Bridging wire and gate cutting with ZX-calculus

TL;DR

This work shows that ZX-calculus provides a natural framework to bridge wire-cutting and gate-cutting in quantum circuit decomposition. It achieves an optimal $1$-norm decomposition for the $n$-qubit MCZ gate, attaining $\gamma=3$ with a single ancilla for any partition, and reproduces the known $R_{ZZ}(\theta)$ gate cut with $\gamma=1+2|\sin(\theta)|$ (without ancilla) by reordering ZX-diagram terms. The results illuminate why classical communication reduces wire-cutting overhead in some cases but not for certain gate cuts, and demonstrate CC-free gate-cut decompositions even when wire cuts employ CC. Overall, ZX-calculus offers new insights and practical tools for decomposing large unitaries, with potential impact on scalable circuit cutting and error-mitigated simulation of distributed quantum computations.

Abstract

Quantum circuit cutting refers to a series of techniques that allow one to partition a quantum computation on a large quantum computer into several quantum computations on smaller devices. This usually comes at the price of a sampling overhead, that is quantified by the $1$-norm of the associated decomposition. The applicability of these techniques relies on the possibility of finding decompositions of the ideal, global unitaries into quantum operations that can be simulated onto each sub-register, which should ideally minimize the $1$-norm. In this work, we show how these decompositions can be obtained diagrammatically using ZX-calculus expanding on the work of Ufrecht et al. [arXiv:2302.00387]. The central idea of our work is that since in ZX-calculus only connectivity matters, it should be possible to cut wires in ZX-diagrams by inserting known decompositions of the identity in standard quantum circuits. We show how, using this basic idea, many of the gate decompositions known in the literature can be re-interpreted as an instance of wire cuts in ZX-diagrams. Furthermore, we obtain improved decompositions for multi-qubit controlled-Z (MCZ) gates with $1$-norm equal to $3$ for any number of qubits and any partition, which we argue to be optimal. Our work gives new ways of thinking about circuit cutting that can be particularly valuable for finding decompositions of large unitary gates. Besides, it sheds light on the question of why exploiting classical communication decreases the 1-norm of a wire cut but does not do so for certain gate decompositions. In particular, using wire cuts with classical communication, we obtain gate decompositions that do not require classical communication.

Figures (10)

• Figure 1: ZX-diagram of an exemplary five-qubit circuit illustrating different cutting techniques. The subsystems A and B are only connected by a CNOT gate. Conventional wire cutting can be used to split the circuit by inserting two wire cuts at the location of the dashed scissors. The solid scissor represents our contribution in this paper and corresponds to applying a wire cut to the wire connecting the two qubits involved in the CNOT in the ZX-diagram representation. This effectively allows us to obtain a gate cut and this idea can potentially be applied to arbitrary ZX-diagrams representing larger unitaries.
• Figure 2: Fundamental ZX-diagrams used in this manuscript. (a) Z- and X-spiders are the main building blocks that characterize ZX-diagrams. The H-box generalizes the Hadamard gate, while Cup and Cap represent Bell states and Bell measurements, respectively. Wires and wire crossing represent the single-qubit identity and the SWAP gate, respectively. (b) Pauli eigenstates are proportional via $\sqrt{2}$ factors to either Z- or X-spiders with a single output and no inputs. (c) Similarly, Pauli measurements are represented via ZX-diagrams with one input and no outputs, and introducing an ancilla bit $b=0,1$ associated with the measurement outcome. (d) Representations of the CNOT gate in ZX-calculus.
• Figure 3: ZX diagrammatic rules used in this manuscript.
• Figure 4: (Top) Representation of an MCZ gate acting on $n=m+m'$ qubits in ZX-calculus. On the right-hand side the ZX-diagram is rewritten using the H-box fusion rule in Fig. \ref{['fig:zx_rules']}f. The scissor indicates where a single wire cut can be inserted to cut the MCZ gate. (Bottom) Representation of a multi-controlled phase gate $\mathrm{MCP}(\theta)$ in ZX-calculus.
• Figure 5: ZX-diagrams resulting from cutting a wire in an MCZ gate. The variables $a_{Y},a_{X}$, $a_{X}'$, $a_{Z}$, $a_{Z}'$ can take values in $\{ 0,1 \}$. The diagrammatic rules shown in Fig. \ref{['fig:zx_rules']} are used to convert the ZX-diagrams to valid quantum circuits. Note that every diagram is correctly normalized apart from the one for the $Y$-term acting on the lower $m'$ qubits which requires an additional factor of $\sqrt{2}$. This means that we need to multiply the probability to sample from the correctly normalized diagram by $1/2$, as explained in the main text.