Scalability of quantum computation with addressable optical lattices

T. R. Beals, J. Vala, and K. B. Whaley

Phys. Rev. A 77, 052309 – Published 7 May 2008

Abstract

We make a detailed analysis of error mechanisms, gate fidelity, and scalability of proposals for quantum computation with neutral atoms in addressable (large lattice constant) optical lattices. We have identified possible limits to the size of quantum computations, arising in three-dimensional (3D) optical lattices from current limitations on the ability to perform single-qubit gates in parallel and in 2D lattices from constraints on laser power. Our results suggest that 3D arrays as large as $100 \times 100 \times 100$ sites (i.e., $\sim 10^{6}$ qubits) may be achievable, provided two-qubit gates can be performed with sufficiently high precision and degree of parallelizability. The parallelizability of long-range interaction-based two-qubit gates is qualitatively compared to that of collisional gates. Different methods of performing single-qubit gates are compared, and a lower bound of $1 \times 10^{- 5}$ is determined on the error rate for the error mechanisms affecting ${}^{133}C s$ in a blue-detuned lattice with Raman-transition-based single-qubit gates, given reasonable limits on experimental parameters.

Received 9 January 2008

DOI:https://doi.org/10.1103/PhysRevA.77.052309

Authors & Affiliations

T. R. Beals

Department of Physics, University of California, Berkeley, California 94720, USA

J. Vala^* and K. B. Whaley^†

Department of Chemistry and Pitzer Center for Theoretical Chemistry, University of California, Berkeley, California 94720, USA

^*Present address: Department of Mathematical Physics, National University of Ireland, Maynooth, County Kildare, Ireland.
^†whaley@berkeley.edu

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Vol. 77, Iss. 5 — May 2008

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Images

Figure 1
(Color online) Solid blue line: Raman scattering cross section $σ (ω_{L})$ for a ${}^{133}C s$ atom in a blue-detuned optical lattice, as a function of lattice light wavelength $λ_{L}$ . Dashed green line: frequency-dependent polarizability $α (ω_{L})$ at the frequency $ω_{L} = 2 π c / λ_{L}$ . Both $σ (ω_{L})$ and $α (ω_{L})$ are calculated for $ϵ_{+ 1}$ -polarized light interacting with the initial state $| F = 3, m_{F} = 0 ⟩$ , with results for the $| F = 4, m_{F} = 0 ⟩$ state or opposite polarization light being similar.Reuse & Permissions
Figure 2
(Color online) Schematics of a single-qubit-flip operation. A focused far-off-resonant addressing laser produces an ac Stark shift $Δ_{ac}$ of the $| F = 4, m_{F} = 1 ⟩$ and $| F = 3, m_{F} = 1 ⟩$ levels in a single target atom. The qubit levels are the $m_{F} = 0$ levels, labeled $| 0 ⟩$ and $| 1 ⟩$ . Levels $| 2 ⟩$ and $| 3 ⟩$ are auxiliary levels involved in the single-qubit gate. (Most of the hyperfine sublevels not involved in the gate are not shown here.) Three global microwave pulses tuned to the transitions $| 0 ⟩ \leftrightarrow | 2 ⟩$ , $| 1 ⟩ \leftrightarrow | 3 ⟩$ , and $| 2 ⟩ \leftrightarrow | 3 ⟩$ drive these transitions in the target atom with transition strengths $Ω_{2}$ , $Ω_{2}$ , and $Ω_{1}$ . Other atoms are not affected to first order (see below) since their transitions are not resonant. Our simulations indicate that best performance is achieved by simultaneously driving all transitions, with the relative transition strengths chosen such that $Ω_{2} = \sqrt{3} / 2 Ω_{1}$ .Reuse & Permissions
Figure 3
Series of “snapshots” of the atomic spatial and internal state wave function for an atom in a 1D trapping potential undergoing a microwave-pulse-based single-qubit $π$ rotation gate, as simulated by QSIMS. The four relevant internal atomic states are labeled $| 0 ⟩$ through $| 3 ⟩$ , as in Fig. 2, and the curves represent the wave function for the atom’s center-of-mass coordinate. Simulation parameters have been chosen to exaggerate the gate errors so that they are visible in this figure. In (a), we see that the initial state of the atom is the internal state $| 0 ⟩$ ( $F = 3$ , $m_{F} = 0$ ), with a Gaussian center-of-mass wave function. In (b) and (c), we see transitions to the auxiliary levels $| 2 ⟩$ and $| 3 ⟩$ . In (d) and (e), vibrationally excited states become visible, particularly for the component of the total wave function in the state $| 3 ⟩$ . In (f), we see the final state with a significant portion of the wave function not in the desired $| 1 ⟩$ state, showing instead noticeable entanglement between spatial and internal degrees of freedom (e.g., the “bumpy” shape of the wave function component for the $| 3 ⟩$ level).Reuse & Permissions
Figure 4
(Color online) 3D plot of gate error $(ϵ)$ versus lattice spacing $(a)$ and beam waist $(w_{0})$ for the Raman single-qubit gate, with lattice depth $U_{L} = 500 μ K$ and detuning $| Δ_{1} | = 2 π \times 5 THz$ .Reuse & Permissions
Figure 5
(Color online) A method for addressing some or all of the atoms in a single 2D plane of a 3D optical lattice. The circles on the right represent atoms in an optical lattice (only a cross-sectional plane of atoms is shown). The stars on the left represent point sources of light, which could consist of lenses on the end of individual optical fibers, with each fiber coupled to a light source and independently controlled. The lens in the center focuses the light from the point sources onto corresponding atoms in a plane (i.e., the vertical plane perpendicular to the page surface). By adjusting either the location of the lens or the array of point sources, different planes of atoms could be targeted.Reuse & Permissions

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