We conduct a systematic study of quantum circuits composed of multiple-control
Z-rotation (mczr) gates as primitives, since they are widely used components in quantum algorithms and also have attracted much experimental interest in recent years. We establish a circuit-polynomial correspondence to characterize the functionality of quantum circuits over the mczr gate set with continuous parameters. An analytic method for exactly synthesizing such quantum circuit to implement any given diagonal unitary matrix with an optimal gate count is proposed, which also enables the circuit depth optimal for specific cases with pairs of complementary gates. Furthermore, we present a gate-exchange strategy together with a flexible iterative algorithm for effectively optimizing the depth of any mczr circuit, which can also be applied to quantum circuits over any other commuting gate set. Besides the theoretical investigation, the practical performance of our circuit synthesis and optimization techniques is further evaluated numerically on two typical examples in quantum computing, including diagonal Hermitian operators and quantum approximate optimization algorithm circuits with tens of qubits, which can demonstrate a reduction in circuit depth by 33.40% and 15.55% on average over relevant prior works, respectively. Therefore, our methods and results provide a pathway for implementing quantum circuits and algorithms on recently developed devices.
We achieve the first constant-depth circuit for arbitrary single-qubit gate synthesis. Unlike prior approaches, the construction is fully unitary and requires no pre-supplied catalyst. For any constant
δ>0, it
ε-approximates an arbitrary single-qubit gate using
O(log1+δ(1/ε)) clean ancillae, Hadamard and
T single-qubit gates,
O(log(1/ε))-width generalized Toffoli gates, and sublogarithmic-width Fan-Out gates. We further eliminate Fan-Out entirely, showing that Hadamard,
T, and generalized Toffoli gates alone suffice for constant-depth synthesis. When restricted to the standard bounded-width gate model, our construction has depth
O(loglog(1/ε)), and we prove a matching
Ω(loglog(1/ε))-depth lower bound. Overall, we establish that
Θ(loglog(1/ε))-depth is unavoidable with only bounded-width gates, yet allowing even logarithmic-width multi-qubit gates suffices to achieve constant-depth synthesis.
These results also reveal new structure in shallow quantum circuit complexity. We give a depth-preserving real simulation of bounded-error decision computation, showing that every depth-
d QAC circuit can be simulated in depth
O(d) using only Hadamard,
X, and generalized Toffoli gates. Thus arbitrary single-qubit rotations and complex amplitudes do not increase the bounded-error decision power of QAC, even at constant depth. In particular, this reduces the long-standing conjecture Parity
∈/QAC
0 to proving a Parity lower bound against circuits consisting only of Hadamard,
X, and generalized Toffoli gates. More generally, this real normal form exposes a direct correspondence between the standard shallow-depth quantum circuit hierarchy and a hierarchy of Forrelation circuits with restricted oracle families.
Diagonal unitary operators are well known to be key building blocks of many quantum algorithms and quantum computing procedures, and thus resource-efficient quantum circuit implementations are in demand. Here we propose a constructive algorithm that can generate a quantum circuit over the primitive gate set {cnot,
RZ} for realizing any given diagonal unitary operator piece by piece. The theoretical analysis reveals that, for the general case, our generated circuit not only ensures the asymptotically optimal gate count, but also nearly halves the circuit depth compared with the previous Welch's method [New J. Phys. 16, 033040 (2014)]. Specifically, this substantial depth optimization originates from the use of a uniform circuit rewriting rule developed here. The performance of our circuit synthesis algorithm is further validated by numerical evaluations on two examples. First, we achieve a nearly 50% depth reduction over Welch's method for synthesizing random diagonal unitary operators with up to 16 qubits. Second, we achieve an average of 22.05% depth reduction for resynthesizing the diagonal part of the specific quantum approximate optimization algorithm (QAOA) circuits with up to 14 qubits.
Uniformly controlled one-qubit gates are quantum gates which can be represented as direct sums of two-dimensional unitary operators acting on a single qubit. We present a quantum gate array which implements any
n-qubit gate of this type using at most ${2}^{n\ensuremath{-}1}\ensuremath{-}1$ controlled-NOT gates, ${2}^{n\ensuremath{-}1}$ one-qubit gates, and a single diagonal
n-qubit gate. To illustrate the versatility of these gates we then apply them to the decomposition of a general
n-qubit gate and a state preparation procedure. Moreover, we study their implementation using only nearest-neighbor gates. We give upper bounds for the one-qubit and controlled-NOT gate counts for all the aforementioned applications. In all four cases, the proposed circuit topologies either improve on or achieve the previously reported upper bounds for the gate counts. Thus, they provide the most efficient method for general gate decompositions currently known.
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.
The preparation of
n-qubit quantum states is a cross-cutting subroutine for many quantum algorithms, and the effort to reduce its circuit complexity is a significant challenge. In the literature, the quantum state preparation algorithm by Sun et al. is known to be optimally bounded, defining the asymptotically optimal width-depth trade-off bounds with and without ancillary qubits. In this work, a simpler algebraic decomposition is proposed to separate the preparation of the real part of the desired state from the complex one, resulting in a reduction in terms of circuit depth, total gates, and CNOT count when
m ancillary qubits are available. The reduction in complexity is due to the use of a single operator
Λ for each uniformly controlled gate, instead of the three in the original decomposition. Using the PennyLane library, this new algorithm for state preparation has been implemented and tested in a simulated environment for both dense and sparse quantum states, including those that are random and of physical interest. Furthermore, its performance has been compared with that of Möttönen et al.'s algorithm, which is a de facto standard for preparing quantum states in cases where no ancillary qubits are used, highlighting interesting lines of development.