quadratic-size circuit that consists of elementary two-qubit controlled-rotation gates around the xxx 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 nnn-qubit Toffoli gate.
To build a general-purpose quantum computer, it is crucial for the quantum devices to implement classical boolean logic. A straightforward realization of quantum boolean logic is to use auxiliary qubits as intermediate storage. This inefficient implementation causes a large number of auxiliary qubits to be used. In this paper, we have derived a systematic way of realizing any general m-to-n bit combinational boolean logic using elementary quantum gates. Our approach transforms the m-to-n bit classical mapping into a t-bit unitary quantum operation with minimum number of auxiliary qubits, then a variation of Toffoli gate is used as the basic building block to construct the unitary operation. Finally, each of these building blocks can be decomposed into one-bit rotation and two-bit control-U gates. The efficiency of the network is taken into consideration by formulating it as a constrained set partitioning problem.
Since an n-qubit circuit consisting of CNOT gates can have up to Ω(n2/log⁡n)Ω(n^2/\log{n})Ω(n2/logn) CNOT gates, it is natural to expect that Ω(n2/log⁡n)Ω(n^2/\log{n})Ω(n2/logn) Toffoli gates are needed to apply a controlled version of such a circuit. We show that the Toffoli count can be reduced to at most n. The Toffoli depth can also be reduced to O(1), at the cost of 2n Toffoli gates, even without using any ancilla or measurement. In fact, using a measurement-based uncomputation, the Toffoli depth can be further reduced to 1. From this, we give two corollaries: any controlled Clifford circuit can be implemented with O(1) T-depth, and any Clifford+T circuit with T-depth D can be controlled with T-depth O(D), even without ancillas. As an application, we show how to catalyze a rotation by any angle up to precision εεε in T-depth exactly 1 using a universal ⌈log⁡2(8/ε)⌉\lceil\log_2(8/ε)\rceil⌈log2​(8/ε)⌉-qubit catalyst state.