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arXiv:cond-mat/0212501 · Physical review. B, Condensed matter

Nonideality of quantum operations with the electron spin of a 31P donor in a Si crystal due to interaction with a nuclear spin system

Semion K. Saikin, Leonid Evgenevich Fedichkin

Published Apr 28, 2003• 25 citations
arXivHTMLPDF

Abstract

We examine a ${}^{31}\mathrm{P}$ donor electron spin in a Si crystal to be used for the purpose of quantum computation. The interaction with an uncontrolled system of ${}^{29}\mathrm{Si}$ nuclear spins influences the electron-spin dynamics appreciably. The hyperfine field at the ${}^{29}\mathrm{Si}$ nuclei positions is noncollinear with the external magnetic field. Quantum operations with the electron wave function, i.e., using magnetic-field pulses or electrical gates, change the orientation of hyperfine field and disturb the nuclear-spin system. This disturbance produces a deviation of the electron spin qubit from an ideal state, at a short-time scale in comparison with the nuclear-spin diffusion time. For H${}_{\mathrm{ext}}\ensuremath{\approx}9$ T, the estimated error rate is comparable to the threshold value required by the quantum error correction algorithms. The rate is lower at higher external magnetic fields.

DOI
10.1103/physrevb.67.161302

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Journal of Physics Condensed Matter•Mar 15, 2000•7 citations
We examine a 31P{}^{31}\mathrm{P}31P donor electron spin in a Si crystal to be used for the purpose of quantum computation. The interaction with an uncontrolled system of 29Si{}^{29}\mathrm{Si}29Si nuclear spins influences the electron-spin dynamics appreciably. The hyperfine field at the 29Si{}^{29}\mathrm{Si}29Si nuclei positions is noncollinear with the external magnetic field. Quantum operations with the electron wave function, i.e., using magnetic-field pulses or electrical gates, change the orientation of hyperfine field and disturb the nuclear-spin system. This disturbance produces a deviation of the electron spin qubit from an ideal state, at a short-time scale in comparison with the nuclear-spin diffusion time. For Hext\ensuremath≈9{}_{\mathrm{ext}}\ensuremath{\approx}9ext​\ensuremath≈9 T, the estimated error rate is comparable to the threshold value required by the quantum error correction algorithms. The rate is lower at higher external magnetic fields.
Donors in silicon, which combine an electron and nuclear spin, are some of the most promising candidates for quantum information. The electron spin has been proposed as a register with fast manipulation and the nuclear spin as a memory with long coherence times. However, this division reduces the complexity of the donor system, in particular behaviors emerging from their interaction. In natural silicon, there is also the presence of 29Si nuclear spins in the donor environment; though they are generally seen as a source of decoherence, they are quantum systems that can be investigated too. The main subject of this thesis is the study of the interactions between these various spins, using different methods to probe and control them. I first concentrate on the coupling between the donor and the 29Si spins. This coupling can be perturbed by the application of dynamical decoupling on the donor electron spin, whose evolution can be made sensitive to the number of 29Si spins interacting together. I then propose an error correction scheme using the donor and 29Si spins, showing key requirements such as coherence times and methods for manipulation and initialization. Secondly, I focus on the donor itself, in a regime where the hyperfine and Zeeman couplings compete with each other. Here, the spin transitions can have different sensitivities to the magnetic environment, and can even be suppressed to first order, resulting in coherence times up to seconds with electron spin-like manipulation times. Controlling this sensitivity was also used to probe the effect of the donor on the 29Si spin bath evolution. Finally, I use electric fields to modulate the hyperfine coupling within the donor. I first characterize the spins’ sensitivity to the electric field, and then demonstrate electrical switching of the nuclear spin response to an external magnetic excitation.
In the Kane silicon-based electron-mediated nuclear spin quantum computer architecture, phosphorous is doped at precise positions in a silicon lattice, and the P donor nuclear spins act as qubits. Logical operations on the nuclear spins are performed using externally applied magnetic and electric fields. There are two important interactions: the hyperfine and exchange interactions, crucial for logical qubit operations. Single qubit operations are performed by applying radio frequency magnetic fields resonant with targeted nuclear spin transition frequencies, tuned by the gate-controlled hyperfine interaction. Two qubit operations are mediated through the exchange interaction between adjacent donor electrons. It is important to examine how these two interactions vary as functions of experimental parameters. Here we provide such an investigation. First, we examine the effects of varying several experimental parameters: gate voltage, inter donor separation, donor depth below the silicon oxide interface and back gate depth, to explore how these variables affect the donor electro density. Second, we calculate the hyperfine interaction and the exchange coupling as a function of these parameters. These calculations were performed using an anisotropic effective mass Hamiltonian. The electric field potential was obtained using Technology Computer Aided Design software, and the interfaces were modelled as a barrier using a step function. We aim to provide relevant information for the experimental design of these devices and highlight the significance of environmental factors other than gate potential that affect the donor electron.
FINDING: Phosphorus-31 (³¹P) in silicon is a leading qubit platform; its nuclear spin-½ coherence is central to quantum error correction and magnetic-field-controlled spin manipulation. MATH: - ³¹P: Z=15, N=16, A=31 → odd-odd nucleus → nuclear spin I = 1/2 (from unpaired proton in 3p³ subshell, [Ne]3s²3p³). - Gyromagnetic ratio γ(³¹P) ≈ 17.235 MHz/T (from NMR data). - Zeeman splitting: ΔE = ħγB₀ → for B₀ = 1 T, ν ≈ 17.235 MHz. - Hyperfine coupling (P donor in Si): A ≈ 117 MHz (bulk) — key for electron-nuclear entanglement. - Coherence time T₂ (nuclear spin) can exceed 1 s at mK temperatures; electron spin T₂ ~ 0.5–1 ms. - Magnetic-field control: Rabi frequency Ω = γB₁ (for nuclear spin) — linear in RF field amplitude. CONNECTION: - Spin-½ is the simplest nontrivial representation of SU(2) — its Bloch sphere is a 2-sphere with intrinsic curvature. - The ratio of hyperfine to Zeeman energy at typical fields (e.g., 0.1–1 T) spans 0.1–1 — near the golden ratio conjugate Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Nuclear spin registers in the vicinity of electron spins in solid state systems offer a powerful resource to address the challenge of scalability in quantum architectures. We investigate here the properties of 29 Si nuclear spins surrounding donor atoms in silicon, and consider the use of such spins, combined with the donor nuclear spin, as a quantum register coupled to the donor electron spin. We find the coherence of the nearby 29 Si nuclear spins is effectively protected by the presence of the donor electron spin, leading to coherence times in the second timescale—over two orders of magnitude greater than the coherence times in bulk silicon. We theoretically investigate the use of such a register for quantum error correction (QEC), including methods to protect nuclear spins from the ionisation/neutralisation of the donor, which is necessary for the re-initialisation of the ancillae qubits. This provides a route for multi-round QEC using donors in silicon.
We study numerically the process of nuclear spin measurement in a solid-state quantum computer of the type proposed by Kane, by calculating the quantum dynamics of two coupled nuclear spins on 31 P donors implanted in 28 Si. We estimate the time of the `quantum swap operation' - the minimum measurement time required for the reliable transfer of quantum information from the nuclear spin subsystem to the electronic subsystem. Our calculations show that for realistic values of the parameters this time is of the order of swap ~5 × 10 -5 s. We also calculate the probability of error for typical values of the external noise.