Techniques for fault-tolerant decomposition of a multicontrolled Toffoli gate

Abstract

Physical implementation of scalable quantum architectures faces an immense challenge in form of fragile quantum states. To overcome it, quantum architectures with fault tolerance is desirable. This is achieved currently by using surface code along with a transversal gate set. This dictates the need for decomposition of universal Multi Control Toffoli (MCT) gates using a transversal gate set. Additionally, the transversal non-Clifford phase gate incurs high latency which makes it an important factor to consider during decomposition. Besides, the decomposition of large Multicontrol Toffoli (MCT) gate without ancilla presents an additional hurdle. In this manuscript, we address both of these issues by introducing Clifford+ZN gate library. We present an ancilla free decomposition of MCT gates with linear phase depth and quadratic phase count. Furthermore, we provide a technique for decomposition of MCT gates in unit phase depth using the Clifford+ZN library, albeit at the cost of ancillary lines and quadratic phase count.

Solovay–Kitaev Approximations of Special Orthogonal Matrices

“…It is worthwhile to consider our version of the Solovay-Kitaev theorem in the context quantum compilation [26,27] in which the conversion of a nonfault-tolerant circuit into a fault tolerant one is investigated. A recent paper introduced several efficient methods for quantum compilation using physical machine descriptions, which included one method based on the Solovay-Kitaev approximations [28].…”

Representation of binary classification trees with binary features by quantum circuits

“…For v > 2, C v NOT(q, Q) is also called (multicontrolled) Toffoli gate. The C v NOT gate can be further decomposed in various ways [37], depending on whether additional ancilla qubits are to be used or not. We particularly consider an ancilla-free decomposition into 8v − 20 two-qubit controlled-rotation gates [38].…”

Finding solutions to the integer case constraint satisfiability problem using Grover's algorithm

“…This is due to the complications involved in decomposing the rotation gates with small angles used in this implementation into fault-tolerant gates. Linear phase-depth (where phase depth is defined as the minimum number of cycles required to execute all the Z N gates) ancillafree decomposition methodology for the MCT gate using the (Clifford+Z N ) quantum gate library (Z N being the N th root of the Pauli's Z-gate, whose action is to flip the phase of the qubit by an angle of π/N ) has been proposed in [34]. However, for achieving low phase-depth, this proposal also has to use ancilla qubits.…”