Scalable quantum computing with qudits on a graph

Abstract

We show a significant reduction of the number of quantum operations and the improvement of the circuit depth for the realization of the Toffoli gate by using qudits. This is done by establishing a general relation between the dimensionality of qudits and their topology of connections for a scalable multi-qudit processor, where higher qudit levels are used for substituting ancillas. The suggested model is of importance for the realization of quantum algorithms and as a method of quantum error correction codes for single-qubit operations.Introduction.-Remarkable progress in realizing controllable quantum systems of an intermediate scale [1][2][3][4][5] makes it realistic to study properties of strongly correlated quantum matter [6-9] and to implement various quantum algorithms [10][11][12][13][14]. However, existing quantum computing systems lack either coherence or controllable interactions between qubits, and this limits their capabilities. A serious obstacle in realizing quantum algorithms is a large number of two-qubit gates, which requires programmable inter-qubit interactions and can cause decoherence. The situation becomes even more challenging in the case of mulit-qubit gates, such as an N -qubit Toffoli gate, which is a basic building block for quantum algorithms like Shor's algorithm [15] and for quantum error corrections schemes [16][17][18]. Its implementation requires 12N − 23 two-qubit gates with N − 2 ancilla qubits or O(N 2 ) gates without them [19], which is of high cost for near-term noisy intermediate-scale quantum devices. Therefore, the reduction of the number of operations that are required for the realization of multi-qubit gates remains a crucial problem.One of possible ways to reduce the number of required operations is to use additional degrees of freedom of quantum systems. This idea has stimulated an extended activity [20,21] in theoretical [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38] and experimental studies [39][40][41][42][43][44][45][46] of quantum computing models with qudits, which are d-dimensional (d > 2) quantum systems. In particular, qudits can be used for substituting ancillas [30,[37][38][39], which allows the reduction of the required number of interactions between information carriers for the realization of multi-qubit gates. In experiments with photonic quantum circuits [39], for a system of an Ndimensional qudit connected with N − 1 qubits, the Nqubit Toffoli gate was realized with 2N − 3 qubit-qudit gates. However, it is hard to expect scalability for such

Qudits and High-Dimensional Quantum Computing

“…The previous method turns the target qubit into a qudit; another method simplifies the Toffoli gate by using only qudits and treating the first two levels of the qudit as qubit levels and other levels as auxiliary levels. The reduction in the complexity of Toffoli gate is accomplished by utilizing the topological relations between the dimensionality of the qudits, where higher qudit levels serve as the ancillas [89].…”

“…The detailed realization of the n-qubit Toffoli gate by the properties and special operations of the tree in topology can be found in Ref. 89. The advantage of this scheme is the scalability and the ability to implement it for the multi-qubit controlled unitary gate C n U.…”

“…Although the qudit system's advantages in various applications and potentials for future development are substantial, this system receives less attention than the conventional qubitbased quantum computing, and a comprehensive review of the qudit-based models and technologies is needed. This review article provides an overview of qudit-based quantum computing covering a variety of topics ranging from circuit building [39,61,71,89,133]; algorithm designs [2,17,26,62,79,119,121]; to experimental methods [2,11,48,60,62,91,99,106]. In this article, high-dimensional generalizations of many widely used quantum gates are presented and the universality of the qudit gates is shown.…”

“…Furthermore, Luo and Wang show that with their proposed universal computation scheme [108]; there is an extra factor of n reduction in the gate requirement, where n is the number of qudits. By introducing qudits to the construction of some wellknown gates such as the Toffoli gate, the elementary gate required are reduced from 12n − 11 gates in the qubit case to 2n − 1 gates by introducing a single (n + 1)-level target carrier [133] and to 2n − 3 gates by utilizing the topological properties [89]. In our discussion of the geometrically quantified qudit-gate efficiency in Section 2.3, the qubit system needs O(n 6 d(I, U) 3 ) one-and two-qubit gates to synthesize a unitary [122] while in the qutrit case the lower bound is O(n k d(I, U) 3 ) where k is an integer that depends on the accuracy of the approximation and can be smaller than 6 [100].…”

Квантово-Оптические Технологии Для Обработки И Передачи Информации

“…Во-вторых, как самостоятельная вычислительная платформа. В докладе будет представлен анализ ряда достижений в области создания квантовых вычислительных систем, использующих квантово-оптические технологии в качестве основного метода экспериментального контроля [3]. В частности, будут представлены предварительные результаты, полученные в рамках проекта по созданию квантового вычислительного устройства на основе ионов.…”