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A 12-CNOT Double Qubit Excitation Gate
Irfansha Shaik
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Abstract
Abstract:Effective implementation of high-level quantum gates is essential for practical quantum computing. To the best of our knowledge, we present the first reported 12-CNOT decomposition of the double qubit excitation operator, improving upon state-of-the-art (SOTA) implementations with 13 CNOTs. Our new circuit has the lowest CNOT count (12), lowest CNOT depth (10), and lowest total circuit depth (16) among all the previous SOTA circuits. Further, we only added 2 extra one-qubit gates compared to the lowest one-qubit gate count (11) among the previous SOTA circuits.
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- Source: https://arxiv.org/abs/2608.11733v1
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Kvantify Aps, Copenhagen, Denmarkirsh@kvantify.dkhttps://orcid.org/0000-0002-7404-348X A 12-CNOT Double Qubit Excitation Gate Irfansha Shaik Abstract Effective implementation of high-level quantum gates is essential for practical quantum computing. To the best of our knowledge, we present the first reported 12-CNOT decomposition of the double qubit excitation operator, improving upon state-of-the-art (SOTA) implementations with 13 CNOTs. Our new circuit has the lowest CNOT count (12), lowest CNOT depth (10), and lowest total circuit depth (16) among all the previous SOTA circuits. Further, we only added 2 extra one-qubit gates compared to the lowest one-qubit gate count (11) among the previous SOTA circuits. †runningauthor: Irfansha Shaik 1 Introduction Effective implementation of important building blocks (such as high-level quantum gates) in quantum algorithms is essential for practical quantum computing. In this work, we look at one such high-level gate, the Double Qubit Excitation Operator, which implements the equation (1) (Eq. 20 of [21]): Uklij(θ)=exp[−iθ8(XiYjXkXl+YiXjXkXl+YiYjYkXl+YiYjXkYl−XiXjYkXl−XiXjXkYl−YiXjYkYl−XiYjYkYl)]. splitU_klij(θ)= [- iθ8 (&\,X_iY_jX_kX_l+Y_iX_jX_kX_l+Y_iY_jY_kX_l+Y_iY_jX_kY_l\\ &-X_iX_jY_kX_l-X_iX_jX_kY_l-Y_iX_jY_kY_l-X_iY_jY_kY_l ) ]. split (1) Using the computational-basis ordering |qiqjqkql⟩ q_iq_jq_kq_l in a 44-qubit system, the operator implements a continuous rotation between the two states |0011⟩ 0011 and |1100⟩ 1100 , as shown in (2): U(θ)|x⟩=cosθ|0011⟩+sinθ|1100⟩,|x⟩=|0011⟩,cosθ|1100⟩−sinθ|0011⟩,|x⟩=|1100⟩,|x⟩,otherwise.U(θ)\, x = cases θ\, 0011 + θ\, 1100 ,& x = 0011 ,\\[2.0pt] θ\, 1100 - θ\, 0011 ,& x = 1100 ,\\[2.0pt] x ,&otherwise. cases (2) The double qubit excitation gate is used as a building block for several quantum algorithms, both near-term and fault-tolerant. In near-term algorithms, for example, it is used as a CNOT efficient alternative to the fermionic double excitation operator in variational ground-state ansätze such as unitary coupled cluster (UCCSD) [9, 10] in the Variational Quantum Eigensolver (VQE) and its adaptive variants such as QEB-ADAPT-VQE [20] and FAST-VQE [5]. In fault-tolerant algorithms, it is used in Trotterized Hamiltonian simulation and time evolution of the electronic-structure Hamiltonian [18]. Further, one can use the operator in state preparation, such as a UCC-type ansatz, for subsequent ground-state energy estimation in fault-tolerant algorithms like quantum phase estimation (QPE) [2]. L R qiq_i Ry(2θ)R_y(2θ) Ry(2θ)R_y(2θ) qjq_j qkq_k qlq_l Figure 1: High-level double-excitation operator. One can implement the double excitation gate simply using a triple controlled RyR_y rotation and some CNOT gates. Figure 1 shows such a high-level 44-qubit circuit with triple controlled Ry(2θ)R_y(2θ) rotation with qiq_i as the target qubit, sandwiched between L and R CNOT circuits. We refer to Yordanov et al. [21] for extended explanation on double qubit excitation operators, which is beyond the scope of this work. In this paper, we present, to the best of our knowledge, the first reported decomposition of the double-excitation operator with 1212 CNOTs, improving on the previously reported SOTA implementations with 1313 CNOTs. The structure of the rest of the paper is as follows. In Subsection 1.1, we will provide more details on the construction of the high-level circuit, including the role of L and R CNOT circuits. We also provide a simple 14-CNOT gate implementation using so-called Gray-code expansion [6, 15] of the C3Ry(2θ)C^3R_y(2θ) rotation. In Subsection 1.2, we will present the current SOTA implementations of the double excitation operator with 13-CNOTs. Finally in Section 2, we present a new 12-CNOT circuit for the double excitation operator. Our new circuit is better in 3 different metrics compared to existing SOTA circuits, i.e., in CNOT count (12), CNOT depth (10), and circuit depth (16). 1.1 A 14-CNOT baseline decomposition using Gray-code expansion Recall that the double-excitation operator implements 8 Pauli strings, as shown in (1). One can naively implement the double-excitation operator of 48 CNOTs, implementing each of the 8 Pauli strings separately using 6 CNOT gates per string (see §4.7.3 and Fig. 4.19 of [8]). As discussed earlier, one can also implement the double-excitation operator using a single C3Ry(2θ)C^3R_y(2θ) rotation and some CNOT gates as shown in Figure 2. Intuitively, we want the controlled rotation to trigger only when the input state is either |0011⟩ 0011 or |1100⟩ 1100 . Precisely, the L CNOT circuit performs this required transformation, as is defined in (3): L|x⟩=|0111⟩,|x⟩=|0011⟩,|1111⟩,|x⟩=|1100⟩,|x′⟩ with qjqkql≠111,otherwise,L\, x = cases 0111 ,& x = 0011 ,\\[2.0pt] 1111 ,& x = 1100 ,\\[2.0pt] x with q_jq_kq_l≠ 111,&otherwise, cases (3) The R CNOT circuit, on the other hand, performs the inverse transformation of L, i.e., LR=ILR=I. Depending on the θ, controlled rotation C3Ry(2θ)C^3R_y(2θ) either flips the input basis states |0011⟩ 0011 and |1100⟩ 1100 or leaves them unchanged. Thus, all the input states excluding |0011⟩ 0011 and |1100⟩ 1100 stay untouched in the output, implementing our desired double-excitation operation. Decomposition of the triple-controlled RyR_y rotation is well studied, and multiple 8-CNOT constructions are known. Here, we use the Gray-code expansion of the C3Ry(2θ)C^3R_y(2θ) rotation [6, 15], with 8 CNOTs and 8 RyR_y rotations, resulting in a baseline 14-CNOT double-excitation gate as shown in Figure 2. L Gray-code C3Ry(2θ)C^3R_y(2θ) R qiq_i Ry(θ4)R_y( θ4) Ry(θ4)R_y( θ4) Ry(-θ4)R_y(- θ4) Ry(-θ4)R_y(- θ4) Ry(θ4)R_y( θ4) Ry(θ4)R_y( θ4) Ry(-θ4)R_y(- θ4) Ry(-θ4)R_y(- θ4) Ry(θ4)R_y( θ4) Ry(θ4)R_y( θ4) Ry(-θ4)R_y(- θ4) Ry(-θ4)R_y(- θ4) Ry(θ4)R_y( θ4) Ry(θ4)R_y( θ4) Ry(-θ4)R_y(- θ4) Ry(-θ4)R_y(- θ4) qjq_j qkq_k qlq_l Figure 2: A 14-CNOT decomposed double-excitation circuit with C3Ry(2θ)C^3R_y(2θ) Gray-code expansion. 1.2 Previous SOTA 13-CNOT double-excitation gate implementations In the literature, several 13-CNOT decompositions of the double-excitation operator have been proposed. The usual approach is to find rewrite rules to absorb some of the CNOTs in the L and R circuits into the various decompositions of the C3Ry(2θ)C^3R_y(2θ) rotation, resulting in 13-CNOT circuits. Later in Table 1, we refer to some well-known examples of these 13-CNOT circuits of the double-excitation operator, and their metrics. In this Subsection, we present two of the best 13-CNOT circuits from the literature. qiq_i Ry(θ4)R_y( θ4) Ry(θ4)R_y( θ4) Ry(-θ4)R_y(- θ4) Ry(-θ4)R_y(- θ4) Ry(θ4)R_y( θ4) Ry(θ4)R_y( θ4) Ry(-θ4)R_y(- θ4) Ry(-θ4)R_y(- θ4) Ry(θ4)R_y( θ4) Ry(θ4)R_y( θ4) Ry(-θ4)R_y(- θ4) Ry(-θ4)R_y(- θ4) Ry(θ4)R_y( θ4) Ry(θ4)R_y( θ4) Ry(-θ4)R_y(- θ4) Ry(-θ4)R_y(- θ4) Rz(π2)R_z( π2) Rz(π2)R_z( π2) qjq_j X X H H H H X X qkq_k H H Rz(-π2)R_z(- π2) Rz(-π2)R_z(- π2) H H Rz(-π2)R_z(- π2) Rz(-π2)R_z(- π2) Ry(-π2)R_y(- π2) Ry(-π2)R_y(- π2) S S qlq_l X X H H H H X X Figure 3: A 13-CNOT double-excitation gate by Yordanov et al. [21, 20] with lowest CNOT depth (11) cf. Table 1. The two highlighted gates on qkq_k are added to correct the original circuit. qiq_i S†S S†S H H Rz(θ4)R_z( θ4) Rz(θ4)R_z( θ4) Rz(θ4)R_z( θ4) Rz(θ4)R_z( θ4) Rz(-θ4)R_z(- θ4) Rz(-θ4)R_z(- θ4) Rz(-θ4)R_z(- θ4) Rz(-θ4)R_z(- θ4) Rz(θ4)R_z( θ4) Rz(θ4)R_z( θ4) Rz(θ4)R_z( θ4) Rz(θ4)R_z( θ4) Rz(-θ4)R_z(- θ4) Rz(-θ4)R_z(- θ4) Rz(-θ4)R_z(- θ4) Rz(-θ4)R_z(- θ4) H H qjq_j qkq_k S†S S†S S S qlq_l Figure 4: Nam [7] double-excitation circuit with 13 CNOTs. Among the 13-CNOT reference circuits it has the lowest one-qubit gate count (11). Cf. Table 1. The 13-CNOT circuit by Yordanov et al. [21, 20] as shown in Figure 3 has the lowest 11 CNOT depth but at the cost of 16 one-qubit gates. Figure 4 on the other hand shows a 13-CNOT circuit by Nam [7] with the lowest one-qubit gate count (11) but at the cost of 13 CNOT depth. Another 13-CNOT circuit by Wang [18] is reported in the Table 1 with 15 one-qubit gates and 13 CNOT depth. 2 A 12-CNOT decomposition of the double-excitation operator qiq_i H H X† X X† X Rz(θ4)R_z( θ4) Rz(θ4)R_z( θ4) Rz(-θ4)R_z(- θ4) Rz(-θ4)R_z(- θ4) Rz(θ4)R_z( θ4) Rz(θ4)R_z( θ4) Rz(-π2-θ4)R_z(- π2- θ4) Rz(-π2-θ4)R_z(- π2- θ4) H H X X X X qjq_j X X X X Rz(θ4)R_z( θ4) Rz(θ4)R_z( θ4) Rz(-θ4)R_z(- θ4) Rz(-θ4)R_z(- θ4) Rz(θ4)R_z( θ4) Rz(θ4)R_z( θ4) Rz(-θ4)R_z(- θ4) Rz(-θ4)R_z(- θ4) X† X X† X qkq_k S†S S†S Rz(π2)R_z( π2) Rz(π2)R_z( π2) qlq_l Figure 5: Our best double-excitation circuit: 12 CNOTs (same-control CNOTs drawn compactly as fan-outs, but expanded when computing depth), CX-depth 10, 13 one-qubit gates. Cf. Table 1. Although several 13-CNOT decompositions of the double-excitation operator have been proposed in the literature, our literature search found no previously reported implementation with fewer than 13 CNOTs. In this section, we present the first 12-CNOT decomposition of the double-excitation operator, as shown in Figure 5. We have used various circuit synthesis and optimization tools such as Q-Synth [11], Qiskit Transpiler [3], Tket [16] to explore different decompositions of the double-excitation operator. Various synthesis techniques such as Clifford Synthesis [13, 14], CNOT+Rz Synthesis [12, 4], KAK decomposition [17] etc. have been useful to extensively explore different decompositions (mainly for excluding dead ends). Table 1 compares our new 12-CNOT circuit with the previous SOTA 13-CNOT circuits, in 44 different metrics. Our new circuit has the lowest CNOT count (12), lowest CNOT depth (10), and lowest total circuit depth (16) among all the previous SOTA circuits. Table 1: CNOT count, CNOT depth, single-qubit gate count, and total circuit depth for double-excitation circuit implementations. Single-qubit gates are counted as universal one-qubit (u3) gates, i.e., each maximal run of consecutive single-qubit gates on a qubit is merged into one u3 gate. Circuit CX count CX depth 1q gates Depth Naive Pauli decomposition [8] 48 48 38 64 Baseline: with Gray-code (Fig. 2) [6, 15] 14 14 8 22 PennyLane 2021 [1, 19] 14 12 14 20 Wang 2021 [18] 13 13 15 22 Nam 2020 [7] 13 13 11 22 Yordanov 2020 [21, 20] 13 11 16 20 This work (Fig. 5) 12 10 13 16 3 Conclusion In this work, we presented, to the best of our knowledge, the first reported 12-CNOT decomposition of the double qubit excitation operator. We compared our new circuit with the previous SOTA 13-CNOT circuits in 4 different metrics. Our new circuit has the lowest CNOT count (12), lowest CNOT depth (10), and lowest total circuit depth (16) among all the previous SOTA circuits. Further, we only added 2 extra one-qubit gates compared to the lowest one-qubit gate count (11) among the previous SOTA circuits. Acknowledgements Author would like to thank Søren Fuglede Jørgensen for the introduction to the problem and the discussions on the topic. Further, author would also like to thank Jaco van de Pol for the corrections on Yordanov’s original circuit in Figure 3. LLMs have been used, mainly Opus 4.8, for setting up experiments, brainstorming, generating Latex figures and Tables, and minor editorial tasks. Finally, author would like to thank Patrick Ettenhuber and Asbjørn Frost Teilmann for feedback and correctness checks. References [1] G. R. Anselmetti, D. Wierichs, C. Gogolin, and R. M. Parrish (2021) Local, expressive, quantum-number-preserving VQE ansätze for fermionic systems. New Journal of Physics 23 (11), p. 113010. External Links: Document, 2104.05695 Cited by: Table 1. [2] S. Fomichev, K. Hejazi, M. S. Zini, M. Kiser, J. Fraxanet, P. A. M. Casares, A. Delgado, J. Huh, A. Voigt, J. E. Mueller, and J. M. 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