Implementation of high-dimensional (HD) quantum gates shows very promising perspectives for HD quantum computation. A bipartite quantum system with arbitrary dimensions
n and
m is termed a quNit-quMit. Here we propose a synthesis scheme to construct the quantum circuit for general quNit-quMit gates with controlled increment (CINC) gates and local gates. This shows that CINC gates combined with local gates form a universal gate set for HD quantum computation. An upper bound of
O(n2) CINC gates is achieved for arbitrary quNit-quMit gate implementation in the proposed scheme, which is the best known result. Especially for the controlled quNit-quMit gates, our scheme requires only 2 CINC gates, whereas the previous scheme required
2n.
Quantum circuits of a general quantum gate acting on multiple
d-level quantum systems play a prominent role in multi-valued quantum computation. We first propose a new recursive Cartan decomposition of semi-simple unitary Lie group
U(3n) (arbitrary
n-qutrit gate). Note that the decomposition completely decomposes an n-qutrit gate into local and non-local operations. We design an explicit quantum circuit for implementing arbitrary two-qutrit gates, and the cost of our construction is 21 generalized controlled X (GCX) and controlled increment (CINC) gates less than the earlier best result of 26 GGXs. Moreover, we extend the program to the
n-qutrit system, and the quantum circuit of generic
n-qutrit gates contained
9641⋅32n−4⋅3n−1−(2n2+4n−3229) GGXs and CINCs is presented. Such asymptotically optimal structure is the best known result so far.
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.
Abstract Quantum circuits of a general quantum gate acting on multiple ‐level quantum systems play a prominent role in multi‐valued quantum computation. A recursive Cartan decomposition of semi‐simple unitary Lie group (arbitrary ‐qutrit gate) is first proposed with a rigorous proof, which completely decomposes an ‐qutrit gate into local and non‐local operations. On this basis, an explicit quantum circuit is designed for implementing arbitrary two‐qutrit gates, and the cost of the construction is 21 generalized controlled (GCX) and controlled increment (CINC) gates less than the earlier best result of 26 GGXs. Furthermore, the program is extended to the ‐qutrit system, and the quantum circuit of generic ‐qutrit gates contained GGXs and CINCs is presented. Such asymptotically optimal structure is the best known result so far and its strength becomes more remarkable as increases, for example, when , the program saves 7146 GCXs compared to the previous best result. In addition, concrete recursive decomposition expressions is given for each non‐local operation instead of only quantum circuit diagrams.
Abstract A high‐dimensional system, so‐called quNit (‐level quantum system with ), provides several unique advantages, and high‐precision manipulation of arbitrary single‐quNit is demonstrated. In this study, a program to construct arbitrary qubit‐quNit quantum computation up to a global phase, and an efficient construction that requires single‐particle and generalized controlled (GCX) gates in the worst case are proposed. Moreover, by encoding the computing basis in polarization and spatial degrees of freedom of a single‐photon state, the optical architectures to implement the elementary‐gate library GCX, single‐partite rotation in a high‐dimension system are designed using linear optics. The necessary single‐photon polarization state and flexibility of linear optical elements make the quantum circuit feasible with current experimental technologies.
Abstract Quantum circuit model is the most popular paradigm for implementing complex quantum computation. Based on Cartan decomposition, it is shown generalized controlled‐X (GCX) gates, 6 single‐qubit rotations about the y‐ and z‐axes, and single‐partite y‐ and z‐rotation‐types which are defined in this paper are sufficient to simulate a controlled‐unitary gate with controlling on . In the scenario of the unitary gate with that is locally equivalent to a diagonal unitary on , GCX gates and single‐partite y‐ and z‐rotation‐types are required to simulate it. The quantum circuit for implementing and are presented. Furthermore, it is found that with controlling has Schmidt rank two, and in other cases the diagonalized form of the target unitaries can be expanded in terms of specific simple types of product unitary operators.