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arXiv:1709.02523 · Physical Review A

Universal Barenco quantum gates via a tunable noncollinear interaction

Xiaofeng Shi

Published Mar 12, 2018• 9 citations
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Abstract

The Barenco gate ($\mathbb{B}$) is a type of two-qubit quantum gate based on which alone universal quantum computation can be achieved. Each $\mathbb{B}$ is characterized by three angles ($\ensuremath{\alpha}$, $\ensuremath{\theta}$, and $\ensuremath{\phi}$), though it works in a two-qubit Hilbert space. Here we design $\mathbb{B}$ 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 $V$ is adjustable. We present two protocols for $\mathbb{B}$. 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.

DOI
10.1103/physreva.97.032310

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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 (\ensuremathα\ensuremath{\alpha}\ensuremathα, \ensuremathθ\ensuremath{\theta}\ensuremathθ, and \ensuremathϕ\ensuremath{\phi}\ensuremathϕ), though it works in a two-qubit Hilbert space. Here we design B\mathbb{B}B via a noncollinear interaction V∣r1r2\ensuremath⟩\ensuremath⟨r1r3∣+H.c.V|{r}_{1}{r}_{2}\ensuremath{\rangle}\ensuremath{\langle}{r}_{1}{r}_{3}|+\mathrm{H}.\mathrm{c}.V∣r1​r2​\ensuremath⟩\ensuremath⟨r1​r3​∣+H.c., where ∣ri\ensuremath⟩|{r}_{i}\ensuremath{\rangle}∣ri​\ensuremath⟩ 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 \ensuremathϕ\ensuremath{\phi}\ensuremathϕ can be tuned by varying the phases of external controls, and the other two variables \ensuremathα\ensuremath{\alpha}\ensuremathα and \ensuremathθ\ensuremath{\theta}\ensuremathθ, 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, \ensuremathα,\ensuremathθ\ensuremath{\alpha},\ensuremath{\theta}\ensuremathα,\ensuremathθ, and \ensuremathϕ\ensuremath{\phi}\ensuremathϕ can be varied by changing the interaction amplitudes and wait durations, and the latter two are dependent on \ensuremathα\ensuremath{\alpha}\ensuremathα nonlinearly. Both protocols can also lead to another universal gate when \ensuremathα,\ensuremathϕ=1/4,1/2\ensuremathπ{\ensuremath{\alpha},\ensuremath{\phi}}={1/4,1/2}\ensuremath{\pi}\ensuremathα,\ensuremathϕ=1/4,1/2\ensuremathπ with appropriate parameters. Implementation of these universal gates is analyzed based on the van der Waals interaction of neutral Rydberg atoms.
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.
Arrays of qubits encoded in the ground-state manifold of neutral atoms trapped in optical (or magnetic) lattices appear to be a promising platform for the realization of a scalable quantum computer. Two-qubit conditional gates between nearest-neighbor qubits in the array can be implemented by exploiting the Rydberg blockade mechanism, as was shown by D. Jaksch et al. [Phys. Rev. Lett. 85, 2208 (2000)]. However, the energy shift due to dipole-dipole interactions causing the blockade falls off rapidly with the interatomic distance, and protocols based on direct Rydberg blockade typically fail to operate between atoms separated by more than one lattice site. In this work, we propose an extension of the protocol of Jaksch et al. for controlled-ZZZ and controlled-not gates which works in the general case where the qubits are not nearest neighbors in the array. Our proposal relies on the Rydberg excitation hopping along a chain of ancilla noncoding atoms connecting the qubits on which the gate is to be applied. The dependence of the gate fidelity on the number of ancilla atoms, the blockade strength, and the decay rates of the Rydberg states is investigated. A comparison between our implementation of a distant controlled-not gate and one based on a sequence of nearest-neighbor two-qubit gates is also provided.
The family of nnn-bit Toffoli gates, with the two-bit Toffoli gate as the figurehead, are of great interest in quantum information as they can be used as universal gates and in quantum error correction, among other things. We present a single-step implementation of arbitrary nnn-bit Toffoli gates (up to a local change of basis), based on resonantly driving a single qubit that has a strong Ising coupling to nnn other qubits. The setup in the two-qubit case turns out to be identical to the universal Barenco gate. The gate time and error are, in theory, independent of the number of control qubits, scaling better than conventional circuit decompositions. We note that our assumptions, namely, strongly coupling n+1n+1n+1 qubits and a driving frequency that scales with nnn, may break down for large systems. Still, our protocol could enhance the capabilities of intermediate-scale quantum computers, and we discuss the prospects of implementing our protocol on trapped ions, Rydberg atoms, and superconducting circuits. Simulations of the latter platform show that the Toffoli gate with two control bits attains fidelities of above 0.98 even in the presence of decoherence. We also show how similar ideas can be used to make a series of controlled-not gates in a single step. We show how these can speed up the implementation of quantum error correcting codes and we simulate the encoding steps of the three-qubit bit-flip code and the seven-qubit Steane code.
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