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arXiv:2603.02387 · International Journal of Theoretical Physics

Barenco gate implementation using driven two- and three-qubit spin chains

Rafael Vieira, Edgard P. M. Amorim

quant-phSubmitted Mar 2, 2026• Announced Mar 4, 2026• 0 citations
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Abstract

We propose a protocol for implementing Barenco-type multi-qubit controlled gates using short driven spin chains. Starting from an Ising interaction with a transverse drive on the last spin, we construct an effective two-qubit Hamiltonian whose time evolution implements the Barenco gate $V_2(\varphi,ω,φ)$ and, in particular, a CNOT gate. We then embed this construction into a three-qubit $XXZ$ chain to realize the three-qubit Barenco gate $V_3(\varphi,ω,φ)$, which includes the Toffoli gate as a special case. The derivation is fully analytical: we perform a sequence of unitary transformations, identify decoupled subspaces, and apply a rotating-wave approximation to obtain simple effective Hamiltonians. We derive explicit conditions on the coupling strengths and driving parameters, provide closed-form expressions for the time-evolution operators in each relevant subspace, and characterize the quality of the implementation using the operator fidelity. Numerical simulations show that the protocol achieves high fidelities over broad parameter ranges, demonstrating its robustness and suitability for quantum information processing in spin-chain platforms.

Journal reference
Int J Theor Phys 65, 81 (2026)
DOI
10.1007/s10773-026-06296-3
Comments
16 pages, 2 figures, one column

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We propose a protocol for implementing Barenco-type multi-qubit controlled gates using short driven spin chains. Starting from an Ising interaction with a transverse drive on the last spin, we construct an effective two-qubit Hamiltonian whose time evolution implements the Barenco gate V2(φ,ω,φ)V_2(\varphi,ω,φ)V2​(φ,ω,φ) and, in particular, a CNOT gate. We then embed this construction into a three-qubit XXZXXZXXZ chain to realize the three-qubit Barenco gate V3(φ,ω,φ)V_3(\varphi,ω,φ)V3​(φ,ω,φ), which includes the Toffoli gate as a special case. The derivation is fully analytical: we perform a sequence of unitary transformations, identify decoupled subspaces, and apply a rotating-wave approximation to obtain simple effective Hamiltonians. We derive explicit conditions on the coupling strengths and driving parameters, provide closed-form expressions for the time-evolution operators in each relevant subspace, and characterize the quality of the implementation using the operator fidelity. Numerical simulations show that the protocol achieves high fidelities over broad parameter ranges, demonstrating its robustness and suitability for quantum information processing in spin-chain platforms.
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Pairwise exchange couplings have long been the standard mechanism for entangling spin qubits in semiconductor systems. However, implementing quantum circuits based on pairwise exchange gates often requires a lengthy sequence of elementary gate operations. In this work, we present an alternative approach: multi-qubit entangling gate operations that simultaneously drive the exchange couplings between multiple pairs of spin qubits. We explore three spin qubit systems in linear or triangular configurations. We derive analytical expressions for these multi-exchange entangling operations and demonstrate how to use the resulting three-qubit gates to construct quantum circuits capable of generating standard entangled states such as GHZ and W states, and the Toffoli gate, by optimizing control parameters. Our results show that this multi-qubit strategy significantly reduces the number of required operations, offering a pathway to more efficient, shallower, and more coherent circuits for spin-qubit processors.
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The Barenco gate (B\mathbb{B}B) is a type of two-qubit quantum gate based on which alone universal quantum computation can be achieved. Each B\mathbb{B}B is characterized by three angles ($\ensuremath{\alpha}$, $\ensuremath{\theta}$, and $\ensuremath{\phi}$), though it works in a two-qubit Hilbert space. Here we design B\mathbb{B}B via a noncollinear interaction $V|{r}_{1}{r}_{2}\ensuremath{\rangle}\ensuremath{\langle}{r}_{1}{r}_{3}|+\mathrm{H}.\mathrm{c}.$, where $|{r}_{i}\ensuremath{\rangle}$ is a state that can be excited from a qubit state and VVV is adjustable. We present two protocols for B\mathbb{B}B. The first (second) protocol consists of two (six) pulses and one (two) wait period(s), where the former causes rotations between qubit states and excited states, and the latter induces gate transformation via the noncollinear interaction. In the first protocol, the variable $\ensuremath{\phi}$ can be tuned by varying the phases of external controls, and the other two variables $\ensuremath{\alpha}$ and $\ensuremath{\theta}$, tunable via adjustment of the wait duration, have a linear dependence on each other. Meanwhile, the first protocol can give rise to cnot and controlled-y gates. In the second protocol, $\ensuremath{\alpha},\ensuremath{\theta}$, and $\ensuremath{\phi}$ can be varied by changing the interaction amplitudes and wait durations, and the latter two are dependent on $\ensuremath{\alpha}$ nonlinearly. Both protocols can also lead to another universal gate when ${\ensuremath{\alpha},\ensuremath{\phi}}={1/4,1/2}\ensuremath{\pi}$ with appropriate parameters. Implementation of these universal gates is analyzed based on the van der Waals interaction of neutral Rydberg atoms.
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