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arXiv:2212.01002 · Physical Review A

Depth-optimized quantum circuit synthesis for diagonal unitary operators with asymptotically optimal gate count

Shihao Zhang, Kai Huang, Lvzhou Li

Published Apr 1, 2024• 12 citations
arXivHTMLPDF

Abstract

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, ${R}_{Z}$} 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.

DOI
10.1103/physreva.109.042601

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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{R}_{Z}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.
The quantum circuit synthesis problem bridges quantum algorithm design and quantum hardware implementation in the Noisy Intermediate-Scale Quantum (NISQ) era. In quantum circuit synthesis problems, diagonal unitary synthesis plays a crucial role due to its fundamental and versatile nature. Meanwhile, experimental results have shown that moderately approximating the original algorithm to conserve quantum resources can improve the fidelity of algorithms during quantum execution. Building on this insight, we propose a quantum circuit synthesis algorithm to design diagonal unitary implementations based on specified quantum resource limits. Our algorithm can synthesize diagonal unitary for quantum circuits with up to 15 qubits on an ordinary laptop. In algorithm efficiency, synthesizing an n-qubit unitary matrix with an exact algorithm requires 2n2^n2n CNOT gates as a baseline. Within the algorithm error $\varepsilon $ range of interest (0\%-12\%), our algorithm achieves a $3.2\varepsilon $ reduction in CNOT gates on average. In runtime, the algorithm efficiently performs, synthesizing 12-qubit diagonal unitary in an average of 6.57 seconds and 15-qubit in approximately 561.71 seconds.
During the noisy intermediate-scale quantum (NISQ) era, it is important to optimize the quantum circuits in circuit depth and gate count, especially entanglement gates, including the CNOT gate. Among all the unitary operators, diagonal unitary matrices form a special class that plays a crucial role in many quantum algorithms/subroutines. Based on a natural gate set {CNOT, Rz}, quantum circuits for general diagonal unitary matrices were discussed in several previous works, and an optimal synthesis algorithm was proposed in terms of circuit depth. In this paper, we are interested in the implementation of diagonal unitary matrices with reflection symmetry, which has promising applications, including the realization of real-time evolution for first quantized Hamiltonians by quantum circuits. Owing to such a symmetric property, we show that the quantum circuit in the existing work can be further simplified and propose a constructive algorithm that optimizes the entanglement gate count. Compared to the previous synthesis methods for general diagonal unitary matrices, the quantum circuit by our proposed algorithm achieves nearly half the reduction in both the gate count and circuit depth.
We theoretically introduce a quantum-circuit synthesis protocol to realize any nnn-qubit transformation U∈U(2n)U\in U(2^n)U∈U(2n), up to a diagonal. The resulting circuit is directly native to and optimal in gate count and depth for systems where bitstring-conditioned rotations with tunable angle and axis, C(n−1)R(θ,φ)C^{(n-1)}R(θ,φ)C(n−1)R(θ,φ), form the set of native operations. Subsequently, we suggest how this protocol can be implemented in a few-mode superconducting circuit, and experimentally demonstrate the method for the smallest nontrivial case, i.e., for two qubits. We report an average gate fidelity of 97.6 % for Haar-random two-qubit unitaries up to phase shifts in the computational basis. Importantly, these results suggest that multimode devices, such as those in the multimon or multiunimon families may become competitive with qubit-array networks for general-unitary implementations for qubit counts as low as three.
We conduct a systematic study of quantum circuits composed of multiple-control ZZZ-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.
Diagonal unitaries are a fundamental but resource-intensive class of quantum operations, arising as the phase separators of QAOA and the time-evolution blocks of Hamiltonian simulation. Under all-to-all connectivity their optimal depth is established, but on nearest-neighbor hardware general-purpose compilers fall back on heuristic search, which yields no analyzable cost bound and becomes intractable at the very sizes where depth is the bottleneck. We address synthesis and compilation jointly. On the synthesis side, we develop a Gray-Path Framework (GPF) that realizes any nnn-qubit diagonal unitary in asymptotically optimal RzR_zRz​ and CNOT depth O(2n/n)O(2^n/n)O(2n/n) without ancillas. Our main result is that compiling GPF onto a two-dimensional nearest-neighbor grid preserves this optimality: routing adds depth Θ(2n/n)Θ(2^n/n)Θ(2n/n) and gate count Θ(2n)Θ(2^n)Θ(2n). Because GPF fixes its entire interaction structure in advance, routing reduces to scheduling a known sequence, with no heuristic search. We give the construction both with and without ancillas: the ancilla-free, cost-optimized layout is a two-row grid, and a 2k2k2k-row layout introduces a space--time tradeoff that cuts depth by 1/k1/k1/k while remaining asymptotically optimal for the enlarged register; both are deterministic and analyzed in closed form. The same complexity is also attained on a linear nearest-neighbor chain, so the preservation is topology-independent, holding on any architecture that contains such a chain. All routing bounds are closed-form, giving the concrete resource estimates that heuristic compilers cannot provide at scale.
Quantum computing faces challenges such as noise, short coherence time, and limited qubit connections. These challenges worsen as quantum circuits become more complex. One major issue is the increasing depth of quantum circuits. This research proposes an optimization framework targeting depth and gate count reduction in quantum circuits, specifically for Noisy Intermediate-Scale Quantum (NISQ) devices. The proposed approach combines unitary merging of single-qubit gates, CNOT cancellation, gate commuting, and rotation gate rewriting strategies. Consecutive gates acting on the same qubit, such asRx(θ1)⋅Rx(θ2)R_{x}\left(\theta_{1}\right) \cdot R_{x}\left(\theta_{2}\right)Rx​(θ1​)⋅Rx​(θ2​), are algebraically merged into a single rotation, while pairs of redundant CNOT gates are eliminated based on gate identity relations. The technique is implemented using Qiskit and validated across five diverse circuits including complex, random, and multi-qubit configurations. Experimental results show an average depth reduction of 33.33% and gate count reduction of 32.14%, with runtime improvement of up to 25%. For instance, an input circuit with a depth of 7 and gate count of 11 was reduced to a depth of 2 and 4 gates. All optimized circuits preserve functional correctness with a fidelityF≥0.99F \geq 0.99F≥0.99. High-resolution circuit diagrams are presented to visually demonstrate improvements before and after optimization. Additionally, global phase shifts such aseiπ/4e^{i \pi / 4}eiπ/4are preserved or analytically characterized where relevant. This work enhances the viability of quantum computations on near-term hardware and opens pathways for future AI-driven quantum optimizations.