We construct an exchange gate with small elementary gates on the space of qudits, which consist of three controlled shift gates and three 'reverse' gates. This is a natural extension of the qubit case. We also consider a similar situation in Fock space, but in this case we find some differences. However, we can construct the exchange gate by making use of a generalized coherent operator based on the Lie algebra su (2), which is a well-known method in quantum optics. We also make a brief comment on 'imperfect clones'.
General n-qubit controlled unitary gates are frequently used in quantum information processing tasks. Barenco, Bennett, Cleve, Di Vincenzo, Margolus and Shor [Phys. Rev. A52 (1995) 3457] have given the general construction methods, and explicit results for up-to-four-qubits controlled unitary gates. We extended their calculation and gave two analytic expressions for the construction of general n-qubit controlled unitary gates in terms of one-qubit and two-qubit CNOT gates. There are two expressions – one is exponential in the qubit number which is efficient for up to ten qubits, and the other is polynomial in the qubit number, which is efficient for more than ten qubits.
Circuit synthesis of multi-controlled gates is crucial for qudit (
d-level) quantum computing. This paper presents efficient synthesis schemes that reduce the elementary gate count for multi-controlled single-qudit gates. For synthesizing general
(n−1)-controlled unitaries on
n qudits, we reduce the controlled-increment (CINC) and generalized controlled-
X (GCX) gate counts to
O(n2), improving upon existing
O(n2+log2d) CINC and
O(n3) GCX bounds. For
(n−1)-controlled special unitaries, this complexity is further reduced to
O(n). Furthermore, we present a method for approximately synthesizing multi-controlled unitary gates with a linear complexity in the number of control qudits. By utilizing the proposed circuit, we present qudit-based circuit constructions for isometries and quantum channels from
n to
m qudits. Moreover, for the first time, we present a circuit synthesis scheme for single-controlled gates using SUM gates and single-qudit gates when
d is prime. This enables all CINC-based circuits for various quantum operations to be converted into SUM-gate circuits while preserving the same asymptotic complexity. Finally, we establish a theoretical lower bound on the number of SUM and CINC gates required to synthesize general
n-qudit unitaries.
We propose the generalized controlled
X (gcx) gate as the two-qudit elementary gate, and based on Cartan decomposition, we also give the one-qudit elementary gates. Then we discuss the physical implementation of these elementary gates and show that it is feasible with current technology. With these elementary gates many important qudit quantum gates can be synthesized conveniently. We provide efficient methods for the synthesis of various kinds of controlled qudit gates and greatly simplify the synthesis of existing generic multi-valued quantum circuits. Moreover, we generalize the quantum Shannon decomposition (QSD), the most powerful technique for the synthesis of generic qubit circuits, to the qudit case. A comparison of ququart (
d=4) circuits and qubit circuits reveals that using ququart circuits may have an advantage over the qubit circuits in the synthesis of quantum circuits.